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\section{Introduction}\label{sintro}
Entanglement turned out to be a crucial resource for quantum
computation. It plays a central r\^ole in quantum communication
and quantum computation. A considerable effort is being put into
quantifying quantum entanglement.
It seems natural to focus the efforts on quantifying entanglement
itself, that is, describing the \emph{impossibility} to prepare a
state by means of LOCC (local operations and classical
communications). One may, although, go another way around and try
to quantify \emph{separability} rather than entanglement: this
turned out to be applicable for building combinatorial
entanglement patterns for multipartite quantum systems
\cite{myjmo}.
In this paper I dwell on the case of bipartite quantum systems. A
state of such system is called {\sc separable} if it can be
prepared by LOCC. In terms of density matrices that means that
$\mathbf{p}$, its density matrix, can be represented as a mixture of
pure product states. According to Carath\'eodory theorem, the
number of this states can be reduced to $n^4$ where $n$ is
the dimension of the state of a single particle.
\medskip
The idea to replace finite sums of projectors by continuous
distributions on the set of unit vectors is put forward making it
possible to provide a geometrical characterization of separable
mixed states of a bipartite quantum system. To consistently
describe the result presented in this paper recall some necessary
definitions.
\paragraph{Basics.} A density matrix $\mathbf{p}$ in the product space
$\mathfrak{B}$ is called {\sc factorizable} if it is a tensor product of
density matrices, $\mathbf{p}=\rho\otimes\rho'$. If $\mathbf{p}$ is a convex
combination of factorizable operators, it is said to be {\sc
separable}
\begin{equation}\label{edefsepar}
\mathbf{p}
\;=\;
\sum_\alpha\limits
\,p_\alpha\,
\rho_\alpha\otimes\rho'_\alpha
\end{equation}
\noindent A crucial feature of quantum mechanics, the phenomenon
of quantum entanglement, stems from the fact that there exist
density operators in the product space which are NOT separable,
they are called {\sc entangled}. A density operator $\mathbf{p}$ is
called {\sc robustly separable} if it has a neighborhood $U$ in
$\mathcal{L}$ such that all operators $\rho'\in{}U$ are separable.
\paragraph{A brief account.} In the Euclidean space $\mathfrak{L}$ of
self-adjoint operators acting in the tensor product space
$\mathfrak{B}=\mathcal{H}\otimes\mathcal{H}'$ we define a real-valued, positive functional
${\mathcal{K}}:\mathfrak{L}\to\mathbb{R}_{+}$ as follows
\[
{\mathcal{K}}(X)
\;=\;
\iint
e^{\bracket{\ppp{\phi}}{X}{\ppp{\phi}}}
\ppp{\,d\mathbf{S}_{\lth}}
\]
\noindent where the integration is taken over the torus---the
Cartesian product of unit spheres in $\mathcal{H},\mathcal{H}'$, respectively, and
consider the hypersurface $\mathcal{K}\subset\mathfrak{L}$
\[
\mathcal{K}
\;=\;
\{X\in\mathfrak{L}
\,\mid\,
{\mathcal{K}}(X)=1
\}
\]
\noindent Then
\begin{itemize}
\item all robustly separable density operators in $\mathcal{H}$ are in 1--1
correspondence with the points of $\mathcal{K}$
\item the density matrix associated with a point $X\in\mathcal{K}$ is
the normal vector to $\mathcal{K}$ at point $X$.
\end{itemize}
\section{Continuous optimal ensembles}\label{scontensemb}
To make the account self-consistent, begin with necessary
definitions. A {\sc density operator} is a non-negative
self-adjoint operator whose trace equals to 1. In particular, for
any unit vector $\ket{\phi}$ the one-dimensional projector
$\raypr{\phi}$ is a density matrix. Note that for any set of
density operators $\rho_\alpha$ the convex combination
$\sum_\alpha{}\rho_\alpha$ is always a density operator.
The set of all self-adjoint operators in $\mathcal{H}=\mathbb{C}^n$ has a
natural structure of a real space $\mathbb{R}^{2n}$, in which the
set of all density matrices is a hypersurface, which is the zero
surface $T=0$ of the affine functional $T=\trc{}X-1$.
In this paper a geometrical characterization of separable
bipartite density operators is provided. It is based on the notion
of continuous ensembles. Generalizing the fact that any convex
combination of density operators is again a density operator, we
represent density operators as probability distributions on the
unit sphere in the state space $\mathcal{H}$ of the system. Let us pass to
a more detailed account of this issue beginning with the case of a
single quantum system.
Let $\mathcal{H}=\mathbb{C}^n$ be a $n$-dimensional Hermitian space,
let $\rho$ be a density matrix in $\mathcal{H}$. We would like to
represent the state whose density operator is $\rho$ by an
ensemble of pure states. We would like this ensemble to be
continuous with the probability density expressed by a function
$\mu(\phi)$ where $\phi$ ranges over all unit vectors in $\mathcal{H}$.
\paragraph{Technical remark.} Pure states form a projective space
rather than the unit sphere in $\mathcal{H}$. On the other hand, one may
integrate over any probabilistic space. Usually distributions of
pure states over the spectrum of observables are studied,
sometimes probability distributions on the projective spaces are
considered \cite{sqprop}. In this paper for technical reasons I
prefer to represent ensembles of pure states by measures on unit
vectors in $\mathcal{H}$. I use the Umegaki measure on $\cfield{B}_\lth$--- the
uniform measure with respect to the action of $U(n)$ normalized
so that $\int_{\cfield{B}_\lth}\,d\mathbf{S}_{\lth}=1$.
\subsection{Effective definition}\label{scontens}
The density operator of a continuous ensemble associated with the
measure $\mu(\phi)$ on the set $\cfield{B}_\lth$ of unit vectors in $\mathcal{H}$ is
calculated as the following (matrix) integral
\begin{equation}\label{e01integral}
\rho
\;=\;
\int_{\phi\in\cfield{B}_\lth}\limits\;
\mu(\phi)\,
\raypr{\phi}
\,\,d\mathbf{S}_{\lth}
\end{equation}
\noindent where $\raypr{\phi}$ is the projector onto the vector
$\bra{\phi}$ and $\,d\mathbf{S}_{\lth}$ is the above mentioned normalized
measure on $\cfield{B}_\lth$:
\begin{equation}\label{einvarmes}
\int_{\phi\in\cfield{B}_\lth}\limits\;
\,\,d\mathbf{S}_{\lth}
\;=\;
1
\end{equation}
\noindent Effectively, the operator integral $\rho$ in
\eqref{e01integral} can be calculated by its matrix elements. In
any fixed basis $\{\ket{\mathbf{e}_i}\}$ in $\mathcal{H}$, each its matrix
element $\rho_{ij}=\bracket{\mathbf{e}_i}{\rho}{\mathbf{e}_j}$ is the following
numerical integral:
\begin{equation}\label{e01basis}
\rho_{ij}
\;=\;
\bracket{\mathbf{e}_i}{\rho}{\mathbf{e}_j}
\;=\;
\int_{\phi\in\cfield{B}_\lth}\limits\;
\mu(\phi)\,
\braket{\mathbf{e}_i}{\phi}
\braket{\phi}{\mathbf{e}_j}
\,\,d\mathbf{S}_{\lth}
\end{equation}
\subsection{Optimal ensembles}\label{soptens}
We need to solve the following variational problem. Given a
functional $Q$ on $L^1(\cfield{B}_\lth)$ and given a density matrix $\rho$ in
$\mathcal{H}$, find the distribution $\mu$ on the set $\cfield{B}_\lth$ of unit
vectors in $\mathcal{H}$ such that
\begin{equation}\label{e03}
\left\lbrace
\begin{array}{l}
\int_{\phi\in\cfield{B}_\lth}\limits\;
\mu(\phi)\,\raypr{\phi}\,d\mathbf{S}_{\lth}
\;=\;\rho
\\
\qquad
\\
Q(\mu)\;\to\; \mbox{extr}
\end{array}
\right.
\end{equation}
\noindent We shall consider functionals $Q$ of the form
\begin{equation}\label{e03q}
Q(\mu)
\;=\;
\int_{\phi\in\cfield{B}_\lth}\limits\;
q(\mu(\phi))\,d\mathbf{S}_{\lth}
\end{equation}
\noindent then, according to \eqref{e01basis}, the variational
problem \eqref{e03} reads
\[
\left\lbrace
\begin{array}{l}
\int_{\phi\in\cfield{B}_\lth}\limits\;
\mu(\phi)
\braket{\mathbf{e}_i}{\phi}
\braket{\phi}{\mathbf{e}_j}
\,d\mathbf{S}_{\lth}\;=\;\rho_{ij}
\\
\int_{\phi\in\cfield{B}_\lth}\limits\;
q(\mu(\phi))\,d\mathbf{S}_{\lth}
\;\to\;
\mbox{extr}
\end{array}\right.
\]
\noindent Solving this variational problem by introducing
Lagrangian multiples $X_{ij}$ we get
\begin{equation}\label{e03a}
q'(\mu(\phi))
\,-\,
\sum_{ij}
X_{ij}
\braket{\mathbf{e}_i}{\phi}
\braket{\phi}{\mathbf{e}_j}
\;=\;
0
\end{equation}
\noindent Combining the Lagrange multiples into the operator
$X=\sum_{ij} X_{ij}\ketbra{\mathbf{e}_j}{\mathbf{e}_i}$ turns the equation
\eqref{e03a} to \(q'(\mu(\phi))
\,=\,
\bracket{\phi}{X}{\phi}
\). Then, denoting by $f$ the inverse of $q'$ we write \eqref{e03a} as
\begin{equation}\label{e01a}
\mu(\phi)
=
f\left(
\bracket{\phi}{X}{\phi}
\right)
\end{equation}
\noindent and the problem reduces to finding $\mu$ from the
condition
\begin{equation}\label{e04ini}
\int_{\phi\in\cfield{B}_\lth}\limits\;
\mu(\phi)
\raypr{\phi}\,d\mathbf{S}_{\lth}
\;=\;\rho
\end{equation}
\noindent which according to \eqref{e01a} and \eqref{e01basis} can
be written as
\begin{equation}\label{e04}
\bracket{\mathbf{e}_i}{\rho}{\mathbf{e}_j}
\;=\;
\int_{\phi\in\cfield{B}_\lth}\limits\;
f\left(
\bracket{\phi}{X}{\phi}
\right)
\raypr{\phi}\,d\mathbf{S}_{\lth}
\end{equation}
\noindent It follows from \eqref{e03a} that the coefficients
$X_{ik}$ can be chosen so that $X_{ik}=\bar{X}_{ki}$. That means
that the problem of finding the optimal ensemble reduces to that
of finding the coefficients of a self-adjoint operator, that is,
to finding $n^2$ numbers from $n^2$ equations.
\subsection{Geometrical interpretation}\label{sgeominterpr}
The equation \eqref{e04} can be given a direct geometrical
meaning. Let $\mathcal{L}\simeq\mathbb{R}^{n^2}$ be the space of all
self-adjoint operators in $\mathcal{H}$. Let $f:\mathbb{R}\to\mathbb{R}$ be a
differentiable function. Consider the real valued functional
$F:\mathcal{L}\to\mathbb{R}$ defined as
\begin{equation}\label{edefderiv}
F(X)
\;=\;
\int_{\phi\in\cfield{B}_\lth}\limits\;
f\left(
\bracket{\phi}{X}{\phi}
\right)
\,d\mathbf{S}_{\lth}
\end{equation}
\noindent which is well-defined as the set $\cfield{B}_\lth$ is compact. Fix
a basis $\{\mathbf{e}_k\}$ in $\mathcal{H}$, then any $X\in\mathcal{L}$ is defined by its
matrix elements $X_{ik}=\bracket{\mathbf{e}_i}{X}{\mathbf{e}_k}$, so
$\bracket{\phi}{X}{\phi}=\sum_{ik}X_{ik}\braket{\phi}{\mathbf{e}_i}\braket{\mathbf{e}_k}{\phi}$.
Then the expression \eqref{edefderiv} can be treated as an
integral depending on the set of parameters $\{X_{ik}\}$. We may
consider the derivatives of $F(X)$ with respect to these
variables, calculate them
\[
\frac{\partial}{\partial X_{ik}}
\,F(X)
\;=\;
\int_{\phi\in\cfield{B}_\lth}\limits\;
\frac{\partial}{\partial X_{ik}}
\left(
\vphantom{\frac{\partial}{\partial X_{ik}}}
\, f\left(
\bracket{\phi}{X}{\phi}
\right)
\right)
\,d\mathbf{S}_{\lth}
\;=\;
\]
\begin{equation}\label{ederivgen}
\;=\;
\int_{\phi\in\cfield{B}_\lth}\limits\;
\, f'\left(
\bracket{\phi}{X}{\phi}
\right)
\braket{\phi}{\mathbf{e}_i}
\,
\braket{\mathbf{e}_k}{\phi}
\,d\mathbf{S}_{\lth}
\;=\;
\end{equation}
\[
\;=\;
\bracket{\mathbf{e}_k}{
\int_{\phi\in\cfield{B}_\lth}\limits\;
f'\left(
\bracket{\phi}{X}{\phi}
\right)
\raypr{\phi}\,d\mathbf{S}_{\lth}
}{\mathbf{e}_i}
\]
\medskip
\noindent So, the gradient of the functional $F$ is the operator
which can be symbolically written as
\begin{equation}\label{ederivsymb}
\nabla F
\;=\;
\int_{\phi\in\cfield{B}_\lth}\limits\;
f'\left(
\bracket{\phi}{X}{\phi}
\right)
\raypr{\phi}\,d\mathbf{S}_{\lth}
\end{equation}
\noindent and effectively calculated using \eqref{ederivgen}.
\subsection{Optimal entropy ensembles}\label{soptentropens}
Let us specify the form of the optimality functional in
\eqref{e03q} assuming it to be the differential entropy of the
appropriate distribution:
\begin{equation}\label{edefq}
q(\mu)
\;=\;
-\mu\,\ln\mu
\end{equation}
\noindent then $q'=-(1+\ln\mu)$ and we have the following $f$ for
\eqref{e04}
\[
f(x)
\;=\;
e^{-(1+x)}
\]
\noindent Introduce, as in \eqref{edefderiv}, the functional
${\mathcal{K}}:\mathcal{L}\to\mathbb{R}$ on the set of all self-adjoint operators in
$\mathcal{H}$ (the minus sign and the unit summand are omitted here being
a matter of renormalization):
\begin{equation}\label{edefk}
{\mathcal{K}}(X)
\;=\;
\int_{\phi\in\cfield{B}_\lth}\limits\;
\, e^{\bracket{\phi}{X}{\phi}
}
\,d\mathbf{S}_{\lth}
\end{equation}
\noindent Note that $\rho(X)=\int_{\phi\in\cfield{B}_\lth}\limits\;
\, e^{\bracket{\phi}{X}{\phi}
}\raypr{\phi}\,d\mathbf{S}_{\lth}$ is always a positive operator, then
\[
{\mathcal{K}}(X)
\;=\;
\trc\int_{\phi\in\cfield{B}_\lth}\limits\;
\, e^{\bracket{\phi}{X}{\phi}
}\raypr{\phi}\,d\mathbf{S}_{\lth}
\;=\;1
\]
\noindent is a condition which defines a full-range density matrix
$\rho(X)$ in $\mathcal{H}$. On the other hand, the condition ${\mathcal{K}}(X)=1$
defines a hypersurface in the Euclidean space $\mathcal{L}$. Together with
the fact that $\left(e^x\right)'=e^x$ and \eqref{ederivsymb} we
come to the following
\paragraph{Statement.} Any full-range density matrix in $\rho$ is
associated with a point on the hypersurface ${\mathcal{K}}(X)=1$ and the
entries of $\rho$ are calculated as the components of the
gradient:
\begin{equation}\label{edefrrhgrad}
\rho
\;=\;
\nabla{}{\mathcal{K}}
\end{equation}
\subsection{The existence}\label{sexist}
Why optimal entropy ensembles do exist for all full-range density
matrices? First note that for any full-range density matrix
$\rho=\sum{}p_k\raypr{\mathbf{e}_k}$ there are infinitely many continuous
ensembles (=probability measures on $\cfield{B}_\lth$ in our setting)
associated with it. An example of such distribution is
$\rho=\sum\,p_k\raypr{\mathbf{e}_k}=\int\mu(\phi)\raypr{\phi}\,d\mathbf{S}_{\lth}$ with
\begin{equation}\label{esmeared}
\mu(\phi)
\;=\;
\frac{
\bigl((L+1)n\bigr)!
}{ L\,n!(L\,n)! }
\;
\sum_{k=1}^{n}\limits
\left(
p_k-
\frac{1}{L(n+1)}
\right)
\,|\braket{\mathbf{e}_k}{\phi}|^{2Ln}
\end{equation}
\noindent as it follows from \cite{mygibbs}. Here $L$ is a
parameter, such that $L>\frac{1}{p_0(n+1)}$ where $p_0>0$ is the
smallest eigenvalue of $\rho$. Any probabilistic density $\mu$
whose support is $\cfield{B}_\lth$ is a point in the interior of the simplex
of all probabilistic measures on $\cfield{B}_\lth$. For each probabilistic
measure on $\cfield{B}_\lth$ its differential entropy can be calculated. The
differential entropy is, in turn, a concave function in the affine
space of probability distributions. Therefore if we have an affine
subset of of probability measure on $\cfield{B}_\lth$, the differential
entropy takes its maximal value in the interior of the simplex of
probability measures. Now return to the condition in
\eqref{e03}---we see that it is affine. Therefore, if we know that
there exist at least one continuous ensemble representing $\rho$
(but we know that as mentioned above), that means that there exist
a maximal entropy ensemble representing $X$, hence it has the
representation
\eqref{edefrrhgrad}.
\section{Bipartite systems}\label{sbipart}
Consider two finite-dimensional quantum systems whose state spaces
are $\mathcal{H},\mathcal{H}'$. The state space of the composite system is the
tensor product $\mathfrak{B}=\mathcal{H}\otimes\mathcal{H}'$. Denote by
$\mathfrak{L}=\mathcal{L}\otimes\mathcal{L}$ the space of all self-adjoint operators in
$\mathfrak{B}$.
\subsection{Continuous ensembles in bipartite case}\label{scontbi}
Let $\mathbf{p}$ be a robustly separable density matrix in the product
space $\mathcal{H}\otimes\mathcal{H}'$. Then it can be represented (in infinitely
many ways) as a continuous ensemble of pure product states.
Carrying out exactly the same reasoning as in section \ref{sexist}
we conclude that among those continuous ensembles there exists one
having the least differential entropy, this will be the ensemble
we are interested in. Like in section \ref{soptentropens},
formulate the variational problem. Let $\mathbf{p}$ be a density
operator in a tensor product space $\mathfrak{B}=\mathcal{H}\otimes\mathcal{H}'$. The task
is to find a probability density $\mu(\ppp\phi)$ defined on the
Cartesian product $\mathfrak{T}=\cfield{B}_\lth\times\cfield{B}_\lth$ of the unit spheres in
$\mathcal{H},\mathcal{H}'$, respectively.
\begin{equation}\label{e03bi}
\left\lbrace
\begin{array}{l}
\int_{\ppp\phi\in\mathfrak{T}}\limits\;
\mu(\ppp\phi)\,\raypr{\ppp\phi}\ppp\,d\mathbf{S}_{\lth}
\;=\;\mathbf{p}
\\
\qquad
\\
Q(\mu)\;\to\; \mbox{extr}
\end{array}
\right.
\end{equation}
Proceeding exactly in the same way as with single particle, we get
the following representation:
\begin{equation}\label{erepbi}
\mathbf{p}
\;=\;
\int_{\ppp\phi\in\mathfrak{T}}\limits\;
e^{\bracket{\ppp\phi}{X}{\ppp\phi}}
\,\raypr{\ppp\phi}\ppp\,d\mathbf{S}_{\lth}
\end{equation}
\noindent for some self-adjoint operator $X$ in $\mathcal{L}$ whose
existence is guaranteed by the same reasons as in section
\ref{sexist}. Why such $X$ does not exist for entangled density
operators? The reason is that the set of probability distributions
among which $e^{\bracket{\ppp\phi}{X}{\ppp\phi}}$ is optimal is
simply void in the entangled case.
\subsection{Geometrical characterization of
robustly separable quantum states}\label{sgeombi}
Now we pass to the main result of this paper. Suppose we deal with
a tensor product of two Hilbert spaces $\mathcal{H},\mathcal{H}'$, each of
dimension $n$. Consider the space $\mathcal{L}$ of all self-adjoint
linear operators in the tensor product $\mathfrak{B}=\mathcal{H}\otimes\mathcal{H}'$,
being a Euclidean space of dimension $n^4$. For any $X\in\mathcal{L}$
we can always calculate the integral
\begin{equation}\label{edefkbi}
{\mathcal{K}}(X)
\;=\;
\int_{\ppp\phi\in\mathfrak{T}}\limits\;
e^{\bracket{\ppp\phi}{X}{\ppp\phi}}
\ppp\,d\mathbf{S}_{\lth}
\end{equation}
\noindent which is always well-defined (as an integral of a
bounded function over a compact set), positive (as the exponent is
always positive) functional from $\mathcal{L}$ to $\mathbb{R}_+$. Consider
the hypersurface $\mathcal{K}$ in $\mathcal{L}$ defined by the equation
\[
\mathcal{K}
\;=\;
\{X\in\mathcal{L}
\,|\,{\mathcal{K}}(X)=1
\}
\]
\noindent In any point of $\mathcal{L}$ the gradient $\nabla {\mathcal{K}}$ can be
calculated. In particular, at any point $X$ of $\mathcal{K}$ the gradient
$\nabla {\mathcal{K}}$ will be a normal vector to $\mathcal{K}$. The surface $\mathcal{K}$
is something given once and forever, it depends only on the
dimensionality of the state space. For any $X$ such that
${\mathcal{K}}(X)=1$, we can calculate the gradient $\mathbf{p}(X)=\nabla
{\mathcal{K}}\left|{}_{X}\right.$ at point $X$ Fix bases $\{\mathbf{e}_i\}$,
$\{\mathbf{e}'_{i'}\}$, then
$X=\sum_{\ppp{i}\,\ppp{k}}\,X_{\ppp{i}\,\ppp{k}}\ketbra{\ppp{i}}{\ppp{k}}$
and the expression \eqref{erepbi} for the operator $\mathbf{p}$ has the
following form:
\begin{equation}\label{edefgradbi}
\mathbf{p}_{\ppp{i}\,\ppp{k}}
\;=\;
\nabla {\mathcal{K}}
\;=\;
\frac{\partial{{\mathcal{K}}}}{\partial{X_{\ppp{i}\,\ppp{k}}}}
\end{equation}
Conversely, given a robustly separable bipartite density matrix
$\mathbf{p}$, we know that it can be represented as a convex combination
of product states: $\mathbf{p}=\sum
p_{\alpha}\rho_{\alpha}\otimes\rho'_{\alpha}$. Each
$\rho_{\alpha}$ can be, in turn, represented as a non-vanishing
probability distribution \eqref{esmeared}. Then exactly the same
reasoning as in section \ref{sexist} can be carried out and there
is a point $X$ on the surface $\mathcal{K}$ associated with $\mathbf{p}$. So,
together with \eqref{edefgradbi}, we have the main result:
\begin{equation}\label{emainresbi}
\bigl\{
\mbox{robustly separable states}
\bigr\}
\quad\leftrightarrow\quad
\bigl\{
\mbox{the points of $\mathcal{K}$}
\bigr\}
\end{equation}
\section*{Summary}
A geometrical interpretation of robustly separable density
operators of a bipartite quantum system with the state space
$\mathfrak{B}=\mathcal{H}\otimes\mathcal{H}'$ is provided. They are represented as normal
vectors to the hypersurface $\mathcal{K}$ in the (Euclidean) space $\mathfrak{L}$
of self-adjoint operators in $\mathfrak{B}$ defined by the following
equation:
\begin{equation}\label{edefkbiconc}
\mathcal{K}
\quad=\quad
\left\{
\vphantom{\int_{\ppp\phi\in\mathfrak{T}}\limits}
\,X\; \right|
\left.
\;
\int_{\ppp\phi\in\mathfrak{T}}\limits\;
e^{\bracket{\ppp\phi}{X}{\ppp\phi}}
\ppp\,d\mathbf{S}_{\lth}
\;=\;1\;
\right\}
\end{equation}
\noindent where the integration is performed over the set of all
unit product vectors $\bra{\ppp\phi}\in\mathfrak{B}$. Each point $X\in\mathcal{K}$
is a self-adjoint operator, the parameter of the probability
distribution on the set of unit vectors which gives a density
operator $\mathbf{p}$. Furthermore, the normal vector to $\mathcal{K}$ at point
$X$ is $\mathbf{p}$ itself:
\begin{equation}\label{erepbiconcl}
\mathbf{p}
\;=\;
\nabla
{\mathcal{K}}\left|{}_{X}\right.
\;=\;
\int_{\ppp\phi\in\mathfrak{T}}\limits\;
e^{\bracket{\ppp\phi}{X}{\ppp\phi}}
\,\raypr{\ppp\phi}\ppp\,d\mathbf{S}_{\lth}
\end{equation}
\paragraph{The final remark.} Given a density matrix $\mathbf{p}$ in
$\mathfrak{B}$, a question arises if it is separable or not. When the
dimension of at least one of spaces $\mathcal{H},\mathcal{H}'$ is 2, this question
was given an effective answer---the positive partial transpose
(PPT) criterion due to Peres-Horodecki was suggested
\cite{perehorod}. The criterion states that $\mathbf{p}$ is separable if
and only if its partial transpose $\mathbf{p}^{T_2}$ remains
non-negative matrix. In higher dimensions PPT is only a necessary
condition for a state to be factorizable as there exist entangled
density matrices whose partial transpose if positive.
Although a geometrical characterization of robustly separable
density matrices is provided, it does not solve (directly, at
least) the `inverse problem'. Nevertheless, the continuous
ensemble method presented in this paper seems to be helpful for
tackling the inverse problem as well. This issue is addressed in
the next paper on continuous ensembles.
\paragraph{Acknowledgments.} The idea to consider continuous
ensemble was inspired by the paper \cite{vidaltarrach}, where the
notion of robustness for entangled states was introduced, I am
grateful to its authors for the inspiration. Much helpful advice
from Serguei Krasnikov is highly appreciated. The financial
support for this research was provided by the research grant No.
04-06-80215a from RFFI (Russian Basic Research Foundation).
Several crucial issues related to this research were intensively
duscussed during the meeting Glafka-2004 `Iconoclastic Approaches
to Quantum Gravity' (15--18 June, 2004, Athens, Greece) supported
by QUALCO Technologies (special thanks to its organizers---Ioannis
Raptis and Orestis Tsakalotos).
|
{
"timestamp": "2005-03-21T11:39:50",
"yymm": "0503",
"arxiv_id": "quant-ph/0503173",
"language": "en",
"url": "https://arxiv.org/abs/quant-ph/0503173"
}
|
\section{Introduction}
The Bloch vector, or vector of coherence \cite{Alicki},
provides a geometric description of the density matrix
of a spin-1/2 particle which is commonly used in nuclear
magnetic resonance.
Mathematically, the Bloch vector may be viewed as the adjoint
representation of an $su(2)$ object in an $so(3)$ basis
\cite{alta3}.
Extension of the notion of vector of coherence to two-spin systems
\cite{fano,quan}, and more generally to quantum spin systems of higher
dimensions \cite{byrd}, has drawn attention in the contexts of
quantum information theory and quantum computation. Specific
motivations include the prospects of a useful quantification
of entanglement for composite systems \cite{mahl,byrd,alta1} and
the quest for equations describing observables in quantum
networks \cite{quan}.
In the present work, the extension of the Bloch formalism to two
spins is used to obtain a geometric representation of the
orbits of the vector of coherence for each spin system in the
case that a nonlocal interaction of the form $\sigma_i\otimes\sigma_j$
is introduced. We propose that this geometric picture will be useful
in devising schemes for control of a quantum state via quantum
interfaces \cite{lloyd}, i.e., through the mediation of an ancillary
system. In this vein, we investigate the limits of control of a
quantum state $S$, mixed or pure, given a nonlocal interaction and
an ancilla $Q$. The simple geometric picture developed below also
applies to another special case of nonlocal interaction, namely the
Heisenberg exchange Hamiltonian. As a second application of our
formal results, we investigate the entanglement power of the
Heisenberg interaction.
\section{Product of operator basis for a density matrix}
\subsection{One qubit}
The density matrix $\rho$ of a two-state system is a
positive semi-definite Hermitian $2 \times 2$
matrix having unit trace. It can always be given expression in terms of
the three trace-free Pauli matrices
$\sigma_i,~i=1,2,3 $, which are
generators of $su(2)$, and $I/{\sqrt{2}}$ ($I$ being the unit matrix):
\begin{equation}
\rho=\frac{1}{2} I +{\bf v}{\bf \sigma}\,.
\label{onequbit}
\end{equation}
Here $\bf v$ is the vector of coherence, whose magnitude
is bounded by $0 \leq\parallel{\bf v}\parallel\leq
1/2$ because
$1/2\leq {\rm Tr}(\rho^2)\leq1$. The two limiting values of
the norm correspond to maximally mixed and pure states, respectively.
The magnitude of the Bloch vector differs by a factor of $1/2$ from
that of the vector of coherence, as a matter of convention.
Unitary operations rotate the Bloch vector without changing its magnitude:
$ SU(2)$ operations on the qubit correspond to $SO(3)$ operations
on the Bloch vector. The dynamical evolution of the Bloch vector
under non-local operations is considered in the next section.
\subsection{Two qubits and the correlation tensor}
In analogy to the representation (\ref{onequbit}), we adopt
the generators of ${\mit G}=SU(4)$, i.e., the elements of
the algebra ${\mit g}=su(4)$ (together with the unit matrix),
as an orthonormal basis for the $4\times4$ density matrix of the
two-qubit system. We employ this basis as it appears in
Ref.~\onlinecite{alta1}, noting that it differs from the basis
used in \cite{byrd,mahl} only in the coefficients.
The dynamical evolution of the system becomes more transparent if we
choose basis elements of the algebra ${\mit g}= su(4)$ in accordance with
its Cartan Decomposition ${\mit g}={\mit p}\oplus{\mit e}$ \cite{Bro,Zhang}.
The algebras ${\mit p}$ and ${\mit e}$ satisfy the commutations relations
\begin{equation}
[{\mit e},{\mit e}] \subset {\mit e}\,, \quad
[{\mit p},{\mit e}] \subset {\mit p}\,, \quad
[{\mit p},{\mit p}] \subset {\mit e}\,.
\end{equation}
The basis elements, $W_j,~j=1,\ldots,15$ of the orthogonal algebra pair
$(e,p)$ are
\begin{equation}
{\mit e}= {\rm span} \frac{i}{2} \{\sigma_x\otimes1,\sigma_y\otimes1,
\sigma_z\otimes1, 1\otimes\sigma_x,1\otimes\sigma_y,1\otimes\sigma_z\}\,,
\label{cart1}
\end{equation}
\begin{equation}
{\mit p}= {\rm span} \frac{i}{2} \{\sigma_x\otimes\sigma_x, \sigma_x\otimes
\sigma_y,\sigma_x\otimes\sigma_z, \\ \sigma_y\otimes\sigma_x, \sigma_y\otimes
\sigma_y,\sigma_y\otimes\sigma_z, \\
\sigma_z\otimes\sigma_x, \sigma_z\otimes
\sigma_y,\sigma_z\otimes\sigma_z \}\,.
\label{cart2}
\end{equation}
The basis defined by Eqs.~(\ref{cart1}) and (\ref{cart2}) is used to
expand the density matrix as
\begin{equation}
\rho=\sum_{j=0}^{15} {\rm Tr}(\rho X_j) X_j
=\sum_{j=0}^{15}\rho_j X_j\,,
\end{equation}
where $X_0=I/\sqrt{4}$, $\rho_0=1/\sqrt{4}$, and $X_j= -i W_j$ ($j=1, \ldots, 15$).
In this representation, the density matrix is specified by three objects,
namely the vectors of coherence
${\bf r}_1$ and ${\bf r}_2$
for the two subsystems along with the spin-spin correlation tensor $T_j^i$.
\begin{equation}
{\bf r}_1=\left( \begin{array}{c}
\rho_1 \\ \rho_2 \\ \rho_3 \end{array}\right)\,, \qquad
{\bf r}_2=\left( \begin{array}{c}
\rho_4 \\ \rho_5 \\ \rho_6 \end{array}\right)\,, \qquad
T_j^i=\left( \begin{array}{ccc}
\rho_7 & \rho_8 & \rho_9 \\
\rho_{10} & \rho_{11} & \rho_{12} \\
\rho_{13} & \rho_{14} & \rho_{15} \end{array}\right)\,.
\end{equation}
We note that the object $ T_j^i$ has other names: Stokes
tensor \cite{Boston}, entanglement tensor \cite{mahl}, and
tensor of coherence (when combined with the coherence vectors
in one object). Details of the properties of $T_j^i$ can
be found in Ref.~\onlinecite{alta1}, where many prior studies
are cited. This tensor contains information on the correlations between
the two subsystems, of both classical and quantum nature. Necessary and
sufficient conditions for separability of a pure state can be stated
in terms of its properties, whereas in the case of a mixed state,
only necessary conditions for separability can be given \cite{alta1}.
\section{Evolution Under Local and non Local Operations}
As we have seen, the Lie algebra ${\mit g}=su(4)$ possesses
a Cartan decomposition $ {\mit g}={\mit e}\oplus {\mit p}$,
which informs us that there exists within the Lie group ${\mit G}=SU(4)$
a subgroup of local operations ${\mit G}_L=SU(2)\otimes SU(2)$
generated by ${\mit e}$. All the other operations are nonlocal
and members of the coset space $SU(4)/SU(2)\otimes SU(2)$, which
does not form a subgroup of $SU(4)$.
It is known (see {\it Proposition 1} of Ref.~\onlinecite{Zhang}) that any
$U\in SU(4)$ can be written as
\begin{equation}
U=k_1Ak_2
\label{decom1}
\end{equation}
with
\begin{equation}
A = \exp\left[\frac{i}{2}(c_1\sigma_{x}^{1}\sigma_{x}^{2}
+c_2\sigma_{y}^{1}\sigma_{y}^{2}+c_3\sigma_{z}^{1}\sigma_{z}^{2})\right]\,,
\label{decom2}
\end{equation}
where $k_1,~k_2 \in SU(2)\otimes SU(2)$ and $c_1,~c_2,~c_3~\in R$.
In the following, we focus on the effect of nonlocal operations generated by
a single operator among the possibilities for
$\sigma_i\otimes\sigma_j$, where $i,j \in\{x,y,z\}$.
(This consideration includes the special case in which two of the
parameters $c_1$, $c_2$, and $c_3$ in the decomposition
(\ref{decom1})-(\ref{decom2}) are zero.)
Such nonlocal operations will be called one-dimensional.
\subsection{Local operations}
Local operations are operations $g\in SU(2)\otimes SU(2)$
generated by the elements of ${\mit e}$. From the commutation
relations $[{\mit e},{\mit e}] \subset {\mit e}$ and
$[{\mit p},{\mit e}] \subset {\mit p}$ it is clear that
the elements of the vectors ${\bf r}_i$ and tensor $(T_i^j)$
do not mix and do not affect one another. Under local operations,
the vectors behave just like ordinary Cartesian vectors.
In particular, a vector of coherence is rotated about
some vector $\hat n$ as illustrated in Fig.~1, i.e.,
\begin{equation}
(r')_{1}^{i}=R^{i}_{j}r_1^j\,,
\qquad (r')_{2}^{i}=R^{i}_{j}r_2^j\,.
\end{equation}
On the other hand, the correlation tensor transforms like
a mixed Cartesian tensor,
\begin{equation}
(T')^{i}_{j}=R^{i}_{m}(R')^{n}_{j} T^{m}_{n}\,.
\end{equation}
The magnitude of each object remains invariant under local
operations. In addition, there exist fifteen more invariants which
can be constructed from the vectors and the tensor \cite{makhl}.
\begin{figure}
\includegraphics[width=12cm]{fig1.eps}
\caption{Local operations on the two spin subsystems produce
a rotation of the corresponding vectors of coherence
around some direction
$\hat{{\bf n}}$.
The effect is the same for both pure states (a) and mixed states (b).}
\end{figure}
\subsection{One-dimensional nonlocal operations }
The nonlocal operations in the coset space $SU(4)/SU(2)\otimes SU(2)$
require, in their construction, exponentiation of at least one of the
elements of ${\mit p}$. Hence, under these operations
the elements of the tensor and vectors of coherence are mixed,
due to the commutation relations $[{\mit p},{\mit e}] \subset {\mit p}$
and $[{\mit p},{\mit p}] \subset {\mit e}$. We shall establish
that the one-dimensional nonlocal operations generated by the
chosen interaction
$\sigma_i\otimes\sigma_j$ give rise to elliptical orbits for
the vectors of coherence of the subsystems. The characteristics
of these elliptic paths depend on the indices $i$ and $j$, on
the initial states of the subsystems, and on the degree of
correlations between
them. These orbits can be described by
non-unitary transformations on each of the individual subsystems
when one traces over the other's degrees of freedom.
Accordingly, we take the interaction Hamiltonian between the two
spins to be $H_I=\sigma_i\otimes\sigma_j/2$, and, for
reasons of simplicity, we suppose that the internal Hamiltonians
for the two spins may be ignored. Assuming that the duration of
the interaction is $\phi$, and appealing to (i) the commutation
relations as summarized in Ref.~\cite{Zhang} and (ii) the identity
\begin{equation}
\exp\left[-i(\phi/2)\sigma_i\otimes\sigma_j\right] =
\cos(\phi/2)I-i\sin(\phi/2)\sigma_i\otimes\sigma_j\,,
\end{equation}
we can make the following observations:
\begin{enumerate}
\item[(i)] The components $r^{i}_1$ and $r^{j}_2$ of the vectors
of coherence remain unaffected; hence the vectors are confined
to planes perpendicular to the $i$-axis and $j$-axis respectively.
\item[(ii)] Of the nine elements of the correlation tensor $T^{k}_{l}$,
only four experience changes.
The five that are unchanged under the
action of $\sigma_i\otimes \sigma_j$ are $T^i_j$ and $T^k_l$ with
$k\neq i$ and $l\neq j$.
\item[(iii)] The vectors $r_1^m+T_j^m$ and $r_2^m+T_m^i$ are rotated, without change
of magnitude, through an angle $\phi$ about the $i$ and $j$ axes,
respectively. (Here $m$ ranges freely over $\{x,y,z\}$).
\item[(iv)] More explicitly, the components of the vectors transform
according to
\begin{equation}
\begin{array}{l}
r_{1}^{i}\rightarrow (r')_{1}^{i}=r_{1}^{i}\,, \nonumber \\
r_{1}^{k}\rightarrow (r')_{1}^{k}=r_1^k \cos \phi- T^{l}_{j}\sin \phi
\,,\nonumber \\
r_{1}^{l}\rightarrow (r')_{1}^{l}= T^{k}_{j}\sin \phi + r_{1}^{l}\cos \phi \,,
\end{array}
\begin{array}{l}
r_{2}^{j}\rightarrow (r')_{2}^{j}=r_{2}^{j}\,,\nonumber \\
r_{2}^{m}\rightarrow (r')_{2}^{m}=r_{2}^{m}\cos \phi- T^{i}_{n}\sin \phi\,,\nonumber\\
r_{2}^{n}\rightarrow (r')_{2}^{n}= T^{i}_{m}\sin \phi +r_{2}^{n}\cos \phi \,,
\end{array}
\end{equation}
and the components of the tensor of coherence, according to
\begin{equation}
\begin{array}{l}
T_{j}^{i}\rightarrow (T')_{j}^{i}=T_{j}^{i}\,,\nonumber \\
T_{j}^{k}\rightarrow (T')_{j}^{k}=T^{k}_{j}\cos \phi-r^{l}_{1}\sin \phi \,,\nonumber\\
T_{j}^{l}\rightarrow (T')_{j}^{l} =r^{k}_{1}\sin \phi + T_{j}^{l} \cos \phi \,,
\end{array},~~
\begin{array}{l}
T_{j}^{i}\rightarrow (T')_{j}^{i}=T_{j}^{i}\,,\nonumber \\
T_{m}^{i}\rightarrow (T')_{m}^{i}=T^{i}_{m}\cos \phi -r^{n}_{2}\sin \phi \,,\nonumber\\
T_{n}^{i}\rightarrow (T')_{n}^{i} =r^{m}_{2}\sin \phi + T_{n}^{i}\cos \phi \,,
\end{array}
\end{equation}
with no change in the tensor's other elements.
The ordered sets of indices $(i,l,k)$ and $(j,m,n)$ belong to
$\{(x,y,z),(y,z,x),(z,x,y)\} $.
\end{enumerate}
Given this behavior, it is not difficult to show that {\it ${\bf r}_1(\phi)$ and
${\bf r}_2(\phi)_2$ follow elliptical orbits}. Since the 1,2 labeling is
arbitrary, it suffices to demonstrate this property for the the
vector ${\bf r}_1(\phi)$.
\smallskip
\noindent
{\it Proof.}
Referring to Fig.~2(a), the coordinates for a vector $\bf s$ tracing
an ellipse in the $x-y$ plane, with principal axes $a$ and $b$ rotated
by an angle $\psi$, are
\begin{equation}
\begin{array}{l}
s_x(\phi)= a ~{\rm cos}\phi~{\rm cos}\psi+ b~{\rm sin}\phi~{\rm sin}\psi\,,\nonumber\\
s_y(\phi)= - a~ {\rm cos}\phi~{\rm sin}\psi+ b~{\rm sin}\phi~{\rm cos}\psi \,.
\end{array}
\end{equation}
The angle $\phi$ is zero when the vector $\bf s$ is aligned with
the principal axis $a$.
The coordinates of the vector of coherence ${\bf r}_1$ moving in the
$k-l$ plane are given by
\begin{equation}
\begin{array}{l}
r_1^k(\phi')=r_1^k(0)\cos \phi' - T_j^l(0) \sin \phi'\,,\nonumber \\
r_1^l(\phi')=T_j^k(0)\sin \phi'+r_1^l(0) \cos \phi' \,.
\end{array}
\end{equation}
Of course, for the vector of coherence, $\phi' = 0 $ does not in general
correspond to the principal axis $a$ (see Fig.~2(a)). In fact,
$\phi'=\phi+\chi$, and the coordinates of ${\bf r}_1 $ can be rewritten
as follows in terms of the phase difference $\chi$:
\begin{equation}
\begin{array}{l}
r_1^k(\phi)=(r_1^k(0)\cos\chi- T_j^l(0)\sin\chi)\cos\phi+
(-T_j^l(0)\cos\chi-r_1^k(0)\sin\chi)\sin\phi \,, \nonumber\\
r_1^l(\phi)=(T_j^k(0)\sin\chi+r_1^l(0)\cos\chi) \cos\phi
+(-r_1^l(0)\sin\chi+T_j^k(0)\cos\chi)\sin\phi \,.
\end{array}
\end{equation}
Comparison of the two sets of coordinates $\{s_x(\phi),s_y(\phi)\}$
and $\{r_1^k(\phi),r_1^l(\phi) \}$ shows that a match can always
be made, such that the parameters $a$, $b$, $\psi$, and $\chi$
can be determined by solving the system of equations
\begin{equation}
\begin{array}{c}
a\cos\psi= r_1^k(0)\cos\chi- T_j^l(0)\sin\chi \,, \nonumber \\
b\sin\psi=-T_j^l(0)\cos\chi-r_1^k(0)\sin\chi \,, \nonumber \\
a\sin\psi=-T_j^k(0)\sin\chi-r_1^l(0)\cos\chi \,, \nonumber \\
b\cos\psi=-r_1^l(0)\sin\chi+T_j^k(0)\cos\chi\,.
\label{system}
\end{array}
\end{equation}
This completes the proof.
It is important to note that the shape of the ellipse depends
explicitly on the spin-spin correlation tensor.
\smallskip
\begin{figure}
\includegraphics[width=12cm]{fig2.eps}
\caption{(a) The vector $\bf s$ describes an ellipse with principal
axes $a$ and $b$ rotated by angle $\psi$ with respect to the $y$-axis.
The angle $\phi$, interpreted as the duration of a group operation, is
measured relative to the $a$ principal axis. (b) The vector of coherence
$\bf r$ corresponding to one of the spins of the two-spin system moves
on an ellipse on the $k-l$ plane, with the angle $\phi'$ measured relative to
the original direction of $\bf r$.}
\end{figure}
Solving Eqs.~(\ref{system}) for the angle $\chi$, we find
\begin{equation}
\tan(2\chi)= \frac{2\left[r^{k}_{1}(0)T^{l}_{j}(0)-r^{l}_{1}(0)T^{k}_{j}(0))\right]}
{-(r^{l}_{1}(0))^2+(T^{k}_{j}(0))^2 -(r^{k}_{1}(0))^2+(T^{l}_{j}(0))^2}\,,
\end{equation}
which specifies the initial orientation of the coherence vector ${\bf r}_1$
with respect to the principal axis $a$.
Suppose now the two-spin system is initially in a {\it product state}.
For this case it is easy to prove these corollaries to our principal result:
\begin{itemize}
\item[(1)] The phase difference $\chi$ is zero. This means that the
initial positions of both coherence vectors lie on the $a$
principal axis (as in Fig.~3(a)).
It follows that the linear entropy of the state of each of the
subsystems (defined by $1 - {\rm Tr}\,\rho^2$) can only decrease,
showing it is possible to increase the entanglement of the system
with this interaction. (This will depend on initial conditions.
See Section \ref{sec:ent-heis}.)
\item[(2)] The length of the semi-minor axis of the ellipse followed by subsystem
1 is given by $ b_1=|r_2^j(0)|[(r_1^l(0))^2+(r_1^k(0))^2]^{1/2}$
(and likewise for subsystem 2 with $1 \rightarrow 2$
and $\{j,k,l\} \rightarrow \{i,n,m\}$).
It follows that for an initially pure state and the assumed
single interaction $\sigma_i \otimes \sigma_j$, the
maximum attainable entanglement is achieved at $\phi$ values of
$\pi/2$ and $3 \pi/2$.
\end{itemize}
For the case of a initial state that is not pure but still
separable, the phase difference $\chi$ does not vanish,
in general (see figure~3(b)).
Accordingly, the implied dynamical behavior of a classically
correlated system distinguishes it from an uncorrelated system,
but not from a system experiencing quantum entanglement.
Moreover, the linear entropy of each subsystem can either
increase or decrease,
showing it is possible to increase or decrease the amount of
entanglement in the system.
\begin{figure}
\includegraphics[width=12cm]{fig3.eps}
\caption{The initial position of the vector of coherence of subsystem
1 or 2 is shown, together with its time evolution under a one-dimensional
nonlocal interaction (dashed line). If the initial state of the two-spin
system is a product state, then the initial position is on the $a$ principal
axis of the elliptical path, as in (a). In general this agreement no longer
occurs if the subsystems are initially correlated, either classically or
quantum mechanically, as in (b).}
\end{figure}
\subsection{General nonlocal operations}
From {\it Proposition 1} of Ref.~\onlinecite{Zhang}, any nonlocal operation
can be decomposed as a product of two local operations and an operation of the form
\begin{equation}
A= \exp\left[ \frac{i}{2}(c_1\sigma_{x}^{1}\sigma_{x}^{2}
+c_2\sigma_{y}^{1}\sigma_{y}^{2}+c_3\sigma_{z}^{1}\sigma_{z}^{2})\right] \,.
\label{decom3}
\end{equation}
The operators $\{Y_i\}=\{i\sigma_{x}^{1}\sigma_{x}^{2}/2,
i\sigma_{y}^{1}\sigma_{y}^{2}/2,i\sigma_{z}^{1}\sigma_{z}^{2}/2 \}$ span
a maximal Abelian subalgebra of ${\mit P}$, and the relations
\begin{equation}
[Y_i,Y_j]=0\,, \qquad
[Y_i,Y_j]_+ = -i|\epsilon_{ijk}|Y_k -\frac{1}{2}\delta_{ij}
\end{equation}
hold, where $[\cdot,\cdot]_+$ denotes the anticommutator.
Consequently, $A$ of Eq.~(\ref{decom3}) can be written
in product form,
\begin{equation}
A =\exp\left[\frac{i}{2}(c_1\sigma_{x}^{1}\sigma_{x}^{2})\right]
\exp\left[\frac{i}{2}(c_2\sigma_{y}^{1}\sigma_{y}^{2})\right]
\exp\left[\frac{i}{2}(c_3\sigma_{z}^{1}\sigma_{z}^{2})\right]\,.
\label{prodform}
\end{equation}
The property (\ref{prodform}) tells us that {\it any nonlocal operation
can be decomposed into a sequence of operations effecting a succession
of circular and elliptic paths in Bloch space}.
This result facilitates the calculation of the final states of
the subsystems, but gives only limited insight into the geometric
characteristics of the coherence vectors' time orbits. For
all $c_i$ distinct, two general observations can be made:
\begin{itemize}
\item[(1)] The motion of the vectors of coherence is no longer restricted
to a plane, since there is no linear combination of
$\{\sigma_x\otimes 1, \sigma_y\otimes 1, \sigma_z\otimes 1\}$
or $\{1\otimes\sigma_x,1\otimes\sigma_y,1\otimes\sigma_z\}$
that is invariant under the action of $A$.
\item[(2)] Characteristics of the trajectories such as
periodicity depend in detail on the parameters $c_1$, $c_2$, and $c_3$.
A trajectory is periodic only if $c_2/c_1$ and $c_3/c_1$ are both rational
numbers. We note also that the set of parameters $\{c_1,c_2,c_3\}$ has
been used to determine the equivalence classes
of nonlocal interactions \cite{Zhang} as well as the invariants of the nonlocal
interactions \cite{makhl}.
\end{itemize}
\subsection{Special case of the Heisenberg Hamiltonian}
The Heisenberg exchange Hamiltonian, corresponding to $c_1=c_2=c_3=-c/2$,
is not included in our general observations on nonlocal interactions
(made for all $c_i$ distinct), but like the one-dimensional Hamiltonians,
it admits a simple geometric picture. This interaction is the primary
two-qubit interaction in several experimental proposals for quantum-dot qubits
\cite{Loss:98, Kane:98, Vrijen:00}. It can also be used for universal
quantum computing on encoded qubits of several types
\cite{Bacon:00, Kempe:01, DiVincenzo:00a, Lidar:02, Wu:02, Byrd:02}.
For these reasons, it warrants special attention.
Introducing the time parameter $\phi$, the operator $A$ of
Eq.~(\ref{prodform}) now takes the form
\begin{eqnarray*}
A(\phi) &=& \exp[-i(c\phi/2)(\sigma_x\otimes \sigma_x
+ \sigma_y\otimes \sigma_y
+ \sigma_z\otimes \sigma_z)] \nonumber \\
&=& \left[\cos^3(c\phi/2) -i\sin^3(c\phi/2)\right]I\otimes I \nonumber \\
&& -(i/2)e^{ic\phi/2}\sin(c\phi)(\sigma_x\otimes \sigma_x
+ \sigma_y\otimes \sigma_y
+ \sigma_z\otimes \sigma_z ).
\end{eqnarray*}
The time development of the density matrix under the operator
$A$ is given $\rho(\phi)= A(\phi)\rho(0)A^{\dagger}(\phi)$
and the corresponding coherence vectors change according to
\begin{equation}
\label{eq:ipart}
r_1^i(\phi) = \frac{1}{2}[r^i_1(0)+r^i_2(0)+(r_1^i(0)-r_2^i(0))\cos(2c\phi)
+(T_{k}^{j}(0)-T_{j}^{k}(0))\sin(2c\phi)] \,,
\end{equation}
where $i,j,k =1,2,3$ and cyclic permutations are implied.
Similarly, for the coherence tensor we have
\begin{equation}
\label{eq:tpart}
T_j^i(\phi) = \frac{1}{2}[T^{i}_{j}(0)+T^{j}_{i}(0)
+(T^{i}_{j}(0)-T^{j}_{i}(0)) \cos(2c\phi)
+(r_1^k(0)-r_2^k(0)) \sin(2c\phi)] \,.
\end{equation}
The elements of
${\bf r}_2(\phi)$
are found by symmetry $1\leftrightarrow 2$.
The quantities $r_1^i+r_2^i$, $T^{i}_{j}+T^{j}_{i}$, and $T^{i}_{i}$
are unchanged by the operation, and the form of the one-parameter
set that describes the time-evolving coherence vector is
\begin{equation}
{\bf r}_1(\phi) = {\bf R} + {\bf S}\cos(2c\phi) + {\bf V}\sin(2c\phi)\,,
\end{equation}
where ${\bf R}={\bf r}_1(0)+{\bf r}_2(0)$,
${\bf S} = {\bf r_1}(0)-{\bf r}_2(0)$, and
\begin{equation}
{\bf V} = \left(\begin{array}{c} T^{3}_{2}(0)-T^{2}_{3}(0) \\
T^{1}_{3}(0)-T^{3}_{1}(0) \\
T^{2}_{1}(0)-T^{1}_{2}(0) \end{array}\right)\,.
\end{equation}
Clearly the vector traces out an ellipse lying in the plane spanned by
${\bf S}$ and ${\bf V}$, defined by
${\bf S}\times{\bf V}$, and
passing through the point ${\bf R}$.
\section{Applications}
We shall now illustrate some of the results of Section III with
two examples. The first provides a controllability result
for nonlocal unitary interactions and the second demonstrates how the orbit
of the coherence vector can be used to describe the entangling power of the
Heisenberg exchange interaction.
\subsection{Quantum control via quantum controllers and one-dimensional
nonlocal interactions}
Let us now consider the implications of the findings of the preceding
sections for the problem of quantum control. To this end, we
adopt the nomenclature of Lloyd \cite{lloyd}
and identify spin 1 with the system $S$ whose quantum state we wish
to control, and spin 2 with the quantum controller or interface $Q$.
It is assumed that (i) only one interaction Hamiltonian $H_I$ is
in play between $S$ and $Q$ and (i) system $Q$ is completely
controllable via control Hamiltonians
$\{H_Q^{m}\}=\{1\otimes \sigma_x, 1\otimes \sigma_y,1\otimes \sigma_z\} $
that span the $su(2)$ algebra. The initial state of the bipartite
system is taken to be a product state in the ensuing analysis.
Suppose that the interaction Hamiltonian is nonlocal, but takes the
one-dimensional form $H_I=\sigma_i\otimes \sigma_j $. Then the set
$\{\{H_Q^{m}\},H_I\}$ $=\{1\otimes \sigma_x, 1\otimes \sigma_y,
1\otimes \sigma_z, \sigma_i\otimes \sigma_x,
\sigma_i\otimes \sigma_y,\sigma_i\otimes \sigma_z\} $
comprises a closed six-element subalgebra ${\mit G}_6$ of ${\mit G} $.
Given this set of operations, the vector of coherence ${\bf r}_S$
of system $S$ remains in the plane perpendicular to the $i$-axis.
It has been established in Sec.~III that when $H_I=\sigma_i\otimes \sigma_j $
is the only element of the algebra $su(4)$ affecting the two-spin system,
the vectors of coherence ${\bf r}_1$ and ${\bf r}_2$ are constrained
to move in elliptical orbits. Now, with the six-element subalgebra ${\mit G}_6$
available to the two-spin system $S+Q$, the reachable set of the system $S$
is enlarged to an {\it elliptical disk} (see Fig.~4). The principle axis of the
disk coincides with the initial coherence vector ${\bf r}_S(0)$ of
the $S$ system, while the length of its semiminor axis is given by
$b=[(r_S^k(0))^2+(r_S^l(0))^2]^{1/2}|{\bf r}_Q(0)|$, where ${\bf r}_Q(0)$ is
the initial coherence vector of system $Q$.
\smallskip
\noindent
{\it Proof.} First, if one implements the two-step sequence of a local
operation $\in 1\otimes su(2)$ on system $Q$ followed by the nonlocal
operation $H_I=\sigma_i\otimes \sigma_j$ on $S+Q$, the orbit
of ${\bf r}_S$ is necessarily an ellipse whose $a$ principle
axis lies along the initial coherence vector ${\bf r}_S(0)$
and whose semimajor axis $b$ is restricted by
$0\le b\le [(r_S^k(0))^2+(r_S^j(0))^2]^{1/2}|{\bf r}_Q(0)|$.
Hence the set reachable by this two-step procedure is the elliptic
disk in question. Second, using the Baker-Campbell-Hausdorff
formula one can show that all the elements of the six-element
subalgebra ${\mit G}_6$ can be constructed by this two step sequence,
so their reachable sets are the same.
From this result we infer that {\it the entropy of
system $S$ cannot be decreased by intervention of the quantum
interface $Q$ if the interaction Hamiltonian is limited to
the form $H_I=\sigma_i\otimes \sigma_j$}. Noting that
$|{\bf r}_Q|\le 1/2$, it follows that
$a\ge b$,
where $a$ and $b$
are respectively the magnitudes of the semimajor and semiminor
axes of the elliptical reachable set. Furthermore, it is
seen that the systems $S$ and $Q$ become maximally entangled
if the initial state of the system $S$ is situated on the equatorial
plane perpendicular to $i$-axis.
\begin{figure}
\includegraphics[width=12cm]{fig4.eps}
\caption{The gray area is the set of reachable states for
the system $S$ if one has full control of the controller $Q$ and
the interaction $\sigma_i\otimes\sigma_j$ is available. This
elliptical disk is characterized by a semimajor axis coincident
with the initial vector of coherence for $S$
and a semiminor axis with $b= [(r_S^k(0))^2+(r_S^j(0))^2]^{1/2}|{\bf r}_Q(0)|$.}
\end{figure}
\subsection{Entanglement power of Heisenberg interaction}
\label{sec:ent-heis}
Upon examining Eq.~(\ref{eq:ipart}), we see that the maximum entanglement,
realized in a maximally entangled pure state, can be achieved if
${\bf {r}}_1(0) = - {\bf {r}}_2(0)$, $|{\bf r}_1|=1/2$, and
$c\phi = \pm\pi/4$. Otherwise, the state is not perfectly entangled
since the linear entropy 1-Tr($\rho^2$) is not minimized. This
conclusion agrees with the result of Zhang {\it et al.} \cite{Zhang}
that the only perfect entanglers that can be achieved with the
Heisenberg Hamiltonian are the square-root of swap and its inverse.
However, suppose that the initial state of the two-spin system
is represented by
$$
\rho(0) = \frac{1}{4}(I + \sigma_z)\otimes (I + \sigma_z) \,,
$$
which is a pure-state density matrix for
which $r_1^z = 1/2 = r_2^z$
and $T^{z}_{z} = 1/2$, all other elements of the coherence vectors
and coherence tensor being zero. Then
$$
r_1^{x}(\phi) = r_1^z(0)\cos^2(c\phi)+r_2^z(0)\sin^2(c\phi)\,,
$$
while all other components of ${\bf r}_1$ and ${\bf r}_2$ vanish at time $\phi$,
and all other $r_1^\alpha(0) =0$.
In this case the ellipse collapses to a line and the coherence vector
simply oscillates between two values along that line.
The only element of the correlation tensor that changes is
$$
T^{1}_{2} = \frac{1}{2}\sin(2c\phi)(r_1^z(0)-r_2^z(0))\,,
$$
which vanishes for an initial tensor product of pure states for which the
subsystems are polarized in the $+z$ direction. Therefore one cannot
create maximally entangled states with these initial conditions.
\section{Conclusions}
In this paper we have developed a geometric representation for the
orbits of the coherence vectors of a two-qubit system. In various
circumstances we have shown that their evolution is described
by elliptical orbits lying within the surface of the Bloch sphere.
Importantly, every two-qubit unitary operation can be expressed as a
combination of one of the evolutions we have considered, together
with ``pre'' and ``post'' local single-qubit rotations.
We anticipate that this geometric picture will be helpful in devising
schemes for control of a quantum state via quantum interfaces, and
we have obtained a controllability result appropriate for such applications.
Given the utility of the coherence-vector picture for modeling
quantum systems and describing their entanglement,
further studies along similar lines may be fruitful. Such
work could include analysis of the orbits of higher-dimensional
quantum states, as well as consideration of the effects of measurement
operations on controllability.
\section*{Acknowledgments}
This research was supported by the U.~S.\ National Science Foundation
under Grant No.~PHY-0140316 (JWC and AM) and by the Nipher Fund.
|
{
"timestamp": "2005-08-19T20:34:04",
"yymm": "0503",
"arxiv_id": "quant-ph/0503208",
"language": "en",
"url": "https://arxiv.org/abs/quant-ph/0503208"
}
|
\section*{The effective polarization density matrix}
In these notes, our first aim is to re-derive the formulas for the
effective density matrix $\rho_\mathrm{eff}$ we introduced in
\cite{Aiello04c}, without using the concept of Stokes parameters
and without using a circular polarization basis.
To begin with, let us establish our notation. In order to make the
results comparable with the ones in Refs. \cite{Peres_et_al}, we
adopt a relativistic notation. All formulas are given in natural
units $(c = \hbar = 1)$ and all quantized fields are transverse.
In this context, single-photon plane-wave states are denoted by
$|\mathbf{k}, \lambda \rangle $ where $\mathbf{k}$ is the spatial part of the four
momentum $k = (k^0, \mathbf{k}), \; k^0 = |\mathbf{k}| \equiv \omega$, and
$\lambda = 1,2$ is the {\em linear} polarization. They are created
from the vacuum state $| 0 \rangle$ by the corresponding creation
operator $\hat{ a} _{\lambda}^\dagger(\mathbf{k} )$ which, together with the
annihilation operator $\hat{ a} _{\lambda}(\mathbf{k} )$, satisfies the
commutation relation
\begin{equation}\label{10} \bigl[ \hat{ a} _{\lambda}(\mathbf{k} ),
\hat{ a} ^\dagger_{\lambda'}(\mathbf{k}' ) \bigr] = (2 \pi)^3 2 k^0
\delta_{\lambda \lambda'} \delta^{(3)}(\mathbf{k} - \mathbf{k}').
\end{equation}
With this convention, the normalization condition between two
plane-wave states reads
\begin{equation}\label{20} \langle \mathbf{k}, \lambda | \mathbf{k}', \lambda'
\rangle = (2 \pi)^3 2 k^0 \delta_{\lambda \lambda'} \delta^{(3)}(\mathbf{k} -
\mathbf{k}'),
\end{equation}
which can be obtained directly by calculating the expectation
value with respect to the vacuum state of the left side of Eq.
(\ref{10}). Then the resolution of the identity can be written as
\begin{equation}\label{30}
1 = |0\rangle \langle 0| + \int \tilde{d} \, \mathbf{k} \sum_{\lambda = 1}^2 |\mathbf{k}, \lambda \rangle \langle \mathbf{k}, \lambda
|+ \sum \left\{ \mathrm{multiparticle} \; \mathrm{states} \right\},
\end{equation}
where $\tilde{d} \, \mathbf{k}$ denotes the Lorentz-invariant measure
\begin{equation}\label{40} \tilde{d} \, \mathbf{k} = \frac{d^3
\mathbf{k}}{(2 \pi)^3} \frac{1}{2 k^0}.
\end{equation}
The plane-wave states $| \mathbf{k}, \lambda \rangle$ are quite special
since they are eigenstates of the linear momentum of the field.
More generally, a single-photon state can be described by the
density operator
\begin{equation}\label{45}
\hat{\rho} = \sum_{\lambda, \lambda'}^{1,2} \int \tilde{d} \, \bk \, \tilde{d} \, \bk'
\rho_{\lambda \lambda'}(\mathbf{k}, \mathbf{k}') | \mathbf{k}, \lambda \rangle \langle \mathbf{k}',
\lambda' |,
\end{equation}
where $\rho_{\lambda \lambda'}(\mathbf{k}, \mathbf{k}')$ is a Hermitian positive
semidefinite $2 \times 2$ matrix. While the linear momentum and
the polarization degrees of freedom are decoupled for photons in
the plane-wave states $|\mathbf{k}, \lambda \rangle $ \cite{Aiello04c}, this
is not the case for photons described by $\hat{\rho}$ which spans
an $\infty^2$-dimensional vector space ($2$ polarization
directions for each wave vector $\mathbf{k}$). Therefore, if one want to
deal with polarization degrees of freedom only, it becomes
necessary to { reduce} $\hat{\rho}$ to an {\em effective}
finite-dimensional representation $\rho_{\mathrm{eff}}$ ($2 \times
2$ or $3 \times 3 $) \cite{Peres_et_al}. Now we present a
derivation of $\rho_\mathrm{eff}$ which enlightens the geometrical
aspects of the problem.
To this end, we have to do some manipulations on the fields. In
the Coulomb gauge, which is assumed to hold throughout this notes,
the free field vector potential operator $\hat{\mathbf{A}}(\mathbf{r},t)$
is written
\begin{equation}\label{50} \hat{\mathbf{A}}(\mathbf{r},t) = \int
\tilde{d} \, \mathbf{k} \sum_{\lambda = 1}^2\bigl[
\bm \epsilon^{(\lambda)}(\mathbf{k})\hat{ a} _{\lambda}(\mathbf{k} ) e^{-i (k^0 t - \mathbf{k}
\cdot \mathbf{r})} + \mathrm{h. c.} \bigr],
\end{equation}
where the real unit vectors $\bm \epsilon^{(\lambda)} (\mathbf{k})$ ({\em linear} polarization) satisfy the
following orthogonality conditions:
\begin{equation}\label{60} \bm \epsilon^{(1)}(\mathbf{k}) \cdot
\bm \epsilon^{(2)}(\mathbf{k}) = 0, \qquad \bm \epsilon^{(1)}(\mathbf{k}) \times
\bm \epsilon^{(2)}(\mathbf{k}) = \frac{\mathbf{k}}{k^0}.
\end{equation}
Since we are working with the {\em transverse} field ($A^0(\mathbf{r}, t)
= 0$), we can assume an Euclidean metric in the spatial
3-dimensional space and make no distinction between high and low
indices.
It is useful to define $\bm \epsilon^{(3)}(\mathbf{k}) \equiv
{\mathbf{k}}/{k^0}$ and to build the $\mathbf{k}$-dependent complete basis
$\mathcal{E}(\mathbf{k})= \{ \bm \epsilon^{(1)}(\mathbf{k}), \bm \epsilon^{(2)}(\mathbf{k}),
\bm \epsilon^{(3)}(\mathbf{k}) \}$.
Let us consider now three unit vectors $\bm x, \bm y, \bm z$ of a
Cartesian coordinate system or, more generally, a set
$\mathcal{R}$ of three real orthogonal unit vectors $\mathcal{R} =
\{ \bm e^{(1)}, \bm e^{(2)}, \bm e^{(3)} \}$:
\begin{equation}\label{70} \bm e^{(a)}\cdot \bm e^{(b)}
= \delta^{ab}, \qquad (a,b =1, \ldots, 3),
\end{equation}
which form a complete basis in the ordinary Euclidean
3-dimensional space:
\begin{equation}\label{80} \sum_{a =1}^3 \bm e^{(a)} : \bm e^{(a)} = \mathbf{1} \quad
\Leftrightarrow \quad\sum_{a =1}^3 e_i^{(a)} e_j^{(a)} =
\delta_{ij}, \qquad (i,j =1, \ldots, 3),
\end{equation}
where the unit dyadic $\mathbf{1}$ has been written as a sum of
dyadic products $\bm e^{(a)}: \bm e^{(a)}$ ($a = 1,2,3$).
The vectors $\bm e^{(a)}$ define three orthogonal {\em spatial}
orientations and they are independent from the {\em momentum}
direction $\mathbf{k}/k^0$. However, for a given $\mathbf{k}$, one can write the
orthogonal transformation $\Lambda(\mathbf{k})$ between the two basis
$\mathcal{E}(\mathbf{k})$ and $\mathcal{R}$ as
\begin{equation}\label{90}
\begin{array}{rcl}
\Lambda_{ab}(\mathbf{k}) & \equiv & \bm e^{(a)} \cdot \bm \epsilon^{(b)}(\mathbf{k})\\
& = & \displaystyle{\sum_{i = 1}^3 e^{(a)}_i
\epsilon^{(b)}_i(\mathbf{k})}.
\end{array}
\end{equation}
Then we can write
\begin{equation}\label{100} \bm \epsilon^{(\lambda)} (\mathbf{k}) = \sum_{b
=1}^3 \bm e^{(b)} \Lambda_{b \lambda} (\mathbf{k}), \qquad (\lambda =
1,2,3),
\end{equation}
and insert this formula in Eq. (\ref{50}) in order to obtain:
\begin{equation}\label{110}
\hat{\mathbf{A}}(\mathbf{r},t)= \sum_{b =1}^3 \bm e^{(b)} \hat{A}_b(\mathbf{r},t),
\end{equation}
where we have defined the $b$-th component of the field
$\hat{\mathbf{A}}(\mathbf{r},t)$ as
\begin{equation}\label{120}
\begin{array}{rcl}
\hat{A}_b(\mathbf{r},t) & = & \displaystyle{ \int \tilde{d} \, \mathbf{k}
\sum_{\lambda = 1}^2\bigl[ \Lambda_{b \lambda} (\mathbf{k})
\hat{ a} _{\lambda}(\mathbf{k} ) e^{-i (k^0 t - \mathbf{k}
\cdot \mathbf{r})} + \mathrm{h. c.} \bigr]}\\\\
& \equiv & \displaystyle{\int \tilde{d} \, \mathbf{k} \,
\hat{\mathcal{A}}_b (\mathbf{k}) e^{-i (k^0 t - \mathbf{k} \cdot \mathbf{r})} +
\mathrm{h. c.}} ,
\end{array}
\end{equation}
where we have defined the transformed annihilation operators
$\hat{\mathcal{A}}_b(\mathbf{k} )$ ($b = 1,2,3$), as:
\begin{equation}\label{130}
\hat{\mathcal{A}}_b(\mathbf{k} ) \equiv \sum_{\lambda = 1}^2 \Lambda_b
\/_\lambda (\mathbf{k}) \hat{ a} _{\lambda}(\mathbf{k} ), \qquad (b=1,2,3).
\end{equation}
It is easy to check that these operators satisfy the following
commutation relations
\begin{equation}\label{140}
\bigl[\hat{\mathcal{A}}_a(\mathbf{k}),
\hat{\mathcal{A}}_{a'}^\dagger(\mathbf{k}') \bigr] = (2 \pi)^3 2 k^0
\Delta_{a a'} \delta^{(3)} (\mathbf{k} - \mathbf{k}'),
\end{equation}
where we have defined the transverse Kronecker symbol $\Delta_{a
a'}$ as
\begin{equation}\label{150}
\Delta_{aa'} \equiv \delta_{aa'} - \frac{k_a k_{a'}}{|\mathbf{k}|^2}.
\end{equation}
As expected, the longitudinal part $- \frac{k_a k_{a'}}{|\mathbf{k}|^2}$
of $\Delta_{aa'}$, spoils the {\em canonical} commutation
relation.
In the quantum theory of photo-detection it is a standard practice
\cite{MandelBook} to define the positive frequency operators
$\hat{V}_b(\mathbf{r},t)$ ($b = 1,2,3$) as
\begin{equation}\label{160}
\hat{V}_b(\mathbf{r},t) \equiv \int \tilde{d} \, \bk \sqrt{2 k^0} \hat{\mathcal{A}}_b
(\mathbf{k}) e^{-i (k^0t - \mathbf{k} \cdot \mathbf{r})},
\end{equation}
and such that $\sum_{b=1}^3 \hat{V}_b^\dagger(\mathbf{r},t)\hat{V}_b
(\mathbf{r},t)$ represent the photon density in $(\mathbf{r},t)$. These
operators can be used to build the {\em polarization correlation}
operators
\begin{equation}\label{170}
\begin{array}{rcl}
\hat{J}_{ab} & \equiv & \displaystyle{\int d^3 \mathbf{r} \, \hat{V}_a^{ \dagger} (\mathbf{r},t) \hat{V}_b}(\mathbf{r},t) \\\\
& = & \displaystyle{ \int \tilde{d} \, \bk \, \hat{\mathcal{A}}_a^{ \dagger}(\mathbf{k})
\hat{\mathcal{A}}_b(\mathbf{k})},
\end{array}
\end{equation}
where the last result follows immediately from Eq. (\ref{160}).
The meaning of the matrix operator $\hat{J} \equiv
||\hat{J}_{ab}||$ becomes clear when we calculate its trace:
\begin{equation}\label{180}
\begin{array}{rcl}
\mathrm{Tr} \hat{J} & = & \displaystyle{\sum_{a = 1}^3 \hat{J}_{aa} } \\\\
& = & \displaystyle{} \int \tilde{d} \, \bk \sum_{\lambda =1}^2
\hat{ a} _\lambda^\dagger (\mathbf{k}) \hat{ a} _\lambda(\mathbf{k}) \equiv \hat{N},
\end{array}
\end{equation}
where $\hat{N}$ denotes the photon-number operator.
Now, by inserting the identity resolution Eq. (\ref{30}) in Eq.
(\ref{170}), we obtain
\begin{equation}\label{190}
\begin{array}{rcl}
\hat{J}_{ab} & = & \displaystyle{ \int \tilde{d} \, \bk \, \hat{\mathcal{A}}_a^{
\dagger}| 0 \rangle \langle 0 |
\hat{\mathcal{A}}_b} + \sum \left\{ \mathrm{multiparticle} \; \mathrm{states} \right\} \\\\
& \equiv & \displaystyle{ \int_\mathcal{D} \tilde{d} \, \mathbf{k} \, | \mathbf{k}, a \rangle \langle \mathbf{k} , b | \qquad (a,b=1,2,3)},
\end{array}
\end{equation}
where the last equality holds {\em only} in the Hilbert spaces
spanned by the one-photon states, $\mathcal{D}$ represents the set
of \emph{detected} modes, and we have defined the single-photon
states $ | \mathbf{k}, a \rangle$ as:
\begin{equation}\label{200}
\hat{\mathcal{A}}_a^{
\dagger}| 0 \rangle = | \mathbf{k}, a \rangle, \qquad (a=1,2,3).
\end{equation}
From Eqs. (\ref{180}-\ref{190}) it is clear that $\hat{J}_{11},
\hat{J}_{22}$ and $\hat{J}_{33}$ form a POVM (positive operator
valued measure \cite{PeresBook}) in the space spanned by the
one-photon states. As we saw previously, the operators
$\hat{J}_{ab}$'s allow us to introduce a $3 \times 3$ correlation
matrix operator
\begin{equation}\label{192}
\hat{J} \equiv \left( \begin{array}{ccc}
\displaystyle{ \hat{J}_{11} } & \displaystyle{ \hat{J}_{21} }
& \displaystyle{ \hat{J}_{31} }
\\
\displaystyle{ \hat{J}_{12} } & \displaystyle{ \hat{J}_{22} }
& \displaystyle{ \hat{J}_{32} }
\\
\displaystyle{ \hat{J}_{13} } & \displaystyle{ \hat{J}_{23} }
& \displaystyle{ \hat{J}_{33} }
\end{array} \right).
\end{equation}
Now, for a light beam properly
collimated around the direction $\bm e^{(3)} = \bm z$, the $2 \times
2$ matrix obtained by extracting the first two rows and two
columns from $\mathbb{J} \equiv \langle \hat{J} \rangle $, coincides with
the well known
{\em coherency matrix} of the beam
\cite{BornWolf}, where the bracket average $\bigl\langle \cdot \bigr\rangle$ is understood with respect
to the state of the field. More generally, three independent $2 \times 2$
matrices can be extracted from $\hat{J}$:
%
\begin{equation}\label{193}
\hat{J}^{(a)} = \left(
\begin{array}{cc}
\displaystyle{ \hat{J}_{bb} } & \displaystyle{ \hat{J}_{cb}
}
\\
\displaystyle{ \hat{J}_{bc} } & \displaystyle{ \hat{J}_{cc}
}
\end{array}
\right),
\quad
\begin{array}{rcl}
a,b,c &\in & \{1,2,3 \},\\
\qquad c & > & b, \\ c,b & \neq & a.
\end{array}
\end{equation}
In principle, each $\mathbb{J}^{(a)} \equiv \langle \hat{J}^{(a)} \rangle$
can be determined by measuring the Stokes parameters of the beam
(either classical or quantum), with a polarization analyzer whose
axis is parallel to $\bm e^{(a)}$. For example, for a beam
propagating along the axis $\bm z$, a set of {\em generalized}
Hermitian Stokes operators \cite{JauchBook,Aiello04c} can be
defined as
\begin{equation}\label{194}
\hat{S}_\mu \equiv \mathrm{Tr}\{ \sigma_{(\mu)} \hat{J}^{(3)} \},
\qquad (\mu = 0, \dots,3),
\end{equation}
where the $\sigma_{(\mu)}$ $(\mu=0,1,2,3)$ are the normalized
Pauli's matrices \cite{Aiello04d}. Then from Eqs.
(\ref{193}-\ref{194}) it readily follows
\begin{equation}\label{195}
\mathbb{J}^{(3)} = \sum_{\mu = 0 }^3 s_\mu \sigma_{(\mu)},
\end{equation}
where $s_\mu \equiv \langle \hat{S}_\mu \rangle$.
Now we are ready to accomplish our initial task by introducing the
$2 \times 2$ effective reduced density matrix
\begin{equation}\label{210}
\rho_\mathrm{eff} \equiv \frac{\mathbb{J}^{(3)}}{\mathrm{Tr}
\mathbb{J}^{(3)}}.
\end{equation}
It is easy to check that when the set $\mathcal{D}$ of the
detected modes reduces to a single mode $\mathbf{k}
\parallel \bm e^{(3)}$, the definition of $\rho_\mathrm{eff}$ above
coincides with the well known polarization density matrix of a
photon \cite{DauFieldRel}. More generally, by using Eq.
(\ref{50}), it is possible to introduce an effective $3 \times 3$
reduced density matrix as $\rho \equiv \mathbb{J} / \mathrm{Tr} [
\mathbb{J}]$, which coincides with the one given by Peres {\em et
al. }\cite{Peres_et_al}.
\section*{Single-photon scattering}
Let us consider now a generic scattering process which transform
the initial single-photon density operator
$\hat{\rho}^{\mathrm{in}}$ in the output density operator
$\hat{\rho}^{\mathrm{out}}$. The most general linear
transformation between $\hat{\rho}^{\mathrm{in}}$ and
$\hat{\rho}^{\mathrm{out}}$, which leaves the density operator
Hermitian and positive semidefinite, can be written
\begin{equation}\label{n1}
\hat{\rho}^{\mathrm{out}} = \sum_{A \in \mathcal{S}} p_A
\mathcal{T}^{A } \hat{\rho}^{\mathrm{in}} \mathcal{T}^{A \dagger},
\end{equation}
where the scattering system has been represented by the ensemble
$\mathcal{S}$ of scattering matrices $\{ \mathcal{T}^{A} \}$, each
of them occurring with probability $p_A \geq 0$. If we insert Eq.
(\ref{45}) into Eq. (\ref{n1}) we obtain
\begin{equation}\label{n2}
\begin{array}{ccl}
\hat{\rho}^{\mathrm{out}}
& = & \displaystyle{ \sum_{\lambda, \lambda'}^{1,2} \int \tilde{d} \, \bk \, \tilde{d} \, \bk'
\rho_{\lambda \lambda'}^\mathrm{in}(\mathbf{k}, \mathbf{k}') \sum_{A \in
\mathcal{S}} p_A \mathcal{T}^{A} | \mathbf{k}, \lambda \rangle \langle \mathbf{k}',
\lambda' |\mathcal{T}^{A \dagger} },
\end{array}
\end{equation}
where
\begin{equation}\label{n4}
\begin{array}{ccl}
\displaystyle{ \sum_{A \in \mathcal{S}} p_A \mathcal{T}^{A } |
\mathbf{k}, \lambda \rangle \langle \mathbf{k}', \lambda' |\mathcal{T}^{A \dagger} } &
= & \displaystyle{ \sum_{\theta, \theta'}^{1,2} \int \tilde{d} \, \bq \, \tilde{d} \, \bq'
\sum_{A \in \mathcal{S}} p_A |\mathbf{q} \theta \rangle \langle \mathbf{q} \theta |
\mathcal{T}^{A} | \mathbf{k}, \lambda \rangle \langle \mathbf{k}', \lambda'
|\mathcal{T}^{A \dagger} | \mathbf{q}' \theta' \rangle \langle \mathbf{q}' \theta' |}\\\\
& = & \displaystyle{ \sum_{\theta, \theta'}^{1,2} \int \tilde{d} \, \bq \,
\tilde{d} \, \bq' \sum_{A \in \mathcal{S}} p_A
\mathcal{T}^{A }_{\theta \lambda}(\mathbf{q},\mathbf{k}) \mathcal{T}^{A \dagger
}_{ \lambda' \theta'}(\mathbf{k}', \mathbf{q}')
|\mathbf{q} \theta \rangle \langle \mathbf{q}' \theta' |}.
\end{array}
\end{equation}
From the equation above, it is straightforward to see that we can
write
\begin{equation}\label{n3}
\hat{\rho}^\mathrm{out} = \displaystyle{ \sum_{\theta,
\theta'}^{1,2} \int \tilde{d} \, \bq \, \tilde{d} \, \bq' \rho^\mathrm{out}_{\theta
\theta'}(\mathbf{q}, \mathbf{q}')
|\mathbf{q} \theta \rangle \langle \mathbf{q}' \theta' |}
\end{equation}
where
\begin{equation}\label{n5}
\rho^\mathrm{out}_{\theta \theta'}(\mathbf{q}, \mathbf{q}') =
\sum_{A \in \mathcal{S}} p_A \sum_{\lambda, \lambda'}^{1,2}
\int \tilde{d} \, \bk \, \tilde{d} \, \bk'
\mathcal{T}^{A}_{\theta \lambda}(\mathbf{q},\mathbf{k}) \rho_{\lambda
\lambda'}^\mathrm{in}(\mathbf{k}, \mathbf{k}') \mathcal{T}^{A \dagger}_{
\lambda' \theta'}(\mathbf{k}', \mathbf{q}').
\end{equation}
Since $\rho^{\mathrm{t}}(\mathbf{k}, \mathbf{k}') = || \rho_{\lambda
\lambda'}^{\mathrm{t}}(\mathbf{k}, \mathbf{k}') ||$ (t = in, out) are $2 \times
2$ matrices, it is always possible to express them in the complete
Pauli basis as \cite{Aiello04d}
\begin{equation}\label{n7}
\rho^{\mathrm{t}}(\mathbf{k}, \mathbf{k}') = \sum_{\mu = 0} ^3
S_\mu^\mathrm{t}(\mathbf{k}, \mathbf{k}') \sigma_{(\mu)},
\end{equation}
where we have introduced the \emph{two-mode Stokes parameters}
$S_\mu^\mathrm{t}(\mathbf{k}, \mathbf{k}') = \mathrm{Tr} \{ \sigma_{(\mu)}
\rho^{\mathrm{t}}(\mathbf{k}, \mathbf{k}') \}$. Since Eq. (\ref{n5}) can be
written in matrix form as
\begin{equation}\label{n7b}
\rho^\mathrm{out}(\mathbf{q}, \mathbf{q}') =
\sum_{A \in \mathcal{S}} p_A
\int \tilde{d} \, \bk \, \tilde{d} \, \bk'
\mathcal{T}^{A}(\mathbf{q},\mathbf{k}) \rho^\mathrm{in}(\mathbf{k}, \mathbf{k}')
\mathcal{T}^{A \dagger}(\mathbf{k}', \mathbf{q}'),
\end{equation}
it is easy to write
\begin{equation}\label{n8}
\begin{array}{rcl}
\displaystyle{ S^\mathrm{out}_\mu (\mathbf{q}, \mathbf{q}')} & = &
\displaystyle{\sum_{\nu = 0} ^3 \int \tilde{d} \, \bk \, \tilde{d} \, \bk'
S^\mathrm{in}_\nu (\mathbf{k}, \mathbf{k}') \sum_{A \in \mathcal{S}} p_A
\mathrm{Tr} \left\{ \sigma_{(\mu)} \mathcal{T}^{A }(\mathbf{q},\mathbf{k})
\sigma_{(\nu)} \mathcal{T}^{A \dagger}(\mathbf{k}',
\mathbf{q}')\right\}}\\\\
& \equiv &
\displaystyle{ \sum_{\nu = 0} ^3 \int \tilde{d} \, \bk \, \tilde{d} \, \bk'
S^\mathrm{in}_\nu (\mathbf{k}, \mathbf{k}') \sum_{A \in \mathcal{S}} p_A \,
m_{\mu \nu}^A}(\mathbf{q}, \mathbf{q}'; \mathbf{k}, \mathbf{k}')\\\\
& \equiv &
\displaystyle{ \sum_{\nu = 0} ^3 \int \tilde{d} \, \bk \, \tilde{d} \, \bk' M_{\mu
\nu}}(\mathbf{q}, \mathbf{q}'; \mathbf{k}, \mathbf{k}') S^\mathrm{in}_\nu (\mathbf{k}, \mathbf{k}'),
\end{array}
\end{equation}
where the four-mode density Mueller matrix
\begin{equation}\label{n9}
m_{\mu \nu}^A (\mathbf{q}, \mathbf{q}'; \mathbf{k}, \mathbf{k}') = \mathrm{Tr} \left\{
\sigma_{(\mu)} \mathcal{T}^{A}(\mathbf{q},\mathbf{k}) \sigma_{(\nu)}
\mathcal{T}^{A \dagger} (\mathbf{k}', \mathbf{q}') \right\},
\end{equation}
is defined for a single ensemble realization $A$, while
\begin{equation}\label{n10}
M_{\mu \nu} (\mathbf{q}, \mathbf{q}'; \mathbf{k}, \mathbf{k}') = \sum_{A \in \mathcal{S}} p_A
\, m_{\mu \nu}^A (\mathbf{q}, \mathbf{q}'; \mathbf{k},
\mathbf{k}'),
\end{equation}
represents the ensemble-averaged four-mode density Mueller
matrix. It is easy to see that when the input state is a
single-mode $\mathbf{k}_0$ state:
\begin{equation}\label{n20}
{\rho}^\mathrm{in}_{\lambda \lambda'} (\mathbf{k} , \mathbf{k}') = \rho_{\lambda \lambda'} \delta^{(3)} (\mathbf{k} -
\mathbf{k}_0)\delta^{(3)} (\mathbf{k}' - \mathbf{k}_0),
\end{equation}
then
\begin{equation}\label{n30}
S^\mathrm{in}_{\mu} (\mathbf{k} , \mathbf{k}') = s^\mathrm{in}_\mu
\delta^{(3)} (\mathbf{k} -
\mathbf{k}_0)\delta^{(3)} (\mathbf{k}' - \mathbf{k}_0),
\end{equation}
where $\rho = || \rho_{\lambda \lambda'} ||$ and
$s^\mathrm{in}_\mu \equiv \mathrm{Tr}\{\sigma_{(\mu)} \rho \}$. In
this case, the single-mode input and output Stokes parameters
have the same functional relation as their
classical counterparts:
\begin{equation}\label{n40}
\begin{array}{rcl}
\displaystyle{ S^\mathrm{out}_\mu (\mathbf{q}_0 , \mathbf{q}_0)} & = &
\displaystyle{ \sum_{\nu = 0} ^3 \int \tilde{d} \, \bk \, \tilde{d} \, \bk' M_{\mu
\nu}(\mathbf{q}_0 , \mathbf{q}_0; \mathbf{k}, \mathbf{k}') s^\mathrm{in}_\nu \delta^{(3)} (\mathbf{k}
-
\mathbf{k}_0)\delta^{(3)} (\mathbf{k}' - \mathbf{k}_0)}\\\\
& = &
\displaystyle{ \sum_{\nu = 0} ^3 M_{\mu \nu}(\mathbf{q}_0 , \mathbf{q}_0;
\mathbf{k}_0, \mathbf{k}_0) s^\mathrm{in}_\nu }\\\\
& \equiv &
\displaystyle{ \sum_{\nu = 0} ^3 M_{\mu \nu} s^\mathrm{in}_\nu }.
\end{array}
\end{equation}
\begin{acknowledgments}
We acknowledge support from the EU under the
IST-ATESIT contract. This project is also supported by FOM.
\end{acknowledgments}
|
{
"timestamp": "2005-03-14T13:39:17",
"yymm": "0503",
"arxiv_id": "quant-ph/0503124",
"language": "en",
"url": "https://arxiv.org/abs/quant-ph/0503124"
}
|
\section{Introduction}
Since Helmholtz \cite{helmholtz77}, it has become natural to describe
a self-sustained\footnote{Self-sustained is a term indicating
oscillation driven by a constant energy input.} musical instrument as
an exciter
coupled to a resonator. More recently, McIntyre et al.\
\cite{mcintyre83} have highlighted that simple models are able to
describe the main functioning of most self-sustained musical
instruments. These models rely on few equations whose
implementation is not \textsc{cpu}-demanding, mainly because the
nonlinearity is spatially localized in an area small compared to the
wavelength. This makes them well adapted for real-time computation
(including both transient and steady states). These models are
particularly popular in the framework of sound synthesis.
On the other hand, calculation in the frequency domain is suitable
for determining periodic solutions of the model (the
values of the harmonics as well as the playing frequency) for a given
set of parameters. Such information can be provided by an iterative
method named the harmonic balance method (HBM). Though the name
``harmonic balance'' seems to date back to 1936 \cite{krylov36}, the
method was popularized nearly forty years ago for electrical and
mechanical engineering purposes, first for forced vibrations
\cite{urabe65}, later for auto-oscillating systems \cite{stokes72}.
The modern version was presented rather shortly after by Nakhla and
Vlach \cite{nakhla76}. In 1978, Schumacher was the first one to use
the HBM for musical acoustics purposes with a focus on the clarinet
\cite{schumacher78}. However in this paper, the playing frequency is
not determined by the HBM. This shortcoming is the major
improvement brought by Gilbert et al.\ \cite{gilbert89} eleven
years later, who proposed a full study of the clarinet including
the playing frequency as an unknown of the problem.
The fact that the HBM can only calculate periodic solutions,
may seem as a drawback. Certainly, transients such as the attack are
impossible to calculate, and the periodic result is boring to listen
to and does not represent the musicality of the instrument. Therefore
the HBM is definitely not intended for sound synthesis. Nevertheless,
self-sustained musical instruments are usually used to generate
harmonic sounds, which are periodic by definition. The HBM
is thus very useful to investigate the behavior of a physical model
of an instrument, depending on its parameter values. This is possible
for both stable and unstable solutions, without care of precise
initial conditions. Moreover, HBM results can be compared to
approximate analytical calculations (like the variable truncation
method (VTM) \cite{kergomard00}), in order to check the validity of
the approximate model considered.
The present paper is based on the work of Gilbert et al \cite{gilbert89}.
Our main contributions are:
extension of the diversity of equations managed, improved convergence
of the method, introduction of basic continuation facilities, and from
a practical point of view, faster calculations.
While the main idea is already described by Gilbert et al.\
\cite{gilbert89},
Section~\ref{s:nummeth} details the principle of the HBM, in
particular the discretization of the problem, both in time and frequency.
Section~\ref{s:harmbal} is devoted to the various contributions of
the current work, which are applied in a computer program called
Harmbal \cite{harmbal}. The framework is defined to include models with three
equations: two linear differential equations, written in the frequency
domain, and a nonlinear coupling equation in the time domain (see
Sec.~\ref{s:self-sustained}). As usual in the HBM, this system of
three equations is solved iteratively. The solving method chosen
(Newton-Raphson, Sec.~\ref{s:harmdet}) has been
investigated and its convergence has been improved through a
backtracking scheme (Secs.~\ref{s:holes}
and~\ref{s:backtracking}).
To illustrate the advantages of the HBM and the improvements, a few
case studies were performed and are presented in Section~\ref{s:case}.
They are based on a classical model of single reed instruments which
is presented in Section~\ref{s:clarinet}. In Sections~\ref{s:verif}
and further, simplifications to each of the three equations are
introduced so that the results could be compared to analytical
calculations, both for cylindrical and stepped-cones bores. Finally
the full model is compared to time-domain simulations. This also
shows the modularity of Harmbal. The comparison is achieved through
the investigation of bifurcation diagrams as the dimensionless blowing
pressure is altered. The derivation of a branch of solution is
obtained thanks to basic continuation with an
auto-adaptative parameter
step.
Finally, various questions are tackled through practical experience
from using Harmbal. Section~\ref{s:practexp} discusses multiplicity of
solutions and poor robustness in the frequency estimation.
\section{Numerical method}\label{s:nummeth}
\subsection{The harmonic balance method}\label{s:HBM}
The harmonic balance method is a numerical method to calculate the
steady-state spectrum of periodic solutions of a
nonlinear dynamical system. In this paper we are only concerned with
periodic solutions. The following provides a detailed and general
description of the method for a nonlinearly coupled exciter-resonator
system.
Let $X(\omega_k)$, $k = 0, \dots, N_t-1$ be the Discrete
Fourier transform (DFT) of one period $x(t)$, $0\le t<T$, of a
$T$-periodic solution of a mathematical system to be defined.
$X(\omega_k)$ will have a number of complex components $N_t$, which
depends on the sampling frequency $f_s=1/T_s$ with which we discretize
$x(t)$ into $N_t=T/T_s$ equidistant samples. Furthermore,
$\omega_k{=}2\pi f_p T_s k$ is the angular frequency of each harmonic of the
fundamental frequency $f_p$ of the oscillation, referred to as the {\em playing
frequency}. Note that the sampling frequency $f_s=N_tf_p$ is automatically
adjusted to the current playing frequency so that we always
consider one period of the oscillation while keeping $N_t$
constant. Note also that $N_t$ should be sufficiently large to avoid
aliasing. Moreover, if it is chosen a power of two, the Fast Fourier
transform (FFT) may be used. Assuming that $N_p<N_t/2$ harmonics
is sufficient to
describe the solution, we define $\vec X\in\mathbb{R}^{2N_p+2}$
as the $N_p+1$ first real components (denoted by $\Re$) of $X(\omega_k)$
followed by their imaginary components ($\Im$):
\begin{align}
\vec X=&\left[\Re\left(X(\omega_0)\right),
\dots,\Re\left(X(\omega_{N_p})\right),\right.\\
&\qquad\left.\Im\left(X(\omega_0)\right), \dots,
\Im\left( X(\omega_{N_p})\right)\right].\nonumber
\end{align}
Note that the components $X_0$ and $X_{N_p+1}$ are the real
and imaginary DC components respectively (and that $X_{N_p+1}$ is
always zero). Our mathematical system can thus be defined by
the nonlinear function $F: \mathbb{R}^{2N_p+3} \to
\mathbb{R}^{2N_p+2}$:
\begin{equation}
\vec X = \vec F(\vec X,f_p).
\label{e:gensyst}
\end{equation}
Until now, the playing frequency has silently been assumed to be a known
quantity. In autonomous systems, however, the frequency is an
additional unknown, so that the $N_p$-harmonic solution seeked is defined by
$2N_p{+}3$ unknowns linked through the $2N_p{+}2$ equations~(\ref{e:gensyst}).
However, it is well known that as
$\vec X$ is a periodic solution of a
dynamical system, any $\vec X^{\prime}$ deduced from
$\vec X$ by a phase rotation (i.e.\ a shift in
the time domain) is also a solution. Thus an additional constraint
has to be added in order to select a single periodic solution among
the infinity of phase-rotated solutions. A common choice (see Ref.\
\onlinecite{gilbert89}) is to consider the solution for which the
first harmonic is real (i.e.\ its imaginary part, $X_{N_p+2}$, is zero). This
additional constraint decreases the number
of unknowns to $2N_p{+}2$ for an $N_p$-harmonic periodic
solution. Thus we get $\vec F: \mathbb{R}^{2N+2} \to \mathbb{R}^{2N+2}$,
and it is now possible to find periodic solutions, if they
exist.
Finally, a simple way of avoiding trivial solutions to
equation~(\ref{e:gensyst}) is to look for roots of the function $\vec G:
\mathbb{R}^{2N_p+2} \to \mathbb{R}^{2N_p+2}$, defined by
\begin{equation}
\vec G(\vec X,f_p)=\frac{\vec X-\vec F(\vec X,f_p)}{X_1},
\label{e:G}
\end{equation}
i.e.\ $\vec G(\vec X,f_p)=0$. This equation is usually solved numerically
through an iteration process, for instance by the Newton-Raphson
method as in our case. How to handle the playing frequency $f_p$ will
be discussed in the following section.
\subsection{Iteration by Newton-Raphson}\label{s:newton}
The equation $\vec G(\vec X,f) = 0$, $\vec G$ being defined by
equation~\eqref{e:G}, is nonlinear and has usually no analytical
solution. (For readability we leave out the index $p$ on the
playing frequency until end of Sec.~\ref{s:harmbal}.) This
section describes the common, iterative Newton-Raphson method.
This is the method used in the program Harmbal (see
Section~\ref{s:harmbal}) although it had to be refined with a
backtracking procedure to improve its convergence, as discussed in
Section~\ref{s:backtracking}.
For the sake of later reference, it is useful to re\-col\-lect the
principles of Newton's method for a one-dimensional problem $g(x)=0$.
Starting with an estimate $x^0$ of the solution, the
next estimate $x^1$ is defined as the intersection
point between the tangent to $g$ at $x_0$ and the
$x$-axis. The method can be summarized as
\begin{equation}
x^{i+1}=x^i-\frac{g(x^i)}{g'(x^i)}.
\end{equation}
This is repeated, as shown in Figure~\ref{f:newton}, while increasing
the iteration index $i$ until $g(x^i)<\varepsilon$, where
$\varepsilon$ is a user-defined threshold value.
\begin{figure}
\ifgalleyfig
\includegraphics[width=1.95in]{eps/newton_g.eps
\else
\ifoutputfig
\includegraphics[width=5.5in]{eps/newton_g.eps
\fi
\fi
\caption
[The iteration process of Newton's method]
{\label{f:newton}
\ifgalleyfig
{The iteration process of Newton's method}
\fi}
\end{figure}
In our $2N_p{+}2$-dimensional case, we have a vector
problem: we search $(\vec X,f)$ for which $\vec G(\vec X,f)=0$. Newton's
method is generalized to the Newton-Raphson method,
which may be written \cite{numrec}:
\begin{equation}
(\vec X^{i+1}, f^{i+1})=(\vec X^i , f^i )
-\left(\mathbf{J}_G^i\right)^{-1}\!\!\cdot \vec G(\vec X^i,f^i),
\label{e:fullstep}
\end{equation}
where $\mathbf{J}^i_G{\triangleq}\nabla G(\vec X^i,f^i)$ is the
{\em Jacobian\/} matrix of $\vec G$ at $(\vec X^i,f^i)$. Note that
all derivatives by $X_{N_p+2}$, which was chosen to be zero, are
ignored. The column $N_p{+}2$ in the Jacobian is thus replaced by the
derivatives with respect to the playing frequency $f$.
$\mathbf{J}^i_G$ is thus a $(2N_p{+}2)$-square
matrix. This means that line number $N_p{+}2$ in
equation~\eqref{e:fullstep} gives the new frequency $f$ instead of
$X_{N_p+2}$.
We define the {\em Newton step\/} $\Delta\vec X{=}\vec X^{i+1}{-}\vec
X^i$ (where $\Delta f{=}f^{i+1}{-}f^i$ replaces $\Delta X_{N_p+2}$),
which follows the local steepest descent direction.
The Jacobian may be found analytically if $\vec G$ is given analytically,
but it is usually sufficient to use the first-order approximation
\begin{equation}
J_{jk}=\frac{\partial G_j}{\partial X_k}
\simeq\frac{G_j(\vec X+\delta\vec X_k,f)-G_j(\vec X,f)}{\delta X},
\label{e:jacobij}
\end{equation}
except for $k=N_p+2$, in which case we use
\begin{equation}
J_{j,N+2}=\frac{\partial G_j}{\partial f}
\simeq\frac{G_j(\vec X,f+\delta f)-G_j(\vec X,f)}{\delta f}.
\label{e:jacobim}
\end{equation}
The components of $\delta\vec X_k$ are zero except for the $k$th
one, which is the tiny perturbation $\delta X$. The iteration has
converged when $|\vec G^i|{\triangleq}|\vec G(\vec
X^i,f^i)|<\varepsilon$.
We found $\varepsilon=10^{-5}$ to be a good compromize between
computation time and solution accuracy.
\section{Implementation and Harmbal}\label{s:harmbal}
\subsection{Equations for self-sustained musical
instruments \label{s:self-sustained} }
Though, to the authors' knowledge, the harmonic balance method in the
context of musical acoustics with unknown playing frequency has only been applied to study models of
clarinet-like instruments, it should be possible to consider many
different classes of self-sustained instruments.
It is well accepted that sound production by a musical instrument
results from the interaction between an exciter and a resonator
through a nonlinear coupling. Moreover, in most playing conditions,
linear modelling of both the exciter and the resonator is a good
approximation.
Therefore, within these hypotheses, any musical instrument could be
modelled by the following three equations:
\begin{equation}
\qquad\quad\left\{
\begin{array}{l@{\qquad}c}
Z_e(\omega) X_e(\omega) = X_c(\omega)& \mathrm{(a)}\\
\phantom{Z_e(\omega)}X_c(\omega) = Z_r(\omega) X_r(\omega) & \mathrm{(b)}\\
\mathcal{F}(x_c(t),x_e(t),x_r(t)) =0 & \mathrm{(c)}
\end{array}
\right.
\label{e:any_instr}
\end{equation}
where $Z_e$ is the dynamic stiffness and $Z_r$ is the input impedance of the exciter and the
resonator, respectively, and $X_e$ and $X_r$ are the spectra
describing the dynamics of the exciter and the resonator during the
steady state (periodicity assumption). $X_c$ is the spectrum of
the coupling variable. All these quantities, and thus
equations~(\ref{e:any_instr}a--b), are defined in the
Fourier domain. Equation~(\ref{e:any_instr}c) is
written in the time domain, where $\mathcal{F}$ is a nonlinear functional of
$x_c$, $x_e$, and $x_r$, which are the inverse Fourier transforms of
$X_c$, $X_e$, and $X_r$, respectively. We apply the discretization as
described in Section~\ref{s:HBM}, implying that
equations~(\ref{e:any_instr}a--b) become vector equations where the
impedances must be written as real $(2N_p{+}2){\times}(2N_p{+}2)$-matrices to accommodate the rules of complex multiplication:
\begin{equation}
Z(f)=\left(
\begin{array}{cc}
\Re(\tilde{Z}(f))&-\Im(\tilde{Z}(f))\\
\Im(\tilde{Z}(f))&\Re(\tilde{Z}(f))\\
\end{array}
\right)
\label{e:impmat1}
\end{equation}
where
\begin{equation}
\tilde{Z}(f)=\left(
\begin{array}{cccc}
Z(0) &0&\cdots&0\\
0& Z(\omega_1)&&0\\
\vdots&&\ddots&\vdots\\
0& 0 &\cdots&Z(\omega_{N_p})
\end{array}
\right)
\label{e:impmat2}
\end{equation}
is complex, and $\Re(\tilde{Z})$ and $\Im(\tilde{Z})$ are the
real and imaginary components of $\tilde{Z}$.
The system~(\ref{e:any_instr}) is solved iteratively
by Harmbal according to the scheme illustrated in Figure~\ref{f:any_instr_solve}.
\begin{figure}
\centerline%
{\setlength{\unitlength}{1em}%
\def$\vec X_r,f$}\def\Xc{$\vec X_c,f$}\def\Xe{$\vec X_e,f${$\vec X_r,f$}\def\Xc{$\vec X_c,f$}\def\Xe{$\vec X_e,f$}
\def$x_r(t)$}\def\xc{$x_c(t)$}\def\xe{$x_e(t)${$x_r(t)$}\def\xc{$x_c(t)$}\def\xe{$x_e(t)$}
\begin{picture}(22.5,12)(-1.5,-0.5)
\put(0,10){\Xc}
\put(1.8,9.3){\vector(1,-1){2}}
\put(3,9){\footnotesize Eq.(\ref{e:any_instr}a)}
\put(3,6.2){\Xe}
\put(6,6.5){\vector(1,0){2.5}}
\put(6,6.9){\footnotesize DFT$^{-1}$}
\put(9,6.2){\xe}
\put(3,10.3){\vector(1,0){5.5}}
\put(4.7,10.7){\footnotesize DFT$^{-1}$}
\put(9,10){\xc}
\put(11.3,8){$\left.\hbox{\vbox to2.8\unitlength{}}\right\}$}
\put(12.8,8.3){\vector(1,0){2.5}}
\put(12.6,8.7){\footnotesize Eq.(\ref{e:any_instr}c)}
\put(16,8){$x_r(t)$}\def\xc{$x_c(t)$}\def\xe{$x_e(t)$}
\put(17,7.5){\vector(0,-1){1.5}}
\put(17.5,6.5){\footnotesize DFT}
\put(15.6,5){$\vec X_r,f$}\def\Xc{$\vec X_c,f$}\def\Xe{$\vec X_e,f$}
\put(17,4.5){\vector(0,-1){1.5}}
\put(17.5,3.5){\footnotesize Eq.(\ref{e:any_instr}b)}
\put(15,2){$\vec F$(\Xc)}
\put(14.7,2.3){\vector(-1,0){3}}
\put(1,9.3){\line(0,-1){7}}
\put(1,2.3){\vector(1,0){8.7}}
\put(10.7,2.3){\circle{2}}
\put(10.1,2){=?}
\put(10.7,1.3){\line(0,-1){1}}
\put(10.7,0.3){\vector(-1,0){5.5}}
\put(1,0){$\Delta\vec X_c,\Delta f$}
\put(7,.6){\small N-R}
\put(0.7,0.3){\line(-1,0){1.7}}
\put(-1,0.3){\line(0,1){8.5}}
\put(-1,8.8){\vector(1,1){1}}
\end{picture}}}
\caption{The iteration loop of the harmonic balance method for a
musical instrument (notations defined in the text)}
\label{f:any_instr_solve}
\end{figure}
In Harmbal, these equations are easily defined by writing new C
functions. Only superficial knowledge of the C language is necessary
to do this.
Three cases related to models of single reed instruments with
cylindrical or stepped-conical bores are studied in particular in
Section~\ref{s:case} in order to validate the code and to illustrate the
modularity of Harmbal.
\subsection{Practical characteristics of Harmbal \label{s:harmdet}}
Both fast calculation, good portability, and independence of
commercial software are easily achieved by programming in C, whose
compiler is freely available for most computer platforms. It is,
however, somewhat difficult to combine portability with easy usage,
because an intuitive usage normally means a graphical and interactive
user interface, while the handling of graphics varies a lot between the
different platforms.
We have chosen to write Harmbal with a nongraphical and
non-interactive\footnote{The term {\em non-interactive\/}
means that the user has no influence on the program while it is
running.} user interface. The major advantage of this is that
independent user interfaces may be further developed depending on need.
Our concept is to save both the parameters and the solution in a
single file. This file also serves as input to Harmbal while individual
parameters can be changed through start-up arguments. The solution
provided by the file works as the initial condition for the harmonic
balance method. Thus the lack of a simple user interface is
compensated by a simple way of re-using an existing solution to solve
the system for a slightly different set of parameters. Solutions for
a range of a parameter values may thereby be calculated by changing the
parameter stepwise and providing the previous solution as an
initial
condition for the next run. The Perl script {\em hbmap\/} provides such zeroth-order continuation facilities.
This procedure may
also be used when searching for a solution where it is difficult to
provide a sufficiently good initial condition, for instance by
successively increasing $N_p$ when wanting many harmonics.
\subsection{Convergence of Newton-Raphson \label{s:holes}}
When merely employing the Newton-Raphson method to determine the
solution of the system at a given set of parameters, we have found
that it is impossible to find a solution at particular combinations of
the parameters. Indeed, for the clarinet model of
Section~\ref{s:helcyl}, no convergence was obtained for particular
values of the parameter $\gamma$ (the dimensionless blowing pressure)
and its neighborhood. This is seen as discontinuities, or
\emph{holes}, in the curves in Figure~\ref{f:holes}
(see Section~\ref{s:case} for the underlying equations and
parameters). Note that the
solutions seem to go continuously through this hole and that the
positions of the holes and their extent vary with the number of
harmonics $N_p$ taken into account.
\begin{figure
\ifgalleyfig%
\includegraphics[width=3.25in]{eps/compmap.eps}%
\else
\ifoutputfig%
\includegraphics[width=5.5in]{eps/compmap.eps}%
\fi
\fi
\caption
[Solution holes: first pressure harmonic
$P_1$ versus blowing pressure $\gamma$ for different $N_p$ with
$N_t=128$, $\zeta=0.5$, and $\eta=10^{-3}$. (Even $N_p$ give the same as
$N_p{-}1$.) Equations and parameters are defined in section \ref{s:case}.]
{\label{f:holes}
\ifgalleyfig
{Solution holes: first pressure harmonic
$P_1$ versus blowing pressure $\gamma$ for different $N_p$ with
$N_t=128$, $\zeta=0.5$, and $\eta=10^{-3}$. (Even $N_p$ give the same as
$N_p{-}1$.) Equations and parameters are defined in section \ref{s:case}.}
\fi}
\end{figure}
The curves were calculated by the program {\em hbmap}. In this case
we have decreased $\gamma$ from 0.5 downward in steps of $10^{-4}$ and
drawn a line between them except across $\gamma$ values where solution failed.
In the holes, the Newton-Raphson
method did not converge, either by alternating between two values of
$\vec P$ (i.e.\ $\vec X_c$) or by starting to diverge.
To study the problem, we simplified the system to a one-dimensional
problem by setting $N_p=1$, thus leaving $P_1$ as
the only nonzero value. $G_1$ thus became the only contributor to
$|\vec G|$, and a simple graph of $G_1$ around the
solution $G_1=0$ could illustrate the problem, as shown in
Figure~\ref{f:GvsP}.
\begin{figure}
\ifgalleyfig%
\includegraphics[width=3.25in]{eps/jumphole128.eps}%
\else
\ifoutputfig%
\includegraphics[width=5.5in]{eps/jumphole128.eps}%
\fi
\fi
\caption
[$G_1$ as $P_1$ varies around the solution $G_1=0$ for various
$\gamma$ around a hole at $\gamma\simeq0.4196$. $N_t=128$ and
$N_p=1$.]
{\label{f:GvsP}
\ifgalleyfig
{$G_1$ as $P_1$ varies around the solution $G_1=0$ for various
$\gamma$ around a hole at $\gamma\simeq0.4196$. $N_t=128$ and
$N_p=1$.}
\fi}
\end{figure}
We see that the curve of $G_1(P_1)$ has inflection points
(visible as ``soft steps'' on the curve) at rather regular
distances. At the centre of a convergence hole,
i.e.\ for $\gamma\simeq0.4196$, an inflection point is located at the
intersection with the horizontal axis. This is a school example of
a situation where Newton's method does not converge because the Newton
step $\Delta P_1$ brings us alternatingly from one side of the
solution to the other, but not closer.
In fact, the existence of inflection points is linked with the digital
sampling of the continuous signal. If the sampling rate is increased,
i.e.\ if $N_t$ is increased, the steps become smaller but occur more
frequently, as shown for $N_t=32$, 128, and 1024 in
Figures~\ref{f:sampling}a--c. The derivative $dG_1/dP_1$ is included
in the figures to quantify the importance of the steps. According to
the Figures~\ref{f:sampling}a--c it seems reasonable to increase $N_t$
to avoid convergence problems. However, this would significantly
increase the computational cost. Another solution is therefore
suggested in the following.
\begin{figure*}
\ifgalleyfig%
\includegraphics[height=.595\width]{eps/jumps32.eps}\hspace{-.21\width}%
\includegraphics[height=.595\width]{eps/jumps128.eps}\hspace{-.21\width}%
\includegraphics[height=.595\width]{eps/jumps1024.eps}\\%}%
\else
\ifoutputfig%
\includegraphics[height=.285\width]{eps/jumps32.eps}\hspace{-.1\width}%
\includegraphics[height=.285\width]{eps/jumps128.eps}\hspace{-.1\width}%
\includegraphics[height=.285\width]{eps/jumps1024.eps}%
\fi
\fi
\caption
[The effect of sampling rate on the ``smoothness'' of $G_1(P_1)$:
(a) $N_t=32$, (b) 128, and (c) 1024. The derivative $dG_1/dP_1$
exhibits the ``roughness''.]
{\label{f:sampling}
\ifgalleyfig
{The effect of sampling rate on the ``smoothness'' of $G_1(P_1)$:
(a) $N_t=32$, (b) 128, and (c) 1024. The derivative $dG_1/dP_1$
exhibits the ``roughness''.}
\fi}
\end{figure*}
\subsection{Backtracking}\label{s:backtracking}
When the Newton-Raphson scheme fails to converge, it often happens
because the Newton step $\Delta\vec X$ leads to a point where $|\vec G(\vec
X,f)|$ is larger than in the previous step. However, acknowledging
that the Newton step points in the direction of the steepest descent, there
must be a point along $\Delta\vec X$ where $|\vec G(\vec X,f)|$ is smaller
than in the previous iteration of the HBM. A backtracking
algorithm described in Numerical Recipes \cite[Sec.9.7]{numrec} solves the
problem elegantly by shortening the Newton step as
described here. The principle is illustrated in the simple
one-dimensional case in Figure~\ref{f:backtracking}, where $g(x)$
replaces $|\vec G(\vec X,f)|$, although we use the multidimensional
notation in the following.
\begin{figure}
\ifgalleyfig%
\includegraphics[width=2.4in]{eps/backtrack3.eps}%
\else
\ifoutputfig%
\includegraphics[width=5.5in]{eps/backtrack3.eps}%
\fi
\fi
\caption
[The principle of backtracking in one dimension. Objective is
to estimate the root of $g(x)$ (solid curve). Broken lines with arrows
show how the Newton step $\Delta x$ from $x$ leads to divergence.
$h(\lambda)$ (dot-dashed curve) is a 2nd order expansion of $g(x)$
along the Newton step, i.e. the $\lambda$ axis. Minimum of
$h(\lambda)$ should be closer to the root of $g(x)$ than $g(x+\Delta
x)$.]
{\label{f:backtracking}
\ifgalleyfig
{The principle of backtracking in one dimension. Objective is
to estimate the root of $g(x)$ (solid curve). Broken lines with arrows
show how the Newton step $\Delta x$ from $x$ leads to divergence.
$h(\lambda)$ (dot-dashed curve) is a 2nd order expansion of $g(x)$
along the Newton step, i.e. the $\lambda$ axis. Minimum of
$h(\lambda)$ should be closer to the root of $g(x)$ than $g(x+\Delta
x)$.}
\fi}
\end{figure}
Defining the $\lambda$ axis along the Newton step, we simply take a
step $\lambda\Delta\vec X$ in the
same direction, where $0<\lambda<1$. The optimal value for $\lambda$
is the one that minimizes the function $h(\lambda)$:
\begin{equation}
h(\lambda)=\textstyle\frac12|\vec G(\vec X^i+\lambda\Delta\vec X)|^2
\end{equation}
with derivative
\begin{equation}
h'(\lambda)=\left(\mathbf{J}_G \cdot \vec G\right)\big|
_{\vec X^i+\lambda\Delta\vec X} \cdot\Delta\vec X.
\end{equation}
During the calculation of the failing Newton step, we computed $\vec
G(\vec X^i)$ and $\vec G(\vec X^{i+1})$, so now it is possible to
calculate with nearly no additional computational effort $h(0) =
\frac12|\vec G(\vec X^i)|^2$, $h'(0) = -|\vec G(\vec X^i)|^2$, and
$h(1) = \frac12|\vec G(\vec X^i+\Delta\vec X)|^2 = \frac12|\vec G(\vec
X^{i+1})|^2$. This allows to propose a quadratic approximation of $h$
for $\lambda$ between $0$ and $1$, for which the minimum is located at
\begin{equation}
\lambda_1=-\frac{\frac12h'(0)}{h(1)-h(0)-h'(0)}.
\end{equation}
It can be shown that $\lambda_1$ should not exceed 0.5, and in practice
$\lambda_1\ge0.1$ is required to avoid a too short step at
this stage.
If $|\vec G(\vec X^i+\lambda_1\Delta\vec X)|$ still is larger than
$|\vec G(\vec X^i)|$, $h(\lambda)$ is then modelled as a cubic function (using
$h(\lambda_1)$ which has just been calculated). The minimum of this
cubic function gives a new value $\lambda_2$, again restricted to
$0.1\lambda_1<\lambda_2<0.5\lambda_1$. This
calculation requires solving a system of two equations, so if also
$\lambda_2$ is not accepted because
$|\vec G(\vec X^i+\lambda_2\Delta\vec X)|$ is still too large, we do
not enhance to a fourth-order model of $h$, which would increase the
computational cost much more. Instead, subsequent cubic modellings are
performed using the most two recent values of $\lambda$. In practice,
however, not many repetitions should be necessary before finding a
better solution, if possible.
\section{Case studies}\label{s:case}
\subsection{The equations for the clarinet \label{s:clarinet}}
The three equations~(\ref{e:any_instr}a--c) may be constructed by
physical modelling. In the case of the clarinet, a common simple model
is described below. We limit the description in the following to a
brief presentation based on dimensionless quantities, {\em
dimensional\/} variables being denoted by a hat ($\hat{\ }$)
hereafter (see Fritz et al.\ \cite{fritz04} for further details).
\medskip
The exciter is an oscillating reed which may be modelled as a spring
with mass and damping:
\begin{equation}
\ddot{\hat{y}}+g_e\dot{\hat{y}}+\omega_e^2\hat{y}
=\frac1{\mu_e}(\hat{p}-p_m),
\label{e:lindiff_dim}
\end{equation}
where $\hat{y}$ is the dynamic reed displacement, and $\hat{p}$
and $p_m$ are the dynamic pressure in the mouthpiece, i.e.\ the {\em
internal\/} pressure, and the static blowing pressure in the player's
mouth, respectively. The constants $\mu_e$, $g_e$, and
$\omega_e$ represent the mass per area, the damping factor, and the
angular resonance frequency of the exciter (the reed). The dots over
$\hat{y}$ denote the time derivative. In dimensionless form,
equation~\eqref{e:lindiff_dim} becomes
\begin{equation}
M\ddot{x}+R\dot{x}+Kx=p,
\label{e:lindiff}
\end{equation}
where $p=\hat p/p_M$ and $x=\hat y/H+\gamma/K$ with
$\gamma=p_m/p_M$. The equilibrium reed opening is $H$ as shown in
Figure~\ref{f:mouthpiece}. In the static regime, when
blowing harder than a maximum pressure $p_M$, i.e.\
$p_m\ge p_M$ ($\gamma\ge1$), the reed blocks the opening, i.e.\
$\hat y=-H$, so we get $\hat p=0$ and can conclude that $K=1$ for
the current reed model.
\begin{figure}
\ifgalleyfig%
\includegraphics[width=1.9in]{eps/mouthpiece.eps}%
\else
\ifoutputfig%
\includegraphics[width=5.5in]{eps/mouthpiece.eps
\fi
\fi
\caption
[Illustration of the mouthpiece]
{\label{f:mouthpiece}
\ifgalleyfig
{Illustration of the mouthpiece}
\fi}
\end{figure}
Like Fritz et al.\ \cite{fritz04} we relate the dimensionless time to
the resonance angular frequency $\omega_r$ of the resonator (the
bore), i.e.\ $t=\omega_r\hat t$, so that the values of the
dimensionless mass $M$, damping $R$, and spring constant $K$ become
\begin{align}
K&=\mu_e H\omega_e^2/p_M=1,\\
R\,&=Kg_e\omega_r/\omega_e^2=g_e\omega_r/\omega_e^2,\\
M&=K\omega_r^2/\omega_e^2=\omega_r^2/\omega_e^2.
\label{e:MRK}
\end{align}
In the Fourier domain, Equation~\eqref{e:lindiff} thus takes the form of
equation~(\ref{e:any_instr}a), $Z_e(\omega)X(\omega)=P(\omega)$, where
\begin{equation}
Z_e(\omega)=1-M\left(\!\frac\omega{2\pi}\!\right)^{\!2}
+iR\left(\!\frac\omega{2\pi}\!\right),
\label{e:reedimp}
\end{equation}
for $i=\sqrt{-1}$ and $\omega =2\pi\hat\omega/\omega_r
=\hat\omega/f_r$ is the dimensionless angular frequency in the Fourier
domain.
A common minimum model for the clarinet assumes a simple reed with no
mass or damping, thus $M=R=0$. Equation~\eqref{e:lindiff} reduces to
$x=p$.
\medskip
The resonator (i.e.\ the air column in the bore of the
instrument) is commonly described by its frequency response $\hat
Z_r(\hat\omega)$. For a simple cylindrical bore of length $l$ with
a closed and an ideal open end, the resonance frequencies are odd
multiples of $f_r=c/4l$, $c$ being the sound speed in the air column
\cite{fletcher91}. The input impedance of the bore may thus be
expressed in dimensionless quantities as
\begin{equation}
Z_r(\omega)=\frac{\hat Z_r(\hat\omega)}{Z_0}
=i\tan\!\left(\frac\omega4 + (1-i)\alpha(\omega)\!\right),
\label{e:freqresponse}
\end{equation}
where $\alpha(\omega)\triangleq\psi\eta\sqrt{\omega/2\pi}$ with
$\psi\simeq1.3$ for common conditions in air and $\eta$ being the
dimensionless loss parameter, which depends on the tube length,
typically 0.02 for a normal clarinet with all holes closed.
$Z_0\triangleq \rho c/S$ is the characteristic impedance of the
cylidrical resonator, $S$ being its cross section, and $\rho$ the
density of air. The last term in the argument of
equation~\eqref{e:freqresponse} includes the dispersion as the real
part and viscous losses as the imaginary part.
Equation~(\ref{e:any_instr}b) becomes
\begin{equation}
P(\omega)=Z_r(\omega)U(\omega),
\label{e:imped}
\end{equation}
where $P(\omega)$ and $U(\omega)\triangleq\hat U(\omega)Z_0/p_M$ are
the dimensionless internal pressure and volume flow of air through the
mouthpiece in the Fourier domain.
\medskip
The coupling equation~(\ref{e:any_instr}c), is given by the Bernoulli
theorem with some supplementary hypotheses applied
between the mouth and the outlet of the reed channel.
The coupling equation is nonlinear and must be calculated
in the time domain. This leads to the following expression for the
dimensionless airflow through the mouthpiece
\cite{kergomard95}:
\begin{equation}
u(p,x)=\zeta\left(1+x-\gamma\right)
\sqrt{|\gamma-p|}\,\mathrm{sign}(\gamma-p)
\label{e:nonlin}
\end{equation}
as long as $x>\gamma-1$, and $u=0$ otherwise.
$\zeta=Z_0wH\sqrt{2/\rho p_M}$ is a dimensionless embouchure parameter
roughly describing the mouthpiece and the position of the player's
mouth, $w$ being the width of the opening and $\rho$ the density of
the air. $\zeta$ is also related to the maximum volume velocity
entering the tube \cite{ollivier04a}.
If the reed dynamics were not taken into account, we had $x=p$ and thus
\begin{equation}
u(p)=\zeta\left(1+p-\gamma\right)
\sqrt{|\gamma-p|}\,\mathrm{sign}(\gamma-p)
\label{e:simplenonlin}
\end{equation}
for $p>\gamma-1$, and, as before, $u=0$ otherwise.
\subsection{Verification of method and models \label{s:verif}}
In the following we want to verify that the HBM (and its
implementation in Harmbal) gives correct results. By using very low
losses in the resonator (small $\eta$) we can compare the results of
the HBM with analytical results. Rising the attenuation in the resonator and
including mass and damping for the exciter, we compare with numerical
results from real-time synthesis of the same system. This also gives us
the opportunity to illustrate the modularity of Harmbal as we change the models
of the resonator and the nonlinear coupling.
\subsubsection{Helmholtz oscillation for cylindrical tubes}\label{s:helcyl}
To compare the HBM results with analytical results, we assume a
nondissipative, nondispersive air column, i.e.\ setting $\eta=0$ and
thus $\alpha=0$ in equation~\eqref{e:freqresponse}. Furthermore, we
assume that the reed has neither mass nor damping and thus use
equation~\eqref{e:simplenonlin}. The resulting square-wave amplitude
(the Helmholtz motion \cite{helmholtz77}) may be found by solving
$u(p)=u(-p)$, which results from the fact that the internal pressure
$p(t)$ and the power $p(t)u(t)$ averaged over a period are zero
according to the lossless hypothesis \cite{kergomard95}. This leads
to the square oscillation with amplitude
\begin{equation}
p(\gamma)=\sqrt{-3\gamma^2+4\gamma-1}.
\label{e:helmotion}
\end{equation}
This result is compared with the results calculated by Harmbal (for the
same set of equations, but $\eta=10^{-5}$ instead of $\eta=0$ to avoid
infinite impedance peaks) for 3, 9, 49, and 299 harmonics
close to the oscillation threshold in
Figure~\ref{f:nearthres-nl}, and at $\gamma=0.4$ in
Figure~\ref{f:largeosc-nl}, which is far from the
threshold.
\begin{figure}[t]
\ifgalleyfig%
\includegraphics[width=3.25in]{eps/HBM-hh-freq-g0.334336.eps}%
\\\includegraphics[width=3.25in]{eps/HBM-hh-time-g0.334336.eps}%
\else
\ifoutputfig%
\includegraphics[width=5.5in]{eps/HBM-hh-freq-g0.334336.eps}%
\\\includegraphics[width=5.5in]{eps/HBM-hh-time-g0.334336.eps}%
\fi
\fi
\caption
[The Helmholtz solution, eq.\
\eqref{e:helmotion} compared with the HBM truncated to 3, 9, 49,
and 299 harmonics close to the oscillation threshold
($\gamma=0.334$, $\zeta=0.5$, $\eta=10^{-5}$). (a) frequency
domain. (b) time domain.]
{\label{f:nearthres-nl}
\ifgalleyfig
{The Helmholtz solution, eq.\
\eqref{e:helmotion} compared with the HBM truncated to 3, 9, 49,
and 299 harmonics close to the oscillation threshold
($\gamma=0.334$, $\zeta=0.5$, $\eta=10^{-5}$). (a) frequency
domain. (b) time domain.}
\fi}
\end{figure}
\begin{figure}
\ifgalleyfig%
\includegraphics[width=3.25in]{eps/HBM-hh-freq-g0.40.eps}%
\else
\ifoutputfig%
\includegraphics[width=5.5in]{eps/HBM-hh-freq-g0.40.eps}%
\fi
\fi
\caption
[The Helmholtz solution, eq.\
\eqref{e:helmotion} compared with the HBM truncated to 3, 9, 49,
and 299 harmonics far from the oscillation threshold ($\gamma=0.40$,
$\zeta=0.5$, $\eta=10^{-5}$) in the frequency domain.]
{\label{f:largeosc-nl}
\ifgalleyfig
{The Helmholtz solution, eq.\
\eqref{e:helmotion} compared with the HBM truncated to 3, 9, 49,
and 299 harmonics far from the oscillation threshold ($\gamma=0.40$,
$\zeta=0.5$, $\eta=10^{-5}$) in the frequency domain.}
\fi}
\end{figure}
As expected, the solution using the HBM shows good convergence towards
the Helmholtz motion as the number of harmonics increases. Note the
deviation for higher
harmonics close to the threshold, even for 299 harmonics.
Dissipation in the resonator ($\eta=10^{-5} \not= 0$)
causes higher harmonics to be damped more in this area of $\gamma$
than for higher blowing pressures (as explained e.g.\ in Ref.\
\onlinecite{kergomard00}). The deviation from a square-wave
signal is thus more noticeable close to the threshold, and as the HBM
calculations imposed a nonzero dissipation, this is probably the
reason for the small deviation in Figure~\ref{f:nearthres-nl}. The
deviation is not visible in the time domain.
A popular simplification of the nonlinear
function~\eqref{e:simplenonlin} is a cubic expansion for small
oscillations (e.g.\ Ref.\ \cite{mcintyre83,grand97,worman71}):
\begin{equation}
\tilde{u}(p) = u_{00}+Ap+Bp^2+Cp^3,
\label{e:cubic}
\end{equation}
where $u_{00}$, $A$, $B$, and $C$ are easily found by expanding
equation~\eqref{e:simplenonlin}. Its Helmholtz solution is easily
calculated like above, yielding
\begin{equation}
p(\gamma)=\sqrt{-\frac AC}=\sqrt{\frac{8\gamma^2(3\gamma-1)}{\gamma+1}}.
\label{e:pcubic}
\end{equation}
The influence of
the difference between the two versions of the nonlinear function is investigated in
Figure~\ref{f:compcub} for the lossless case.
\begin{figure}
\ifgalleyfig%
\includegraphics[width=3.25in]{eps/compcub-freq-g0.334336.eps}%
\\\includegraphics[width=3.25in]{eps/compcub-freq-g0.40.eps}%
\else
\ifoutputfig%
\includegraphics[width=5.5in]{eps/compcub-freq-g0.334336.eps}%
\\\includegraphics[width=5.5in]{eps/compcub-freq-g0.40.eps}%
\fi
\fi
\caption
[The (lossless) Helmholtz motion and the
(almost lossless) HBM for 299 harmonics using the full
nonlinearity~\eqref{e:simplenonlin} and
the cubic expansion~\eqref{e:cubic} (a) close to the
oscillation threshold ($\gamma=0.334$) and (b) far from it
($\gamma=0.40$) for $\zeta=0.5$, $\eta=10^{-5}$. Above the 11th
harmonic only every 10th harmonic is shown.]
{\label{f:compcub}
\ifgalleyfig
{The (lossless) Helmholtz motion and the
(almost lossless) HBM for 299 harmonics using the full
nonlinearity~\eqref{e:simplenonlin} and
the cubic expansion~\eqref{e:cubic} (a) close to the
oscillation threshold ($\gamma=0.334$) and (b) far from it
($\gamma=0.40$) for $\zeta=0.5$, $\eta=10^{-5}$. Above the 11th
harmonic only every 10th harmonic is shown.}
\fi}
\end{figure}
Close to the oscillation threshold, Figure~\ref{f:compcub}a, we see
that there is no significant difference between the two versions of
the nonlinear equation, as expected. The fact that the HBM is lower
for higher harmonics is as before due to the small attenuation we had to
include to perform the numerical calculations. Far from the
threshold, however, Figure~\ref{f:compcub}b, we see that the cubic
expansion fails to approximate the nonlinear equation. For lower
harmonics this error is larger than the attenuation effect in the HBM
calculations. This is further discussed by Fritz et al \cite{fritz04}.
In Figure~\ref{f:P1-gamma} we have completed some of the curves that
we failed to make in Figure~\ref{f:holes}, and even increased the number
of harmonics, owing to the backtracking mechanism. Admittedly, at
$N_p=49$, a few holes can still be seen, but the convergence is
significantly improved.
\begin{figure}
\ifgalleyfig%
\includegraphics[width=3.25in]{eps/HBM-hh-gamma.eps}%
\else
\ifoutputfig%
\includegraphics[width=5.5in]{eps/HBM-hh-gamma.eps}%
\fi
\fi
\caption
[Amplitude of first harmonic as the blowing pressure increases for
the Helmholtz solution \eqref{e:helmotion} and the HBM truncated to
1, 3, 9, and 49 harmonics, the last coinciding with Helmholtz
($\zeta=0.5$, $\eta=10^{-5}$)]
{\label{f:P1-gamma}
\ifgalleyfig
{Amplitude of first harmonic as the blowing pressure increases for
the Helmholtz solution \eqref{e:helmotion} and the HBM truncated to
1, 3, 9, and 49 harmonics, the last coinciding with Helmholtz
($\zeta=0.5$, $\eta=10^{-5}$)}
\fi}
\end{figure}
Here the amplitude of the first
harmonic is plotted for different numbers of harmonics as a function
of the blowing pressure $\gamma$ together with first harmonic of the
Helmholtz solution, deduced from equation~\eqref{e:helmotion}. In
practice, the solution at $\gamma = 0.4$ was found and then
\emph{hbmap} was used to make Harmbal calculate solution for each of a large
number of subsequent values of $\gamma$ down to the oscillation threshold
by using the previous solution as initial value. The procedure was
repeated from $\gamma=0.4$ up to the point where the reed started to
beat, i.e.\ for $p<\gamma-1$ in equation~\eqref{e:simplenonlin}. Without losses
(Helmholtz solution) the beating threshold does not arrive
before $\gamma=0.5$, and this should be expected for the nearly
lossless case studied with the HBM also. However, the number of harmonics
$N_p$ taken into account in the HBM calculations is too small to
follow the sharp edges of the square signal. The resulting overshoots
in $p(t)$, as seen in Figure~\ref{f:nearthres-nl}b, cause $p$ to
prematurely exceed the criterion for beating. The beating threshold
converges to 0.5 as $N_p$ increases (see also Ref.\ \onlinecite{fritz04}). Note
that, for the
chosen value of $\zeta$, it can be calculated following
Hirschberg \cite[eq.(45)]{hirschberg95} that above $\gamma \simeq
0.45$, the Helmholtz solution loses its
stability through a
subharmonic bifurcation (a period-doubling occuring).
By Figure~\ref{f:P1-gamma} we can also verify that the model
experiences a direct Hopf bifurcation (which is known since the work
of Grand et al.\ \cite{grand97}). Thus, a single harmonic is enough to
study the solution around the threshold. Far from the threshold, more
harmonics have to be taken into account for $P_1$ to converge toward the
Helmholtz solution. This is not obvious and for example
contradictory with the hypothesis made for the VTM
\cite{kergomard00}. Thus Harmbal
appears as an interesting tool to evaluate the relevance of
approximate methods according to the parameter values.
\subsubsection{Helmholtz oscillation for a stepped conical tube}
The saxophone works similarly to the clarinet, but the bore has a
conic form. In this section we compare the HBM calculations with
analytical results, and in order to calculate the Helmholtz motion
when losses are ignored, we need to simplify the cone by assuming that
it consists of a sequence of $N$ sylinders of length $l$ and cross section
$S_i=\frac12i(i+1)S_1$, $S_1=S$ being the cross section of the smallest
cylinder, and $i=1,\dots,N$ (see Ref.\ \onlinecite{dalmont00}). The
total length of the instrument is thus $L=Nl$. The input impedance of
such a {\em stepped cone\/} may be written as
\begin{equation}
Z_r(\omega)=\frac{2i}
{\cot\!\left(\!\frac{\omega'}{4} - i\alpha(\omega')\right)
+ \cot\!\left(\!\frac{\omega'}{4N}- i\alpha(\frac{\omega'}N)\right)},
\label{e:coneimp}
\end{equation}
where $\omega'\triangleq2\omega/(N+1)$ when $\omega=2\pi\hat
f/f_r$, where $f_r$ is the first eigenfrequency of
this resonator. We have ignored the dispersion term
here. Equation~\eqref{e:coneimp} is used instead of
equation~\eqref{e:freqresponse}, and the damping
$\alpha(\omega)=\psi\eta\sqrt{\omega/2\pi}$ is zero in the analytic
Helmholtz case and very small ($\eta=2\cdot10^{-5}$ below which
convergence became difficult) for the calculations with the HBM.
As before, the pressure amplitude of the ideal lossless case is
calculated by solving $u(p)=u(-Np)$, and two solutions are
possible:\footnote{This result corrects equation~(14) in
ref.~\onlinecite{dalmont00}}
\begin{equation}
\begin{array}{rl
\llap{$p^{\pm}$}(\gamma)\!\!\!\!&=\displaystyle\frac{(N{-}1)(2{-}3\gamma)}{2(N^2-N+1)}\\
&\displaystyle\pm\, \frac{ \sqrt{(N{-}1)^2+(N{+}1)^2(-3\gamma^2{+}4\gamma{-}1)}}{2(N^2-N+1)}
\end{array}
\label{e:coneN}
\end{equation}
as long as $\gamma<1/(N+1)$ for the standard Helmholtz motion ($p^+$) and
$\gamma<N/(N+1)$ for the inverted one ($p^-$), which is unstable.
Above these limits $p^+=\gamma$ and $p^-=-\gamma/N$. The magnitude of
the first harmonic of a square or rectangular wave is then given by
\begin{equation}
P_1^{\pm}(\gamma)=\frac{\sin\frac\pi{N+1}}{\frac\pi{N+1}}p^{\pm}(\gamma).
\label{e:P1p}
\end{equation}
For $N=1$, equation~\eqref{e:coneN} reduces to
equation~\eqref{e:helmotion}. For higher $N$, the pressure oscillation
becomes asymmetric.
We take the case $N=2$ and get
\begin{equation}
p^{\pm}(\gamma)=\frac16\left(2-3\gamma\pm\sqrt{-27\gamma^2+36\gamma-8}\right).
\label{e:coneN2}
\end{equation}
This result is compared with HBM calculations in
Figure~\ref{f:coneN2A} for $\gamma=0.31$.
\begin{figure}
\ifgalleyfig%
\includegraphics[width=3.25in]{eps/freq-n2e-5g0.31A.eps}%
\\\includegraphics[width=3.25in]{eps/time-n2e-5g0.31A.eps}%
\else
\ifoutputfig%
\includegraphics[width=5.5in]{eps/freq-n2e-5g0.31A.eps}%
\\\includegraphics[width=5.5in]{eps/time-n2e-5g0.31A.eps}%
\fi
\fi
\caption
[Comparison between the standartd Helmholtz motion of a stepped cone
($N=2$) and the HBM for various $N_p$ at $\gamma=0.31$, $\zeta=0.2$, and
$\eta=2\cdot10^{-5}$. (a) The magnitude of the harmonics and (b) one
oscillation period. $N_t$ varies from 128 for $N_p{=}5$ to
1024 for $N_p{=}180$.]
{\label{f:coneN2A}
\ifgalleyfig
{Comparison between the standartd Helmholtz motion of a stepped cone
($N=2$) and the HBM for various $N_p$ at $\gamma=0.31$, $\zeta=0.2$, and
$\eta=2\cdot10^{-5}$. (a) The magnitude of the harmonics and (b) one
oscillation period. $N_t$ varies from 128 for $N_p{=}5$ to
1024 for $N_p{=}180$.}
\fi}
\end{figure}
Theoretically, the spectrum of the Helmholtz solution,
Figure~\ref{f:coneN2A}a, shows that every third component is missing
(actually zero) while the remaining components decrease in
magnitude thus
forming the asymmetric pressure oscillation as shown in
Figure~\ref{f:coneN2A}a. The HBM, on the other hand, suggests that
the first component in each pair be smaller than the second component.
This results in a {\em dip\/} at the middle of the long, positive part
of the period (i.e. on both extremities $t=0$ and $t=1024$ of the
curve in Figure \ref{f:coneN2A}). The same was observed for $N=3$ and $N=4$, where the long
part of the period was divided by similar dips into three and four parts,
respectively (not shown). The number of
time samples, $N_t$ did not change this fact, but as
Figure~\ref{f:coneN2A} indicates, the dips gradually become narrower
as the number of harmonics $N_p$ increases. This indicates that the
HBM approaches the Helmholtz solution as $N_p$ approaches infinity.
\begin{figure}
\ifgalleyfig%
\includegraphics[width=3.25in]{eps/map-n2e-5P1.eps}%
\else
\ifoutputfig%
\includegraphics[width=5.5in]{eps/map-n2e-5P1.eps}%
\fi
\fi
\caption
[Amplitude of first harmonic $P_1$ as a function
of the blowing pressure $\gamma$ for
the Helmholtz solution \eqref{e:coneN2} for 2-stepped cone and the
HBM truncated to 2, 5,\dots, and 63 harmonics, the last coinciding
with Helmholtz ($\zeta=0.2$, $\eta=2\cdot10^{-5}$). Only nonbeating
regimes are shown.]
{\label{f:P1-gammaN2}
\ifgalleyfig
{Amplitude of first harmonic $P_1$ as a function
of the blowing pressure $\gamma$ for
the Helmholtz solution \eqref{e:coneN2} for 2-stepped cone and the
HBM truncated to 2, 5,\dots, and 63 harmonics, the last coinciding
with Helmholtz ($\zeta=0.2$, $\eta=2\cdot10^{-5}$). Only nonbeating
regimes are shown.}
\fi}
\end{figure}
A bifurcation diagram is plotted in Figure~\ref{f:P1-gammaN2}.
Similarly to Figure~\ref{f:P1-gamma} for the cylindrical bore, the
amplitude of the first harmonic is plotted for different number of
harmonics as a function of the blowing pressure $\gamma$. The Helmholtz
solution (equation~\eqref{e:P1p} with $N=2$) is also plotted.
As shown by Ollivier et al.\ \cite{ollivier04b}, the lower part of the upper branch and the branch of the inverted
Helmholtz motion are unstable.
In practice, these curves are more difficult to obtain with
{\em hbmap\/} than for the cylindrical bore, especially close to the
subcritical oscillation threshold around $\gamma=0.28$, where computation was not
possible at this low losses. More sophisticated continuation schemes should be
considered to obtain complete curves. However, it is obvious from the
diagram that the model experiences a sub-critical Hopf bifurcation,
which agrees with the conclusion of Grand et al.\ \cite{grand97}. This means that a
single-harmonic approximation is not enough to study the solution
around this threshold, since the small-amplitude hypothesis does not
hold. Further from the threshold, convergence toward the Helmholtz
motion is ensured as the number of harmonics $N_p$ is increased.
Only the nonbeating reed regime is considered in the figure and,
similarly to Figure~\ref{f:P1-gamma}, it can be noted that the beating
threshold for the model with $N_p$ harmonics depends on $N_p$ but
converges toward the Helmholtz threshold $\gamma=1/3$ (corresponding
to the lossless, continuous system) as $N_p$ is increased.
\subsubsection{Validation with time-domain model}
When adding a mass and damping to the reed or viscous losses and
dispersion to the pipe, it is more difficult to compare Harmbal
results with analytic solutions. This has been done by Fritz et al.\
\cite{fritz04}
as far as the playing frequency is concerned, by comparison with
approximate analytical formula. Here, we propose to confront both the
playing frequency and the amplitude of the first partial with numerical results obtained with a time-domain method. We use a
newly developed (real-time) time-domain method (here called TDM) by
Guillemain et al.\ \cite{guillemain03a}. It is based on the same set
of equations as presented in Section~\ref{s:clarinet} except that the
impedance of the bore is slightly modified to be expressed as an
infinite impulse response. In the Fourier domain, it can be expressed
as
\begin{equation}
Z_r(\tilde\omega)=\frac{1-a_1e^{-i\tilde\omega}-b_0e^{-i\tilde\omega D}}
{1 - a_1e^{-i\tilde\omega} + b_0 e^{-i\tilde\omega D}}.
\label{e:philimp}
\end{equation}
where $\tilde\omega=\hat\omega/f_s$, $f_s$ being the sampling
frequency, and the integer $D=\mathrm{round}(f_s/2f_r)$ the time delay
in samples for the sound
wave to propagate to the end of the bore and back.
The constants $a_1$ and $b_0$ are to be adjusted so that the two
first peaks of resonance have the same
amplitude as the two first peaks of
equation~\eqref{e:freqresponse}.
To express equation~\eqref{e:philimp} using our terminology,
we remember that $\omega=2\pi\hat f/f_r$ and obtain
\begin{equation}
Z_r(\omega)=\frac{1 - a_1e^{-i\omega\frac{f_r}{f_s}} - b_0 e^{-i\omega/2}}
{1 - a_1e^{-i\omega\frac{f_r}{f_s}} + b_0 e^{-i\omega/2}}.
\label{e:philimp2}
\end{equation}
In this section, we also include the mass and damping of the reed,
so $M$ and $R$ are no longer zero. The TDM does not work for
$M=R=0$, or even for values close to this, so we have used a reed with
weak interaction with the pipe resonance as well as one with close to
normal reed impedance. The corresponding values for $\omega_e$ and
$q_e\triangleq g_e/\omega_e$ are shown in Table~\ref{t:phil}.
\begin{table}
\caption{The values of $M$ and $R$ for three strengths of reed
interaction. The bore parameters are $D=247$ ($f_r=103.4$\,Hz),
$a_1=0.899$, and $b_0=0.0946$ for sampling frequency
$f_s=51100$\,Hz.}
\smallskip
\centerline{%
\begin{tabular}{l|cccc}\hline
Reed &$\omega_e$/Hz&$q_e$ &$M$&$R$\\\hline
Weak &10000 & 0.1 &$1.070\e{-4}$ &$1.034\e{-3\vphantom{^1}}$\\
Normal &\ph02500 & 0.2 &$1.712\e{-3}$ &$\ph08.28\e{-3}$\\
\hline
\end{tabular}}
\label{t:phil}
\end{table}
Figure~\ref{f:phil-gamma}a shows the bifurcation diagram for two values
of $\zeta$ and for weak and normal reed impedance,
while Figure~\ref{f:phil-gamma}b shows the corresponding variation in
the dimensionless playing frequency $f_p/f_r$. The lines represent the
continuous solutions of the HBM, and the symbols show a set of results
derived from the steady-state part of the TDM signal.
\begin{figure}
\ifgalleyfig%
\includegraphics[width=3.25in]{eps/phil-gamma-p15.eps}%
\\\includegraphics[width=3.25in]{eps/freqcomp.eps}%
\else
\ifoutputfig%
\includegraphics[width=5.5in]{eps/phil-gamma-p15.eps}%
\\\includegraphics[width=5.5in]{eps/freqcomp.eps}%
\fi
\fi
\caption
[Comparison between HBM and TDM of the amplitude of (a) the
first harmonic $P_1$ and (b) the dimensionless playing frequency
$f_p/f_r$ as the blowing pressure $\gamma$ increases for a
clarinet-like system with viscous losses and weak and normal reed
interaction. TDM values for $\zeta=0.50$ and $\gamma>0.48$ are
omitted due to period doubling. So are the beating regimes of HBM
calculations.
($f_s\,{=}\,51100$Hz, $N_t\,{=}\,512$, $f_r\,=\,103.4$\,Hz, $N_p\,{=}\,15$)]
{\label{f:phil-gamma}
\ifgalleyfig
{Comparison between HBM and TDM of the amplitude of (a) the
first harmonic $P_1$ and (b) the dimensionless playing frequency
$f_p/f_r$ as the blowing pressure $\gamma$ increases for a
clarinet-like system with viscous losses and weak and normal reed
interaction. TDM values for $\zeta=0.50$ and $\gamma>0.48$ are
omitted due to period doubling. So are the beating regimes of HBM
calculations.
($f_s\,{=}\,51100$Hz, $N_t\,{=}\,512$, $f_r\,=\,103.4$\,Hz, $N_p\,{=}\,15$)}
\fi}
\end{figure}
The TDM symbols fall well on the lines of the HBM, except for
$\zeta=0.50$ when $\gamma$ approaches 0.5. Then the TDM experiences
period doubling, i.e.\ two subsequent periods of the signal differ.
At the same time, not being able to show subharmonics, the HBM shows
signs of a beating reed, possibly a solution that is unstable and thus
not attainable by time-domain methods.
Note that three points have to be verified before comparing results
from the HBM and the TDM:
The numerical scheme used in the TDM to approximate the time
derivatives in the reed equation~\eqref{e:lindiff} requires
discretization. Depending on the sampling
frequency $f_s$, the peak of resonance of the reed deviates more
or less from the one given by the continuous equation. For normal
reed interaction
($f_e$=2500 Hz), the deviation is negligible, but it may
become significant in the case of weak reed interaction, where the peak is at
10000\,Hz. However, the
fact that the reed and the bore interact weakly in the latter case, implies
that the exact position of the peak has little importance. Therefore,
at the used sampling frequency, the discretization in the TDM is not
compensated for in the HBM calculations.
Then there should be agreement between the sampling frequency $f_s$ in the
TDM and the number of samples $N_t$ per period in the HBM. Their
relation is given by
\begin{equation}
N_t=\frac{f_s}{f_p}.
\end{equation}
In order to have a sufficiently high sampling rate, we have chosen
$N_t=512$. The playing frequency $f_p$ is plotted in
Figure~\ref{f:phil-gamma}b, and we used an average
$f_s=51100$\,Hz for both the HBM and the TDM.
Finally, it seems also necessary that $N_p$ and $N_t$ are chosen so that
\begin{equation}
N_p+1 = \frac{N_t}{2}.
\end{equation}
In practice, however, when comparing bifurcation diagrams of the first
harmonic $P_1$, as in Figure \ref{f:phil-gamma}, rather low values of
$N_p$ give good results. Nevertheless, more harmonics are obviously
needed to compare waveforms in the time domain, especially far from the
oscillation threshold.
\section{Practical experiences}\label{s:practexp}\label{s:disc}
\subsection{Multiple solutions}
As we consider a nonlinear problem, we cannot anticipate the number of
solutions. Therefore, it should not be surprising that it is possible
to obtain multiple solutions for a given set of parameter values.
When searching for a particular solution, this may be a practical
problem. Fritz et al.\ \cite{fritz04} have discovered that some
solutions seem to disappear when increasing the number of harmonics
$N_p$, implying that solutions may arise from the truncation to a
finite $N_p$. We have now discovered alternative solutions that
persist even at very high $N_p$.
Let us illustrate this with the simple model of the clarinet used in
Section~\ref{s:helcyl}, where the reed is a spring without mass or
damping, the nonlinearity is given by equation~\eqref{e:simplenonlin},
and the bore is an ideal cylinder with nearly lossless propagation
and no dispersion. Figure~\ref{f:bizsol} shows a three-level
sister solution together with the related Helmholtz solution for a
large number of harmonics, $N_p=2000$.
\begin{figure
\ifgalleyfig%
\includegraphics[width=3.25in]{eps/comp_p_helmbiz2.eps}%
\\\includegraphics[width=3.25in]{eps/comp_u_helmbiz2.eps}%
\else
\ifoutputfig%
\includegraphics[width=5.5in]{eps/comp_p_helmbiz2.eps
\\\includegraphics[width=5.5in]{eps/comp_u_helmbiz2.eps
\fi
\fi
\caption
[The pressure (a) and volume-flow (b) wave form of the
Helmholtz solution and a 3-level sister solution calculated by
the HBM employing the simple clarinet model with $\zeta=0.5$, $\gamma=0.4$,
$N_p=2000$, $\eta=10^{-5}$.]
{\label{f:bizsol}
\ifgalleyfig
{The pressure (a) and volume-flow (b) wave form of the
Helmholtz solution and a 3-level sister solution calculated by
the HBM employing the simple clarinet model with $\zeta=0.5$, $\gamma=0.4$,
$N_p=99$, $\eta=10^{-5}$.}
\fi}
\end{figure}
A solution of the lossless problem should satisfy the criteria
\cite{kergomard95}
\begin{equation}
\left\{
\begin{array}{l}
p(t+\pi)=-p(t)\\
u(t+\pi)=u(t)\\
\end{array}
\right.
\label{e:pu_temp}
\end{equation}
(the dimensionless period being $2\pi$),
as well as the conditions stated before equation~\eqref{e:helmotion},
noting that $p(t)\,{=}\,u(t)\,{=}\,0$ for all $t$ is the static solution. It
is easily verified graphically that both of the solutions in
Figure~\ref{f:bizsol} satisfy these conditions. Moreover, since they
also satisfy equation~\eqref{e:simplenonlin}, the three-level
solution is a solution of the lossless model.
Whereas the system of time-domain equations~\eqref{e:pu_temp} has an
infinity of solutions, truncation in frequency-domain limits the number
of solutions. The unique solution of the HBM with only one harmonic is
obviously a sine. Let us analyse the situation in the
simplest nontrivial case of the lossless problem with two odd
harmonics and a cubic
expansion for nonlinear coupling. Ignoring even harmonics, the HBM gives a
system of two equations (see Kergomard et al.\ \cite{kergomard00}):
\begin{equation}
\left\{
\begin{array}{rclr}
\alpha & = & 3P_12(1+x+2|x|^2) &
\mathrm{(a)}\\
\alpha x & = &
P_12(1+3x|x|^2+6x), &
\mathrm{(b)}\\
\end{array}
\right.
\label{e:3h_P}
\end{equation}
where $\alpha=-A/C$ and $x=P_3/P_1$. As
equation~(\ref{e:3h_P}a) imposes $P_3$ to be
real, solving this system amounts to solving
\begin{equation}
x^3+x^2-x=1/3.
\label{e:3h_x}
\end{equation}
This equation has three real solutions $x \simeq -1.5151$,
$-0.2776$ and $0.7926$.
All of them are found by Harmbal for negligible losses ($\eta=10^{-5}$),
and the corresponding waveforms are presented in
Figure~\ref{f:helm_biz_3h}. We note that the second solution leads to the Helmholtz motion
when increasing the number of harmonics (with the theoretical value
known to be $x=-1/3$) whereas the third one corresponds to the three-level solution in Figure \ref{f:bizsol}. We can also easily imagine that these three solutions of the truncated problem are three-harmonic approximations of square waves that are distributed on three levels: $p^\pm \simeq \pm 0.5$ and $p=0$. Respectively, they have two, zero, and one steps at the zero-level.
It should be noted that the conditions (\ref{e:pu_temp}) for the continuous problem do not constrain the duration of each step. Figure \ref{f:sol_np100} shows two such twin solutions for $N_p=99$ corresponding to the three-level solution in Figure \ref{f:helm_biz_3h}. This has to be kept in mind when increasing $N_p$ using the HBM.
\begin{figure}
\ifgalleyfig%
\includegraphics[width=3.25in]{eps/helm_biz_3h_bis.eps}%
\else
\ifoutputfig%
\includegraphics[width=5.5in]{eps/helm_biz_3h_bis.eps}%
\fi
\fi
\caption
[The pressure waveform of the
three solutions found by
the HBM with $N_p=3$ employing the simple clarinet model with
$\zeta=0.5$, $\gamma=0.4$, $\eta=10^{-5}$.]
{\label{f:helm_biz_3h}
\ifgalleyfig
{The pressure waveform of the
three solutions found by
the HBM with $N_p=3$ employing the simple clarinet model with
$\zeta=0.5$, $\gamma=0.4$, $\eta=10^{-5}$.}
\fi}
\end{figure}
\begin{figure}
\ifgalleyfig%
\includegraphics[width=3.25in]{eps/deux_sol_np100.eps}%
\else
\ifoutputfig%
\includegraphics[width=5.5in]{eps/deux_sol_np100.eps}%
\fi
\fi
\caption
[The pressure waveform of two solutions that differ by the duration of their steps, found by
the HBM employing the simple clarinet model with
$\zeta=0.5$, $\gamma=0.4$, $\eta=10^{-5}$, $N_p=99$.]
{\label{f:sol_np100}
\ifgalleyfig
{The pressure waveform of two solutions that differ by the duration of their steps, found by
the HBM employing the simple clarinet model with
$\zeta=0.5$, $\gamma=0.4$, $\eta=10^{-5}$, $N_p=99$.}
\fi}
\end{figure}
While the Helmholtz motion is known to be stable \cite{kergomard95},
the two three-level solutions can be considered as a combination of the
static solution (the zero level) and the square wave (two levels with
opposite values). Since we know from Kergomard \cite{kergomard95} that
in the case of ideal propagation (neither losses nor dispersion), the
stability domain of these two solutions are mutually exclusive, it can
be concluded that the three-level solutions are unstable.
Taking into account losses in the propagation does not make the
three-level solutions vanish. But a
simple reasoning to determine the stability of this
solution is not possible in this case. To
the authors knowledge,
however, such a solution has never been observed experimentally at low
level of excitation.
\subsection{Initial value of the playing frequency}
A practical difficulty encountered is the convergence of the playing
frequency $f_p$. If its initial value is not close enough to the
solution, divergence is almost inevitable.
This occurs because the resonator impedance $Z_r$ tends to vanish
outside the immediate surroundings of the resonance peaks of the resonator,
rendering $\vec F(\vec P,f_p)$ very small and thereby $\vec G\simeq
\vec P/P_1$ nearly constant with respect to $f_p$. The slope
$\partial\vec G/\partial f_p$ thus becomes close to
zero, the Newton step leads far away from the solution, and
convergence fails. Dissipation widens the resonance peaks and thus
also the convergence range.
For a simple system where the playing frequency is known to correspond
to a resonance peak of the tube, initializing $f_p$ is easy. However,
with dispersion or other inharmonic effects, choosing an initial value
for $f_p$ may be difficult. In Harmbal the problem may to some extent
be avoided by the possibility of gradually adding the
dispersion (or other inharmonic effects), so that the playing frequency can be
followed quasi-continuously from a known solution without dispersion,
for instance by using {\em hbmap}.
\section{Conclusions}\label{s:concl}
The harmonic balance method (HBM) is suited for studies of
self-sustained oscillations of musical instruments, and the
computer program {\em Harmbal\/} has been developed for this application.
It is available with its source
code \cite{harmbal}, has a free licence, and is already in use by
several researchers. It is programmed in C, runs fast, and is easily
used by other application, such as for continuation purposes.
Some difficulties are related to the digital sampling of the signal
and can be solved by introducing a backtracking mechanism. When using
a large number of harmonics, the extreme case of the (lossless)
Helmholtz motion can be solved for different shapes of
resonators. Nevertheless, the value of the first harmonic $P_1$
seems to be well predicted by lower values of $N_p$, in particular
close to the threshold of a direct bifurcation.
For the saxophone we used a stepped-cone bore and observed one or more
dips during the longest part of the period, depending on the number of
steps. These dips approach pure impulses
as $N_p$ increases. The number of samples $N_t$ in a period showed to
be insignificant for these dips.
The HBM can lead to some alternative solutions for a unique
set of parameters. The nondissipative versions of these
solutions satisfy the continuous model
equations, but they are not stable and thus cannot be attained by ab
initio time-domain calculations. Another
problem is the great sensitivity to the guessed playing frequency.
As a consequence, a certain expertise is needed in order to use the
method, but, thanks to an automatic continuation procedure, the
calculation is easy. We note that also experimental
results can be used for the impedance of the resonator.
\begin{acknowledgments}
The Europeen Union through the MOSART project is acknowledged for
financial support. We would also like to thank Claudia Fritz at IRCAM
in Paris for thorough testing and valuable feedback, Jo\"el Gilbert at
Laboratoire d'Acoustique de l'Universit\'e du Maine (LAUM) in Le Mans,
and Philippe Guillemain at the Laboratoire de M\'ecanique et
d'Acoustique at CNRS in Marseille for fruitful discussions during the
work, and the latter also for kindly providing some Matlab code for the
time-domain model.
\end{acknowledgments}
\bibliographystyle{jasasty}
|
{
"timestamp": "2005-03-07T11:23:46",
"yymm": "0503",
"arxiv_id": "physics/0503047",
"language": "en",
"url": "https://arxiv.org/abs/physics/0503047"
}
|
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\rlvec(0.866025403784439 .5) \rlvec(0.866025403784439 -.5)
\rlvec(-0.866025403784439 -.5) \rlvec(0 1)
\rmove(0 -1) \rlvec(-0.866025403784439 .5)
\savepos(0.866025403784439 -.5)(*ex *ey)
\esegment
\move(*ex *ey)
}
\def\RhombusA{\bsegment
\rlvec(0.866025403784439 .5) \rlvec(0.866025403784439 -.5)
\rlvec(-0.866025403784439 -.5) \rlvec(-0.866025403784439 .5)
\savepos(0.866025403784439 -.5)(*ex *ey)
\esegment
\move(*ex *ey)
}
\def\RhombusB{\bsegment
\rlvec(0.866025403784439 .5) \rlvec(0 -1)
\rlvec(-0.866025403784439 -.5) \rlvec(0 1)
\savepos(0 -1)(*ex *ey)
\esegment
\move(*ex *ey)
}
\def\RhombusC{\bsegment
\rlvec(0.866025403784439 -.5) \rlvec(0 -1)
\rlvec(-0.866025403784439 .5) \rlvec(0 1)
\savepos(0.866025403784439 -.5)(*ex *ey)
\esegment
\move(*ex *ey)
}
\def\alpha{\alpha}
\def\beta{\beta}
\def\gamma{\gamma}
\def\delta{\delta}
\def\varepsilon{\varepsilon}
\def\zeta{\zeta}
\def\eta{\eta}
\def\theta{\theta}
\def\vartheta{\vartheta}
\def\iota{\iota}
\def\kappa{\kappa}
\def\lambda{\lambda}
\def\rho{\rho}
\def\sigma{\sigma}
\def\tau{\tau}
\def\varphi{\varphi}
\def\chi{\chi}
\def\psi{\psi}
\def\omega{\omega}
\def\Gamma{\Gamma}
\def\Delta{\Delta}
\def\Theta{\Theta}
\def\Lambda{\Lambda}
\def\Sigma{\Sigma}
\def\Phi{\Phi}
\def\Psi{\Psi}
\def\Omega{\Omega}
\def\row#1#2#3{#1_{#2},\ldots,#1_{#3}}
\def\rowup#1#2#3{#1^{#2},\ldots,#1^{#3}}
\def\times{\times}
\def\crf{}
\def\rf{}
\def\rfnew{}
\def{\mathbb P}{{\mathbb P}}
\def{\mathbb R}{{\mathbb R}}
\def{\mathcal X}{{\mathcal X}}
\def{\mathbb C}{{\mathbb C}}
\def{\mathcal Mf}{{\mathcal Mf}}
\def{\mathcal F\mathcal M}{{\mathcal F\mathcal M}}
\def{\mathcal F}{{\mathcal F}}
\def{\mathcal G}{{\mathcal G}}
\def{\mathcal V}{{\mathcal V}}
\def{\mathcal T}{{\mathcal T}}
\def{\mathcal A}{{\mathcal A}}
\def{\mathbb N}{{\mathbb N}}
\def{\mathbb Z}{{\mathbb Z}}
\def{\mathbb Q}{{\mathbb Q}}
\def\left.\tfrac \partial{\partial t}\right\vert_0{\left.\tfrac \partial{\partial t}\right\vert_0}
\def\dd#1{\tfrac \partial{\partial #1}}
\def\today{\ifcase\month\or
January\or February\or March\or April\or May\or June\or
July\or August\or September\or October\or November\or December\fi
\space\number\day, \number\year}
\def\nmb#1#2{#2}
\def\iprod#1#2{\langle#1,#2\rangle}
\def\pder#1#2{\frac{\partial #1}{\partial #2}}
\def\int\!\!\int{\int\!\!\int}
\def\({\left(}
\def\){\right)}
\def\[{\left[}
\def\]{\right]}
\def\operatorname{supp}{\operatorname{supp}}
\def\operatorname{Comp}{\operatorname{Comp}}
\def\operatorname{Part}{\operatorname{Part}}
\def\operatorname{Df}{\operatorname{Df}}
\def\operatorname{dom}{\operatorname{dom}}
\def\operatorname{Ker}{\operatorname{Ker}}
\def\operatorname{Per}{\operatorname{Per}}
\def\operatorname{Tr}{\operatorname{Tr}}
\def\operatorname{Res}{\operatorname{Res}}
\def\operatorname{Aut}{\operatorname{Aut}}
\def\operatorname{kgV}{\operatorname{kgV}}
\def\operatorname{ggT}{\operatorname{ggT}}
\def\operatorname{diam}{\operatorname{diam}}
\def\operatorname{Im}{\operatorname{Im}}
\def\operatorname{Re}{\operatorname{Re}}
\def\operatorname{ord}{\operatorname{ord}}
\def\operatorname{rang}{\operatorname{rang}}
\def\operatorname{rng}{\operatorname{rng}}
\def\operatorname{grd}{\operatorname{grd}}
\def\operatorname{inv}{\operatorname{inv}}
\def\operatorname{maj}{\operatorname{maj}}
\def\operatorname{fmaj}{\operatorname{fmaj}}
\def\operatorname{nmaj}{\operatorname{nmaj}}
\def\operatorname{neg}{\operatorname{neg}}
\def\operatorname{sneg}{\operatorname{sneg}}
\def\operatorname{des}{\operatorname{des}}
\def\operatorname{\overline{maj}}{\operatorname{\overline{maj}}}
\def\operatorname{\overline{des}}{\operatorname{\overline{des}}}
\def\operatorname{\overline{maj}'}{\operatorname{\overline{maj}'}}
\def\operatorname{maj'}{\operatorname{maj'}}
\def\operatorname{zbk}{\operatorname{zbk}}
\def\operatorname{nzbk}{\operatorname{nzbk}}
\def\operatorname{NC}{\operatorname{NC}}
\def\operatorname{NCmatch}{\operatorname{NCmatch}}
\def\operatorname{ln}{\operatorname{ln}}
\def\operatorname{der}{\operatorname{der}}
\def\operatorname{Hom}{\operatorname{Hom}}
\def\operatorname{tr}{\operatorname{tr}}
\def\operatorname{Span}{\operatorname{Span}}
\def\operatorname{grad}{\operatorname{grad}}
\def\operatorname{div}{\operatorname{div}}
\def\operatorname{rot}{\operatorname{rot}}
\def\operatorname{Sp}{\operatorname{Sp}}
\def\operatorname{sgn}{\operatorname{sgn}}
\def\lim\limits{\lim\limits}
\def\sup\limits{\sup\limits}
\def\bigcup\limits{\bigcup\limits}
\def\bigcap\limits{\bigcap\limits}
\def\limsup\limits{\limsup\limits}
\def\liminf\limits{\liminf\limits}
\def\int\limits{\int\limits}
\def\sum\limits{\sum\limits}
\def\max\limits{\max\limits}
\def\min\limits{\min\limits}
\def\prod\limits{\prod\limits}
\def\operatorname{tan}{\operatorname{tan}}
\def\operatorname{cot}{\operatorname{cot}}
\def\operatorname{arctan}{\operatorname{arctan}}
\def\operatorname{arccot}{\operatorname{arccot}}
\def\operatorname{arccot}{\operatorname{arccot}}
\def\operatorname{tanh}{\operatorname{tanh}}
\def\operatorname{coth}{\operatorname{coth}}
\def\operatorname{arcsinh}{\operatorname{arcsinh}}
\def\operatorname{arccosh}{\operatorname{arccosh}}
\def\operatorname{arctanh}{\operatorname{arctanh}}
\def\operatorname{arccoth}{\operatorname{arccoth}}
\def\ss{\ss}
\let\vv\v
\def\operatorname{Tr}{\operatorname{Tr}}
\def\po#1#2{(#1)_{#2}}
\def\fl#1{\left\lfloor#1\right\rfloor}
\def\cl#1{\left\lceil#1\right\rceil}
\def\coef#1{\left\langle#1\right\rangle}
\def{\bar X}{{\bar X}}
\def{\sqrt{-1}}{{\sqrt{-1}}}
\def{(\sqrt{-1})}{{(\sqrt{-1})}}
\def\operatorname{bk}{\operatorname{bk}}
\def\operatorname{CT}{\operatorname{CT}}
\def\operatorname{NC}{\operatorname{NC}}
\def\operatorname{rank}{\operatorname{rank}}
\def\operatorname{stat}{\operatorname{stat}}
\def\operatorname{NCmatch}{\operatorname{NCmatch}}
\def\operatorname{Pf}{\operatorname{Pf}}
\DeclareMathOperator{\h}{H}
\def\raise-15pt\hbox{{\Huge$\square$}}{\raise-15pt\hbox{{\Huge$\square$}}}
\def\qbinom#1#2{\left[\begin{smallmatrix} #1\\#2\end{smallmatrix}\right]_q}
\begin{document}
\newbox\Adr
\setbox\Adr\vbox{
\centerline{\sc C.~Krattenthaler$^\dagger$}
\vskip18pt
\centerline{Institut Camille Jordan, Universit\'e Claude Bernard
Lyon-I,}
\centerline{21, avenue Claude Bernard, F-69622 Villeurbanne Cedex,
France.}
\centerline{E-mail: {\tt\footnotesize kratt@euler.univ-lyon1.fr}}
\centerline{WWW: \footnotesize\tt http://igd.univ-lyon1.fr/\~{}kratt}
}
\title{Advanced Determinant Calculus: A Complement}
\author[C.~Krattenthaler]{\box\Adr}
\address{Institut Girard
Desargues, Universit\'e Claude Bernard Lyon-I,
21, avenue Claude Bernard, F-69622 Villeurbanne Cedex, France.}
\email{kratt@euler.univ-lyon1.fr}
\thanks{$^\dagger$ Research partially supported by EC's IHRP Programme,
grant HPRN-CT-2001-00272, ``Algebraic Combinatorics in Europe", and by
the ``Algebraic Combinatorics" Programme during Spring 2005
of the Institut Mittag--Leffler of the Royal Swedish Academy of Sciences}
\subjclass[2000]{Primary 05A19;
Secondary 05A10 05A15 05A17 05A18 05A30 05E10 05E15 11B68 11B73 11C20
11Y60 15A15 33C45 33D45 33E05}
\keywords{Determinants, Vandermonde determinant, Cauchy's double
alternant, skew circulant matrix, confluent alternant, confluent
Cauchy determinant, Pfaffian,
Hankel determinants, orthogonal polynomials,
Chebyshev polynomials, Meixner polynomials, Laguerre
polynomials, continued fractions,
binomial coefficient, Catalan numbers, Fibonacci numbers, Bernoulli numbers,
Stirling numbers, non-intersecting lattice paths,
plane partitions, tableaux, rhombus tilings, lozenge tilings,
alternating sign matrices,
non-crossing partitions, perfect matchings, permutations,
signed permutations,
inversion number, major index, compositions, integer partitions,
descent algebra, non-commutative symmetric functions,
elliptic functions,
the number $\pi$, LLL-algorithm}
\begin{abstract}
This is a complement to my previous article {``Advanced
Determinant Calculus"} ({\it S\'eminaire Lotharingien Combin.}\ {\bf
42} (1999), Article~B42q, 67~pp.). In the present article, I
share with the reader my experience of applying the
methods described in
the previous article in order to solve a particular problem
from number theory (G.~Almkvist, J.~Petersson and the author,
{\it Experiment.\ Math.}\ {\bf 12} (2003), 441--456).
Moreover, I add a list of determinant evaluations which I consider as
interesting, which have been found since
the appearance of the previous article, or which I failed to mention
there, including several conjectures and open problems.
\end{abstract}
\maketitle
\section{Introduction}
In the previous article
\machSeite{KratBN}\cite{KratBN},
I described several methods to evaluate determinants, and
I provided a long list of known determinant evaluations.
The present article is meant as a complement to
\machSeite{KratBN}\cite{KratBN}.
Its purpose is three-fold: first, I want to shed light on
the problem of evaluating determinants from a slightly different
angle, by sharing with the reader my experience of applying the
methods from
\machSeite{KratBN}\cite{KratBN} in order to solve a particular problem
from number theory (see Sections~\ref{sec:det} and \ref{sec:eval});
second, I shall address the question
why it is apparently in the first case combinatorialists
(such as myself) who are so interested in determinant evaluations
and get so easily excited about them (see Section~\ref{sec:comb});
and, finally third,
I add a list of determinant evaluations, which I consider
as interesting, which have been found since
the appearance of
\machSeite{KratBN}\cite{KratBN}, or which I failed to mention in
the list given in Section~3 of
\machSeite{KratBN}\cite{KratBN} (see
Section~\ref{sec:detlist}), including several conjectures and open
problems.
\section{Enumerative combinatorics, nice formulae, and determinants}
\label{sec:comb}
Why are combinatorialists so fascinated by determinant
evaluations?
A simplistic answer to this question goes
as follows. Clearly, binomial coefficients $\binom nk$ or
Stirling numbers (of the second kind) $S(n,k)$ are basic
objects in (enumerative) combinatorics; after all they count the
subsets of cardinality $k$ of a set with $n$ elements,
respectively the ways of partitioning such a set of $n$
elements into $k$ pairwise disjoint non-empty subsets. Thus, if one
sees an identity such as\footnote{For more information on this
determinant see Theorems~\ref{thm:MM} and \ref{thm:MM2} in this section
and
\machSeite{KratBN}\cite[Sections~2.2, 2.3 and 2.5]{KratBN}.}
\begin{equation} \label{eq:M1}
\det_{1\le i,j\le n}\(\binom {a+b}{a-i+j} \)=
\prod _{i=1} ^{n}\frac {(a+b+i-1)!\,(i-1)!} {(a+i-1)!\,(b+i-1)!},
\end{equation}
or\footnote{This determinant evaluation follows easily from the matrix
factorisation
$$\(S(i+j,i) \)_{1\le i,j\le n}=\((-1)^kk^i/(k!\,(i-k)!)\)_{1\le
i,k\le n}\cdot\(k^j\)_{1\le k,j\le n},$$
application of
\machSeite{KratBN}\cite[Theorem~26, (3.14)]{KratBN} to the first
determinant, and
application of the Vandermonde determinant evaluation to the
second.}
\begin{equation} \label{eq:S2}
\det_{1\le i,j\le n}\(S(i+j,i) \)=\prod _{i=1} ^{n}i^i
\end{equation}
(and there are many more of that kind; see
\machSeite{KratBN}\cite{KratBN} and Section~\ref{sec:detlist}),
there is an obvious excitement that one cannot escape.
Although this is indeed an explanation which applies in many cases,
there is also an answer on a more substantial level, which brings us
to the reason why {\it I\/} like (and need) determinant evaluations.
The {\it favourite question} for an enumerative combinatorialist (such as
myself) is
$$\text {\it How many $\langle\dots\rangle$ are there?}$$
Here, $\langle\dots\rangle$ can be permutations with certain properties,
certain
partitions, certain paths, certain trees, etc. The {\it
favourite theorem} then is:
\begin{Theorem} \label{thm:fav}
The number of $\langle\dots\rangle$ of size $n$ is equal to
$$NICE(n).$$
\end{Theorem}
I have already explained the meaning of $\langle\dots\rangle$. What does
$NICE(n)$ stand for?
Typical examples for $NICE(n)$ are formulae
such as
\begin{equation} \label{eq:Cat}
\frac {1} {n+1}\binom {2n}n
\end{equation}
(\!{\it Catalan numbers}; cf.\
\machSeite{StanBI}\cite[Ex.~6.19]{StanBI}) or
\begin{equation} \label{eq:alt}
\prod _{i=0} ^{n-1}\frac {(3i+1)!}{(n+i)!}
\end{equation}
(the number of $n\times n$
alternating sign matrices and several other combinatorial objects;
cf.\
\machSeite{BresAO}\cite{BresAO}). Let us be more precise.
\begin{DefinitionA} \label{def:1}
The symbol $NICE(n)$ is a formula of the type
\begin{equation} \label{eq:NICE}
\xi^n\cdot\text{\em Rat}(n)\cdot
\prod _{i=1} ^{k}\frac {(a_in+b_i)!} {(c_in+d_i)!},
\end{equation}
where $\text{\em Rat}(n)$ is a rational function in $n$,
and where $a_i,c_i\in {\mathbb Z}$ for $i=1,2,\dots,k$,
${\mathbb Z}$ denoting the set of integers.
The parameters $b_i,c_i,\xi$ can be arbitrary real or complex
numbers. {\em(}If necessary, $(a_in+b_i)!$ has to be interpreted as
$\Gamma(a_in+b_i+1)$, where $\Gamma(x)$ is the Euler gamma function,
and similarly for $(c_in+d_i)!$.{\em)}
\end{DefinitionA}
Clearly, the formulae \eqref{eq:Cat} and \eqref{eq:alt} fit this
``Definition".\footnote{The writing $NICE(n)$ is borrowed from
Doron Zeilberger
\machSeite{ZeilAP}\cite[Recitation~III]{ZeilAP}.
The technical term for a formula of the type \eqref{eq:NICE} is
{\it``hypergeometric term"}, see
\machSeite{PeWZAA}\cite[Sec.~3.2]{PeWZAA}, whereas, most often,
the colloquial terms {\it``closed form"} or {\it``nice formula"}
are used for it, see
\machSeite{ZeilAP}\cite[Recitation~II]{ZeilAP}.
More recently, some authors call sequences given by
formulae of that type sequences of ``round" numbers, see
\machSeite{KupeAH}\cite[Sec.~6]{KupeAH}.}
If one is working on a particular problem,
how can one recognise that one is looking at a sequence of numbers
given by $NICE(n)$? The key observation is that, if we factorise
$(an+b)!$ into its prime factors, where $a$ and $b$ are integers,
then, as $n$ runs through the positive integers, the numbers
$(an+b)!$ explode quickly, whereas the prime factors
occurring in the factorisation will grow only moderately, more
precisely, they will grow roughly linearly. Thus, if we encounter a
sequence the prime factorisation of which has this property, we can
be sure that there is a formula $NICE(n)$ for this sequence.
Even better, as I explain in Appendix~A of
\machSeite{KratBN}\cite{KratBN}, the program
{\tt Rate}\footnote{\label{foot:Rate}{\tt Rate} is available from {\tt
http://igd.univ-lyon1.fr/\~{}kratt}. It is
based on a rather simple algorithm which involves rational interpolation.
In contrast to what I read, with great surprise, in
\machSeite{CoBWAA}\cite{CoBWAA},
the explanations of how {\tt Rate} works in Appendix~A of
\machSeite{KratBN}\cite{KratBN} can be read and understood without any
knowledge about
determinants and, in particular, without any knowledge of the fifty
or so pages that precede Appendix~A in
\machSeite{KratBN}\cite{KratBN}.}
will (normally\footnote{\label{foot:normally}{\tt Rate}
will {\it always} be able to guess
a formula of the type \eqref{eq:NICE} if there are enough initial
terms of the sequence available. However, there is a larger class of
sequences which have the property that the size of the primes in the
prime factorisation of the
terms of the sequence grows only slowly with $n$. These are sequences
given by formulae containing ``Abelian" factors, such as
$n^n$. Unfortunately, {\tt Rate} does not know how to handle
such factors. Recently, Rubey
\machSeite{RubeAD}\cite{RubeAD} proposed an algorithm for covering
Abelian factors as well. His implementation {\tt Guess} is written
in {\sl Axiom} and is available
at {\tt http://www.mat.univie.ac.at/\~{}rubey/martin.html}.})
be able to guess the formula.
To illustrate this, let us look at a particular example.
Let us suppose that the first few values of our sequence are the
following:
\begin{multline*}
1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900,
2674440, \\
9694845, 35357670, 129644790, 477638700, 1767263190,
6564120420.
\end{multline*}
The prime factorisation of the second-to-last number is
(we are using {\sl
Mathematica} here)
\MATH
\goodbreakpoint%
In[1]:= FactorInteger[477638700]
\goodbreakpoint%
Out[1]= %
\MATHlbrace %
\MATHlbrace 2, 2%
\MATHrbrace , %
\MATHlbrace 3, 1%
\MATHrbrace , %
\MATHlbrace 5, 2%
\MATHrbrace , %
\MATHlbrace 7, 1%
\MATHrbrace , %
\MATHlbrace 11, 1%
\MATHrbrace , %
\MATHlbrace 23, 1%
\MATHrbrace , %
\MATHlbrace 29, 1%
\MATHrbrace , %
\MATHlbrace 31, 1%
\MATHrbrace %
\MATHrbrace
\goodbreakpoint%
\endgroup
whereas the prime factorisations of the next-to-last and the last
number in this sequence are
\MATH
\goodbreakpoint%
In[2]:= FactorInteger[1767263190]
\goodbreakpoint%
Out[2]= %
\MATHlbrace %
\MATHlbrace 2, 1%
\MATHrbrace , %
\MATHlbrace 3, 1%
\MATHrbrace , %
\MATHlbrace 5, 1%
\MATHrbrace , %
\MATHlbrace 7, 1%
\MATHrbrace , %
\MATHlbrace 11, 1%
\MATHrbrace , %
\MATHlbrace 23, 1%
\MATHrbrace , %
\MATHlbrace 29, 1%
\MATHrbrace , %
\MATHlbrace 31, 1%
\MATHrbrace ,
> %
\MATHlbrace 37, 1%
\MATHrbrace %
\MATHrbrace
\goodbreakpoint%
In[3]:= FactorInteger[6564120420]
\goodbreakpoint%
Out[3]= %
\MATHlbrace %
\MATHlbrace 2, 2%
\MATHrbrace , %
\MATHlbrace 3, 1%
\MATHrbrace , %
\MATHlbrace 5, 1%
\MATHrbrace , %
\MATHlbrace 11, 1%
\MATHrbrace , %
\MATHlbrace 13, 1%
\MATHrbrace , %
\MATHlbrace 23, 1%
\MATHrbrace , %
\MATHlbrace 29, 1%
\MATHrbrace , %
\MATHlbrace 31, 1%
\MATHrbrace ,
> %
\MATHlbrace 37, 1%
\MATHrbrace %
\MATHrbrace
\goodbreakpoint%
\endgroup
\noindent
(To decipher this for the reader unfamiliar with {\sl Mathematica}:
the prime factorisation of the last number is
$ 2^ 2 3^ 1 5^ 1 11^ 1 13^ 1 23^ 1 29^ 1 31^ 1 37^ 1$.)
One observes, first of all, that the occurring prime factors are
rather small in comparison to the numbers of which they are factors,
and, second, that the size of the prime factors grows only very
slowly (from 31 to 37). Thus, we {\it can be sure} that there is a
``nice" formula $NICE(n)$ for this sequence. Indeed, {\tt Rate} needs only
the first five members of the sequence to come up with a guess for
$NICE(n)$:
\MATH
\goodbreakpoint%
In[4]:= <{<}rate.m
\goodbreakpoint%
In[5]:= Rate[1,2,5,14,42]
\goodbreakpoint%
\leavevmode%
i0 1
\leavevmode%
4 Gamma[- + i0]
\leavevmode%
2
Out[5]= ----------------------
\leavevmode%
Sqrt[Pi] Gamma[2 + i0]
\goodbreakpoint%
\endgroup
As the reader will have guessed, {\tt Rate} uses the parameter $i_0$
instead of $n$. In fact, the formula is a fancy way to write
$\frac {1} {i_0+1}\binom {2i_0}{i_0}$, that is, we were looking at
the sequence of Catalan numbers \eqref{eq:Cat}.
To see the sharp contrast, here are the first few terms of another sequence:
$$1,2,9,272,589185.$$
(Also these are combinatorial numbers. They count the perfect
matchings of the $n$-dimensional hypercube; cf.\
\machSeite{PropAH}\cite[Problem~19]{PropAH}.)
Let us factorise the last two numbers:
\MATH
\goodbreakpoint%
In[6]:= FactorInteger[272]
\goodbreakpoint%
Out[6]= %
\MATHlbrace %
\MATHlbrace 2, 4%
\MATHrbrace , %
\MATHlbrace 17, 1%
\MATHrbrace %
\MATHrbrace
\goodbreakpoint%
In[7]:= FactorInteger[589185]
\goodbreakpoint%
Out[7]= %
\MATHlbrace %
\MATHlbrace 3, 2%
\MATHrbrace , %
\MATHlbrace 5, 1%
\MATHrbrace , %
\MATHlbrace 13093, 1%
\MATHrbrace %
\MATHrbrace
\goodbreakpoint%
\endgroup
The presence of the big prime factor 13093 in the last factorisation is
a sure sign that we cannot expect a formula $NICE(n)$ as described
in the ``Definition" for this sequence of numbers. (There may well be a
simple formula of a different kind.
It is not very likely, though. In any case, such a formula has not
been found up to this date.)
\medskip
Now, that I have sufficiently explained all the ingredients in the
``prototype theorem'' Theorem~\ref{thm:fav}, I can explain why theorems of this
form are so attractive (at least to me): the objects (i.e., the
permutations, partitions, paths, trees, etc.) that it deals
with are usually very simple to explain, the statement is very simple
and can be understood by anybody, the result $NICE(n)$ has a very
elegant form, and yet, very often it is not easy at all to give a
proof (not to mention a true {\it explanation} why such an elegant
result occurs.)
Here are two examples. They concern
{\it rhombus tilings}, by which I mean tilings
of a region by rhombi with side lengths 1 and angles of
$60^\circ$ and $120^\circ$. The first one is a one century old theorem
due to MacMahon
{\machSeite{MacMAA}\cite[Sec.~429, $q\rightarrow 1$; proof in
Sec.~494]{MacMAA}}.\footnote{To be correct, MacMahon did not know
anything about rhombus tilings, they did not exist in enumerative
combinatorics at the time. The objects that he considered were
{\it plane partitions}. However, there is a very simple bijection
between plane partitions contained in an $a\times b\times c$ box
and rhombus tilings of a hexagon
with side lengths $a,b,c,a,b,c$, as explained for example in
\machSeite{DT}\cite{DT}.}
\begin{Theorem}
\label{thm:MM}
The number of rhombus tilings of a hexagon with
side lengths $a,b,c,a,b,c$ whose angles are $120^\circ$
{\em(}see Figure~\ref{fig:1}.a for an example of such a hexagon, and
Figure~\ref{fig:1}.b for an example of a rhombus tiling{\em)} is equal to
\begin{equation} \label{eq:M2}
\prod _{i=1} ^{c}\frac {(a+b+i-1)!\,(i-1)!} {(a+i-1)!\,(b+i-1)!}.
\end{equation}
\quad \quad \qed
\end{Theorem}
\begin{figure}[h]
\centertexdraw{
\drawdim truecm \linewd.02
\rhombus \rhombus \rhombus \rhombus \ldreieck
\move (-0.866025403784439 -.5)
\rhombus \rhombus \rhombus \rhombus \rhombus \ldreieck
\move (-1.732050807568877 -1)
\rhombus \rhombus \rhombus \rhombus \rhombus \rhombus \ldreieck
\move (-1.732050807568877 -1)
\rdreieck
\rhombus \rhombus \rhombus \rhombus \rhombus \rhombus \ldreieck
\move (-1.732050807568877 -2)
\rdreieck
\rhombus \rhombus \rhombus \rhombus \rhombus \rhombus \ldreieck
\move (-1.732050807568877 -3)
\rdreieck
\rhombus \rhombus \rhombus \rhombus \rhombus \rhombus
\move (-1.732050807568877 -4)
\rdreieck
\rhombus \rhombus \rhombus \rhombus \rhombus
\move (-1.732050807568877 -5)
\rdreieck
\rhombus \rhombus \rhombus \rhombus
\move(8 0)
\bsegment
\drawdim truecm \linewd.02
\rhombus \rhombus \rhombus \rhombus \ldreieck
\move (-0.866025403784439 -.5)
\rhombus \rhombus \rhombus \rhombus \rhombus \ldreieck
\move (-1.732050807568877 -1)
\rhombus \rhombus \rhombus \rhombus \rhombus \rhombus \ldreieck
\move (-1.732050807568877 -1)
\rdreieck
\rhombus \rhombus \rhombus \rhombus \rhombus \rhombus \ldreieck
\move (-1.732050807568877 -2)
\rdreieck
\rhombus \rhombus \rhombus \rhombus \rhombus \rhombus \ldreieck
\move (-1.732050807568877 -3)
\rdreieck
\rhombus \rhombus \rhombus \rhombus \rhombus \rhombus
\move (-1.732050807568877 -4)
\rdreieck
\rhombus \rhombus \rhombus \rhombus \rhombus
\move (-1.732050807568877 -5)
\rdreieck
\rhombus \rhombus \rhombus \rhombus
\linewd.12
\move(0 0)
\RhombusA \RhombusB \RhombusB
\RhombusA \RhombusA \RhombusB \RhombusA \RhombusB \RhombusB
\move (-0.866025403784439 -.5)
\RhombusA \RhombusB \RhombusB \RhombusB \RhombusB
\RhombusA \RhombusA \RhombusB \RhombusA
\move (-1.732050807568877 -1)
\RhombusB \RhombusB \RhombusA \RhombusB \RhombusB \RhombusA
\RhombusB \RhombusA \RhombusA
\move (1.732050807568877 0)
\RhombusC \RhombusC \RhombusC
\move (1.732050807568877 -1)
\RhombusC \RhombusC \RhombusC
\move (3.464101615137755 -3)
\RhombusC
\move (-0.866025403784439 -.5)
\RhombusC
\move (-0.866025403784439 -1.5)
\RhombusC
\move (0.866025403784439 -2.5)
\RhombusC \RhombusC
\move (0.866025403784439 -3.5)
\RhombusC \RhombusC \RhombusC
\move (2.598076211353316 -5.5)
\RhombusC
\move (0.866025403784439 -5.5)
\RhombusC
\move (-1.732050807568877 -3)
\RhombusC
\move (-1.732050807568877 -4)
\RhombusC
\move (-1.732050807568877 -5)
\RhombusC \RhombusC
\esegment
\htext (-1.5 -9){\small a. A hexagon with sides $a,b,c,a,b,c$,}
\htext (-1.5 -9.5){\small \hphantom{a. }where $a=3$, $b=4$, $c=5$}
\htext (6.8 -9){\small b. A rhombus tiling of a hexagon}
\htext (6.8 -9.5){\small \hphantom{b. }with sides $a,b,c,a,b,c$}
\rtext td:0 (4.3 -4.1){$\sideset {} c
{\left.\vbox{\vskip2.6cm}\right\}}$}
\rtext td:60 (2.6 -.55){$\sideset {} {}
{\left.\vbox{\vskip2cm}\right\}}$}
\rtext td:120 (-.34 -.2){$\sideset {} {}
{\left.\vbox{\vskip1.7cm}\right\}}$}
\rtext td:0 (-2.4 -3.6){$\sideset {c} {}
{\left\{\vbox{\vskip2.6cm}\right.}$}
\rtext td:240 (-0.1 -6.9){$\sideset {} {}
{\left.\vbox{\vskip2cm}\right\}}$}
\rtext td:300 (2.9 -7.3){$\sideset {} {}
{\left.\vbox{\vskip1.7cm}\right\}}$}
\htext (-.9 0.2){$a$}
\htext (2.8 -.1){$b$}
\htext (3.2 -7.9){$a$}
\htext (-0.4 -7.65){$b$}
}
\caption{}
\label{fig:1}
\end{figure}
The second one is more recent, and is due to Ciucu, Eisenk\"olbl, Zare
and the author {\machSeite{CiEKAA}\cite[Theorem~1]{CiEKAA}}.
\begin{Theorem} \label{enum}
If $a,b,c$ have the same parity, then
the number of lozenge tilings of a hexagon with side lengths
$a,b+m,c,a+m,b,c+m$, with an equilateral triangle of
side length $m$ removed from its centre
{\em(}see Figure~\ref{hex} for an example{\em)}
is given by
\begin{multline} \label{eq:enum}
\frac {\h(a + m)\h(b + m)\h(c + m)\h(a + b + c + m)
}
{\h(a + b + m)\h(a + c + m)\h(b + c + m)
}
\frac {\h(m + \left \lceil {\frac{a + b + c}{2}} \right \rceil)
\h(m + \left \lfloor {\frac{a + b + c}{2}} \right \rfloor)
} {\h({\frac{a + b}{2}} + m) \h({\frac{a + c}{2}} + m)\h({\frac{b + c}{2}} + m)
}
\\
\times\frac {\h(\left \lceil {\frac{a}{2}} \right \rceil)
\h(\left \lceil {\frac{b}{2}} \right \rceil)
\h(\left \lceil {\frac{c}{2}} \right \rceil)
\h(\left \lfloor {\frac{a}{2}} \right \rfloor)\,
\h(\left \lfloor {\frac{b}{2}} \right \rfloor)\,
\h(\left \lfloor {\frac{c}{2}} \right \rfloor)\,
}
{\h({\frac{m}{2}} + \left \lceil {\frac{a}{2}} \right \rceil)\,
\h({\frac{m}{2}} + \left \lceil {\frac{b}{2}} \right \rceil)\,
\h({\frac{m}{2}} + \left \lceil {\frac{c}{2}} \right \rceil)\,
\h({\frac{m}{2}} + \left \lfloor {\frac{a}{2}} \right \rfloor)\,
\h({\frac{m}{2}} + \left \lfloor {\frac{b}{2}} \right \rfloor)\,
\h({\frac{m}{2}} + \left \lfloor {\frac{c}{2}} \right \rfloor)\,
}\\
\times
\frac {\h(\frac{m}{2})^2 \h({\frac{a + b + m}{2}})^2
\h({\frac{a + c + m}{2}})^2 \h({\frac{b + c +
m}{2}})^2
}
{\h({\frac{m}{2}} + \left \lceil {\frac{a + b + c}{2}} \right \rceil)
\h({\frac{m}{2}} + \left \lfloor {\frac{a + b + c}{2}} \right \rfloor)
\h({\frac{a + b}{2}})\h({\frac{a + c}{2}})\h({\frac{b + c}{2}})
},
\end{multline}
where
\begin{equation} \label{eq:hyperfac}
\h(n):=\begin{cases}
\prod _{k=0} ^{n-1}{\Gamma(k+1)}\quad &\text {for $n$ an integer,}\\
\prod _{k=0} ^{n-\frac {1} {2}}{\Gamma(k+\frac {1} {2})} \quad &\text
{for $n$ a half-integer}.
\end{cases}
\end{equation}
\quad \quad \qed
\end{Theorem}
(There is a similar theorem if the parities of $a,b,c$ should not be the
same, see
\machSeite{CiEKAA}\cite[Theorem~2]{CiEKAA}. Together, the two theorems
generalise MacMahon's Theorem~\ref{thm:MM}.\footnote{Bijective proofs
of Theorem~\ref{thm:MM} which ``explain" the ``nice" formula are
known
\machSeite{KratAY}%
\machSeite{KratBK}%
\cite{KratAY,KratBK}. I do not ask for a bijective proof of
Theorem~\ref{enum} because I consider the task of finding one as daunting.})
\begin{figure}
\centertexdraw{
\drawdim truecm \linewd.02
\rhombus \rhombus \rhombus \rhombus \rhombus \rhombus \rhombus
\ldreieck
\move(-.866025 -.5)
\rhombus \rhombus \rhombus \rhombus \rhombus \rhombus \rhombus \rhombus
\move(-1.7305 -1)
\rhombus \rhombus \rhombus \ldreieck \rmove(.866025 -.5)
\rhombus \rhombus \rhombus
\move(-1.7305 -1)
\rdreieck \rhombus \rhombus \rhombus \ldreieck \rhombus \rhombus \rhombus
\move(-1.7305 -2)
\rdreieck \rhombus \rhombus \rhombus \rhombus \rhombus \rhombus
\move(-1.7305 -3)
\rdreieck \rhombus \rhombus \rhombus \rhombus \rhombus
\htext(-.8 0){$a$}
\htext(4 -.7){$b+m$}
\htext(6.9 -3.95){$\left. \vbox{\vskip.6cm} \right\} c$}
\htext(-3.2 -4){$c+m \left\{ \vbox{\vskip1.6cm} \right.$}
\htext(0 -6){$b$}
\htext(4.9 -6){$a+m$}
\htext(1.7 -3.95){$\left. \vbox{\vskip1cm} \right\}m$}
\rtext td:60 (4 -1.3) {$\left. \vbox{\vskip3.6cm} \right\} $}
\rtext td:-60 (-.8 0){$\left\{ \vbox{\vskip1.6cm} \right. $}
\rtext td:-60 (4.6 -5.3) {$\left. \vbox{\vskip2.5cm} \right\} $}
\rtext td:60 (0.3 -5.6){$\left\{ \vbox{\vskip2.7cm}\right. $}
}
\caption{\protect\small A hexagon with triangular hole}
\label{hex}
\end{figure}
The reader should notice that the right-hand side of \eqref{eq:M2} is
indeed of the form $NICE(a)$, while the right-hand side of
\eqref{eq:enum} is of the form $NICE(m/2)$.
\medskip
Where is the connexion to determinants? As it turns out, these two
theorems are in fact {\it determinant evaluation theorems}. More
precisely, Theorem~\ref{thm:MM} is equivalent to the following
theorem.
\begin{Theorem} \label{thm:MM2}
\begin{equation} \label{eq:M3}
\det_{1\le i,j\le c}\(\binom {a+b}{a-i+j} \)=
\prod _{i=1} ^{c}\frac {(a+b+i-1)!\,(i-1)!} {(a+i-1)!\,(b+i-1)!}.
\end{equation}
\quad \quad \qed
\end{Theorem}
(The reader should notice that this is exactly \eqref{eq:M1} with $n$
replaced by $c$.) On the
other hand, Theorem~\ref{enum} is equivalent to the theorem
below.\footnote{To be correct, this is a little bit oversimplified.
The truth is that equivalence holds only if $m$ is even. An
additional argument is necessary for proving the result for the case
that $m$ is odd. We refer the reader who is interested in these
details to
\machSeite{CiEKAA}\cite[Sec.~2]{CiEKAA}.}
\begin{Theorem} \label{enum2}
If $m$ is even, the determinant
\begin{equation} \label{mat1}
\det_{1\le i,j\le a+m} \begin{pmatrix} \dbinom{b+c+m}{b-i+j}&
\text {\scriptsize $1\le i\le a$}\\
\dbinom{\frac {b+c} {2}}{\frac {b+a} {2}-i+j}&
\text {\scriptsize $a+1\le i \le a+m$}
\end{pmatrix}
\end{equation}
is equal to \eqref{eq:enum}.\quad \quad \qed
\end{Theorem}
\begin{figure}
\centertexdraw{
\drawdim cm \setunitscale.7
\linewd.01
\rhombus \rhombus \rhombus \rhombus \rhombus \rhombus \rhombus
\ldreieck
\move(-.866025 -.5)
\rhombus \rhombus \rhombus \rhombus \rhombus \rhombus \rhombus \rhombus
\move(-1.73205 -1)
\rhombus \rhombus \rhombus \ldreieck \rmove(.866025 -.5)
\rhombus \rhombus \rhombus
\move(-1.73205 -1)
\rdreieck \rhombus \rhombus \rhombus \ldreieck \rhombus \rhombus \rhombus
\move(-1.73205 -2)
\rdreieck \rhombus \rhombus \rhombus \rhombus \rhombus \rhombus
\move(-1.73205 -3)
\rdreieck \rhombus \rhombus \rhombus \rhombus \rhombus
\linewd.08
\move(0 0)
\RhombusA \RhombusA \RhombusA \RhombusA \RhombusA \RhombusA \RhombusA
\RhombusB
\move(-.866025 -.5)
\RhombusA \RhombusA \RhombusA \RhombusA \RhombusA \RhombusA
\RhombusB \RhombusA
\move(-1.73205 -1)
\RhombusA \RhombusA \RhombusB \RhombusB \RhombusA \RhombusB \RhombusA
\RhombusA
\move(2.598 -3.5)
\RhombusA \RhombusB \RhombusA
\move(1.73205 -4)
\RhombusB \RhombusA \RhombusA
\move(-1.73205 -1)
\RhombusC \RhombusC
\move(-1.73205 -2) \RhombusC
\move(-1.73205 -3)\RhombusC \RhombusC \RhombusC
\move(.866025 -1.5) \RhombusC
\move(.866025 -2.5) \RhombusC
\htext(-4 -8){\small
a. A lozenge tiling of the cored hexagon in Figure~\ref{hex}}
\htext(-1 0.2){$a$}
\htext(4.4 -1){$b+m$}
\htext(6.9 -3.9){$\left. \vbox{\vskip.35cm} \right\} c$}
\htext(-3.8 -4){$c+m \left\{ \vbox{\vskip1.2cm} \right.$}
\htext(0 -6.2){$b$}
\htext(5 -6){$a+m$}
\htext(1.7 -4){$\left. \vbox{\vskip0.77cm} \right\}m$}
\rtext td:60 (4 -1.3) {$\left. \vbox{\vskip2.52cm} \right\} $}
\rtext td:-60 (-.8 0.2){$\left\{ \vbox{\vskip1.02cm} \right. $}
\rtext td:-60 (4.6 -5.3) {$\left. \vbox{\vskip1.75cm} \right\} $}
\rtext td:60 (0.3 -5.8){$\left\{ \vbox{\vskip1.89cm}\right. $}
\move(11 0)
\bsegment
\linewd.05
\move(0 0)
\RhombusA \RhombusA \RhombusA \RhombusA \RhombusA \RhombusA \RhombusA
\RhombusB
\move(-.866025 -.5)
\RhombusA \RhombusA \RhombusA \RhombusA \RhombusA \RhombusA
\RhombusB \RhombusA
\move(-1.73205 -1)
\RhombusA \RhombusA \RhombusB \RhombusB \RhombusA \RhombusB \RhombusA
\RhombusA
\move(2.598 -3.5)
\RhombusA \RhombusB \RhombusA
\move(1.73205 -4)
\RhombusB \RhombusA \RhombusA
\move(-1.73205 -1)
\RhombusC \RhombusC
\move(-1.73205 -2) \RhombusC
\move(-1.73205 -3)\RhombusC \RhombusC \RhombusC
\move(.866025 -1.5) \RhombusC
\move(.866025 -2.5) \RhombusC
\ringerl(.433012 .25)
\hdSchritt \hdSchritt \hdSchritt \hdSchritt \hdSchritt \hdSchritt \hdSchritt \vdSchritt
\ringerl(-.433012 -.25)
\hdSchritt \hdSchritt \hdSchritt \hdSchritt \hdSchritt \hdSchritt \vdSchritt \hdSchritt
\ringerl(-1.299037 -.75) \hdSchritt \hdSchritt
\vdSchritt \vdSchritt \hdSchritt \vdSchritt \hdSchritt \hdSchritt
\ringerl(3.031 -3.25) \hdSchritt \vdSchritt \hdSchritt
\ringerl(2.165 -3.75) \vdSchritt \hdSchritt \hdSchritt
\ringerl(6.4952 -4.25)
\ringerl(5.6292 -4.75)
\ringerl(4.7632 -5.25)
\ringerl(3.8971 -5.75)
\ringerl(3.031 -6.25)
\htext(-2 -8){\small b. The corresponding path family}
\esegment
\htext(4 -18){\small c. The path family made orthogonal}}
\vskip-7cm
$$
\Einheit=.7cm
\Gitter(11,7)(0,0)
\Koordinatenachsen(11,7)(0,0)
\Pfad(0,3),11\endPfad
\Pfad(2,1),22\endPfad
\Pfad(2,1),1\endPfad
\Pfad(3,0),2\endPfad
\Pfad(3,0),11\endPfad
\Pfad(1,4),111111\endPfad
\Pfad(7,3),2\endPfad
\Pfad(7,3),1\endPfad
\Pfad(2,5),1111111\endPfad
\Pfad(9,4),2\endPfad
\Pfad(4,1),2\endPfad
\Pfad(4,1),11\endPfad
\Pfad(5,3),1\endPfad
\Pfad(6,2),2\endPfad
\Pfad(6,2),1\endPfad
\Kreis(0,3.02) \Kreis(5,0)
\Label\lo{A_1}(0,3) \Label\ru{E_1}(5,0)
\Kreis(1,4.02) \Kreis(8,3)
\Label\lo{A_2}(1,4) \Label\ru{E_4}(8,3)
\Kreis(2,5.02) \Kreis(9,4)
\Label\lo{A_3}(2,5) \Label\ru{E_5}(9,4)
\Kreis(4,2.02) \Kreis(6,1)
\Label\lo{A_4}(4,2) \Label\ru{E_2}(6,1)
\Kreis(5,3.02) \Kreis(7,2)
\Label\lo{A_5}(5,3) \Label\ru{E_3}(7,2)
\hskip6cm
$$
\vskip1cm
\caption{}
\label{tiling}
\end{figure}
The link between rhombus tilings (and equivalent objects such as
plane partitions, semistandard tableaux, etc.) and determinants
which explains the above two equivalence statements is {\it
non-intersecting lattice paths}.\footnote{There exists in fact a
second link between rhombus tilings and determinants which is not
less interesting or less important. It is a well-known fact that
rhombus tilings are in bijection with perfect matchings of certain
hexagonal graphs. (See for example
\machSeite{KupeAG}\cite[Figures~13 and 14]{KupeAG}.) In view of this fact,
this second link is given by Kasteleyn's
theorem \machSeite{KastAA}\cite{KastAA} saying that the number of
perfect matchings of a planar graph is given by the Pfaffian of
a slight perturbation of the adjacency matrix of the graph.
See \machSeite{KupeAG}\cite{KupeAG} for an exposition of
Kasteleyn's result, including historical notes, and for
adaptations taking symmetries of the graph into
account.} The latter are families of paths in a
lattice with the property that no two paths in the family have a
point in common. Indeed, rhombus tilings are (usually) in bijection
with families of non-intersecting paths in the integer lattice
${\mathbb Z}^2$ which consist of unit horizontal and vertical steps.
(Figure~\ref{tiling} illustrates
the bijection for the rhombus tilings which appear in
Theorem~\ref{enum} in an example.
In that bijection, all horizontal steps of the paths are in the positive
direction, and all vertical steps are in the negative direction.
See the explanations that accompany
\machSeite{CiEKAA}\cite[Figure~8]{CiEKAA}
for a detailed description.
Since, as I explained, Theorem~\ref{enum}
essentially is a generalisation of Theorem~\ref{thm:MM}, this gives
also an idea for the bijection for the rhombus tilings which appear in
the latter theorem. For other instances of bijections between rhombus
tilings and non-intersecting lattice paths see
\machSeite{CiKrAA}%
\machSeite{CiKrAC}%
\machSeite{CiKrAD}%
\machSeite{EisTAA}%
\machSeite{EisTAB}%
\machSeite{EisTAF}%
\machSeite{FiscAA}%
\machSeite{KratBY}%
\machSeite{OkKrAA}%
\cite{CiKrAA,CiKrAC,CiKrAD,EisTAA,EisTAB,EisTAF,FiscAA,KratBY,OkKrAA}.).
In the case that the starting points and the end points of the lattice
paths are fixed, the following
many-author-theorem applies.\footnote{%
This result was discovered and rediscovered several times.
In a probabilistic form, it occurs for the
first time in work by Karlin and McGregor
\machSeite{KaMGAB}%
\machSeite{KaMGAC}%
\cite{KaMGAB,KaMGAC}.
In matroid theory, it is discovered in its discrete form by Lindstr\"om
\machSeite{LindAA}\cite[Lemma~1]{LindAA}. Then, in the 1980s the theorem is
rediscovered at about the same time in three different
communities, not knowing from each other at the time:
in statistical physics by Fisher
\machSeite{FishAA}\cite[Sec.~5.3]{FishAA}
in order to apply it to
the analysis of vicious walkers as a model of wetting and melting,
in combinatorial chemistry by John and Sachs
\machSeite{JoSaAB}\cite{JoSaAB} and
Gro\-nau, Just, Schade, Scheffler and Wojciechowski
\machSeite{GrJSAA}\cite{GrJSAA}
in order to compute Pauling's bond order
in benzenoid hydrocarbon molecules, and in enumerative combinatorics
by Gessel and Viennot
\machSeite{GeViAA}%
\machSeite{GeViAB}%
\cite{GeViAA,GeViAB} in order to count
tableaux and plane partitions. Since only Lindstr\"om, and then
Gessel and Viennot state the result in its most general form (not
reproduced here), I call this theorem most often
the ``Lindstr\"om--Gessel--Viennot theorem." It must be also
mentioned that the so-called ``Slater determinant"
in quantum mechanics (cf.\
\machSeite{SlatZY}\cite{SlatZY} and
\machSeite{SlatZZ}\cite[Ch.~11]{SlatZZ})
may qualify as an ``ancestor" of
the Lindstr\"om--Gessel--Viennot determinant.}
\begin{Theorem}[\sc Karlin--McGregor, Lindstr\"om, Gessel--Viennot,
Fisher,\break John--Sachs, Gronau--Just--Schade--Scheffler--Wojciechowski]
\label{thm:nonint}
Let $A_1,A_2,\break \dots,A_n$ and $E_1,E_2,\dots,E_n$ be lattice points
such that for $i<j$ and $k<l$ any lattice path between $A_i$ and $E_l$ has a
common point with any lattice path between $A_j$ and $E_k$.
Then the number of all families $(P_1,P_2,\dots,P_n)$ of
non-intersecting lattice paths, $P_i$ running from $A_i$ to $E_i$,
$i=1,2,\dots,n$, is given by
$$\det_{1\le i,j\le n}\big(P(A_j\to E_i)\big),$$
where $P(A\to E)$ denotes the number of all lattice paths from $A$ to
$E$.\quad \quad \qed
\end{Theorem}
It goes beyond the scope of this article to include the proof of this
theorem here. However, I cannot help telling that it is an extremely
beautiful and simple proof that {\it every} mathematician
should have seen once, even if (s)he
does not have any use for it in her/his own research. I refer the
reader to
\machSeite{GeViAA}%
\machSeite{GeViAB}%
\machSeite{StemAE}%
\cite{GeViAA,GeViAB,StemAE}.
\medskip
Now the origin of the determinants becomes evident. In particular,
since, for rhombus tilings, we have to deal with lattice paths
in the integer lattice consisting of unit horizontal and vertical
steps, and since the number of such lattice paths
which connect two lattice points is given by a binomial coefficient,
we see that the enumeration of rhombus tilings must be a rich
source for binomial determinants. This is indeed the case, and there
are several instances in which such determinants can be evaluated in the
form $NICE(.)$ (see
\machSeite{CiucAH}%
\machSeite{CiEKAA}%
\machSeite{CiKrAA}%
\machSeite{CiKrA}%
\machSeite{CiKrAD}%
\machSeite{EisTAA}%
\machSeite{EisTAB}%
\machSeite{EisTAF}%
\machSeite{FiscAA}%
\machSeite{FuKrAC}%
\machSeite{KratBN}%
\cite{CiucAH,CiEKAA,CiKrAA,CiKrAC,CiKrAD,EisTAA,EisTAB,EisTAF,FiscAA,FuKrAC,KratBN}
and Section~\ref{sec:detlist}).
Often the evaluation part is highly non-trivial.
The evaluation of the determinant \eqref{eq:M3} is not very difficult
(see
\machSeite{KratBN}\cite[Sections~2.2, 2.3, 2.5]{KratBN} for 3 different ways
to evaluate it).
On the other hand, the
evaluation of the determinant \eqref{mat1} requires some effort (see
\machSeite{CiEKAA}\cite[Sec.~7]{CiEKAA}).
\medskip
To conclude this section, I state another determinant evaluation,
to which I shall come back later. Its origin lies as well in the
enumeration of rhombus tilings and plane partitions (see
\machSeite{KratBD}\cite[Theorem~10]{KratBD} and
\machSeite{CiKrAB}\cite[Theorem~2.1]{CiKrAB}).
\begin{Theorem} \label{thm:xy}
For any complex numbers $x$ and $y$ there holds
\begin{multline} \label{eq:Krat}
\det_{0\le i,j\le n-1}\(\frac {(x+y+i+j-1)!}
{(x+2i-j)!\,(y+2j-i)!}\)\\
=\prod _{i=0} ^{n-1}\frac {i!\,(x+y+i-1)!\,(2x+y+2i)_i\,(x+2y+2i)_i}
{(x+2i)!\,(y+2i)!},
\end{multline}
where the {\em shifted factorials} or
{\em Pochhammer symbols} $(a)_k$ are defined by
$(a)_k:=a(a+1)\cdots(a+k-1)$, $k\ge1$, and $(a)_0:=1$.
{\em(}In this formula,
a factorial $m!$ has to be interpreted as $\Gamma(m+1)$ if $m$ is not a
non-negative integer.{\em)}
\end{Theorem}
\section{A determinant from number theory}\label{sec:det}
However, determinants do not only arise in combinatorics, they also
arise in other fields. In this section, I want to present
a determinant which arose in number theory, explain in some detail
its origin, and then outline the steps which led to its evaluation,
thereby giving the reader an opportunity to look ``behind the scenes"
while one tries to make the determinant evaluation methods described
in
\machSeite{KratBN}\cite{KratBN} work.
The story begins with the following two series expansions for $\pi$.
The first one is due to Bill Gosper
\machSeite{Gosper}\cite{Gosper},
\begin{equation} \label{eq:Gosper}
\pi =\sum_{n=0}^\infty \frac{50n-6}{\binom{3n}n2^n} ,
\end{equation}
and was used by Fabrice Bellard
\machSeite{Bellard}\cite[file {\tt pi1.c}]{Bellard}
to find an algorithm for computing the $n$-th decimal of $\pi $ without
computing the earlier ones, thus improving an earlier algorithm due to
Simon Plouffe
\machSeite{Plouffe}\cite{Plouffe}.
The second one,
\begin{equation} \label{eq:Bellard}
\pi=\frac {1} {740025}\(\sum _{n=1} ^{\infty}\frac {3P(n)} {\binom
{7n}{2n}2^{n-1}}-20379280\),
\end{equation}
where
\begin{multline*}
P(n)=-885673181n^5+3125347237n^4-2942969225n^3\\
+1031962795n^2-196882274n+10996648,
\end{multline*}
is due to Fabrice Bellard
\machSeite{Bellard}\cite{Bellard}, and was used by him in his
world record setting
computation of the 1000 billionth {\it binary} digit of $\pi$, being based
on the algorithm in
\machSeite{BaBoPl}\cite{BaBoPl}.
Going beyond that, my co-authors from
\machSeite{AlKPAA}\cite{AlKPAA},
Gert Almkvist and Joakim Petersson, asked themselves the following
question:
{\em Are there more expansions of the type
$$\pi=\sum_{n=0}^\infty \frac {S(n)}{\binom{mn}{pn}a^n},$$
where $S(n)$ is some polynomial in $n$ {\em(}depending on $m,p,a${\em)}?}
How can one go about to get some intuition about this question?
One chooses some specific
$m,p,a$, goes to the computer, computes
$$p(k)=\sum _{n=0} ^{\infty}\frac {n^k} {\binom {mn}{pn}a^n}$$
to many, many digits for $k=0,1,2,\dots$, puts
$$\pi,p(0),p(1),p(2),\dots$$
into the LLL-algorithm (which comes, for example, with the {\sl Maple}
computer algebra package), and one waits whether the algorithm
comes up with an integral linear combination of
$\pi,p(0),p(1),p(2),\dots$.\footnote{For readers unfamiliar with the
LLL-algorithm: in this particular application,
it takes as an input rational numbers $r_1,r_2,\dots,r_m$ (which, in
our case, will be the numbers $1$ and the rational approximations of $\pi$,
$p(0)$, $p(1)$, \dots \ which we computed), and, if
successful, outputs {\it small\/} integers $c_1,c_2,\dots,c_m$ such
that $c_1r_1+c_2r_2+\dots+c_mr_m$ is {\it very small}. Thus, if $r_i$
was a good approximation for the real number $x_i$, $i=1,2,\dots,m$,
one can expect that actually $c_1x_1+c_2x_2+\dots+c_mx_m=0$.
See
\machSeite{LeLLAA}\cite[Sec.~1, in particular
the last paragraph]{LeLLAA} and
\machSeite{CohHAA}\cite[Ch.~2]{CohHAA}
for the description of and more information on this important algorithm.
In particular, also here, the output of the
algorithm (if there is) is just a (very guided) {\it guess}. Thus, a
proof is still needed, although the probability that the guess is
wrong is infinitesimal. As a matter of fact, it is very likely that
Bellard had no proof of his \hbox{formula \eqref{eq:Bellard} \dots}}
Indeed, Table~\ref{tab:2} shows the parameter values, where
the LLL-algorithm gave a result.
\begin{table}[h]
\begin{tabular}{c|c|l|cl}
$m$ & $p$ & $\hphantom{-}a$ & $\deg(S)$ & \\
\cline{1-4}
\hphantom{1}3 & \hphantom{1}1 & $\hphantom{-}2$ & \hphantom{1}1 & (Gosper)
\\
\hphantom{1}7 & \hphantom{1}2 & $\hphantom{-}2$ & \hphantom{1}5 & (Bellard)
\\
\hphantom{1}8 & \hphantom{1}4 & $-4$ & \hphantom{1}4 & \\
10 & \hphantom{1}4 & $\hphantom{-}4$ & \hphantom{1}8 & \\
12 & \hphantom{1}4 & $-4$ & \hphantom{1}8 & \\
16 & \hphantom{1}8 & $\hphantom{-}16$ & \hphantom{1}8 & \\
24 & 12 & $-64$ & 12 & \\
32 & 16 & $\hphantom{-}256$ & 16 & \\
40 & 20 & $-4^5$ & 20 & \\
48 & 24 & $\hphantom{-}4^6$ & 24 & \\
56 & 28 & $-4^7$ & 28 & \\
64 & 32 & $\hphantom{-}4^8$ & 32 & \\
72 & 36 & $-4^9$ & 36 & \\
80 & 40 & $\hphantom{-}4^{10}$ & 40 &
\end{tabular}
\vskip10pt
\caption{}
\label{tab:2}
\end{table}
For example, it found
$$
\pi =\frac 1r\sum_{n=0}^\infty \frac{S(n)}{\binom{16n}{8n}16^n},
$$
where
$$
r=3^65^37^211^213^2
$$
and
\begin{multline*}
S(n)=-869897157255-3524219363487888n+112466777263118189n^2
\\
-1242789726208374386n^3+6693196178751930680n^4-19768094496651298112n^5
\\
+32808347163463348736n^6-28892659596072587264n^7+10530503748472012800n^8,
\end{multline*}
and
$$
\pi =\frac 1r\sum_{n=0}^\infty \frac{S(n)}{\binom{32n}{16n}256^n} ,
$$
where
$$
r=2^33^{10}5^67^311^1 13^217^219^223^229^231^2
$$
and
{\allowdisplaybreaks
\begin{align*}
S(n)=&-2062111884756347479085709280875
\\
&+1505491740302839023753569717261882091900n
\\
&-112401149404087658213839386716211975291975n^2
\\
&+3257881651942682891818557726225840674110002n^3
\\
&-51677309510890630500607898599463036267961280n^4
\\
&+517337977987354819322786909541179043148522720n^5
\\
&-3526396494329560718758086392841258152390245120n^6
\\
&+171145766235995166227501216110074805943799363584n^7
\\
&-60739416613228219940886539658145904402068029440n^8
\\
&+159935882563435860391195903248596461569183580160n^9
\\
&-313951952615028230229958218839819183812205608960n^{10}
\\
&+457341091673257198565533286493831205566468325376n^{11}
\\
&-486846784774707448105420279985074159657397780480n^{12}
\\
&+367314505118245777241612044490633887668208926720n^{13}
\\
&-185647326591648164598342857319777582801297080320n^{14}
\\
&+56224688035707015687999128994324690418467340288n^{15}
\\
&-7687255778816557786073977795149360408612044800n^{16} .
\end{align*}}%
Of course, there could be many more.
If one looks more closely at Table~\ref{tab:2}, then, if one
disregards the first, second and fourth line, one cannot escape to
notice a pattern: {\it apparently, for each $k=1,2,\dots$, there is a
formula
$$
\pi =\sum_{n=0}^\infty \frac{S_k(n)}{\binom{8kn}{4kn}(-4)^{kn}} ,
$$
where $S_k(n)$ is some polynomial in $n$ of degree $4k$.}
In order to make progress on this observation, we have to first
see how one can prove such an identity, once it is found. In fact,
this is not difficult at all. To illustrate the idea, let us go
through a proof of Gosper's identity \eqref{eq:Gosper}.
The beta integral evaluation (cf.\
\machSeite{AAR}\cite[Theorem~1.1.4]{AAR}) gives
$$
\frac 1{\binom{3n}n}=(3n+1)\int_0^1x^{2n}(1-x)^ndx .
$$
Hence the right hand side of the formula will be
$$
\int_0^1\sum_{n=0}^\infty (50n-6)(3n+1)\left(\frac {x^2(1-x)}{2}\right)^ndx
.
$$
We have
\begin{equation} \label{eq:rational}
\sum_{n=0}^\infty (50n-6)(3n+1)y^n=\frac{2(56y^2+97y-3)}{(1-y)^3} .
\end{equation}
Thus, if substituted, we obtain
\begin{align}\notag
RHS&=8\int_0^1\frac{28x^6-56x^5+28x^4-97x^3+97x^2-6}{(x^3-x^2+2)^3}dx\\
&=
\left[ \frac{4x(x-1)(x^3-28x^2+9x+8)}{(x^3-x^2+2)^2}+4\arctan (x-1)\right]
_0^1=\pi .
\label{eq:RHS}
\end{align}
(Clearly, both \eqref{eq:rational} and \eqref{eq:RHS} are routine
calculations, and therefore we did not do it by hand, but let them be
worked out by {\sl Maple}.)
Now let us fix $k\ge1$. We apply the same procedure to
$
\sum_{n=0}^\infty {S_k(n)}\big/{\binom{8kn}{4kn}(-4)^{kn}} ,
$
where $S_k(n)$ is (hopefully) some (unknown) polynomial in $n$.
The beta integral evaluation gives
$$
\frac 1{\binom{8kn}{4kn}}=(8kn+1)\int_0^1x^{4kn}(1-x)^{4kn}dx .
$$
Hence, if $S_k(n)$ should have degree $d$ in $n$,
\begin{align}\notag
\sum _{n=0} ^{\infty}\frac {S_k(n)}{\binom{8kn}{4kn}(-4)^{kn}}
&=\int_0^1\sum_{n=0}^\infty (8kn+1)S_k(n)\left(\frac
{x^{4k}(1-x)^{4k}}{(-4)^k}\right)^n\,dx
\notag\\
&=\int_0^1 \frac {P_k(x)} {\(x^{4k}(1-x)^{4k}-(-4)^k\)^{d+2}}\,dx,
\label{eq:sumint}
\end{align}
where $P_k(x)$ is some polynomial in $x$. For convenience, let us
write $P$ as a short-hand for $P_k$.
Let $Q(x):=x^{4k}(1-x)^{4k}-(-4)^k$. Now we make the wild assumption
that
$$
\int \frac{P(x)}{Q(x)^{d+2}}\,dx=\frac{R(x)}{Q(x)^{d+1}}+2\arctan (x)+2\arctan
(x-1) ,
$$
for some polynomial $R(x)$ with $R(0)=R(1)=0$. Then the original sum
would indeed be equal to $\pi$. The last equality is equivalent to
$$
\frac P{Q^{d+2}}=\frac{R^{\prime }}{Q^{d+1}}-(d+1)\frac{Q^{\prime }R}{Q^{d+2}%
}+2\left(\frac 1{x^2+1}+\frac 1{x^2-2x+2}\right),
$$
or
$$QR^{\prime }-(d+1)Q^{\prime }R=P-2Q^{d+2}\left(\frac 1{x^2+1}+
\frac 1{x^2-2x+2}\right) .
$$
In our examples, we observed that
$$
R(x)=(2x-1)\check{R}\big(x(1-x)\big)
$$
for a polynomial $\check R$. So, let us make the substitution
$$t=x(1-x).$$
Then, after some simplification, the above differential equation
becomes
\begin{equation} \label{eq:diff}
-(1-4t)Q\frac{d\check{R}}{dt}+(2Q+4k(4k+1)(1-4t)t^{4k-1})\check{R}-P+2(3-2t)%
\frac{Q^{4k+2}}{t^2-2t+2}=0 ,
\end{equation}
where $Q(t)=t^{4k}-(-4)^{k}$.
Now, writing $N(k)=4k(4k+1)$, we make the Ansatz
\begin{align*}
\check R(t)&=\sum _{j=1} ^{N(k)-1}a(j)t^j,\\
S_k(n)&=\sum _{j=0} ^{4k}a(N(k)+j)n^j.
\end{align*}
(The reader should recall that $S_k(n)$ defines $P_k(t)=P(t)$
through \eqref{eq:sumint}.) Comparing
coefficients of powers of $t$ on both sides of \eqref{eq:diff}, we
get a system of $N(k)+4k$ linear equations for the unknowns
$a(1),a(2),\dots,a(N(k)+4k)$.
Hence: {\it If the determinant of this system
of linear equations is non-zero, then
there does indeed exist a representation
$$\pi=\sum_{n=0}^\infty \frac {S_k(n)}{\binom{8kn}{4kn}(-4)^n}.$$
}
To see whether we could indeed hope for the determinant to be
non-zero, we went again to the computer and looked at the values of
the determinant in some small
instances. (Obviously, we do not want to do this by
hand, since for $k=1$ the matrix is already a $24\times 24$ matrix!)
So, let us program the matrix. (We shall see
the mathematical definition of the matrix in just a
moment, see \eqref{eq:Det}.\footnote{To tell the truth, this is the form of the
matrix after some simplifications have already been carried out.
(In particular, we are looking at a matrix which is slightly smaller
than the original one.)
See
\machSeite{AlKPAA}\cite[beginning of Section~4]{AlKPAA} for these details.
There, the matrix in \eqref{eq:Det} is called $M'''$.})
\MATH
\goodbreakpoint%
In[8]:= a[k\MATHtief ,j\MATHtief ]:=Module[%
\MATHlbrace Var=j/(4k)%
\MATHrbrace ,
\leavevmode%
(-1)\MATHhoch (Var-1)*8k(4k+1)(-4)\MATHhoch (k*(Var+1))*
\leavevmode%
Product[4k*l-1,%
\MATHlbrace l,1,4k-Var%
\MATHrbrace ]*Product[4k*l+1,%
\MATHlbrace l,1,Var-1%
\MATHrbrace ]
\leavevmode%
]
In[9]:= A[k\MATHtief , i\MATHtief , j\MATHtief ] := Module[%
\MATHlbrace Var%
\MATHrbrace ,
\leavevmode%
Var = %
\MATHlbrace Floor[(i - 2)/(4*k - 1)],
\leavevmode%
Floor[(j - 1)/(4*k)], Mod[i - 2, 4*k - 1],
\leavevmode%
Mod[j - 1, 4*k]%
\MATHrbrace ;
\leavevmode%
If[i == 1,
\leavevmode%
If[Mod[j, 4*k] === 0, a[k, j], 0],
\leavevmode%
If[Var[[1]] - Var[[2]] == 0,
\leavevmode%
Switch[Var[[3]] - Var[[4]], 0, f1[k, Var[[3]]+1, j], -1,
\leavevmode%
f0[k, Var[[3]]+1, j], \MATHtief , 0],
\leavevmode%
If[Var[[1]] - Var[[2]] == 1,
\leavevmode%
Switch[Var[[3]] - Var[[4]], 0, g1[k, Var[[3]]+1, j], -1,
\leavevmode%
g0[k, Var[[3]]+1, j], \MATHtief , 0], 0]]]]
\goodbreakpoint%
In[10]:= A[k\MATHtief ] := Table[A[k, i, j], %
\MATHlbrace i, 1, 16*k\MATHhoch 2%
\MATHrbrace , %
\MATHlbrace j, 1, 16*k\MATHhoch 2%
\MATHrbrace ]
\goodbreakpoint%
In[11]:= f0[k\MATHtief , t\MATHtief , j\MATHtief ] := j*(-4)\MATHhoch k;
\leavevmode%
f1[k\MATHtief , t\MATHtief , j\MATHtief ] := -(2 + 4*j)*(-4)\MATHhoch k;
\leavevmode%
g0[k\MATHtief , t\MATHtief , j\MATHtief ] := (4*k*(4*k + 1) - j);
\leavevmode%
g1[k\MATHtief , t\MATHtief , j\MATHtief ] := (-4*4*k*(4*k + 1) + 2 + 4*j)
\goodbreakpoint%
\endgroup
We shall not try to digest this at this point. Let us accept the
program as a black box, and let us compute the determinant for $k=2$.
\MATH
\goodbreakpoint%
In[12]:= Det[A[2]]
\goodbreakpoint%
Out[12]= -601576375580370166777074138698518196031142518971568946712\MATHbackslash
> 2204136674781038302774231725971306459064075121023092662279814\MATHbackslash
> 015195545600000000000
\goodbreakpoint%
\endgroup
Magnificent! This is certainly {\it not\/} zero. However, what are we going to
do with this gigantic number? Remembering our discussion about
``nice" numbers and ``nice" formulae in the preceding section,
let us factorise it in its prime factors.
\MATH
\goodbreakpoint%
In[13]:= FactorInteger[\%]
\goodbreakpoint%
Out[13]= %
\MATHlbrace %
\MATHlbrace -1, 1%
\MATHrbrace , %
\MATHlbrace 2, 325%
\MATHrbrace , %
\MATHlbrace 3, 39%
\MATHrbrace , %
\MATHlbrace 5, 11%
\MATHrbrace , %
\MATHlbrace 7, 11%
\MATHrbrace , %
\MATHlbrace 11, 3%
\MATHrbrace , %
\MATHlbrace 13, 2%
\MATHrbrace %
\MATHrbrace
\goodbreakpoint%
\endgroup
I would say that this is sensational: a number with 139 digits,
and the biggest prime factor is 13! As a matter of fact, this is not just
a rare exception. Table~\ref{tab:1} shows the factorisations of the first five
determinants. (We could not go further because of the exploding size
of the matrix of which the determinant is taken.)
\begin{table}[h]
\vskip10pt
\begin{tabular}{l|l}
\hphantom{1}$k$ & $\hphantom{-}\det(A(k))$ \\
\hline\\[-8pt]
\hphantom{1}1 & $\hphantom{-}2^{59}3^55^67^1$ \\
\hphantom{1}2 & $-2^{325}3^{39}5^{11}7^{11}11^313^2$ \\
\hphantom{1}3 & $\hphantom{-}2^{772}3^{146}5^{28}7^{17}11^{17}13^{18}17^419^323^1$ \\
\hphantom{1}4 & $-2^{1913}3^{111}5^{58}7^{38}11^{21}13^{22}17^{24}19^723^529^231^1$ \\
\hphantom{1}5 & $\hphantom{-}
2^{2932}3^{202}5^{306}7^{69}11^{29}13^{27}17^{28}19^{29}23^{9}29^631^537^2$%
\end{tabular}
\vskip10pt
\caption{}
\label{tab:1}
\end{table}
Thus, these
experimental results {\it make us sure} that there must be a ``nice"
formula for the determinant. Indeed, we prove in
\machSeite{AlKPAA}\cite{AlKPAA}
that\footnote{Strictly speaking, this is not a formula
$NICE(k)$ according to my ``Definition" in the preceding section,
because of the presence of the ``Abelian" factors $k^{8k^2+2k}$ and
$(4k+1)!^{4k}$, see Footnote~\ref{foot:normally}. Nevertheless, the
reader will certainly admit that this is a {\it nice} and {\it
closed\/} formula.}
\begin{equation} \label{eq:det(A(k))}
\det (A(k))=(-1)^{k-1}
2^{16k^3+20k^2+6k}k^{8k^2+2k}(4k+1)!^{4k}
\prod_{j=1}^{4k}\frac{(2j)!}{j!^2}.
\end{equation}
Hence the desired theorem follows.
\begin{Theorem} \label{T1}
For all $k\geq 1$ there is a formula
$$
\pi =\sum_{n=0}^\infty \frac{S_k(n)}{\binom{8kn}{4kn}(-4)^{kn}},
$$
where $S_k(n)$ is a polynomial in $n$
of degree $4k$ with rational coefficients. The polynomial $S_k(n)$ can be
found by solving the previously described system of linear equations.
\end{Theorem}
I must admit that we were extremely lucky that
it was indeed possible to {\it evaluate} the determinant {\it
explicitly}. To recall,
``all" we needed to prove our theorem (Theorem~\ref{T1})
was to show that the determinant was
{\it non-zero}. To be honest, I would not have the slightest idea how to
do this here without finding the exact value of the determinant.
\medskip
Now, after all this somewhat ``dry" discussion, let me present the
determinant.
We had to determine the
determinant of the $16k^2\times 16k^2$ matrix
\begin{equation} \label{eq:Det}
\begin{pmatrix}
0\dots0\,*&
0\dots0\,*&
0\dots0\,*&
\dots&
\dots&
\dots&
0\dots0\,*\\
\hbox{\Large$F_1$}&\hbox{\Large$0$}&\hbox{\Large$0$}&
\dots&\dots&\dots&\hbox{\Large$0$}\\
\hbox{\Large$G_1$}&\hbox{\Large$F_2$}& \hbox{\Large$0$}&
\dots&\dots&\dots&\hbox{\Large$0$}\\
\hbox{\Large$0$}&\hbox{\Large$G_2$}&\hbox{\Large$F_3$}& &&&\vdots\\
\hbox{\Large$0$}&\hbox{\Large$0$}&\hbox{\Large$G_3$}& &
\ddots& &\vdots\\
\vdots&\ddots&\ddots&\ddots&\ddots&\ddots&\vdots\\
\vdots&&\ddots&\ddots&\ddots&\hbox{\Large$F_{4k-1}$}&\hbox{\Large$0$}\\
\vdots&&&\hbox{\Large$0$}&\hbox{\Large$0$}&
\hbox{\Large$G_{4k-1}$}&\hbox{\Large$F_{4k}$}\\
\hbox{\Large$0$}&\dots&\dots&\dots&\hbox{\Large$0$}&\hbox{\Large$0$}&\hbox{\Large$G_{4k}$}
\end{pmatrix},
\end{equation}
where the $\ell$-th non-zero entry in the first row
(these are marked by $*$) is
$$
(-1)^{\ell -1}(-4)^{(\ell +1)k}8k(4k+1)\left(\prod _{i=1} ^{4k-\ell }(4ik-1)\right)
\left(\prod _{i=1} ^{\ell -1}(4ik+1)\right),
$$
and where each block $F_t$ and $G_t$ is a $(4k-1)\times(4k)$ matrix (that
is, these are {\it rectangular} blocks!) with non-zero entries only
on its (two) main diagonals,
$$
F_t =\left(\smallmatrix f_1(4(t -1)k+1)&f_0(4(t -1)k+2)&0&\dots\\
0&f_1(4(t -1)k+2)&f_0(4(t -1)k+3)&0&\dots\\
&\ddots&\ddots&\\
&&\ddots&\ddots&\\
&&0&f_1(4t k-2)&f_0(4t k-1)&0\\
&&&0&f_1(4t k-1)&f_0(4t k)\\
\endsmallmatrix\right)
$$
and
$$
G_t =\left(\smallmatrix g_1(4(t -1)k+1)&g_0(4(t -1)k+2)&0&\dots\\
0&g_1(4(t -1)k+2)&g_0(4(t -1)k+3)&0&\dots\\
&\ddots&\ddots&\\
&&\ddots&\ddots&\\
&&0&g_1(4t k-2)&g_0(4t k-1)&0\\
&&&0&g_1(4t k-1)&g_0(4t k)\\
\endsmallmatrix\right).
$$
We have almost worked our way through the definition of the
determinant. The only missing piece is the definition of the
functions $f_0,f_1,g_0,g_1$ in the blocks $F_t$ and $G_t$. Here
it is:
\begin{align}
\notag
f_0(j)&=j(-4)^k,\\
\notag
f_1(j)&=-(4j+2)(-4)^k,\\
\notag
g_0(j)&=(N(k)-j),\\
\label{eq:fg}
g_1(j)&=-(4N(k)-4j-2),
\end{align}
where, as before, we write $N(k)=4k(4k+1)$ for short.
\section{The evaluation of the determinant}
\label{sec:eval}
I now describe how the determinant of \eqref{eq:Det} was evaluated by
applying to it the methods described in
\machSeite{KratBN}\cite{KratBN}. To make this
section as self-contained as possible, for each of them
I briefly recall how it works before putting it into action.
\medskip
{\it ``Method" 0}: {\it Do row and column operations until the determinant
reduces to something manageable.}
In fact, at a first glance, this does not look too bad.
Our matrix \eqref{eq:Det}, of which we want to compute
the determinant and show that it is non-zero, is a very sparse
matrix. Moreover, it looks almost like a two-diagonal matrix.
It seems that one should be able to do a few row and
column manipulations and thus reduce the matrix to a matrix of a
simpler form of which we can evaluate the determinant.
Well, we tried that. Unfortunately, the above impression is
deceiving. First of all, the diagonals of the blocks do not really
fit together to form diagonals which run from one end of the matrix
to the other. Second, there remains still the first row which does
not fit the pattern of the rest of the matrix.
So, whatever we did, we ended up nowhere.
Maybe we should try something more \hbox{sophisticated \dots}
\medskip
{\it Method 1}
\machSeite{KratBN}\cite[Sec.~2.6]{KratBN}: {\it LU-factorisation}.
Suppose we are given a family of matrices $A(1), A(2),A(3),\dots$ of
which we want to compute the determinants. Suppose further that we
can write
$$A(k)\cdot U(k)=L(k),$$
where $U(k)$ is an upper triangular matrix with 1s on the diagonal,
and where $L(k)$ is a lower triangular matrix. Then, clearly,
$$\det(A(k))=\text {product of the diagonal entries of $L(k)$}.$$
But how do we find $U(k)$ and $L(k)$? We go to the computer, crank
out $U(k)$ and $L(k)$ for $k=1,2,3,\dots$, until we are able to make
a guess. Afterwards we prove the guess by proving the corresponding
identities.
Well, we programmed that, we stared at the output on the computer
screen, but we could not make any sense of it.
\medskip
{\it Method 2}
\machSeite{KratBN}\cite[Sec.~2.3]{KratBN}: {\it Condensation}.
This is based on a determinant formula due to Jacobi (see
\machSeite{BresAO}\cite[Ch.~4]{BresAO}
and
\machSeite{KnutAF}\cite[Sec.~3]{KnutAF}). Let $A$ be an $n\times n$ matrix. Let
$A_{i_1,i_2,\dots,i_\ell}^{j_1,j_2,\dots,j_\ell}$ denote the
submatrix of $A$ in which
rows $i_1,i_2,\dots,i_\ell$ and columns $j_1,j_2,\dots,j_\ell$ are
omitted. Then there holds
\begin{equation} \label{eq:cond}
\hfill \det A\cdot \det A_{1,n}^{1,n}=\det A_{1}^{1}\cdot \det A_n^n-
\det A_1^n\cdot \det A_n^1.
\end{equation}
If we consider a family of matrices $A(1),A(2),\dots$, and if all the
consecutive minors of $A(n)$ belong to the same family, then this
allows one to give an inductive proof of a conjectured determinant
evaluation for $A(n)$.
Let me illustrate this by reproducing Amdeberhan's condensation proof
\machSeite{AmdeAD}\cite{AmdeAD} of \eqref{eq:Krat}. Let $M_n(x,y)$ denote the
determinant in \eqref{eq:Krat}. Then we have
\begin{align} \notag
\big(M_n(x,y)\big)_n^n&=M_{n-1}(x,y), \\
\notag
\big(M_n(x,y)\big)_1^1&=M_{n-1}(x+1,y+1), \\
\notag
\big(M_n(x,y)\big)_n^1&=M_{n-1}(x-1,y+2), \\
\notag
\big(M_n(x,y)\big)_1^n&=M_{n-1}(x+2,y-1), \\
\label{eq:minors}
\big(M_n(x,y)\big)_{1,n}^{1,n}&=M_{n-2}(x+1,y+1).
\end{align}
Thus, we know that Equation \eqref{eq:cond} is satisfied
with $A$ replaced by $M_n(x,y)$,
where the minors appearing in \eqref{eq:cond} are given by
\eqref{eq:minors}. This can be interpreted as a recurrence for
the sequence $\big(M_n(x,y)\big)_{n\ge0}$.
Indeed, given $M_0(x,y)$ and $M_1(x,y)$, the
equation \eqref{eq:cond} determines $M_n(x,y)$ uniquely
for all $n\ge0$ (given that
$M_n(x,y)$ never vanishes).
Thus, since the right-hand side of \eqref{eq:Krat} is indeed never
zero, for the proof of \eqref{eq:Krat}
it suffices to check \eqref{eq:Krat} for
$n=0$ and $n=1$, and that the right-hand side of \eqref{eq:Krat}
also satisfies \eqref{eq:cond},
all of which is a routine task.
\medskip
Now, a short glance at the definition of our matrix \eqref{eq:Det}
will convince us
quickly that application of
this method to it is rather hopeless. For example, omission of
the first row already brings us outside of our family of matrices.
So, also this method is not much help to solve our problem, which is
really a pity, because it is the most painless of \hbox{all \dots}
\medskip
{\it Method 3}
\machSeite{KratBN}\cite[Sec.~2.4]{KratBN}: {\it Identification of
factors}. In order to sketch the idea, let us quickly go through a
(standard) proof of the {\it Vandermonde determinant evaluation},
\begin{equation} \label{eq:Vandermonde}
\det_{1\le i,j\le n}\(X_i^{j-1}\)=\prod _{1\le i<j\le n}
^{}(X_j-X_i).
\end{equation}
\begin{proof}If $X_{i_1}=X_{i_2}$ with $i_1\ne i_2$, then the Vandermonde
determinant \eqref{eq:Vandermonde} certainly vanishes because in that
case two rows of the determinant are identical. Hence,
$(X_{i_1}-X_{i_2})$ divides the determinant as a polynomial
in the $X_i$'s. But that means that the complete product $\prod
_{1\le i<j\le n} (X_j-X_i)$ (which is exactly the right-hand side
of \eqref{eq:Vandermonde}) must divide the determinant.
On the other hand, the determinant is a polynomial in the $X_i$'s of
degree at most $\binom n2$. Combined with the previous observation,
this implies that the determinant equals the right-hand side product
times, possibly, some constant. To compute the constant, compare
coefficients of $X_1^0X_2^1\cdots X_n^{n-1}$ on both sides of
\eqref{eq:Vandermonde}. This completes the proof of
\eqref{eq:Vandermonde}.
\end{proof}
At this point, let us extract the essence of this proof.
The basic steps are:
{\em
\begin{enumerate}
\item[(S1)] Identification of factors
\item[(S2)] Determination of degree bound
\item[(S3)] Computation of the multiplicative constant.
\end{enumerate}
}
As I report in
\machSeite{KratBN}\cite{KratBN}, this turns out to be an extremely
powerful method which has numerous applications. To given an idea of
the flavour of the method, I show a few steps when it is applied
to the determinant in
\eqref{eq:Krat} (ignoring the fact that we have already found a very
simple proof of its evaluation; see
\machSeite{KratBD}\cite[proof of Theorem~10]{KratBD}
for the complete proof using the ``identification of factors" method).
\medskip
To get started, we have to transform the assertion \eqref{eq:Krat} into
an assertion about polynomials. This is easily done, we just have to
factor
$$(x+y+i-1)!/(x+2i)!/(y+2n-i-2)!$$
out of the $i$-th row of the determinant. If we subsequently cancel
common factors on both sides of \eqref{eq:Krat}, we arrive at the
equivalent assertion
\begin{multline} \label{eq:Krat1}
\det_{0\le i,j\le
n-1}\big((x+y+i)_{j}\,(x+2i-j+1)_j\,(y+2j-i+1)_{2n-2j-2}\big)\\
=\prod _{i=0}
^{n-1}\big(i!\,(y+2i+1)_{n-i-1}\,(2x+y+2i)_i\,(x+2y+2i)_i\big),
\end{multline}
where, as before,
$(\alpha)_k$ is the standard notation for shifted factorials
(Pochhammer symbols) explained in the statement of Theorem~\ref{thm:xy}.
In order to apply the same idea as in the above evaluation of the
Vandermonde determinant, as a first step we have to show that the
right-hand side of \eqref{eq:Krat1} divides the determinant on the
left-hand side as a polynomial in $x$ and $y$. For example, we would
have to prove that $(x+2y+2n-2)$
(actually,
$(x+2y+2n-2)^{\fl{(n+1)/3}}$, we will come to that in a moment)
divides the determinant. Equivalently,
if we set $x=-2y-2n+2$ in the determinant, then it should vanish. How
could we prove that? Well, if it vanishes then there must be a linear
combination of the columns, or of the rows, that vanishes. Equivalently, for
$x=-2y-2n+2$ we find a vector in the kernel of the matrix in \eqref{eq:Krat1},
respectively of its transpose. More generally (and this addresses the
fact that
we actually want to prove that $(x+2y+2n-2)^{\fl{(n+1)/3}}$ divides the
determinant):
\medskip
{\em \leftskip1cm
\rightskip1cm
\noindent
For proving that $(x+2y+2n-2)^E$ divides the
determinant, we find $E$ linear independent vectors in the
kernel.
\par}
\medskip
\noindent
(For a formal justification that this does indeed suffice,
see Section~2 of
\machSeite{KratBI}\cite{KratBI}, and in particular the Lemma in that
section.)
Okay, how is this done in practice? You go to your computer, crank
out these vectors in the kernel, for
$n=1,2,3,\dots$, and try to make a guess what they are in general.
To see how this works, let us do it in our example.
First of all, we program the kernel of the matrix in \eqref{eq:Krat1}
with $x=-2y-2n+2$ (again, we are using {\sl Mathematica}
here).\footnote{In the program, {\tt V[n]} represents the kernel,
which
is clearly a vector space. In the computer output, it is given in
parametric form, the parameters being the {\tt c[i]}'s.}
\MATH
\goodbreakpoint%
In[14]:= p=Pochhammer;
\leavevmode%
m[i\MATHtief ,j\MATHtief ,n\MATHtief ]:=p[x+y+i, j]*p[y+2*j+1-i, 2*n-2*j-2]*p[x-j+1+2i,j];
\leavevmode%
V[n\MATHtief ]:=(x=-2y-2n+2;
\leavevmode%
Var=Sum[c[j]*Table[m[i,j,n],\MATHlbrace i,0,n-1\MATHrbrace ],\MATHlbrace j,0,n-1\MATHrbrace ];
\leavevmode%
Var=Solve[Var==Table[0,\MATHlbrace n\MATHrbrace ],Table[c[i],\MATHlbrace i,0,n-1\MATHrbrace ]];
\leavevmode%
Factor[Table[c[i],\MATHlbrace i,0,n-1\MATHrbrace ]/.Var])
\goodbreakpoin
\endgroup
What the computer gives is the following:
\MATH
\goodbreakpoint%
In[15]:= V[2]
Out[15]= \MATHlbrace \MATHlbrace -2 c[1], c[1]\MATHrbrace \MATHrbrace
\goodbreakpoin
In[16]:= V[3]
Out[16]= \MATHlbrace \MATHlbrace -2 c[2], -c[2], c[2]\MATHrbrace \MATHrbrace
\goodbreakpoin
In[17]:= V[4]
Out[17]= \MATHlbrace \MATHlbrace -2 c[3], -3 c[3], 0, c[3]\MATHrbrace \MATHrbrace
\goodbreakpoin
In[18]:= V[5]
Out[18]= \MATHlbrace \MATHlbrace -2 c[4], -5 c[4], -2 c[3] - c[4], c[3], c[4]\MATHrbrace \MATHrbrace
\goodbreakpoin
In[19]:= V[6]
Out[19]= \MATHlbrace \MATHlbrace -2 c[5], -7 c[5], -2 (c[4] + 2 c[5]), -c[4], c[4], c[5]\MATHrbrace \MATHrbrace
\goodbreakpoin
In[20]:= V[7]
Out[20]= \MATHlbrace \MATHlbrace -2 c[6], -9 c[6], -2 c[5] - 9 c[6], %
-3 c[5] - c[6], 0, %
> c[5], c[6]\MATHrbrace \MATHrbrace
\goodbreakpoin
\endgroup
At this point, the computations become somewhat slow.
So we should help our computer. Indeed, on the basis of what we have
obtained so far, it is ``obvious" that, somewhat unexpectedly, $y$ does not
appear in the result. Therefore we simply set $y$ equal to some random
number, and then the computer can go much further without any effort.
\MATH
\goodbreakpoin
In[21]:= y=101
\goodbreakpoin
In[22]:= V[8]
Out[22]= \MATHlbrace \MATHlbrace -2 c[7], -11 c[7], -2 (c[6] + 8 c[7]), -5 (c[6] + c[7]),
> -2 c[5] - c[6], c[5], c[6], c[7]\MATHrbrace \MATHrbrace
\goodbreakpoin
In[23]:= V[9]
Out[23]= \MATHlbrace \MATHlbrace -2 c[8], -13 c[8], -2 c[7] - 25 c[8], -7 (c[7] + 2 c[8]),
> -2 c[6] - 4 c[7] - c[8], -c[6], c[6], c[7], c[8]\MATHrbrace \MATHrbrace
\goodbreakpoin
In[24]:= V[10]
Out[24]= \MATHlbrace \MATHlbrace -2 c[9], -15 c[9], -2 (c[8] + 18 c[9]), -3 (3 c[8] + 10 c[9]),
> -2 c[7] - 9 c[8] - 6 c[9], -3 c[7] - c[8], 0, c[7], c[8], c[9]\MATHrbrace \MATHrbrace
\goodbreakpoin
In[25]:= V[11]
Out[25]= \MATHlbrace \MATHlbrace -2 c[10], -17 c[10], -2 c[9] - 49 c[10], -11 (c[9] + 5 c[10]),
> -2 (c[8] + 8 c[9] + 10 c[10]), -5 c[8] - 5 c[9] - c[10],
> -2 c[7] - c[8], c[7], c[8], c[9], c[10]\MATHrbrace \MATHrbrace
\goodbreakpoin
\endgroup
Let us extract some information out of these data.
For convenience, we write $M_n$ for the
matrix in \eqref{eq:Krat1} in the sequel. For example, by
just looking at the coefficients of $c[n-1]$ appearing in $V[n]$, we
extract that
\begin{enumerate}
\item [] the vector $(-2,1)$ is in the kernel of $M_2$,
\item [] the vector $(-2,-1,1)$ is in the kernel of $M_3$,
\item [] the vector $(-2,-3,0,1)$ is in the kernel of $M_4$,
\item [] the vector $(-2,-5,-1,0,1)$ is in the kernel of $M_5$,
\item [] the vector $(-2,-7,-4,0,0,1)$ is in the kernel of $M_6$,
\item [] the vector $(-2,-9,-9,-1,0,0,1)$ is in the kernel of $M_7$,
\item [] the vector $(-2,-11,-16,-5,0,0,0,1)$ is in the kernel of $M_8$,
\item [] the vector $(-2,-13,-25,-14,-1,0,0,0,1)$ is in the kernel of $M_9$,
\item [] the vector $(-2,-15,-36,-30,-6,0,0,0,0,1)$ is in the kernel of $M_{10}$,
\item [] the vector $(-2,-17,-49,-55,-20,-1,0,0,0,0,1)$ is in the kernel of $M_{11}$.
\end{enumerate}
Okay, now we have to make sense out of this.
Our vectors in the kernel have the following structure: first, there
are some negative numbers, then follow a few zeroes, and finally there
is a trailing 1. I believe that we do not have any problem to
guess what the zeroeth\footnote{The indexing convention in the matrix
in \eqref{eq:Krat1} of which the determinant is taken is that rows and
columns are indexed by $0,1,\dots,n-1$. We keep this convention here.}
or the first coordinate of our vector is. Since
the second coordinates are always negatives of
squares, there is also no problem
there. What about the third coordinates? Starting with the vector for
$M_7$, these are $-1,-5,-14,-30,-55,\dots$. I guess, rather than
thinking hard, we should consult {\tt Rate} (see Footnote~\ref{foot:Rate}):
\MATH
\goodbreakpoint%
In[26]:= Rate[-1,-5,-14,-30,-55]
\leavevmode%
-(i0 (1 + i0) (1 + 2 i0))
Out[26]= \MATHlbrace -------------------------\MATHrbrace
\leavevmode%
6
\goodbreakpoint%
\endgroup
After replacing {\tt i0} by $n-6$ (as we should), this becomes
$-(n-6)(n-5)(2n-11)/6$. An interesting feature of this formula is that
it also works well for $n=6$ and $n=5$. Equipped with this experience,
we let {\tt Rate} work out the fourth coordinate:
\MATH
\goodbreakpoint%
In[27]:= Rate[0,0,0,-1,-6,-20]
\leavevmode%
2
\leavevmode%
-((-3 + i0) (-2 + i0) (-1 + i0))
Out[27]= \MATHlbrace ---------------------------------\MATHrbrace
\leavevmode%
12
\goodbreakpoint%
\endgroup
After replacement of {\tt i0} by $n-5$, this is
$-(n-8)(n-7)^2(n-6)/12$. Let us summarise our results so far: the
first five coordinates of our vector in the kernel of $M_n$ are
\begin{multline*}
-2,\ -(2n-5),\ -\frac {(n-4)(2n-8)} {2},
\ -\frac {(n-6)(n-5)(2n-11)} {6},
\\
-\frac {(n-8)(n-7)(n-6)(2n-14)} {12}.
\end{multline*}
I would say, there is a clear pattern emerging: the $s$-th coordinate
is equal to
$$-\frac {(n-2s)_{s-1}\,(2n-3s-2)} {s!}=
-\frac {(2n-3s-2)} {(n-s-1)}\frac {(n-2s)_{s}} {s!}.$$
Denoting the above expression by $f(n,s)$, the vector
$$(f(n,0),f(n,1),\dots,f(n,n-2),1)$$
is apparently in the kernel of $M_n$ for $n\ge2$. To prove this, we
have to show that
\begin{multline*}
\sum _{s=0} ^{n-2}\frac
{(2n-3s-2)} {(n-s-1)}\frac {(n-2s)_{s}} {(s)!}\\
\cdot(-y-2n+i+2)_{s}\,(-2y-2n+2i-s+3)_{s}\,
(y+2s-i+1)_{2n-2s-2}\\
=(-y-2n+i+2)_{n-1}\,(-2y-3n+2i+4)_{n-1}.
\end{multline*}
In
\machSeite{KratBD}\cite{KratBD} it was argued that this identity follows from
a certain hypergeometric identity due to Singh
\machSeite{SinVAA}\cite{SinVAA}. However,
for just having {\it some} proof of this identity, this careful
literature search was not necessary. In fact, nowadays, {\it once you
write down a binomial or hypergeometric identity, it is already
proved!} One simply puts the binomial/hypergeometric sum into the
{\it Gosper--Zeilberger algorithm}
(see
\machSeite{PeWZAA}%
\machSeite{ZeilAP}%
\machSeite{ZeilAM}%
\machSeite{ZeilAV}%
\cite{PeWZAA,ZeilAP,ZeilAM,ZeilAV}),
which outputs a recurrence for it, and
then the only task is to verify that the (conjectured) right-hand side
also satisfies the same recurrence, and to check the identity
for sufficiently many
initial values (which one has already done anyway while producing the
conjecture).\footnote{As you may have suspected, this is again a little
bit oversimplified. But not much. The Gosper--Zeilberger algorithm
applies {\it always} to hypergeometric sums, and there are only very few
binomial sums where it does not apply. (For the sake of completeness, I
mention that there are also several algorithms available to deal with
multi-sums, see
\machSeite{ChSaAA}%
\machSeite{WiZeAC}%
\cite{ChSaAA,WiZeAC}. These do, however, rather quickly
challenge the resources of today's computers.) {\em Maple}
implementations written by Doron
Zeilberger are available from {\tt
http://www.math.temple.edu/\~{}zeilberg},
those written by Fr\'ed\'eric Chyzak are available from
{\tt http://algo.inria.fr/chyzak/mgfun.html},
{\sl Mathematica} implementations
written by Peter Paule, Axel Riese, Markus Schorn, Kurt Wegschaider
are available
from {\tt
http://www.risc.uni-linz.ac.at/research/combinat/risc/software}.}
As I mentioned earlier, actually
we need more vectors in the kernel. However, this is not difficult.
Take a closer look, and you will see that the pattern
persists (set $c[n-1]=0$ in the vector for $V[n]$, etc.).
It will take you no time to work
out a full-fledged conjecture for $\fl{(n+1)/3}$ linear independent
vectors in the kernel of $M_n$.
I do not want to go through Steps (S2) and (S3), that is, the degree
calculation and the computation of the constant. As it turns out, to
do this conveniently you need to introduce more variables in the
determinant in \eqref{eq:Krat1}. Once you do this, everything works
out very smoothly. I refer the reader to
\machSeite{KratBD}\cite{KratBD} for these details.
\medskip
Now, let us come back to our determinant, the
determinant of \eqref{eq:Det}, and apply
``identification of factors" to it.%
\footnote{What I describe in the sequel is, except for very few
excursions that ended up in a dead end, and which are therefore omitted here,
the way how the determinant evaluation was found.}
To begin with, here is bad news: ``identification of factors"
crucially requires the existence of indeterminates.
But, where are they in \eqref{eq:Det}? If we look at the definition
of the matrix \eqref{eq:Det}, which, in the end, depends on the
auxiliary functions $f_0,f_1,g_0,g_1$ defined in \eqref{eq:fg},
then we see that there are no
indeterminates at all. Everything is (integral) numbers. So,
to get even started, we need
to introduce indeterminates in a way such that the more general
determinant would still factor ``nicely." We do not have much
guidance. Maybe, since we already made the abbreviation
$N(k)=4k(4k+1)$, we should replace $N(k)$ by $X$? Okay, let us try
this, that is, let us put
\begin{align}
\notag
f_0(j)&=j(-4)^k,\\
\notag
f_1(j)&=-(4j+2)(-4)^k,\\
\notag
g_0(j)&=(X-j),\\
\label{eq:fg1}
g_1(j)&=-(4X-4j-2)
\end{align}
instead of \eqref{eq:fg}. Let us compute the new determinant for
$k=2$. We program the new functions $f_0,f_1,g_0,g_1$,
\MATH
\goodbreakpoint%
In[28]:= f0[k\MATHtief , t\MATHtief , j\MATHtief ] := j*(-4)\MATHhoch k;
\leavevmode%
f1[k\MATHtief , t\MATHtief , j\MATHtief ] := -(2 + 4*j)*(-4)\MATHhoch k;
\leavevmode%
g0[k\MATHtief , t\MATHtief , j\MATHtief ] := (X - j);
\leavevmode%
g1[k\MATHtief , t\MATHtief , j\MATHtief ] := (-4*X + 2 + 4*j)
\goodbreakpoint%
\endgroup
we enter the new determinant for $k=2$,
\MATH
\goodbreakpoint%
In[29]:= Factor[Det[A[2]]]
\goodbreakpoint%
\endgroup
and, after a waiting time of more than 15
minutes,\footnote{\label{foot:kompl}which I use
to explain why our computer needs so long to calculate this
determinant of a very sparse matrix of size $16\cdot 2^2=64$: isn't it
true that, nowadays, determinants of matrices with several hundreds of
rows and columns can be calculated without the slightest difficulty
(particularly if they are very sparse)?
Well, we should not forget that this is true for determinants of
matrices with {\it numerical\/} entries. However, our matrix
\eqref{eq:Det} with the modified definitions \eqref{eq:fg1} of
$f_0,f_1,g_0,g_1$ has now entries which are {\it polynomials} in
$X$. Hence, when our computer algebra program applies (internally)
some elimination algorithm to compute the determinant, huge rational
expressions will slowly build up and will slow down the calculations
(and, at times, will make our computer crash \dots). As I learn from
Dave Saunders, {\sl Maple} and {\sl Mathematica} do currently in fact
not use the best known algorithms for dealing with determinants of
matrices with polynomial entries. (This may have to do with the fact
that the developers try to optimise the algorithms for numerical
determinants in the first case.) It is known how to avoid the
expression swell and compute polynomial matrix determinants in time
about $mn^3$, where $n$ is the dimension of the matrix
and $m$ is the bit length of the determinant
(roughly, in univariate case, $m$ is degree times
maximum coefficient length).}
we obtain
\MATH
\goodbreakpoint%
Out[29]= -1406399608474882323154910525986578515918369681041517636\MATHbackslash
\leavevmode%
2
> 11783762359972003840000000 (-64 + X) (-48 + X) (-40 + X)
\leavevmode%
3 4 5 6 7
> (-32 + X) (-24 + X) (-16 + X) (-8 + X) X
\leavevmode%
2
> (9653078694297600 - 916000657637376 X + 36130368757760 X
\leavevmode%
3 4 5 6
> - 758218948608 X + 8928558848 X - 55938432 X + 145673 X )
\goodbreakpoint%
\endgroup
Not bad. There are many factors which are linear in $X$. (This is what
we were after.) However, the
irreducible polynomial of degree 6 gives us some headache. (The
degrees of the irreducible part of the polynomial
will grow quickly with $k$.) How are we going to guess what this
factor could be, and, even more daunting, even if we should be
able to come up with a guess, how would we go about to prove it?
So, maybe we should modify our choice of how to introduce
indeterminates into the matrix. In fact, we overlooked something:
maybe, in a hidden manner, the variable $X$ is also there at other
places in \eqref{eq:fg}, that is, when $X$ is specialised to
$N(k)=4k(4k+1)$ at these places it becomes invisible.
More specifically, maybe we should insert the
difference $X-4k(4k+1)$ in the definitions of $f_0$ and $f_1$ (which
would disappear for $X=4k(4k+1)$). So, maybe we should try:
\begin{align*}
\notag
f_0(j)&=(4k(4k + 1) - X + j)(-4)^k,\\
\notag
f_1(j)&=-(16k(4k + 1) - 4X + 2 + 4j)(-4)^k,\\
\notag
g_0(j)&=(X-j),\\
g_1(j)&=-(4X-4j-2),
\end{align*}
Okay, let us modify our computer program accordingly,
\MATH
\goodbreakpoint%
In[30]:= f0[k\MATHtief , t\MATHtief , j\MATHtief ] := (4*k*(4*k + 1) - X + j)*(-4)\MATHhoch k;
\leavevmode%
f1[k\MATHtief , t\MATHtief , j\MATHtief ] := -(4*4*k*(4*k + 1) - 4*X + 2 + 4*j)*(-4)\MATHhoch k;
\leavevmode%
g0[k\MATHtief , t\MATHtief , j\MATHtief ] := (X - j);
\leavevmode%
g1[k\MATHtief , t\MATHtief , j\MATHtief ] := (-4*X + 2 + 4*j)
\goodbreakpoint%
\endgroup
and let us compute the new determinant for $k=2$:
\MATH
\goodbreakpoint%
In[31]:= Factor[Det[A[2]]]
\goodbreakpoint%
\endgroup
This makes us wait for
another 15 minutes, after which we are rewarded with:
\MATH
\goodbreakpoint%
Out[31]= -296777975397624679901369809794412104454134763494070841\MATHbackslash
> 1155365196124754770317472271790417634937439881166252558632\MATHbackslash
> 616674197504000000000 (-141 + 2 X) (-139 + 2 X) (-137 + 2 X)
> (-135 + 2 X) (-133 + 2 X) (-131 + 2 X) (-129 + 2 X)
\goodbreakpoint%
\endgroup
Excellent! There is no big irreducible polynomial anymore. Everything
is linear factors in $X$. But, wait, there is still a problem: in the
end (recall Step~(S2)!)
we will have to compare the degrees of the determinant and of the
right-hand side as polynomials in $X$. If we expand the determinant
according to its definition, then the conclusion is that the degree
of the determinant is bounded above by $16k^2-1$, which, for $k=2$ is
equal to $31$. The right-hand side polynomial however which we
computed above has degree 7. This is a big gap!
I skip some other things (ending up in dead ends \dots) that we tried
at this point.
Altogether they pointed to the
fact that, apparently, {\it one} indeterminate is not sufficient. Perhaps it
is a good idea to ``diversify" the variable $X$, that is, to make two
variables, $X_1$ and $X_2$, out of $X$:
\begin{align*}
\notag
f_0(j)&=(4k(4k + 1) - X_2 + j)(-4)^k,\\
\notag
f_1(j)&=-(16k(4k + 1) - 4X_1 + 2 + 4j)(-4)^k,\\
\notag
g_0(j)&=(X_2-j),\\
g_1(j)&=-(4X_1-4j-2).
\end{align*}
We program this,
\MATH
\goodbreakpoint%
In[32]:= f0[k\MATHtief , t\MATHtief , j\MATHtief ] := (4*k*(4*k + 1) - X[2] + j)*(-4)\MATHhoch k;
\leavevmode%
f1[k\MATHtief , t\MATHtief , j\MATHtief ] := -(4*4*k*(4*k + 1) - 4*X[1] + 2 + 4*j)*(-4)\MATHhoch k;
\leavevmode%
g0[k\MATHtief , t\MATHtief , j\MATHtief ] := (X[2] - j);
\leavevmode%
g1[k\MATHtief , t\MATHtief , j\MATHtief ] := (-4*X[1] + 2 + 4*j)
\goodbreakpoint%
\endgroup
and, in order to avoid overstraining our computer, compute this
time the new determinant for $k=1$:
\MATH
\goodbreakpoint%
In[33]:= Factor[Det[A[1]]]
\goodbreakpoint%
\endgroup
After some minutes there appears
\MATH
\goodbreakpoint%
Out[33]= 3242591731706757120000 (-37 + 2 X[1]) (-35 + 2 X[1])
\leavevmode%
3 2
> (-33 + 2 X[1]) (1 + 2 X[1] - 2 X[2]) (3 + 2 X[1] - 2 X[2])
> (5 + 2 X[1] - 2 X[2])
\goodbreakpoint%
\endgroup
on the computer screen.
On the positive side: the determinant still factors completely into
linear factors, something which we could not expect a priori.
Moreover, the degree (in $X_1$ and $X_2$)
has increased, it is now equal to 9 although we were
only computing the determinant for $k=1$. However, a gap remains, the
degree should be $16k^2-1=15$ if $k=1$.
Thus, it may be wise to introduce another genuine variable, $Y$.
For example, we may think of simply homogenising the definitions of
$f_0,f_1,g_0,g_1$:
\begin{align*}
\notag
f_0(j)&=(4k(4k + 1)Y - X_2 + jY)(-4)^k,\\
\notag
f_1(j)&=-(16k(4k + 1)Y - 4X_1 + (2 + 4j)Y)(-4)^k,\\
\notag
g_0(j)&=(X_2-jY),\\
g_1(j)&=-(4X_1-(4j+2)Y).
\end{align*}
We program this,
\MATH
\goodbreakpoint%
In[34]:= f0[k\MATHtief , t\MATHtief , j\MATHtief ] := (4*k*(4*k + 1)*Y - X[2] + j*Y)*(-4)\MATHhoch k;
\leavevmode%
f1[k\MATHtief , t\MATHtief , j\MATHtief ] := -(4*4*k*(4*k + 1)*Y - 4*X[1] +
\leavevmode%
(2 + 4*j)*Y)*(-4)\MATHhoch k;
\leavevmode%
g0[k\MATHtief , t\MATHtief , j\MATHtief ] := (X[2] - j*Y);
\leavevmode%
g1[k\MATHtief , t\MATHtief , j\MATHtief ] := (-4*X[1] + (2 + 4*j)*Y)
\goodbreakpoint%
In[35]:= Factor[Det[A[1]]]
\goodbreakpoint%
\endgroup
we wait for some more minutes,
and we obtain
\MATH
\leavevmode%
6
Out[35]= -3242591731706757120000 Y (33 Y - 2 X[1]) (35 Y - 2 X[1])
\leavevmode%
3 2
> (37 Y - 2 X[1]) (Y + 2 X[1] - 2 X[2]) (3 Y + 2 X[1] - 2 X[2])
> (5 Y + 2 X[1] - 2 X[2])
\goodbreakpoint%
\endgroup
Great! The degree in $X_1,X_2,Y$ is 15, as it should
be!
At this point, one becomes greedy. The more variables we have, the
easier will be the proof. We ``diversify" the variables $X_1,X_2,Y$,
that is, we make them $X_{1,t},X_{2,t},Y_t$ if they appear in the
blocks $F_t$ or $G_t$, respectively, $t=1,2,\dots,4k$ (cf.\
\eqref{eq:Det} and the {\sl Mathematica} code for the precise meaning
of this definition):
\begin{align}
\notag
f_0(j)&=(4k(4k + 1)Y_t - X_{2,t} + jY_t)(-4)^k,\\
\notag
f_1(j)&=-(16k(4k + 1)Y_t - 4X_{1,t} + (2 + 4j)Y_t)(-4)^k,\\
\notag
g_0(j)&=(X_{2,t}-jY_t),\\
\label{eq:fg5}
g_1(j)&=-(4X_{1,t}-(4j+2)Y_t).
\end{align}
Now there are so many variables so that there is no way to do the
factorisation of the new determinant for $k=1$ on the computer unless
one plays tricks (which we\break did).%
\footnote{\label{foot:tricks}See
Footnote~\ref{foot:kompl} for the explanation of the complexity problem.
``Playing tricks" would mean to compute the
determinant for various special choices of the variables
$X_{1,t},X_{2,t},Y_t$,
and then reconstruct the general result by interpolation.
This is possible because we know an a priori degree bound
(namely $15$) for the polynomial.
However, this would become infeasible for $k=3$, for example. ``Playing
tricks" then would mean to be content with an
``almost sure" guess, the latter being based on features of the (unknown)
general result that are already visible in the earlier results,
and on calculations done for special values of the variables.
For example, if we encounter determinants $\det M(k)$, where the $M(k)$'s
are some square matrices, $k=1,2,\dots$, and the results for
$k=1,2,\dots,k_0-1$
show that $x-y$ must be a factor of $\det M(k)$ to some power, then
one would specialise $y$ to some value that would make $x-y$ distinct
from any other linear factors containing $x$, and, supposing that
$y=17$ is such a choice, compute $\det M(k_0)$ with $y=17$. The exact
power of $x-y$ in the unspecialised determinant $\det M(k_0)$
can then be read off from
the exponent of $x-17$ in the specialised one.
If it should happen that it is also infeasible
to calculate $\det M(k_0)$ with $x$ still unspecialised, then there is
still a way out. In that case, one specialises $y$ {\it and\/} $x$, in
such a way that $x-y$ would be a prime $p$ that one expects not to occur
as a prime factor in any other factor of the determinant $\det
M(k_0)$. The exact
power of $x-y$ in the unspecialised determinant $\det M(k_0)$
can then be read off from the exponent of $p$ in the prime
factorisation of the specialised determinant. See
Subsections~\ref{sec:signed} and \ref{sec:poset}, and in particular
Footnote~\ref{foot:maj} for further instances where this trick was applied.}
But let us pretend that we are able to do it:
\MATH
\goodbreakpoint%
In[36]:= f0[k\MATHtief , t\MATHtief , j\MATHtief ] := (4*k*(4*k + 1)*Y[t] - X[2, t]
\leavevmode%
+ j*Y[t])*(-4)\MATHhoch k;
\leavevmode%
f1[k\MATHtief , t\MATHtief , j\MATHtief ] := -(4*4*k (4*k + 1)*Y[t] - 4*X[1, t] +
\leavevmode%
(2 + 4*j)*Y[t])*(-4)\MATHhoch k;
\leavevmode%
g0[k\MATHtief , t\MATHtief , j\MATHtief ] := (X[2, t] - j*Y[t]);
\leavevmode%
g1[k\MATHtief , t\MATHtief , j\MATHtief ] := (-4*X[1, t] + (2 + 4*j)*Y[t])
\goodbreakpoint%
In[37]:= Factor[Det[A[1]]]
\goodbreakpoint%
Out[37]= 3242591731706757120000 (2 X[1, 1] - 33 Y[1]) Y[1]
> (2 X[1, 1] - 2 X[2, 1] + Y[1]) (2 X[1, 2] - 35 Y[2]) Y[2]
> (2 X[1, 2] - 2 X[2, 2] + Y[2]) (-2 X[2, 2] Y[1] + 2 X[1, 1]
> Y[2] + 3 Y[1] Y[2]) (2 X[1, 3] - 37 Y[3]) Y[3] (2 X[1, 3] -
> 2 X[2, 3] + Y[3]) (-2 X[2, 3] Y[1] + 2 X[1, 1] Y[3] +
> 5 Y[1] Y[3]) (-2 X[2, 3] Y[2] + 2 X[1, 2] Y[3] + 3 Y[2] Y[3])
\goodbreakpoint%
\endgroup
By staring a little bit at this result (and the one that we computed
for $k=2$), we extracted that,
apparently, we have
\begin{multline} \label{eq:CK4}
\det A^X=(-1)^{k-1}4^{2k(4k^2+7k+2)}k^{2k(4k+1)}
\prod _{i=1} ^{4k}(i+1)_{4k-i+1}\\
\times
\prod _{a=1}
^{4k-1}\left(2X_{1,a}-(32k^2+2a-1)Y_a\right)\\
\times
\prod _{1\le a\le b\le 4k-1} ^{}
(2X_{2,b}Y_a-2X_{1,a}Y_b-(2b-2a+1)Y_aY_b),
\end{multline}
where $A^X$ denotes the new general matrix given through
\eqref{eq:Det} and \eqref{eq:fg5}, and
where, as before,
$(\alpha)_k$ is the standard notation for shifted factorials
(Pochhammer symbols) explained in the statement of Theorem~\ref{thm:xy}.
The special case that we need in the end to prove our Theorem~\ref{T1} is
$X_{1,t}=X_{2,t}=N(k)$ and $Y_t=1$.
\medskip
Now we are in business. Here is the {\it Sketch of the proof of
\eqref{eq:CK4}}:
\medskip
Re (S1): For each factor of the (conjectured) result \eqref{eq:CK4}, we
find a linear combination of the rows which vanishes if the factor
vanishes. (In other terms: if the indeterminates in the matrix are
specialised so that a particular factor vanishes, we find a vector in
the kernel of the transpose of the specialised matrix.)
For example, to explain the factor
$(2X_{1,1}-(32k^2+1)Y_1)$, we found:
If $X_{1,1}=\frac {32k^2+1} {2}Y_1$, then
\begin{multline} \label{eq:combin}
\frac {2(X_{2,4k-1}-(N(k)-1)Y_{4k-1})} {(-4)^{k(4k+1)+1}(16k^2+1)\prod
_{\ell=1} ^{4k-1}(4\ell k+1)}\cdot(\text {row 0 of $A^X$})
\\+
\sum _{s =0} ^{4k}\sum _{t =0} ^{4k-2}\Bigg(\frac {(-1)^{s (k-1)}2^t} {4^{s k}}
\prod _{\ell=0} ^{s -1}\frac {4k-1+4\ell k} {16k^2+1-4\ell k}
\prod _{\ell=4k-t } ^{4k-1}\frac
{2X_{1,\ell}-(32k^2+2\ell-1)Y_\ell}
{X_{2,\ell-1}-(16k^2+\ell-1)Y_{\ell-1}}\Bigg)\\
\cdot(\text {row $(16k^2-(4k-1)s -t -1)$ of $A^X$})=0,
\end{multline}
as is easy to verify. (Since the coefficients of the various rows in
\eqref{eq:combin} are rational functions in the indeterminates
$X_{1,t},X_{2,t},Y_t$, they
are rather easy to work out from computer data.
One does not even need {\tt Rate} \dots)
\medskip
Re (S2): The total degree in the $X_{1,t}$'s, $X_{2,t}$'s, $Y_{t}$'s of the
product on the right-hand side of \eqref{eq:CK4} is $16k^2-1$. As we
already remarked earlier, the degree of the determinant is at most
$16k^2-1$. Hence, the determinant is equal to the product times,
possibly, a constant.
\medskip
Re (S3): For the evaluation of the constant, we compare coefficients of
$$
X_{1,1}^{4k}X_{1,2}^{4k-1}\cdots X_{1,4k-1}^2Y_1^1Y_2^2\cdots
Y_{4k-1}^{4k-1}.
$$
After some reflection,
it turns out that the constant is equal to a determinant of the same
form, that is, of the form \eqref{eq:Det}, but with auxiliary
functions
\begin{align}
\notag
f_0(j)&=(N(k)+j)(-4)^k,\\
\notag
f_1(j)&=4(-4)^k,\\
\notag
g_0(j)&=-j,\\
\label{eq:fg6}
g_1(j)&=-4.
\end{align}
\medskip
What a set-back!
It seems that we are in the same situation as at the very beginning.
We started with the determinant
of the matrix
\eqref{eq:Det} with auxiliary functions \eqref{eq:fg}, and we ended up
with the same type of determinant, with auxiliary functions
\eqref{eq:fg6}. There is little hope though: the functions in
\eqref{eq:fg6} are somewhat simpler as those in \eqref{eq:fg}.
Nevertheless, we have to play the same game again; that is, if we
want to apply the method of identification of factors, then we have
to introduce indeterminates. Skipping the experimental part, we came up
with
\begin{align*} f_0(j)&=(Z_t+j)(-4)^k,\\
f_1(j)&=4(-4)^kX_t,\\
g_0(j)&=-j,\\
g_1(j)&=-4X_t,
\end{align*}
where $t$ has the same meaning as before in \eqref{eq:fg5}.
Denoting the new matrix by $A^Z$, computer calculations suggested
that apparently
\begin{equation} \label{eq:CK9}
\det A^Z=(-1)^{k-1}2^{16k^3+20k^2+14k-1}k^{4k}(4k+1)!
\prod _{a=1} ^{4k-1}\Bigg(X_a^{4k+1-a}
\prod _{b=0} ^{a-1}(Z_a-4bk)\Bigg).
\end{equation}
The special case that we need is $Z_t=N(k)$ and $X_t=1$.
So, we apply again the method of identification of factors.
Everything runs smoothly (except that the details of the
verification of the factors are somewhat more unpleasant here).
When we come finally to the point that we want to determine the
constant, it turns out that the constant is
equal to --- no surprise anymore ---
the determinant of a matrix of the form \eqref{eq:Det} with
auxiliary functions
\begin{align*} f_0(j)&=(-4)^k,\\
f_1(j)&=4(-4)^k,\\
g_0(j)&=0,\\
g_1(j)&=-4.
\end{align*}
Now, is this good or bad news? In other words, while painfully working
through the steps of ``identification of factors," will we
forever continue
producing new determinants of the form \eqref{eq:Det}, which we must
again handle by the same method? To give it away: this is indeed {\it
very good} news. The function $g_0(j)$ vanishes identically!
It makes it possible that now Method~0 (= do some row and column
manipulations) works. (See
\machSeite{AlKPAA}\cite{AlKPAA} for the details.)
We are --- finally --- done with the proof of
\eqref{eq:CK4}, and, since the right-hand side {\it does not\/} vanish
for $X_{1,t}=X_{2,t}=N(k)$ and $Y_t=1$, with the proof of
Theorem~\ref{T1}!
\raise-15pt\hbox{{\Huge$\square$}}
\section{More determinant evaluations}\label{sec:detlist}
This section complements the list of known determinant evaluations
given in Section~3 of
\machSeite{KratBN}\cite{KratBN}. I list here several determinant
evaluations which I believe are interesting or attractive
(and, in the ideal case, both), that have appeared
since
\machSeite{KratBN}\cite{KratBN}, or that I failed to mention in
\machSeite{KratBN}\cite{KratBN}.
I also include several conjectures and open problems, some of them old,
some of them new. As in
\machSeite{KratBN}\cite{KratBN}, each evaluation is accompanied by some
remarks providing information on the context in which it arose.
Again, the selection of determinant evaluations presented
reflects totally my taste, which must be blamed in the case
of any shortcomings. The order of presentation follows loosely the
order of presentation of determinants in
\machSeite{KratBN}\cite{KratBN}.
\subsection{More basic determinant evaluations}
I begin with two determinant evaluations belonging to the category ``standard
determinants" (see Section~2.1 in
\machSeite{KratBN}\cite{KratBN}). They are among
those which I missed to state in
\machSeite{KratBN}\cite{KratBN}.
The reminder for inclusion here is the paper
\machSeite{AmZeAB}\cite{AmZeAB}.
There, Amdeberhan and Zeilberger propose an {\it automated approach} towards
determinant evaluations via the condensation method (see ``Method~2" in
Section~\ref{sec:eval}). They provide a list of examples which
can be obtained in that way. As they remark at the end of the paper,
all of these are special cases of Lemma~5 in
\machSeite{KratBN}\cite{KratBN}, with the
exception of three, namely Eqs.~(8)--(10) in
\machSeite{AmZeAB}\cite{AmZeAB}.
In their turn, two of them, namely (8) and (9), are special cases of
the following evaluation. (For (10), see Lemma~\ref{prop:AmZe} below.)
\begin{Lemma} \label{lem:AmZe}
Let $P(Z)$ be a polynomial in $Z$ of degree $n-1$
with leading coefficient $L$. Then
\begin{equation} \label{eq:D1}
\det_{1\le i,j\le n}\(P(X_i+Y_j)\)=L^n
\prod _{i=1} ^{n}\binom {n-1}i\prod _{1\le i<j\le n}
^{}(X_i-X_j)(Y_j-Y_i).
\end{equation}
\quad \quad \qed
\end{Lemma}
This lemma is easily proved along the lines of
the standard proof of the Vandermonde determinant evaluation which
we recalled in Section~\ref{sec:eval} (see the proof of
\eqref{eq:Vandermonde}) or by condensation.
A multiplicative version of Lemma~\ref{lem:AmZe} is the following.
\begin{Lemma} \label{lem:AmZe2}
Let $P(Z)=p_{n-1}Z^{n-1}+p_{n-2}Z^{n-2}+\dots+p_0$. Then
\begin{equation} \label{eq:D2}
\det_{1\le i,j\le n}\(P(X_iY_j)\)=
\prod _{i=0} ^{n-1}p_i\prod _{1\le i<j\le n}
^{}(X_i-X_j)(Y_i-Y_j).
\end{equation}
\quad \quad \qed
\end{Lemma}
On the other hand, identity~(10) from
\machSeite{AmZeAB}\cite{AmZeAB} can be generalised to the following
Cauchy-type determinant evaluation. As all the identities from
\machSeite{AmZeAB}\cite{AmZeAB}, it can also be proved by the
condensation method.
\begin{Lemma} \label{prop:AmZe}
Let $a_0,a_1,\dots,a_{n-1}$, $c_0,c_1,\dots,c_{n-1}$, $b$, $x$ and $y$
be indeterminates. Then, for any positive integer $n$, there holds
\begin{multline} \label{eq:AmZe}
\det_{0\le i,j\le n-1}\(\frac{(x+a_i+c_j)(y+bi+c_j)}
{ (x+a_i+bi+c_j)}\)\\=b^{n-1}\,(n-1)!
\(\binom n2b+(n-1)x+y+\sum_{i=1}^{n-1}a_i +
\sum_{i=0}^{n-1}c_i\)\\
\times
\frac{\displaystyle\prod_{0\le i<j\le n-1}^{}(c_j-c_i)\
\prod_{i=1}^{n-1}(y-x-a_i)
\prod_{1\le i<j\le n-1}^{}((j-i)b-a_i+a_j)}
{\displaystyle\prod_{i=1}^{n-1}\prod_{j=0}^{n-1}
{(x+a_i+bi+c_j)}}.
\end{multline}
\quad \quad \qed
\end{Lemma}
Speaking of Cauchy-type determinant evaluations, this brings us to a
whole family of such evaluations which were instrumental in
Kuperberg's recent advance
\machSeite{KupeAH}\cite{KupeAH}
on the enumeration of (symmetry classes of) {\it alternating sign matrices}.
The reason that determinants, and also Pfaffians, play an important
role in this context is due to Propp's discovery
(described for the first time in
\machSeite{ElKLAB}\cite[Sec.~7]{ElKLAB} and exploited in
\machSeite{KupeAD}%
\machSeite{KupeAH}\cite{KupeAD,KupeAH}) that alternating sign matrices
are in bijection with
{\it configurations in the six vertex model}, and due to
determinant and Pfaffian formulae due to Izergin
\machSeite{IzerAA}\cite{IzerAA} and Kuperberg
\machSeite{KupeAH}\cite{KupeAH} for certain multivariable
partition functions of the six vertex model under various
boundary conditions. In many cases,
this leads to determinants which are, or are similar to,
{\it Cauchy's evaluation of the double alternant\/} (see
\machSeite{MuirAB}%
\cite[vol.~III, p.~311]{MuirAB} and \eqref{eq:Cauchy} below)
or {\it Schur's Pfaffian version}
\machSeite{SchuAA}\cite[pp.~226/227]{SchuAA}
of it (see \eqref{eq:Schur} below).
Let me recall that
the {\it Pfaffian} $\operatorname{Pf}(A)$ of a
skew-symmetric $(2n)\times(2n)$ matrix $A$ is defined by
\begin{equation} \label{eq:Pfaff}
\operatorname{Pf}(A)=\sum _{\pi} ^{}(-1)^{c(\pi)}\prod _{(ij)\in \pi} ^{}A_{ij},
\end{equation}
where the sum is over all perfect matchings $\pi$ of the complete
graph on $2n$ vertices, where $c(\pi)$ is the {\em crossing
number} of $\pi$, and where the product is over all edges $(ij)$,
$i<j$, in the matching $\pi$ (see e.g.\
\machSeite{StemAE}\cite[Sec.~2]{StemAE}).
What links Pfaffians
so closely to determinants is (aside from similarity of definitions)
the fact that the Pfaffian of a skew-symmetric matrix is, up to sign,
the square root of its determinant. That is,
$\det(A)=\operatorname{Pf}(A)^2$ for any skew-symmetric $(2n)\times(2n)$ matrix $A$
(cf.\
\machSeite{StemAE}\cite[Prop.~2.2]{StemAE}). See the corresponding remarks and
additional references in
\machSeite{KratBN}\cite[Sec.~2.8]{KratBN}.
The following three theorems present the relevant evaluations.
They are Theorems~15--17 from
\machSeite{KupeAH}\cite{KupeAH}. All of them are proved using
identification of factors (see ``Method~3" in Section~\ref{sec:eval}).
The results in Theorem~\ref{thm:Kup1}
contain whole sets of indeterminates, whereas the results in
Theorems~\ref{thm:Kup2} and \ref{thm:Kup3}
only have two indeterminates $p$ and $q$, respectively three
indeterminates $p$, $q$ and $r$, in them.
Identity \eqref{eq:Kup4} is originally due to
Laksov, Lascoux and Thorup
\machSeite{LaLTAA}\cite{LaLTAA} and Stembridge
\machSeite{StemAE}\cite{StemAE}, independently.
The reader must be warned that the statements in
\machSeite{KupeAH}\cite[Theorems~15--17]{KupeAH} are often blurred
by typos.
\begin{Theorem} \label{thm:Kup1}
Let $x_1,x_2,\dots$ and $y_1,y_2,\dots$ be indeterminates. Then, for
any positive integer $n$, there hold
\begin{equation}
\label{eq:Cauchy}
\det_{1\leq i,j\leq n}\left(
\frac{1}{x_i+y_j}
\right) =
\frac{\displaystyle
\prod_{1\leq i<j\leq n}(x_i-x_j)(y_i-y_j)
}{\displaystyle
\prod_{1\leq i, j\leq n}(x_i+y_j)
},
\end{equation}
\begin{multline}
\label{eq:Kup2}
\det_{1\le i,j\le n}\(\frac1{x_i+y_j} - \frac1{1+x_iy_j}\)
= \frac{\displaystyle\prod_{1\le i<j\le n}
(1-x_ix_j)(1-y_iy_j)(x_j-x_i)(y_j-y_i)}
{\displaystyle\prod_{1\le i,j\le n} (x_i+y_j)(1+x_iy_j)}\\
\times \prod_{i=1}^n (1-x_i)(1-y_i) ,
\end{multline}
\begin{equation} \label{eq:Schur}
\underset{1\le i,j\le 2n}\operatorname{Pf}\(\frac {x_i-x_j} {x_i+x_j}\)=
\prod _{1\le i<j\le 2n} ^{}\frac {x_i-x_j} {x_i+x_j}.
\end{equation}
\begin{equation}
\label{eq:Kup4}
\underset{1\le i,j\le 2n}\operatorname{Pf}\(\frac{x_i-x_j}{1-x_ix_j}\)
= \prod_{1\le i<j\le 2n} \frac{x_i-x_j}{1-x_ix_j}.
\end{equation}
\quad \quad \qed
\end{Theorem}
\begin{Theorem} \label{thm:Kup2}
Let $p$ and $q$ be indeterminates. Then, for
any positive integer $n$, there hold
\begin{equation} \label{eq:Kup5}
\det_{1\le i,j\le n}\(\frac{q^{n+j-i}-q^{-(n+j-i)}}
{p^{n+j-i}-p^{-(n+j-i)}}\)
= \frac{\displaystyle\prod_{1\le i \ne j\le n} (p^{j-i}-p^{-(j-i)})
\prod_{1\le i,j\le n} (qp^{j-i}-q^{-1}p^{-(j-i)})}
{\displaystyle\prod_{1\le i,j\le n} (p^{n+j-i}-p^{-(n+j-i)})},
\end{equation}
\begin{equation} \label{eq:Kup6}
\det_{1\le i,j\le n}\(\frac{q^{j-i}+q^{-(j-i)}}{p^{j-i}+p^{-(j-i)}}\)
= (-1)^{\binom{n}2}
\frac{\displaystyle
2^n\prod_{\substack{1\le i \ne j\le n \\ 2\mid j-i}}(p^{j-i}-p^{-j+i})
\prod_{\substack{1\le i,j\le n \\ 2\nmid j-i}}(qp^{j-i}-q^{-1}p^{-j+i})}
{\displaystyle\prod_{1\le i,j\le n} (p^{j-i}+p^{-j+i})} ,
\end{equation}
\begin{multline} \label{eq:Kup7}
\det_{1\le i,i\le n}\(\frac{q^{n+j+i}-q^{-(n+j+i)}}{p^{n+j+i}-p^{-(n+j+i)}} -
\frac{q^{n+j-i}-q^{-(n+j-i)}}{p^{n+j-i}-p^{-(n+j-i)}}\) \\=
\frac{\prod_{1\le i<j\le 2n}(p^{j-i}-p^{-(j-i)})
\prod_{\displaystyle\substack{1\le i,j \le 2n+1 \\ 2|j}}
(qp^{j-i}-q^{-1}p^{-(j-i)})}
{\displaystyle\prod_{1\le i,j\le n} (p^{n+j-i}-p^{-(n+j-i)})
(p^{n+j+i}-p^{-(n+j+i)})},
\end{multline}
\begin{multline} \label{eq:Kup8}
\det_{1\le i,j\le n}\(\frac{q^{j+i}+q^{-(j+i)}}{p^{j+i}+p^{-(j+i)}} -
\frac{\displaystyle q^{j-i}+q^{-j+i}}{p^{j-i}+p^{-(j-i)}}\)\\
=(-1)^{\binom n2}
\frac{\displaystyle 2^n\prod_{1\le i<j \le n} (p^{2(j-i)}-p^{-2(j-i)})^2
\prod_{\substack{1\le i,j \le 2n+1 \\ 2\nmid i,\,2|j}} (qp^{j-i}-q^{-1}p^{-(j-i)})}
{\displaystyle\prod_{1\le i,j\le n} (p^{j-i}+p^{-(j-i)})(p^{j+i}+p^{-(j+i)})}.
\end{multline}
\quad \quad \qed
\end{Theorem}
\begin{Theorem} \label{thm:Kup3}
Let $p$, $q$, and $r$ be indeterminates. Then, for
any positive integer $n$, there hold
\begin{multline} \label{eq:Kup9}
\underset{1\le i,j\le 2n}\operatorname{Pf} \(\frac{(q^{j-i}-q^{-(j-i)})(r^{j-i}-r^{-(j-i)})}
{(p^{j-i}-p^{-(j-i)})}\) \\
= \frac{\displaystyle\prod_{1\le i<j\le n} (p^{j-i}-p^{-(j-i)})^2
\prod_{1\le i,j\le n}
(qp^{j-i}-q^{-1}p^{-(j-i)})(rp^{j-i}-r^{-1}p^{-(j-i)})}
{\displaystyle\prod_{1\le i,j \le n} (p^{n+j-i}-p^{-(n+j-i)})}
\end{multline}
\begin{multline} \label{eq:Kup10}
\underset{1\le i,j\le 2n}\operatorname{Pf} \Bigg((p^{j+i}-p^{-(j+i)})(p^{j-i}-p^{-(j-i)})
\biggl(\frac{q^{j+i}-q^{-(j+i)}}{p^{j+i}-p^{-(j+i)}} -
\frac{q^{j-i}-q^{-(j-i)}}{p^{j-i}-p^{-(j-i)}}\biggr)\\
\cdot
\biggl(\frac{r^{j+i}-r^{-(j+i)}}
{p^{j+i}-p^{-(j+i)}} -
\frac{r^{j-i}-r^{-(j-i)}}{p^{j-i}-p^{-(j-i)}}\biggr)\Bigg) \\
= \frac{\displaystyle\prod_{1\le i<j \le 2n}(p^{j-i}-p^{-(j-i)})
\prod_{\substack{1\le i,j \le 2n+1 \\ 2|j}}
(qp^{j-i}-q^{-1}p^{-(j-i)})(rp^{j-i}-r^{-1}p^{-(j-i)})}
{\displaystyle\prod_{1\le i<j \le 2n}(p^{j+i}-p^{-(j+i)})}.
\end{multline}
\quad \quad \qed
\end{Theorem}
Subsequent to Kuperberg's work,
Okada \machSeite{OkadAJ}\cite{OkadAJ}
related Kuperberg's determinants and Pfaffians to characters of
classical groups, by coming up with rather complex, but still
beautiful determinant identities. In particular, this allowed him to
settle one more of the conjectured enumeration formulae on symmetry
classes of alternating sign matrices.
Generalising even further, Ishikawa, Okada, Tagawa and
Zeng
\machSeite{IsOTAA}\cite{IsOTAA} have found more such determinant
identities. Putting them into the framework of certain special
representations of the symmetric group, Lascoux
\machSeite{LascAT}\cite{LascAT} has clarified the mechanism which
gives rise to these identities.
\medskip
The next six determinant lemmas are corollaries of
{\it elliptic determinant evaluations}
due to Rosengren and Schlosser
\machSeite{RoScAC}\cite{RoScAC}.
(The latter will be addressed later in
Subsection~\ref{sec:ell}.) They partly extend the fundamental
determinant lemmas in
\machSeite{KratBN}\cite[Sec.~2.2]{KratBN}.
For the statements of the lemmas, we need the notion
of a {\it norm} of a polynomial
$a_0+a_1z+\dots+a_kz^k$, which we define to be the reciprocal of the
product of its roots, or, more explicitly,
as $(-1)^ka_k/a_0$.
If we specialise $p=0$ in Lemma~\ref{wp}, \eqref{awpi}, then we obtain a
determinant identity which generalises at the same time
the Vandermonde determinant evaluation, Lemma~\ref{lem:AmZe} and
Lemma~\ref{lem:AmZe2}.
\begin{Lemma} \label{lem:Van1}
Let $P_1,P_2,\dots,P_n$ be polynomials of degree $n$ and norm
$t$, given by
$$P_j(x)=(-1)^nta_{j,0}x^n+
\sum _{k=0} ^{n-1}a_{j,k}x^k.$$
Then
\begin{equation} \label{eq:Van1}
\det_{1\leq i,j\leq n}\left(P_j(x_i)\right)=(1-tx_1\dotsm x_n)
\bigg(\prod_{1\leq i<j\leq n}(x_j-x_i)\bigg)
\underset{0\le j\le n-1}{\det_{1\le i\le n}}(a_{i,j}).
\end{equation}
\quad \quad \qed
\end{Lemma}
Further determinant identities which generalise other {\it Weyl
denominator formulae} (cf.\
\machSeite{KratBN}\cite[Lemma~2]{KratBN}) could be obtained from
the special case $p=0$ of the other determinant evaluations in
Lemma~\ref{wp}.
A generalisation of Lemma~6 from
\machSeite{KratBN}\cite{KratBN} in the same spirit
can be obtained by setting $p=0$ in
Theorem~\ref{adet}. It is given as Corollary~5.1 in
\machSeite{RoScAC}\cite{RoScAC}.
\begin{Lemma} \label{lem:RS1}
Let $x_1,\dots,x_n$, $a_1,\dots,a_n$, and $t$ be indeterminates.
For each $j=1,\dots,n$, let $P_j$ be a polynomial
of degree $j$ and norm $ta_1\dotsm a_j$. Then there holds
\begin{multline} \label{eq:RS1}
\det_{1\le i,j\le n}\left(P_j(x_i)
\prod_{k=j+1}^n(1-a_kx_i)\right)\\
=\frac{1-ta_1\dotsm a_nx_1\dotsm x_n}{1-t}
\prod_{i=1}^nP_i(1/a_i)
\prod_{1\le i<j\le n}a_j(x_j-x_i).
\end{multline}
\quad \quad \qed
\end{Lemma}
We continue with a consequence of Theorem~\ref{adetcor} (see Corollary~5.3
in \machSeite{RoScAC}\cite{RoScAC}).
The special case $P_{j-1}(x)=1$, $j=1,\dots,n$,
is Lemma~A.1 of
\machSeite{SchlAB}\cite{SchlAB}, which was needed in order to obtain an
{\it $A_n$ matrix inversion} that played a crucial role in the derivation
of {\it multiple basic hypergeometric series identities}.
A slight generalisation was given in
\machSeite{SchlAF}\cite[Lemma~A.1]{SchlAF}.
\begin{Lemma}\label{adetcorr}
Let $x_1,\dots,x_n$ and $b$ be indeterminates.
For each $j=1,\dots,n$, let $P_{j-1}(x)$ be a polynomial in
$x$ of degree at most $j-1$ with constant term $1$,
and let $Q(x)=(1-y_1x)\dotsm (1-y_{n+1}x)$.
Then there holds
\begin{multline}\label{adetcorrid}
Q(b)\;\det_{1\le i,j\le n}\left(x_i^{n+1-j}P_{j-1}(x_i)
-b^{n+1-j}P_{j-1}(b)\frac{Q(x_i)}{Q(b)}\right)\\
=(1-bx_1\cdots x_ny_1\dotsm y_{n+1})
\prod_{i=1}^n(x_i-b)\prod_{1\le i<j\le n}(x_i-x_j).
\end{multline}
\quad \quad \qed
\end{Lemma}
Pairing the $(i,j)$-entry in the determinant in \eqref{eq:RS1} with
itself, but with $x_i$ replaced by $1/x_i$, one can construct another
determinant which evaluates in closed form. The result given below is
Corollary~5.5 in
\machSeite{RoScAC}\cite{RoScAC}. It is the special case
$p=0$ of Theorem~\ref{cdet}.
\begin{Lemma}\label{cdetr}
Let $x_1,\dots,x_n$, $a_1,\dots,a_n$, and $c_1,\dots,c_{n+2}$ be
indeterminates. For each $j=1,\dots,n$, let $P_j$ be a polynomial
of degree $j$ with norm
$(c_1\dotsm c_{n+2}a_{j+1}\dotsm a_n)^{-1}$.
Then there holds
\begin{multline}\label{cdetrid}
\det_{1\leq i,j\leq n}\left(x_i^{-n-1}
\prod_{k=1}^{n+2}(1-c_kx_i)\,
P_j(x_i)\prod_{k=j+1}^n(1-a_kx_i)\right.\\
\left.-x_i^{n+1}\prod_{k=1}^{n+2}(1-c_kx_i^{-1})\,
P_j(x_i^{-1})\prod_{k=j+1}^n(1-a_kx_i^{-1})\right)\\
=\frac{a_1\dotsm a_n}
{x_1\dotsm x_n\,(1-c_1\dotsm c_{n+2}a_1\dotsm a_n)}
\prod_{i=1}^nP_i(1/a_i)\\\times
\prod_{1\leq i<j\leq n+2}(1-c_ic_j)\prod_{i=1}^n(1-x_i^2)
\prod_{1\le i<j\le n}a_j(x_i-x_j)(1-1/x_ix_j).
\end{multline}
\quad \quad \qed
\end{Lemma}
It is worthwhile to state the limit case $c_{n+2}\to\infty$ of this
lemma separately, in which case the norm constraint on the polynomials
$P_j$ drops out, but, in return, the degree of $P_j$ gets lowered by
one (see
\machSeite{RoScAC}\cite[Cor.~5.8]{RoScAC}).
\begin{Lemma}\label{cdetr1}
Let $x_1,\dots,x_n$, $a_2,\dots,a_n$, and $c_1,\dots,c_{n+1}$ be
indeterminates. For each $j=1,\dots,n$, let $P_{j-1}$ be a polynomial
of degree at most $j-1$. Then there holds
\begin{multline}\label{cdetr1id}
\det_{1\leq i,j\leq n}\left(x_i^{-n}
\prod_{k=1}^{n+1}(1-c_kx_i)\,
P_{j-1}(x_i)\prod_{k=j+1}^n(1-a_kx_i)\right.\\
\left.-x_i^n\prod_{k=1}^{n+1}(1-c_kx_i^{-1})\,
P_{j-1}(x_i^{-1})\prod_{k=j+1}^n(1-a_kx_i^{-1})\right)\\
=\prod_{i=1}^nP_{i-1}(1/a_i)\prod_{1\leq i<j\leq n+1}(1-c_ic_j)\\\times
\prod_{i=1}^nx_i^{-1}(1-x_i^2)
\prod_{1\le i<j\le n}a_j(x_i-x_j)(1-1/x_ix_j).
\end{multline}
\quad \quad \qed
\end{Lemma}
Dividing both sides of \eqref{cdetr1id} by $\prod_{i=2}^n a_i^{i-1}$
and then letting $a_i$ tend to $\infty$, $i=2,3,\dots,n$,
we arrive at the determinant evaluation below (see
\machSeite{RoScAC}\cite[Cor.~5.11]{RoScAC}).
Its special case $P_{j-1}(x)=1$, $j=1,\dots,n$,
is Lemma~A.11 of
\machSeite{SchlAB}\cite{SchlAB}, needed there in order to obtain a
{\it $C_n$ matrix
inversion}, which was later applied in
\machSeite{SchlAG}\cite{SchlAG} to derive
{\it multiple q-Abel and q-Rothe summations}.
\begin{Lemma}\label{cdetr1cor}
Let $x_1,\dots,x_n$, and $c_1,\dots,c_{n+1}$ be
indeterminates. For each $j=1,\dots,n$, let $P_{j-1}$ be a polynomial
of degree at most $j-1$. Then there holds
\begin{multline}\label{cdetr1corid}
\det_{1\leq i,j\leq n}\left(x_i^{-j}
\prod_{k=1}^{n+1}(1-c_kx_i)\,P_{j-1}(x_i)
-x_i^j\prod_{k=1}^{n+1}(1-c_kx_i^{-1})\,P_{j-1}(x_i^{-1})\right)\\
=\prod_{i=1}^n P_{i-1}(0)
\prod_{1\leq i<j\leq n+1}(1-c_ic_j)
\prod_{i=1}^nx_i^{-1}(1-x_i^2)
\prod_{1\le i<j\le n}(x_j-x_i)(1-1/x_ix_j).
\end{multline}
\quad \quad \qed
\end{Lemma}
It is an attractive feature of this determinant identity that it
contains, at the same time, the {\it Weyl denominator formulae} for the
classical root systems $B_n$, $C_n$ and $D_n$ as special cases (cf.\
\machSeite{KratBN}\cite[Lemma~2]{KratBN}). This is seen by setting
$P_j(x)=1$ for all $j$, $c_1=c_2=\dots=c_{n-1}=0$,
and then $c_n=0$, $c_{n+1}=-1$ for the type $B_n$ case,
$c_n=c_{n+1}=0$ for the type $C_n$ case,
and $c_n=1$, $c_{n+1}=-1$ for the type $D_n$ case, respectively.
\medskip
A determinant which is of completely different type, but
which also belongs to the category of basic determinant evaluations, is the
determinant of a matrix where only two (circular) diagonals are filled
with non-zero elements. It was applied with advantage in
\machSeite{HaKrAA}\cite{HaKrAA}
to evaluate {\it Scott-type permanents}.
\begin{Lemma} \label{prop:2diag}
Let $n$ and $r$ be positive integers, $r\le n$, and
$x_1,x_2,\dots,x_n$, $y_1,y_2,\dots,y_n$ be indeterminates. Then, with
$d=\gcd(r,n)$, we have
\begin{multline}
\det\begin{pmatrix}
x_1&0&\dots&0&y_{n-r+1}&0&\\
0&x_2&0&&0&y_{n-r+2}&0\\
&&\ddots&&&&\ddots&0\\
0&&&&&&0&y_n\\
y_1&0&\\
0&y_2&0\\
&0&\ddots&0&&&\ddots&0\\
&&0&y_{n-r}&0&&0&x_n\end{pmatrix}\\
\hskip3.6cm=\prod _{i=1} ^{d}\bigg(\prod _{j=1} ^{n/d}x_{i+(j-1)d}-
(-1)^{n/d}\prod _{j=1} ^{n/d}y_{i+(j-1)d}\bigg).
\end{multline}
{\em(}I.e., in the matrix there are only nonzero entries along two
diagonals, one of which is a broken diagonal.{\em)}
\quad \quad \qed
\end{Lemma}
A further basic determinant evaluation which I missed to state in
\machSeite{KratBN}\cite{KratBN} is the evaluation of the
determinant of a {\it skew circulant matrix} attributed to Scott
\machSeite{ScotAB}\cite{ScotAB} in
\machSeite{MuirAB}\cite[p.~356]{MuirAB}. It was in fact
recently used by Fulmek in
\machSeite{FulmAF}\cite{FulmAF}
to find a closed form formula for the number of {\it non-intersecting
lattice paths with equally spaced starting and end points living
on a cylinder}, improving on earlier results by Forrester
\machSeite{ForrAC}\cite{ForrAC} on the {\it vicious walker model\/}
in {\it statistical mechanics}, see
\machSeite{FulmAF}\cite[Lemma~9]{FulmAF}.
\begin{Theorem} \label{thm:circulant1}
Let $n$ by a fixed positive integer, and let
$a_0,a_1,\dots,a_{n-1}$ be indeterminates. Then
\begin{multline} \label{eq:circulant1}
\det\begin{pmatrix} a_0&a_1&a_2&\dots&a_{n-2}&a_{n-1}\\
-a_{n-1}&a_0&a_1&\dots&a_{n-3}&a_{n-2}\\
-a_{n-2}&-a_{n-1}&a_0&\dots&a_{n-4}&a_{n-3}\\
\hdotsfor6\\
-a_{1}&-a_2&-a_3&\dots&-a_{n-1}&a_{0}
\end{pmatrix}\\=\prod _{i=0}
^{n-1}(a_0+\omega^{2i+1}a_1+\omega^{2(2i+1)}a_2+\dots+\omega^{(n-1)(2i+1)}a_{n-1}),
\end{multline}
where $\omega$ is a primitive $(2n)$-th root of unity.\quad \quad \qed
\end{Theorem}
\subsection{More confluent determinants}
Here I continue the discussion from the beginning of Section~3 in
\machSeite{KratBN}\cite[Theorems~20--24]{KratBN}. There I presented
determinant evaluations of matrices which, essentially,
consist of several vertical strips, each of which is formed by
taking a certain column vector and gluing it together with its
derivative, its second derivative, etc., respectively by a similar
construction where the derivative is replaced by a difference or
$q$-difference operator.
Since most of this subsection will be under the influence of the
so-called {\it ``$q$-disease'',}\footnote{\label{foot:q}The
distinctive symptom of this
disease is to invariably raise the question ``Is there also a
$q$-analogue?" My epidemiological research on {\textsf MathSciNet} revealed
that, while basically non-existent during the 1970s, this disease slowly
spread during the 1980s, and then had a sharp increase around 1990,
when it jumped from about 20 papers per year published with the word
``$q$-analog$*$" in it to over 80 in 1995, and since then it has been
roughly stable at 60--70 papers per year. In its
simplest form, somebody who is infected by this disease takes a
combinatorial identity, and replaces every occurrence of a positive
integer $n$
by its {\it ``$q$-analogue"} $1+q+q^2+\dots+q^{n-1}$, inserts some powers of
$q$ at the appropriate places, and hopes that the result of these
manipulations would be again
an identity, thus constituting a ``$q$-analogue" of the original
equation. I refer the reader to the bible
\machSeite{GaRaAA}\cite{GaRaAA} for a rich
source of $q$-identities, and for the right way to look at (most)
combinatorial $q$-identities. In another form, given a certain set of
objects of which one knows the exact number, one defines a {\it statistics}
stat on these objects and now tries to evaluate
$\sum _{O\text{ an object}} ^{}q^{\operatorname{stat}(O)}$. For a very instructive
text following these lines see
\machSeite{FoHaAL}\cite{FoHaAL}, with emphasis on the
objects being permutations. There is also an important third form of
the disease in which one works in the ring of polynomials in variables
$x,y,\dots$ with coefficients being rational functions in $q$,
and in which some pairs of variables satisfy commutation
relations of the type $xy=qxy$. The study of such polynomial rings
and algebras is often motivated by {\it quantum groups} and {\it
quantum algebras}.
The reader may want to consult
\machSeite{KoorAG}\cite{KoorAG} to learn more about this
direction.
While my description did not make this clear,
the three described forms of the $q$-disease are indeed strongly
inter-related.}
we shall need the standard $q$-notations
$(a;q)_k$, denoting the {\em $q$-shifted factorial\/} and being given by
$(a;q)_0:=1$ and
$$(a;q)_k:=(1-a)(1-aq)\cdots(1-aq^{k-1})$$
if $k$ is a positive integer, as well as
$\left[\begin{smallmatrix}\alpha\\k\end{smallmatrix}\right]_q$, denoting
the {\em $q$-binomial coefficient\/} and being defined by
$\left[\begin{smallmatrix}\alpha\\k\end{smallmatrix}\right]_q=0$ if $k<0$,
$\left[\begin{smallmatrix}\alpha\\0\end{smallmatrix}\right]_q=1$, and
$$\begin{bmatrix} \alpha\\k\end{bmatrix}_q:=
\frac {(1-q^\alpha)(1-q^{\alpha-1})\cdots(1-q^{\alpha-k+1})}
{(1-q^k)(1-q^{k-1})\cdots(1-q)}$$
if $k$ is a positive integer.
Clearly we have $\lim_{q\to1}\[\smallmatrix \alpha\\k\endsmallmatrix\]_q=
\binom \alpha k$.
The first result that I present is a $q$-extension of the evaluation
of the {\it
confluent alternant\/} due to Schendel
\machSeite{ScheAA}\cite{ScheAA} (cf.\
\machSeite{KratBN}\cite[paragraph before Theorem~20]{KratBN}). In fact,
Theorem~23 of \machSeite{KratBN}\cite{KratBN} already provided a
$q$-extension of (a generalisation of) Schendel's formula.
However, in
\machSeite{JohWAF}\cite[Theorem~1]{JohWAF}, Johnson found a
different $q$-extension. The theorem below is a slight generalisation
of it. (The theorem below reduces to Johnson's theorem if one puts
$C=0$. For $q=1$, the theorem below and
\machSeite{KratBN}\cite[Theorem~23]{KratBN} become equivalent. To go
from one determinant to the other in this special case, one would have
to take a certain factor out of each column.)
\begin{Theorem} \label{thm:Johnson1}
Let $n$ be a non-negative integer, and
let $A_m(X)$ denote the $n\times m$ matrix
$$\(\begin{bmatrix} C+i\\i-j\end{bmatrix}_q
(X;q)_{i-j}\)_{0\le i\le n-1,\,0\le j\le m-1}.$$
Given a composition of $n$, $n=m_1+\dots+m_\ell$, there holds
\begin{multline} \label{eq:Johnson1}
\det_{n\times n}\big(A_{m_1}(X_1)\,A_{m_2}(X_2)\dots
A_{m_\ell}(X_\ell)\big)\\=
q^{\sum_{1\le i<j<k\le \ell}m_im_jm_k}
\prod _{1\le i<j\le \ell} ^{}
\prod _{g=1} ^{m_i}
\prod _{h=1} ^{m_j}
\dfrac{(q^{h-1}X_i-q^{g-1}X_j)
(1-q^{C+g+h-1+\sum_{r=1}^{i-1}m_r)}}
{(1-q^{g+h-1+\sum_{r=1}^{i-1}m_r)}}.
\end{multline}
\quad \quad \qed
\end{Theorem}
In
\machSeite{JohWAF}\cite[Theorem~2]{JohWAF}, Johnson provides as well a
confluent $q$-extension of the evaluation of Cauchy's double
alternant \eqref{eq:Cauchy}.
Already the case $q=1$ seems to not have appeared in the literature earlier.
Here, I was not able to introduce an additional parameter (as, for
example, the $C$ in Theorem~\ref{thm:Johnson1}).
\begin{Theorem} \label{thm:Johnson2}
Let $n$ be a non-negative integer, and
let $A_m(X)$ denote the $n\times m$ matrix
$$\(\frac {1} {(Y_i-X)(Y_i-qX)(Y_i-q^2X)\cdots(Y_i-q^{j-1}X)}\)
_{1\le i\le n,\,1\le j\le m}.$$
Given a composition of $n$, $n=m_1+\dots+m_\ell$, there holds
\begin{multline} \label{eq:Johnson2}
\det_{n\times n}\big(A_{m_1}(X_1)\,A_{m_2}(X_2)\dots
A_{m_\ell}(X_\ell)\big)\\=
\frac{\displaystyle
\(\prod _{1\le i<j\le n} ^{}(Y_i-Y_j)\)
\(\prod _{1\le i<j\le \ell} ^{}
\prod _{g=1} ^{m_i}
\prod _{h=1} ^{m_j}
(q^{h-1}X_j-q^{g-1}X_i)
\)}
{\displaystyle
\prod _{i=1} ^{n}
\prod _{j=1} ^{\ell}(Y_i-X_j)(Y_i-qX_j)(Y_i-q^2X_j)\cdots(Y_i-q^{m_j-1}X_j)}.
\end{multline}
\quad \quad \qed
\end{Theorem}
A surprising mixture between the confluent alternant and a confluent
double alternant appears in
\machSeite{CiucAL}\cite[Theorem~A.1]{CiucAL}. Ciucu used it there in
order to prove a {\it Coulomb gas law} and a {\it superposition
principle} for the joint correlation of certain
collections of holes for the {\it rhombus tiling model on the triangular
lattice}. (His main result is in fact based on an even more general, and
more complex, determinant evaluation, see
\machSeite{CiucAL}\cite[Theorem~8.1]{CiucAL}.)
\begin{Theorem} \label{thm:Ciucu}
Let $s_1,s_2, \dots, s_m\geq1$ and
$t_1, t_2,\dots, t_n\geq1$
be integers.
Write $S=\sum_{i=1}^m s_i$, $T=\sum_{j=1}^n t_j$, and assume $S\geq T$.
Let $x_1,x_2,\dots,x_m$ and $y_1,y_2,\dots,y_n$ be indeterminates. Define $N$ to be the $S\times S$ matrix
\begin{equation} \label{eq:Ciuc}
N=\left[\begin{matrix} A&B \end{matrix}\right]
\end{equation}
whose blocks are given by
\begin{multline}
A=
\\
\left(\!
\begin{matrix}
{\scriptscriptstyle \frac{{\binom 0 0}}{y_1-x_1}}\!\!\!\!&
{\scriptscriptstyle \frac{{\binom 1 0}}{(y_1-x_1)^2}}\!\!\!\!&{\scriptscriptstyle \cdots}\!\!\!\!&
{\scriptscriptstyle \frac{{\binom {t_1-1} 0}}{(y_1-x_1)^{t_1}}}&
\ &
{\scriptscriptstyle \frac{{\binom 0 0}}{y_n-x_1}}\!\!\!\!&
{\scriptscriptstyle \frac{{\binom 1 0}}{(y_n-x_1)^2}}\!\!\!\!&{\scriptscriptstyle \cdots}\!\!\!\!&
{\scriptscriptstyle \frac{{\binom {t_n-1} 0}}{(y_n-x_1)^{t_n}}}
\\
{\scriptscriptstyle \frac{{\binom 1 1}}{(y_1-x_1)^2}}\!\!\!\!&
{\scriptscriptstyle \frac{{\binom 2 1}}{(y_1-x_1)^3}}\!\!\!\!&{\scriptscriptstyle \cdots}\!\!\!\!&
{\scriptscriptstyle \frac{{\binom {t_1} 1}}{(y_1-x_1)^{t_1+1}}}&
\ &
{\scriptscriptstyle \frac{{\binom 1 1}}{(y_n-x_1)^2}}\!\!\!\!&
{\scriptscriptstyle \frac{{\binom 2 1}}{(y_n-x_1)^3}}\!\!\!\!&{\scriptscriptstyle \cdots}\!\!\!\!&
{\scriptscriptstyle \frac{{\binom {t_n} 1}}{(y_n-x_1)^{t_n+1}}}
\\
{\scriptstyle\cdot}\!\!\!\!&
{\scriptstyle\cdot}\!\!\!\!&\ \!\!\!\!&
{\scriptstyle\cdot}&
\!\!\!\!\cdots\!\!\! &
{\scriptstyle\cdot}\!\!\!\!&
{\scriptstyle\cdot}\!\!\!\!&\ \!\!\!\!&
{\scriptstyle\cdot}
\\
{\scriptstyle\cdot}\!\!\!\!&
{\scriptstyle\cdot}\!\!\!\!&\ \!\!\!\!&
{\scriptstyle\cdot}&
\ &
{\scriptstyle\cdot}\!\!\!\!&
{\scriptstyle\cdot}\!\!\!\!&\ \!\!\!\!&
{\scriptstyle\cdot}
\\
{\scriptstyle\cdot}\!\!\!\!&
{\scriptstyle\cdot}\!\!\!\!&\ \!\!\!\!&
{\scriptstyle\cdot}&
\ &
{\scriptstyle\cdot}\!\!\!\!&
{\scriptstyle\cdot}\!\!\!\!&\ \!\!\!\!&
{\scriptstyle\cdot}
\\
{\scriptscriptstyle \frac{{\binom {s_1-1} {s_1-1}}}{(y_1-x_1)^{s_1}}}\!\!\!\!&
{\scriptscriptstyle \frac{{\binom {s_1} {s_1-1}}}{(y_1-x_1)^{s_1+1}}}\!\!\!\!&{\scriptscriptstyle \cdots}\!\!\!\!&
{\scriptscriptstyle \frac{{\binom {s_1+t_1-2} {s_1-1}}}{(y_1-x_1)^{s_1+t_1-1}}}&
\ &
{\scriptscriptstyle \frac{{\binom {s_1-1} {s_1-1}}}{(y_n-x_1)^{s_1}}}\!\!\!\!&
{\scriptscriptstyle \frac{{\binom {s_1} {s_1-1}}}{(y_n-x_1)^{s_1+1}}}\!\!\!\!&{\scriptscriptstyle \cdots}\!\!\!\!&
{\scriptscriptstyle \frac{{\binom {s_1+t_n-2} {s_1-1}}}{(y_n-x_1)^{s_1+t_n-1}}}
\\
\ \!\!\!\!&
\ \!\!\!\!&\cdot\!\!\!\!&
\ &
\ &
\ \!\!\!\!&
\ \!\!\!\!&\cdot\!\!\!\!&
\
\\
\ \!\!\!\!&
\ \!\!\!\!&\cdot\!\!\!\!&
\ &
\ &
\ \!\!\!\!&
\ \!\!\!\!&\cdot\!\!\!\!&
\
\\
\ \!\!\!\!&
\ \!\!\!\!&\cdot\!\!\!\!&
\ &
\ &
\ \!\!\!\!&
\ \!\!\!\!&\cdot\!\!\!\!&
\
\\
{\scriptscriptstyle \frac{{\binom 0 0}}{y_1-x_m}}\!\!\!\!&
{\scriptscriptstyle \frac{{\binom 1 0}}{(y_1-x_m)^2}}\!\!\!\!&{\scriptscriptstyle \cdots}\!\!\!\!&
{\scriptscriptstyle \frac{{\binom {t_1-1} 0}}{(y_1-x_m)^{t_1}}}&
\ &
{\scriptscriptstyle \frac{{\binom 0 0}}{y_n-x_m}}\!\!\!\!&
{\scriptscriptstyle \frac{{\binom 1 0}}{(y_n-x_m)^2}}\!\!\!\!&{\scriptscriptstyle \cdots}\!\!\!\!&
{\scriptscriptstyle \frac{{\binom {t_n-1} 0}}{(y_n-x_m)^{t_n}}}
\\
{\scriptscriptstyle \frac{{\binom 1 1}}{(y_1-x_m)^2}}\!\!\!\!&
{\scriptscriptstyle \frac{{\binom 2 1}}{(y_1-x_m)^3}}\!\!\!\!&{\scriptscriptstyle \cdots}\!\!\!\!&
{\scriptscriptstyle \frac{{\binom {t_1} 1}}{(y_1-x_m)^{t_1+1}}}&
\ &
{\scriptscriptstyle \frac{{\binom 1 1}}{(y_n-x_m)^2}}\!\!\!\!&
{\scriptscriptstyle \frac{{\binom 2 1}}{(y_n-x_m)^3}}\!\!\!\!&{\scriptscriptstyle \cdots}\!\!\!\!&
{\scriptscriptstyle \frac{{\binom {t_n} 1}}{(y_n-x_m)^{t_n+1}}}
\\
{\scriptstyle\cdot}\!\!\!\!&
{\scriptstyle\cdot}\!\!\!\!&\ \!\!\!\!&
{\scriptstyle\cdot}&
\!\!\!\!\cdots\!\!\! &
{\scriptstyle\cdot}\!\!\!\!&
{\scriptstyle\cdot}\!\!\!\!&\ \!\!\!\!&
{\scriptstyle\cdot}
\\
{\scriptstyle\cdot}\!\!\!\!&
{\scriptstyle\cdot}\!\!\!\!&\ \!\!\!\!&
{\scriptstyle\cdot}&
\ &
{\scriptstyle\cdot}\!\!\!\!&
{\scriptstyle\cdot}\!\!\!\!&\ \!\!\!\!&
{\scriptstyle\cdot}
\\
{\scriptstyle\cdot}\!\!\!\!&
{\scriptstyle\cdot}\!\!\!\!&\ \!\!\!\!&
{\scriptstyle\cdot}&
\ &
{\scriptstyle\cdot}\!\!\!\!&
{\scriptstyle\cdot}\!\!\!\!&\ \!\!\!\!&
{\scriptstyle\cdot}
\\
{\scriptscriptstyle \frac{{\binom {s_m-1} {s_m-1}}}{(y_1-x_m)^{s_m}}}\!\!\!\!&
{\scriptscriptstyle \frac{{\binom {s_m} {s_m-1}}}{(y_1-x_m)^{s_m+1}}}\!\!\!\!&{\scriptscriptstyle \cdots}\!\!\!\!&
{\scriptscriptstyle \frac{{\binom {s_m+t_1-2} {s_m-1}}}{(y_1-x_m)^{s_m+t_1-1}}}&
\ &
{\scriptscriptstyle \frac{{\binom {s_m-1} {s_m-1}}}{(y_n-x_m)^{s_m}}}\!\!\!\!&
{\scriptscriptstyle \frac{{\binom {s_m} {s_m-1}}}{(y_n-x_m)^{s_m+1}}}\!\!\!\!&{\scriptscriptstyle \cdots}\!\!\!\!&
{\scriptscriptstyle \frac{{\binom {s_m+t_n-2} {s_m-1}}}{(y_n-x_m)^{s_m+t_n-1}}}
\end{matrix}
\!\right)
\end{multline}
and
\begin{equation}
B=
\left(\begin{matrix}
{\scriptstyle {\binom 0 0}x_1^0}\!\!\!&
{\scriptstyle {\binom 1 0}x_1}\!\!\!&{\scriptstyle \cdots}\!\!\!&
{\scriptstyle {\binom {S-T-1} {0}}x_1^{S-T-1}}
\\
{\scriptstyle {\binom {0} {1}}x_1^{-1}}\!\!\!&
{\scriptstyle {\binom {1} {1}}x_1^0}\!\!\!&{\scriptstyle \cdots}\!\!\!&
{\scriptstyle {\binom {S-T-1} {1}}x_1^{S-T-2}}
\\
{\scriptstyle \cdot}\!\!\!&
{\scriptstyle \cdot}\!\!\!&{\scriptstyle \ }\!\!\!&
{\scriptstyle \cdot}
\\
{\scriptstyle \cdot}\!\!\!&
{\scriptstyle \cdot}\!\!\!&{\scriptstyle \ }\!\!\!&
{\scriptstyle \cdot}
\\
{\scriptstyle \cdot}\!\!\!&
{\scriptstyle \cdot}\!\!\!&{\scriptstyle \ }\!\!\!&
{\scriptstyle \cdot}
\\
{\scriptstyle {\binom {0} {s_1-1}}x_1^{1-s_1}}\!\!\!&
{\scriptstyle {\binom {1} {s_1-1}}x_1^{2-s_1}}\!\!\!&{\scriptstyle \cdots}\!\!\!&
{\scriptstyle {\binom {S-T-1} {s_1-1}}x_1^{S-T-s_1}}
\\
{\scriptstyle \ }\!\!\!&
{\scriptstyle \ }\!\!\!&\cdot \!\!\!&
{\scriptstyle \ }
\\
{\scriptstyle \ }\!\!\!&
{\scriptstyle \ }\!\!\!&\cdot \!\!\!&
{\scriptstyle \ }
\\
{\scriptstyle \ }\!\!\!&
{\scriptstyle \ }\!\!\!&\cdot \!\!\!&
{\scriptstyle \ }
\\
{\scriptstyle {\binom {0} {0}}x_m^0}\!\!\!&
{\scriptstyle {\binom {1} {0}}x_m}\!\!\!&{\scriptstyle \cdots}\!\!\!&
{\scriptstyle {\binom {S-T-1} {0}}x_m^{S-T-1}}
\\
{\scriptstyle {\binom {0} {1}}x_m^{-1}}\!\!\!&
{\scriptstyle {\binom {1} {1}}x_m^0}\!\!\!&{\scriptstyle \cdots}\!\!\!&
{\scriptstyle {\binom {S-T-1} {1}}x_m^{S-T-2}}
\\
{\scriptstyle \cdot}\!\!\!&
{\scriptstyle \cdot}\!\!\!&{\scriptstyle \ }\!\!\!&
{\scriptstyle \cdot}
\\
{\scriptstyle \cdot}\!\!\!&
{\scriptstyle \cdot}\!\!\!&{\scriptstyle \ }\!\!\!&
{\scriptstyle \cdot}
\\
{\scriptstyle \cdot}\!\!\!&
{\scriptstyle \cdot}\!\!\!&{\scriptstyle \ }\!\!\!&
{\scriptstyle \cdot}
\\
{\scriptstyle {\binom {0} {s_m-1}}x_m^{1-s_m}}\!\!\!&
{\scriptstyle {\binom {1} {s_m-1}}x_m^{2-s_m}}\!\!\!&{\scriptstyle \cdots}\!\!\!&
{\scriptstyle {\binom {S-T-1} {s_m-1}}x_m^{S-T-s_m}}
\end{matrix}\right).
\end{equation}
Then we have
\vbox{\noindent
\begin{equation}
\det N =\frac{\prod_{1\leq i<j\leq m}(x_j-x_i)^{s_is_j}\prod_{1\leq i<j\leq n}(y_i-y_j)^{t_it_j}}
{\prod_{i=1}^m\prod_{j=1}^n(y_j-x_i)^{s_it_j}}.
\end{equation}
\quad \quad \qed}
\end{Theorem}
This theorem generalises at the same time numerous previously obtained
determinant evaluations. It reduces of course to Cauchy's double
alternant when $m=n$ and $s_1=s_2=\dots=s_m=t_1=t_2=\dots=t_n=1$.
(In that case, the submatrix $B$ is empty.)
It reduces to the confluent alternant for
$t_1=t_2=\dots=t_n=0$ (i.e., in the case where the submatrix $A$ is
empty). The case $m=rn$, $s_1=s_2=\dots=s_m=1$, $t_1=t_2=\dots=t_n=r$
is stated as an exercise in
\machSeite{MuirAD}\cite[Ex.~42, p.~360]{MuirAD}. Finally, a mixture
of the double alternant and the Vandermonde determinant appeared already
in \machSeite{HaKrAA}\cite[Theorem~(Cauchy+)]{HaKrAA} where it was
used to
evaluate {\it Scott-type permanents}. This mixture turns out to be the
special case $s_1=s_2=\dots=s_m=t_1=t_2=\dots=t_n=1$ (but not
necessarily $m=n$) of Theorem~\ref{thm:Ciucu}.
If $S=T$ (i.e., in the case where the submatrix $B$ is empty),
Theorem~\ref{thm:Ciucu}
provides the evaluation of a confluent double alternant which
is different from the one in Theorem~\ref{thm:Johnson2} for $C=0$ and
$q=1$. While, for the general form of Theorem~\ref{thm:Ciucu}, I was
not able to find a $q$-analogue, I was able to find one for this
special case, that is, for the case where $B$ is empty.
In view of the
fact that there are also
$q$-analogues for the other extreme
case where the submatrix $A$ is empty (namely Theorem~\ref{thm:Johnson1} and
\machSeite{KratBN}\cite[Theorem~23]{KratBN}), I still suspect that a
$q$-analogue of the general form of Theorem~\ref{thm:Ciucu} should exist.
\begin{Theorem} \label{thm:Ciucu1}
Let $s_1, s_2,\dots, s_m\geq1$ and $t_1, t_2,\dots, t_n\geq1$
be integers such that $s_1+s_2+\dots+s_m=t_1+t_2+\dots+t_n$.
Let $x_1,x_2,\dots,x_m$ and $y_1,y_2,\dots,y_n$ be indeterminates.
Let $A$ be the matrix
\begin{multline}
A=
\\
\left(\!
\begin{matrix}
{\scriptscriptstyle \frac{{\qbinom 0 0}}{\coef{x_1,y_1}^1}}\!\!\!\!&
{\scriptscriptstyle \frac{{\qbinom 1 0}}{\coef{x_1,y_1}^2}}\!\!\!\!&{\scriptscriptstyle \cdots}\!\!\!\!&
{\scriptscriptstyle \frac{{\qbinom {t_1-1} 0}}{\coef{x_1,y_1}^{t_1}}}&
\ &
{\scriptscriptstyle \frac{{\qbinom 0 0}}{\coef{x_1,y_n}^1}}\!\!\!\!&
{\scriptscriptstyle \frac{{\qbinom 1 0}}{\coef{x_1,y_n}^2}}\!\!\!\!&{\scriptscriptstyle \cdots}\!\!\!\!&
{\scriptscriptstyle \frac{{\qbinom {t_n-1} 0}}{\coef{x_1,y_n}^{t_n}}}
\\
{\scriptscriptstyle \frac{{\qbinom 1 1}}{\coef{x_1,y_1}^2}}\!\!\!\!&
{\scriptscriptstyle \frac{{\qbinom 2 1}}{\coef{x_1,y_1}^3}}\!\!\!\!&{\scriptscriptstyle \cdots}\!\!\!\!&
{\scriptscriptstyle \frac{{\qbinom {t_1} 1}}{\coef{x_1,y_1}^{t_1+1}}}&
\ &
{\scriptscriptstyle \frac{{\qbinom 1 1}}{\coef{x_1,y_n}^2}}\!\!\!\!&
{\scriptscriptstyle \frac{{\qbinom 2 1}}{\coef{x_1,y_n}^3}}\!\!\!\!&{\scriptscriptstyle \cdots}\!\!\!\!&
{\scriptscriptstyle \frac{{\qbinom {t_n} 1}}{\coef{x_1,y_n}^{t_n+1}}}
\\
{\scriptstyle\cdot}\!\!\!\!&
{\scriptstyle\cdot}\!\!\!\!&\ \!\!\!\!&
{\scriptstyle\cdot}&
\!\!\!\!\cdots\!\!\! &
{\scriptstyle\cdot}\!\!\!\!&
{\scriptstyle\cdot}\!\!\!\!&\ \!\!\!\!&
{\scriptstyle\cdot}
\\
{\scriptstyle\cdot}\!\!\!\!&
{\scriptstyle\cdot}\!\!\!\!&\ \!\!\!\!&
{\scriptstyle\cdot}&
\ &
{\scriptstyle\cdot}\!\!\!\!&
{\scriptstyle\cdot}\!\!\!\!&\ \!\!\!\!&
{\scriptstyle\cdot}
\\
{\scriptstyle\cdot}\!\!\!\!&
{\scriptstyle\cdot}\!\!\!\!&\ \!\!\!\!&
{\scriptstyle\cdot}&
\ &
{\scriptstyle\cdot}\!\!\!\!&
{\scriptstyle\cdot}\!\!\!\!&\ \!\!\!\!&
{\scriptstyle\cdot}
\\
{\scriptscriptstyle \frac{{\qbinom {s_1-1} {s_1-1}}}{\coef{x_1,y_1}^{s_1}}}\!\!\!\!&
{\scriptscriptstyle \frac{{\qbinom {s_1} {s_1-1}}}{\coef{x_1,y_1}^{s_1+1}}}\!\!\!\!&{\scriptscriptstyle \cdots}\!\!\!\!&
{\scriptscriptstyle \frac{{\qbinom {s_1+t_1-2} {s_1-1}}}{\coef{x_1,y_1}^{s_1+t_1-1}}}&
\ &
{\scriptscriptstyle \frac{{\qbinom {s_1-1} {s_1-1}}}{\coef{x_1,y_n}^{s_1}}}\!\!\!\!&
{\scriptscriptstyle \frac{{\qbinom {s_1} {s_1-1}}}{\coef{x_1,y_n}^{s_1+1}}}\!\!\!\!&{\scriptscriptstyle \cdots}\!\!\!\!&
{\scriptscriptstyle \frac{{\qbinom {s_1+t_n-2} {s_1-1}}}{\coef{x_1,y_n}^{s_1+t_n-1}}}
\\
\ \!\!\!\!&
\ \!\!\!\!&\cdot\!\!\!\!&
\ &
\ &
\ \!\!\!\!&
\ \!\!\!\!&\cdot\!\!\!\!&
\
\\
\ \!\!\!\!&
\ \!\!\!\!&\cdot\!\!\!\!&
\ &
\ &
\ \!\!\!\!&
\ \!\!\!\!&\cdot\!\!\!\!&
\
\\
\ \!\!\!\!&
\ \!\!\!\!&\cdot\!\!\!\!&
\ &
\ &
\ \!\!\!\!&
\ \!\!\!\!&\cdot\!\!\!\!&
\
\\
{\scriptscriptstyle \frac{{\qbinom 0 0}}{\coef{x_m,y_1}^1}}\!\!\!\!&
{\scriptscriptstyle \frac{{\qbinom 1 0}}{\coef{x_m,y_1}^2}}\!\!\!\!&{\scriptscriptstyle \cdots}\!\!\!\!&
{\scriptscriptstyle \frac{{\qbinom {t_1-1} 0}}{\coef{x_m,y_1}^{t_1}}}&
\ &
{\scriptscriptstyle \frac{{\qbinom 0 0}}{\coef{x_m,y_n}^1}}\!\!\!\!&
{\scriptscriptstyle \frac{{\qbinom 1 0}}{\coef{x_m,y_n}^2}}\!\!\!\!&{\scriptscriptstyle \cdots}\!\!\!\!&
{\scriptscriptstyle \frac{{\qbinom {t_n-1} 0}}{\coef{x_m,y_n}^{t_n}}}
\\
{\scriptscriptstyle \frac{{\qbinom 1 1}}{\coef{x_m,y_1}^2}}\!\!\!\!&
{\scriptscriptstyle \frac{{\qbinom 2 1}}{\coef{x_m,y_1}^3}}\!\!\!\!&{\scriptscriptstyle \cdots}\!\!\!\!&
{\scriptscriptstyle \frac{{\qbinom {t_1} 1}}{\coef{x_m,y_1}^{t_1+1}}}&
\ &
{\scriptscriptstyle \frac{{\qbinom 1 1}}{\coef{x_m,y_n}^2}}\!\!\!\!&
{\scriptscriptstyle \frac{{\qbinom 2 1}}{\coef{x_m,y_n}^3}}\!\!\!\!&{\scriptscriptstyle \cdots}\!\!\!\!&
{\scriptscriptstyle \frac{{\qbinom {t_n} 1}}{\coef{x_m,y_n}^{t_n+1}}}
\\
{\scriptstyle\cdot}\!\!\!\!&
{\scriptstyle\cdot}\!\!\!\!&\ \!\!\!\!&
{\scriptstyle\cdot}&
\!\!\!\!\cdots\!\!\! &
{\scriptstyle\cdot}\!\!\!\!&
{\scriptstyle\cdot}\!\!\!\!&\ \!\!\!\!&
{\scriptstyle\cdot}
\\
{\scriptstyle\cdot}\!\!\!\!&
{\scriptstyle\cdot}\!\!\!\!&\ \!\!\!\!&
{\scriptstyle\cdot}&
\ &
{\scriptstyle\cdot}\!\!\!\!&
{\scriptstyle\cdot}\!\!\!\!&\ \!\!\!\!&
{\scriptstyle\cdot}
\\
{\scriptstyle\cdot}\!\!\!\!&
{\scriptstyle\cdot}\!\!\!\!&\ \!\!\!\!&
{\scriptstyle\cdot}&
\ &
{\scriptstyle\cdot}\!\!\!\!&
{\scriptstyle\cdot}\!\!\!\!&\ \!\!\!\!&
{\scriptstyle\cdot}
\\
{\scriptscriptstyle \frac{{\qbinom {s_m-1} {s_m-1}}}{\coef{x_m,y_1}^{s_m}}}\!\!\!\!&
{\scriptscriptstyle \frac{{\qbinom {s_m} {s_m-1}}}{\coef{x_m,y_1}^{s_m+1}}}\!\!\!\!&{\scriptscriptstyle \cdots}\!\!\!\!&
{\scriptscriptstyle \frac{{\qbinom {s_m+t_1-2} {s_m-1}}}{\coef{x_m,y_1}^{s_m+t_1-1}}}&
\ &
{\scriptscriptstyle \frac{{\qbinom {s_m-1} {s_m-1}}}{\coef{x_m,y_n}^{s_m}}}\!\!\!\!&
{\scriptscriptstyle \frac{{\qbinom {s_m} {s_m-1}}}{\coef{x_m,y_n}^{s_m+1}}}\!\!\!\!&{\scriptscriptstyle \cdots}\!\!\!\!&
{\scriptscriptstyle \frac{{\qbinom {s_m+t_n-2} {s_m-1}}}{\coef{x_m,y_n}^{s_m+t_n-1}}}
\end{matrix}
\!\right),
\end{multline}
where $\coef{x,y}^e:=(y-x)(qy-x)(q^2y-x)\cdots(q^{e-1}y-x)$.
Then we have
\vbox{
\begin{multline}
\det A =
q^{\frac {1} {6}
\sum _{i=1} ^{m}(s_i-1)s_i(2s_i-1)}
\(\prod _{1\le i<j\le m}
\prod _{g=1} ^{s_i}
\prod _{h=1} ^{s_j} (q^{g-1}x_j-q^{h-1}x_i)\)\\
\times
\(\prod _{1\le i<j\le n}
\prod _{g=1} ^{t_i}
\prod _{h=1} ^{t_j}
(q^{g-1}y_i-q^{h-1}y_j)\)
\(\prod _{j=1} ^{m}
\prod _{i=1} ^{n}
\prod _{g=1} ^{t_i}
\prod _{h=1} ^{s_j}
\frac 1{(q^{g+h-2}y_i-x_j)}\).
\end{multline}
\quad \quad \qed}
\end{Theorem}
\subsection{More determinants containing derivatives and compositions
of series}
Inspired by formulae of Mina
\machSeite{MinaAA}\cite{MinaAA}, Kedlaya
\machSeite{KedlAA}\cite{KedlAA} and Strehl and Wilf
\machSeite{StWiAA}\cite{StWiAA} for determinants of
matrices the entries of which being given by (coefficients of)
{\it multiple derivatives} and {\it compositions of formal power
series} (see also
\machSeite{KratBN}\cite[Lemma~16]{KratBN}),
Chu embedded all these in a larger
context in the remarkable systematic study
\machSeite{ChuWBG}\cite{ChuWBG}.
He shows that, at the heart of these formulae, there is the
{\it Fa\`a di Bruno formula}\footnote{As one can read in
\machSeite{JohWAE}\cite{JohWAE}, ``Fa\`a di Bruno was neither the
first to state the formula that bears his name nor the first to prove
it." In Section~4 of that article, the author tries to trace back the
roots of the formula. It is apparently impossible to find the author
of the formula with certainty. In his book
\machSeite{ArboAA}\cite[p.~312]{ArboAA}, Arbogast describes a recursive rule
how, from the top term, to generate all other terms in the
formula. However, the explicit formula is never written down.
(I am not able to verify the conclusions in
\machSeite{CraiAA}\cite{CraiAA}. It seems to
me that the author mixes the knowledge that we have today with what is
really written in
\machSeite{ArboAA}\cite{ArboAA}.)
The formula appears explicitly in Lacroix's book
\machSeite{LacrAA}\cite[p.~629]{LacrAA}, but Lacroix's precise
sources remain unknown. I refer the reader to
\machSeite{JohWAE}\cite[Sec.~4]{JohWAE} and
\machSeite{CraiAA}\cite{CraiAA}
for more detailed remarks on the history of the formula.}
for multiple derivatives of a composition of
two formal power series. Using it, he derives the following
determinant reduction formulae
\machSeite{ChuWBG}\cite[Theorems~4.1 and 4.2]{ChuWBG} for determinants
of matrices containing multiple derivatives of compositions of formal
power series.
\begin{Theorem} \label{thm:Chu1}
Let $f(x)$ and $\phi_k(x)$ and $w_k(x)$, $k=0,1,\dots,n$,
be formal power series in $x$
with coefficients in a commutative ring. Then
\begin{equation} \label{eq:Chu1}
\det_{0\le i,j,\le n}\(\frac {d^j}
{dx^j}\bigg(w_j(x)\phi_i\big(f(x)\big)\bigg)\)=
\big(f'(x)\big)^{\binom {n+1}2}
\(\prod _{k=0} ^{n}w_k(x)\)\det_{0\le i,j,\le
n}\(\phi_i^{(j)}\big(f(x)\big)\),
\end{equation}
where $\phi^{(j)}(x)$ is short for $\frac {d^j} {dx^j}\phi(x)$. If, in
addition, $w_k(x)$ is a polynomial of degree at most $k$,
$k=1,2,\dots,n$, then
\begin{equation} \label{eq:Chu2}
\det_{1\le i,j,\le n}\(\frac {d^j}
{dx^j}\bigg(w_j(x)\phi_i\big(f(x)\big)\bigg)\)=
\big(f'(x)\big)^{\binom {n+1}2}
\(\prod _{k=1} ^{n}w_k(x)\)\det_{1\le i,j,\le
n}\(\phi_i^{(j)}\big(f(x)\big)\).
\end{equation}
\quad \quad \qed
\end{Theorem}
Specialising the series $\phi_i(x)$ so that the determinants
on the right-hand sides of \eqref{eq:Chu1} or \eqref{eq:Chu2} can be
evaluated, he obtains numerous nice corollaries.
Possible choices are $\phi_i(x)=\exp(y_ix)$,
$\phi_i(x)=\log(1+y_ix)$,
$\phi_i(x)=x^{y_i}$, or
$\phi_i(x)=(a_i+b_ix)/(c_i+d_ix)$. See
\machSeite{ChuWBG}\cite[Cor.~4.3 and 4.4]{ChuWBG} for the
corresponding results.
Further reduction formulae and determinant evaluations from
\machSeite{ChuWBG}\cite{ChuWBG}
address determinants of matrices
formed out of coefficients of iterated compositions of formal power
series. In order to have a convenient notation, let us write
$f^{\coef{n}}(x)$ for the $n$-fold composition of $f$ with itself,
$$f^{\coef{n}}(x)=f(f(\cdots(f(x)))),$$
with $n$ occurrences of $f$ on the right-hand side. Chu shows
\machSeite{ChuWBG}\cite[Sec.~1.4]{ChuWBG} that it
is possible to extend this $n$-fold composition to values of $n$ other
than non-negative integers. This given, Theorems~4.6 and 4.7 from
\machSeite{ChuWBG}\cite{ChuWBG} read as follows.
\begin{Theorem} \label{thm:Chu3}
Let $f(x)=x+
\sum _{m=2} ^{\infty}f_mx^m$, $g(x)=
\sum _{m=1} ^{\infty}g_mx^m$ and $w_k(x)$, $k=1,2,\dots,n$,
be formal power series with
coefficients in some commutative ring. Then
\begin{equation} \label{eq:Chu3}
\det_{1\le i,j\le n}\Big([x^j]w_j(x)f^{\coef{y_i}}\big(g(x)\big)\Big)=
f_2^{\binom n2}g_1^{\binom {n+1}2} \(
\prod _{k=1} ^{n}w_k(0)\)\(
\prod _{1\le i<j\le n} ^{}(y_j-y_i)\),
\end{equation}
where $[x^j]h(x)$ denotes the coefficient of $x^j$ in the series $h(x)$.
If, in addition, $w_n(0)=0$, then
\begin{multline} \label{eq:Chu4}
\det_{1\le i,j\le
n}\Big([x^{j+1}]w_j(x)f^{\coef{y_i}}\big(g(x)\big)\Big)
\\=
f_2^{\binom n2}g_1^{\binom {n+1}2}
\(\prod _{1\le i<j\le n} ^{}(y_j-y_i)\)
\det_{1\le i,j\le n}\Big([x^{1+j-i}]w_j(x)\Big).
\end{multline}
Furthermore, we have
\begin{equation} \label{eq:Chu5}
\det_{1\le i,j\le n}\Big([x^{j+1}]f^{\coef{y_i}}(x)\Big)=
f_2^{\binom {n+1}2}\(\prod _{k=1} ^{n}y_k\)\(
\prod _{1\le i<j\le n} ^{}(y_j-y_i)\).
\end{equation}
\quad \quad \qed
\end{Theorem}
\subsection{More on Hankel determinant evaluations}
Section~2.7 of
\machSeite{KratBN}\cite{KratBN} was devoted to {\it Hankel
determinants}. There, I tried to convince the reader that, whenever
you think that a certain Hankel determinant evaluates nicely, then
the explanation will be (sometimes more sometimes less) hidden
in the theory of {\em continued fractions} and
{\em orthogonal polynomials}. In retrospect,
it seems that the success of this try
was mixed. Since readers are always right, this has to be blamed
entirely on myself, and, indeed, the purpose of the present subsection
is to rectify some shortcomings from then.
Roughly speaking, I explained in
\machSeite{KratBN}\cite{KratBN} that, given a Hankel determinant
\begin{equation} \label{eq:Hankel1}
\det_{0\le i,j\le n-1}(\mu_{i+j}),
\end{equation}
to find its evaluation one should expand the generating function of
the sequence of coefficients $(\mu_k)_{k\ge0}$ in terms of a continued
fraction, respectively find the sequence of orthogonal polynomials
$(p_n(x))_{n\ge0}$ with moments $\mu_k$, $k=0,1,\dots$, and then the
value of the Hankel determinant \eqref{eq:Hankel1} can be read off the
coefficients of the continued fraction, respectively from the
recursion coefficients of the orthogonal polynomials. What I missed to
state is that the knowledge of the
orthogonal polynomials makes it also possible to find the value of the
Hankel determinants which start with $\mu_1$ and $\mu_2$, respectively
(instead of $\mu_0$). In the theorem below I summarise the results
that were already discussed in
\machSeite{KratBN}\cite{KratBN}
(for which classical references are
\machSeite{WallCF}%
\cite[Theorem 51.1]{WallCF} or
\machSeite{VienAE}%
\cite[Cor.~6, (19), on p.~IV-17; Proposition~1, (7),
on p.~V-5]{VienAE}), and I add the two missing ones.
\begin{Theorem}
\label{cor:cfracHankel}
Let $(\mu_k)_{k\ge0}$ be a sequence with generating
function $\sum_{k=0}^\infty{\mu_k}x^k$ written in the form
\begin{equation}
\label{eq:momentgf}
\sum_{k=0}^\infty{\mu_k}x^k=\cfrac{
\mu_0}
{1+a_0x-\cfrac{
b_1x^2}
{1+a_1x-\cfrac{
b_2x^2}
{1+a_2x-\cdots}}}\quad .
\end{equation}
Then
\begin{equation} \label{eq:Hankel2}
\det_{0\le i,j\le n-1}(\mu_{i+j})=\mu_0^nb_1^{n-1}b_2^{n-2}\cdots
b_{n-2}^2b_{n-1}.
\end{equation}
Let $(p_n(x))_{n\ge0}$ be a sequence of monic polynomials, the
polynomial $p_n(x)$ having degree $n$, which is orthogonal with
respect to some functional $L$, that is, $L(p_m(x)p_n(x))=\delta_{m,n}c_{n}$,
where the $c_n$'s are some non-zero constants and $\delta_{m,n}$ is the
Kronecker delta. Let
\begin{equation}
p_{n+1}(x)=(a_{n}+x)p_{n}(x)-b_{n}p_{n-1}(x)
\label{eq:three-term2}
\end{equation}
be the corresponding three-term recurrence which is guaranteed by
Favard's theorem. Then the generating function $\sum _{k=0}
^{\infty}\mu_kx^k$ for the moments
$\mu_k=L(x^k)$ satisfies \eqref{eq:momentgf} with the $a_i$'s and
$b_i$'s being the coefficients in the three-term recurrence
\eqref{eq:three-term2}. In particular, the Hankel determinant
evaluation \eqref{eq:Hankel2} holds, with the $b_i$'s from the
three-term recurrence \eqref{eq:three-term2}.
If $(q_n)_{n\ge0}$ is the sequence recursively defined by $q_0=1$,
$q_1=-a_0$, and
$$q_{n+1}=-a_n q_n-b_nq_{n-1},$$
then in the situation above we also have
\begin{equation} \label{eq:Hankel3}
\det_{0\le i,j\le n-1}(\mu_{i+j+1})=\mu_0^nb_1^{n-1}b_2^{n-2}\cdots
b_{n-2}^2b_{n-1}q_n
\end{equation}
and
\begin{equation} \label{eq:Hankel4}
\det_{0\le i,j\le n-1}(\mu_{i+j+2})=\mu_0^nb_1^{n-1}b_2^{n-2}\cdots
b_{n-2}^2b_{n-1}
\sum _{k=0} ^{n}q_k^2b_{k+1}\cdots b_{n-1}b_n.
\end{equation}
\quad \quad \qed
\end{Theorem}
I did not find a reference for \eqref{eq:Hankel3} and
\eqref{eq:Hankel4}. These two identities follow however easily from
Viennot's combinatorial model
\machSeite{VienAE}\cite{VienAE} for orthogonal polynomials and Hankel
determinants of moments. More precisely, in this theory the moments
$\mu_k$ are certain generating functions for {\it Motzkin paths}, and,
due to Theorem~\ref{thm:nonint}, the
Hankel determinants $\det_{0\le i,j\le n-1}(\mu_{i+j+m})$ are
generating functions for families $(P_1,P_2,\dots,P_n)$
of non-intersecting Motzkin paths, $P_i$ running from $(-i,0)$ to
$(j+m,0)$. In the case $m=0$, it is explained in
\machSeite{VienAE}\cite[Ch.~IV]{VienAE} how to find the corresponding
Hankel determinant evaluation \eqref{eq:Hankel2}
using this combinatorial model. The idea is that in that case there is
a unique family of non-intersecting Motzkin paths, and its weight
gives the right-hand side of \eqref{eq:Hankel2}.
If $m=1$ or $m=2$ one can argue similarly. The paths are uniquely
determined with the exception of their portions in
the strip $0\le x\le m$. The various possibilities that one has there
then yield the right-hand sides of \eqref{eq:Hankel3} and
\eqref{eq:Hankel4}.
Since there are so many explicit families of orthogonal polynomials,
and, hence, so many ways to apply the above theorem, I listed only a
few standard Hankel determinant evaluations explicitly in
\machSeite{KratBN}\cite{KratBN}. I did append a long list of
references and sketched in which ways these give rise to more Hankel
determinant evaluations. Apparently, these remarks were at times too
cryptic, in particular concerning the theme {\it ``orthogonal polynomials as
moments}.'' This is treated systematically in the two papers
\machSeite{IsStAB}\machSeite{IsStAC}\cite{IsStAB,IsStAC} by Ismail and
Stanton. There it is shown that certain classical polynomials
$(r_n(x))_{n\ge0}$, such as, for example,
the {\em Laguerre polynomials}, the {\em Meixner polynomials}, or
the {\em Al-Salam--Chihara polynomials} (but there are others as well,
see
\machSeite{IsStAB}\machSeite{IsStAC}\cite{IsStAB,IsStAC}), are {\it
moments} of other families of classical orthogonal polynomials. Thus,
application of Theorem~\ref{cor:cfracHankel} with $\mu_n=r_n(x)$
immediately tells that
the evaluations of the corresponding Hankel determinants
\begin{equation} \label{eq:pn}
\det_{0\le i,j\le n-1}\big(r_{i+j}(x)\big)
\end{equation}
(and also the higher ones in \eqref{eq:Hankel3} and
\eqref{eq:Hankel4}) are known. In particular, the explicit forms can
be extracted from the coefficients of the three-term recursions for
these other families of orthogonal polynomials. Thus, whenever you encounter a
determinant of the form \eqref{eq:pn}, you must check whether
$(r_n(x))_{n\ge0}$ is a family of orthogonal polynomials (which, as I
explained in
\machSeite{KratBN}\cite{KratBN}, one does by consulting the
compendium \machSeite{KoSwAA}\cite{KoSwAA} of hypergeometric
orthogonal polynomials compiled by Koekoek and Swarttouw), and if the
answer is ``yes", you will find the solution of your determinant
evaluation through the results in
\machSeite{IsStAB}\machSeite{IsStAC}\cite{IsStAB,IsStAC}
by applying Theorem~\ref{cor:cfracHankel}.
\medskip
While Theorem~\ref{cor:cfracHankel} describes in detail the connexion
between Hankel determinants and the continued fractions of the type
\eqref{eq:momentgf}, which are
often called {\it $J$-fractions} (which is short for {\it Jacobi continued
fractions}), I missed to tell in
\machSeite{KratBN}\cite{KratBN} that there is also a close relation
between Hankel determinants and so-called {\it $S$-fractions}
(which is short for {\it Stieltjes continued fractions}). I try to remedy this
by the theorem below (cf.\ for example
\machSeite{JoThAA}\cite[Theorem~7.2]{JoThAA}, where
$S$-fractions are called {\it regular $C$-fractions}).
In principle, since $S$-fractions are special cases of $J$-fractions
\eqref{eq:momentgf} in which the coefficients $a_i$ are all zero,
the corresponding result for the Hankel determinants is in fact implied by
Theorem~\ref{cor:cfracHankel}. Nevertheless, it is useful to state it
separately. I am not
able to give a reference for \eqref{eq:Hankel7}, but, again, it is not too
difficult to derive it from Viennot's combinatorial model
\machSeite{VienAE}\cite{VienAE} for orthogonal
polynomials and moments that was mentioned above.
\begin{Theorem} \label{thm:cfrac2}
Let $(\mu_k)_{k\ge0}$ be a sequence with generating
function $\sum_{k=0}^\infty{\mu_k}x^k$ written in the form
\begin{equation}
\label{eq:momentgfS}
\sum_{k=0}^\infty{\mu_k}x^k=\cfrac{
\mu_0}
{1+\cfrac{
a_1x}
{1+\cfrac{
a_2x}
{1+\cdots}}}\quad .
\end{equation}
Then
\begin{align} \label{eq:Hankel5}
\det_{0\le i,j\le n-1}(\mu_{i+j})&=\mu_0^n(a_1a_2)^{n-1}(a_3a_4)^{n-2}\cdots
(a_{2n-5}a_{2n-4})^2(a_{2n-3}a_{2n-2}),\\
\label{eq:Hankel6}
\det_{0\le i,j\le n-1}(\mu_{i+j+1})&=(-1)^n\mu_0^n
a_1^n(a_2a_3)^{n-1}(a_4a_5)^{n-2}\cdots
(a_{2n-4}a_{2n-3})^2(a_{2n-2}a_{2n-1}),
\end{align}
and
\vbox{
\begin{multline} \label{eq:Hankel7}
\det_{0\le i,j\le n-1}(\mu_{i+j+2})=\mu_0^n
a_1^n(a_2a_3)^{n-1}(a_4a_5)^{n-2}\cdots
(a_{2n-4}a_{2n-3})^2(a_{2n-2}a_{2n-1})\\
\times
\sum _{0\le i_1-1<i_2-2<\dots<i_n-n\le n} ^{}a_{i_1}a_{i_2}\cdots a_{i_n}.
\end{multline}
\quad \quad \qed}
\end{Theorem}
Using this theorem, Tamm
\machSeite{TammAA}\cite[Theorem~3.1]{TammAA}
observed that from {\it Gau\ss' continued fraction for the ratio of two
contiguous $_2F_1$-series} one can deduce several interesting
binomial Hankel determinant evaluations, some of them had already
been found earlier by E\u gecio\u glu, Redmond and Ryavec
\machSeite{EgRiAA}\cite[Theorem~4]{EgRiAA}
while working on {\it polynomial Riemann hypotheses}.
Gessel and Xin
\machSeite{GeXiAB}\cite{GeXiAB} undertook a systematic analysis of
this approach, and they arrived at a set of 18 Hankel determinant
evaluations, which I list as
\eqref{eq:3nA}--\eqref{eq:3nR} in the theorem below. They are preceded by the
Hankel determinant evaluation \eqref{eq:3nAAAA}, which appears only in
\machSeite{EgRiAA}\cite[Theorem~4]{EgRiAA}.
\begin{Theorem} \label{thm:Tamm}
For any positive integer $n$, there hold
\begin{equation}
\label{eq:3nAAAA}
\det_{0\le i,j\le n-1}\(\binom {3i+3j+2}{i+j}\)
=
\prod _{i=0} ^{n-1}\frac {(\frac {4} {3})_i\,(\frac {5} {6})_i\,
(\frac {5} {3})_i\,(\frac {7} {6})_i}
{(\frac {3} {2})_{2i}\,(\frac {5} {2})_{2i}}\(\frac {27} {4}\)^{2i},
\end{equation}
\begin{equation} \label{eq:3nA}
\det_{0\le i,j\le n-1}\(\frac {1} {3i+3j+1}\binom {3i+3j+1}{i+j}\)
=
\prod _{i=0} ^{n-1}\frac {(\frac {2} {3})_i\,(\frac {1} {6})_i\,
(\frac {4} {3})_i\,(\frac {5} {6})_i}
{(\frac {1} {2})_{2i}\,(\frac {3} {2})_{2i}}\(\frac {27} {4}\)^{2i},
\end{equation}
\begin{equation}
\label{eq:3nB}
\det_{0\le i,j\le n-1}\(\frac {1} {3i+3j+4}\binom {3i+3j+4}{i+j+1}\)
=
\prod _{i=0} ^{n-1}\frac {(\frac {4} {3})_i\,(\frac {5} {6})_i\,
(\frac {5} {3})_i\,(\frac {7} {6})_i}
{(\frac {3} {2})_{2i}\,(\frac {5} {2})_{2i}}\(\frac {27} {4}\)^{2i},
\end{equation}
\begin{equation}
\label{eq:3nC}
\det_{0\le i,j\le n-1}\(\frac {1} {3i+3j+2}\binom {3i+3j+2}{i+j+1}\)
=
\prod _{i=0} ^{n-1}\frac {(\frac {4} {3})_i\,(\frac {5} {6})_i\,
(\frac {5} {3})_i\,(\frac {7} {6})_i}
{(\frac {3} {2})_{2i}\,(\frac {5} {2})_{2i}}\(\frac {27} {4}\)^{2i},
\end{equation}
\begin{equation}
\label{eq:3nD}
\det_{0\le i,j\le n-1}\(\frac {1} {3i+3j+5}\binom {3i+3j+5}{i+j+2}\)
=
\prod _{i=0} ^{n}\frac {(\frac {2} {3})_i\,(\frac {1} {6})_i\,
(\frac {4} {3})_i\,(\frac {5} {6})_i}
{(\frac {1} {2})_{2i}\,(\frac {3} {2})_{2i}}\(\frac {27} {4}\)^{2i},
\end{equation}
\begin{equation}
\label{eq:3nE}
\det_{0\le i,j\le n-1}\(\frac {2} {3i+3j+1}\binom {3i+3j+1}{i+j+1}\)
=
\prod _{i=0} ^{n}\frac {(\frac {2} {3})_i\,(\frac {1} {6})_i\,
(\frac {4} {3})_i\,(\frac {5} {6})_i}
{(\frac {1} {2})_{2i}\,(\frac {3} {2})_{2i}}\(\frac {27} {4}\)^{2i},
\end{equation}
\begin{equation}
\label{eq:3nF}
\det_{0\le i,j\le n-1}\(\frac {2} {3i+3j+4}\binom {3i+3j+4}{i+j+2}\)
=
\prod _{i=0} ^{n}\frac {(\frac {4} {3})_i\,(\frac {5} {6})_i\,
(\frac {5} {3})_i\,(\frac {7} {6})_i}
{(\frac {3} {2})_{2i}\,(\frac {5} {2})_{2i}}\(\frac {27} {4}\)^{2i},
\end{equation}
\begin{equation}
\label{eq:3nG}
\det_{0\le i,j\le n-1}\(\frac {2} {(3i+3j+1)(3i+3j+2)}
\binom {3i+3j+2}{i+j+1}\)
=
\prod _{i=0} ^{n-1}2\frac {(\frac {5} {3})_i\,(\frac {1} {6})_i\,
(\frac {7} {3})_i\,(\frac {5} {6})_i}
{(\frac {3} {2})_{2i}\,(\frac {5} {2})_{2i}}\(\frac {27} {4}\)^{2i},
\end{equation}
\begin{multline}
\label{eq:3nH}
\det_{0\le i,j\le n-1}\(\frac {2} {(3i+3j+4)(3i+3j+5)}
\binom {3i+3j+5}{i+j+2}\) \\
=(-1)^n
\prod _{i=1} ^{n}\frac {(\frac {5} {3})_i\,(\frac {1} {6})_i\,
(\frac {4} {3})_i\,(-\frac {1} {6})_i}
{(\frac {1} {2})_{2i}\,(\frac {3} {2})_{2i}}\(\frac {27} {4}\)^{2i},
\end{multline}
\begin{equation}
\label{eq:3nI}
\det_{0\le i,j\le n-1}\(\frac {(9i+9j+5)} {(3i+3j+1)(3i+3j+2)}
\binom {3i+3j+2}{i+j+1}\)
=
\prod _{i=0} ^{n-1}5\frac {(\frac {2} {3})_i\,(\frac {7} {6})_i\,
(\frac {4} {3})_i\,(\frac {11} {6})_i}
{(\frac {3} {2})_{2i}\,(\frac {5} {2})_{2i}}\(\frac {27} {4}\)^{2i},
\end{equation}
\begin{equation}
\label{eq:3nJ}
\det_{0\le i,j\le n-1}\(\frac {(9i+9j+14)} {(3i+3j+4)(3i+3j+5)}
\binom {3i+3j+5}{i+j+2}\) =
\prod _{i=1} ^{n}2\frac {(\frac {2} {3})_i\,(\frac {7} {6})_i\,
(\frac {1} {3})_i\,(\frac {5} {6})_i}
{(\frac {1} {2})_{2i}\,(\frac {3} {2})_{2i}}\(\frac {27} {4}\)^{2i}.
\end{equation}
Let $a_0=-2$ and $a_m=\frac {1} {3m+1}\binom {3m+1}m$ for $m\ge1$. Then
\begin{equation} \label{eq:3nK}
\det_{0\le i,j\le n-1}(a_{i+j})=
\prod _{i=0} ^{n-1}(-2)\frac {(\frac {1} {3})_i\,(-\frac {1} {6})_i\,
(\frac {5} {3})_i\,(\frac {7} {6})_i}
{(\frac {1} {2})_{2i}\,(\frac {3} {2})_{2i}}\(\frac {27} {4}\)^{2i}.
\end{equation}
Let $b_0=10$ and $b_m=\frac {2} {3m+2}\binom {3m+2}m$ for $m\ge1$. Then
\begin{equation} \label{eq:3nL}
\det_{0\le i,j\le n-1}(b_{i+j})=
\prod _{i=0} ^{n-1}10\frac {(\frac {2} {3})_i\,(\frac {1} {6})_i\,
(\frac {7} {3})_i\,(\frac {11} {6})_i}
{(\frac {3} {2})_{2i}\,(\frac {5} {2})_{2i}}\(\frac {27} {4}\)^{2i}.
\end{equation}
Furthermore,
\begin{equation} \label{eq:3nM}
\det_{0\le i,j\le n-1}\(\frac {2} {3i+3j+5}\binom {3i+3j+5}{i+j+1}\)
=
\prod _{i=0} ^{n}\frac {(\frac {4} {3})_i\,(\frac {5} {6})_i\,
(\frac {5} {3})_i\,(\frac {7} {6})_i}
{(\frac {3} {2})_{2i}\,(\frac {5} {2})_{2i}}\(\frac {27} {4}\)^{2i}.
\end{equation}
Let $c_0=\frac {7} {2}$ and
$c_m=\frac {2} {3m+1}\binom {3m+1}{m+1}$ for $m\ge1$. Then
\begin{equation} \label{eq:3nN}
\det_{0\le i,j\le n-1}(c_{i+j})=
\prod _{i=0} ^{n}\frac {(\frac {4} {3})_i\,(\frac {5} {6})_i\,
(\frac {5} {3})_i\,(\frac {7} {6})_i}
{(\frac {3} {2})_{2i}\,(\frac {5} {2})_{2i}}\(\frac {27} {4}\)^{2i}.
\end{equation}
Let $d_0=-5$ and $d_m=\frac {8} {(3m+1)(3m+2)}\binom {3m+2}m$ for $m\ge1$. Then
\begin{equation} \label{eq:3nO}
\det_{0\le i,j\le n-1}(d_{i+j})=
\prod _{i=0} ^{n-1}(-5)\frac {(\frac {4} {3})_i\,(-\frac {1} {6})_i\,
(\frac {8} {3})_i\,(\frac {7} {6})_i}
{(\frac {3} {2})_{2i}\,(\frac {5} {2})_{2i}}\(\frac {27} {4}\)^{2i}.
\end{equation}
Furthermore,
\begin{equation} \label{eq:3nP}
\det_{0\le i,j\le n-1}\(\frac {8} {(3i+3j+4)(3i+3j+5)}
\binom {3i+3j+5}{i+j+1}\)
=
\prod _{i=0} ^{n-1}2\frac {(\frac {7} {3})_i\,(\frac {5} {6})_i\,
(\frac {8} {3})_i\,(\frac {7} {6})_i}
{(\frac {5} {2})_{2i}\,(\frac {7} {2})_{2i}}\(\frac {27} {4}\)^{2i}.
\end{equation}
Let $e_0=14$ and
$e_m=\frac {2(9m+5)} {(3m+1)(3m+2)}\binom {3m+2}m$ for $m\ge1$. Then
\begin{equation} \label{eq:3nQ}
\det_{0\le i,j\le n-1}(e_{i+j})=
\prod _{i=0} ^{n-1}14\frac {(\frac {1} {3})_i\,(\frac {5} {6})_i\,
(\frac {5} {3})_i\,(\frac {13} {6})_i}
{(\frac {3} {2})_{2i}\,(\frac {5} {2})_{2i}}\(\frac {27} {4}\)^{2i}.
\end{equation}
Furthermore,
\begin{equation} \label{eq:3nR}
\det_{0\le i,j\le n-1}\(\frac {2(9i+9j+14)} {(3i+3j+4)(3i+3j+5)}
\binom {3i+3j+5}{i+j+1}\)
=
\prod _{i=1} ^{n}2\frac {(\frac {2} {3})_i\,(\frac {7} {6})_i\,
(\frac {1} {3})_i\,(\frac {5} {6})_i}
{(\frac {1} {2})_{2i}\,(\frac {3} {2})_{2i}}\(\frac {27} {4}\)^{2i}.
\end{equation}
\quad \quad \qed
\end{Theorem}
Some of the numbers appearing on the right-hand sides of the formulae
in this theorem have combinatorial significance, although no intrinsic
explanation is known why this is the case. More precisely, the numbers on the
right-hand sides of \eqref{eq:3nA},
\eqref{eq:3nD} and \eqref{eq:3nE} count {\it cyclically symmetric
transpose-complementary plane partitions} (cf.\
\machSeite{MiRRAD}\cite{MiRRAD} and
\machSeite{BresAO}\cite{BresAO}), whereas those on the right-hand
sides of \eqref{eq:3nAAAA},
\eqref{eq:3nB}, \eqref{eq:3nC} and \eqref{eq:3nF} count {\em vertically
symmetric alternating sign matrices} (cf.\
\machSeite{KupeAH}\cite{KupeAH} and
\machSeite{BresAO}\cite{BresAO}).
In \machSeite{EgRiAA}\cite[Theorem~4]{EgRiAA},
E\u gecio\u glu, Redmond and Ryavec prove also the following
common generalisation of \eqref{eq:3nB} and \eqref{eq:3nC}.
(The first identity is the special case $x=0$, while the second is the
special case $x=1$ of the following theorem.)
\begin{Theorem} \label{thm:EgRR}
For $m\ge0$, let $s_m(x)=
\sum _{k=0} ^{m}\frac {k+1} {m+1}\binom {3m-k+1}{m-k}x^k$. Then, for
any positive integer $n$, there holds
\begin{equation} \label{eq:EgRR}
\det_{0\le i,j\le n-1}(s_{i+j}(x))=
\prod _{i=0} ^{n-1}\frac {(\frac {4} {3})_i\,(\frac {5} {6})_i\,
(\frac {5} {3})_i\,(\frac {7} {6})_i}
{(\frac {3} {2})_{2i}\,(\frac {5} {2})_{2i}}\(\frac {27} {4}\)^{2i}.
\end{equation}
\quad \quad \qed
\end{Theorem}
As I mentioned above,
in \machSeite{KratBN}\cite{KratBN} I only stated a few special Hankel
determinant evaluations explicitly, because there are too many ways to
apply Theorems~\ref{cor:cfracHankel} and \ref{thm:cfrac2}.
I realise, however, that I should have stated the
evaluation of the {\it Hankel determinant of Catalan numbers} there.
I make this up now by doing this in the theorem below. I did not do it
then because orthogonal polynomials are not needed for its evaluation
(the orthogonal polynomials which are tied to Catalan numbers as
moments are {\it Chebyshev polynomials}, but, via
Theorems~\ref{cor:cfracHankel} and \ref{thm:cfrac2}, one would only
cover the cases $m=0,1,2$ in the theorem below). In fact, the Catalan number
$C_n=\frac {1} {n+1}\binom {2n}n$ can be alternatively written as
$C_n=(-1)^n2^{2n+1}\binom {1/2}{n+1}$, and therefore the Hankel
determinant evaluation below follows from
\machSeite{KratBN}\cite[Theorem~26, (3.12)]{KratBN}.
This latter observation shows that
even a more general determinant, namely
$\det_{0\le i,j\le n-1}(C_{\lambda_i+j})$, can be evaluated in closed form.
For historical remarks on this ubiquitous determinant see
\machSeite{GhKrAA}\cite[paragraph before the Appendix]{GhKrAA}.
\begin{Theorem} \label{thm:Catalan}
\begin{equation} \label{eq:Catalan}
\det_{0\le i,j\le n-1}(C_{m+i+j})=
\prod _{1\le i\le j\le m-1} ^{}\frac {2n+i+j} {i+j}.
\end{equation}
\quad \quad \qed
\end{Theorem}
As in
\machSeite{KratBN}\cite{KratBN},
let me conclude the part on Hankel determinants
by pointing the reader to further
papers containing interesting results on them, high-lighting
sometimes the point of view of orthogonal polynomials that I
explained above, sometimes a combinatorial point of view.
The first point of view is put forward in
\machSeite{WimpAB}\cite{WimpAB} (see
\machSeite{KratZZ}\cite{KratZZ} for the solution of the conjectures in
that paper) in order to present Hankel determinant evaluations of
matrices with {\it hypergeometric $_2F_1$-series} as entries.
The orthogonal polynomials approach is also used in
\machSeite{CvRIAA}\cite{CvRIAA} to show that a certain Hankel determinant
defined by {\it Catalan numbers} evaluates to {\it Fibonacci numbers}.
In \machSeite{AnWiAA}\cite{AnWiAA}, one finds Hankel determinant
evaluations involving generalisations of the {\it Bernoulli numbers}.
The combinatorial point of view dominates in
\machSeite{AignAA}%
\machSeite{CiglAM}%
\machSeite{CiglAO}%
\machSeite{CiglAV}%
\machSeite{EhreAB}\cite{AignAA,CiglAM,CiglAO,CiglAV,EhreAB}, where Hankel
determinants involving {\it $q$-Catalan numbers, $q$-Stirling numbers}, and
{\it $q$-Fibonacci numbers} are considered.
A very interesting new direction, which seems to have much potential,
is opened up by Luque and Thibon in
\machSeite{LuThAB}\cite{LuThAB}. They show that {\it Selberg-type
integrals} can be evaluated by means of {\it Hankel
hyperdeterminants}, and they prove many hyperdeterminant
generalisations of classical Hankel determinant evaluations.
At last, (but certainly not least!),
I want to draw the reader's attention to Lascoux's
``unorthodox"\footnote{It could easily be that it
is the ``modern" treatment of the theory which must be labelled with
the attribute ``unorthodox." As Lascoux documents in
\machSeite{LascAZ}\cite{LascAZ}, in his treatment {\it he} follows the
tradition of great masters such as Cauchy, Jacobi or Wro\'nski \dots}
approach to Hankel determinants and orthogonal polynomials through
symmetric functions which he presents in detail in
\machSeite{LascAZ}\cite[Ch.~4, 5, 8]{LascAZ}.
In particular, Theorem~\ref{cor:cfracHankel}, Eq.~\eqref{eq:Hankel2}
are the contents of Theorem~8.3.1 in
\machSeite{LascAZ}\cite{LascAZ} (see also the end of Section~5.3 there), and
Theorem~\ref{thm:cfrac2}, Eqs.~\eqref{eq:Hankel5} and
\eqref{eq:Hankel6} are the contents of Theorem~4.2.1 in
\machSeite{LascAZ}\cite{LascAZ}.
The usefulness of this symmetric function approach is,
for example, demonstrated in
\machSeite{HoLMAA}%
\machSeite{HoLMAB}\cite{HoLMAA,HoLMAB} in order to evaluate Hankel
determinants of matrices the entries of which are {\it Rogers--Szeg\H
o}, respectively {\it Meixner polynomials}.
\subsection{More binomial determinants}
A vast part of Section~3 in
\machSeite{KratBN}\cite{KratBN} is occupied by binomial
determinants. As I mentioned in Section~\ref{sec:comb} of the present article,
an extremely rich source for binomial determinants is rhombus tiling
enumeration. I want to present here some which did not already appear in
\machSeite{KratBN}\cite{KratBN}.
To begin with, I want to remind the reader of an old problem posed by
Andrews in
\machSeite{AndrAO}\cite[p.~105]{AndrAO}. The determinant in this problem is a
variation of a determinant which enumerates {\it cyclically
symmetric plane partitions} and {\it descending plane partitions},
which was evaluated by Andrews in
\machSeite{AndrAN}\cite{AndrAN} (see also
\machSeite{KratBN}\cite[Theorem~32]{KratBN}; the latter determinant
arises from the one in \eqref{eq:desc-var} by replacing $j+1$ by $j$
in the bottom of the binomial coefficient).
\begin{Problem} \label{prob:7}
Evaluate the determinant
\begin{equation} \label{eq:desc-var}
D_1(n):=\det_{0\le i,j\le n-1}\left(\delta_{ij}+\binom {\mu+i+j}{j+1}\right),
\end{equation}
where $\delta_{ij}$ is the Kronecker delta.
In particular, show that
\begin{equation} \label{eq:fmu}
\frac {D_1(2n)} {D_1(2n-1)}=(-1)^{\binom {n-1}2}
\frac {2^n\,\(\frac\mu2+n\)_{\cl{n/2}}\,\(\frac {\mu}
{2}+2n+\frac {1} {2}\)_{n-1}} {(n)_n\,\(-\frac {\mu} {2}-2n+\frac {3}
{2}\)_{\cl{(n-2)/2}}}.
\end{equation}
\end{Problem}
The determinants $D_1(n)$ are rather intriguing. Here are the first
few values:
\begin{align*}
D_1(1)&=\mu+1,\\
D_1(2)&=(\mu+1)(\mu+2),\\
D_1(3)&= \frac {1} {12}{( \mu + 1 ) ( \mu + 2 ) ( \mu + 3 )
( \mu + 14 ) },\\
D_1(4)&=\frac {1} {72}
{( \mu + 1 ) ( \mu + 2 ) ( \mu + 3 )
( \mu + 4 ) ( \mu + 9 ) ( \mu + 14 ) },\\
D_1(5)&= \frac {1} {8640}
( \mu + 1 ) ( \mu + 2 ) ( \mu + 3 )
( \mu + 4 ) ( \mu + 5 ) ( \mu + 9
) \\
&\kern3cm
\times
( 3432 + 722 \mu + 45 \mu^2 + \mu^3 ) ,\\
D_1(6)&=\frac {1} {518400}
( \mu + 1 ) ( \mu + 2 ) ( \mu + 3 )
( \mu + 4 ) ( \mu + 5 ) ( \mu + 6 )
( \mu + 8 ) ( \mu + 13 ) \\
&\kern3cm
\times
( \mu + 15 )
( 3432 + 722 \mu + 45 \mu^2 + \mu^3 ) ,\\
D_1(7)&=\frac {1} {870912000}
( \mu + 1 ) ( \mu + 2 ) ( \mu + 3 )
( \mu + 4 ) ( \mu + 5 ) ( \mu + 6 )
( \mu + 7 ) ( \mu + 8 ) ^2\\
&\kern3cm
\times
( \mu + 13 ) ( \mu + 15 ) ^2
( \mu + 34 ) ( \mu^3 + 47 \mu^2 + 954 \mu + 5928)
,\\
D_1(8)&=\frac {1} {731566080000}
( \mu + 1 ) ( \mu + 2 ) ( \mu + 3 )
( \mu + 4 ) ( \mu + 5 ) ( \mu + 6 )
( \mu + 7 ) ( \mu + 8 ) ^3 \\
&\kern3cm\times
( \mu + 10 ) ( \mu + 15 ) ^2
( \mu + 17 ) ( \mu + 19 ) \\
&\kern3cm\times
( \mu + 21 )
( \mu + 34 ) ( \mu^3 + 47 \mu^2 + 954 \mu + 5928 )
.
\end{align*}
``So," these determinants factor almost completely, there is only a
relatively small (in degree) irreducible factor which is not
linear. (For example, this factor is of degree $6$ for $D_1(9)$ and
$D_1(10)$, and of degree $7$ for $D_1(11)$ and $D_1(12)$.)
Moreover, this ``bigger" factor is always the same for $D_1(2n-1)$ and
$D_1(2n)$. Not only that, the quotient which is predicted in
\eqref{eq:fmu} is at the same time a building block in the result
of the evaluation
of the determinant which enumerates the cyclically symmetric and
descending plane partitions (see
\machSeite{AndrAO}\cite{AndrAO}).
All this begs for an explanation in terms
of a factorisation of the matrix of which the determinant is taken
from. In fact, for the plane partition matrix there is such a
factorisation, due to Mills, Robbins and Rumsey
\machSeite{MiRRAD}\cite[Theorem~5]{MiRRAD} (see also
\machSeite{KratBN}\cite[Theorem~36]{KratBN}).
The question is whether there is a similar one for the matrix in
\eqref{eq:desc-var}.
Inspired by this conjecture and by the variations in
\machSeite{CiEKAA}\cite[Theorems~11--13]{CiEKAA} (see
\machSeite{KratBN}\cite[Theorem~35]{KratBN}) on Andrews' original
determinant evaluation in
\machSeite{AndrAN}\cite{AndrAN},
Guoce Xin (private communication)
observed that, if we change the sign in front of the
Kronecker delta in \eqref{eq:desc-var}, then the resulting determinant
factors completely into linear factors.
\begin{Conjecture} \label{conj:Xin1}
Let $\mu$ be an indeterminate and $n$ be a non-negative integer.
The determinant
\begin{equation} \label{eq:Xin-det1}
\det_{0\le i,j\le n-1}\left(-\delta_{ij}+\binom {\mu+i+j}{j+1}\right)
\end{equation}
is equal to
\begin{multline} \label{eq:Xin-erg1a}
(-1)^{n/2}2^{n(n+2)/4}
\frac {\(\frac \mu2\)_{n/2}} {\(\frac n2\)!}
\( \prod_{i=0}^{(n-2)/2}\frac {i!^2}{(2i)!^2}
\)\\
\times
\( \prod_{i=0}^{\fl{(n-4)/4}} \(\frac\mu2+3i+\frac52\)_{(n-4i-2)/2}^2
\(-\frac\mu2-\frac{3n}2+3i+3\)_{(n-4i-4)/2}^2\)
\end{multline}
if $n$ is even, and it is equal to
\begin{multline} \label{eq:Xin-erg1b}
(-1)^{(n-1)/2}2^{(n+3)(n+1)/4}\,\(\frac {\mu-1} {2}\)_{(n+1)/2}
\( \prod_{i=0}^{(n-1)/2}\frac{i!\,(i+1)!} {(2i)!\,(2i+2)!}\)\\
\times
\(\prod_{i=0}^{\fl{(n-3)/4}}\(\frac\mu2+3i+\frac52\)_{(n-4i-3)/2}^2
\(-\frac\mu2-\frac{3n}2+\frac32+3i\)_{(n-4i-1)/2}^2\)
\end{multline}
if $n$ is odd.
\end{Conjecture}
In fact, it seems that also the ``next" determinant, the determinant
where one replaces $j+1$ at the bottom of the binomial coefficient in
\eqref{eq:Xin-det1} by $j+2$ factors completely when $n$ is odd.
(It does not when $n$ is even, though.)
\begin{Conjecture} \label{conj:Xin2}
Let $\mu$ be an indeterminate. For any odd non-negative integer $n$
there holds
\begin{multline} \label{eq:Xin-det2}
\det_{0\le i,j\le n-1}\left(-\delta_{ij}+\binom
{\mu+i+j}{j+2}\right)\\=
(-1)^{(n-1)/2}2^{(n-1)(n+5)/4}(\mu+1)
\frac {\(\frac \mu2-1\)_{(n+1)/2}} {\(\frac{n+1}2\)!}
\(\prod_{i=0}^{(n-1)/2}\frac {i!^2} {(2i)!^2}
\(\frac\mu2+3i+\frac32\)_{(n-4i-1)/2}^2\)\\
\times
\( \prod_{i=0}^{\fl{(n-3)/4}}
\(-\frac\mu2-\frac{3n}2+3i+\frac52\)_{(n-4i-3)/2}^2\) .
\end{multline}
\end{Conjecture}
For the combinatorialist
I add that all the determinants in Problem~\ref{prob:7}
and Conjectures~\ref{conj:Xin1} and \ref{conj:Xin2} count
certain rhombus tilings, as do the original determinants in
\machSeite{AndrAN}%
\machSeite{AndrAO}%
\machSeite{CiEKAA}\cite{AndrAN,AndrAO,CiEKAA}.
Alain Lascoux (private communication)
did not understand why we should stop here, and he
hinted at a parametric family of determinant evaluations into which the
case of odd $n$ of Conjecture~\ref{conj:Xin1} is embedded as a special
case.
\begin{Conjecture} \label{conj:Xin3}
Let $\mu$ be an indeterminate. For any odd non-negative integers $n$
and $r$ there holds
\begin{multline} \label{eq:Xin-det3}
\det_{0\le i,j\le n-1}\left(-\delta_{i,j+r-1}+\binom
{\mu+i+j}{j+r}\right)\\\kern-1pt
=
(-1)^{(n-r)/2}2^{(n^2+6n-2nr+r^2-4r+2)/4}
\(
\prod _{i=0} ^{r-2}i!\)
\(
\prod _{i=0} ^{(r-3)/2}\frac {(n-2i-2)!^2} {\(\frac {n-2i-3} {2}\)!^2\,
(n+2i)!\, (n+2i+2)!}\)\kern-1pt\\
\times
(\mu-r)\(\frac {\mu+1} {2}\)_{(n-r)/2}\(
\prod _{i=1} ^{r-1}(m-r+i)_{n+r-2i+1}\)
\( \prod_{i=0}^{(n-1)/2}\frac{i!\,(i+1)!} {(2i)!\,(2i+2)!}\)\\
\times
\(\prod_{i=0}^{\fl{(n-r-2)/4}}\(\frac\mu2+3i+r+\frac32\)_{(n-4i-r-2)/2}^2
\(-\frac\mu2-\frac{3n}2+\frac {r} {2}+3i+1\)_{(n-4i-r)/2}^2\).
\end{multline}
\end{Conjecture}
\medskip
The next binomial determinant that I want to mention is, strictly
speaking, not a determinant but a Pfaffian (see \eqref{eq:Pfaff}
for the definition).
While doing {\it $(-1)$-enumeration of sef-complementary plane partitions},
Eisenk\"olbl
\machSeite{EisTAF}\cite{EisTAF} encountered an, I admit, complicated looking
Pfaffian,
\begin{equation} \label{eq:mn}
\underset{1\le i,j\le a}\operatorname{Pf}\big(M(m_1,m_2,n_1,n_2,a,b)\big),
\end{equation}
where $a$ is even and $b$ is odd, and where
\begin{multline*}
M_{ij}(m_1, m_2, n_1, n_2, a, b) \\
= \sum_{l=1} ^{\fl{(a + b - 1)/4)}}
(-1)^{i + j} \(\binom {n_1}{ (b - 1)/2 + \fl{(i - 1)/2} - l + 1}
\binom {m_1} { -a/2 + \fl{(j - 1)/2} + l} \right.\\
-
\left. \binom {n_1} {(b - 1)/2 + \fl{(j - 1)/2} - l + 1}
\binom {m_1} {-a/2 + \fl{(i - 1)/2} + l}\) \\
+
\sum _{l= 1} ^{\cl{(a + b - 1)/4}}
\( \binom {n_2} { (b - 1)/2 + \fl{i/2} - l +1}
\binom {m_2} {-a/2 + \fl{j/2} + l - 1}
\right. \\
-
\left. \binom {n_2} {(b - 1)/2 + \fl{j/2} - l+1}
\binom {m_2} {-a/2 + \fl{i/2} + l-1}\).
\end{multline*}
Remarkably however, experimentally this Pfaffian, first of all,
factors completely into factors which are linear in the variables
$m_1,m_2,n_1,n_2$, but not only that, there seems to be complete
separation, that is, each linear factor contains only one of
$m_1,m_2,n_1,n_2$. One has the impression that this phenomenon should
have an explanation in a factorisation of the matrix in
\eqref{eq:mn}. However, the task of finding one does not seem to be an
easy one in view of the ``entangledness" of the parameters in the sums
of the matrix entries.
\begin{Problem} \label{prob:6}
\leavevmode
\kern-5pt\footnote{Theresia Eisenk\"olbl has recently
solved this problem in
\machSeite{EisTAG}\cite{EisTAG}.}
Find and prove the closed form evaluation of the Pfaffian in
\eqref{eq:mn}.
\end{Problem}
Our next determinants can be considered as {\it shuffles} of two
binomial determinants. Let us first consider
\begin{equation} \label{det-ep}
\det_{1\le i,j\le a+m} \begin{pmatrix} \dbinom{b+c+m}{b-i+j}&
\text {\scriptsize $1\le i\le a$}\\
\dbinom{\frac {b+c} {2}}{\frac {b+a} {2}-i+j+\varepsilon}&
\text {\scriptsize $a+1\le i \le a+m$}
\end{pmatrix}.
\end{equation}
In fact, if $\varepsilon=0$, and if $a,b,c$ all have the same parity,
then this is exactly the determinant in \eqref{mat1}, the evaluation
of which proves Theorem~\ref{enum}, as we explained in
Section~\ref{sec:comb}. If $\varepsilon=1/2$ and $a$ has parity
different from that of $b$ and $c$, then the corresponding determinant
was also evaluated in
\machSeite{CiEKAA}\cite{CiEKAA}, and this evaluation implied the
companion result to Theorem~\ref{enum} that we mentioned immediately
after the statement of the theorem. In the last section of
\machSeite{CiEKAA}\cite{CiEKAA},
it is reported that, apparently, there are also nice
closed forms for the determinant in \eqref{det-ep} for $\varepsilon=1$
and $\varepsilon=3/2$, both of which imply as well enumeration
theorems for {\it rhombus tilings of a hexagon with an equilateral
triangle removed from
its interior} (see Conjectures~1 and 2 in
\machSeite{CiEKAA}\cite{CiEKAA}). We reproduce the
conjecture for $\varepsilon=1$ here, the one for $\varepsilon=3/2$ is
very similar in form.
\begin{Conjecture} \label{conj:loch1}
Let $a,b,c,m$ be non-negative integers, $a,b,c$ having the same parity.
Then for $\varepsilon=1$ the determinant in \eqref{det-ep} is equal to
\begin{multline} \label{eq:1-step}
\frac {1} {4}\frac {\h(a + m)\h(b + m)\h(c + m)\h(a + b + c + m)
}
{\h(a + b + m)\h(a + c + m)\h(b + c + m)
}\\
\times
\frac {\h(m + \left \lceil {\frac{a + b + c}{2}} \right \rceil)
\h(m + \left \lfloor {\frac{a + b + c}{2}} \right \rfloor)
} {\h({\frac{a + b}{2}} + m+1) \h({\frac{a + c}{2}} + m-1)\h({\frac{b + c}{2}} + m)
}
\\
\times\frac {\h(\left \lceil {\frac{a}{2}} \right \rceil)
\h(\left \lceil {\frac{b}{2}} \right \rceil)
\h(\left \lceil {\frac{c}{2}} \right \rceil)
\h(\left \lfloor {\frac{a}{2}} \right \rfloor)\,
\h(\left \lfloor {\frac{b}{2}} \right \rfloor)\,
\h(\left \lfloor {\frac{c}{2}} \right \rfloor)\,
}
{\h({\frac{m}{2}} + \left \lceil {\frac{a}{2}} \right \rceil)\,
\h({\frac{m}{2}} + \left \lceil {\frac{b}{2}} \right \rceil)\,
\h({\frac{m}{2}} + \left \lceil {\frac{c}{2}} \right \rceil)\,
\h({\frac{m}{2}} + \left \lfloor {\frac{a}{2}} \right \rfloor)\,
\h({\frac{m}{2}} + \left \lfloor {\frac{b}{2}} \right \rfloor)\,
\h({\frac{m}{2}} + \left \lfloor {\frac{c}{2}} \right \rfloor)\,
}\\
\times
\frac {\h(\frac{m}{2})^2 \h({\frac{a + b + m}{2}})^2
\h({\frac{a + c + m}{2}})^2 \h({\frac{b + c +
m}{2}})^2
}
{\h({\frac{m}{2}} + \left \lceil {\frac{a + b + c}{2}} \right \rceil)
\h({\frac{m}{2}} + \left \lfloor {\frac{a + b + c}{2}} \right \rfloor)
\h({\frac{a + b}{2}}-1)\h({\frac{a + c}{2}}+1)\h({\frac{b + c}{2}})
}P_1(a,b,c,m),
\end{multline}
where $P_1(a,b,c,m)$ is the polynomial given by
$$P_1(a,b,c,m)=\begin{cases} (a+b)(a+c)+2am&\text {if $a$ is
even,}\\
(a+b)(a+c)+2(a+b+c+m)m&\text {if $a$ is odd,}\end{cases}$$
and where $\h(n)$ is the hyperfactorial defined in \eqref{eq:hyperfac}.
\end{Conjecture}
Two other examples of determinants in which the upper part is given
by one binomial matrix, while the lower part is given by a different
one, arose in
\machSeite{CiKrAC}\cite[Conjectures~A.1 and A.2]{CiKrAC}. Again, both of
them seem to factor completely into linear factors, and both of them
imply enumeration results for {\it rhombus tilings of a certain V-shaped
region}. The right-hand sides of the (conjectured)
results are the weirdest ``closed" forms in enumeration that I am
aware of.\footnote{No non-trivial simplifications seem to be possible.}
We state just the first of the two conjectures, the other is very similar.
\begin{Conjecture}
Let $x,y,m$ be non-negative integers. Then the determinant
\begin{equation} \label{eq:A.2}
\det_{1\le i,j\le m+y}\left(\left\{\begin{matrix} \binom
{x+i}{x-i+j}&i=1,\dots,m\hfill\\
\binom{x+2m-i+1}{m+y-2i+j+1}&i=m+1,\dots,m+y
\end{matrix}\right\}\right).
\end{equation}
is equal to
\begin{multline} \label{eq:A.1}
\prod _{i=1} ^{m}\frac {(x+i)!} {(x-i+m+y+1)!\,(2i-1)!}
\prod _{i=m+1} ^{m+y}\frac {(x+2m-i+1)!}
{(2m+2y-2i+1)!\,(m+x-y+i-1)!}\\
\times
{2^{\binom {m}2 + \binom y2 }}
\prod_{i = 1}^{m-1}i!
\prod_{i = 1}^{y-1}i!
\prod_{i \ge 0}^{}
({ \textstyle x+i+{\frac{3}{2}} }) _{m-2i-1}
\prod_{i \ge 0}^{}
({ \textstyle x - y+{\frac{5}{2}} + 3 i}) _{ \left \lfloor
{\frac{3 y}{2}} -{\frac{9 i}{2}} \right \rfloor-2}\\
\times \prod_{i \ge 0}^{}
({ \textstyle x + {\frac{3 m}{2}} - y + \left \lceil
{\frac{3 i}{2}} \right \rceil+\frac {3} {2}})
_{ 3 \left \lceil {\frac{y}{2}}
\right \rceil - \left \lceil
{\frac{9 i}{2}} \right \rceil -2}
\prod_{i \ge 0}
^{}
({ \textstyle { x+ {\frac{3 m}{2}} - y + \left \lfloor
{\frac{3 i}{2}} \right \rfloor+2}}) _{ 3 \left \lfloor
{\frac{y}{2}} \right \rfloor - \left \lfloor {\frac{9 i}{2}} \right
\rfloor-1}\\
\times \prod_{i \ge 0}^{}
({ \textstyle x+m - \left \lfloor {\frac{y}{2}} \right
\rfloor}+i+1) _{ 2 \left \lfloor {\frac{y}{2}} \right
\rfloor-m - 2 i }
\prod_{i \ge 0}
^{}
({ \textstyle x + \left \lfloor {\frac{y}{2}} \right \rfloor+i+2})
_{m - 2 \left \lfloor {\frac{y}{2}} \right \rfloor-2i-2}
\\\times
{\frac{ \displaystyle
\prod_{i = 0}^{y}
({ \textstyle x - y+3i+1}) _{m + 2 y-4i}
\prod_{i = 0}^{ \left \lceil {\frac{y}{2}} \right \rceil-1}
({ \textstyle x+m - y+i+1}) _{3 y-m-4i}
}
{\displaystyle
\prod_{i \ge 0}^{}
({ \textstyle x+ {\frac{m}{2}} - {\frac{y}{2}}+i+1}) _{y-2i}\,
({ \textstyle x + {\frac{m}{2}}-
{\frac{y}{2}}+i+{\frac{3}{2}}}) _{y-2i-1} }}\\
\times\frac {\displaystyle
\prod_{i = 0}^{y}
({ \textstyle x+i+2 }) _{2m - 2 i - 1} }
{ ({ \textstyle x + y+2}) _{ m - y-1} \,(m+x-y+1)_{m+y} }.
\end{multline}
Here, shifted factorials occur with positive as well as with negative
indices. The convention with respect to which these have
to be interpreted is
$$(\alpha)_k:=\begin{cases} \alpha(\alpha+1)\cdots(\alpha+k-1)&\text {if
}k>0,\\
1&\text {if }k=0,\\
1/(\alpha-1)(\alpha-2)\cdots(\alpha+k)&\text {if }k<0.
\end{cases}$$
All products $\prod
_{i\ge0} ^{}(f(i))_{g(i)}$ in \eqref{eq:A.1}
have to interpreted as the products over
all $i\ge0$ for which $g(i)\ge0$.
\end{Conjecture}
For further conjectures of determinants of shuffles of two binomial
matrices I refer the reader to Conjectures~1--3 in Section~4 of
\machSeite{FuKrAC}\cite{FuKrAC}.
All of them imply also enumeration results for rhombus
tilings of hexagons. This time, these would be results about the number of
{\it rhombus tilings of a symmetric hexagon with some fixed rhombi on the
symmetry axis}.
\subsection{Determinants of matrices with recursive entries}
Binomial coefficients $\binom {i+j}i$
satisfy the basic recurrence of the Pascal triangle,
\begin{equation} \label{eq:Pascal}
p_{i,j}=p_{i,j-1}+p_{i-1,j}.
\end{equation}
We have seen many determinants of matrices with entries containing
binomial coefficients in the preceding subsection and in
\machSeite{KratBN}\cite[Sec.~3]{KratBN}.
In \machSeite{BacRAA}\cite{BacRAA}, Bacher reports an experimental
study of determinants of matrices $(p_{i,j})_{0\le i,j\le n-1}$, where
the coefficients $p_{i,j}$ satisfy the recurrence \eqref{eq:Pascal}
(and sometimes more general recurrences), but where the initial
conditions for $p_{i,0}$ and $p_{0,i}$, $i\ge 0$, are different from
the ones for binomial coefficients. He makes many interesting
observations. The most intriguing one says that these determinants
satisfy also a linear recurrence (albeit a much longer one).
It is intriguing because it
points towards the possibility of {\it automatising determinant
evaluations}\footnote{The reader should recall that the successful
automatisation
\machSeite{PeWZAA}%
\machSeite{WegsAA}%
\machSeite{WiZeAC}%
\machSeite{ZeilAM}%
\machSeite{ZeilAV}%
\cite{PeWZAA,WegsAA,WiZeAC,ZeilAM,ZeilAV}
of the evaluation of binomial and hypergeometric sums
is fundamentally based on producing recurrences by the computer.},
something that several authors (cf.\ e.g.\
\machSeite{AmZeAB}%
\machSeite{KratBN}%
\machSeite{PeWiAA}%
\cite{AmZeAB,KratBN,PeWiAA}) have been aiming at (albeit, with only
limited success up to now). The
conjecture (and, in fact, a generalisation thereof) has been proved by
Petkov\v sek and Zakraj\v sek in
\machSeite{ZaPeAA}\cite{ZaPeAA}. Still, there remains a large gap to
fill until computers will replace humans doing determinant evaluations.
The paper
\machSeite{BacRAA}\cite{BacRAA} contained as well several
pretty conjectures
on closed form evaluations of special cases of such
determinants. These were subsequently proved in
\machSeite{KratBU}\cite{KratBU}. We state three of them in the
following three theorems. The first two are proved in
\machSeite{KratBU}\cite{KratBU} by working out the
LU-factorisation (see ``Method~1" in Section~\ref{sec:eval})
for the matrices of
which the determinant is computed. The third one is derived by simple
row and column operations.
\begin{Theorem} \label{thm:Pasc1}
Let $(a_{i,j})_{i,j\ge0}$ be the sequence given by the
recurrence
$$a_{i,j}=a_{i-1,j}+a_{i,j-1}+x\,a_{i-1,j-1},\quad \quad i,j\ge1,
$$
and the initial conditions $a_{i,0}=\rho^i$ and $a_{0,i}=\sigma^i$,
$i\ge0$. Then
\begin{equation} \label{eq:Pasc1}
\det_{0\le i,j\le n-1}(a_{i,j})=(1+x)^{\binom
{n-1}2}(x+\rho+\sigma-\rho\sigma)^{n-1}.
\end{equation}
\quad \quad \qed
\end{Theorem}
\begin{Theorem} \label{thm:Pasc2}
Let $(a_{i,j})_{i,j\ge0}$ be the sequence given by the
recurrence
$$a_{i,j}=a_{i-1,j}+a_{i,j-1}+x\,a_{i-1,j-1},\quad \quad i,j\ge1,
$$
and the initial conditions $a_{i,i}=0$, $i\ge0$, $a_{i,0}=\rho^{i-1}$ and
$a_{0,i}=-\rho^{i-1}$, $i\ge1$. Then
\begin{equation} \label{eq:Pasc2}
\det_{0\le i,j\le 2n-1}(a_{i,j})=(1+x)^{2(n-1)^2}
(x+\rho)^{2n-2}.
\end{equation}
\quad \quad \qed
\end{Theorem}
\begin{Theorem}\label{thm:Pasc3}
Let $(a_{i,j})_{i,j\ge0}$ be the sequence given by the recurrence
$$a_{i,j}=a_{i-1,j}+a_{i,j-1}+x\,a_{i-1,j-1},\quad \quad i,j\ge1,
$$
and the initial conditions $a_{i,0}=i$ and $a_{0,i}=-i$,
$i\ge0$. Then
\begin{equation} \label{eq:Pasc3}
\det_{0\le i,j\le 2n-1}(a_{i,j})=(1+x)^{2n(n-1)}.
\end{equation}
\quad \quad \qed
\end{Theorem}
Certainly, the proofs in
\machSeite{KratBU}\cite{KratBU} are not very illuminating. Neuwirth
\machSeite{NeuwAE}\cite{NeuwAE} has looked more carefully into the
structure of recursive sequences of the type as those in
Theorems~\ref{thm:Pasc1}--\ref{thm:Pasc3}. Even more generally, he
looks at sequences $(f_{i,j})_{i,j\ge0}$
satisfying the recurrence relation
\begin{equation} \label{eq:rec}
f_{i,j}=c_{j}f_{i-1,j}+d_jf_{i,j-1}+e_jf_{i-1,j-1},\quad i,j\ge1,
\end{equation}
for some given sequences $(c_j)_{j\ge1}$, $(d_j)_{j\ge1}$,
$(e_j)_{j\ge1}$. He approaches the problem by finding appropriate
{\it matrix decompositions} for the (infinite) matrix
$(f_{i,j})_{i,j\ge0}$. In two special
cases, he is able to apply his
decomposition results to work out the LU-factorisation of the matrix
$(f_{i,j})_{i,j\ge0}$ explicitly, which then yields an elegant
determinant evaluation in both of these cases. Neuwirth's first
result
\machSeite{NeuwAE}\cite[Theorem~5]{NeuwAE}
addresses the case where the initial values $f_{0,j}$ satisfy a first
order recurrence determined by the coefficients $d_j$ from
\eqref{eq:rec}. It generalises Theorem~\ref{thm:Pasc1}. There is no
restriction on the initial values $f_{i,0}$ for $i\ge1$.
\begin{Theorem} \label{thm:Neuw1}
Let $(c_j)_{j\ge1}$, $(d_j)_{j\ge1}$ and $(e_j)_{j\ge1}$ be given
sequences, and let $(f_{i,j})_{i,j\ge0}$ be the doubly indexed
sequence given by the recurrence
\eqref{eq:rec} and the initial conditions $f_{0,0}=1$ and
$f_{0,j}=d_jf_{0,j-1}$, $j\ge1$. Then
\begin{equation} \label{eq:Neuw1}
\det_{0\le i,j\le n-1}(f_{i,j})=
\prod _{0\le i<j\le n-1} ^{}(e_{i+1}+c_jd_{i+1}).
\end{equation}
\quad \quad \qed
\end{Theorem}
Neuwirth's second
result
\machSeite{NeuwAE}\cite[Theorem~6]{NeuwAE}
also generalises Theorem~\ref{thm:Pasc1}, but in a different way.
This time, the initial values $f_{0,j}$, $j\ge1$, are free, whereas
the initial values $f_{i,0}$ satisfy a first order recurrence
determined by
the coefficients $c_j$ from the recurrence \eqref{eq:rec}.
Below, we state its most attractive special case, in which all the
$c_j$'s are identical.
\begin{Theorem} \label{thm:Neuw2}
Let $(d_j)_{j\ge1}$ and $(e_j)_{j\ge1}$ be given
sequences, and let $(f_{i,j})_{i,j\ge0}$ be the doubly indexed
sequence given by the recurrence
\eqref{eq:rec} and the initial conditions $f_{0,0}=1$ and
$f_{i,0}=cf_{i-1,0}$, $i\ge1$. Then
\begin{equation} \label{eq:Neuw2}
\det_{0\le i,j\le n-1}(f_{i,j})=
\prod _{i=1} ^{n-1}(cd_{i}+e_{i})^{n-i}.
\end{equation}
\quad \quad \qed
\end{Theorem}
\subsection{Determinants for signed permutations} \label{sec:signed}
The next class of determinants that we consider are determinants of
matrices in which rows and columns are indexed by
{\it elements of reflection groups}
(the latter being groups generated by reflections of hyperplanes in real
$n$-dimensional space; see
\machSeite{HumpAC}\cite{HumpAC} for more information on these
groups, and, more generally, on Coxeter groups).
The prototypical example of a reflection group is the {\it symmetric group}
$\mathfrak S_n$ of permutations of an $n$-element set. In
\machSeite{KratBN}\cite{KratBN}, there
appeared two determinant evaluations associated to the symmetric group,
see Theorems~55 and 56 in
\machSeite{KratBN}\cite{KratBN}. They concerned evaluations of
determinants of the type
\begin{equation} \label{eq:stat}
\det_{\sigma,\pi\in \mathfrak S_n}\(q^{\operatorname{stat}(\sigma\pi^{-1})}\),
\end{equation}
due to Varchenko, Zagier, and Thibon, respectively, in which stat is
the statistic {\it ``number of inversions,"} respectively {\it``major index."}
We know that in many fields of mathematics there exist certain
diseases which are typical for that field. Algebraic combinatorics is
no exception. Here, I am {\it not\/} talking
of the earlier mentioned ``$q$-disease"
(see Footnote~\ref{foot:q};
although, due to the presence of $q$ we might also count it as a case
of $q$-disease), but of the disease which manifests itself by the
question ``And what about the other types?"\footnote{\label{foot:root}%
In order
to give a reader who is not acquainted with the language and theory of
reflection groups an idea what this question is referring to,
I mention that all finite reflection groups have
been classified, each having been assigned a certain ``type." So,
usually one proves something for the symmetric group $\mathfrak S_n$,
which, according to this classification, has type $A_{n-1}$, and
then somebody (which could be oneself) will ask the question ``Can you
also do this for the other types?", meaning whether or not there exists
an analogous result for the other finite reflection groups.}
So let us ask this question, that is,
are there theorems similar to the two theorems
which we mentioned above for other
reflection groups?
So, first of all we need analogues of the statistics ``number of
inversions" and ``major index" for other reflection groups. Indeed,
these are available in the literature. The analogue of ``number of
inversions" is the so-called {\it length} of an element in a Coxeter
group (see
\machSeite{HumpAC}\cite{HumpAC} for the definition).
As a matter of fact, a closed form evaluation
of the determinant \eqref{eq:stat}, where $\mathfrak S_n$ is replaced
by any finite or affine reflection group, and where stat is the
length, is known (and was already implicitly mentioned in
\machSeite{KratBN}\cite{KratBN}). This result is due to Varchenko
\machSeite{VarcAC}\cite[Theorem~(1.1),
where $a(H)$ is specialised to $q$]{VarcAC}.
His result is actually much more general, as it is valid for real
hyperplane arrangements in which each hyperplane is assigned a different
weight. I will not state it here explicitly because I do
not want to go through the definitions and notations which would be
necessary for doing that.
So, what about analogues of the ``major index" for other reflection
groups? These are also available, and there are in fact several of
them. The first person to introduce a major index for reflection groups
other than the symmetric groups was Reiner in
\machSeite{ReivAC}\cite{ReivAC}.
He proposed a major index for the {\it hyperoctahedral group} $B_n$, which
arose naturally in his study of {\it $P$-partitions for signed
posets}. The elements of $B_n$ are often called {\it signed
permutations}, and they are all elements of the form
$\pi_1\pi_2\dots\pi_n$, where
$\pi_i\in\{\pm1,\pm2,\dots,\pm n\}$, $i=1,2,\dots,n$, and where
$\vert\pi_1\vert\vert\pi_2\vert\dots\vert\pi_n\vert$ is a permutation
in $\mathfrak S_n$. To define their multiplicative structure, it is
most convenient to view $\pi=\pi_1\pi_2\dots\pi_n$ as a linear operator on
${\mathbb R}^n$ acting by permutation and sign changes of the co-ordinates. To
be precise, the action is given by
$\pi(e_i)=(\operatorname{sgn}\pi_i)e_{\vert\pi_i\vert}$, where $e_i$ is the $i$-th
standard basis vector in ${\mathbb R}^n$, $i=1,2,\dots,n$. The multiplication
of two signed permutations is then simply the composition of the
corresponding linear operators.
The major index $\operatorname{maj}_B\pi$ of an element $\pi\in B_n$
which Reiner defined is, as in the symmetric group case, the sum of
all positions of {\it descents} in $\pi$. (There is a natural notion
of ``descent" for any Coxeter group.) Concretely, it is
$$\operatorname{maj}_B\pi:=\chi(\pi_n<0)+
\sum _{i=1} ^{n-1}i\cdot\chi(\pi_i>_B\pi_{i+1}),$$
where we impose the order $1<_B2<_B\cdots<_Bn<_B-n<_B\cdots<_B-2<_B-1$
on our ground set $\{\pm1,\pm2,\dots,\pm n\}$,
and where $\chi(\mathcal A)=1$ if $\mathcal A$ is true and
$\chi(\mathcal A)=0$ otherwise.
There is overwhelming computational
evidence\footnote{\label{foot:maj}The
reader may wonder what this computational evidence
could actually be. After all, we are talking about a determinant of a
matrix of size $2^nn!$. More concretely, for $n=1,2,3,4,5$ these are
matrices of size $2$, $8$, $48$, $384$, $3840$, respectively.
While {\sl Maple} or {\sl Mathematica} have no problem to compute
these determinants for $n=1$ and $n=2$, it takes already considerable time to
do the computation for $n=3$, and it is, of course, completely hopeless to
let them compute the one for $n=4$, a determinant of a matrix of size
$384$ which has polynomial entries (cf.\
Footnote~\ref{foot:kompl}). However, the results for $n=1,2,3$
already ``show" that the determinant will factor completely into
factors of the form $1-q^i$, $i=1,2,\dots,2n$. One starts to expect
the same to be true for higher $n$. To get a formula for $n=4$, one
would then apply the tricks explained in
Footnote~\ref{foot:tricks}. That is, one specialises $q$ to $4$, at
which value the first $8$ cyclotomic polynomials (in fact, even more)
are clearly distinguishable by their prime factorisations, and one computes
the determinant. The exponents of the various factors $1-q^i$ can then
be extracted from the exponents of the prime factors in the prime
factorisation of the determinant with $q=4$. Unfortunately, the data
collected for $n=1,2,3,4$ do not suffice to come up with a
guess, and, on the other hand, {\sl Maple} and {\sl Mathematica} will
certainly be incapable to compute a determinant of a matrix of size
$3840$ (which, just to store it on the disk, occupies already 10
megabytes \dots). So then, what did I mean when I said that the conjecture is
based on data including $n=5$? This turned out to become a ``test
case" for {\sl LinBox}, a C++ template library for exact
high-performance linear algebra
\machSeite{DGGGAA}\cite{DGGGAA}, which is freely available under
{\tt http://linalg.org}. To be honest,
I was helped by Dave Saunders and
Zhendong Wan (two of the developers) who applied {\sl LinBox}
to do rank and Smith normal form computations for the specialised matrix with
respect to various prime powers (each of which taking several hours).
The specific computational approach that worked here is quite recent
(thus, it came just in time for our purpose),
and is documented in
\machSeite{SaWaAA}\cite{SaWaAA}.
The results of the computations made it possible to come
up with a ``sure" prediction for the exponents with which the various
prime factors occur in the prime factorisation of the specialised determinant.
As in the case $n=4$, the exponents of the various factors $1-q^i$,
$i=1,2,\dots,2n$ can then easily be extracted. (The guesses were
subsequently also
tested with special values of $q$ other than $q=4$.)}
that the ``major-de\-ter\-min\-ant" for $B_n$, i.e., the
determinant \eqref{eq:stat} with $\text{stat}=\operatorname{maj}_B$ and with
$\mathfrak S_n$ replaced by $B_n$, factors completely into cyclotomic
polynomials.
\begin{Conjecture} \label{prob:1}
For any positive integer $n$, we have
\begin{equation} \label{eq:maj-Bn}
\det_{\sigma,\pi\in B_n}\(q^{\operatorname{maj}_B(\sigma\pi^{-1})}\)=
\prod_{i=1}^n (1-q^{2i})^{2^{n-1}n!/i}
\prod_{i=2}^n (1-q^i)^{2^n n!(i-1)/i}.
\end{equation}
\end{Conjecture}
A different major index for $B_n$ was proposed by Adin and Roichman in
\machSeite{AdRoAC}\cite{AdRoAC}.
It arises there naturally in a combinatorial study of
{\it polynomial algebras which are diagonally invariant under $B_n$}.
(In fact, more generally, {\it wreath products} of the form $C_m\wr
\mathfrak S_n$, where $C_m$ is the cyclic group of order $m$,
and their diagonal actions on polynomial algebras are
studied in
\machSeite{AdRoAC}\cite{AdRoAC}. These groups are also sometimes called {\it
generalised reflection groups}.
In this context, $B_n$ is the special case
$C_2\wr\mathfrak S_n$.)
If we write $\operatorname{neg}\pi$ for the number of $i$ for which $\pi_i$ is
negative, then the {\it flag-major index} fmaj of Adin and Roichman is
defined by
\begin{equation} \label{eq:fmaj-def}
\operatorname{fmaj}\pi:=2\operatorname{maj}_A\pi+\operatorname{neg}\pi,
\end{equation}
where $\operatorname{maj}_A$ is the ``ordinary" major index due to MacMahon,
$$\operatorname{maj}_A\pi:=\sum _{i=1} ^{n-1}i\cdot\chi(\pi_i>\pi_{i+1}).$$
If one now goes to the computer and calculates the determinant on the
left-hand side of
\eqref{eq:maj-Bn} with $\operatorname{maj}_B$ replaced by $\operatorname{fmaj}$
for $n=1,2,3,4,5$ (see
Footnote~\ref{foot:maj} for the precise meaning of ``calculating the
determinant for $n=1,2,3,4,5$"), then again the results factor
completely into cyclotomic factors. Even more generally, it seems that
one can treat the two parts on the right-hand side of
\eqref{eq:fmaj-def}, that is ``major index" and
``number of negative letters," separately.
\begin{Conjecture} \label{prob:2}
For any positive integer $n$, we have
\begin{equation} \label{eq:fmaj-Bn}
\det_{\sigma,\pi\in B_n}\(q^{\operatorname{maj}_A(\sigma\pi^{-1})}p^{\operatorname{neg}(\sigma\pi^{-1})}\)=
\prod _{i=1} ^{n}(1-p^{2i})^{2^{n-1}n!/i}
\prod _{i=2} ^{n}(1-q^i)^{2^{n}n!(i-1)/i}.
\end{equation}
\end{Conjecture}
I should remark that Adin and Roichman have shown in
\machSeite{AdRoAC}\cite{AdRoAC}
that the statistics fmaj is equidistributed with the statistics length
on $B_n$. However, even in the case where we just look at the
flag-major determinant (that is, the case where $q=p^2$ in
Conjecture~\ref{prob:2}), this does not seem to help. (Neither length nor
flag-major index satisfy a simple law with respect to multiplication
of signed permutations.) In fact, from the
data one sees that the flag-major determinants are different from the
length determinants (that is, the determinants \eqref{eq:stat}, where
$\mathfrak S_n$ is replaced by $B_n$ and stat is flag-major,
respectively length).
Initially, I had my program wrong, and,
instead of taking the (ordinary) major index $\operatorname{maj}_A$ of the signed
permutation $\pi=\pi_1\pi_2\dots\pi_n$ in \eqref{eq:fmaj-def},
I computed taking the major index of the {\it absolute value}
of $\pi$. This absolute value is obtained
by forgetting all signs of the letters of $\pi$, that is,
writing $\vert\pi\vert$ for the absolute value of $\pi$,
$\vert\pi\vert=\vert\pi_1\vert\,\vert\pi_2\vert\,\dots\,\vert\pi_n\vert$.
Curiously, it seems that also this ``wrong"
determinant factors nicely. (Again, the evidence for this
conjecture is based on data which were obtained in the way described in
Footnote~\ref{foot:maj}.)
\begin{Conjecture} \label{prob:3}
For any positive integer $n$, we have
\begin{equation} \label{eq:fabsmaj-Bn}
\det_{\sigma,\pi\in B_n}\(q^{\operatorname{maj}\vert\sigma\pi^{-1}\vert}p^{\operatorname{neg}(\sigma\pi^{-1})}\)=
(1-p^{2})^{2^{n-1}n\cdot n!}
\prod _{i=2} ^{n}(1-q^i)^{2^{n}n!(i-1)/i}.
\end{equation}
\end{Conjecture}
Since, as I indicated earlier, Adin and Roichman actually define a
flag-major index for wreath
products $C_m\wr\mathfrak S_n$, a question that suggests itself is
whether or not we can expect closed product formulae for the
corresponding determinants. Clearly, since we are now dealing with
determinants of the size $m^nn!$, computer computations will
exhaust our computer's resources even faster if $m>2$. The
calculations that I was able to do suggest strongly that there is indeed an
extension of the statement in Conjecture~\ref{prob:2} to the case
of arbitrary $m$, if one uses the definition of major index
and the ``negative" statistics for $C_m\wr\mathfrak S_n$ as
in \machSeite{AdRoAC}\cite{AdRoAC}. (See
\machSeite{AdRoAC}\cite[Section~3]{AdRoAC}
for the definition of the major index.
The sum on the right-hand side of
\machSeite{AdRoAC}\cite[(3.1)]{AdRoAC} must be taken as the extension of
the ``negative" statistics neg to $C_m\wr\mathfrak S_n$.)
\begin{Problem} \label{prob:4}
Find and prove the closed form evaluation of
\begin{equation} \label{eq:maj-wreath}
\det_{\sigma,\pi\in C_m\wr \mathfrak S_n}\(q^{\operatorname{maj}(\sigma\pi^{-1})}
p^{\operatorname{neg}(\sigma\pi^{-1})}\),
\end{equation}
where $\operatorname{maj}$ and $\operatorname{neg}$ are the extensions to $C_m\wr \mathfrak S_n$
of the statistics $\operatorname{maj}_A$ and
$\operatorname{neg}$ in Conjecture~{\em\ref{prob:2}}, as described in
the paragaph above.
\end{Problem}
Together with Brenti, Adin and Roichman proposed another major
statistics for signed permutations in
\machSeite{AdBRAA}\cite{AdBRAA}. They call it the
{\it negative major index}, denoted nmaj, and it is defined as the sum of the
ordinary major index and the sum of the absolute values of the
negative letters, that is,
$$\operatorname{nmaj}\pi:=\operatorname{maj}_A\pi+\operatorname{sneg}\pi,$$
where $\operatorname{sneg}\pi:=-\sum _{i=1} ^{n}\chi(\pi_i<0)\pi_i$.
Also for this statistics, the corresponding determinant seems to
factor nicely. In fact, it seems that one can again treat the two
components of the definition of the statistics, that is, ``major index" and
``sum of negative letters," separately. (Once more,
the evidence for this
conjecture is based on data which were obtained in the way described in
Footnote~\ref{foot:maj}.)
\begin{Conjecture} \label{prob:8}
For any positive integer $n$, we have
\begin{equation} \label{eq:nmaj-Bn}
\det_{\sigma,\pi\in
B_n}\(q^{\operatorname{maj}_A(\sigma\pi^{-1})}p^{\operatorname{sneg}(\sigma\pi^{-1})}\)=
\prod _{i=1} ^{n}(1-p^{2i^2})^{2^{n-1}n!/i}
\prod _{i=2} ^{n}(1-q^i)^{2^{n}n!(i-1)/i}.
\end{equation}
\end{Conjecture}
If one compares the (conjectured) result with the (conjectured) one
for the ``flag-major determinant" in Conjecture~\ref{prob:2},
then one notices the somewhat mind-boggling fact that
one obtains the right-hand side of \eqref{eq:nmaj-Bn} from the one
of \eqref{eq:fmaj-Bn} by simply replacing (in the factored form
of the latter) $1-p^{2i}$ by $1-p^{2i^2}$, everything else, the
exponents, the ``$q$-part", is identical. It is difficult to imagine
an intrinsic explanation why this should be the case.
Since Thibon's proof of the evaluation of the determinant
\eqref{eq:stat} with stat being the (ordinary) major index for
permutations (see
\machSeite{KratBN}\cite[Appendix~C]{KratBN})
involved the {\em descent algebra} of the symmetric group, viewed
in terms of {\em non-commutative symmetric functions}, one might
speculate that to prove Conjectures~\ref{prob:1}--\ref{prob:3} and
\ref{prob:8} it may be necessary to work with $B_n$ versions of descent
algebras (which exist, see
\machSeite{SoloAA}\cite{SoloAA})
and non-commutative symmetric functions (which also exist, see
\machSeite{ChoCAA}\cite{ChoCAA}).
Adriano Garsia points out that all the determinants in \eqref{eq:stat},
\eqref{eq:maj-Bn}, \eqref{eq:fmaj-Bn}--\eqref{eq:nmaj-Bn} are special
instances of {\it group determinants}. (See the excellent survey article
\machSeite{LamTAA}\cite{LamTAA} for information on group
determinants.)
The main theorem on group determinants, due to Frobenius, says that
a general group determinant factorises into irreducible factors, each
of which corresponding to an irreducible representation of the group,
and the exact exponent to which the irreducible factor is raised is
the degree of the corresponding irreducible representation. In view of
this, Garsia poses the following problem, a solution of which
would refine the above
conjectures, Problem~\ref{prob:4},
and the earlier mentioned results of Varchenko, Zagier,
and Thibon.
\begin{Problem} \label{prob:Frob}
For each of the above special group determinants, determine
the closed formula for the value of the irreducible factor corresponding
to a fixed irreducible representation of the involved group
{\em(}$\mathfrak S_n$, $B_n$, $C_m\wr \mathfrak S_n$, respectively{\em)}.
\end{Problem}
It seems that a solution to this problem has not even been worked out
for the determinant which is the subject of the results of Varchenko
and Zagier, that is, for the determinant
\eqref{eq:stat} with stat being the number of inversions.
For further work on statistics for (generalised) reflection groups
(thus providing further prospective candidates for forming interesting
determinants), I refer the reader to
\machSeite{AdBRAA}%
\machSeite{AdBRAB}%
\machSeite{BagnAA}%
\machSeite{BernAA}%
\machSeite{BernAB}%
\machSeite{BiagAA}%
\machSeite{BiCaAA}%
\machSeite{FoHaAM}%
\machSeite{FoHaAN}%
\machSeite{FoHaAO}%
\machSeite{HaLRAA}%
\machSeite{ReRoAA}%
\cite{AdBRAA,AdBRAB,BagnAA,BernAA,BernAB,BiagAA,BiCaAA,FoHaAM,FoHaAN,FoHaAO,HaLRAA,ReRoAA}.
I must report that, somewhat disappointingly, it seems that the
various major indices proposed for the group $D_n$ of even signed
permutations (see
\machSeite{BiagAA}%
\machSeite{BiCaAA}%
\machSeite{ReivAC}%
\cite{BiagAA,BiCaAA,ReivAC})
apparently do not give rise to determinants in the same
way as above that have nice product formulae. This remark seems to
also apply to determinants formed in an anlogous way by using
the various statistics proposed for the alternating group in
\machSeite{ReRoAB}\cite{ReRoAB}.
\subsection{More poset and lattice determinants} \label{sec:poset}
Continuing the discussion of determinants which arise under the
influence of the above-mentioned ``reflection group disease," we turn
our attention to two miraculous
determinants which were among the last things Rodica Simion was able
to look at. Some of her considerations in this direction are reported in
\machSeite{SchFAA}\cite{SchFAA}.
The first of the two is a determinant of a matrix the rows and columns of
which are indexed by {\it type $B$ non-crossing partitions}.
This determinant is inspired by the evaluation of
an analogous one for {\it ordinary}
non-crossing partitions (that is, in ``reflection group language,"
{\it type $A$ non-crossing
partitions}), due to Dahab
\machSeite{DahaAA}\cite{DahaAA} and Tutte
\machSeite{TuttAC}\cite{TuttAC} (see
\machSeite{KratBN}\cite[Theorem~57, (3.69)]{KratBN}). Recall (see
\machSeite{StanAP}\cite[Ch.~1 and
3]{StanAP} for more information) that a {\it {\rm(}set{\rm)}
partition}
of a set $S$ is a collection $\{B_1,B_2,\dots,B_k\}$ of
pairwise disjoint non-empty subsets of $S$ such that
their union is equal to $S$. The subsets $B_i$ are also called {\it
blocks} of the partition. One partially orders
partitions by refinement. With respect to this partial order, the
partitions form a lattice. We write $\pi\lor_{A}\gamma$ (the $A$ stands
for the fact that, in ``reflection group language", we are looking at
``type $A$ partitions") for
the join of $\pi$ and $\gamma$ in this lattice. Roughly speaking, the
join of $\pi$ and $\gamma$ is formed by considering altogether
all the blocks of $\pi$ and $\gamma$. Subsequently,
whenever we find two blocks which have a non-empty
intersection, we merge them into a bigger block, and we keep doing
this until all the (merged) blocks are pairwise disjoint.
If $S=\{1,2,\dots,n\}$, we call a
partition {\it non-crossing} if for any $i<j<k<l$ the elements $i$ and
$k$ are in the same block at the same time as the elements $j$ and $l$
are in the same block only if these two blocks are the same. (I refer
the reader to
\machSeite{SimiAD}\cite{SimiAD} for a survey on non-crossing partitions.)
Reiner
\machSeite{ReivAD}\cite{ReivAD}
introduced non-crossing partitions
in type $B$. Partitions of type $B_n$ are (ordinary) partitions of
$\{1,2,\dots,n,-1,-2,\dots,-n\}$ with the property that whenever $B$
is a block then so is $-B:=\{-b:b\in B\}$, and that there is at most
one block $B$ with $B=-B$. A block $B$ with $B=-B$, if present,
is called the {\it zero-block} of the partition. We denote the set of
all type $B_n$ partitions by $\Pi_n^B$, the number of zero blocks of a
partition $\pi$by $\operatorname{zbk} \pi$, and we write $\operatorname{nzbk}\pi$ for half of the
number of the non-zero blocks. Type $B_n$
non-crossing partitions are a subset of type $B_n$ partitions.
Imposing the order $1<2<\dots<n<-1<-2<\dots<-n$ on our ground-set,
the definition of type $B_n$ non-crossing partitions is identical with
the one for type $A$ non-crossing partitions, that is, given this
order on the ground-set, a $B_n$ partition is called {\it non-crossing}
if for any $i<j<k<l$ the elements $i$ and
$k$ are in the same block at the same time as the elements $j$ and $l$
are in the same block only if these two blocks are the same.
We write $\operatorname{NC}_n^B$ for the set of all $B_n$ non-crossing partitions.
The determinant defined by type $B_n$ non-crossing partitions that
Simion tried to evaluate was the one in
\eqref{eq:Simion1} below.\footnote{In fact, instead of $\lor_A$, the
``ordinary" join, she
used the join in the type $B_n$ partition lattice $\Pi_n^B$. However,
the number of {\it non-zero} blocks will be the same regardless of whether
we take the join of two type $B_n$ non-crossing partitions with
respect to ``ordinary" join or with respect to ``type $B_n$" join. This
is in contrast to the numbers of zero blocks, which can differ
largely. (To be more precise, one way to form the ``type $B_n$" join is
to first form the ``ordinary" join, and then merge all zero blocks
into one big block.) The reason that I insist on using $\lor_A$ is
that this is crucial for the more general
Conjecture~\ref{conj:Simion2}. To tell the truth, the discovery of the
latter conjecture is due to a programming error on my behalf (that is,
originally I aimed to program the ``type $B$" join, but it happened
to be the ``ordinary" join \dots).}
The use of the ``type $A$" join $\lor_A$ for two type $B$ non-crossing
partitions in \eqref{eq:Simion1} may seem strange. However, this is
certainly a well-defined operation. The result may neither be a type
$B$ partition nor a non-crossing one, it will just be an ordinary partition
of the ground-set $\{1,2,\dots,n,-1,-2,\dots,-n\}$. We extend the
notion of ``zero block" and ``non-zero block" to these objects in the
obvious way.
Simion observed that, as in the case of the
analogous type $A_n$ partition determinant due to Dahab and Tutte, it
factors apparently completely
into factors which are Chebyshev polynomials. Based on some
additional numerical calculations,\footnote{\label{foot:Simion}%
Evidently, more than
five years later, thanks to technical progress since then,
one can go much farther when doing computer
calculations. The evidence for Conjecture~\ref{conj:Simion1} which I
have is based on, similar to the conjectures and calculations
on determinants for signed permutations in Subsection~\ref{sec:signed} (see
Footnote~\ref{foot:maj}), the exact form of the determinants for $n=1,2,3,4$,
which were already computed by Simion, and, essentially, the exact
form of the determinants for $n=5$ and $6$. By ``essentially"
I mean, as earlier, that I computed the determinant for many special values of
$q$, which then let me make a guess on the basis of comparison of the prime
factors in the factorised results with the prime factors of the
candidate factors, that is the irreducible factors of the Chebyshev
polynomials.
Finally, for guessing the general form of the exponents, the available
data were not sufficient for {\tt Rate} (see
Footnote~\ref{foot:Rate}). So I consulted the
fabulous {\it On-Line Encyclopedia of Integer Sequences}
({\tt http://www.research.att.com/\~{}njas/sequences/Seis.html}),
originally created by Neil Sloane and Simon Plouffe
\machSeite{SlPlAA}\cite{SlPlAA},
and since many years continuously further developed by Sloane and his
team
\machSeite{SloaAA}\cite{SloaAA}. An appropriate
selection from the results turned up by the Encyclopedia then
led to the exponents on the right-hand side of
\eqref{eq:Simion1}.} I propose the following conjecture.
\begin{Conjecture} \label{conj:Simion1}
For any positive integer $n$, we have
\begin{equation} \label{eq:Simion1}
\det_{\pi,\gamma\in\operatorname{NC}_n^B}\(q^{\operatorname{nzbk} (\pi\lor_{A}\gamma)}\)=
\prod _{i=1} ^{n}\left(\dfrac {U_{3i-1}(\sqrt q/2)}
{U_{i-1}(\sqrt q/2)}\)^{\binom {2n}{n-i}},
\end{equation}
where $U_{m}(x):=\sum _{j\ge0} ^{}(-1)^j\binom
{m-j}j(2x)^{m-2j}$ is the $m$-th
{\em Chebyshev polynomial of the second kind}.
\end{Conjecture}
If proved, this would solve Problem~1 in
\machSeite{SchFAA}\cite{SchFAA}. It would also
solve Problem~2 from
\machSeite{SchFAA}\cite{SchFAA},
because $U_{m-1}(\sqrt q/2)$ is, up to multiplication by a power of
$q$, equal to the product $\prod _{j\mid m} ^{}f_j(q)$, where the
polynomials $f_j(q)$ are the ones of
\machSeite{SchFAA}\cite{SchFAA}. A simple
computation then shows that, when the right-hand side product of
\eqref{eq:Simion1} is expressed in terms of the $f_j(q)$'s, one obtains
\begin{equation} \label{eq:fprod}
\prod _{k=1} ^{n}f_{3k}(q)^{e_{n,k}},
\end{equation}
where
$$e_{n,k}=\underset {\ell\not\equiv 0\,(\text{mod }3)}
{\sum _{\ell=1} ^{\fl{n/k}}}\binom
{2n}{n-\ell k}.$$
This agrees with the data in
\machSeite{SchFAA}\cite{SchFAA} and with the further ones I
have computed (see Footnote~\ref{foot:Simion}).
Even more seems to be true. The following conjecture predicts the
evaluation of the more general determinant where we also keep track of
the zero blocks.
\begin{Conjecture} \label{conj:Simion2}
For any positive integer $n$, we have
\begin{equation} \label{eq:Simion2}
\det_{\pi,\gamma\in\operatorname{NC}_n^B}\(q^{\operatorname{nzbk} (\pi\lor_{A}\gamma)}z^{\operatorname{zbk}
(\pi\lor_{A}\gamma)}\)=z^{\frac {1} {2}\binom {2n}{n}}
\prod _{i=1} ^{n}\Big(2T_{2i}(\sqrt q/2)+2-z\Big)^{\binom {2n}{n-i}},
\end{equation}
where $T_{m}(x):=\frac {1} {2}\sum _{j\ge0} ^{}(-1)^j\frac {m} {m-j}\binom
{m-j}j(2x)^{m-2j}$ is the $m$-th
{\em Chebyshev polynomial of the first kind}.
\end{Conjecture}
Again, the conjecture is supported by extensive numerical
calculations. It is not too difficult to show, by using some
identities for Chebyshev polynomials, that
Conjecture~\ref{conj:Simion2} implies Conjecture~\ref{conj:Simion1}.
The other determinant which Simion looked at (cf.\
\machSeite{SchFAA}\cite[Problem~9ff]{SchFAA}),
was the $B_n$ analogue
of a determinant of a matrix the rows and columns of which are indexed
by {\it non-crossing matchings}, due to Lickorish
\machSeite{LickAA}\cite{LickAA}, and
evaluated by Ko and Smolinsky
\machSeite{KoSmAA}\cite{KoSmAA} and independently by Di~Francesco
\machSeite{DiFrAA}\cite{DiFrAA}
(see
\machSeite{KratBN}\cite[Theorem~58]{KratBN}). As we may regard (ordinary)
non-crossing matchings as partitions all the blocks of which consist of two
elements, we {\it define} a {\it $B_n$ non-crossing matching} to be a $B_n$
non-crossing partition all the blocks of which consist of two elements.
We shall be concerned with $B_{2n}$ non-crossing matchings, which we
denote by $\operatorname{NCmatch}(2n)$. With this notation, the following seems to be
true.
\begin{Conjecture} \label{conj:Simion3}
For any positive integer $n$, we have
\begin{equation} \label{eq:Simion3}
\det_{\pi,\gamma\in\operatorname{NCmatch}(2n)}\(q^{\operatorname{nzbk} (\pi\lor_{A}\gamma)}z^{\operatorname{zbk}
(\pi\lor_{A}\gamma)}\)=
\prod _{i=1} ^{n}\Big(2T_{2i}(q/2)+2-z^2\Big)^{\binom {2n}{n-i}}.
\end{equation}
\end{Conjecture}
The reader should notice the remarkable fact that,
in the case that Conjectures~\ref{conj:Simion2}
and \ref{conj:Simion3} are true, the right-hand side of \eqref{eq:Simion3}
is, up to a power of $z$, equal to the right-hand side of
\eqref{eq:Simion2} with $q$ replaced by $q^2$ and $z$ replaced by
$z^2$. An intrinsic explanation why this should be the
case is not known.
An analogous relation between the determinants of
Tutte and of Lickorish, respectively, was observed, and proved, in
\machSeite{CoSSAA}\cite{CoSSAA}. Also here, no intrinsic
explanation is known.
The reader is referred to
\machSeite{SchFAA}\cite{SchFAA} for further open problems
related to the determinants in
Conjectures~\ref{conj:Simion1}--\ref{conj:Simion3}.
Finally, it may also be worthwhile to look at determinants defined
using $D_n$ non-crossing partitions and non-crossing matchings,
see
\machSeite{AtReAA}\cite{AtReAA} and
\machSeite{ReivAD}\cite{ReivAD} for two possible definitions of those.
\medskip
The reader may have wondered why in
Conjectures~\ref{conj:Simion1} and \ref{conj:Simion2} we considered
determinants defined by type $B_n$ non-crossing partitions, which form
in fact a lattice, but used the extraneous type $A$ join\footnote{and
not even the one in the type $A$ {\it non-crossing} partition lattice!} in the
definition of the determinant, instead of the join which is
intrinsic to the lattice of type $B_n$ non-crossing partitions.
In particular, what would happen if we would make the latter choice?
As it turns out, for that situation there exists an elegant general
theorem due to Lindstr\"om
\machSeite{LindAB}\cite{LindAB}, which I missed to state in
\machSeite{KratBN}\cite{KratBN}. I refer to
\machSeite{StanAP}\cite[Ch.~3]{StanAP} for the explanation of the
poset terminology used in the statement.
\begin{Theorem} \label{thm:Lind}
Let $L$ be a finite meet semilattice, $R$ be a commutative ring, and
$f:L\times L\to R$ be an incidence function, that is, $f(x,y)=0$ unless
$x\land y=x$. Then
\begin{equation} \label{eq:Lind}
\det_{x,y\in L}\big(f(x\land y,x)\big)=
\prod _{y\in L} ^{}
\(\sum _{x\in L} ^{}\mu(x,y)f(x,y) \),
\end{equation}
where $\mu$ is the M\"obius function of $L$.\quad \quad \qed
\end{Theorem}
Clearly, this does indeed answer our question, we just have to
specialise $f(x,y)=h(x)$ for $x\land y=x$, where $h$ is
some function from $L$ to $R$. The fact that the above
theorem talks about meets instead of joins is of course
no problem because this is just a matter
of convention.
Having an answer in such a great generality, one is tempted to pose
the problem of finding a general theorem that would encompass the
above-mentioned determinant evaluations due to Tutte, Dahab, Ko and
Smolinsky, Di~Francesco, as well as Conjectures~\ref{conj:Simion1} and
\ref{conj:Simion2}. This problem is essentially Problem~6 in
\machSeite{SchFAA}\cite{SchFAA}.
\begin{Problem} \label{prob:9}
Let $L$ and $L'$ be two lattices {\em(}semilattices?{\em)} with
$L'\subseteq L$. Furthermore, let $R$ be a commutative ring, and
let $f$ be a function from $L$ to $R$.
Under which conditions is there a compact formula for the determinant
\begin{equation} \label{eq:det9}
\det_{x,y\in L'}\big(f(x\land_L y)\big) ,
\end{equation}
where $\land_L$ is the meet operation in $L$?
\end{Problem}
By specialisation in Theorem~\ref{thm:Lind},
one can derive numerous corollaries. For example, a
very attractive one is the evaluation of the ``GCD determinant" due to Smith
\machSeite{SmitAA}\cite{SmitAA}. (In fact, Smith's result is a more
general one for {\it factor closed subsets} of the positive integers.)
\begin{Theorem} \label{thm:Smith}
For any positive integer $n$, we have
$$\det_{1\le i,j\le n}\big(\gcd(i,j)\big)=
\prod _{i=1} ^{n}\phi(i),$$
where $\phi$ denotes the Euler totient function.\quad \quad \qed
\end{Theorem}
An interesting generalisation of Theorem~\ref{thm:Lind}
to posets was given by
Altini\c sik, Sagan and Tu\u glu
\machSeite{AlSTAA}\cite{AlSTAA}.
Again, all undefined terminology can be found in
\machSeite{StanAP}\cite[Ch.~3]{StanAP}.
\begin{Theorem} \label{thm:AlSaTu}
Let $P$ be a finite poset, $R$ be a commutative ring, and $f,g:P\times
P\to R$ be two incidence functions, that is, $f(x,y)=0$ unless $x\le y$
in $P$, the same being true for $g$. Then
$$\det_{x,y\in P}\(
\sum _{z\in P} ^{}f(z,x)g(z,y)\)=
\prod _{x\in P} ^{}f(x,x)g(x,x).$$
\quad \quad \qed
\end{Theorem}
The reader is referred to Section~3 of
\machSeite{AlSTAA}\cite{AlSTAA} for the
explanation why this theorem implies Lindstr\"om's.
\subsection{Determinants for compositions}
Our next family of determinants consists of determinants of matrices
the rows and columns of which are indexed by {\it
compositions}. Recall that a composition of a non-negative integer $n$
is a vector $(\alpha_1,\alpha_2,\dots,\alpha_k)$ of non-negative integers such that
$\alpha_1+\alpha_2+\dots+\alpha_k=n$, for some $k$. For a fixed $k$,
let $\mathcal C(n,k)$ denote the corresponding set of compositions of $n$.
While working on a problem in global optimisation,
Brunat and Montes
\machSeite{BrMoAA}\cite{BrMoAA}
discovered the following surprising determinant evaluation.
It allowed them to show how to explicitly express a multivariable
polynomial as a {\it difference of convex functions}.
In the statement, we use standard multi-index notation:
if $\boldsymbol\alpha=(\alpha_1,\alpha_2,\dots,\alpha_k)$ and
$\boldsymbol \beta=(\beta_1,\beta_2,\dots,\beta_k)$ are two compositions, we
let
$$\boldsymbol\alpha^{\boldsymbol \beta}:=\alpha_1^{\beta_1}\alpha_2^{\beta_2}\cdots
\alpha_k^{\beta_k}.$$
\begin{Theorem} \label{thm:Brunat}
For any positive integers $n$ and $k$, we have
\begin{equation} \label{eq:Brunat}
\det_{\boldsymbol\alpha,\boldsymbol\beta\in\mathcal
C(n,k)}\left(\boldsymbol\alpha^{\boldsymbol \beta}\right)=
n^{{\binom{ n + k-1} k}+k-1}\,
\prod_{i = 1}^{ n-1}
i^{( n -i+1) {\binom {n + k-i-1} { k-2}}}.
\end{equation}
\quad \quad \qed
\end{Theorem}
In recent joint work
\machSeite{BrKMAA}\machSeite{BrMoAB}\cite{BrKMAA,BrMoAB},
Brunat, Montes and the author showed that there is in fact
a polynomial generalisation of this determinant evaluation.
\begin{Theorem} \label{conj:Brunat1}
Let $\mathbf x=(x_1,x_2,\dots,x_k)$ be a vector of indeterminates,
and let $\lambda$ be an indeterminate.
Then, for any non-negative integers $n$ and $k$, we have
\begin{equation} \label{eq:Brunat1}
\det_{\boldsymbol\alpha,\boldsymbol\beta\in\mathcal
C(n,k)}\left((\mathbf x+\lambda\boldsymbol\alpha)^
{\boldsymbol \beta}\right)=\lambda^{(k-1)\binom {n+k-1}k}
\left({ \vert \mathbf x\vert+\lambda n }\right)
^{\binom{n+k-1} {k}}
\prod _{i=1} ^{n}i^{(k-1)\binom {n+k-i-1}{k-1}},
\end{equation}
where $\mathbf x+\lambda\boldsymbol\alpha$ is short for
$(x_1+\lambda\alpha_1,x_2+\lambda\alpha_2,\dots,x_k+\lambda\alpha_k)$, and
where $\vert \mathbf x\vert=x_1+x_2+\dots+x_k$.
\end{Theorem}
As a matter of fact,
there is actually a binomial variant which implies the
above theorem. Extending our
multi-index notation, let
$$\binom{\boldsymbol\alpha}{\boldsymbol \beta}:=
\binom{\alpha_1}{\beta_1}\binom{\alpha_2}{\beta_2}\cdots
\binom{\alpha_k}{\beta_k}.$$
\begin{Theorem} \label{conj:Brunat2}
Let $\mathbf x=(x_1,x_2,\dots,x_k)$ be a vector of indeterminates,
and let $\lambda$ be an indeterminate.
Then, using the notation from Theorem~{\em\ref{conj:Brunat1}},
for any non-negative integers $n$ and $k$, we have
\begin{equation} \label{eq:Brunat2}
\det_{\boldsymbol\alpha,\boldsymbol\beta\in\mathcal
C(n,k)}\left(\binom{\mathbf x+\lambda\boldsymbol\alpha}
{\boldsymbol \beta}\right)=\lambda^{(k-1)\binom {n+k-1}k}
\prod _{i=1} ^{n} \left(\frac{{ \vert \mathbf x\vert+(\lambda-1)n+i }}
i\right)
^{\binom{n+k-i-1} {k-1}}.
\end{equation}
\end{Theorem}
The above theorem is proved in
\machSeite{BrKMAA}\cite{BrKMAA} by
identification of factors (see ``Method~3" in Section~\ref{sec:eval}).
Theorem~\ref{conj:Brunat2} follows by extracting the
highest homogeneous component in \eqref{eq:Brunat1}.
I report that, if one naively replaces ``compositions" by ``integer
partitions" in the above considerations, then the arising determinants
do not have nice product formulae.
\medskip
Another interesting determinant of a matrix with rows indexed by
compositions appears in the work of Bergeron, Reutenauer, Rosas and
Zabrocki
\machSeite{BeRRAA}\cite[Theorem~4.8]{BeRRAA} on
{\it Hopf algebras of non-commutative symmetric functions}. It
was used there to show that a certain set of generators of
non-commutative symmetric functions were algebraically independent.
To state their determinant evaluation, we need to introduce some
notation. Given a composition
$\boldsymbol\alpha=(\alpha_1,\alpha_2,\dots,\alpha_k)$ of $n$
with all summands $\alpha_i$ positive,
we let
$D(\boldsymbol\alpha)=\{\alpha_1,\alpha_1+\alpha_2,\dots,\alpha_1+\alpha_2+\dots+\alpha_{k-1}\}$.
Furthermore, for two compositions $\boldsymbol\alpha$ and
$\boldsymbol\beta$ of $n$, we write
$\boldsymbol\alpha\cup\boldsymbol\beta$ for the (unique) composition
$\boldsymbol\gamma$ of $n$ with
$D(\boldsymbol\gamma)=D(\boldsymbol\alpha)\cup D(\boldsymbol\beta)$.
Finally, $\boldsymbol\alpha!$ is short for
$\alpha_1!\,\alpha_2!\cdots\alpha_k!$.
\begin{Theorem} \label{thm:BRRZ}
Let $\operatorname{Comp}(n)$ denote the set of all compositions of $n$ all summands
of which are positive. Then
\begin{equation} \label{eq:BRRZ}
\det_{\boldsymbol\alpha,\boldsymbol\beta\in\operatorname{Comp}(n)}
\big((\boldsymbol\alpha\cup\boldsymbol\beta)!\big)=
\prod _{\boldsymbol\gamma\in\operatorname{Comp}(n)} ^{}
\prod _{i=1} ^{\ell(\boldsymbol\gamma)}a_{\gamma_{i}},
\end{equation}
where $\ell(\boldsymbol\gamma)$ is the number of summands
{\em(}components{\em)} of
the composition $\boldsymbol\gamma$, and where $a_m$ denotes the number of
{\em indecomposable permutations} of $m$ {\em(}cf.\
\machSeite{StanBI}\cite[Ex.~5.13(b)]{StanBI}{\em)}.
These numbers can be computed
recursively by $a_1=1$ and
$$a_n=n!-\sum _{i=1} ^{n-1}a_i(n-1)!, \quad \quad n>1.$$
\quad \quad \qed
\end{Theorem}
As explained in
\machSeite{BeRRAA}\cite{BeRRAA}, one proves the theorem by factoring the matrix
in \eqref{eq:BRRZ} in
the form $CDC^t$, where $C$ is the ``incidence matrix" of ``refinement of
compositions," and where $D$ is a diagonal matrix. Thus, in
particular, the LU-factorisation of the matrix is determined.
\subsection{Two partition determinants}
On the surface,
{\it integer partitions} (see below for their definition) seem to be
very closely related to compositions, as they can be considered as
``compositions where the order of the summands is without importance."
However, experience shows that integer partitions are much more
complex combinatorial objects than compositions. This may be the
reason that the ``composition determinants" from the preceding
subsection do not seem to have analogues for integer partitions.
Leaving aside this disappointment,
here {\it is} a determinant of a matrix
in which rows and columns are indexed by integer
partitions. This determinant arose in work on {\it linear forms of values
of the Riemann zeta function evaluated at positive integers},
although the traces of it have now been completely erased in the
final version of the article
\machSeite{KrRiAA}\cite{KrRiAA}. (The symmetric function
calculus in Section~12 of the earlier version
\machSeite{KrRiAAA}\cite{KrRiAAA}
gives a vague idea where it may have come from.)
Recall that the {\it power symmetric function} of degree $d$ in
$x_1,x_2,\dots,x_k$ is given by $x_1^d+x_2^d+\dots+x_k^d$, and is
denoted by $p_d(x_1,x_2,\dots,x_k)$. (See
\machSeite{LascAZ}\cite[Ch.~1 and 2]{LascAZ},
\machSeite{MacdAC}\cite[Ch.~I]{MacdAC} and
\machSeite{StanBI}\cite[Ch.~7]{StanBI} for
in-depth expositions of the theory of
{\it symmetric functions}.) Then, while working on
\machSeite{KrRiAA}\cite{KrRiAA},
Rivoal and the author needed to evaluate the determinant
\begin{equation} \label{eq:Part}
\det_{\lambda,\mu\in\operatorname{Part}(n,k)}\big(p_{\lambda}(\mu_1,\mu_2,\dots,\mu_k)\big),
\end{equation}
where $\operatorname{Part}(n,k)$ is the set of integer partitions of $n$ with at
most $k$ parts, that is, the set of all possibilities to write $n$ as
a sum of non-negative integers,
$n=\lambda_1+\lambda_2+\dots+\lambda_k$, with $\lambda_1\ge\lambda_2\ge\dots\ge\lambda_k\ge0$
(the non-zero $\lambda_i$'s being called the {\it parts of $\lambda$}),
and where
$$p_\lambda(x_1,x_2,\dots,x_k)=p_{\lambda_1}(x_1,\dots,x_k)
p_{\lambda_1}(x_1,\dots,x_k)\cdots p_{\lambda_k}(x_1,\dots,x_k).
$$
Following the advice given in Section~\ref{sec:comb}, we went to the
computer and let it calculate the prime factorisations of the values
of this determinant for small values of $n$ and $k$. Indeed, the prime
factors turned out be always small so that we were sure that a
``nice" formula exists for the determinant. However, even more seemed
to be true. Recall that, in order to facilitate a proof of a (still
unknown) formula, it is (almost) always a good idea to try to
introduce more parameters (see
\machSeite{KratBN}\cite[Sec.~2]{KratBN}).
This is what we did. It led us consider the following determinant,
\begin{equation} \label{eq:Part-gen}
\det_{\lambda,\mu\in\operatorname{Part}(n,k)}\big(p_{\lambda}(\mu_1+X_1,\mu_2+X_2,\dots,\mu_k+X_k)
\big),
\end{equation}
where $X_1,X_2,\dots,X_k$ are indeterminates.
Here are some values of the determinant \eqref{eq:Part-gen} for
special values of $n$ and $k$. For $n=k=3$ we obtain
$$ 6 \,( X_1 - X_2+2 ) ( X_1 - X_3+1 )
( X_2 - X_3+1 ) ( X_1 + X_2 + X_3+3) ^4,
$$
for $n=4$ and $k=3$ we obtain
\begin{multline*}
8 \,( X_1 - X_2 +1) ( X_1 - X_2+2 )
( X_1 - X_2+3 ) ( X_1 - X_3+2 ) \\
\times
( X_2 - X_3+1 ) ( X_1 + X_2 + X_3+4) ^7,
\end{multline*}
for $n=5$ and $k=3$ we get
\begin{multline*}
8 \,( X_1 - X_2+1) ( X_1 - X_2+2) ( X_1 - X_2+3)
( X_1 - X_2+4) ( X_1 - X_3+2)\\
\times ( X_1 - X_3+3) ( X_2 - X_3+1)
( X_2 - X_3+2) ( X_1 + X_2 + X_3+5)^{11},
\end{multline*}
for $n=6$ and $k=3$ we get
\begin{multline*}
576 \,( X_1 - X_2+1 ) ( X_1 - X_2+2 ) ^2
( X_1 - X_2+3 ) ^2 ( X_1 - X_2+4 )
( X_1 - X_2+5 )\\
\times ( X_1 - X_3+1 )
( X_1 - X_3+2 ) ( X_1 - X_3+3 )
( X_1 - X_3+4 ) ( X_2 - X_3+1 ) ^2 \\
\times
( X_2 - X_3+2 ) ^2
( X_1 + X_2 + X_3+6 ) ^{16},
\end{multline*}
for $n=4$ and $k=4$ we get
\begin{multline*}
192\,( X_1 - X_2 +1) ( X_1 - X_2+2 )
( X_1 - X_2+3 ) ( X_1 - X_3+2 )
( X_2 - X_3+1 ) \\
\times ( X_1 - X_4+1 )
( X_2 - X_4+1 ) ( X_3 - X_4+1 )
( X_1 + X_2 + X_3 + X_4+4 ) ^7
\end{multline*}
while for $n=5$ and $k=4$ we get
\begin{multline*}
48 \,( X_1 - X_2+1 ) ( X_1 - X_2+2 )
( X_1 - X_2+3 ) ( X_1 - X_2+4 )
( X_1 - X_3+2 ) \\
\times ( X_1 - X_3+3 ) ( X_2 - X_3+1 ) ( X_2 - X_3+2 )
( X_1 - X_4+2 ) \\
\times ( X_2 - X_4+1) ( X_3 - X_4+1 )
( X_1 + X_2 + X_3 + X_4+5 ) ^{12}.
\end{multline*}
It is ``therefore" evident that there will be one factor which is a
power of $n+\sum _{i=1} ^{k}X_i$, whereas the other factors will be of
the form $X_i-X_j+c_{i,j}$, again raised to some power.
In fact, the determinant is easy to compute for $k=1$ because, in that
case, it reduces to a special case of the
Vandermonde determinant evaluation. However, we were not able to come even
close to computing \eqref{eq:Part-gen} in general.
(As I already indicated, finally we managed to avoid
the determinant evaluation in our work in
\machSeite{KrRiAA}\cite{KrRiAA}.)
To proceed further, we remark that evaluating \eqref{eq:Part-gen} is
equivalent to evaluating the same determinant,
but with the power symmetric functions
$p_\lambda$ replaced by the {\it Schur functions} $s_\lambda$, because the
transition matrix between these two bases of symmetric functions is
the {\it character table of the symmetric group} of the corresponding
order (cf.\
\machSeite{MacdAC}\cite[Ch.~I, Sec.~7]{MacdAC}), the determinant of
which is known (see Theorem~\ref{thm:chi} below;
since in our determinants \eqref{eq:Part} and \eqref{eq:Part-gen} the
indices range over all partitions of $n$ {\it with at most $k$ parts}, it is
in fact the refinement given in Theorem~\ref{thm:chik}
which we have to apply). The determinant with
Schur functions has some advantages over the one with power symmetric
functions since the former decomposes into a finer block structure.
Alain Lascoux observed that, in fact, there is a
generalisation of the Schur function determinant to {\it
Gra\3mannian Schubert polynomials}, which contains another set of variables,
$Y_1,Y_2,\dots,Y_{n+k-1}$. More precisely, given
$\lambda=(\lambda_1,\lambda_2,\dots,\lambda_k)$, let $\mathbb
Y_\lambda(X_1,X_2,\dots,X_k;Y_1,Y_2,\dots)$ denote the polynomial in the
variables $X_1,X_2,\dots,X_k$ and $Y_1,Y_2,\dots$, defined by (see
\machSeite{LascAZ}\cite[Sections~1.4 and 9.7; the order of the
$B_{k_i}$ should be reversed in (9.7.2) and analogous places]{LascAZ})
$$\mathbb
Y_\lambda(X_1,X_2,\dots,X_k;Y_1,Y_2,\dots):=\det_{1\le i,j\le k}(S_{\lambda_i-i+j}
(X_1,\dots,X_k;Y_1,\dots,Y_{\lambda_{i}+k-i})),$$
where the entries of the determinant are given by
\begin{equation} \label{eq:XY}
\sum _{m=0} ^{\infty}S_m
(X_1,\dots,X_k;Y_1,\dots,Y_l)x^m=
\frac {\prod _{i=1} ^{l}(1-Y_ix)} {\prod _{i=1} ^{k}(1-X_ix)}.
\end{equation}
The Gra\3mannian Schubert polynomial $\mathbb Y_\lambda$ reduces to the Schur
function $s_\lambda$ when all the variables $Y_i$, $i=1,2,\dots$, are set equal
to $0$. Given these definitions, Alain Lascoux (private communication)
established the following result.
\begin{Theorem} \label{thm:Lascoux}
Let $X_1,X_2,\dots,X_k,Y_1,Y_2,\dots,Y_{n+k-1}$ be indeterminates.
Then,
\begin{multline} \label{eq:Part-gena}
\det_{\lambda,\mu\in\operatorname{Part}(n,k)}
\big(\mathbb Y_{\lambda}(\mu_1+X_1,\mu_2+X_2,\dots,\mu_k+X_k;
Y_1,Y_2,\dots)
\big)\\=\Pi(n,k)
\prod _{\sigma\in G(n,k)} ^{}\(n+\sum _{i=1} ^{k}X_i-\sum _{i=1}
^{k}Y_{\sigma(i)}\),
\end{multline}
where $G(n,k)$ denotes the set of
all {\it Gra\3mannian permutations},\footnote{A permutation $\sigma$ in
$\mathfrak S_\infty$ (the set of all permutations of the natural
numbers $\mathbb N$ which fix all but a finite number of elements of
$\mathbb N$)
is called {\it Gra\3mannian} if $\sigma(i)<\sigma(i+1)$ for all $i$ except
possibly for one, the latter being called the {\it descent} of $\sigma$
(see \machSeite{MacdAE}\cite[p.~13]{MacdAE}). An inversion of $\sigma$ is
a pair $(i,j)$ such that $i<j$ but $\sigma(i)>\sigma(j)$. We remark that the
number of Gra\3mannian permutations with descent (if existent) at $k$
and at most $n-1$ inversions is equal to the number of
partitions of at most $n-1$
with at most $k$ parts (including the empty partition).
More concretely, if we denote this number by $g_{n,k}$, the
generating function of the numbers $g_{n,k}$ is given by
$$
\sum _{n=1} ^{\infty}g_{n,k}x^{n-1}=
\frac {1} {1-q}\prod _{i=1} ^{k}\frac {1} {1-q^i}.$$
}
the descent of which {\em(}if existent{\em)} is at $k$, and
which contain at most $n-1$ inversions, and where $\Pi(n,k)$ is given
recursively by
\begin{equation} \label{eq:Pi}
\Pi(n,k)=\Pi(n,k-1)\Pi(n-k,k)
\prod _{\mu\in\operatorname{Part}^+(n,k)} ^{}
\prod _{j=1} ^{k}(X_j+\mu_j-X_k),
\end{equation}
$\operatorname{Part}^+(n,k)$ denoting the set of partitions of $n$ into {\em exactly}
$k$ {\em(}positive{\em)} parts,
with initial conditions $\Pi(n,k)=1$ if $k=1$ or $n\le1$. Explicitly,
\begin{equation} \label{eq:Piexpl}
\Pi(n,k)=\underset{\vert w\vert_B<k-1}
{\prod _{w} ^{}}
\prod _{\mu\in\operatorname{Part}(n-\vert w\vert_A k+\operatorname{inv} w,k-\vert w\vert_B)} ^{}
\prod _{j=1} ^{k-\vert w\vert_B}(X_j+\mu_j-X_{k-\vert w\vert_B}),
\end{equation}
where the product over $w$ runs over all finite-length words $w$ with letters
from $\{A,B\}$, including the empty word.
The notation $\vert w\vert_B$ means the number of
occurrences of $B$ in $w$, with the analogous meaning for $\vert
w\vert_A$. The quantity $\operatorname{inv} w$ denotes the number of inversions of
$w=w_1w_2\dots$,
which is the number of pairs of letters $(w_i,w_j)$, $i<j$, such
that $w_i=B$ and $w_j=A$.
\end{Theorem}
We obtain the evaluation of the determinant \eqref{eq:Part} if we set
$X_i=Y_i=0$ for all $i$ in the above theorem and multiply by \eqref{eq:chik}.
Likewise, we obtain the evaluation of the determinant
\eqref{eq:Part-gen} if we set $Y_i=0$ for all $i$ in the
above theorem and multiply the result by \eqref{eq:chik}.
For example, here is the determinant \eqref{eq:Part-gena} for $n=3$ and $k=2$,
\begin{multline*}
(X_1-X_2+2)(X_1+X_2-Y_1-Y_2+3)(X_1+X_2-Y_1-Y_3+3)\\
\times
(X_1+X_2-Y_2-Y_3+3)(X_1+X_2-Y_1-Y_4+3),
\end{multline*}
and the following is the one for $n=k=3$,
\begin{multline*}
(X_1-X_2+2)(X_1-X_3+1)(X_2-X_3+1)\\
\times
(X_1+X_2+X_3-Y_1-Y_2-Y_3+3)
(X_1+X_2+X_3-Y_1-Y_2-Y_4+3)\\
\times
(X_1+X_2+X_3-Y_1-Y_3-Y_4+3)
(X_1+X_2+X_3-Y_1-Y_2-Y_5+3).
\end{multline*}
In the sequel, I sketch Lascoux's proof of Theorem~\ref{thm:Lascoux}.
The first step consists in applying the
{\it Monk formula for double Schubert polynomials}
in the case of Gra\3mannian Schubert polynomials (see
\machSeite{KoVeAA}\cite{KoVeAA}),
$$ \(\sum _{i=1} ^{k}X_i-\sum _{i=1} ^{k}Y_{\lambda_{k-i+1}+i}\) \mathbb Y_\lambda
= \mathbb Y_{\lambda+(1,0,0,\dots)} +
\sum_\mu \mathbb Y_\mu,
$$
where $\mathbb Y_\lambda$ is short for $\mathbb
Y_\lambda(X_1,X_2,\dots,X_k;Y_1,Y_2,\dots)$,
and where the sum on the right-hand side
is over all partitions $\mu$ of the same size as
$\lambda+(1,0,0,\dots)$ but lexicographically smaller. Clearly, by using
this identity, appropriate row operations
in the determinant \eqref{eq:Part-gena} show that
\begin{equation} \label{eq:factor}
n+\sum _{i=1} ^{k}X_i-\sum _{i=1} ^{k}Y_{\sigma(i)},
\end{equation}
is one of its factors, the relation between $\sigma$ and $\lambda$
being $\sigma(i)=\lambda_{k-i+1}+i$, $i=1,2,\dots,k$.
In particular, $\sigma$ can be extended (in a unique way)
to a Gra\3mannian permutation.
Moreover, doing these row operations, and taking out the
factors of the form \eqref{eq:factor}, we collect on the one hand the
product in \eqref{eq:Part-gena}, and
we may on the other hand reduce
the determinant \eqref{eq:Part-gena} to a determinant of the same
form, in which, however, the partitions
$\lambda=(\lambda_1,\lambda_2,\dots,\lambda_k)$ run over all partitions
of size {\it at most\/} $n$
with the {\it additional property that\/} $\lambda_1=\lambda_2$ (instead of over all
partitions from
$\operatorname{Part}(n,k)$).
As it turns out, the determinant
thus obtained is independent of the variables $Y_1,Y_2,\break
\dots,Y_{n+k-1}$.
Indeed, if we expand each Schubert polynomial
$\mathbb Y_\lambda(\mu_1+X_1,\mu_2+X_2,\dots,\mu_k+X_k;Y_1,Y_2,\dots)$
in the determinant as a
linear combination of Schur functions in $X_1,X_2,\break
\dots,X_k$ with
coefficients being polynomials in the $Y_1,Y_2,\dots,Y_{n+k-1}$, then, by also
using that
$$S_1(\mu_1+X_1,\mu_2+X_2,\dots,\mu_k+X_k)=
n+\sum _{i=1} ^{k}X_i$$
is independent of $\mu$, it is not difficult to see that one can
eliminate all the $Y_i$'s by appropriate row operations.
To summarise the current state of the discussion: we have already
explained the occurrence of the product on the right-hand side of
\eqref{eq:Part-gena} as a factor of the determinant. Moreover, the
remaining factor is given by a determinant of the same form as in
\eqref{eq:Part-gena},
\begin{equation} \label{eq:det-Y}
\det_{\lambda,\mu}
\big(\mathbb Y_{\lambda}(\mu_1+X_1,\mu_2+X_2,\dots,\mu_k+X_k;
Y_1,Y_2,\dots)
\big),
\end{equation}
where, as before, $\mu$ runs over
all partitions in $\operatorname{Part}(n,k)$, but where $\lambda$ runs over partitions
$(\lambda_1,\lambda_2,\dots,\lambda_k)$
of size at most $n$, with the additional
restriction that $\lambda_1=\lambda_2$. The final observation was that this
latter determinant is in fact independent of the $Y_i$'s. This allows
us to specify $Y_1$ arbitrarily, say $Y_1=X_k$. Now another property
of double Schubert polynomials, namely that (this follows from the
definition of double Schubert polynomials by means of divided
differences, see \machSeite{LascAZ}\cite[(10.2.3)]{LascAZ},
and standard properties of divided differences)
\begin{equation} \label{eq:Y1}
\mathbb Y_{\rho+(1,1,\dots,1)}(X_1,X_2,\dots,X_k;Y_1,Y_2,\dots)=
\mathbb Y_{\rho}(X_1,X_2,\dots,X_k;Y_2,\dots)\prod _{j=1} ^{k}(X_j-Y_1)
\end{equation}
comes in handy. (In the vector $(1,1,\dots,1)$ on the left-hand side
of \eqref{eq:Y1} there are $k$ occurrences of $1$.)
Namely, if $Y_1=X_k$, the matrix in \eqref{eq:det-Y}
(of which the determinant is taken),
$M(n,k;Y_1,Y_2,\dots)$ say,
decomposes in block form. If $\lambda$ is a partition with $\lambda_k>0$ and
$\mu$ is a partition with $\mu_k=0$, then, because of \eqref{eq:Y1},
the corresponding entry in \eqref{eq:det-Y} vanishes. Furthermore, in
the block where $\lambda$ and $\mu$ are partitions with $\lambda_k=\mu_k=0$,
because of the definition \eqref{eq:XY} of the quantities $S_m(.)$
the corresponding entry reduces to
\begin{multline*}
\mathbb Y_{\lambda}(\mu_1+X_1,\mu_2+X_2,\dots,\mu_{k-1}+X_{k-1},X_k;
X_k,Y_2,\dots)\\
=\mathbb Y_{\lambda}(\mu_1+X_1,\mu_2+X_2,\dots,\mu_{k-1}+X_{k-1};
Y_2,\dots).
\end{multline*}
In other words, this block is identical with $M(n,k-1;Y_2,\dots)$.
Finally, in
the block indexed by partitions $\lambda$ and $\mu$ with $\lambda_k>0$ and
$\mu_k>0$, we may use \eqref{eq:Y1} to factor $
\prod _{j=1} ^{k}(X_j+\mu_j-X_k)$ out of the column indexed by
$\mu$, for all such $\mu$.
What remains is identical with $M(n-k,k;Y_2,\dots)$. Taking determinants,
we obtain the recurrence \eqref{eq:Pi}. (Here we use again that the
determinants in \eqref{eq:det-Y}, that is,
in particular, the determinants of
$M(n,k;Y_1,\dots)$, $M(n,k-1;Y_2,\dots)$, and of $M(n-k,k;Y_2,\dots)$, are
all independent of the $Y_i$'s.) The explicit form
\eqref{eq:Piexpl} for $\Pi(n,k)$ can be easily derived by induction on
$n$ and $k$.
\medskip
A few paragraphs above, we mentioned in passing another interesting determinant
of a matrix the rows and columns of which are indexed by (integer) partitions:
the {\it determinant of the character table of the symmetric group $\mathfrak
S_n$} (cf.\
\machSeite{JameAA}\cite[Cor.~6.5]{JameAA}).
Since this is a classical and beautiful determinant evaluation which I missed
to state in \machSeite{KratBN}\cite{KratBN}, I present it now in the
theorem below. There, the notation $\lambda\vdash n$ stands for ``$\lambda$ is
a partition of $n$."
For all undefined notation, I refer the reader to standard
texts on the representation theory of symmetric groups, as for example
\machSeite{JameAA}%
\machSeite{JaKeAA}%
\machSeite{SagaAQ}%
\cite{JameAA,JaKeAA,SagaAQ}.
\begin{Theorem} \label{thm:chi}
For partitions $\lambda$ and $\rho$ of $n$, let $\chi^\lambda(\rho)$ denote the
value of the irreducible character $\chi^\lambda$ evaluated at a permutation of
cycle type $\rho$. Then
\begin{equation} \label{eq:chi}
\det_{\lambda,\rho\,\vdash n}\(\chi^\lambda(\rho)\)=
\prod _{\mu\,\vdash n} ^{}
\prod _{i\ge1} ^{}\mu_i.
\end{equation}
In words: the determinant of the character table of the symmetric
group $\mathfrak S_n$ is equal to the products of all the parts of all
the partitions of $n$.\quad \quad \qed
\end{Theorem}
A refinement of this statement, where we restrict to partitions
of $n$ with at most $k$ parts, is the following.
\begin{Theorem} \label{thm:chik}
With the notation of Theorem~{\em\ref{thm:chi}}, for all positive
integers $n$ and $k$, $n\ge k$, we have
\begin{equation} \label{eq:chik}
\det_{\lambda,\rho\in\operatorname{Part}(n,k)}\(\chi^\lambda(\rho)\)=
\prod _{\mu\in\operatorname{Part}(n,k)} ^{}
\prod _{i\ge1} ^{}m_i(\mu)!,
\end{equation}
where $m_i(\mu)$ is the number of times $i$ occurs as a part
in the partition $\mu$.\quad \quad \qed
\end{Theorem}
This determinant evaluation follows from the decomposition of the {\it
full\/} character table of $\mathfrak S_n$ in the form
\begin{equation} \label{eq:LK}
\(\chi^\lambda(\rho)\)_{\lambda,\rho\,\vdash n}=L\cdot K^{-1},
\end{equation}
where $L=(L_{\lambda,\mu})_{\lambda,\mu\,\vdash n}$
is the transition matrix from power symmetric functions
to monomial symmetric functions, and where
$K=(K_{\lambda,\mu})_{\lambda,\mu\,\vdash n}$ is the {\it Kostka matrix},
the transition matrix from Schur functions to monomial symmetric
functions (see
\machSeite{MacdAC}\cite[Ch.~I, (6.12)]{MacdAC}). For, if we order the
partitions of $n$ so that the partitions in $\operatorname{Part}(n,k)$ come before
the partitions in $\operatorname{Part}(n,k+1)$, $k=1,2,\dots,n-1$, and within
$\operatorname{Part}(n,k)$ lexicographically, then, with respect to this order, the
matrix $L$ is lower triangular and the matrix $K$, and hence also
$K^{-1}$, is upper triangular. Furthermore,
the matrix $K$, and hence also $K^{-1}$, is even block diagonal,
the blocks along the diagonal being the ones which are formed by the
rows and columns indexed by the partitions in $\operatorname{Part}(n,k)$,
$k=1,2,\dots,n$. These facts together imply that the decomposition
\eqref{eq:LK} restricts to the submatrices indexed by partitions in
$\operatorname{Part}(n,k)$,
\begin{equation*
\(\chi^\lambda(\rho)\)_{\lambda,\rho\in \operatorname{Part}(n,k)}=
(L_{\lambda,\mu})_{\lambda,\mu\in\operatorname{Part}(n,k)}\cdot
(K_{\lambda,\mu})_{\lambda,\mu\in\operatorname{Part}(n,k)}^{-1}.
\end{equation*}
If we now take determinants on both sides, then, in view of
$K_{\mu,\mu}=1$ and of
$$L_{\mu,\mu}=\prod _{i\ge1} ^{}m_i(\mu)!$$
for all $\mu$, the theorem follows.
\medskip
Further examples of nice
determinant evaluations of tables of
{\it characters of representations
of symmetric groups and their double covers} can be found in
\machSeite{BeOSAA}%
\machSeite{OlssAB}%
\cite{BeOSAA,OlssAB}.
Determinants of tables of {\it characters of the alternating group}
can be found in
\machSeite{BeOlAE}\cite{BeOlAE}.
\subsection{Elliptic determinant evaluations}\label{sec:ell}
In special functions theory there is currently a disease rapidly
spreading, generalising the earlier mentioned $q$-disease (see
Footnote~\ref{foot:q}). It could be called the {\it ``elliptic
disease}." Recall that, during the $q$-disease, we replaced every
positive integer $n$ by $1+q+q^2+\dots+q^{n-1}=(1-q^n) /(1-q)$,
and, more generally, shifted factorials $a(a+1)\cdots(a+k-1)$ by
$q$-shifted factorials $(1-\alpha)(1-q\alpha)\cdots(1-\alpha q^{k-1})$. (Here,
$\alpha$ takes the role of $q^a$, and one drops the powers of $1-q$ in
order to ease notation.) Doing this with some ``ordinary" identity, we
arrived (hopefully) at its {\it $q$-analogue}.
Now, once infected by the elliptic disease, we would replace every
occurrence of a term $1-x$ (and, looking at the definition of
$q$-shifted factorials, we can see that there will be many) by its
{\it elliptic analogue} $\theta(x;p)$:
$$\theta(x)=\theta(x;p)=\prod_{j=0}^\infty(1-p^jx)(1-p^{j+1}/x). $$
Here, $p$ is a complex number with $|p|<1$, which will be fixed
throughout. Up to a trivial factor,
$\theta(e^{2\pi ix};e^{2\pi i \tau})$ equals
the {\it Jacobi theta function} $\theta_1(x|\tau)$ (cf.\
\machSeite{WW}\cite{WW}). Clearly, $\theta(x)$ reduces to $1-x$ if $p=0$.
At first sight, one will be sceptical if this is a fruitful
thing to do. After all, for working with the functions $\theta(x)$,
the only identities which are available are
the (trivial) {\it inversion formula}
\begin{equation}\label{ti}\theta(1/x)=-\frac1 x\,\theta(x), \end{equation}
the (trivial) {\it quasi-periodicity}
\begin{equation}\label{tp}\theta(px)=-\frac1 x\,\theta(x),\end{equation}
and {\it Riemann's} (highly non-trivial) {\it addition formula} (cf.\
\machSeite{WW}\cite[p.~451, Example~5]{WW})
\begin{equation}\label{tadd}
\theta(xy)\,\theta(x/y)\,\theta(uv)\,\theta(u/v)-
\theta(xv)\,\theta(x/v)\,\theta(uy)\,\theta(u/y)
=\frac uy\,\theta(yv)\,\theta(y/v)\,\theta(xu)\,\theta(x/u).
\end{equation}
Nevertheless, it has turned out recently
that a surprising number of identities from
the ``ordinary" and from the ``$q$-world" can be lifted to the
elliptic level. This is particularly true for series of hypergeometric
nature. We refer the reader to Chapter~11 of
\machSeite{GaRaAA}\cite{GaRaAA} for an account of the current state of
the art
in the theory of, as they are called now, {\it elliptic hypergeometric
series}.
On the following pages,
I give elliptic determinant evaluations a rather extensive coverage
because, first of all, they were non-existent in
\machSeite{KratBN}\cite{KratBN} (with the exception of the mention of
the papers
\machSeite{MilnAO}%
\machSeite{MilnAP}\cite{MilnAP,MilnAO} by Milne), and, second, because
I believe that the ``elliptic research" is a research direction that
will further prosper in the next future and will have
numerous applications in
many fields, also outside of just special functions theory and number
theory. I further believe that the determinant evaluations presented
in this subsection will turn out to be as fundamental as the
determinant evaluations in Sections~2.1 and 2.2 in
\machSeite{KratBN}\cite{KratBN}. For some of them this belief is
already a fact. For example,
determinant evaluations involving elliptic functions have
come into the picture in the theory of {\it
multiple elliptic hypergeometric series}, see
\machSeite{KN}%
\machSeite{RainAA}%
\machSeite{RoseAA}%
\machSeite{Ro}%
\machSeite{RoScAB}%
\machSeite{Sp}%
\machSeite{WarnAG}%
\cite{KN,RainAA,RoseAA,Ro,RoScAB,Sp,WarnAG}.
They have also an important role in the study of
\emph{Ruijsenaars operators} and related {\it integrable systems}
\machSeite{H}%
\machSeite{Ru}%
\cite{H,Ru}. Furthermore, they have recently found applications
in number theory
to the problem of counting the number of {\it representations of an integer
as a sum of triangular numbers}
\machSeite{RoseAB}\cite{RoseAB}.
Probably the first elliptic determinant evaluation is due to Frobenius
\machSeite{Fr}\cite[(12)]{Fr}.
This identity has found applications to {\it Ruijsenaars
operators}
\machSeite{Ru}\cite{Ru}, to {\it multidimensional elliptic hypergeometric
series} and {\it integrals}
\machSeite{KN}\cite{KN},
\machSeite{RainAA}\cite{RainAA} and to {\it number theory}
\machSeite{RoseAB}\cite{RoseAB}. For a generalisation
to {\it higher genus Riemann surfaces}, see
\machSeite{F}\cite[Cor.~2.19]{F}.
Amdeberhan \machSeite{AmdeAC}\cite{AmdeAC} observed that it can be
easily proved using the condensation method (see ``Method~2" in
Section~\ref{sec:eval}).
\begin{Theorem}
\label{froa}
Let $x_1,x_2,\dots,x_n$, $a_1,a_2,\dots,a_n$ and $t$ be indeterminates.
Then there holds
\begin{equation}\label{froaid}
\det_{1\leq i,j\leq n}\left(\frac{\theta(ta_jx_i)}{\theta(t)\,
\theta(a_jx_i)}\right)
=\frac{\theta(ta_1\dotsm a_nx_1\dotsm x_n)}{\theta(t)}
\frac{\displaystyle\prod_{1\leq i<j\leq n}
a_jx_j\,\theta(a_i/a_j)\,\theta(x_i/x_j)}
{\displaystyle\prod_{i,j=1}^n\theta(a_jx_i)}.
\end{equation}
\quad \quad \qed
\end{Theorem}
For $p=0$ and $t\to\infty$, this determinant identity reduces to
Cauchy's evaluation \eqref{eq:Cauchy} of the double alternant, and,
thus, may be regarded as an ``elliptic analogue" of the latter.
Okada
\machSeite{OkadAK}\cite[Theorem~1.1]{OkadAK} has recently found an
elliptic extension of Schur's evaluation
\eqref{eq:Schur} of a Cauchy-type Pfaffian.
His proof works by the Pfaffian version of the condensation method.
\begin{Theorem} \label{thm:Okada}
Let $x_1,x_2,\dots,x_n$, $t$ and $w$ be indeterminates.
Then there holds
\begin{multline}\label{eq:Okada}
\underset{1\leq i,j\leq 2n}\operatorname{Pf}\left(\frac{\theta(x_j/x_i)}{\theta(x_ix_j)}
\frac{\theta(tx_ix_j)}{\theta(t)}
\frac{\theta(wx_ix_j)}{\theta(w)}
\right)
\\=
\frac{\theta(tx_1\dotsm x_{2n})}{\theta(t)}
\frac{\theta(wx_1\dotsm x_{2n})}{\theta(w)}
\prod _{1\le i<j\le 2n} ^{}
\frac{\theta(x_j/x_i)}
{x_j\,\theta(x_ix_j)}.
\end{multline}
\quad \quad \qed
\end{Theorem}
The next group of determinant evaluations is from
\machSeite{RoScAC}\cite[Sec.~3]{RoScAC}. As the Vandermonde determinant
evaluation, or the other Weyl denominator formulae (cf.\
\machSeite{KratBN}\cite[Lemma~2]{KratBN}), are fundamental {\it
polynomial} determinant evaluations, the evaluations in
Lemma~\ref{wp} below are equally
fundamental in the elliptic domain as they can be considered as the
elliptic analogues of the former. Indeed, Rosengren and Schlosser
show that they {\it imply} the {\it Macdonald identities associated to
affine root systems}
\machSeite{MacdAA}\cite{MacdAA}, which are the affine analogues of the Weyl
denominator formulae. In particular, in this way they obtain new
proofs of the Macdonald identities.
In order to conveniently formulate Rosengren and
Schlosser's determinant evaluations, we shall adopt the following
terminology from
\machSeite{RoScAC}\cite{RoScAC}.
For $0<|p|<1$ and $t\neq 0$,
an {\it $A_{n-1}$ theta function $f$ of norm $t$} is
a holomorphic function for $x\neq 0$ such that
\begin{equation}\label{ade}f(px)=\frac{(-1)^n}{tx^n}\,f(x).\end{equation}
Moreover, if $R$ denotes either of the root systems $B_n$,
$B^\vee_n$, $C_n$, $C^\vee_n$, $BC_n$ or $D_n$ (see
Footnote~\ref{foot:root} and \machSeite{HumpAC}\cite{HumpAC} for
information on root systems),
we call $f$ an {\it$R$ theta function} if
{\allowdisplaybreaks
\begin{align*}
f(px)&=-\frac{1}{p^{n-1}x^{2n-1}}\,f(x),& f(1/x)&=-\frac
1x\,f(x),& R&=B_n,\\
f(px)&=-\frac{1}{p^{n}x^{2n}}\,f(x),& f(1/x)&=-f(x),&
R&=B_n^\vee,\\
f(px)&=\frac{1}{p^{n+1}x^{2n+2}}\,f(x),& f(1/x)&=-f(x),&
R&=C_n,\\
f(px)&=\frac{1}{p^{n-\frac12}x^{2n}}\,f(x),& f(1/x)&=-\frac
1x\,f(x),& R&=C_n^\vee,\\
f(px)&=\frac{1}{p^{n}x^{2n+1}}\,f(x),& f(1/x)&=-\frac
1x\,f(x),& R&=BC_n,\\
f(px)&=\frac{1}{p^{n-1}x^{2n-2}}\, f(x),& f(1/x)&=f(x),&
R&=D_n.
\end{align*}
}
Given this definition,
Rosengren and Schlosser \machSeite{RoScAC}\cite[Lemma~3.2]{RoScAC}
show that a function $f$ is an $A_{n-1}$ theta function of norm $t$
if and only if there exist constants $C$, $b_1,\dots,b_{n}$ such that
$b_1\dotsm b_n=t$ and
$$f(x)=C\,\theta(b_1x)\cdots\theta(b_nx), $$
and for the other six cases, they show that $f$ is an
$R$ theta function if and only if there exist
constants $C$, $b_1,\dots,b_{n-1}$ such that
\begin{align*}f(x)&=C\,\theta(x)\,\theta(b_1x)\,\theta(b_1/x)
\cdots\theta(b_{n-1}x)\,\theta(b_{n-1}/x),&
R&=B_n,\\
f(x)&=C\,x^{-1}\theta(x^2;p^2)\,\theta(b_1x)\,\theta(b_1/x)
\cdots\theta(b_{n-1}x)\,\theta(b_{n-1}/x),&
R&=B_n^\vee,\\
f(x)&=C\,x^{-1}\theta(x^2)\,\theta(b_1x)\,\theta(b_1/x)
\cdots\theta(b_{n-1}x)\,\theta(b_{n-1}/x),& R&=C_n,\\
f(x)&=C\,\theta(x;p^{\frac12})\,\theta(b_1x)\,\theta(b_1/x)
\cdots\theta(b_{n-1}x)\,\theta(b_{n-1}/x),&
R&=C_n^\vee,\\
f(x)&=C\,\theta(x)\,\theta(px^2;p^2)\,\theta(b_1x)\,\theta(b_1/x)
\cdots\theta(b_{n-1}x)\,\theta(b_{n-1}/x),&
R&=BC_n,\\
f(x)&=C\,\theta(b_1x)\,\theta(b_1/x)
\cdots\theta(b_{n-1}x)\,\theta(b_{n-1}/x),& R&=D_n,
\end{align*}
where $\theta(x)=\theta(x;p)$.
If one puts $p=0$, then an $A_{n-1}$ theta function
of norm $t$ becomes a polynomial of degree $n$ such that the
reciprocal of the product of its roots is equal to $t$. Similarly,
if one puts $p=0$, then a $D_n$ theta function
becomes a polynomial in $(x+1/x)$ of degree $n$.
This is the
specialisation of some of the following results which is relevant for
obtaining the earlier Lemmas~\ref{lem:RS1}--\ref{cdetr1cor}.
The elliptic extension of the Weyl denominator formulae
is the following formula.
(See
\machSeite{RoScAC}\cite[Prop.~3.4]{RoScAC}.)
\begin{Lemma}\label{wp}
Let $f_1,\dots,f_n$ be $A_{n-1}$ theta functions of norm
$t$. Then,
\begin{equation}\label{awpi}\det_{1\leq i,j\leq n}\left(f_j(x_i)\right)=
C\,W_{A_{n-1}}(x),
\end{equation}
for some constant $C$, where
$$W_{A_{n-1}}(x)=\theta(tx_1\dotsm x_n)
\,\prod_{1\leq i<j\leq n}x_j\theta(x_i/x_j).$$
Moreover, if $R$ denotes either $B_n$,
$B^\vee_n$, $C_n$, $C^\vee_n$, $BC_n$ or $D_n$ and $f_1,\dots,f_n$
are $R$ theta functions, we have
\begin{equation}\label{wpi}
\det_{1\leq i,j\leq n}\left(f_j(x_i)\right)=C\,W_R(x) ,
\end{equation}
for some constant $C$, where
{\allowdisplaybreaks
\begin{align*}
W_{B_n}(x)&=\prod_{i=1}^n\theta(x_i)
\prod_{1\leq i<j\leq n}x_i^{-1}\theta(x_ix_j)\,\theta(x_i/x_j), \\
W_{B_n^\vee}(x)&=\prod_{i=1}^n x_i^{-1}\theta(x_i^2;p^2)
\prod_{1\leq i<j\leq n}x_i^{-1}\theta(x_ix_j)\,\theta(x_i/x_j), \\
W_{C_n}(x)&=\prod_{i=1}^n x_i^{-1}\theta(x_i^2)
\prod_{1\leq i<j\leq n}x_i^{-1}\theta(x_ix_j)\,\theta(x_i/x_j),\\
W_{C_n^\vee}(x)&=\prod_{i=1}^n\theta(x_i;p^{\frac 12})
\prod_{1\leq i<j\leq n}x_i^{-1}\theta(x_ix_j)\,\theta(x_i/x_j), \\
W_{BC_n}(x)&=\prod_{i=1}^n\theta(x_i)\,\theta(px_i^2;p^2)
\prod_{1\leq i<j\leq n}x_i^{-1}\theta(x_ix_j)\,\theta(x_i/x_j), \\
W_{D_n}(x)&=\prod_{1\leq i<j\leq n}x_i^{-1}\theta(x_ix_j)\,\theta(x_i/x_j).
\end{align*}}%
\quad \quad \qed
\end{Lemma}
Rosengren and Schlosser show in
\machSeite{RoScAC}\cite[Prop.~6.1]{RoScAC}
that the famous Macdonald identities for
affine root systems
\machSeite{MacdAA}\cite{MacdAA} are equivalent to special cases of
this lemma. We state the corresponding results below.
\begin{Theorem}\label{mdp}
The following determinant evaluations hold:
$$
\det_{1\leq i,j\leq n}\left(x_i^{j-1}
\theta((-1)^{n-1}p^{j-1}tx_i^n;p^n)\right)=\frac{(p;p)_\infty^{n}}{(p^n;p^n)_\infty^n}
\,W_{A_{n-1}}(x),
$$
\begin{multline*}
\det_{1\leq i,j\leq n}\left(x_i^{j-n}
\theta(p^{j-1}x_i^{2n-1};p^{2n-1})
-x_i^{n+1-j}
\theta(p^{j-1}x_i^{1-2n};p^{2n-1})
\right)\\
=\frac{2(p;p)_\infty^{n}}
{(p^{2n-1};p^{2n-1})_\infty^n}\,W_{B_{n}}(x),
\end{multline*}
\begin{multline*}
\det_{1\leq i,j\leq n}\left(x_i^{j-n-1}
\theta(p^{j-1}x_i^{2n};p^{2n})
-x_i^{n+1-j}
\theta(p^{j-1}x_i^{-2n};p^{2n})
\right)\\
=\frac{2(p^2;p^2)_\infty(p;p)_\infty^{n-1}}
{(p^{2n};p^{2n})_\infty^n}\,W_{B_{n}^\vee}(x),
\end{multline*}
\begin{multline*}
\det_{1\leq i,j\leq n}\left(x_i^{j-n-1}
\theta(-p^{j}x_i^{2n+2};p^{2n+2})
-x_i^{n+1-j}
\theta(-p^{j}x_i^{-2n-2};p^{2n+2})
\right)\\
=\frac{(p;p)_\infty^{n}}
{(p^{2n+2};p^{2n+2})_\infty^n}\,
W_{C_{n}}(x),
\end{multline*}
\begin{multline*}
\det_{1\leq i,j\leq n}\left(x_i^{j-n}
\theta(-p^{j-\frac12}x_i^{2n};p^{2n})
-x_i^{n+1-j}
\theta(-p^{j-\frac12}x_i^{-2n};p^{2n})
\right)\\
=\frac{(p^{\frac12};p^{\frac12})_\infty(p;p)_\infty^{n-1}}
{(p^{2n};p^{2n})_\infty^n}\,
W_{C_{n}^\vee}(x),
\end{multline*}
\begin{multline*}
\det_{1\leq i,j\leq n}\left(x_i^{j-n}
\theta(-p^{j}x_i^{2n+1};p^{2n+1})
-x_i^{n+1-j}
\theta(-p^{j}x_i^{-2n-1};p^{2n+1})
\right)\\
=\frac{(p;p)_\infty^n}{(p^{2n+1};p^{2n+1})_\infty^n}\,
W_{BC_{n}}(x),
\end{multline*}
\begin{multline*}
\det_{1\leq i,j\leq n}\left(x_i^{j-n}
\theta(-p^{j-1}x_i^{2n-2};p^{2n-2})
+x_i^{n-j}
\theta(-p^{j-1}x_i^{2-2n};p^{2n-2})
\right)\\
=\frac{4(p;p)_\infty^n}{(p^{2n-2};p^{2n-2})_\infty^n}
\,W_{D_{n}}(x),
\qquad n\geq 2.
\end{multline*}
\quad \quad \qed
\end{Theorem}
Historically, aside from Frobenius' elliptic Cauchy identity \eqref{froaid},
the subject of elliptic determinant evaluations begins with Warnaar's
remarkable paper \machSeite{WarnAG}\cite{WarnAG}. While the main subject of
this paper is {\it elliptic hypergeometric series}, some elliptic
determinant evaluations turn out to be crucial for the proofs of
the results. Lemma~5.3 from
\machSeite{WarnAG}\cite{WarnAG} extends one of the basic
determinant lemmas listed in
\machSeite{KratBN}\cite{KratBN}, namely
\machSeite{KratBN}\cite[Lemma~5]{KratBN}, to the elliptic world, to
which it reduces in the case $p=0$. We present this important
elliptic determinant evaluation in the theorem below.
\begin{Theorem}
\label{bcdet}
Let $x_1,x_2,\dots,x_n$, $a_1,a_2,\dots,a_n$ be indeterminates.
For each $j=1,\dots,n$, let $P_j(x)$ be a $D_j$ theta function.
Then there holds
\begin{multline}\label{bcdetid}
\det_{1\le i,j\le n}\left(P_{j}(x_i)
\prod_{k=j+1}^n\theta(a_kx_i)\,\theta(a_k/x_i)\right)\\
=\prod_{i=1}^nP_{i}(a_i)
\prod_{1\le i<j\le n}a_jx_j^{-1}\,\theta(x_jx_i)\,\theta(x_j/x_i).
\end{multline}
\quad \quad \qed
\end{Theorem}
Warnaar used this identity to obtain a summation formula for a
{\it multidimensional elliptic hyper\-geometric
series}. Further related applications may be found in
\machSeite{RoseAA}%
\machSeite{Ro}%
\machSeite{RoScAB}%
\machSeite{Sp}%
\cite{RoseAA,Ro,RoScAB,Sp}.
The relevant special case of the above theorem is the following
(see \machSeite{WarnAG}\cite[Cor.~5.4]{WarnAG}).
It is the elliptic generalisation of
\machSeite{KratBN}\cite[Theorem~28]{KratBN}.
In the
statement, we use the notation
\begin{equation} \label{eq:pqell}
(a;q,p)_m=\theta(a;p)\,\theta(aq;p)\cdots \theta(aq^{m-1};p),
\end{equation}
which extends the notation for $q$-shifted factorials to the elliptic
world.
\begin{Theorem} \label{thm:Warn1}
Let $X_1,X_2,\dots,X_n$, $A$, $B$ and $C$ be indeterminates. Then, for
any non-negative integer $n$, there holds
\begin{multline} \label{eq:Warn1}
\det_{1\le i,j\le n}\(\frac {(AX_i;q,p)_{n-j}\,(AC/X_i;q,p)_{n-j}}
{(BX_i;q,p)_{n-j}\,(BC/X_i;q,p)_{n-j}}\)\\=(Aq)^{\binom n2}
\prod _{1\le i<j\le n} ^{}X_j\,\theta(X_i/X_j)\,\theta(C/X_iX_j)
\prod _{i=1} ^{n}\frac {(B/A;q,p)_{i-1}\,(ABCq^{2n-2i};q,p)_{i-1}}
{(BX_i;q,p)_{n-1}\,(BC/X_i;q,p)_{n-1}}.
\end{multline}
\quad \quad \qed
\end{Theorem}
Theorem~29 from \machSeite{KratBN}\cite{KratBN}, which is slightly
more general than \machSeite{KratBN}\cite[Theorem~28]{KratBN}, can
also be extended to an elliptic theorem by suitably specialising
the variables in Theorem~\ref{bcdet}.
\begin{Theorem} \label{thm:Warn1a}
Let $X_1,X_2,\dots,X_n$, $Y_1,Y_2,\dots,Y_n$, $A$ and $B$ be
indeterminates. Then, for
any non-negative integer $n$, there holds
\vbox{\noindent
\begin{multline} \label{eq:Warn1a}
\det_{1\le i,j\le n}\(\frac {(X_iY_j;q,p)_{j}\,(AY_j/X_i;q,p)_{j}}
{(BX_i;q,p)_{j}\,(AB/X_i;q,p)_{j}}\)\\=
q^{2\binom n3}(AB)^{\binom n2}
\prod _{1\le i<j\le n} ^{}\theta(X_jX_i/A)\,\theta(X_j/X_i)\\
\times
\prod _{i=1} ^{n}\frac {(ABY_iq^{i-2};q,p)_{i-1}\,(Y_i/Bq^{i-1};q,p)_{i-1}}
{X_i^{i-1}\,(BX_i;q,p)_{n-1}\,(AB/X_i;q,p)_{n-1}}.
\end{multline}
\quad \quad \qed}
\end{Theorem}
Another, very elegant, special case of Theorem~\ref{bcdet}
is the following elliptic Cauchy-type
determinant evaluation.
It was used by Rains
\machSeite{RainAA}\cite[Sec.~3]{RainAA} in the course
of deriving a {\it $BC_n\leftrightarrow BC_m$ integral transformation}.
\begin{Lemma}
\label{frobc}
Let $x_1,x_2,\dots,x_n$ and $a_1,a_2,\dots,a_n$ be indeterminates.
Then there holds
\begin{equation}\label{frobcid}
\det_{1\leq i,j\leq n}\left(\frac 1{\theta(a_jx_i)\,\theta(a_j/x_i)}\right)
=\frac{\displaystyle\prod_{1\leq i<j\leq n}
a_jx_j^{-1}\,\theta(x_jx_i)\,\theta(x_j/x_i)\,\theta(a_ia_j)\,\theta(a_i/a_j)}
{\displaystyle\prod_{i,j=1}^n\theta(a_jx_i)\,\theta(a_j/x_i)}.
\end{equation}
\quad \quad \qed
\end{Lemma}
The remaining determinant evaluations in the current subsection,
with the exception of the very last one, are all due to Rosengren and
Schlosser
\machSeite{RoScAC}\cite{RoScAC}. The first one is
a further (however non-obvious) consequence of Theorem~\ref{bcdet}
(see \machSeite{RoScAC}\cite[Cor.~4.3]{RoScAC}).
Two related determinant evaluations,
restricted to the polynomial case, were applied in
\machSeite{SchlAB}\cite{SchlAB} and
\machSeite{SchlAF}\cite{SchlAF}
to obtain {\it multidimensional matrix inversions} that played a major
role in the derivation of new {\it summation formulae for multidimensional
basic hypergeometric series}.
\begin{Theorem} \label{thm:cdet}
Let $x_1,x_2,\dots,x_n$, $a_1,a_2,\dots,a_{n+1}$, and
$b$ be indeterminates.
For each $j=1,\dots,n+1$, let $P_j(x)$ be a $D_j$ theta function.
Then there holds
\begin{multline}
P_{n+1}(b)\det_{1\le i,j\le n}\!\Bigg(P_{j}(x_i)
\prod_{k=j+1}^{n+1}\big(\theta(a_kx_i)\,\theta(a_k/x_i)\big)\\
-\frac{P_{n+1}(x_i)}{P_{n+1}(b)}P_{j}(b)
\prod_{k=j+1}^{n+1}\big(\theta(a_kb)\,\theta(a_k/b)\big)\Bigg)\\
=\prod_{i=1}^{n+1}P_{i}(a_i)
\prod_{1\le i<j\le n+1}a_jx_j^{-1}\,\theta(x_jx_i)\,\theta(x_j/x_i),
\end{multline}
where $x_{n+1}=b$.
\quad \quad \qed
\end{Theorem}
The next determinant evaluation is Theorem~4.4 from
\machSeite{RoScAC}\cite{RoScAC}. It generalises another basic
determinant lemma listed in
\machSeite{KratBN}\cite{KratBN}, namely Lemma~6 from
\machSeite{KratBN}\cite{KratBN}, to the elliptic case.
It looks as if it is a limit case of
Warnaar's in Theorem~\ref{bcdet}. However, limits are very problematic
in the elliptic world, and therefore it does not seem that
Theorem~\ref{bcdet} implies the theorem below.
For a generalisation in a different direction than Theorem~\ref{bcdet}
see \machSeite{TV}\cite[Appendix B]{TV} (cf.\ also
\machSeite{RoScAC}\cite[Remark~4.6]{RoScAC}).
\begin{Theorem}\label{adet}
Let $x_1,x_2,\dots,x_n$, $a_1,a_2,\dots,a_n$, and $t$ be indeterminates.
For each $j=1,\dots,n$, let $P_j(x)$ be an $A_{j-1}$ theta function of
norm $ta_1\dotsm a_j$. Then there holds
\begin{multline}\label{adetid}
\det_{1\le i,j\le n}\left(P_j(x_i)
\prod_{k=j+1}^n\theta(a_kx_i)\right)\\
=\frac{\theta(ta_1\dotsm a_nx_1\dotsm x_n)}{\theta(t)}
\prod_{i=1}^nP_i(1/a_i)
\prod_{1\le i<j\le n}a_jx_j\,\theta(x_i/x_j).
\end{multline}
\quad \quad \qed
\end{Theorem}
As is shown in \machSeite{RoScAC}\cite[Cor.~4.8]{RoScAC}, this
identity implies the following determinant evaluation.
\begin{Theorem} \label{adetcor}
Let $x_1,x_2,\dots,x_n$, $a_1,a_2,\dots,a_{n+1}$ and $b$ be indeterminates.
For each $j=1,\dots,n+1$, let $P_j(x)$ be an $A_{j-1}$ theta function of
norm $ta_1\dotsm a_j$. Then there holds
\begin{multline}
P_{n+1}(b)\;\det_{1\le i,j\le n}\left(P_j(x_i)
\prod_{k=j+1}^{n+1}\theta(a_kx_i)-\frac{P_{n+1}(x_i)}{P_{n+1}(b)}
P_j(b)\prod_{k=j+1}^{n+1}\theta(a_kb)\right)\\
=\frac{\theta(tba_1\dotsm a_{n+1}x_1\dotsm x_n)}{\theta(t)}
\prod_{i=1}^{n+1}P_i(1/a_i)
\prod_{1\le i<j\le n+1}a_jx_j\,\theta(x_i/x_j),
\end{multline}
where $x_{n+1}=b$.
\quad \quad \qed
\end{Theorem}
By combining Lemma~\ref{frobc} and Theorem~\ref{adet},
a determinant evaluation similar to the one in Theorem~\ref{thm:cdet},
but different, is obtained in
\machSeite{RoScAC}\cite[Theorem~4.9]{RoScAC}.
\begin{Theorem}\label{cdet}
Let $x_1,x_2,\dots,x_n$, $a_1,a_2,\dots,a_n$, and $c_1,\dots, c_{n+2}$ be
indeterminates. For each $j=1,\dots,n$, let $P_j(x)$ be an $A_{j-1}$
theta function of norm
$(c_1\dotsm c_{n+2}a_{j+1}\dotsm a_n)^{-1}$. Then there holds
\begin{multline}\label{dv}
\det_{1\leq i,j\leq n}\left(x_i^{-n-1}
P_j(x_i)\prod_{k=1}^{n+2}\theta(c_kx_i)\,
\prod_{k=j+1}^n\theta(a_kx_i)\right.\\
\left.-x_i^{n+1}P_j(x_i^{-1})\prod_{k=1}^{n+2}\theta(c_kx_i^{-1})\,
\prod_{k=j+1}^n\theta(a_kx_i^{-1})\right)\\
\kern-4cm=\frac{a_1\dotsm a_n}
{x_1\dotsm x_n\,\theta(c_1\dotsm c_{n+2}a_1\dotsm a_n)}
\prod_{i=1}^nP_i(1/a_i)\\\times
\prod_{1\leq i<j\leq n+2}\theta(c_ic_j)\prod_{i=1}^n\theta(x_i^2)
\prod_{1\le i<j\le n}a_jx_i^{-1}\,\theta(x_ix_j)\,\theta(x_i/x_j).
\end{multline}
\quad \quad \qed
\end{Theorem}
The last elliptic determinant evaluation which I present here is a
surprising elliptic extension of a determinant evaluation due to
Andrews and Stanton
\machSeite{AnStAA}\cite[Theorem~8]{AnStAA}
(see \machSeite{KratBN}\cite[Theorem~42]{KratBN})
due to Warnaar \machSeite{WarnAG}\cite[Theorem~4.17]{WarnAG}.
It is surprising because in the former there appear $q$-shifted
factorials {\it and\/} $q^2$-shifted factorials at the same time,
but nevertheless there exists
an elliptic analogue, and to obtain it one only has to add the $p$
everywhere in
the shifted factorials to convert them to elliptic ones.
\begin{Theorem} \label{thm:Warn2}
Let $x$ and $y$ be indeterminates. Then, for any non-negative integer
$n$, there holds
\begin{multline} \label{eq:Warn2}
\det_{0\le i,j\le n-1}\(\frac {(y/xq^{i};q^2,p)_{i-j}\,
(q/yxq^i;q^2,p)_{i-j}\, (1/x^2q^{2+4i};q^2,p)_{i-j}} {(q;q,p)_{2i+1-j}\,
(1/yxq^{2i};q,p)_{i-j}\, (y/xq^{1+2i};q,p)_{i-j}}\)\\
=\prod _{i=0} ^{n-1}\frac {(x^2q^{2i+1};q,p)_i\, (xq^{3+i}/y;q^2,p)_i\,
(yxq^{2+i};q^2,p)_i} {(x^2q^{2i+2};q^2,p)_i\, (q;q^2,p)_{i+1}\,
(yxq^{1+i};q,p)_i\, (xq^{2+i}/y;q,p)_i}.
\end{multline}
\quad \quad \qed
\end{Theorem}
In closing this final subsection, I remind the reader that,
as was already said before, many Hankel determinant
evaluations involving elliptic functions can be found in
\machSeite{MilnAO}\cite{MilnAP} and
\machSeite{MilnAP}\cite{MilnAO}.
\section*{Acknowledgments}
I would like to thank Anders Bj\"orner and Richard
Stanley, and the Institut Mittag--Leffler, for giving me the opportunity
to work in a relaxed and inspiring atmosphere during the
``Algebraic Combinatorics" programme in Spring 2005 at the Institut,
without which this article would never have reached its present form.
Moreover, I am extremely grateful to Dave Saunders and Zhendong Wan
who performed the {\sl LinBox} computations of determinants of size $3840$,
without which it would have been impossible for me to formulate
Conjectures~\ref{prob:1}--\ref{prob:3} and \ref{prob:8}.
Furthermore I wish to thank Josep Brunat, Adriano Garsia, Greg Kuperberg,
Antonio Montes,
Yuval Roichman, Hjalmar Rosengren, Michael Schlosser,
Guoce Xin, and especially Alain
Lascoux, for the many useful comments and discussions which helped to
improve the contents of this paper considerably.
|
{
"timestamp": "2005-08-10T16:39:40",
"yymm": "0503",
"arxiv_id": "math/0503507",
"language": "en",
"url": "https://arxiv.org/abs/math/0503507"
}
|
\section{INTRODUCTION}
The theory of holonomic modules over the Weyl algebra and more
general algebras of differential or $q$-difference operators is
becoming increasingly important, both as a crucial part of the
general theory of D-modules and in view of various applications
(see, for example, \cite{BK,Cart,Gue,S}). Well-known pathological
properties of differential operators over fields of positive
characteristic make the available, for this case, analogs of the
theory of D-modules much more complicated \cite{Bog,Lyub}. More
importantly, the resulting structures are not connected with the
existing analysis in positive characteristic based on a completely
different algebraic foundation.
Any non-discrete locally compact field of a positive
characteristic $p$ is isomorphic to the field $K$ of formal Laurent
series with coefficients from the Galois field $\mathbb F_q$, $q=p^\nu$, $\nu \in \mathbb
Z_+$. The field $K$ is endowed with a non-Archimedean absolute
value as follows. If $z\in K$,
$$
z=\sum\limits_{i=m}^\infty \zeta_ix^i,\quad m\in \mathbb
Z,\ \zeta_i\in \mathbb F_q ,\ \zeta_m\ne 0,
$$
then $|z|=q^{-m}$. This valuation can be extended onto the field
$\overline{K}_c$, the completion of an algebraic closure of $K$.
Analysis over $K$ and $\overline{K}_c$, which was initiated in the great paper
by Carlitz \cite{Carl} and developed subsequently by Wagner, Goss,
Thakur, the author, and many others (see the bibliography in
\cite{G2,Th3}) is very different from the classical calculus. An
important feature is the availability of many non-trivial additive
(actually, $\mathbb F_q$-linear) polynomials and power series of the form
$u(t)=\sum\limits a_kt^{q^k}$.
Taking into account the fact that the usual factorial $i!$, seen as
an element of $K$, vanishes for $i\ge p$, Carlitz introduced the new factorial
\begin{equation}
D_i=[i][i-1]^q\ldots [1]^{q^{i-1}},\quad [i]=x^{q^i}-x\ (i\ge 1),\
D_0=1,
\end{equation}
the $\mathbb F_q$-linear logarithm and exponential (which obtained a wide
generalization later, in the theory of Drinfeld modules), as well
as an important polynomial system, the Carlitz polynomials.
Subsequently many other $\mathbb F_q$-linear special functions, such as
Thakur's hypergeometric function \cite{Th1,Th2,Th3} and further special
polynomial systems, were introduced and investigated. The
difference operator
\begin{equation}
\Delta u(t)=u(xt)-xu(t)
\end{equation}
introduced in \cite{Carl} became the main ingredient of the
$\mathbb F_q$-linear calculus and analytic theory of differential equations
over $K$ developed in \cite{K2,K3,K4}. The role of a derivative is
played by the $\mathbb F_q$-linear operator $d=\sqrt[q]{}\circ \Delta$ ({\it the
Carlitz derivative}). The latter appears also in the
$\mathbb F_q$-linear umbral calculus \cite{K5} where an important role
belongs to the following new analog of binomial coefficients
\begin{equation}
\binom{k}{m}_K=\frac{D_k}{D_mD_{k-m}^{q^m}},\quad 0\le m\le k.
\end{equation}
The meaning of a polynomial coefficient in a differential equation
of the above type is not a usual multiplication by a polynomial,
but the action of a polynomial in the Frobenius operator $\tau$,
$\tau u=u^q$. With this notation, $d=\tau^{-1}\Delta$. The
operator $d$ is defined on any $\mathbb F_q$-linear $\overline{K}_c$-valued continuous
function; in particular, it decreases by one the ``$\mathbb F_q$-linear
degree'' of any $\mathbb F_q$-linear polynomial (see the relation (8)
below).
The above developments show that in the positive characteristic
case a natural counterpart of the Weyl algebra is, for the case of
a single variable, the ring $\mathfrak A_1$ generated by $\tau,d$, and
scalars from $\overline{K}_c$, with the relations \cite{K1}
\begin{equation}
d\tau -\tau d=[1]^{1/q},\quad \tau \lambda =\lambda^q\tau ,\quad ,
d\lambda =\lambda^{1/q}d\ (\lambda \in \overline{K}_c).
\end{equation}
Some algebraic properties of $\mathfrak A_1$ were studied in \cite{K3} --
it is left and right Noetherian, with no zero divisors.
The aim of this paper is to initiate the dimension theory for
modules over $\mathfrak A_1$ and more general ``several variable'' rings.
The definition of the latter is not straightforward. If, for
example, we consider the natural action of the Carlitz derivatives
$d_s$ and $d_t$ on an $\mathbb F_q$-linear monomial
$f(s,t)=s^{q^m}t^{q^n}$, we notice immediately that $d_s^mf$ is
not a polynomial, nor even a holomorphic function in $t$, if
$m>n$ (since the action of $d$ is not linear and involves taking
the $q$-th root). Moreover, it follows from the relation
$d(s^{q^m})=[m]^{1/q}s^{q^{m-1}}$ and the last commutation
relation in (4) that $d_s$ and $d_t$ do not commute even on
monomials $f$ with $m<n$.
A reasonable generalization is inspired by Zeilberger's idea (see
\cite{Cart}) to study holonomic properties of sequences of
functions making a transform with respect to the discrete
variables, which reduces the continuous-discrete case to the
purely continuous one (simultaneously in all the variables). In
our situation, if $\{ P_k(s)\}$ is a sequence of $\mathbb F_q$-linear
polynomials with $\deg P_k\le q^k$, we set
\begin{equation*}
f(s,t)=\sum \limits_{k=0}^\infty P_k(s)t^{q^k},\tag{$*$}
\end{equation*}
and $d_s$ is well-defined. In the variable $t$, we consider not
$d_t$ but the linear operator $\Delta_t$. The latter does not
commute with $d_s$ either, but satisfies the commutation relations
$$
d_s\Delta_t-\Delta_td_s=[1]^{1/q}d_s,\quad \Delta_t\tau -\tau
\Delta_t=[1]\tau ,
$$
so that the resulting ring $\mathfrak A_2$ resembles a universal
enveloping algebra of a solvable Lie algebra. Similarly we define
$\mathfrak A_{n+1}$ for $n>1$.
Introducing in $\mathfrak A_{n+1}$ an analog of the Bernstein filtration
and considering filtered modules over $\mathfrak A_{n+1}$, we find that
basic principles of the theory of algebraic D-modules \cite{Cout}
carry over to this case without serious complications. However,
the nonlinearity of $\tau$ and $d$ brings new phenomena. In
particular, already the ring $\mathfrak A_1$ possesses non-trivial
finite-dimensional representations. Therefore an analog of the
Bernstein inequality does not hold here without some additional
assumptions.
In spite of this fact, the notion of a holonomic module (that is a
module with the minimal possible GK dimension) seems to have a
reasonable sense for the case of $\mathfrak A_{n+1}$-modules. The examples
considered in this paper (both for $\mathfrak A_1$-modules and
$\mathfrak A_{n+1}$-modules with $n\ge 1$) show that the cases of an
anomalously small GK dimension may be seen as degenerate ones. In
terms of applications to analysis, it appears that a remarkable
phenomenon discovered by Zeilberger (see \cite{Cart}) -- that
virtually all important special functions and sequences of
classical analysis generate holonomic modules -- is maintained in
the positive characteristic case, if a holonomic module is defined
as a one with a minimal ``generic'' GK dimension, with degenerate
cases excluded. In the author's opinion, such applications provide
a sufficient justification for the definition of a quasi-holonomic
module given in this paper (Sect. 3.2).
Accordingly, the case we study in a greater detail is that of quasi-holonomic
submodules of the $\mathfrak A_{n+1}$-module of $\mathbb F_q$-linear functions
$u(s,t_1,\ldots ,t_n)$, polynomial in $s$ and holomorphic near the
origin in $t_1,\ldots ,t_n$. Following \cite{Cart} we call a
function $f$ quasi-holonomic if such is the module $\mathfrak A_{n+1}f$. We
prove general conditions for a function $f$ to be quasi-holonomic and
verify them for basic objects of this branch of analysis -- the
Carlitz polynomials, Thakur's hypergeometric polynomials, and the
$K$-binomial coefficients (3), making the above transformation (*) from
discrete variables to continuous ones.
Considering the $K$-binomial coefficients we use this occasion to
prove also the fact that they belong to the ring of integers not
only for the field $K$, but for any place of the global function
field $\mathbb F_q (x)$. Together with the results of \cite{K5}, this
property supports the case for considering the expressions (3) as
``proper'' analogs of the classical binomial coefficients. For
other analogs of the latter see \cite{Th3}.
\section{The Carlitz Ring}
{\bf 2.1.} Denote by $\mathcal F_{n+1}$ the set of all germs of functions of the
form
\begin{equation}
f(s,t_1,\ldots ,t_n)=\sum\limits_{k_1=0}^\infty \ldots \sum\limits_{k_n=0}^\infty
\sum\limits_{m=0}^{\min (k_1,\ldots ,k_n)}a_{m,k_1,\ldots
,k_n}s^{q^m}t_1^{q^{k_1}}\ldots t_n^{q^{k_n}}
\end{equation}
where $a_{m,k_1,\ldots ,k_n}\in \overline{K}_c$ are such that all the series
are convergent on some neighbourhoods of the origin. We do not
exclude the case $n=0$ where $\mathcal F_1$ will mean the set of all
$\mathbb F_q$-linear power series $\sum\limits_ma_ms^{q^m}$ convergent on
a neighbourhood of the origin. $\widehat{\mathcal F}_{n+1}$ will denote
the set of all polynomials from $\mathcal F_{n+1}$, that is the series
(5) in which only a finite number of coefficients is different
from zero.
The ring $\mathfrak A_{n+1}$ is generated by the operators $\tau
,d_s,\Delta_{t_1},\ldots \Delta_{t_n}$ on $\mathcal F_{n+1}$ defined in
the Introduction, and the operators of multiplication by scalars
from $\overline{K}_c$. To simplify the notation, we will write $\Delta_j$
instead of $\Delta_{t_j}$ and identify a scalar $\lambda \in \overline{K}_c$
with the operator of multiplication by $\lambda$. The operators
$\Delta_j$ are $\overline{K}_c$-linear, so that
\begin{equation}
\Delta_j\lambda =\lambda \Delta_j,\quad \lambda \in \overline{K}_c ,
\end{equation}
while the operators $\tau ,d_s$ satisfy the commutation relations
(4). In the action of each operator $d_s,\Delta_j$ (acting in a
single variable), other variables are treated as scalars. The
operator $\tau$ acts simultaneously on all the variables and
coefficients, so that
$$
\tau f=\sum a^q_{m,k_1,\ldots
,k_n}s^{q^{m+1}}t_1^{q^{k_1+1}}\ldots t_n^{q^{k_n+1}}.
$$
It follows from (2) that
\begin{equation}
\Delta_jt_j^{q^k}=\begin{cases}
[k]t_j^{q^k}, & \text{if $k\ge 1$};\\
0, & \text{if $k=0$};
\end{cases}
\end{equation}
the second equality can be included in the first one, if we set $[0]=0$.
Similarly
\begin{equation}
d_ss^{q^m}=[m]^{1/q}s^{q^{m-1}},\quad m\ge 0.
\end{equation}
Since $|[m]|=q^{-1}$ for any $m\ge 1$, the action of operators
from $\mathfrak A_{n+1}$ does not spoil convergence of the series (5).
The identity $[k+1]-[k]^q=[1]$, together with (7) and (8), implies
the commutation relations
\begin{equation}
\Delta_j\tau -\tau \Delta_j=[1]\tau ,\quad
d_s\Delta_j-\Delta_jd_s=[1]^{1/q}d_s,\quad j=1,\ldots ,n,
\end{equation}
verified by applying both sides of each equality to an arbitrary
monomial.
Using the commutation relations (4), (6), and (9), we can write
any element $a\in \mathfrak A_{n+1}$ as a finite sum
\begin{equation}
a=\sum c_{l,\mu,i_1,\ldots ,i_n}\tau^ld_s^\mu \Delta_1^{i_1}\ldots
\Delta_n^{i_n}.
\end{equation}
\medskip
\begin{prop}
The representation (10) of an element $a\in \mathfrak A_{n+1}$ is unique.
\end{prop}
\medskip
{\it Proof}. Suppose that
\begin{equation}
\sum\limits_{l,\mu,i_1,\ldots ,i_n}c_{l,\mu,i_1,\ldots ,i_n}\tau^ld_s^\mu
\Delta_1^{i_1}\ldots \Delta_n^{i_n}=0.
\end{equation}
Applying the left-hand side of (11) to the function
$st_1^{q^{k_1}}\ldots t_n^{q^{k_n}}$ with $k_1,\ldots ,k_n>0$ we
find that
$$
\sum\limits_l\left( \sum\limits_{i_1,\ldots ,i_n}c_{l,0,i_1,\ldots ,i_n}
[k_1]^{i_1q^l}\ldots [k_n]^{i_nq^l}\right)
s^{q^l}t_1^{q^{k_1+l}}\ldots t_n^{q^{k_n+l}}=0
$$
whence
$$
\sum\limits_{i_1,\ldots ,i_n}c_{l,0,i_1,\ldots ,i_n}
[k_1]^{i_1q^l}\ldots [k_n]^{i_nq^l}=0
$$
for each $l$. Writing this in the form
\begin{equation}
\sum\limits_{i_n}\rho (i_n)y^{i_n}=0
\end{equation}
where
$$
\rho (i_n)=\sum\limits_{i_1,\ldots ,i_{n-1}}c_{l,0,i_1,\ldots ,i_n}
[k_1]^{i_1q^l}\ldots [k_{n-1}]^{i_{n-1}q^l},\quad y=[k_n]^{q^l},
$$
and taking into account that (12) holds for arbitrary $k_n\ge 1$,
that is for an infinite set of values of $y$, we find that $\rho
(i_n)=0$. Repeating this reasoning we get the equality $c_{l,0,i_1,\ldots
,i_n}=0$ for all $l,0,i_1,\ldots ,i_n$.
Suppose that $c_{l,\mu ,i_1,\ldots ,i_n}=0$ for $\mu \le \mu_0$
and arbitrary $l,i_1,\ldots ,i_n$. Then we apply the left-hand
side of (11) to the function $s^{q^{\mu_0+1}}t_1^{q^{k_1}}\ldots
t_n^{q^{k_n}}$ and proceed as before coming to the equality
$c_{l,\mu_0+1,i_1,\ldots ,i_n}=0$ for all $l,i_1,\ldots ,i_n$.
$\qquad \blacksquare$
\medskip
It is easy to prove by induction with respect to $n$ (using the
commutation relations (9) and the result from \cite{K3} regarding
the case $n=0$) that $\mathfrak A_{n+1}$ has no zero-divisors.
\medskip
{\bf 2.2.} Let us introduce a filtration in $\mathfrak A_{n+1}$ denoting
by $\Gamma_\nu$, $\nu \in \mathbb Z_+$, the $\overline{K}_c$-vector space of
operators (10) with $\max \{ l+\mu +i_1+\cdots +i_n\} \le \nu$
where the maximum is taken over all the terms contained in the
representation (10). It is clear that $\mathfrak A_{n+1}$ is a filtered
ring (for the definitions see \cite{MR}). Setting $T_0=\overline{K}_c$, $T_\nu
=\Gamma_\nu /\Gamma_{\nu -1}$, $\nu \ge 1$, we introduce the
associated graded ring
$$
\gr (\mathfrak A_{n+1})=\bigoplus\limits_{\nu=0}^\infty T_\nu .
$$
It is generated by scalars $\lambda \in T_0$ and the images
$\bar \tau,\bar d_s,\bar \Delta_1,\ldots
,\bar \Delta_n\in T_1$ of the elements $\tau
,d_s,\Delta_1,\ldots ,\Delta_n\in \Gamma_1$ respectively, which
satisfy, by virtue of (4), (6), and (9), the relations
\begin{gather*}
\bar d_s\bar\tau-\bar\tau \bar d_s=0,
\bar\tau \lambda
=\lambda^q\bar\tau,\bar d_s\lambda
=\lambda^{1/q}\bar d_s,\\
\bar d_s\bar \Delta_j-\bar \Delta_j\bar d_s=0,
\bar \Delta_j\bar \tau -\bar\tau \bar \Delta_j=0,
\bar \Delta_j\lambda =\lambda \bar \Delta_j\quad
(j=1,\ldots ,n).
\end{gather*}
It is clear that $\mathfrak A_{n+1}$ is a (left and right) almost
normalizing extension of the field $\overline{K}_c$ (see Chapter 1, \S 6 in
\cite{MR}), so that the rings $\mathfrak A_{n+1}$ and $\gr (\mathfrak A_{n+1})$
are left and right Noetherian.
Let us compute the dimension of the $\overline{K}_c$-vector space
$\Gamma_\nu$. Note that
$$
\dim \Gamma_\nu =\dim \bigoplus\limits_{j=1}^\nu T_j,
$$
so that $\dim \Gamma_\nu$ coincides with the dimension of the
appropriate space appearing in the natural filtration in $\gr (\mathfrak A_{n+1})$.
\medskip
\begin{lem}
For any $\nu \in \mathbb N$
$$
\dim \Gamma_\nu = \binom{\nu +n+2}{n+2}.
$$
\end{lem}
\medskip
{\it Proof}. The number $\dim \Gamma_\nu$ coincides with the
number of non-negative integral solutions $(l,\mu ,i_1,\ldots
,i_n)$ of the inequality $l+\mu +i_1+\cdots +i_n\le \nu$, so that
$$
\dim \Gamma_\nu =\sum\limits_{j=0}^\nu N(j,n+2)
$$
where $N(j,k)$ is the number of different representations of $j$
as sums of $k$ non-negative integers. It is known (Proposition 6.1 in
\cite{Lan}) that $N(j,k)=\dbinom{j+k-1}{k-1}$. Then (see Sect. 1.3
from \cite{Ri})
$$
\dim \Gamma_\nu =\sum\limits_{j=0}^\nu \binom{j+n+1}{n+1}=
\sum\limits_{i=0}^\nu \binom{\nu+n+1-i}{n+1}=\binom{\nu
+n+2}{n+2},
$$
as desired. $\qquad \blacksquare$
\section{Filtered Modules}
{\bf 3.1.} Let $M$ be a left module over the Carlitz ring
$\mathfrak A_{n+1}$. Suppose we have a filtration $\{ \mathfrak M_j\}$ of $M$,
that is
\begin{equation}
\mathfrak M_0\subset \mathfrak M_1\subset \ldots \subset M,\quad
M=\bigcup\limits_{j\ge 0}\mathfrak M_j,
\end{equation}
and $\Gamma_\nu \mathfrak M_j\subset \mathfrak M_{\nu +j}$ for any $\nu ,j\in
\mathbb Z_+$. We assume that each $\mathfrak M_j$ is a finite-dimensional
vector space over $\overline{K}_c$. Below we write $\mathfrak M_j=\{ 0\}$ and
$\Gamma_\nu =\{ 0\}$ if $j<0$ and $\nu <0$.
In a standard way \cite{Cout} we define the graded module
$$
\gr (M)=\bigoplus\limits_{j\ge 0}\left( \mathfrak M_j/\mathfrak M_{j-1}\right)
$$
over $\gr (\mathfrak A_{n+1})$, associated with the filtration (13). As
usual, the filtration (13) is called {\it good}, if $\gr (M)$ is
finitely generated.
Main properties of filtered modules over the Weyl algebra (see
\cite{Bj,Cout}) carry over to our situation without any
substantial changes, both in their formulations and proofs. In
fact, the only technical difference is that the operators $\tau$
and $d_s$ are semilinear, not linear. However, as it is explained
in Appendix I to Chapter 2 of \cite{Bour}, basic notions of linear
algebra remain valid for semilinear mappings -- a semilinear
mapping of a vector space into itself can be interpreted as a
linear mapping between two different vector spaces, and, for
instance, dimensions of the kernel and cokernel are not changed in
this interpretation. Note that everywhere in this paper we
consider vector spaces over the algebraically closed field $\overline{K}_c$,
on which $\tau$ induces an automorphism. Below, as before, $\dim$
means the dimension over $\overline{K}_c$.
In particular, for a good filtration there exist a polynomial
$\chi \in \mathbb Q[t]$ and a number $N\in \mathbb N$, such that
$$
\dim \mathfrak M_s=\sum\limits_{i=0}^s\dim (\mathfrak M_i/\mathfrak M_{i-1})=\chi
(s)\text{ for }s\ge N.
$$
The number $d(M)=\deg \chi$, called the (Gelfand-Kirillov) {\it
dimension} of $M$, and the leading coefficient of $\chi$
multiplied by $d(M)!$, called the {\it multiplicity} $m(M)$ of
$M$, do not depend on the choice of a good filtration on $M$. A
filtration $\{ \mathfrak M_i\}$ is good if and only if there exists such
$k_0\in \mathbb N$ that
$$
\mathfrak M_{i+k}=\Gamma_i\mathfrak M_k\text{ \ for all }k\ge k_0.
$$
If $N$ and $M/N$ are a submodule and the corresponding quotient
module, with the induced filtrations, then $d(M)=\max \{
d(N),d(M/N)\}$, and if $d(N)=d(M/N)$, then $m(M)=m(N)+m(M/N)$. For
a direct sum $M=M_1\oplus \cdots \oplus M_k$ we have $d(M)=\max \{
d(M_1),\ldots ,d(M_k)\}$.
In particular, if we consider $\mathfrak A_{n+1}$ as a left module over
itself, then by Lemma 1
\begin{equation}
d(\mathfrak A_{n+1})=n+2,\quad m(\mathfrak A_{n+1})=1.
\end{equation}
It follows from (14) and the above general facts that for any
finitely generated left $\mathfrak A_{n+1}$-module
\begin{equation}
d(M)\le n+2.
\end{equation}
By (14), the bound in (15) in general cannot be improved. However,
if $I$ is a non-zero left ideal in $\mathfrak A_{n+1}$, then
\begin{equation}
d(\mathfrak A_{n+1}/I)\le n+1.
\end{equation}
The proof of (16) is identical to the proof of Corollary 9.3.5
from \cite{Cout}.
\medskip
{\bf 3.2.} Let us consider the set $\widehat{\mathcal F}_{n+1}$ of
polynomials (5) as a $\mathfrak A_{n+1}$-module. A filtration
$$
\mathcal F^{(0)}_{n+1}\subset \mathcal F^{(1)}_{n+1}\subset \ldots \subset
\widehat{\mathcal F}_{n+1}
$$
can be introduced by setting $\mathcal F^{(j)}_{n+1}$ to be the
collection of all the polynomials (5), in which the maximal
indices $k_1,\ldots ,k_n$ corresponding to non-zero coefficients
$a_{m,k_1,\ldots ,k_n}$ do not exceed $j$. This filtration is
obviously good.
\medskip
\begin{prop}
For the module $\widehat{\mathcal F}_{n+1}$,
\begin{equation}
d\left( \widehat{\mathcal F}_{n+1}\right) =n+1,\quad m\left( \widehat{\mathcal F}_{n+1}\right)
=n!
\end{equation}
\end{prop}
\medskip
{\it Proof}. Let us compute $\dim \mathcal F^{(j)}_{n+1}$. For a fixed
$\mu$, the quantity of $n$-tuples $(k_1,\ldots ,k_n)$ of
non-negative integers, for which $\min (k_1,\ldots ,k_n)=\mu$, is
added up from those $n$-tuples where $i$ numbers are equal to
$\mu$ while $n-i$ numbers are strictly larger and can take $j-\mu$
values. Therefore the above quantity equals
$\sum\limits_{i=1}^n\dbinom{n}{i}(j-\mu )^{n-i}$. Next, $\mu +1$
possible values of $m$ in (5) correspond to each $n$-tuple. Thus,
$$
\dim \mathcal F^{(j)}_{n+1}=\sum\limits_{\mu =0}^j(\mu +1)
\sum\limits_{i=1}^n\dbinom{n}{i}(j-\mu )^{n-i}=\sum\limits_{\mu =0}^j(\mu
+1)\left\{ (j-\mu +1)^n-(j-\mu )^n\right\} .
$$
Denote $r_\mu =(j-\mu +1)^n-(j-\mu )^n$, $R_i=r_0+r_1+\cdots
+r_i=(j+1)^n-(j-i)^n$. Performing the Abel transformation we get
\begin{multline*}
\dim \mathcal F^{(j)}_{n+1}=(j+1)R_j-\sum\limits_{i=0}^{j-1}R_i
=(j+1)^{n+1}-j(j+1)^n+\sum\limits_{i=0}^{j-1}(j-i)^n\\
=(j+1)^n+\sum\limits_{k=1}^jk^n=(j+1)^n+S_n(j+1)
\end{multline*}
where $S_n(N)=1^n+2^n+\cdots +(N-1)^n$.
It is known (\cite{IR}, Chapter 15) that
$$
S_n(N)=\frac{1}{n+1}\sum\limits_{k=0}^n\binom{n+1}{k}B_kN^{n+1-k}
$$
where $B_k$ are the Bernoulli numbers. Therefore we find that
$$
\dim \mathcal F^{(j)}_{n+1}=\frac{(j+1)^{n+1}}{n+1}+P_n(j)
$$
where $P_n$ is a polynomial of the degree $n$. This implies (17).
$\qquad \blacksquare$
\medskip
It is natural to call an $\mathfrak A_{n+1}$-module $M$ {\it quasi-holonomic} if
$d(M)=n+1$. Thus, $\widehat{\mathcal F}_{n+1}$ is an example of a
quasi-holonomic module.
\medskip
{\bf 3.3.} Let us look at possible values of $d(M)$ for
$\mathfrak A_1$-modules. The next result demonstrates a sharp difference
from the case of modules over the Weyl algebras.
\medskip
\begin{teo}
\begin{description}
\item{{\rm (i)}} For any $k=1,2,\ldots$, there exists such a
nontrivial $\mathfrak A_1$-module $M$ that $\dim M=k$ ($\dim$ means the
dimension over $\overline{K}_c$), that is $d(M)=0$.
\item{{\rm (ii)}} Let $M$ be a finitely generated $\mathfrak A_1$-module
with a good filtration. Suppose that there exists a ``vacuum vector'' $v\in
M$, such that $d_sv=0$ and $\tau^m(v)\ne 0$ for all
$m=0,1,2,\ldots$. Then $d(M)\ge 1$.
\end{description}
\end{teo}
\medskip
{\it Proof}. (i) Let $M=(\overline{K}_c )^k$. Denote by $\mathbf e_1,\ldots
,\mathbf e_k$ the standard basis in $M$, that is $\mathbf
e_j=(0,\ldots ,0,1,0,\ldots ,0)$, with 1 at the $j$-th place. Let
$(\lambda_{ij})$ be a $k\times k$ matrix over $\overline{K}_c$, such that
$\lambda_{ij}\in \mathbb F_q$ if $i\ne j$, while the diagonal elements
satisfy the equation $\lambda^q-\lambda +[1]^{1/q}=0$. We define
the action of $\tau$ and $d_s$ on $M$ as follows:
$$
\tau (c\mathbf e_j)=c^q\mathbf e_j;\ d_s(\mathbf
e_j)=\sum\limits_{i=1}^n\lambda_{ij}\mathbf e_i;\ d_s(c\mathbf
e_j)=c^{1/q}\mathbf e_j,\quad c\in \overline{K}_c ,j=1,\ldots ,k,
$$
with subsequent additive continuation onto $M$.
If $x=\sum\limits_{j=1}^kc_j\mathbf e_j$, $c_j\in \overline{K}_c$, then we
have
$$
\tau d_s(x)=\sum\limits_{j=1}^kc_j\sum\limits_{i=1}^n\lambda_{ij}^q\mathbf
e_i,\quad d_s\tau (x)=\sum\limits_{j=1}^kc_j\sum\limits_{i=1}^n\lambda_{ij}\mathbf
e_i,
$$
so that
$$
d_s\tau (x)-\tau d_s(x)=[1]^{1/q}x,
$$
and we have indeed an $\mathfrak A_1$-module.
(ii) It follows from the relation
$[d_s,\tau^m]=[m]^{1/q}\tau^{m-1}$ (see \cite{K3}) that
$$
d_s\tau^mv=[m]^{1/q}\tau^{m-1}v,\quad m=1,2,\ldots ,
$$
that is $\tau^{m-1}v$ is an eigenvector of a linear operator
$d_s\tau$ on $M$ (considered as a $\overline{K}_c$-vector space) corresponding
to the eigenvalue $[m]^{1/q}$. Therefore the vectors $\tau^{m-1}v$
are linearly independent. It follows from the existence of the
Hilbert polynomial $\chi$ implementing the dimension $d(M)$ that
$d(M)\ge 1$. $\qquad \blacksquare$
\medskip
\section{Holonomic Functions}
{\bf 4.1.} Let $0\ne f\in \mathcal F_{n+1}$,
$$
I_f=\left\{ \varphi \in \mathfrak A_{n+1}:\ \varphi (f)=0\right\} .
$$
$I_f$ is a left ideal in $\mathfrak A_{n+1}$. The left $\mathfrak A_{n+1}$-module
$M_f=\mathfrak A_{n+1}/I_f$ is isomorphic to the submodule
$\mathfrak A_{n+1}f\subset \mathcal F_{n+1}$ -- an element $\varphi (f)\in
\mathfrak A_{n+1}f$ corresponds to the class of $\varphi \in \mathfrak A_{n+1}$ in
$M_f$. A natural good filtration in $M_f$ is induced from that in
$\mathfrak A_{n+1}$ -- the subspace $\mathfrak M_j$ is generated by elements
$\tau^ld_s^\mu \Delta_1^{i_1}\ldots \Delta_n^{i_n}f$ with $l+\mu
+i_1+\cdots +i_n\le j$.
As we know (see (16)), if $I_f\ne \{ 0\}$, then $d(M_f)\le n+1$.
We call a function $f$ {\it quasi-holonomic} if the module $M_f$ is
quasi-holonomic, that is $d(M_f)=n+1$. The condition $I_f\ne \{ 0\}$
means that $f$ is a solution of a ``differential equation''
$\varphi (f)=0$, $\varphi \in \mathfrak A_{n+1}$. For $n=0$, we have the
following easy result.
\medskip
\begin{teo}
If a non-zero function $f\in \mathcal F_1$ satisfies an equation
$\varphi (f)=0$, $0\ne \varphi \in \mathfrak A_1$, then $f$ is quasi-holonomic.
\end{teo}
\medskip
{\it Proof}. It is sufficient to show that $\dim M_f=\infty$. In
fact, the sequence $\left\{ \tau^lf\right\}_{l=0}^\infty$ is
linearly independent because otherwise we would have such a
finite collection of elements $c_0,c_1,\ldots ,c_N\in \overline{K}_c$, some of
which are different from zero, that
\begin{equation}
c_0f(s)+c_1f^q(s)+\cdots +c_Nf^{q^N}(s)=0
\end{equation}
for all $s$ from a neighbourhood of the origin in $\overline{K}_c$. It follows
from (18) that $f$ takes only a finite number of values. By the
uniqueness theorem for non-Archimedean holomorphic functions,
$f(s)\equiv \text{const}$ on some neighbourhood of the origin. Due
to the $\mathbb F_q$-linearity, $f(s)\equiv 0$, and we have come to a
contradiction. $\qquad \blacksquare$
\medskip
In particular, any $\mathbb F_q$-linear polynomial of $s$ is quasi-holonomic,
since it is annihilated by $d_s^m$, with a sufficiently large $m$.
\medskip
{\bf 4.2.} If $n>0$, the situation is more complicated. We call
the module $M_f$ (and the corresponding function $f$) {\it
degenerate} if $d(M_f)<n+1$ (by the Bernstein inequality, there is
no degeneracy phenomena for modules over the complex Weyl
algebra). We give an example of degeneracy for the case $n=1$.
Let $f(s,t_1)=g(st_1)\in \mathcal F_2$ where the function $g$ belongs to
$\mathcal F_1$ and satisfies an equation $\varphi (g)=0$, $\varphi \in
\mathfrak A_1$. Then $f$ is degenerate.
Indeed, by the general rule, $\mathfrak M_j$ is spanned by elements
$\tau^ld_s^\mu \Delta_1^{i_i}f$ with $l+\mu +i_1\le j$. In the present
situation,
$$
\Delta_1f=g(xst_1)-xg(st_1)=\tau d_sg,
$$
so that an element $\tau^ld_s^\mu \Delta_1^{i_i}f$ is a linear
combination of elements $\left( \tau^{l+\lambda}d_s^{\mu
+\nu}g\right) (s,t)$ with $\lambda \le i_1$, $\nu \le i_1$.
Therefore $\mathfrak M_j$ is contained in the linear hull of elements
$\tau^kd_s^mg$, $k+m\le 2j$. By Theorem 2, the $\overline{K}_c$-dimension of
the latter does not exceed a linear function of $2j$, so that
$d(M_f)\le 1$. On the other hand, since, as in the proof of
Theorem 2, the system of functions $\left\{
\tau^lf\right\}_{l=0}^\infty$ is linearly independent, we find that
$d(M_f)=1$.
In order to exclude the degenerate case, we introduce the notion
of a non-sparse function.
A function $f\in \mathcal F_{n+1}$ of the form (5) is called {\it
non-sparse} if there exists such a sequence $m_l\to \infty$ that,
for any $l$, there exist sequences $k_1^{(i)},k_2^{(i)},\ldots
,k_n^{(i)}\ge m_l$ (depending on $l$), such that $k_\nu^{(i)}\to
\infty$ as $i\to \infty$ ($\nu =1,\ldots ,n$), and $a_{m,k_1^{(i)},\ldots
,k_n^{(i)}}\ne 0$.
\medskip
\begin{lem}
If a function $f$ is non-sparse, then the system of functions
$(\tau d_s)^\lambda \Delta_1^{j_1}\ldots \Delta_n^{j_n}f$
($\lambda ,j_1,\ldots ,j_n=0,1,2,\ldots$) is linearly independent
over $\overline{K}_c$.
\end{lem}
\medskip
{\it Proof}. Suppose that
\begin{equation}
\sum\limits_{\lambda =0}^\Lambda \sum\limits_{j_1=0}^{J_1}\ldots
\sum\limits_{j_n=0}^{J_n}c_{\lambda ,j_1,\ldots ,j_n}
(\tau d_s)^\lambda \Delta_1^{j_1}\ldots \Delta_n^{j_n}f=0
\end{equation}
for some $c_{\lambda ,j_1,\ldots ,j_n}\in \overline{K}_c$, $\Lambda ,J_1,\ldots
,J_n\in \mathbb N$. Substituting (5) into (19) and collecting
coefficients of the power series we find that
\begin{equation}
\sum\limits_{\lambda =0}^\Lambda \sum\limits_{j_1=0}^{J_1}\ldots
\sum\limits_{j_n=0}^{J_n}c_{\lambda ,j_1,\ldots ,j_n}
[m_l]^\lambda [k_1^{(i)}]^{j_1}\ldots[k_n^{(i)}]^{j_n}=0
\end{equation}
for all $l,i$.
We see from (20) that the polynomial
$$
\sum\limits_{j_n=0}^{J_n}\left\{ \sum\limits_{\lambda =0}^\Lambda
\sum\limits_{j_1=0}^{J_1}\ldots
\sum\limits_{j_{n-1}=0}^{J_{n-1}}c_{\lambda ,j_1,\ldots ,j_n}
[m_l]^\lambda [k_1^{(i)}]^{j_1}\ldots[k_{n-1}^{(i)}]^{j_{n-1}}
\right\} z^{j_n}
$$
has an infinite sequence of different roots, so that
$$
\sum\limits_{\lambda =0}^\Lambda
\sum\limits_{j_1=0}^{J_1}\ldots
\sum\limits_{j_{n-1}=0}^{J_{n-1}}c_{\lambda ,j_1,\ldots ,j_n}
[m_l]^\lambda [k_1^{(i)}]^{j_1}\ldots[k_{n-1}^{(i)}]^{j_{n-1}}=0
$$
for all $l,i$, and for each $j_n=0,1,\ldots ,J_n$. Repeating this
reasoning we find that all the coefficients $c_{\lambda ,j_1,\ldots
,j_n}$ are equal to zero. $\qquad\blacksquare$
\medskip
Now the above arguments regarding $d(M_f)$ yield the following
result.
\medskip
\begin{teo}
If a function $f$ is non-sparse, then $d(M_f)\ge n+1$. If, in
addition, $f$ satisfies an equation $\varphi (f)=0$, $0\ne \varphi \in
\mathfrak A_{n+1}$, then $f$ is quasi-holonomic.
\end{teo}
\medskip
As in the classical situation, one can construct quasi-holonomic
functions by addition.
\medskip
\begin{prop}
If the functions $f,g\in \mathcal F_{n+1}$ are quasi-holonomic, and $f+g$ is
non-sparse, then $f+g$ is quasi-holonomic.
\end{prop}
\medskip
{\it Proof}. Consider the $\mathfrak A_{n+1}$-module
$M_2=(\mathfrak A_{n+1}f)\oplus (\mathfrak A_{n+1}g)$. Since $f$ and $g$ are
both quasi-holonomic, we have $d(M_2)=n+1$. Next, let $N_2$ be a
submodule of $M_2$ consisting of such pairs $(\varphi (f),\varphi
(g))$ that $\varphi (f)+\varphi (g)=0$. Then $d(M_2)=\max \{
d(N_2),d(M_2/N_2)\}$, so that $d(M_2/N_2)\le n+1$.
On the other hand, we have an injective mapping $\mathfrak A_{n+1}(f+g)\to
M_2/N_2$, which maps $\varphi (f+g)$ to the image of $(\varphi (f),\varphi
(g))$ in $M_2/N_2$. Therefore $d(\mathfrak A_{n+1}(f+g))\le d(M_2/N_2)\le
n+1$. It remains to use Theorem 3. $\qquad \blacksquare$
\medskip
{\bf 4.3.} We use Theorem 3 to prove that the functions (5)
obtained via the sequence-to-function transform ($*$) or its
multi-index generalizations, from some well-known sequences of
polynomials over $K$ are quasi-holonomic.
a) {\it The Carlitz polynomials}. The sequence
$$
f_k(s)=D_k^{-1}\prod \limits _{\genfrac{}{}{0pt}{1}{m\in
\mathbb F_q [x]}{\deg m<k}}(s-m) \quad (k\ge 1),\quad f_0(s)=s,
$$
of normalized Carlitz polynomials forms an orthonormal basis of
the space of all $\mathbb F_q$-linear continuous functions on the ring of
integers of the field $K$. Its transform ($*$), the function
\begin{equation}
C_s(t)=\sum\limits_{k=0}^\infty f_k(s)t^{q^k}
\end{equation}
called the {\it Carlitz module}, is one of the main objects of the
function field arithmetic \cite{G2,Th3}.
It is known \cite{Carl,G1} that
$$
f_k(s)=\sum\limits_{i=0}^k\frac{(-1)^{k-i}}{D_iL_{k-i}^{q^i}}s^{q^i}
$$
where $L_i=[i][i-1]\ldots [1]$ ($i\ge 1$), $L_0=1$. By (1), we
have
\begin{equation}
|D_i|=q^{-\frac{q^i-1}{q-1}},\quad |L_i|=q^{-i},
\end{equation}
so that
$$
\left| D_iL_{k-i}^{q^i}\right| =q^{-\left(
\frac{q^i-1}{q-1}+(k-i)q^i\right) },\quad 0\le i\le k.
$$
For large values of $k$, an elementary investigation of the
function $z\mapsto (k-z)q^z$, $z\le k$, shows that
$$
\max\limits_{0\le i\le k}(k-i)q^i\le \alpha q^k,\quad \alpha >0,
$$
so that
$$
|f_k(s)|\le q^{\alpha q^k}
$$
for all $s\in \overline{K}_c$ with $|s|\le q^{-1}$. Therefore the series (21)
converges for small $|t|$, so that the Carlitz module function
belongs to $\mathcal F_2$.
Since $d_sf_i=f_{i-1}$ for $i\ge 1$, and $d_sf_0=0$ \cite{G1}, we
see that $d_sC_s(t)=C_s(t)$. Clearly, the function $C_s(t)$ is
non-sparse. Therefore the Carlitz module function is quasi-holonomic,
jointly in both its variables.
\medskip
b) {\it Thakur's hypergeometric polynomials}. We consider the
polynomial case of Thakur's hypergeometric function
\cite{Th1,Th2,Th3}:
\begin{equation}
{}_lF_\lambda (-a_1,\ldots ,-a_l;-b_1,\ldots ,-b_\lambda
;z)=\sum\limits_m\frac{(-a_1)_m\ldots (-a_l)_m}{(-b_1)_m\ldots (-b_\lambda
)_mD_m}z^{q^m}
\end{equation}
where $a_1,\ldots ,a_l,b_1,\ldots ,b_\lambda \in\mathbb Z_+$,
\begin{equation}
(-a)_m=\begin{cases}
(-1)^{a-m}L_{a-m}^{-q^m}, & \text{if $m\le a$},\\
0, & \text{if $m>a$},\end{cases},\quad a\in \mathbb Z_+.
\end{equation}
It is seen from (24) that the terms in (23), which make sense and
do not vanish, are those with $m\le \min (a_1,\ldots ,a_l,b_1,\ldots
,b_\lambda )$. Let
\begin{multline}
f(s,t_1,\ldots ,t_l,u_1,\ldots ,u_\lambda )\\ =
\sum\limits_{k_1=0}^\infty \ldots \sum\limits_{k_l=0}^\infty
\sum\limits_{\nu_1=0}^\infty \ldots \sum\limits_{\nu_\lambda =0}^\infty
{}_lF_\lambda (-k_1,\ldots ,-k_l;-\nu_1,\ldots ,-\nu_\lambda ;s)
t_1^{q^{k_1}}\ldots t_l^{q^{k_l}}u_1^{q^{\nu_1}}\ldots
u_\lambda^{q^{\nu_\lambda}}.
\end{multline}
We prove as above that all the series in (25) converge near the
origin. Thus, $f\in \mathcal F_{l+\lambda +1}$.
It is known (\cite{Th3}, Sect. 6.5) that
\begin{equation}
d_s{}_lF_\lambda (-k_1,\ldots ,-k_l;-\nu_1,\ldots ,-\nu_\lambda
;s)={}_lF_\lambda (-k_1+1,\ldots ,-k_l+1;-\nu_1+1,\ldots ,-\nu_\lambda +1;s)
\end{equation}
if all the parameters $k_1,\ldots ,k_l,\nu_1,\ldots ,\nu_\lambda$
are different from zero. If at least one of them is equal to zero,
then the left-hand side of (26) equals zero. This property implies
the identity $d_sf=f$, the same as that for the Carlitz module
function. Since $f$ is non-sparse, it is quasi-holonomic.
In the next section we will see that the $K$-binomial coefficients
(3) correspond to a quasi-holonomic function satisfying a more
complicated equation containing also the operator $\Delta_t$.
\section{$K$-Binomial Coefficients}
{\bf 5.1.} Let us consider the $K$-binomial coefficients (3). It
follows from (22) that
$$
\left| \binom{k}{m}_K\right| =1,\quad 0\le m\le k.
$$
Since $\binom{k}{m}_K\in \mathbb F_q (x)$, it is natural to consider also
other places of $\mathbb F_q (x)$, that is other non-equivalent absolute
values on $\mathbb F_q (x)$. It is well known (\cite{Weil}, Sect. 3.1) that
they are parametrized by monic irreducible polynomials $\pi \in \mathbb F_q
[x]$. The absolute value $|t|_\pi$, $t\in \mathbb F_q (x)$, is defined as
follows. We write $t=\pi^\nu \alpha /\alpha'$ where $m\in \mathbb
Z$, $\alpha ,\alpha'\in \mathbb F_q [x]$, and $\pi$ does not divide $\alpha
,\alpha'$. Then $|t|_\pi =|\pi |_\pi^\nu$, $|\pi |_\pi
=q^{-\delta}$ where $\delta =\deg \pi$; as usual, $|0|_\pi =0$.
The absolute value $|\cdot |$ used elsewhere in this paper
corresponds to $\pi (x)=x$.
\medskip
\begin{prop}
For any monic irreducible polynomial $\pi \in \mathbb F_q[x]$,
the $K$-binomial coefficients (3) satisfy the inequality
$$
\left| \binom{k}{m}_K\right|_\pi \le 1,\quad 0\le m\le k.
$$
\end{prop}
\medskip
{\it Proof}. First we compute $|D_m|_\pi$. It follows from Lemma
2.13 of \cite{LN} that
$$
|[i]|_\pi =\begin{cases}
q^{-\delta }, & \text{if $\delta$ divides $i$},\\
1, & \text{otherwise}.\end{cases}
$$
Writing $m=j\delta +i$, with $i,j\in \mathbb Z_+$, $0\le
i<\delta$, we find that
\begin{multline*}
|D_m|_\pi =|[j\delta ]|_\pi^{q^i}|[(j-1)\delta ]|_\pi^{q^{\delta
+i}}\ldots |[\delta ]|_\pi^{q^{(j-1)\delta +i}}=\left\{ q^{-\delta
}\cdot \left( q^{-\delta}\right)^{q^\delta }\cdot \ldots \cdot
\left( q^{-\delta}\right)^{q^{(j-1)\delta }}\right\}^{q^i} \\
=\left\{ \left( q^{-\delta}\right)^{1+q^\delta +\cdots
+q^{(j-1)\delta}}\right\}^{q^i}=q^{-\delta q^i\frac{q^{j\delta }
-1}{q^\delta -1}}.
\end{multline*}
Similarly we can write $k-m=\varkappa \delta +\lambda$, with
$\varkappa ,\lambda \in \mathbb Z_+$, $0\le \lambda <\delta$, and
get that
$$
|D_{k-m}|=q^{-\delta q^\lambda \frac{q^{\varkappa \delta }-1}{q^\delta -1}}.
$$
If $i+\lambda <\delta$, then we obtain a similar representation
for $k$ simply by adding those for $m$ and $k-m$, so that
\begin{multline*}
\log_q\left| \binom{k}{m}_K\right|_\pi =-\frac{\delta}{q^\delta
-1}\left\{ q^{i+\lambda }\left( q^{(j+\varkappa )\delta }-1\right)
-q^i\left( q^{j\delta }-1\right) -q^\lambda \left( q^{\varkappa \delta
}-1\right) q^{j\delta +i}\right\} \\
=-\frac{\delta}{q^\delta -1}q^i\left( 1+q^{\lambda +j\delta
}-q^\lambda -q^{j\delta }\right) =-\frac{\delta}{q^\delta
-1}q^i\left( q^\lambda -1\right) \left( q^{j\delta }-1\right) \le 0.
\end{multline*}
If $i+\lambda \ge \delta$, then $k=(j+\varkappa +1)\delta +\nu$
where $0\le \nu =i+\lambda -\delta <\delta$. In this case
\begin{multline*}
\log_q\left| \binom{k}{m}_K\right|_\pi =-\frac{\delta}{q^\delta
-1}\left\{ q^\nu \left( q^{(j+\varkappa +1)\delta }-1\right)
-q^i\left( q^{j\delta }-1\right) -q^\lambda \left( q^{\varkappa \delta
}-1\right) q^{j\delta +i}\right\} \\
=-\frac{\delta}{q^\delta -1}\left( q^i+q^{\lambda +j\delta +i
}-q^{i+j\delta }-q^\nu \right) <0,
\end{multline*}
since $\nu <i+\lambda$. $\qquad \blacksquare$
\medskip
Below we will use only the valuation with $\pi (x)=x$, that is, as
above, consider the field $K$.
\medskip
{\bf 5.2.} Let us derive, for the $K$-binomial coefficients
(3), analogs of the classical Pascal and Vandermonde identities.
\medskip
\begin{prop}
The identity
\begin{equation}
\binom{k}{m}_K=\binom{k-1}{m-1}_K^q+\binom{k-1}{m}_K^qD_m^{q-1}
\end{equation}
holds, if $0\le m\le k$ and it is assumed that
$\dbinom{k}{-1}_K=\dbinom{k-1}{k}_K=0$.
\end{prop}
\medskip
{\it Proof}. Let $e_m(t)=D_mf_m(t)$ be the ``non-normalized''
Carlitz polynomials. They satisfy the main $K$-binomial identity
\cite{Carl,K5}
\begin{equation}
e_k(st)=\sum\limits_{m=0}^k\binom{k}{m}_Ke_m(s)\left\{
e_{k-m}(t)\right\}^{q^m},
\end{equation}
which holds, for example, for any $s,t\in \mathbb F_q [x]$.
It is known \cite{Carl,G1} that
\begin{equation}
e_k=e_{k-1}^q-D_{k-1}^{q-1}e_{k-1}.
\end{equation}
Let us rewrite the left-hand side of (28) in accordance with (29),
and apply to each term the identity (28) with $k-1$ substituted
for $k$. We have
$$
e_{k-1}^q(st)=\sum\limits_{i=0}^{k-1}\binom{k-1}{i}_K^qe_i^q(s)
e_{k-i-1}^{q^{i+1}}(t).
$$
By (29), $e_i^q=e_{i+1}+D_i^{q-1}e_i$,
$e_{k-i-1}^q=e_{k-i}+D_{k-i-1}^{q-1}e_{k-i-1}$, whence
\begin{multline*}
e_{k-1}^q(st)
=\sum\limits_{j=1}^k\binom{k-1}{j-1}_K^qe_j(s)e_{k-j}^{q^j}(t)
+\sum\limits_{i=0}^{k-1}\binom{k-1}{i}_K^q
D_i^{q-1}e_i(s)e_{k-i}^{q^i}(t)\\
+\sum\limits_{i=0}^{k-1}\binom{k-1}{i}_K^q
D_i^{q-1}D_{k-i-1}^{q^i(q-1)}e_i(s)e_{k-i-1}^{q^i}(t).
\end{multline*}
Note that
\begin{equation}
\binom{k-1}{i}_K^qD_i^{q-1}D_{k-i-1}^{q^i(q-1)}=D_{k-1}^{q-1}\binom{k-1}{i}_K.
\end{equation}
Indeed, the left-hand side of (30) equals
$$
\frac{D_{k-1}^q}{D_i^qD_{k-i-1}^{q^{i+1}}}D_i^{q-1}D_{k-i-1}^{q^{i+1}-q^i}=
\frac{D_{k-1}}{D_iD_{k-i-1}^{q^i}}D_{k-1}^{q-1}
$$
and coincides with the right-hand side. Therefore the last sum in
the expression for $e_{k-1}^q(st)$ equals
$$
D_{k-1}^{q-1}\sum\limits_{i=0}^{k-1}\binom{k-1}{i}_Ke_i(s)e_{k-i-1}^{q^i}(t)
=D_{k-1}^{q-1}e_{k-1}(st).
$$
Using (29) again we find that
$$
e_k(st)=\sum\limits_{i=0}^k\binom{k-1}{i-1}_K^qe_i(s)e_{k-i}^{q^i}(t)
+\sum\limits_{i=0}^k\binom{k-1}{i}_K^qD_i^{q-1}e_i(s)e_{k-i}^{q^i}(t),
$$
and the comparison with (28) yields
$$
\sum\limits_{m=0}^k\left\{ \binom{k}{m}_K-\binom{k-1}{m-1}_K^q-
\binom{k-1}{m}_K^qD_m^{q-1}\right\} e_m(s)e_{k-m}^{q^m}(t)=0
$$
for any $s,t$.
Since the Carlitz polynomials are linearly independent, we obtain
that
$$
\left\{ \binom{k}{m}_K-\binom{k-1}{m-1}_K^q-
\binom{k-1}{m}_K^qD_m^{q-1}\right\} e_{k-m}^{q^m}(t)=0
$$
for any $t$, and it remains to note that $e_{k-m}(t)\ne 0$ if
$t\in \mathbb F_q [x]$, $\deg t\ge k$, by the definition of the Carlitz
polynomials. $\qquad \blacksquare$
\medskip
More generally, we have the following Vandermonde-type identity.
Let $k,m$ be integers, $0\le m\le k$.
\begin{prop}
Define $c_{li}^{(m)}\in K$ by the recurrent relation
\begin{equation}
c_{l+1,i}^{(m)}=c_{l,i-1}^{(m)}+c_{li}^{(m)}D_{m-i}^{q-1}
\end{equation}
and the initial conditions $c_{li}^{(m)}=0$ for $i<0$ and $i>l$,
$c_{00}^{(m)}=1$. Then, for any $l\le m$,
\begin{equation}
\binom{k}{m}_K=\sum\limits_{i=0}^lc_{li}^{(m)}\binom{k-l}{m-i}_K^{q^l}.
\end{equation}
\end{prop}
\medskip
{\it Proof}. The identity (32) is trivial for $l=0$. Suppose it
has been proved for some $l$. Let us transform the right-hand side
of (32) using the identity (27). Then we have
\begin{multline*}
\binom{k}{m}_K=\sum\limits_{i=0}^lc_{li}^{(m)}\binom{k-l-1}{m-i-1}_K^{q^{l+1}}
+\sum\limits_{i=0}^lc_{li}^{(m)}\binom{k-l-1}{m-i}_K^{q^{l+1}}D_{m-i}^{q-1}\\
=\sum\limits_{j=1}^{l+1}c_{l,j-1}^{(m)}\binom{k-l-1}{m-j}_K^{q^{l+1}}
+\sum\limits_{i=0}^lc_{li}^{(m)}\binom{k-l-1}{m-i}_K^{q^{l+1}}D_{m-i}^{q-1}.
\end{multline*}
Since we assume that $c_{l,-1}^{(m)}=c_{l,l+1}^{(m)}=0$, the
summation in both the above sums can be performed from 0 to $l+1$.
Using (31) we obtain the required identity (32) with $l+1$
substituted for $l$. $\qquad \blacksquare$
\medskip
{\bf 5.3.} Now we consider a function $f\in \mathcal F_2$ associated with
the $K$-binomial coefficients, that is
\begin{equation}
f(s,t)=\sum\limits_{k=0}^\infty
\sum\limits_{m=0}^k\binom{k}{m}_Ks^{q^m}t^{q^k}.
\end{equation}
Obviously, $f$ is non-sparse.
\medskip
\begin{prop}
The function (33) satisfies the equation
\begin{equation}
d_sf(s,t)=\Delta_tf(s,t)+[1]^{1/q}f(s,t),
\end{equation}
so that $f$ is quasi-holonomic.
\end{prop}
\medskip
{\it Proof}. Let us compute $d_sf$. We have
$$
d_sf(s,t)=\sum\limits_{k=1}^\infty
\sum\limits_{m=1}^k\binom{k}{m}_K^{1/q}[m]^{1/q}s^{q^{m-1}}t^{q^{k-1}}
=\sum\limits_{\nu =0}^\infty
\sum\limits_{\mu =0}^\nu \binom{\nu +1}{\mu +1}_K^{1/q}[\mu
+1]^{1/q}s^{q^\mu}t^{q^\nu}.
$$
Using Proposition 5 we find that $d_sf=\Sigma_1+\Sigma_2$ where
$$
\Sigma_1=\sum\limits_{\nu =0}^\infty
\sum\limits_{\mu =0}^\nu \binom{\nu}{\mu}_K[\mu
+1]^{1/q}s^{q^\mu}t^{q^\nu},
$$
$$
\Sigma_2=\sum\limits_{\nu =0}^\infty
\sum\limits_{\mu =0}^\nu \binom{\nu}{\mu +1}_K[\mu
+1]^{1/q}D_{\mu +1}^{1-q^{-1}}s^{q^\mu}t^{q^\nu}.
$$
Note that
$$
[\mu +1]^{1/q}=\left( x^{q^{\mu +1}}-x\right)^{1/q}=\left(
x^{q^\mu }-x\right) +\left( x^q-x\right)^{1/q}=[\mu ]+[1]^{1/q},
$$
so that
\begin{equation}
\Sigma_1=\sum\limits_{\nu =0}^\infty
\sum\limits_{\mu =0}^\nu \binom{\nu}{\mu}_K[\mu ]
s^{q^\mu}t^{q^\nu}+[1]^{1/q}f(s,t).
\end{equation}
Next, we have
$$
\binom{\nu}{\mu +1}_K[\mu +1]^{1/q}D_{\mu
+1}^{1-q^{-1}}=\frac{D_\nu}{D_{\mu +1}D_{\nu -\mu -1}^{q^{\mu
+1}}}D_{\mu +1}\left( \frac{[\mu +1]}{D_{\mu +1}}\right)^{1/q}
=\frac{D_\nu}{D_\mu D_{\nu -\mu -1}^{q^{\mu +1}}},
$$
and also
$$
D_{\nu -\mu -1}^q=\frac{1}{[\nu -\mu ]}[\nu -\mu ]D_{\nu -\mu -1}^q
=\frac{D_{\nu -\mu}}{[\nu -\mu ]},
$$
whence
$$
D_{\nu -\mu -1}^{q^{\mu +1}}=\frac{D_{\nu -\mu }^{q^\mu}}{[\nu -\mu
]^{q^\mu }}.
$$
Therefore
$$
\Sigma_2=\sum\limits_{\nu =0}^\infty
\sum\limits_{\mu =0}^\nu \binom{\nu}{\mu}_K[\nu -\mu
]^{q^\mu}s^{q^\mu}t^{q^\nu}.
$$
As above, $[\nu -\mu ]^{q^\mu }=\left( x^{q^{\nu
-\mu}}-x\right)^{q^\mu}=[\nu ]-[\mu ]$, so that
$$
\Sigma_2=\sum\limits_{\nu =0}^\infty
\sum\limits_{\mu =0}^\nu \binom{\nu}{\mu}_K([\nu ]-[\mu
])s^{q^\mu}t^{q^\nu}.
$$
Together with (35), this implies (34). $\qquad \blacksquare$
\newpage
|
{
"timestamp": "2006-05-01T11:20:17",
"yymm": "0503",
"arxiv_id": "math/0503398",
"language": "en",
"url": "https://arxiv.org/abs/math/0503398"
}
|
\section{Introduction}
In this paper we discuss the limiting theory for a novel, unifying class of
non-parametric measures of the variation of financial prices. The theory
covers commonly used estimators of variation such as realised volatility,
but it also encompasses more recently suggested quantities like realised
power variation and realised bipower variation. We considerably strengthen
existing results on the latter two quantities, deepening our understanding
and unifying their treatment. We will outline the proofs of these theorems,
referring for the very technical, detailed formal proofs of the general
results to a companion probability theory paper \cite%
{BarndorffNielsenGraversenJacodPodolskyShephard(04shiryaev)}. Our emphasis
is on exposition, explaining where the results come from and how they sit
within the econometrics literature.
Our theoretical development is motivated by the advent of complete records
of quotes or transaction prices for many financial assets. Although market
microstructure effects (e.g. discreteness of prices, bid/ask bounce,
irregular trading etc.) mean that there is a mismatch between asset pricing
theory based on semimartingales and the data at very fine time intervals it
does suggest the desirability of establishing an asymptotic distribution
theory for estimators as we use more and more highly frequent observations.
Papers which directly model the impact of market microstructure noise on
realised variance include \cite{BandiRussell(03)}, \cite{HansenLunde(03)},
\cite{ZhangMyklandAitSahalia(03)}, \cite%
{BarndorffNielsenHansenLundeShephard(04)} and \cite{Zhang(04)}. Related work
in the probability literature on the impact of noise on discretely observed
diffusions can be found in \cite{GloterJacod(01a)} and \cite%
{GloterJacod(01b)}, while \cite{DelattreJacod(97)} report results on the
impact of rounding on sums of functions of discretely observed diffusions.
In this paper we ignore these effects.
Let the $d$-dimensional vector of the log-prices of a set of assets follow
the process
\begin{equation*}
Y=\left( Y^{1},...,Y^{d}\right) ^{\prime }.
\end{equation*}
At time $t\geq 0$ we denote the log-prices as $Y_{t}$. Our aim is to
calculate measures of the variation of the price process (e.g. realised
volatility) over discrete time intervals (e.g. a day or a month). Without
loss of generality we can study the mathematics of this by simply looking at
what happens when we have $n$ high frequency observations on the time
interval $t=0$ to $t=1$ and study what happens to our measures of variation
as $n\rightarrow \infty $ (e.g., for introductions to this, \cite%
{BarndorffNielsenShephard(02realised)}). In this case returns will be
measured over intervals of length $n^{-1}$ as
\begin{equation}
\Delta _{i}^{n}Y=Y_{i/n}-Y_{(i-1)/n},\quad i=1,2,...,n, \label{return}
\end{equation}%
where $n$ is a positive integer.
We will study the behaviour of the realised generalised bipower variation
process
\begin{equation}
\frac{1}{n}\sum_{i=1}^{\left\lfloor nt\right\rfloor }g(\sqrt{n}~\Delta
_{i}^{n}Y)h(\sqrt{n}~\Delta _{i+1}^{n}Y), \label{RGBP}
\end{equation}%
as $n$ becomes large and where $g$ and $h$ are two given, matrix functions
of dimensions $d_{1}\times d_{2}$ and $d_{2}\times d_{3}$ respectively,
whose elements have at most polynomial growth. Here $\left\lfloor
x\right\rfloor $ denotes the largest integer less than or equal to $x$.
Although (\ref{RGBP}) looks initially rather odd, in fact most of the
non-parametric volatility measures used in financial econometrics fall
within this class (a measure not included in this setup is the range
statistic studied in, for example, \cite{Parkinson(80)}). Here we give an
extensive list of examples and link them to the existing literature. More
detailed discussion of the literature on the properties of these special
cases will be given later.
\begin{example}
\label{Example: 1}\textbf{(a)} Suppose $g(y)=\left( y^{j}\right) ^{2}$ and $%
h(y)=1$, then (\ref{RGBP}) becomes%
\begin{equation*}
\sum_{i=1}^{\left\lfloor nt\right\rfloor }\left( \Delta _{i}^{n}Y^{j}\right)
^{2},
\end{equation*}%
which is called the realised quadratic variation process of $Y^{j}$ in
econometrics, e.g. \cite{Jacod(94)}, \cite{JacodProtter(98)}, \cite%
{BarndorffNielsenShephard(02realised)}, \cite%
{BarndorffNielsenShephard(04multi)} and \cite{MyklandZhang(05)}. The
increments of this quantity, typically calculated over a day or a week, are
often called the realised variances in financial economics and have been
highlighted by \cite{AndersenBollerslevDieboldLabys(01)} and \cite%
{AndersenBollerslevDiebold(05)} in the context of volatility measurement and
forecasting.
\noindent \textbf{(b)} Suppose $g(y)=yy^{\prime }$ and $h(y)=I$, then (\ref%
{RGBP}) becomes, after some simplification,
\begin{equation*}
\sum_{i=1}^{\left\lfloor nt\right\rfloor }\left( \Delta _{i}^{n}Y\right)
\left( \Delta _{i}^{n}Y\right) ^{\prime }.
\end{equation*}%
\newline
This is the realised covariation process. It has been studied by \cite%
{JacodProtter(98)}, \cite{BarndorffNielsenShephard(04multi)} and \cite%
{MyklandZhang(05)}. \cite{AndersenBollerslevDieboldLabys(03model)} study the
increments of this process to produce forecast distributions for vectors of
returns. \
\noindent \textbf{(c)} Suppose $g(y)=\left\vert y^{j}\right\vert ^{r}$ for $%
r>0$ and $h(y)=1$, then (\ref{RGBP}) becomes
\begin{equation*}
n^{-1+r/2}\sum_{i=1}^{\left\lfloor nt\right\rfloor }\left\vert \Delta
_{i}^{n}Y^{j}\right\vert ^{r},
\end{equation*}%
which is called the realised $r$-th order power variation. When $r$ is an
integer it has been studied from a probabilistic viewpoint by \cite%
{Jacod(94)} while \cite{BarndorffNielsenShephard(03bernoulli)} look at the
econometrics of the case where $r>0$. The increments of these types of high
frequency volatility measures have been informally used in the financial
econometrics literature for some time when $r=1$, but until recently without
a strong understanding of their properties. Examples of their use include
\cite{Schwert(90JB)}, \cite{AndersenBollerslev(98)} and \cite%
{AndersenBollerslev(97jef)}, while they have also been informally discussed
by \cite[pp. 349--350]{Shiryaev(99)}\ and \cite{MaheswaranSims(93)}.
Following the work by \cite{BarndorffNielsenShephard(03bernoulli)}, \cite%
{GhyselsSantaClaraValkoanov(04)} and \cite{ForsbergGhysels(04)} have
successfully used realised power variation as an input into volatility
forecasting competitions.
\noindent \textbf{(d)} Suppose $g(y)=\left\vert y^{j}\right\vert ^{r}$ and $%
h(y)=\left\vert y^{j}\right\vert ^{s}$ for $r,s>0$, then (\ref{RGBP})
becomes
\begin{equation*}
n^{-1+(r+s)/2}\sum_{i=1}^{\left\lfloor nt\right\rfloor }\left\vert \Delta
_{i}^{n}Y^{j}\right\vert ^{r}\left\vert \Delta _{i+1}^{n}Y^{j}\right\vert
^{s},
\end{equation*}%
which is called the realised $r,s$-th order bipower variation process. This
measure of variation was introduced by \cite{BarndorffNielsenShephard(04jfe)}%
, while a more formal discussion of its behaviour in the $r=s=1$ case was
developed by \cite{BarndorffNielsenShephard(03test)}. These authors'
interest in this quantity was motivated by its virtue of being resistant to
finite activity jumps so long as $\max (r,s)<2$. Recently \cite%
{BarndorffNielsenShephardWinkel(04)} and \cite{Woerner(04power)} have
studied how these results on jumps extend to infinite activity processes,
while \cite{CorradiDistaso(04)} have used these statistics to test the
specification of parametric volatility models.
\noindent \textbf{(e)} Suppose
\begin{equation*}
g(y)=\left(
\begin{array}{cc}
\left\vert y^{j}\right\vert & 0 \\
0 & \left( y^{j}\right) ^{2}%
\end{array}%
\right) ,\quad h(y)=\left(
\begin{array}{c}
\left\vert y^{j}\right\vert \\
1%
\end{array}%
\right) .
\end{equation*}%
Then (\ref{RGBP}) becomes,%
\begin{equation*}
\left(
\begin{array}{c}
\displaystyle\sum_{i=1}^{\left\lfloor nt\right\rfloor }\left\vert \Delta
_{i}^{n}Y^{j}\right\vert \left\vert \Delta _{i+1}^{n}Y^{j}\right\vert \\
\displaystyle\sum_{i=1}^{\left\lfloor nt\right\rfloor }\left( \Delta
_{i}^{n}Y^{j}\right) ^{2}%
\end{array}%
\right) .
\end{equation*}%
\cite{BarndorffNielsenShephard(03test)} used the joint behaviour of the
increments of these two statistics to test for jumps in price processes. \
\cite{HuangTauchen(03)} have empirically studied the finite sample
properties of these types of jump tests. \cite%
{AndersenBollerslevDiebold(03bipower)} \ and \cite{ForsbergGhysels(04)} use
bipower variation as an input into volatility forecasting. \
\end{example}
We will derive the probability limit of (\ref{RGBP}) under a general
Brownian semimartingale, the workhorse process of modern continuous time
asset pricing theory. Only the case of realised quadratic variation, where
the limit is the usual quadratic variation QV (defined for general
semimartingales), has been previously been studied under such wide
conditions. Further, under some stronger but realistic conditions, we will
derive a limiting distribution theory for (\ref{RGBP}), so extending a
number of results previously given in the literature on special cases of
this framework.
The outline of this paper is as follows. Section 2 contains a detailed
listing of the assumptions used in our analysis. Section 3 gives a statement
of a weak law of large numbers for these statistics and the corresponding
central limit theory is presented in Section 4. Extensions of the results to
higher order variations is briefly indicated in Section 5. Section 6
illustrates the theory by discussing how it gives rise to tests for jumps in
the price processes, using bipower and tripower variation. The corresponding
literature which discusses various special cases of these results is also
given in these sections. Section 8 concludes, while there is an Appendix
which provides an outline of the proofs of the results discussed in this
paper. For detailed, quite lengthy and highly technical formal proofs we
refer to our companion probability theory paper \cite%
{BarndorffNielsenGraversenJacodPodolskyShephard(04shiryaev)}.
\section{Notation and models}
We start with $Y$ on some filtered probability space $\left( \Omega ,%
\mathcal{F},\left( \mathcal{F}_{t}\right) _{t\geq 0},P\right) $. In most of
our analysis we will assume that $Y$ follows a $d$-dimensional Brownian
semimartingale (written $Y\in \mathcal{BSM}$). It is given in the following
statement.
\noindent \textbf{Assumption (H): }We have
\begin{equation}
Y_{t}=Y_{0}+\int_{0}^{t}a_{u}\mathrm{d}u+\int_{0}^{t}\sigma _{u-}\mathrm{d}%
W_{u}, \label{H}
\end{equation}%
where $W$ is a $d^{\prime }$-dimensional standard Brownian motion (BM), $a$
is a $d$-dimensional process whose elements are predictable and has locally
bounded sample paths, and the spot covolatility $d,d^{\prime }$-dimensional
matrix $\sigma $ has elements which have c\`{a}dl\`{a}g sample paths.
Throughout we will write
\begin{equation*}
\Sigma _{t}=\sigma _{t}\sigma _{t}^{\prime },
\end{equation*}%
the spot covariance matrix. Typically $\Sigma _{t}$ will be full rank, but
we do not assume that here. We will write $\Sigma _{t}^{jk}$ to denote the $%
j,k$-th element of $\Sigma _{t}$, while we write%
\begin{equation*}
\sigma _{j,t}^{2}=\Sigma _{t}^{jj}.
\end{equation*}
\begin{remark}
Due to the fact that $t\mapsto \sigma _{t}^{jk}$ is c\`{a}dl\`{a}g all
powers of $\sigma _{t}^{jk}$ are locally integrable with respect to the
Lebesgue measure. \ In particular then $\int_{0}^{t}\Sigma _{u}^{jj}\mathrm{d%
}u<\infty $ for all $t$ and $j$.
\end{remark}
\begin{remark}
Both $a$ and $\sigma $ can have, for example, jumps, intraday seasonality
and long-memory.
\end{remark}
\begin{remark}
The stochastic volatility (e.g. \cite{GhyselsHarveyRenault(96)} and \cite%
{Shephard(05)}) component of $Y$,
\begin{equation*}
\int_{0}^{t}\sigma _{u-}\mathrm{d}W_{u},
\end{equation*}%
is always a vector of local martingales each with continuous sample paths,
as $\int_{0}^{t}\Sigma _{u}^{jj}\mathrm{d}u<\infty $ for all $t$ and $j$.
All continuous local martingales with absolutely continuous quadratic
variation can be written in the form of a stochastic volatility process.
This result, which is due to \cite{Doob(53)}, is discussed in, for example,
\cite[p. 170--172]{KaratzasShreve(91)}. Using the Dambis-Dubins-Schwartz
Theorem, we know that the difference between the entire continuous local
martingale class and the SV class are the local martingales which have only
continuous, not absolutely continuous\footnote{%
An example of a continuous local martingale which has no SV representation
is a time-change Brownian motion where the time-change takes the form of the
so-called \textquotedblleft devil's staircase,\textquotedblright\ which is
continuous and non-decreasing but not absolutely continuous (see, for
example, \cite[Section 27]{Munroe(53)}). This relates to the work of, for
example, \cite{CalvetFisher(02)} on multifractals.}, QV. The drift $%
\int_{0}^{t}a_{u}\mathrm{d}u$ has elements which are absolutely continuous.
This assumption looks ad hoc, however if we impose a lack of arbitrage
opportunities and model the local martingale component as a SV process then
this property must hold (\cite[p. 3]{KaratzasShreve(98)} and \cite[p. 583]%
{AndersenBollerslevDieboldLabys(03model)}). Hence (\ref{H}) is a rather
canonical model in the finance theory of continuous sample path processes.
\end{remark}
We are interested in the asymptotic behaviour, for $n\rightarrow \infty $,
of the following volatility measuring process:
\begin{equation}
Y^{n}(g,h)_{t}=\frac{1}{n}\sum_{i=1}^{\left\lfloor nt\right\rfloor }g(\sqrt{n%
}~\Delta _{i}^{n}Y)h(\sqrt{n}~\Delta _{i+1}^{n}Y), \label{XP}
\end{equation}%
where $g$ and $h$ are two given conformable matrix functions and recalling
the definition of $\Delta _{i}^{n}Y$ given in (\ref{return}).
\section{Law of large numbers}
To build a weak law of large numbers for $Y^{n}(g,h)_{t}$ we need to make
the pair $(g,h)$ satisfy the following assumption.
\noindent \textbf{Assumption (K):} All the elements of $f$ on $\mathbf{R}%
^{d} $ are continuous with at most polynomial growth.
This amounts to there being suitable constants $C>0$ and $p\geq 2$ such that
\begin{equation}
x\in \mathbf{R}^{d}\quad \Rightarrow \quad \left\Vert f(x)\right\Vert \leq
C(1+\Vert x\Vert ^{p}). \label{G1}
\end{equation}
We also need the following notation.
\begin{equation*}
\rho _{\sigma }(g)=\mathrm{E}\left\{ g(X)\right\} ,\quad \text{where\quad }%
X|\sigma \sim N(0,\sigma \sigma ^{\prime }),
\end{equation*}%
and
\begin{equation*}
\rho _{\sigma }(gh)=\mathrm{E}\left\{ g(X)h(X)\right\} .
\end{equation*}
\begin{example}
\label{Example: second}\textbf{(a)} Let $g(y)=yy^{\prime }$ and $h(y)=I$,
then $\rho _{\sigma }(g)=\Sigma $ and $\rho _{\sigma }(h)=I$.
\noindent \textbf{(b)} Suppose $g(y)=\left\vert y^{j}\right\vert ^{r}$ then $%
\rho _{\sigma }(g)=\mu _{r}\sigma _{j}^{r}$, where $\sigma _{j}^{2}$ is the $%
j,j$-th element of $\Sigma $, $\mu _{r}=\mathrm{E}(\left\vert u\right\vert
^{r})$ and $u\sim N(0,1)$.
\end{example}
This setup is sufficient for the proof of Theorem 1.2 of \cite%
{BarndorffNielsenGraversenJacodPodolskyShephard(04shiryaev)}, which is
restated here.
\begin{theorem}
\label{TT1} Under (H) and assuming $g$ and $h$ satisfy (K) we have that
\begin{equation}
Y^{n}(g,h)_{t}~\rightarrow ~Y(g,h)_{t}:=\int_{0}^{t}\rho _{\sigma
_{u}}(g)\rho _{\sigma _{u}}(h)\mathrm{d}u, \label{WLLN}
\end{equation}%
where the convergence is in probability, locally uniform in time. \newline
\end{theorem}
The result is quite clean as it is requires no additional assumptions on $Y$
and so is very close to dealing with the whole class of financially coherent
continuous sample path processes.
This Theorem covers a number of existing setups which are currently
receiving a great deal of attention as measures of variation in financial
econometrics. Here we briefly discuss some of the work which has studied the
limiting behaviour of these objects.
\begin{example}
\textbf{(Example \ref{Example: 1}(a) continued)}. Then $g(y)=\left(
y^{j}\right) ^{2}$ and $h(y)=1$, so (\ref{WLLN}) becomes%
\begin{equation*}
\sum_{i=1}^{\left\lfloor nt\right\rfloor }\left( \Delta _{i}^{n}Y^{j}\right)
^{2}\rightarrow ~\int_{0}^{t}\sigma _{j,u}^{2}\mathrm{d}u=[Y^{j}]_{t},
\end{equation*}%
the quadratic variation (QV) of $Y^{j}$. This well known result in
probability theory is behind much of the modern work on realised volatility,
which is compactly reviewed in \cite{AndersenBollerslevDiebold(05)}.
\noindent (\textbf{Example \ref{Example: 1}(b) continued}). As $%
g(y)=yy^{\prime }$ and $h(y)=I$, then
\begin{equation*}
\sum_{i=1}^{\left\lfloor nt\right\rfloor }\left( \Delta _{i}^{n}Y\right)
\left( \Delta _{i}^{n}Y\right) ^{\prime }\rightarrow ~\int_{0}^{t}\Sigma _{u}%
\mathrm{d}u=[Y]_{t},
\end{equation*}%
the well known multivariate version of QV.
\noindent \textbf{(Example \ref{Example: 1}(c) continued).} Then $%
g(y)=\left\vert y^{j}\right\vert ^{r}$ and $h(y)=1$ so
\begin{equation*}
n^{-1+r/2}\sum_{i=1}^{\left\lfloor nt\right\rfloor }\left\vert \Delta
_{i}^{n}Y^{j}\right\vert ^{r}\rightarrow ~\mu _{r}\int_{0}^{t}\sigma
_{j,u}^{r}\mathrm{d}u.
\end{equation*}%
This result is due to \cite{Jacod(94)} and \cite%
{BarndorffNielsenShephard(03bernoulli)}.
\noindent \textbf{(Example \ref{Example: 1}(d) continued).} Then $%
g(y)=\left\vert y^{j}\right\vert ^{r}$ and $h(y)=\left\vert y^{j}\right\vert
^{s}$ for $r,s>0$, so
\begin{equation*}
n^{-1+(r+s)/2}\sum_{i=1}^{\left\lfloor nt\right\rfloor }\left\vert \Delta
_{i}^{n}Y^{j}\right\vert ^{r}\left\vert \Delta _{i+1}^{n}Y^{j}\right\vert
^{s}\rightarrow ~\mu _{r}\mu _{s}\int_{0}^{t}\sigma _{j,u}^{r+s}\mathrm{d}u,
\end{equation*}%
a result due to \cite{BarndorffNielsenShephard(04jfe)}, who derived it under
stronger conditions than those used here.
\noindent \textbf{(Example \ref{Example: 1}(e) continued).} Then
\begin{equation*}
g(y)=\left(
\begin{array}{cc}
\left\vert y^{j}\right\vert & 0 \\
0 & \left( y^{j}\right) ^{2}%
\end{array}%
\right) ,\quad h(y)=\left(
\begin{array}{c}
\left\vert y^{j}\right\vert \\
1%
\end{array}%
\right) ,
\end{equation*}%
so
\begin{equation*}
\left(
\begin{array}{c}
\displaystyle\sum_{i=1}^{\left\lfloor nt\right\rfloor }\left\vert \Delta
_{i}^{n}Y^{j}\right\vert \left\vert \Delta _{i+1}^{n}Y^{j}\right\vert \\
\displaystyle\sum_{i=1}^{\left\lfloor nt\right\rfloor }\left( \Delta
_{i}^{n}Y^{j}\right) ^{2}%
\end{array}%
\right) \rightarrow \left( ~%
\begin{array}{c}
\mu _{1}^{2} \\
1%
\end{array}%
\right) \int_{0}^{t}\sigma _{j,u}^{2}\mathrm{d}u.
\end{equation*}%
\cite{BarndorffNielsenShephard(03test)} used this type of result to test for
jumps as this particular bipower variation is robust to jumps.
\end{example}
\section{Central limit theorem\label{sect:CLT}}
\subsection{Motivation}
It is important to be able to quantify the difference between the estimator $%
Y^{n}(g,h)$ and $Y(g,h)$. In this subsection we do this by giving a central
limit theorem for $\sqrt{n}(Y^{n}(g,h)-Y(g,h))$. We have to make some
stronger assumptions both on the process $Y$ and on the pair $(g,h)$ in
order to derive this result.
\subsection{Assumptions on the process}
We start with a variety of assumptions which strengthen (H) and (K) given in
the previous subsection.
\noindent \textbf{Assumption (H0):} We have (H) with
\begin{equation}
\sigma _{t}=\sigma _{0}+\int_{0}^{t}a_{u}^{\ast }\mathrm{d}%
u+\int_{0}^{t}\sigma _{u-}^{\ast }\mathrm{d}W_{u}+\int_{0}^{t}v_{u-}^{\ast }%
\mathrm{d}Z_{u}, \label{H'}
\end{equation}%
where $Z$ is a $d^{\prime \prime }$-dimensional L\'{e}vy process,
independent of $W$. Further, the processes $a^{\ast }$, $\sigma ^{\ast }$, $%
v^{\ast }$ are adapted c\`{a}dl\`{a}g arrays, with $a^{\ast }$ also being
predictable and locally bounded.
\noindent \textbf{Assumption (H1):} We have (H) with
\begin{eqnarray}
\sigma _{t} &=&\sigma _{0}+\int_{0}^{t}a_{u}^{\ast }\mathrm{d}%
u+\int_{0}^{t}\sigma _{u-}^{\ast }\mathrm{d}W_{u}+\int_{0}^{t}v_{u-}^{\ast }%
\mathrm{d}V_{u} \label{assumption (V)} \\
&&+\int_{0}^{t}\int_{E}\varphi \circ w(u-,x)\left( \mu -\nu \right) \left(
\mathrm{d}u,\mathrm{d}x\right) +\int_{0}^{t}\int_{E}\left( w-\varphi \circ
w\right) \left( u-,x\right) \mu \left( \mathrm{d}u,\mathrm{d}x\right) .
\notag
\end{eqnarray}%
Here $a^{\ast }$, $\sigma ^{\ast }$, $v^{\ast }$ are adapted c\`{a}dl\`{a}g
arrays, with $a^{\ast }$ also being predictable and locally bounded. $V$ is
a $d^{\prime \prime }$-dimensional Brownian motion independent of $W$. $\mu $
is a Poisson measure on $\left( 0,\infty \right) \times E$ independent of $W$
and $V$, with intensity measure $\nu (\mathrm{d}t,\mathrm{d}x)=\mathrm{d}%
t\otimes F(\mathrm{d}x)$ and $F$ is a $\sigma $-finite measure on the Polish
space $\left( E,\mathcal{E}\right) $. $\varphi $ is a continuous truncation
function on $R^{dd^{\prime }}$ (a function with compact support, which
coincide with the identity map on the neighbourhood of $0$). Finally $%
w(\omega ,u,x)$ is a map $\Omega \times \lbrack 0,\infty )\times E$ into the
space of $d\times d^{\prime }$arrays which is $\mathcal{F}_{u}\otimes $ $%
\mathcal{E}-$measurable in $(\omega ,x)$ for all $u$ and c\`{a}dl\`{a}g in $%
u $, and such that for some sequences $\left( S_{k}\right) $ of stopping
times increasing to $+\infty $ we have%
\begin{equation*}
\sup_{\omega \in \Omega ,u<S_{k}(\omega )}\left\Vert w(\omega
,u,x)\right\Vert \leq \psi _{k}(x)\quad \text{where\quad }\int_{E}\left(
1\wedge \psi _{k}(x)^{2}\right) F(\mathrm{d}x)<\infty .
\end{equation*}
\noindent \textbf{Assumption (H2): }$\Sigma =\sigma \sigma ^{\prime }$ is
everywhere invertible.
\begin{remark}
Assumption (H1) looks quite complicated but has been setup so that the same
conditions on the coefficients can be applied both to $\sigma $ and $\Sigma
=\sigma \sigma ^{\prime }$. If there were no jumps then it would be
sufficient to employ the first line of (\ref{assumption (V)}). The
assumption (H1) is rather general from an econometric viewpoint as it allows
for flexible leverage effects, multifactor volatility effects, jumps,
non-stationarities, intraday effects, etc.
\end{remark}
\subsection{Assumptions on $g$ and $h$}
In order to derive a central limit theorem we need to impose some regularity
on $g$ and $h$.
\noindent \textbf{Assumption (K1): }$f$ is even (that is $f(x)=f(-x)$ for $%
x\in R^{d}$) and continuously differentiable, with derivatives having at
most polynomial growth.
In order to handle some of the most interesting cases of bipower variation,
where we are mostly interested in taking low powers of absolute values of
returns which may not be differentiable at zero, we sometimes need to relax
(K1). The resulting condition is quite technical and is called (K2). It is
discussed in the Appendix.
\noindent \textbf{Assumption (K2):} $f$ is even and continuously
differentiable on the complement $B^{c}$\ of a closed subset $B\subset
\mathbb{R}^{d}$ and satisfies%
\begin{equation*}
||y||\leq 1\Longrightarrow |f(x+y)-f(x)|\leq C(1+||x||^{p})||y||^{r}
\end{equation*}%
for some constants $C$, $p\geq 0$ and $r\in \left( 0,1\right] $. Moreover
a) If $r=1$ then $B$ has Lebesgue measure $0$.
b) If $r<1$ then $B$ satisfies
\begin{equation}
\left.
\begin{array}{l}
\text{for any positive definite }d\times d\text{ matrix }C\text{ and } \\
\text{any }N(0,C)\text{-random vector }U\text{ the distance }d(U,B) \\
\text{from }U\text{ to }B\text{ has a density }\psi _{C}\text{ on }R_{+},%
\text{ such that } \\
sup_{x\in R_{+},|C|+|C^{-1}|\leq A}\psi _{C}(x)<\infty \text{ for all }%
A<\infty ,%
\end{array}%
\right\} \label{K13}
\end{equation}
\qquad\ and we have
\begin{equation}
x\in B^{c},~\Vert y\Vert \leq 1\bigwedge {\frac{d(x,B)}{2}}~~\Rightarrow
~~\left\{
\begin{array}{l}
\Vert \nabla f(x)\Vert \leq {\frac{C(1+\Vert x\Vert ^{p})}{d(x,B)^{1-r}}}, \\%
[2.5mm]
\Vert \nabla f(x+y)-\nabla f(x)\Vert \leq {\frac{C(1+\Vert x\Vert ^{p})\Vert
y\Vert }{d(x,B)^{2-r}}}.%
\end{array}%
\right. \label{K11}
\end{equation}
\begin{remark}
These conditions accommodate the case where $f$ equals $\left\vert
x^{j}\right\vert ^{r}$: this function satisfies (K1) when $r>1$, and (K2)
when $r\in (0,1]$ (with the same $r$ of course). When $B$ is a finite union
of hyperplanes it satisfies (\ref{K13}). Also, observe that (K1) implies
(K2) with $r=1$ and $B=\emptyset $.
\end{remark}
\subsection{Central limit theorem}
Each of the following assumptions (J1) and (J2) are sufficient for the
statement of Theorem 1.3 of \cite%
{BarndorffNielsenGraversenJacodPodolskyShephard(04shiryaev)} to hold.
\noindent \textbf{Assumption (J1):} We have (H1) and $g$ and $h$ satisfy
(K1).
\noindent \textbf{Assumption (J2):} We have (H1), (H2) and $g$ and $h$
satisfy (K2).\newline
The result of the Theorem is restated in the following.
\begin{theorem}
\label{TT3}Assume at least one of (J1) and (J2) holds, then the process
\begin{equation*}
\sqrt{n}~(Y^{n}(g,h)_{t}-Y(g,h)_{t})
\end{equation*}%
converges stably in law towards a limiting process $U(g,h)$ having the form%
\begin{equation}
U(g,h)_{t}^{jk}=\sum_{j^{\prime }=1}^{d_{1}}\sum_{k^{\prime
}=1}^{d_{3}}\int_{0}^{t}\alpha (\sigma _{u},g,h)^{jk,j^{\prime }k^{\prime }}~%
\mathrm{d}B_{u}^{j^{\prime },k^{\prime }},
\end{equation}%
where%
\begin{equation*}
\sum_{l=1}^{d_{1}}\sum_{m=1}^{d_{3}}\alpha (\sigma ,g,h)^{jk,lm}\alpha
(\sigma ,g,h)^{j^{\prime }k^{\prime },lm}=A(\sigma ,g,h)^{jk,j^{\prime
}k^{\prime }},
\end{equation*}%
and%
\begin{eqnarray*}
A(\sigma ,g,h)^{jk,j^{\prime }k^{\prime }} &=&\displaystyle%
\sum_{l=1}^{d_{2}}\sum_{l^{\prime }=1}^{d_{2}}\left\{ \rho _{\sigma }\left(
g^{jl}g^{j^{\prime }l^{\prime }}\right) \rho _{\sigma }\left(
h^{lk}h^{l^{\prime }k^{\prime }}\right) +\rho _{\sigma }\left( g^{jl}\right)
\rho _{\sigma }\left( h^{l^{\prime }k^{\prime }}\right) \rho _{\sigma
}\left( g^{j^{\prime }l^{\prime }}h^{lk}\right) \right. \\
&&\displaystyle+\rho _{\sigma }\left( g^{j^{\prime }l^{\prime }}\right) \rho
_{\sigma }\left( h^{lk}\right) \rho _{\sigma }\left( g^{jl}h^{l^{\prime
}k^{\prime }}\right) \\
&&\displaystyle\left. -3\rho _{\sigma }\left( g^{jl}\right) \rho _{\sigma
}\left( g^{j^{\prime }l^{\prime }}\right) \rho _{\sigma }\left(
h^{lk}\right) \rho _{\sigma }\left( h^{l^{\prime }k^{\prime }}\right)
\right\} .
\end{eqnarray*}%
Furthermore, $B$ is a standard Wiener process which is defined on an
extension of $\left( \Omega ,\mathcal{F},\left( \mathcal{F}_{t}\right)
_{t\geq 0},P\right) $ and is independent of the $\sigma $--field $\mathcal{F}
$.
\end{theorem}
\begin{remark}
Convergence stably in law is slightly stronger than convergence in law. It
is discussed in, for example, \cite[pp. 512-518]{JacodShiryaev(03)}.
\end{remark}
\begin{remark}
Suppose $d_{3}=1$, which is the situation looked at in Example \ref{Example:
1}(e). Then $Y^{n}(g,h)_{t}$ is a vector and so the limiting law of $\sqrt{n}%
(Y^{n}(g,h)-Y(g,h))$ simplifies. It takes on the form of
\begin{equation}
U(g,h)_{t}^{j}=\sum_{j^{\prime }=1}^{d_{1}}\int_{0}^{t}\alpha (\sigma
_{u},g,h)^{j,j^{\prime }}~\mathrm{d}B_{u}^{j^{\prime }},
\end{equation}%
where%
\begin{equation*}
\sum_{l=1}^{d_{1}}\alpha (\sigma ,g,h)^{j,l}\alpha (\sigma ,g,h)^{j^{\prime
},l}=A(\sigma ,g,h)^{j,j^{\prime }}.
\end{equation*}%
Here%
\begin{eqnarray*}
A(\sigma ,g,h)^{j,j^{\prime }} &=&\displaystyle\sum_{l=1}^{d_{2}}\sum_{l^{%
\prime }=1}^{d_{2}}\left\{ \rho _{\sigma }(g^{jl}g^{j^{\prime }l^{\prime
}})\rho _{\sigma }(h^{l}h^{l^{\prime }})+\rho _{\sigma }(g^{jl})\rho
_{\sigma }(h^{l^{\prime }})\rho _{\sigma }(g^{j^{\prime }l^{\prime
}}h^{l})\right. \\
&&\displaystyle+\left. \rho _{\sigma }(g^{j^{\prime }l^{\prime }})\rho
_{\sigma }(h^{l})\rho _{\sigma }(g^{jl}h^{l^{\prime }})-3\rho _{\sigma
}(g^{jl})\rho _{\sigma }(g^{j^{\prime }l^{\prime }})\rho _{\sigma
}(h^{l})\rho _{\sigma }(h^{l^{\prime }})\right\} .
\end{eqnarray*}%
In particular, for a single point in time $t$,
\begin{equation*}
\sqrt{n}~(Y^{n}(g,h)_{t}-Y(g,h)_{t})\rightarrow MN\left(
0,\int_{0}^{t}A(\sigma _{u},g,h)\mathrm{d}u\right) ,
\end{equation*}%
where $MN$ denotes a mixed Gaussian distribution. and $A(\sigma ,g,h)$
denotes a matrix whose $j,j^{\prime }$-th element is $A(\sigma
,g,h)^{j,j^{\prime }}$.
\end{remark}
\begin{remark}
Suppose $g(y)=I$, then $A$ becomes
\begin{equation*}
A(\sigma ,g,h)^{jk,j^{\prime }k^{\prime }}=\rho _{\sigma
}(h^{jk}h^{j^{\prime }k^{\prime }})-\rho _{\sigma }(h^{jk})\rho _{\sigma
}(h^{j^{\prime }k^{\prime }}).
\end{equation*}
\end{remark}
\subsection{Leading examples of this result}
\begin{example}
Suppose $d_{1}=d_{2}=d_{3}=1$, then
\begin{equation}
U(g,h)_{t}=\int_{0}^{t}\sqrt{A(\Sigma _{u},g,h)}~\mathrm{d}B_{u},
\end{equation}%
where%
\begin{equation*}
A(\sigma ,g,h)=\rho _{\sigma }(gg)\rho _{\sigma }(hh)+2\rho _{\sigma
}(g)\rho _{\sigma }(h)\rho _{\sigma }(gh)-3\left\{ \rho _{\sigma }(g)\rho
_{\sigma }(h)\right\} ^{2}.
\end{equation*}%
We consider two concrete examples of this setup.
\noindent \textbf{(i)} Power variation. Suppose $g(y)=1$ and $%
h(y)=\left\vert y^{j}\right\vert ^{r}$ where $r>0$, then $\rho _{\sigma
}(g)=1$,
\begin{equation*}
\rho _{\sigma }(h)=\rho _{\sigma }(gh)=\mu _{r}\sigma _{j}^{r},\quad \rho
_{\sigma }(hh)=\mu _{2r}\sigma _{j}^{2r}.
\end{equation*}%
This implies that%
\begin{eqnarray*}
A(\sigma ,g,h) &=&\mu _{2r}\sigma _{j}^{2r}+2\mu _{r}^{2}\sigma
_{j}^{2r}-3\mu _{r}^{2}\sigma _{j}^{2r} \\
&=&\left( \mu _{2r}-\mu _{r}^{2}\right) \sigma _{j}^{2r} \\
&=&v_{r}\sigma _{j}^{2r},
\end{eqnarray*}%
where $v_{r}=\mathrm{Var}(\left\vert u\right\vert ^{r})$ and $u\sim N(0,1)$.
When $r=2$, this yields a central limit theorem for the realised quadratic
variation process, with
\begin{equation*}
U(g,h)_{t}=\int_{0}^{t}\sqrt{2\sigma _{j,u}^{4}}~\mathrm{d}B_{u},
\end{equation*}%
a result which appears in \cite{Jacod(94)}, \cite{MyklandZhang(05)} and,
implicitly, \cite{JacodProtter(98)}, while the case of a single value of $t$
appears in \cite{BarndorffNielsenShephard(02realised)}. For the more general
case of $r>0$ \cite{BarndorffNielsenShephard(03bernoulli)} derived, under
much stronger conditions, a central limit theorem for $U(g,h)_{1}$. Their
result ruled out leverage effects, which are allowed under Theorem \ref{TT3}%
. The finite sample behaviour of this type of limit theory is studied in,
for example, \cite{BarndorffNielsenShephard(05tom)}, \cite%
{GoncalvesMeddahi(04)} and \cite{NielsenFrederiksen(05)}.
\noindent \textbf{(ii)} Bipower variation. Suppose $g(y)=\left\vert
y^{j}\right\vert ^{r}$ and $h(y)=\left\vert y^{j}\right\vert ^{s}$ where $%
r,s>0$, then%
\begin{eqnarray*}
\rho _{\sigma }(g) &=&\mu _{r}\sigma _{j}^{r},\quad \rho _{\sigma }(h)=\mu
_{s}\sigma _{j}^{s},\quad \rho _{\sigma }(gg)=\mu _{2r}\sigma _{j}^{2r},\quad
\\
\rho _{\sigma }(hh) &=&\mu _{2s}\sigma _{j}^{2s},\quad \rho _{\sigma
}(gh)=\mu _{r+s}\sigma _{j}^{r+s}.
\end{eqnarray*}%
This implies that
\begin{eqnarray*}
A(\sigma ,g,h) &=&\mu _{2r}\sigma _{j}^{2r}\mu _{2s}\sigma _{j}^{2s}+2\mu
_{r}\sigma _{j}^{r}\mu _{s}\sigma _{j}^{s}\mu _{r+s}\sigma _{j}^{r+s}-3\mu
_{r}^{2}\sigma _{j}^{2r}\mu _{s}^{2}\sigma _{j}^{2s} \\
&=&\left( \mu _{2r}\mu _{2s}+2\mu _{r+s}\mu _{r}\mu _{s}-3\mu _{r}^{2}\mu
_{s}^{2}\right) \sigma _{j}^{2r+2s}.
\end{eqnarray*}%
In the $r=s=1$ case \cite{BarndorffNielsenShephard(03test)} derived, under
much stronger conditions, a central limit theorem for $U(g,h)_{1}$. Their
result ruled out leverage effects, which are allowed under Theorem \ref{TT3}%
. In that special case, writing
\begin{equation*}
\vartheta =\frac{\pi ^{2}}{4}+\pi -5,
\end{equation*}%
we have
\begin{equation*}
U(g,h)_{t}=\mu _{1}^{2}\int_{0}^{t}\sqrt{\left( 2+\vartheta \right) \sigma
_{j,u}^{4}}~\mathrm{d}B_{u}.
\end{equation*}
\end{example}
\begin{example}
Suppose $g=I$, $h(y)=yy^{\prime }$. Then we have to calculate%
\begin{equation*}
A(\sigma ,g,h)^{jk,j^{\prime }k^{\prime }}=\rho _{\sigma
}(h^{jk}h^{j^{\prime }k^{\prime }})-\rho _{\sigma }(h^{jk})\rho _{\sigma
}(h^{j^{\prime }k^{\prime }}).
\end{equation*}%
However,
\begin{equation*}
\rho _{\sigma }(h^{jk})=\Sigma ^{jk},\quad \rho _{\sigma
}(h^{jk}h^{j^{\prime }k^{\prime }})=\Sigma ^{jk}\Sigma ^{j^{\prime
}k^{\prime }}+\Sigma ^{jj^{\prime }}\Sigma ^{kk^{\prime }}+\Sigma
^{jk^{\prime }}\Sigma ^{kj^{\prime }},
\end{equation*}%
so
\begin{eqnarray*}
A(\sigma ,g,h)^{jk,j^{\prime }k^{\prime }} &=&\Sigma ^{jk}\Sigma ^{j^{\prime
}k^{\prime }}+\Sigma ^{jj^{\prime }}\Sigma ^{kk^{\prime }}+\Sigma
^{jk^{\prime }}\Sigma ^{kj^{\prime }}-\Sigma ^{jk}\Sigma ^{j^{\prime
}k^{\prime }} \\
&=&\Sigma ^{jj^{\prime }}\Sigma ^{kk^{\prime }}+\Sigma ^{jk^{\prime }}\Sigma
^{kj^{\prime }}.
\end{eqnarray*}%
This is the result found in \cite{BarndorffNielsenShephard(04multi)}, but
proved under stronger conditions, and is implicit in the work of \cite%
{JacodProtter(98)}.
\end{example}
\begin{example}
\label{Example:vector case}Suppose $d_{1}=d_{2}=2$, $d_{3}=1$ and $g$ is
diagonal. Then
\begin{equation}
U(g,h)_{t}^{j}=\sum_{j^{\prime }=1}^{2}\int_{0}^{t}\alpha (\sigma
_{u},g,h)^{j,j^{\prime }}~\mathrm{d}B_{u}^{j^{\prime }},
\end{equation}%
where%
\begin{equation*}
\sum_{l=1}^{2}\alpha (\sigma ,g,h)^{j,l}\alpha (\sigma ,g,h)^{j^{\prime
},l}=A(\sigma ,g,h)^{j,j^{\prime }}.
\end{equation*}%
Here%
\begin{eqnarray*}
A(\sigma ,g,h)^{j,j^{\prime }} &=&\rho _{\sigma }(g^{jj}g^{j^{\prime
}j^{\prime }})\rho _{\sigma }(h^{j}h^{j^{\prime }})+\rho _{\sigma
}(g^{jj})\rho _{\sigma }(h^{j^{\prime }})\rho _{\sigma }(g^{j^{\prime
}j^{\prime }}h^{j}) \\
&&+\rho _{\sigma }(g^{j^{\prime }j^{\prime }})\rho _{\sigma }(h^{j})\rho
_{\sigma }(g^{jj}h^{j^{\prime }})-3\rho _{\sigma }(g^{jj})\rho _{\sigma
}(g^{j^{\prime }j^{\prime }})\rho _{\sigma }(h^{j})\rho _{\sigma
}(h^{j^{\prime }}).
\end{eqnarray*}
\end{example}
\begin{example}
\label{Example:joint BPV and RV}Joint behaviour of realised QV and realised
bipower variation. This sets%
\begin{equation*}
g(y)=\left(
\begin{array}{cc}
\left\vert y^{j}\right\vert & 0 \\
0 & 1%
\end{array}%
\right) ,\quad h(y)=\left(
\begin{array}{c}
\left\vert y^{j}\right\vert \\
\left( y^{j}\right) ^{2}%
\end{array}%
\right) .
\end{equation*}%
The implication is that
\begin{equation*}
\rho _{\sigma }(g^{11})=\rho _{\sigma }(g^{22}g^{11})=\rho _{\sigma
}(g^{11}g^{22})=\mu _{1}\sigma _{j},\ \rho _{\sigma }(g^{22})=1,\ \rho
_{\sigma }(g^{11}g^{11})=\sigma _{j}^{2},\ \rho _{\sigma }(g^{22}g^{22})=1,
\end{equation*}%
\begin{equation*}
\rho _{\sigma }(h^{1})=\mu _{1}\sigma _{j},\ \rho _{\sigma }(h^{2})=\rho
_{\sigma }(h^{1}h^{1})=\sigma _{j}^{2},\ \rho _{\sigma }(h^{1}h^{2})=\rho
_{\sigma }(h^{2}h^{1})=\mu _{3}\sigma _{j}^{3},\ \rho _{\sigma
}(h^{2}h^{2})=3\sigma _{j}^{4},
\end{equation*}%
\begin{equation*}
\rho _{\sigma }(g^{11}h^{1})=\sigma _{j}^{2},\ \rho _{\sigma
}(g^{11}h^{2})=\mu _{3}\sigma _{j}^{3},\ \rho _{\sigma }(g^{22}h^{1})=\mu
_{1}\sigma _{j},\ \rho _{\sigma }(g^{22}h^{2})=\sigma _{j}^{2}.
\end{equation*}%
Thus
\begin{eqnarray*}
A(\sigma ,g,h)^{1,1} &=&\sigma _{j}^{2}\sigma _{j}^{2}+2\mu _{1}\sigma
_{j}\mu _{1}\sigma _{j}\sigma _{j}^{2}-3\mu _{1}\sigma _{j}\mu _{1}\sigma
_{j}\mu _{1}\sigma _{j}\mu _{1}\sigma _{j} \\
&=&\sigma _{j}^{4}\left( 1+2\mu _{1}^{2}-3\mu _{1}^{4}\right) =\mu
_{1}^{4}(2+\vartheta )\sigma _{j}^{4},
\end{eqnarray*}%
while%
\begin{equation*}
A(\sigma ,g,h)^{2,2}=3\sigma _{j}^{4}+2\sigma _{j}^{4}-3\sigma
_{j}^{4}=2\sigma _{j}^{4},
\end{equation*}%
and%
\begin{eqnarray*}
A(\sigma ,g,h)^{1,2} &=&\mu _{1}\sigma _{j}\mu _{3}\sigma _{j}^{3}+\mu
_{1}\sigma _{j}\sigma _{j}^{2}\mu _{1}\sigma _{j}+\mu _{1}\sigma _{j}\mu
_{3}\sigma _{j}^{3}-3\mu _{1}\sigma _{j}\mu _{1}\sigma _{j}\sigma _{j}^{2} \\
&=&2\sigma _{j}^{4}\left( \mu _{1}\mu _{3}-\mu _{1}^{2}\right) =2\mu
_{1}^{2}\sigma _{j}^{4}.
\end{eqnarray*}%
This generalises the result given in \cite{BarndorffNielsenShephard(03test)}
to the leverage case. In particular we have that
\begin{equation*}
\left(
\begin{array}{c}
U(g,h)_{t}^{1} \\
U(g,h)_{t}^{2}%
\end{array}%
\right) =\left(
\begin{array}{l}
\displaystyle\mu _{1}^{2}\int_{0}^{t}\sqrt{2\sigma _{u}^{4}}\mathrm{d}%
B_{u}^{1}+\mu _{1}^{2}\int_{0}^{t}\sqrt{\vartheta \sigma _{u}^{4}}\mathrm{d}%
B_{u}^{2} \\
\displaystyle\int_{0}^{t}\sqrt{2\sigma _{u}^{4}}\mathrm{d}B_{u}^{1}.%
\end{array}%
\right)
\end{equation*}
\end{example}
\section{Multipower variation\label{sect:multipower variation}}
A natural extension of generalised bipower variation is to generalised
multipower variation%
\begin{equation*}
Y^{n}(g)_{t}=\frac{1}{n}\sum_{i=1}^{\left\lfloor nt\right\rfloor }\left\{
\dprod\limits_{i^{\prime }=1}^{I\wedge \left( i+1\right) }g_{i^{\prime }}(%
\sqrt{n}~\Delta _{i-i^{\prime }+1}^{n}Y)\right\} .
\end{equation*}%
This measure of variation, for the $g_{i^{\prime }}$ being absolute powers,
was introduced by \cite{BarndorffNielsenShephard(03test)}.
We will be interested in studying the properties of $Y^{n}(g)_{t}$ for given
functions $\left\{ g_{i}\right\} $ with the following properties.
\noindent \textbf{Assumption (K}$^{\ast }$\textbf{):} All the $\left\{
g_{i}\right\} $ are continuous with at most polynomial growth.
The previous results suggests that if $Y$ is a Brownian semimartingale and
Assumption (K$^{\ast }$) holds then
\begin{equation*}
Y^{n}(g)_{t}\rightarrow Y(g)_{t}:=\int_{0}^{t}\dprod\limits_{i=0}^{I}\rho
_{\sigma _{u}}(g_{i})\mathrm{d}u.
\end{equation*}
\begin{example}
\textbf{(a)} Suppose $I=4$ and $g_{i}(y)=\left\vert y^{j}\right\vert $, then
$\rho _{\sigma }(g_{i})=\mu _{1}\sigma _{j}$ so
\begin{equation*}
Y(g)_{t}=\mu _{1}^{4}\int_{0}^{t}\sigma _{j,u}^{4}\mathrm{d}u,
\end{equation*}%
a scaled version of integrated quarticity. \newline
\noindent \textbf{(b)} Suppose $I=3$ and $g_{i}(y)=\left\vert
y^{j}\right\vert ^{4/3}$, then
\begin{equation*}
\rho _{\sigma }(g_{i})=\mu _{4/3}\sigma _{j}^{4/3}
\end{equation*}%
so
\begin{equation*}
Y(g)_{t}=\mu _{4/3}^{3}\int_{0}^{t}\sigma _{j,u}^{4}\mathrm{d}u.
\end{equation*}
\end{example}
\begin{example}
Of some importance is the generic case where $g_{i}(y)=\left\vert
y^{j}\right\vert ^{2/I}$, which implies
\begin{equation*}
Y(g)_{t}=\mu _{2/I}^{I}\int_{0}^{t}\sigma _{j,u}^{2}\mathrm{d}u.
\end{equation*}%
Thus this class provides an interesting alternative to realised variance as
an estimator of integrated variance. Of course it is important to know a
central limit theory for these types of quantities. \ \cite%
{BarndorffNielsenGraversenJacodPodolskyShephard(04shiryaev)} show that when
(H1) and (H2) hold then
\begin{equation*}
\sqrt{n}\left[ Y^{n}(g)_{t}-Y(g)_{t}\right] \rightarrow \int_{0}^{t}\sqrt{%
\omega _{I}^{2}\sigma _{j,u}^{4}}~\mathrm{d}B_{u},
\end{equation*}%
where%
\begin{equation*}
\omega _{I}^{2}=\mathrm{Var}\left( \dprod\limits_{i=1}^{I}\left\vert
u_{i}\right\vert ^{2/I}\right) +2\sum_{j=1}^{I-1}\mathrm{Cov}\left(
\dprod\limits_{i=1}^{I}\left\vert u_{i}\right\vert
^{2/I},\dprod\limits_{i=1}^{I}\left\vert u_{i-j}\right\vert ^{2/I}\right) ,
\end{equation*}%
with $u_{i}\sim NID(0,1)$. Clearly $\omega _{1}^{2}=2$, while recalling that
$\mu _{1}=\sqrt{2/\pi }$,
\begin{eqnarray*}
\omega _{2}^{2} &=&\mathrm{Var}(\left\vert u_{1}\right\vert \left\vert
u_{2}\right\vert )+2\mathrm{Cov}(\left\vert u_{1}\right\vert \left\vert
u_{2}\right\vert ,\left\vert u_{2}\right\vert \left\vert u_{3}\right\vert )
\\
&=&1+2\mu _{1}^{2}-3\mu _{1}^{4},
\end{eqnarray*}
\noindent and
\begin{eqnarray*}
\omega _{3}^{2} &=&\mathrm{Var}(\left( \left\vert u_{1}\right\vert
\left\vert u_{2}\right\vert \left\vert u_{3}\right\vert \right) ^{2/3})+2%
\mathrm{Cov}(\left( \left\vert u_{1}\right\vert \left\vert u_{2}\right\vert
\left\vert u_{3}\right\vert \right) ^{2/3},\left( \left\vert
u_{2}\right\vert \left\vert u_{3}\right\vert \left\vert u_{4}\right\vert
\right) ^{2/3}) \\
&&+2\mathrm{Cov}(\left( \left\vert u_{1}\right\vert \left\vert
u_{2}\right\vert \left\vert u_{3}\right\vert \right) ^{2/3},\left(
\left\vert u_{3}\right\vert \left\vert u_{4}\right\vert \left\vert
u_{5}\right\vert \right) ^{2/3}) \\
&=&\left( \mu _{4/3}^{3}-\mu _{2/3}^{6}\right) +2\left( \mu _{4/3}^{2}\mu
_{2/3}^{2}-\mu _{2/3}^{6}\right) +2\left( \mu _{4/3}\mu _{2/3}^{4}-\mu
_{2/3}^{6}\right) .
\end{eqnarray*}
\end{example}
\begin{example}
The law of large numbers and the central limit theorem also hold for linear
combinations of processes like $Y(g)$ above. For example one may denote by $%
\zeta^n_i$ the $d\times d$ matrix whose $(k,l)$ entry is $%
\sum_{j=0}^{d-1}\Delta^n_{i+j}Y^k\Delta^n_{i+j}Y^l$. Then
\begin{equation*}
Z^n_t=\frac{n^{d-1}}{d!}\sum_{i=1}^{[nt]}\det(\zeta^n_i)
\end{equation*}
is a linear combinations of processes $Y^n(g)$ for functions $g_l$ being of
the form $g_l(y)=y^jy^k$. It is proved in \cite{JacodLejayTalay(05)} that
under (H)
\begin{equation*}
Z^n_t\rightarrow Z_t:=\int_0^t \det(\sigma_u\sigma^{\prime}_u)du
\end{equation*}
in probability, whereas under (H1) and (H2) the associated CLT is the
following convergence in law:
\begin{equation*}
\sqrt{n}(Z^n_t-Z_t)\rightarrow \int_0^t\sqrt{\Gamma(\sigma_u)}~dB_u,
\end{equation*}
where $\Gamma(\sigma)$ denotes the covariance of the variable $%
\det(\zeta)/d! $, and $\zeta$ is a $d\times d$ matrix whose $(k,l)$ entry is
$\sum_{j=0}^{d-1}U_j^kU_j^l$ and the $U_j$'s are i.i.d. centered Gaussian
vectors with covariance $\sigma\sigma^{\prime}$.
This kind of result may be used for testing whether the rank of the
diffusion coefficient is everywhere smaller than $d$ (in which case one
could use a model with a $d^{\prime}<d$ for the dimension of the driving
Wiener process $W$).
\end{example}
\section{Conclusion}
This paper provides some rather general limit results for realised
generalised bipower variation. In the case of power variation and bipower
variation the results are proved under much weaker assumptions than those
which have previously appeared in the literature. In particular the
no-leverage assumption is removed, which is important in the application of
these results to stock data.
There are a number of open questions. It is rather unclear how
econometricians might exploit the generality of the $g$ and $h$ functions to
learn about interesting features of the variation of price processes. It
would be interesting to know what properties $g$ and $h$ must possess in
order for these statistics to be robust to finite activity and infinite
activity jumps. A challenging extension is to construct a version of
realised generalised bipower variation which is robust to market
microstructure effects. Following the work on the realised volatility there
are two leading strategies which may be able to help: the kernel based
approach, studied in detailed by \cite%
{BarndorffNielsenHansenLundeShephard(04)}, and the subsampling approach of
\cite{ZhangMyklandAitSahalia(03)} and \cite{Zhang(04)}. In the realised
volatility case these methods are basically equivalent, however it is
perhaps the case that the subsampling method is easier to extend to the
non-quadratic case.
\section{Acknowledgments}
Ole E. Barndorff-Nielsen's work is supported by CAF (\texttt{www.caf.dk}),
which is funded by the Danish Social Science Research Council. Neil
Shephard's research is supported by the UK's ESRC through the grant
\textquotedblleft High frequency financial econometrics based upon power
variation.\textquotedblright\
\section{Proof of Theorem \protect\ref{TT3}}
\subsection{Strategy for the proof}
Below we give a fairly detailed account of the basic techniques in the proof
of Theorem \ref{TT3}, in the one-dimensional case and under some relatively
minor simplifying assumptions. Throughout we set $h=1$ for the main
difficulty in the proof is being able to deal with the generality in the $g$
function. Once that has been mastered the extension to the bipower measure
is not a large obstacle. We refer the reader to \cite%
{BarndorffNielsenGraversenJacodPodolskyShephard(04shiryaev)} for readers who
wish to see the more general case. In this subsection we provide a brief
outline of the content of the Section.
The aim of this Section is to show that%
\begin{equation}
\sqrt{n}\left( \frac{1}{n}\,\sum_{i=1}^{[nt]}g\left( \sqrt{n}\,\triangle
_{i}^{n}Y\right) -\int_{0}^{t}\rho _{\sigma _{u}}(g)\right) \rightarrow
\int_{0}^{t}\sqrt{\rho _{\sigma _{u}}(g^{2})-\rho _{\sigma _{u}}(g)^{2}}\;%
\mathrm{d}B_{u} \label{eqn 0}
\end{equation}%
where $B$\ is a Brownian motion independent of the process $Y$\ and the
convergence is (stably) in law. This case is important for the extension to
realised generalised bipower (and multipower) variation is relatively simple
once this fundamental result is established.
The proof of this result is done\ in a number of steps, some of them
following fairly standard reasoning, others requiring special techniques.
The first step is to rewrite the left hand side of (\ref{eqn 0}) as follows%
\begin{eqnarray*}
&&\sqrt{n}\left( \frac{1}{n}\,\sum_{i=1}^{[nt]}g(\sqrt{n}\,\triangle
_{i}^{n}Y)-\int_{0}^{t}\rho _{\sigma _{u}}(g)\mathrm{d}u\right) \\
&=&\frac{1}{\sqrt{n}}\,\sum_{i=1}^{[nt]}\left\{ g(\sqrt{n}\,\triangle
_{i}^{n}Y)-\mathrm{E}\left[ g(\triangle _{i}^{n}Y)\,|\,\mathcal{F}_{\frac{i-1%
}{n}}\right] \right\} \, \\
&&+\sqrt{n}\left( \frac{1}{n}\,\sum_{i=1}^{[nt]}\mathrm{E}\left[ g(\triangle
_{i}^{n}Y)\,|\,\mathcal{F}_{\frac{i-1}{n}}\right] \,-\int_{0}^{t}\rho
_{\sigma _{u}}(g)\mathrm{d}u\right) .
\end{eqnarray*}%
It is rather straightforward to show that the first term of the right hand
side satisfies
\begin{equation*}
\frac{1}{\sqrt{n}}\,\sum_{i=1}^{[nt]}\left\{ \,g(\sqrt{n}\,\triangle
_{i}^{n}Y)-\mathrm{E}\left[ g(\triangle _{i}^{n}Y)\,|\,\mathcal{F}_{\frac{i-1%
}{n}}\right] \right\} \rightarrow \int_{0}^{t}\sqrt{\rho _{\sigma
_{u}}(g^{2})-\rho _{\sigma _{u}}(g)^{2}}\mathrm{d}B_{u}.
\end{equation*}%
Hence what remains is to verify that%
\begin{equation}
\sqrt{n}\left( \frac{1}{n}\,\sum_{i=1}^{[nt]}\mathrm{E}\left[ g(\triangle
_{i}^{n}Y)\,|\,\mathcal{F}_{\frac{i-1}{n}}\right] \,-\int_{0}^{t}\rho
_{\sigma _{u}}(g)\mathrm{d}u\right) \rightarrow 0. \label{2}
\end{equation}%
We have%
\begin{eqnarray}
&&\sqrt{n}\left( \frac{1}{n}\,\sum_{i=1}^{[nt]}\mathrm{E}\left[ g(\triangle
_{i}^{n}Y)\,|\,\mathcal{F}_{\frac{i-1}{n}}\right] \,-\int_{0}^{t}\rho
_{\sigma _{u}}(g)\mathrm{d}u\right) \notag \\
&=&\frac{1}{\sqrt{n}}\,\sum_{i=1}^{[nt]}\mathrm{E}\left[ g(\triangle
_{i}^{n}Y)\,|\,\mathcal{F}_{\frac{i-1}{n}}\right] \,-\sqrt{n}%
\sum_{i=1}^{[nt]}\int_{(i-1)/n}^{i/n}\rho _{\sigma _{u}}(g)\mathrm{d}u
\notag \\
&&+\sqrt{n}\left( \sum_{i=1}^{[nt]}\int_{(i-1)/n}^{i/n}\rho _{\sigma _{u}}(g)%
\mathrm{d}u-\int_{0}^{t}\rho _{\sigma _{u}}(g)\mathrm{d}u\right) \label{3}
\end{eqnarray}%
where%
\begin{equation*}
\sqrt{n}\left\{ \sum_{i=1}^{[nt]}\int_{(i-1)/n}^{i/n}\rho _{\sigma _{u}}(g)%
\mathrm{d}u-\int_{0}^{t}\rho _{\sigma _{u}}(g)\mathrm{d}u\right\}
\rightarrow 0.
\end{equation*}%
The first term on the right hand side of (\ref{3}) is now split into the
difference of%
\begin{equation}
\frac{1}{\sqrt{n}}\,\sum_{i=1}^{[nt]}\left\{ \mathrm{E}\left[ g(\triangle
_{i}^{n}Y)\,|\,\mathcal{F}_{\frac{i-1}{n}}\right] \,-\rho _{\frac{i-1}{n}%
}\right\} \label{4}
\end{equation}%
where
\begin{equation*}
\rho _{\frac{i-1}{n}}=\rho _{\sigma _{\frac{i-1}{n}}}(g)=\mathrm{E}\left[
g(\sigma _{\frac{i-1}{n}}\triangle _{i}^{n}W)\,|\,\mathcal{F}_{\frac{i-1}{n}}%
\right]
\end{equation*}%
and%
\begin{equation}
\sqrt{n}\sum_{i=1}^{[nt]}\int_{(i-1)/n}^{i/n}\left\{ \rho _{\sigma _{u}}(g)%
\mathrm{d}u-\rho _{\frac{i-1}{n}}\right\} \mathrm{d}u. \label{4b}
\end{equation}%
It is rather easy to show that (\ref{4}) tends to $0$ in probability
uniformly in $t$. The challenge is thus to show the same result holds for (%
\ref{4b}).
To handle (\ref{4b}) one splits the individual terms in the sum into%
\begin{equation}
\sqrt{n}\ \Phi ^{\prime }\left( \sigma _{\frac{i-1}{n}}\right)
\int_{(i-1)/n}^{i/n}\left( \sigma _{u}-\sigma _{\frac{i-1}{n}}\right)
\mathrm{\,d}u \label{5}
\end{equation}%
plus%
\begin{equation}
\sqrt{n}\,\int_{(i-1)/n}^{i/n}\,\left\{ \Phi (\sigma _{u})-\Phi \left(
\sigma _{\frac{i-1}{n}}\right) -\Phi ^{\prime }\left( \sigma _{\frac{i-1}{n}%
}\right) \cdot \left( \sigma _{u}-\sigma _{\frac{i-1}{n}}\right) \right\}
\,\,\mathrm{d}u, \label{6}
\end{equation}%
where $\Phi (x)$ is a shorthand for $\rho _{x}(g)$ and $\Phi ^{\prime }(x)$
denotes the derivative with respect to $x$.\ That (\ref{6})\ tends to $0$\
may be shown via splitting it into two terms, each of which tends to $0$\ as
is verified by a sequence of inequalities, using in particular Doob's
inequality. To prove that (\ref{5}) converges to $0$, again one splits, this
time into three terms, using the differentiability of $g$\ in the relevant
regions and the mean value theorem for differentiable functions. The two
first of these terms can be handled by relatively simple means, the third
poses the most difficult part of the whole proof and is treated via
splitting it into seven parts. It is at this stage that the assumption that $%
g$\ be even comes into play and is crucial.
This section has six other subsections. In subsection \ref%
{subsection:conventions} we introduce our basic notation, while in \ref%
{subsection:model assumptions} we set out the model and review the
assumptions we use. In subsection \ref{subsection:main result} we state the
theorem we will prove. Subsections \ref{subsection: intermediate limiting
results}, \ref{subsection:13b} and \ref{subsection:proof of 13a} give the
proofs of the successive steps.
\subsection{Notational conventions \label{subsection:conventions}}
All processes mentioned in the following are defined on a given filtered
probability space $(\Omega ,\mathcal{F},(\mathcal{F}_{t}),P)$. We shall in
general use standard notation and conventions. For instance, given a process
$(Z_{t})$ we write
\begin{equation*}
\triangle _{i}^{n}Z:=Z_{\frac{i}{n}}-Z_{\frac{i-1}{n}},\ \ \ i,n\geq 1.
\end{equation*}
We are mainly interested in convergence in law of sequences of c\`{a}dl\`{a}%
g processes. In fact all results to be proved will imply convergence `stably
in law' which is a slightly stronger notion. For this we shall use the
notation
\begin{equation*}
(Z_{t}^{n})\rightarrow (Z_{t}),
\end{equation*}%
where $(Z_{t}^{n})$ and $(Z_{t})$ are given c\`{a}dl\`{a}g processes.
Furthermore we shall write
\begin{equation*}
(Z_{t}^{n})\overset{P}{\rightarrow }0\ \ \ \text{meaning}\ \ \sup_{0\leq
s\leq t}|Z_{s}^{n}|\rightarrow 0\ \ \mbox{\rm in\ probability\ for\ all}\
t\geq 0,
\end{equation*}%
\begin{equation*}
(Z_{t}^{n})\overset{P}{\rightarrow }(Z_{t})\ \ \ \text{meaning}\ \
(Z_{t}^{n}-Z_{t})\overset{P}{\rightarrow }0.
\end{equation*}%
Often
\begin{equation*}
Z_{t}^{n}=\sum_{i=1}^{[nt]}a_{i}^{n}\ \ \ \text{for all}\ t\geq 0,
\end{equation*}%
where the $a_{i}^{n}$'s are $\mathcal{F}_{\frac{i-1}{n}}$-measurable. Recall
here that given c\`{a}dl\`{a}g processes $(Z_{t}^{n}),\,(Y_{t}^{n})$ and $%
(Z_{t})$ we have
\begin{equation*}
(Z_{t}^{n})\rightarrow (Z_{t})\ \ \text{if}\ \ (Z_{t}^{n}-Y_{t}^{n})\overset{%
P}{\rightarrow }0\ \ \text{and}\ \ (Y_{t}^{n})\rightarrow (Z_{t}).\vspace{1mm%
}
\end{equation*}
Moreover, for $h:\mathbf{R}\rightarrow \mathbf{R}$ Borel measurable of at
most polynomial growth we note that $x\mapsto \rho _{x}(h)$ is locally
bounded and continuous if $h$ is continuous at $0$.\vspace{1mm}\newline
In what follows many arguments will consist of a series of estimates of
terms indexed by $i,n$ and $t$. In these estimates we shall denote by $C$ a
finite constant which may vary from place to place. Its value will depend on
the constants and quantities appearing in the assumptions of the model but
it is always independent of $i,n$ and $t$.
\subsection{Model and basic assumptions \label{subsection:model assumptions}}
Throughout the following $(W_{t})$ denotes a $((\mathcal{F}_{t}),P)$-Wiener
process and $(\sigma _{t})$ a given c\`{a}dl\`{a}g $(\mathcal{F}_{t})$%
-adapted process. Define
\begin{equation*}
Y_{t}:=\int_{0}^{t}\sigma _{s-}\,\mathrm{d}W_{s}\ \ \ \ \ t\geq 0,
\end{equation*}%
implying that is $(Y_{t})$ is a continuous local martingale. We have deleted
the drift of the $\left( Y_{t}\right) $ process as taking care of it is a
simple technical task, while its presence increase the clutter of the
notation. Our aim is to study the asymptotic behaviour of the processes
\begin{equation*}
\{(X_{t}^{n}(g))\,|\,n\geq 1\,\}
\end{equation*}%
where%
\begin{equation*}
X_{t}^{n}(g)=\frac{1}{n}\,\sum_{i=1}^{[nt]}g(\sqrt{n}\,\triangle
_{i}^{n}Y),\ \ \ t\geq 0,\,n\geq 1.
\end{equation*}%
Here $g:\mathbf{R}\rightarrow \mathbf{R}$ is a given continuous function of
at most polynomial growth. We are especially interested in $g$'s of the form
$x\mapsto |x|^{r}\ (r>0)$ but we shall keep the general notation since
nothing is gained in simplicity by assuming that $g$ is of power form. We
shall throughout the following assume that $g$ furthermore satisfies the
following.
\begin{assumption}
\textbf{(K)}: $g$ is an even function and continuously differentiable in $%
B^{c}$ where $B\subseteq \mathbf{R}$ is a closed Lebesgue null-set and $%
\exists \ M,\,p\geq 1$ such that
\begin{equation*}
|g(x+y)-g(x)|\leq M(1+|x|^{p}+|y|^{p})\cdot |y|\ ,
\end{equation*}%
for all $x,y\in \mathbf{R}$.
\end{assumption}
\begin{remark}
The assumption (K) implies, in particular, that if $x\in B^{c}$ then
\begin{equation*}
|g^{\prime }(x)|\leq M(1+|x|^{p}).\vspace{1mm}
\end{equation*}%
Observe that only power functions corresponding to $r\geq 1$ do satisfy (K).
The remaining case $0<r<1$ requires special arguments which will be omitted
here\vspace{1mm} (for details see \cite%
{BarndorffNielsenGraversenJacodPodolskyShephard(04shiryaev)}).
\end{remark}
In order to prove the CLT-theorem we need some additional structure on the
volatility process $(\sigma _{t})$. A natural set of assumptions would be
the following.
\begin{assumption}
\textbf{(H0)}: $(\sigma _{t})$ can be written as
\begin{equation*}
\sigma _{t}=\sigma _{0}+\int_{0}^{t}a_{s}^{\ast }\,\mathrm{d}%
s+\int_{0}^{t}\sigma _{s}^{\ast }\,\mathrm{d}W_{s}+\int_{0}^{t}v_{s-}^{\ast
}\,\mathrm{d}Z_{s}
\end{equation*}%
where $(Z_{t})$ is a $((\mathcal{F}_{t}),P)$-L\'{e}vy process independent of
$(W_{t})$ and $(\sigma _{t}^{\ast })$ and $(v_{t}^{\ast })$ are adapted c%
\`{a}dl\`{a}g processes and $(a_{t}^{\ast })$ a predictable locally bounded
process.
\end{assumption}
However, in modelling volatility it is often more natural to define $(\sigma
_{t}^{2})$ as being of the above form, i.e.%
\begin{equation*}
\sigma _{t}^{2}=\sigma _{0}^{2}+\int_{0}^{t}a_{s}^{\ast }\,\mathrm{d}%
s+\int_{0}^{t}\sigma _{s}^{\ast }\,\mathrm{d}W_{s}+\int_{0}^{t}v_{s-}^{\ast
}\,\mathrm{d}Z_{s}.
\end{equation*}%
Now this does not in general imply that $(\sigma _{t})$ has the same form;
therefore we shall replace (H0) by the more general structure given by the
following assumption.
\begin{assumption}
\textbf{(H1)}: $(\sigma _{t})$ can be written, for $t\geq 0$, as%
\begin{equation*}
\begin{array}{lll}
\sigma _{t} & = & \displaystyle\sigma _{0}+\int_{0}^{t}a_{s}^{\ast }\,%
\mathrm{d}s+\int_{0}^{t}\sigma _{s}^{\ast }\,\mathrm{d}W_{s}+%
\int_{0}^{t}v_{s-}^{\ast }\,\mathrm{d}V_{s} \\
& & \displaystyle+\int_{0}^{t}\int_{E}q\circ \phi (s-,x)\,(\mu -\nu )(%
\mathrm{d}s\,\mathrm{d}x) \\
& & \displaystyle+\int_{0}^{t}\int_{E}\ \left\{ \phi (s-,x)-q\circ \phi
(s-,x)\right\} \,\mu (\mathrm{d}s\,\mathrm{d}x).%
\end{array}%
\end{equation*}%
Here $(a_{t}^{\ast }),\,(\sigma _{t}^{\ast })$ and $(v_{t}^{\ast })$ are as
in (H0) and $(V_{t})$ is another $((\mathcal{F}_{t}),P)$-Wiener process
independent of $(W_{t})$ while $q$ is a continuous truncation function on $%
\mathbf{R}$, i.e. a function with compact support coinciding with the
identity on a neighbourhood of $0$. Further $\mu $ is a Poisson random
measure on $(0,\infty )\times E$ independent of $(W_{t})$ and $(V_{t})$ with
intensity measure $\nu (\mathrm{d}s\,\mathrm{d}x)=\mathrm{d}s\otimes F(%
\mathrm{d}x)$, $F$ being a $\sigma $-finite mea\-sure on a measurable space $%
(E,\mathcal{E})$ and
\begin{equation*}
(\omega ,s,x)\mapsto \phi (\omega ,s,x)
\end{equation*}%
is a map from $\Omega \times \,[\,0,\infty )\times E$ into $\mathbf{R}$
which is $\mathcal{F}_{s}\otimes \mathcal{E}$ measurable in $(\omega ,x)$
for all $s$ and c\`{a}dl\`{a}g in $s$, satisfying furthermore that for some
sequence of stopping times $(S_{k})$ increasing to $+\infty $ we have for
all $k\geq 1$
\begin{equation*}
\int_{E}\left\{ 1\wedge \psi _{k}(x)^{2}\right\} \,F(\mathrm{d}x)<\infty ,
\end{equation*}%
where%
\begin{equation*}
\psi _{k}(x)=\sup_{\omega \in \Omega ,\,s<S_{k}(\omega )}|\phi (\omega
,s,x)|.
\end{equation*}
\end{assumption}
\begin{remark}
(H1) is weaker than (H0), and if $(\sigma _{t}^{2})$ satisfies (H1) then so
does $(\sigma _{t})$.\newline
\end{remark}
Finally we shall also assume a non-degeneracy in the model.
\begin{assumption}
\textbf{(H2)}: $(\sigma _{t})$ satisfies
\begin{equation*}
0<\sigma _{t}^{2}(\omega )\ \text{for all}\ (t,\omega ).\vspace{1mm}
\end{equation*}
\end{assumption}
According to general stochastic analysis theory it is known that to prove
convergence in law of a sequence $(Z_{t}^{n})$ of c\`{a}dl\`{a}g processes
it suffices to prove the convergence of each of the stopped processes $%
(Z_{T_{k}\wedge t}^{n})$ for at least one sequence of stopping times $%
(T_{k}) $ increasing to $+\infty $. Applying this together with standard
localisation techniques (for details see \cite%
{BarndorffNielsenGraversenJacodPodolskyShephard(04shiryaev)}), we may assume
that the following more restrictive assumptions are satisfied.
\begin{assumption}
\textbf{(H1a)}: $(\sigma _{t})$ can be written as
\begin{equation*}
\sigma _{t}=\sigma _{0}+\int_{0}^{t}a_{s}^{\ast }\,ds+\int_{0}^{t}\sigma
_{s-}^{\ast }\,\mathrm{d}W_{s}+\int_{0}^{t}v_{s-}^{\ast }\,\mathrm{d}%
V_{s}+\int_{0}^{t}\int_{E}\phi (s-,x)(\mu -\nu )(\mathrm{d}s\,\mathrm{d}x)\
\ \ t\geq 0.\vspace{1mm}
\end{equation*}%
Here $(a_{t}^{\ast }),\,(\sigma _{t}^{\ast })$ and $(v_{t}^{\ast })$ are
real valued uniformly bounded c\`{a}dl\`{a}g $(\mathcal{F}_{t})$-adapted
proces\-ses; $(V_{t})$ is another $((\mathcal{F}_{t}),P)$-Wiener process
independent of $(W_{t})$. Further $\mu $ is a Poisson random measure on $%
(0,\infty )\times E$ independent of $(W_{t})$ and $(V_{t})$ with intensity
measure $\nu (\mathrm{d}s\,\mathrm{d}x)=\mathrm{d}s\otimes F(\mathrm{d}x)$, $%
F$ being a $\sigma $-finite mea\-sure on a measurable space $(E,\mathcal{E})$
and
\begin{equation*}
(\omega ,s,x)\mapsto \phi (\omega ,s,x)
\end{equation*}%
is a map from $\Omega \times \,[\,0,\infty )\times E$ into $\mathbf{R}$
which is $\mathcal{F}_{s}\otimes \mathcal{E}$ measurable in $(\omega ,x)$
for all $s$ and c\`{a}dl\`{a}g in $s$, satisfying furthermore
\begin{equation*}
\psi (x)=\sup_{\omega \in \Omega ,\,s\geq 0}|\phi (\omega ,s,x)|\leq
M<\infty \ \ \text{and}\ \ \int \psi (x)^{2}\,F(\mathrm{d}x)<\infty .\vspace{%
1mm}
\end{equation*}
\end{assumption}
Likewise, by a localisation argument, we may assume
\begin{assumption}
\textbf{(H2a)}: $(\sigma _{t})$ satisfies
\begin{equation*}
a<\sigma _{t}^{2}(\omega )<b\ \ \ \text{for all}\ (t,\omega )\ \text{for some%
}\ a,b\in (0,\infty ).\vspace{1mm}
\end{equation*}
\end{assumption}
Observe that under the more restricted assumptions $(Y_{t})$ is a continuous
martingale having moments of all orders and $(\sigma _{t})$ is represented
as a sum of three square integrable martingales plus a continuous process of
bounded variation. Furthermore, the increments of the increasing processes
corresponding to the three martingales and of the bounded variation process
are dominated by a constant times $\triangle t$, implying in particular that
\begin{equation}
\mathrm{E}\left[ \,\left\vert \sigma _{v}-\sigma _{u}\right\vert ^{2}\right]
\leq C\,(v-u),\ \ \ \ \text{for all}\ 0\leq u<v.\vspace{2mm} \label{8}
\end{equation}
\subsection{Main result \label{subsection:main result}\newline
}
As already mentioned, our aim is to show the following special version of
the general CLT-result given as Theorem \ref{TT3}.
\begin{theorem}
\vspace{2mm}Under assumptions (K), (H1a) and (H2a), there exists a Wiener
process $(B_{t})$ defined on some extension of $(\Omega ,\mathcal{F},(%
\mathcal{F}_{t}),P)$ and independent of $\mathcal{F}$ such that%
\begin{equation}
\left( \sqrt{n}\left( \,\ \frac{1}{n}\,\sum_{i=1}^{[nt]}g(\sqrt{n}%
\,\triangle _{i}^{n}Y)-\int_{0}^{t}\rho _{\sigma _{u}}(g)\,\mathrm{d}%
u\,\right) \right) \rightarrow \int_{0}^{t}\sqrt{\rho _{\sigma
_{u-}}(g^{2})-\rho _{\sigma _{u-}}(g)^{2}}\,\mathrm{d}B_{u}.
\label{Main result}
\end{equation}%
\emph{\ }
\end{theorem}
Introducing the notation%
\begin{equation*}
U_{t}(g)=\int_{0}^{t}\sqrt{\rho _{\sigma _{u-}}(g^{2})-\rho _{\sigma
_{u-}}(g)^{2}}\,\mathrm{d}B_{u}\ \ \ t\geq 0\vspace{1mm}
\end{equation*}%
we may reexpress (\ref{Main result}) as
\begin{equation}
\left( \sqrt{n}\,\left( X_{t}^{n}(g)-\int_{0}^{t}\sigma _{u}(g)\,\mathrm{d}%
u\right) \,\right) \rightarrow (U_{t}(g)). \label{Main result reform}
\end{equation}%
To prove this, introduce the set of variables $\{\beta
_{i}^{n}\,|\,i,\,n\geq 1\}$ given by
\begin{equation*}
\beta _{i}^{n}=\sqrt{n}\cdot \sigma _{\frac{i-1}{n}}\cdot \triangle
_{i}^{n}W,\ \ \ i,\,n\geq 1.
\end{equation*}
The $\beta _{i}^{n}$'s should be seen as approximations to $\sqrt{n}%
\,\triangle _{i}^{n}Y$. In fact, since
\begin{equation*}
\sqrt{n}\,\triangle _{i}^{n}Y-\beta _{i}^{n}=\sqrt{n}\,\int_{(i-1)/n}^{i/n}(%
\sigma _{s}-\sigma _{\frac{i-1}{n}})\,\mathrm{d}W_{s}
\end{equation*}%
and $(\sigma _{t})$ is uniformly bounded, a straightforward application of (%
\ref{8}) and the Burkholder-Davis-Gundy-inequalities (e.g. \cite[pp. 160-171]%
{RevuzYor(99)}) gives for every $p>0$ the following simple estimates.
\begin{equation}
\mathrm{E}\left[ \,|\sqrt{n}\,\triangle _{i}^{n}Y-\beta _{i}^{n}|^{p}\,|\,%
\mathcal{F}_{\frac{i-1}{n}}\right] \leq \frac{C_{p}}{n^{p\wedge 1}}
\end{equation}%
and
\begin{equation}
\mathrm{E}\left[ \,|\sqrt{n}\,\triangle _{i}^{n}Y|^{p}+|\beta
_{i}^{n}|^{p}\,|\,\mathcal{F}_{\frac{i-1}{n}}\right] \leq C_{p}\vspace{1mm}
\label{12}
\end{equation}%
for all $i,n\geq 1$. Observe furthermore that
\begin{equation*}
\mathrm{E}\left[ g(\beta _{i}^{n})\,|\,\mathcal{F}_{\frac{i-1}{n}}\right]
=\rho _{\sigma _{\frac{i-1}{n}}}(g),\ \ \ \text{for all}\ i,\,n\geq 1.%
\vspace{1mm}
\end{equation*}
Introduce for convenience, for each $t>0$ and $n\geq 1$, the shorthand
notation
\begin{equation*}
U_{t}^{n}(g)=\frac{1}{\sqrt{n}}\,\sum_{i=1}^{[nt]}\,\,\left\{ g(\sqrt{n}%
\,\triangle _{i}^{n}Y)-\mathrm{E}\left[ g(\sqrt{n}\,\triangle _{i}^{n}Y)\,|\,%
\mathcal{F}_{\frac{i-1}{n}}\right] \right\} \,
\end{equation*}%
and
\begin{equation*}
\tilde{U}_{t}^{n}(g)=\frac{1}{\sqrt{n}}\,\sum_{i=1}^{[nt]}\,\,\left\{
g(\beta _{i}^{n})-\rho _{\sigma _{\frac{i-1}{n}}}(g)\right\} =\frac{1}{\sqrt{%
n}}\,\sum_{i=1}^{[nt]}\,\,\left\{ g(\beta _{i}^{n})-\mathrm{E}\left[ g(\beta
_{i}^{n})\,|\,\mathcal{F}_{\frac{i-1}{n}}\right] \right\} \,.\vspace{1mm}
\end{equation*}%
The asymptotic behaviour of $(\tilde{U}_{t}^{n}(g))$ is well known. More
precisely under the the given assumptions\thinspace (\thinspace in fact much
less is needed\thinspace ) we have
\begin{equation*}
(U_{t}^{n}(g))\rightarrow (U_{t}(g)).\vspace{1mm}
\end{equation*}%
This result is a rather straightforward consequence of \cite[Theorem IX.7.28]%
{JacodShiryaev(03)}. Thus, if $(U_{t}^{n}(g)-\tilde{U}_{t}^{n}(g))\overset{P}%
{\rightarrow }0$ we may deduce the following result.
\begin{theorem}
\label{theorem B}\ \ \emph{Let $(B_{t})$ and $(U_{t}(g))$ be as above. Then}
\begin{equation*}
(\tilde{U}_{t}^{n}(g))\rightarrow (U_{t}(g)).\vspace{1mm}
\end{equation*}
\end{theorem}
\noindent \textbf{Proof.}
As pointed out just above it is enough to prove that
\begin{equation*}
(U_{t}^{n}(g)-\tilde{U}_{t}^{n}(g))\overset{P}{\rightarrow }0.
\end{equation*}%
But for $t\geq 0$ and $n\geq 1$
\begin{equation*}
U_{t}^{n}(g)-\tilde{U}_{t}^{n}(g)=\sum_{i=1}^{[nt]}\,\left( \xi _{i}^{n}-%
\mathrm{E}\left[ \xi _{i}^{n}\,|\,\mathcal{F}_{\frac{i-1}{n}}\right] \right)
\end{equation*}%
where
\begin{equation*}
\xi _{i}^{n}=\frac{1}{\sqrt{n}}\left\{ g(\sqrt{n}\triangle
_{i}^{n}Y)-g(\beta _{i}^{n})\right\} ,\ \ \ i,n\geq 1.
\end{equation*}%
Thus we have to prove
\begin{equation*}
\left( \,\sum_{i=1}^{[nt]}\,\left\{ \xi _{i}^{n}-\mathrm{E}\left[ \xi
_{i}^{n}\,|\,\mathcal{F}_{\frac{i-1}{n}}\right] \right\} \right) \overset{P}{%
\rightarrow }0.
\end{equation*}%
But, as the left hand side of this relation is a sum of martingale
differences, this is implied by Doob's inequality (e.g. \cite[pp. 54-55]%
{RevuzYor(99)}) if for all $t>0$
\begin{equation*}
\sum_{i=1}^{[nt]}\,\mathrm{E}[(\xi _{i}^{n})^{2}]=\mathrm{E}%
[\,\sum_{i=1}^{[nt]}\,\mathrm{E}[(\xi _{i}^{n})^{2}\,|\,\mathcal{F}_{\frac{%
i-1}{n}}]\,]\rightarrow 0\ \ \ \text{as}\ n\rightarrow \infty .\vspace{1mm}
\end{equation*}%
Fix $t>0$. Using the Cauchy-Schwarz inequality and the
Burkholder-Davis-Gundy inequalities we have for all $i,n\geq 1$.
\begin{eqnarray*}
\mathrm{E}\left[ (\xi _{i}^{n})^{2}\,|\,\mathcal{F}_{\frac{i-1}{n}}\right]
&=&\frac{1}{n}\,\mathrm{E}\left[ \left\{ g(\sqrt{n}\triangle
_{i}^{n}Y)-\beta _{i}^{n}+\beta _{i}^{n}-g(\beta _{i}^{n})\right\} ^{2}\,|\,%
\mathcal{F}_{\frac{i-1}{n}}\right] \\
&\leq &\frac{C}{n}\,\mathrm{E}\left[ \,(1+|\sqrt{n}\triangle
_{i}^{n}Y|^{p}+|\beta _{i}^{n}|^{p})^{2}\cdot (\sqrt{n}\triangle
_{i}^{n}Y-\beta _{i}^{n})^{2}\,|\,\mathcal{F}_{\frac{i-1}{n}}\right] \\
&\leq &\frac{C}{n}\,\sqrt{\mathrm{E}\left[ \,(1+|\sqrt{n}\triangle
_{i}^{n}Y|^{2p}+|\beta _{i}^{n}|^{2p})\,|\,\mathcal{F}_{\frac{i-1}{n}}\right]
}\cdot \sqrt{\mathrm{E}\left[ (\sqrt{n}\triangle _{i}^{n}Y-\beta
_{i}^{n})^{4}\,|\,\mathcal{F}_{\frac{i-1}{n}}\right] } \\
&\leq &C\,\sqrt{\mathrm{E}\left[ \,\left( \int_{(i-1)/n}^{i/n}\left( \sigma
_{u-}-\sigma _{\frac{i-1}{n}}\right) \,\mathrm{d}W_{u}\right) ^{4}\,|\,%
\mathcal{F}_{\frac{i-1}{n}}\right] } \\
&\leq &C\,\sqrt{\mathrm{E}\left[ \left( \int_{(i-1)/n}^{i/n}\left( \sigma
_{u-}-\sigma _{\frac{i-1}{n}}\right) ^{2}\,\mathrm{d}u\right) ^{2}\,|\,%
\mathcal{F}_{\frac{i-1}{n}}\right] }.
\end{eqnarray*}%
Thus
\begin{eqnarray*}
\sum_{i=1}^{[nt]}\,\mathrm{E}[(\xi _{i}^{n})^{2}] &\leq &Cn\,\frac{t}{n}%
\,\sum_{i=1}^{[nt]}\mathrm{E}\,\left[ \sqrt{\mathrm{E}\,\left[ \left(
\int_{(i-1)/n}^{i/n}\left( \sigma _{u-}-\sigma _{\frac{i-1}{n}}\right) ^{2}\,%
\mathrm{d}u\right) ^{2}\,|\,\mathcal{F}_{\frac{i-1}{n}}\right] }\right] \, \\
&\leq &C\,tn\,\sqrt{\frac{1}{n}\,\sum_{i=1}^{[nt]}\mathrm{E}\left[ \left(
\int_{(i-1)/n}^{i/n}\left( \sigma _{u-}-\sigma _{\frac{i-1}{n}}\right) ^{2}\,%
\mathrm{d}u\right) ^{2}\right] } \\
&\leq &Ctn\,\sqrt{\frac{1}{n^{2}}\,\sum_{i=1}^{[nt]}\mathrm{E}\left[
\,\int_{(i-1)/n}^{i/n}\left( \sigma _{u-}-\sigma _{\frac{i-1}{n}}\right)
^{4}\,\mathrm{d}u\right] \,} \\
&\leq &Ct\,\sqrt{\,\sum_{i=1}^{[nt]}\int_{(i-1)/n}^{i/n}\mathrm{E}\left[
\left( \sigma _{u-}-\sigma _{\frac{i-1}{n}}\right) ^{2}\,\right] \mathrm{d}%
u\,} \\
&\rightarrow &\,0\vspace{2mm},
\end{eqnarray*}%
as $n\rightarrow \infty $ by Lebesgue's Theorem and the boundedness of $%
(\sigma _{t})$.
\noindent $\square $
To prove the convergence (\ref{Main result reform}) it suffices, using
Theorem \ref{theorem B} above, to prove that
\begin{equation*}
\left( U_{t}^{n}(g)-\sqrt{n}\,\left\{ \,X_{t}^{n}(g)-\int_{0}^{t}\rho
_{\sigma _{u}}(g)\,\mathrm{d}u\right\} \,\right) \overset{P}{\rightarrow }0.%
\vspace{1mm}
\end{equation*}%
But as
\begin{equation*}
U_{t}^{n}(g)-\sqrt{n}\,X_{t}^{n}(g)=-\frac{1}{\sqrt{n}}\,\sum_{i=1}^{[nt]}%
\mathrm{E}\left[ \,g(\sqrt{n}\,\triangle _{i}^{n}Y)\,|\,\mathcal{F}_{\frac{%
i-1}{n}}\right]
\end{equation*}%
and, as is easily seen,
\begin{equation*}
\left( \sqrt{n}\,\int_{0}^{t}\rho _{\sigma _{u}}\,\left( g\right) \mathrm{d}%
u-\,\sum_{i=1}^{[nt]}\sqrt{n}\,\int_{(i-1)/n}^{i/n}\rho _{\sigma _{u}}(g)\,%
\mathrm{d}u\right) \overset{P}{\rightarrow }0,\vspace{2mm}
\end{equation*}%
the job is to prove that
\begin{equation*}
\,\sum_{i=1}^{[nt]}\eta _{i}^{n}\,\overset{P}{\rightarrow }0\ \ \ \text{for
all}\ t>0,
\end{equation*}%
where for $i,n\geq 1$
\begin{equation*}
\eta _{i}^{n}=\,\frac{1}{\sqrt{n}}\,\mathrm{E}\,\left[ g(\sqrt{n}\,\triangle
_{i}^{n}Y)\,|\,\mathcal{F}_{\frac{i-1}{n}}\right] \,-\sqrt{n}%
\,\int_{(i-1)/n}^{i/n}\rho _{\sigma _{u}}(g)\,\mathrm{d}u.\vspace{1mm}
\end{equation*}%
Fix $t>0$ and write, for all $i,n\geq 1$,
\begin{equation*}
\eta _{i}^{n}=\eta (1)_{i}^{n}+\eta (2)_{i}^{n}
\end{equation*}%
where
\begin{equation}
\eta (1)_{i}^{n}=\frac{1}{\sqrt{n}}\,\,\left\{ \mathrm{E}\left[ \,g(\sqrt{n}%
\,\triangle _{i}^{n}Y)\,|\,\mathcal{F}_{\frac{i-1}{n}}\right] -\rho _{\sigma
_{\frac{i-1}{n}}}(g)\,\right\}
\end{equation}%
and
\begin{equation}
\eta (2)_{i}^{n}=\sqrt{n}\,\int_{(i-1)/n}^{i/n}\left\{ \rho _{\sigma
_{u}}(g)-\rho _{\sigma _{\frac{i-1}{n}}}(g)\right\} \mathrm{d}u.\vspace{1mm}
\end{equation}
We will now separately prove
\begin{equation}
\,\eta (1)^{n}=\sum_{i=1}^{[nt]}\eta (1)_{i}^{n}\,\overset{P}{\rightarrow }0
\label{13a}
\end{equation}%
and%
\begin{equation}
\,\eta (2)^{n}=\,\sum_{i=1}^{[nt]}\eta (2)_{i}^{n}\,\overset{P}{\rightarrow }%
0\vspace{1mm}. \label{13b}
\end{equation}
\subsection{Some auxiliary estimates\label{subsection: intermediate limiting
results}}
In order to show (\ref{13a}) and (\ref{13b}) we need some refinements of the
estimate (\ref{8}) above. To state these we split up $(\sqrt{n}\,\triangle
_{i}^{n}Y-\beta _{i}^{n})$ into several terms. By definition
\begin{equation*}
\sqrt{n}\,\triangle _{i}^{n}Y-\beta _{i}^{n}=\sqrt{n}\,\int_{(i-1)/n}^{i/n}%
\,\left( \sigma _{u-}-\sigma _{\frac{i-1}{n}}\right) \,\mathrm{d}W_{u}%
\vspace{1mm}
\end{equation*}%
for all $i,n\geq 1$. Writing
\begin{equation*}
E_{n}=\{x\in \mathrm{E}\,|\,|\Psi (x)|>1/\sqrt{n}\,\}
\end{equation*}%
the difference $\sigma _{u}-\sigma _{\frac{i-1}{n}}$ equals
\begin{eqnarray*}
&&\int_{(i-1)/n}^{u}a_{s}^{\ast }\,ds+\int_{(i-1)/n}^{u}\sigma _{s-}^{\ast
}\,\mathrm{d}W_{s}+\int_{(i-1)/n}^{u}v_{s-}^{\ast }\,\mathrm{d}%
V_{s}+\int_{(i-1)/n}^{u}\int_{E}\phi (s-,x)\,(\mu -\nu )(\mathrm{d}s\,%
\mathrm{d}x)\vspace{1mm} \\
&=&\displaystyle\sum_{j=1}^{5}\xi (j)_{i}^{n}(u),
\end{eqnarray*}%
for $i,n\geq 1$ and $u\geq (i-1)/n$ where
\begin{eqnarray*}
\displaystyle\xi (1)_{i}^{n}(u) &=&\int_{(i-1)/n}^{u}a_{s}^{\ast }\,\mathrm{d%
}s+\int_{(i-1)/n}^{u}\,\left( \sigma _{s-}^{\ast }-\sigma _{\frac{i-1}{n}%
}^{\ast }\right) \,\mathrm{d}W_{s}+\int_{(i-1)/n}^{u}\,\left( v_{s-}^{\ast
}-v_{\frac{i-1}{n}}^{\ast }\right) \,\mathrm{d}V_{s} \\
\displaystyle\xi (2)_{i}^{n}(u) &=&\sigma _{\frac{i-1}{n}}^{\ast }\,\left(
W_{u}-W_{\frac{i-1}{n}}\right) +v_{\frac{i-1}{n}}^{\ast }\,\left( V_{u}-V_{%
\frac{i-1}{n}}\right) \\
\displaystyle\xi (3)_{i}^{n}(u) &=&\int_{(i-1)/n}^{u}\int_{E_{n}^{c}}\phi
(s-,x)\,(\mu -\nu )(\mathrm{d}s\,\mathrm{d}x) \\
\displaystyle\xi (4)_{i}^{n}(u) &=&\int_{(i-1)/n}^{u}\int_{E_{n}}\left\{
\phi (s-,x)-\phi \left( \frac{i-1}{n},x\right) \right\} \,(\mu -\nu )(%
\mathrm{d}s\,\mathrm{d}x) \\
\displaystyle\xi (5)_{i}^{n}(u) &=&\int_{(i-1)/n}^{u}\int_{E_{n}}\phi \left(
\frac{i-1}{n},x\right) \,(\mu -\nu )(\mathrm{d}s\,\mathrm{d}x)\vspace{1mm}
\end{eqnarray*}%
That is, for $i,n\geq 1$,
\begin{equation}
\sqrt{n}\,\triangle _{i}^{n}Y-\beta _{i}^{n}=\sum_{j=1}^{5}\xi (j)_{i}^{n}
\label{17}
\end{equation}%
where
\begin{equation*}
\ \xi (j)_{i}^{n}=\sqrt{n}\,\int_{(i-1)/n}^{i/n}\,\xi (j)_{i}^{n}(u-)\,%
\mathrm{d}W_{u}\ \ \ \ \mbox{\rm for}\ j=1,2,3,4,5.\vspace{1mm}
\end{equation*}%
The specific form of the variables implies, using Burkholder-Davis-Gundy
inequalities, that for every $q\geq 2$ we have
\begin{eqnarray*}
\mathrm{E}[\,|\xi (j)_{i}^{n}|^{q}\,] &\leq &C_{q}\,n^{q/2}\,\mathrm{E}\,%
\left[ \left( \int_{(i-1)/n}^{i/n}\,\xi (j)_{i}^{n}(u)^{2}\,\mathrm{d}%
u\right) ^{q/2}\right] \\
&\leq &\displaystyle n\int_{(i-1)/n}^{i/n}\,\mathrm{E}[\,|\xi
(j)_{i}^{n}(u)|^{q}\,]\mathrm{\,d}u \\
&\leq &\displaystyle\sup_{(i-1)/n\leq u\leq i/n}\,\mathrm{E}[\,|\xi
(j)_{i}^{n}(u)|^{q}\,]
\end{eqnarray*}%
for all $i,n\geq 1$ and all $j$. These terms will now be estimated. This is
done in the following series of lemmas where $i$ and $n$ are arbitrary and
we use the notation
\begin{equation*}
d_{i}^{n}=\int_{(i-1)/n}^{i/n}\mathrm{E}\,\left[ \left( \sigma _{s-}^{\ast
}-\sigma _{\frac{i-1}{n}}^{\ast }\right) ^{2}+\left( v_{s-}^{\ast }-v_{\frac{%
i-1}{n}}^{\ast }\right) ^{2}+\int_{E}\left\{ \phi (s-,x)-\phi \left( \frac{%
i-1}{n},x\right) \right\} ^{2}\,F(\mathrm{d}x)\right] \mathrm{d}s.\vspace{1mm%
}
\end{equation*}
\begin{lemma}
\label{lemma 1st}
\begin{equation*}
\mathrm{E}[\,(\xi (1)_{i}^{n})^{2}]\leq C_{1}\cdot (1/n^{2}+d_{i}^{n}).
\end{equation*}
\end{lemma}
\begin{lemma}
\label{lemma 2nd}%
\begin{equation*}
\mathrm{E}[\,(\xi (2)_{i}^{n})^{2}]\leq C_{2}/n.
\end{equation*}
\end{lemma}
\begin{lemma}
\begin{equation*}
\mathrm{E}[\,(\xi (3)_{i}^{n})^{2}]\leq C_{3}\,\varphi (1/\sqrt{n})/n,
\end{equation*}%
where%
\begin{equation*}
\varphi (\epsilon )=\int_{\{\,|\Psi |\leq \epsilon \,\}}\Psi (x)^{2}\,F(%
\mathrm{d}x).
\end{equation*}
\end{lemma}
\begin{lemma}
\begin{equation*}
\mathrm{E}[\,(\xi (4)_{i}^{n})^{2}]\leq C_{4}\,d_{i}^{n}.
\end{equation*}
\end{lemma}
\begin{lemma}
\label{lemma 5th}
\begin{equation*}
\mathrm{E}[\,(\xi (5)_{i}^{n})^{2}]\leq C_{5}/n.
\end{equation*}
\end{lemma}
The proofs of these five Lemmas rely on straightforward martingale
inequalities.
Observe that Lebesgue's Theorem ensures, since the processes involved are
assumed c\`{a}dl\`{a}g and uniformly bounded, that as $n\rightarrow \infty $
\begin{equation*}
\sum_{i=1}^{[nt]}d_{i}^{n}\,\rightarrow 0\ \ \ \ \text{for all}\ t>0.\vspace{%
1mm}
\end{equation*}
Taken together these statements imply the following result.
\begin{corollary}
\noindent \emph{For\ all\ $t>0$ as }$n\rightarrow \infty $\emph{\ }
\begin{equation*}
\sum_{i=1}^{[nt]}\,\left\{ \mathrm{E}[\,(\xi (1)_{i}^{n})^{2}]+\mathrm{E}%
[\,(\xi (3)_{i}^{n})^{2}]+\mathrm{E}[\,(\xi (4)_{i}^{n})^{2}]\right\}
\,)\,\rightarrow 0.\vspace{1mm}
\end{equation*}
\end{corollary}
Below we shall invoke this Corollary as well as Lemmas \ref{lemma 2nd} and %
\ref{lemma 5th}.\ \newline
\subsection{Proof of $\,\protect\eta (2)^{n}\protect\overset{P}{\rightarrow }%
0$ \label{subsection:13b}}
Recall we wish to show that
\begin{equation}
\,\eta (2)^{n}=\sum_{i=1}^{[nt]}\eta (2)_{i}^{n}\,\overset{P}{\rightarrow }0.
\label{eqn 7}
\end{equation}%
>From now on let $t>0$ be fixed. We split the $\eta (2)_{i}^{n}$'s according
to
\begin{equation*}
\eta (2)_{i}^{n}=\eta ^{\prime }(2)_{i}^{n}+\eta ^{\prime \prime
}(2)_{i}^{n}\ \ \ \ i,n\geq 1
\end{equation*}%
where, writing $\Phi (x)$ for $\rho _{x}(g)$,
\begin{equation*}
\eta ^{\prime }(2)_{i}^{n}=\sqrt{n}\ \Phi ^{\prime }\left( \sigma _{\frac{i-1%
}{n}}\right) \int_{(i-1)/n}^{i/n}\left( \sigma _{u}-\sigma _{\frac{i-1}{n}%
}\right) \,\mathrm{d}u
\end{equation*}%
and
\begin{equation*}
\eta ^{\prime \prime }(2)_{i}^{n}=\sqrt{n}\,\int_{(i-1)/n}^{i/n}\,\,\left\{
\Phi (\sigma _{u})-\Phi \left( \sigma _{\frac{i-1}{n}}\right) -\Phi ^{\prime
}\left( \sigma _{\frac{i-1}{n}}\right) \cdot \left( \sigma _{u}-\sigma _{%
\frac{i-1}{n}}\right) \right\} \,\,\mathrm{d}u.
\end{equation*}%
Observe that the assumptions on $g$ imply that $x\mapsto \Phi (x)$ is
differentiable with a bounded derivative on any bounded interval not
including $0$; in particular\thinspace (see (H2a))
\begin{equation}
|\,\Phi (x)-\Phi (y)-\Phi ^{\prime }(y)\cdot (x-y)\,|\leq \Psi (|x-y|)\cdot
|x-y|,\ \ \ x^{2},y^{2}\in (a,b), \label{eqn 8}
\end{equation}%
where $\Psi :\mathbf{R}_{+}\rightarrow \mathbf{R}_{+}$ is continuous,
increasing and $\Psi (0)=0$. \vspace{1mm}
With this notation we shall prove (\ref{eqn 7}) by showing%
\begin{equation*}
\,\sum_{i=1}^{[nt]}\eta ^{\prime }(2)_{i}^{n}\,\overset{P}{\rightarrow }0
\end{equation*}%
and
\begin{equation*}
\ \,\sum_{i=1}^{[nt]}\eta ^{\prime \prime }(2)_{i}^{n}\,\overset{P}{%
\rightarrow }0.\vspace{1mm}
\end{equation*}%
Inserting the description of $(\sigma _{t})$\thinspace (see (H1a)) we may
write
\begin{equation*}
\eta ^{\prime }(2)_{i}^{n}=\eta ^{\prime }(2,1)_{i}^{n}+\eta ^{\prime
}(2,2)_{i}^{n}
\end{equation*}%
where for all $i,n\geq 1$
\begin{equation*}
\eta ^{\prime }(2,1)_{i}^{n}=\sqrt{n}\ \Phi ^{\prime }\left( \sigma _{\frac{%
i-1}{n}}\right) \int_{(i-1)/n}^{i/n}\left( \int_{(i-1)/n}^{u}a_{s}^{\ast }\,%
\mathrm{d}s\right) \,\,\mathrm{d}u
\end{equation*}%
and
\begin{eqnarray*}
\eta ^{\prime }(2,2)_{i}^{n} &=&\displaystyle\sqrt{n}\ \Phi ^{\prime }\left(
\sigma _{\frac{i-1}{n}}\right) \int_{(i-1)/n}^{i/n}\,\left[
\int_{(i-1)/n}^{u}\,\sigma _{s-}^{\ast }\,\mathrm{d}W_{s}+\int_{(i-1)/n}^{u}%
\,v_{s-}^{\ast }\,\mathrm{d}V_{s}\right. \, \\
&&+\displaystyle\left. \int_{E}\phi (s-,x)\,(\mu -\nu )(\mathrm{d}s\,\mathrm{%
d}x)\right] \mathrm{d}u.\vspace{1mm}
\end{eqnarray*}%
By (H2a) and (\ref{eqn 8}) and the uniform boundedness of $(a_{t}^{\ast })$
we have
\begin{equation*}
|\eta ^{\prime }(2,1)_{i}^{n}|\leq C\,\sqrt{n}\,\int_{(i-1)/n}^{i/n}\left\{
u-(i-1)/n\right\} \,\mathrm{d}u\leq C/n^{3/2}
\end{equation*}%
for all $i,n\geq 1$ and thus
\begin{equation*}
\,\sum_{i=1}^{[nt]}\eta ^{\prime }(2,1)_{i}^{n}\,\overset{P}{\rightarrow }0.%
\vspace{1mm}
\end{equation*}%
Since
\begin{equation*}
(W_{t}),\ (V_{t})\ \text{and}\ \left( \int_{0}^{t}\int_{E}\phi (s-,x)(\mu
-\nu )(\mathrm{d}s\,\mathrm{d}x)\right) \vspace{1mm}
\end{equation*}%
are all martingales we have
\begin{equation*}
\mathrm{E}\left[ \eta ^{\prime }(2,2)_{i}^{n}\,|\,\mathcal{F}_{\frac{i-1}{n}}%
\right] =0\ \ \ \text{for all}\quad i,n\geq 1.\vspace{1mm}
\end{equation*}
By Doob's inequality it is therefore feasible to estimate
\begin{equation*}
\sum_{i=1}^{[nt]}\,\mathrm{E}[\,(\eta ^{\prime }(2,2)_{i}^{n})^{2}].\vspace{%
1mm}
\end{equation*}%
Inserting again the description of $(\sigma _{t})$ we find, applying simple
inequalities, in particular Jensen's, that
\begin{eqnarray*}
&&(\eta ^{\prime }(2,2)_{i}^{n})^{2} \\
&\leq &\displaystyle C\,n\,\left( \int_{(i-1)/n}^{i/n}\left\{
\int_{(i-1)/n}^{u}\,\sigma _{s-}^{\ast }\,\mathrm{d}W_{s}\right\} \,\mathrm{d%
}u\right) ^{2}+C\,n\,\left( \,\int_{(i-1)/n}^{i/n}\left\{
\int_{(i-1)/n}^{u}\,v_{s-}^{\ast }\,\mathrm{d}V_{s}\right\} \,\mathrm{d}%
u\right) ^{2} \\
&&\displaystyle+C\,n\,\left( \,\int_{(i-1)/n}^{i/n}\int_{(i-1)/n}^{u}\left\{
\int_{E}\phi (s-,x)\,(\mu -\nu )(\mathrm{d}s\,\mathrm{d}x)\right\} \mathrm{d}%
u\right) ^{2} \\
&\leq &\displaystyle C\,\int_{(i-1)/n}^{i/n}\left(
\,\int_{(i-1)/n}^{u}\,\sigma _{s-}^{\ast }\,\mathrm{d}W_{s}\,\right) ^{2}\,%
\mathrm{d}u+C\,\int_{(i-1)/n}^{i/n}\left( \,\int_{(i-1)/n}^{u}\,v_{s-}^{\ast
}\,\mathrm{d}V_{s}\,\right) ^{2}\,\mathrm{d}u \\
&&\displaystyle+C\,\int_{(i-1)/n}^{i/n}\left(
\,\int_{(i-1)/n}^{u}\int_{E}\phi (s-,x)\,(\mu -\nu )(\mathrm{d}s\,\mathrm{d}%
x)\,\right) ^{2}\,\mathrm{d}u.\vspace{1mm}
\end{eqnarray*}%
The properties of the Wiener integrals and the uniform boundedness of $%
(\sigma _{t}^{\ast })$ and $(v_{t}^{\ast })$ ensure that
\begin{equation*}
\mathrm{E}\left[ \left( \,\int_{(i-1)/n}^{u}\,\sigma _{s-}^{\ast }\,\mathrm{d%
}W_{s}\,\right) ^{2}\,|\,\mathcal{F}_{\frac{i-1}{n}}\right] \leq C\cdot
\left( u-\frac{i-1}{n}\right)
\end{equation*}%
and likewise
\begin{equation*}
\mathrm{E}\left[ \left( \,\int_{(i-1)/n}^{u}\,v_{s-}^{\ast }\,\mathrm{d}%
V_{s}\,\right) ^{2}\,|\,\mathcal{F}_{\frac{i-1}{n}}\right] \leq C\cdot
\left( u-\frac{i-1}{n}\right) \vspace{1mm}
\end{equation*}%
for all $i,n\geq 1$. Likewise for the Poisson part we have
\begin{eqnarray*}
&&\displaystyle\mathrm{E}\left[ \left( \,\int_{(i-1)/n}^{u}\int_{E}\phi
(s-,x)\,(\mu -\nu )(\mathrm{d}s\,\mathrm{d}x)\,\right) ^{2}\,|\,\mathcal{F}_{%
\frac{i-1}{n}}\right] \\
&\leq &\displaystyle C\int_{(i-1)/n}^{u}\int_{E}\mathrm{E}[\phi
^{2}(s,x)\,|\,\mathcal{F}_{\frac{i-1}{n}}]\,F(\mathrm{d}x)\,\mathrm{d}s%
\vspace{1mm}
\end{eqnarray*}%
yielding a similar bound. Putting all this together we have for all $i,n\geq
1$
\begin{eqnarray*}
\mathrm{E}[\,(\eta ^{\prime }(2,2)_{i}^{n})^{2}\,|\,\mathcal{F}_{\frac{i-1}{n%
}}] &\leq &C\,\int_{(i-1)/n}^{i/n}(u-(i-1)/n)\,\mathrm{d}u \\
&\leq &C/n^{2}.
\end{eqnarray*}%
Thus as $n\rightarrow \infty $ so
\begin{equation*}
\sum_{i=1}^{[nt]}\mathrm{E}[\,(\eta ^{\prime }(2,2)_{i}^{n})^{2}]\rightarrow
0.\vspace{1mm}
\end{equation*}%
and since
\begin{equation*}
\mathrm{E}\left[ \eta ^{\prime }(2,2)_{i}^{n}\,|\,\mathcal{F}_{\frac{i-1}{n}}%
\right] =0\ \ \ \ \text{for all}\quad i,n\geq 1\vspace{1mm}
\end{equation*}%
we deduce from Doob's inequality that
\begin{equation*}
\,\sum_{i=1}^{[nt]}\eta ^{\prime }(2,2)_{i}^{n}\,\overset{P}{\rightarrow }0
\end{equation*}%
proving\ altogether
\begin{equation*}
\,\sum_{i=1}^{[nt]}\eta ^{\prime }(2)_{i}^{n}\,\overset{P}{\rightarrow }0.%
\vspace{1mm}
\end{equation*}%
Applying once more (H2a) and (\ref{eqn 8}) we have for every $\epsilon >0$
and every $i,n$ that
\begin{eqnarray*}
|\eta ^{\prime \prime }(2)_{i}^{n}| &\leq &\sqrt{n}\int_{(i-1)/n}^{i/n}\,%
\Psi \left( \left\vert \sigma _{u}-\sigma _{\frac{i-1}{n}}\right\vert
\right) \cdot \left\vert \sigma _{u}-\sigma _{\frac{i-1}{n}}\right\vert \,%
\mathrm{d}u \\
&\leq &\sqrt{n}\,\Psi (\epsilon )\int_{(i-1)/n}^{i/n}\,\left\vert \sigma
_{u}-\sigma _{\frac{i-1}{n}}\right\vert \,\mathrm{d}u+\sqrt{n}\,\Psi (2\sqrt{%
b})/\epsilon \int_{(i-1)/n}^{i/n}\,\left\vert \sigma _{u}-\sigma _{\frac{i-1%
}{n}}\right\vert ^{2}\,\mathrm{d}u.\vspace{1mm}
\end{eqnarray*}%
Thus from (\ref{8}) and its consequence
\begin{equation*}
\mathrm{E}\,\left[ \left\vert \sigma _{u}-\sigma _{\frac{i-1}{n}}\right\vert
\,\right] \leq C/\sqrt{n}
\end{equation*}%
we get
\begin{equation*}
\sum_{i=1}^{[nt]}\mathrm{E}[\,|\eta ^{\prime \prime }(2)_{i}^{n}|\,]\leq
Ct\,\Psi (\epsilon )+\frac{C\,\Psi (b)}{\sqrt{n}\,\epsilon }
\end{equation*}%
for all $n$ and all $\epsilon $. Letting here first $n\rightarrow \infty $
and then $\epsilon \rightarrow 0$ we may conclude that as $n\rightarrow
\infty $
\begin{equation*}
\sum_{i=1}^{[nt]}\mathrm{E}[\,|\eta ^{\prime \prime
}(2)_{i}^{n}|\,]\rightarrow 0\
\end{equation*}%
implying the convergence
\begin{equation*}
\,\sum_{i=1}^{[nt]}\eta (2)_{i}^{n}\,\overset{P}{\rightarrow }0.\vspace{1mm}
\end{equation*}%
Thus ending the proof of (\ref{13b}).
\noindent $\square $
\subsection{Proof of $\protect\eta (1)^{n}\protect\overset{P}{\rightarrow }0$
\label{subsection:proof of 13a}}
Recall we are to show that \textbf{\ }
\begin{equation}
\eta (1)^{n}=\,\sum_{i=1}^{[nt]}\eta (1)_{i}^{n}\,\overset{P}{\rightarrow }0.
\end{equation}%
Let still $t>0$ be fixed. Recall that
\begin{eqnarray*}
\eta (1)_{i}^{n} &=&\frac{1}{\sqrt{n}}\,\left\{ \mathrm{E}\left[ \,g(\sqrt{n}%
\,\triangle _{i}^{n}Y)\,|\,\mathcal{F}_{\frac{i-1}{n}}\right] \,-\rho
_{\sigma _{\frac{i-1}{n}}}(g)\right\} \\
&=&\frac{1}{\sqrt{n}}\,\mathrm{E}\,\left[ g(\sqrt{n}\,\triangle
_{i}^{n}Y)-g(\beta _{i}^{n})\,|\,\mathcal{F}_{\frac{i-1}{n}}\right] .\vspace{%
1mm}
\end{eqnarray*}%
Introduce the notation\thinspace (recall the assumption (K))
\begin{equation*}
A_{i}^{n}=\{\,|\sqrt{n}\,\triangle _{i}^{n}Y-\beta _{i}^{n}|>\,d(\beta
_{i}^{n},B)/2\,\}.\vspace{1mm}
\end{equation*}%
Since $B$ is a Lebesgue null set and $\beta _{i}^{n}$ is absolutely
continuous, $g^{\prime }(\beta _{i}^{n})$ is defined $a.s.$ and, by
assumption, $g$ is differentiable on the interval joining $\triangle
_{i}^{n}Y(\omega )$ and $\beta _{i}^{n}(\omega )$ for all $\omega \in
A_{i}^{n\,c}$. Thus, using the Mean Value Theorem, we may for all $i,n\geq 1$
write
\begin{eqnarray*}
&&g(\sqrt{n}\,\triangle _{i}^{n}Y)-g(\beta _{i}^{n}) \\
&=&\left\{ g(\sqrt{n}\,\triangle _{i}^{n}Y)-g(\beta _{i}^{n})\right\} \cdot
\mathbf{1}_{A_{i}^{n}} \\
&&+g^{\prime }(\beta _{i}^{n})\cdot (\sqrt{n}\,\triangle _{i}^{n}Y-\beta
_{i}^{n})\cdot \mathbf{1}_{A_{i}^{n\,c}} \\
&&+\left\{ g^{\prime }(\alpha _{i}^{n})-g^{\prime }(\beta _{i}^{n})\right\}
\cdot (\sqrt{n}\,\triangle _{i}^{n}Y-\beta _{i}^{n})\cdot \mathbf{1}%
_{A_{i}^{n\,c}} \\
&=&\sqrt{n}\,\left\{ \delta (1)_{i}^{n}+\delta (2)_{i}^{n}+\delta
(3)_{i}^{n}\right\} ,\vspace{1mm}
\end{eqnarray*}%
where $\alpha _{i}^{n}$ are random points lying in between $\sqrt{n}%
\,\triangle _{i}^{n}Y$ and $\beta _{i}^{n}$, i.e.
\begin{equation*}
\sqrt{n}\,\triangle _{i}^{n}Y\wedge \beta _{i}^{n}\leq \alpha _{i}^{n}\leq
\sqrt{n}\,\triangle _{i}^{n}Y\vee \beta _{i}^{n},
\end{equation*}%
and%
\begin{equation*}
\begin{array}{lll}
\delta (1)_{i}^{n} & = & \left[ \,\left\{ g(\sqrt{n}\,\triangle
_{i}^{n}Y)-g(\beta _{i}^{n})\right\} -g^{\prime }(\beta _{i}^{n})\cdot (%
\sqrt{n}\,\triangle _{i}^{n}Y-\beta _{i}^{n})\,\right] \cdot \mathbf{1}%
_{A_{i}^{n}}/\sqrt{n} \\
\delta (2)_{i}^{n} & = & \left\{ g^{\prime }(\alpha _{i}^{n})-g^{\prime
}(\beta _{i}^{n})\right\} \cdot (\sqrt{n}\,\triangle _{i}^{n}Y-\beta
_{i}^{n})\cdot \mathbf{1}_{A_{i}^{n\,c}}/\sqrt{n} \\
\delta (3)_{i}^{n} & = & g^{\prime }(\beta _{i}^{n})\cdot (\sqrt{n}%
\,\triangle _{i}^{n}Y-\beta _{i}^{n})/\sqrt{n}.%
\end{array}%
\end{equation*}%
Thus it suffices to prove
\begin{equation*}
\,\sum_{i=1}^{[nt]}\mathrm{E}\,\left[ \delta (k)_{i}^{n}\,|\,\mathcal{F}_{%
\frac{i-1}{n}}\right] \,\,\overset{P}{\rightarrow }0,\ \ \ k=1,2,3.
\end{equation*}
Consider the case $k=1$. Using (K) and the fact that $\beta _{i}^{n}$ is
absolutely continuous we have a.s.%
\begin{eqnarray*}
&&|g(\sqrt{n}\,\triangle _{i}^{n}Y)-g(\beta _{i}^{n})| \\
&\leq &M(1+|\sqrt{n}\,\triangle _{i}^{n}Y-\beta _{i}^{n}|^{p}+|\beta
_{i}^{n}|^{p})\cdot |\sqrt{n}\,\triangle _{i}^{n}Y-\beta _{i}^{n}| \\
&\leq &(2^{p}+1)M(1+|\sqrt{n}\,\triangle _{i}^{n}Y|^{p}+|\beta
_{i}^{n}|^{p})\cdot |\sqrt{n}\,\triangle _{i}^{n}Y-\beta _{i}^{n}|,
\end{eqnarray*}%
and
\begin{equation*}
|\,g^{\prime }(\beta _{i}^{n})\cdot (\sqrt{n}\,\triangle _{i}^{n}Y-\beta
_{i}^{n})\,|\leq M(1+|\beta _{i}^{n}|^{p})\cdot |\sqrt{n}\,\triangle
_{i}^{n}Y-\beta _{i}^{n}|.\vspace{1mm}
\end{equation*}%
By Cauchy-Schwarz's inequality $\mathrm{E}[\,|\delta (1)_{i}^{n}|\,]$ is
therefore for all $i,n\geq 1$ less than
\begin{equation*}
C\cdot \mathrm{E}[\,1+|\sqrt{n}\,\triangle _{i}^{n}Y|^{3p}+|\beta
_{i}^{n}|^{3p}]^{1/3}\cdot \mathrm{E}[\,(\sqrt{n}\,\triangle _{i}^{n}Y-\beta
_{i}^{n})^{2}/n\,]^{1/2}\cdot P(A_{i}^{n})^{1/6}\vspace{1mm}
\end{equation*}%
implying for fixed $t,$ by means of (\ref{2}), that
\begin{eqnarray*}
\mathrm{E}[\left[ \sum_{i=1}^{[nt]}|\,\delta (1)_{i}^{n}|\right] \, &\leq
&C\cdot \sup_{i\geq 1}P(A_{i}^{n})^{1/6}\,\sum_{i=1}^{[nt]}\mathrm{E}%
[\,(\triangle _{i}^{n}Y-\beta _{i}^{n})^{2}/n\,]^{1/2}\vspace{1mm} \\
&\leq &C\cdot \sup_{i\geq 1}P(A_{i}^{n})^{1/6}\,\sum_{i=1}^{[nt]}1/n \\
&\leq &Ct\cdot \sup_{i\geq 1}P(A_{i}^{n})^{1/6}.\vspace{1mm}
\end{eqnarray*}%
For all $i,n\geq 1$ we have for every $\epsilon >0$
\begin{eqnarray*}
P(A_{i}^{n}) &\leq &P(A_{i}^{n}\cap \{d(\beta _{i}^{n},B)\leq \epsilon
\})+P(A_{i}^{n}\cap \{d(\beta _{i}^{n},B)>\epsilon \})\vspace{1mm} \\
&\leq &P(d(\beta _{i}^{n},B)\leq \epsilon )+P(|\sqrt{n}\,\triangle
_{i}^{n}Y-\beta _{i}^{n}|>\epsilon /2)\vspace{1mm} \\
&\leq &P(d(\beta _{i}^{n},B)\leq \epsilon )+\frac{4}{\epsilon ^{2}}\cdot
\mathrm{E}[\,(\sqrt{n}\,\triangle _{i}^{n}Y-\beta _{i}^{n})^{2}]\vspace{1mm}
\\
&\leq &P(d(\beta _{i}^{n},B)\leq \epsilon )+\frac{C}{n\,\epsilon ^{2}}.
\end{eqnarray*}%
But (H2a) implies that the densities of $\beta _{i}^{n}$ are pointwise
dominated by a Lebesgue integrable function $h_{a,b}$ providing, for all $%
i,n\geq 1$, the estimate
\begin{eqnarray}
P(A_{i}^{n}) &\leq &\int_{\{x\,|\,d(x,B)\leq \epsilon \}}h_{a,b}\,\mathrm{d}%
\lambda _{1}+\frac{C}{n\,\epsilon ^{2}} \label{eqn 10} \\
&=&\alpha _{\epsilon }+\frac{C}{n\,\epsilon ^{2}}.\vspace{1mm} \notag
\end{eqnarray}%
Observe $\lim_{\epsilon \rightarrow 0}\alpha _{\epsilon }=0$. Taking now in (%
\ref{eqn 10}) $\sup $ over $i$ and then letting first $n\rightarrow \infty $
and then $\epsilon \downarrow 0$ we get
\begin{equation*}
\lim_{n}\,\sup_{i\geq 1}\,P(A_{i}^{n})=0
\end{equation*}%
proving that
\begin{equation*}
\mathrm{E}\left[ \,\sum_{i=1}^{[nt]}|\,\delta (1)_{i}^{n}|\right]
\,\rightarrow 0
\end{equation*}%
and\ thus
\begin{equation*}
\,\sum_{i=1}^{[nt]}\mathrm{E}\left[ \,\delta (1)_{i}^{n}\,|\,\mathcal{F}_{%
\frac{i-1}{n}}\right] \,\overset{P}{\rightarrow }0.\vspace{1mm}
\end{equation*}
Consider next the case $k=2$. As assumed in (K), $g$ is continuously
differentiable outside of $B$. Thus for each $A>1$ and $\epsilon >0$ there
exists a function $G_{A,\,\epsilon }:(0,1)\rightarrow \mathbf{R}_{+}$ such
that for given $0<\epsilon ^{\prime }<\epsilon /2$
\begin{equation*}
\left\vert g^{\prime }(x+y)-g^{\prime }(x)\right\vert \leq G_{A,\,\epsilon
}(\epsilon ^{\prime })\ \ \text{for all}\ |x|\leq A,\ |y|\leq \epsilon
^{\prime }<\epsilon <d(x,B).\vspace{1mm}
\end{equation*}%
Observe that $\lim_{\epsilon ^{\prime }\downarrow 0}G_{A,\,\epsilon
}(\epsilon ^{\prime })=0$ for all $A$ and $\epsilon $.\vspace{1mm} Fix $A>1$
and $\epsilon \in (0,1)$. For all $i,n\geq 1$ we have
\begin{eqnarray*}
&&|g^{\prime }(\alpha _{i}^{n})-g^{\prime }(\beta _{i}^{n})|\cdot \mathbf{1}%
_{A_{i}^{n\,c}} \\
&=&\displaystyle|g^{\prime }(\alpha _{i}^{n})-g^{\prime }(\beta
_{i}^{n})|\cdot \mathbf{1}_{A_{i}^{n\,c}}\,(\mathbf{1}_{\{|\alpha
_{i}^{n}|+|\beta _{i}^{n}|>A\}}+\mathbf{1}_{\{|\alpha _{i}^{n}|+|\beta
_{i}^{n}|\leq A\}})\vspace{1mm} \\
&\leq &\displaystyle|g^{\prime }(\alpha _{i}^{n})-g^{\prime }(\beta
_{i}^{n})|\cdot \frac{|\alpha _{i}^{n}|+|\beta _{i}^{n}|}{A}+|g^{\prime
}(\alpha _{i}^{n})-g^{\prime }(\beta _{i}^{n})|\cdot \mathbf{1}%
_{A_{i}^{n\,c}\,\cap \,\{|\alpha _{i}^{n}|+|\beta _{i}^{n}|\leq A\}} \\
&\leq &\displaystyle\frac{C}{A}\cdot (1+|\alpha _{i}^{n}|^{p}+|\beta
_{i}^{n}|^{p})^{2}+|g^{\prime }(\alpha _{i}^{n})-g^{\prime }(\beta
_{i}^{n})|\cdot \mathbf{1}_{A_{i}^{n\,c}\,\cap \,\{|\alpha _{i}^{n}|+|\beta
_{i}^{n}|\leq A\}} \\
&\leq &\displaystyle\frac{C}{A}\cdot (1+|\sqrt{n}\,\triangle
_{i}^{n}Y|^{2p}+|\beta _{i}^{n}|^{2p})+|g^{\prime }(\alpha
_{i}^{n})-g^{\prime }(\beta _{i}^{n})|\cdot \mathbf{1}_{A_{i}^{n\,c}\,\cap
\,\{|\alpha _{i}^{n}|+|\beta _{i}^{n}|\leq A\}}.\vspace{1mm}
\end{eqnarray*}%
Now writing
\begin{eqnarray*}
1 &=&\mathbf{1}_{\{d(\beta _{i}^{n},B)\leq \epsilon \}}+\mathbf{1}%
_{\{d(\beta _{i}^{n},B)>\epsilon \}}\vspace{1mm} \\
&=&\mathbf{1}_{\{d(\beta _{i}^{n},B)\leq \epsilon \}} \\
&&+\mathbf{1}_{\{d(\beta _{i}^{n},B)>\epsilon \}\,\cap \,\{|\alpha
_{i}^{n}-\beta _{i}^{n}|\leq \epsilon ^{\prime }\}} \\
&&+\mathbf{1}_{\{d(\beta _{i}^{n},B)>\epsilon \}\,\cap \,\{|\alpha
_{i}^{n}-\beta _{i}^{n}|>\epsilon ^{\prime }\}}
\end{eqnarray*}%
for all $0<\epsilon ^{\prime }<\epsilon /2$ we have
\begin{eqnarray*}
\mathbf{1}_{A_{i}^{n\,c}\,\cap \,\{|\alpha _{i}^{n}|+|\beta _{i}^{n}|\leq
A\}} &\leq &\mathbf{1}_{\{d(\beta _{i}^{n},B)\leq \epsilon \}\,\cap
\,A_{i}^{n\,c}\,\cap \,\{|\alpha _{i}^{n}|+|\beta _{i}^{n}|\leq A\}} \\
&&+\mathbf{1}_{A_{i}^{n\,c}\,\cap \,\{|\alpha _{i}^{n}|+|\beta _{i}^{n}|\leq
A\}\,\cap \,\{d(\beta _{i}^{n},B)>\epsilon \}\,\cap \,\{|\alpha
_{i}^{n}-\beta _{i}^{n}|\leq \epsilon ^{\prime }\}} \\
&&+\mathbf{1}_{A_{i}^{n\,c}\,\cap \,\{|\alpha _{i}^{n}|+|\beta _{i}^{n}|\leq
A\}\,\cap \,\{d(\beta _{i}^{n},B)>\epsilon \}}\cdot \frac{|\alpha
_{i}^{n}-\beta _{i}^{n}|}{\epsilon ^{\prime }}.
\end{eqnarray*}%
Combining this with the fact that
\begin{eqnarray*}
|g^{\prime }(\alpha _{i}^{n})-g^{\prime }(\beta _{i}^{n})| &\leq
&C(1+|\alpha _{i}^{n}|^{p}+|\beta _{i}^{n}|^{p}) \\
&\leq &CA^{p}
\end{eqnarray*}%
on $A_{i}^{n\,c}\,\cap \,\{|\alpha _{i}^{n}|+|\beta _{i}^{n}|\leq A\}$ we
obtain that
\begin{eqnarray*}
&&|g^{\prime }(\alpha _{i}^{n})-g^{\prime }(\beta _{i}^{n})|\cdot \mathbf{1}%
_{A_{i}^{n\,c}\,\cap \,\{|\alpha _{i}^{n}|+|\beta _{i}^{n}|\leq A\}} \\
&\leq &CA^{p}\cdot \left( \,\mathbf{1}_{\{d(\beta _{i}^{n},B)\leq \epsilon
\}}+\frac{|\alpha _{i}^{n}-\beta _{i}^{n}|}{\epsilon ^{\prime }}\right)
\,+G_{A,\,\epsilon }(\epsilon ^{\prime })\vspace{1mm} \\
&\leq &CA^{p}\cdot (\,\mathbf{1}_{\{d(\beta _{i}^{n},B)\leq \epsilon \}}+%
\frac{|\sqrt{n}\,\triangle _{i}^{n}Y-\beta _{i}^{n}|}{\epsilon ^{\prime }}%
\,)+G_{A,\,\epsilon }(\epsilon ^{\prime }).\vspace{1mm}
\end{eqnarray*}
Putting this together means that
\begin{eqnarray*}
\sqrt{n}\,|\delta (2)_{i}^{n}| &=&|g^{\prime }(\alpha _{i}^{n})-g^{\prime
}(\beta _{i}^{n})|\cdot |\sqrt{n}\,\triangle _{i}^{n}Y-\beta _{i}^{n}|\cdot
\mathbf{1}_{A_{i}^{n\,c}} \\
&\leq &\left\{ \frac{C}{A}\cdot (1+|\sqrt{n}\,\triangle
_{i}^{n}Y|^{2p}+|\beta _{i}^{n}|^{2p})+G_{A,\,\epsilon }(\epsilon ^{\prime
})\right\} \cdot |\sqrt{n}\,\triangle _{i}^{n}Y-\beta _{i}^{n}|\vspace{1mm}
\\
&&+\,CA^{p}\cdot \left( \mathbf{1}_{\{d(\beta _{i}^{n},B)\leq \epsilon
\}}\cdot |\sqrt{n}\,\triangle _{i}^{n}Y-\beta _{i}^{n}|+\frac{|\sqrt{n}%
\,\triangle _{i}^{n}Y-\beta _{i}^{n}|^{2}}{\epsilon ^{\prime }}\right) .%
\vspace{1mm}
\end{eqnarray*}%
Exploiting here the inequalities (\ref{2}) and (\ref{3}) we obtain, for all $%
A>1$ and $0<2\epsilon ^{\prime }<\epsilon <1$ and all $i,n\geq 1$, using H%
\"{o}lder's inequality, the following estimate
\begin{equation*}
\mathrm{E}[\,|\delta (2)_{i}^{n}|\,]\leq C\left( \frac{1}{A\,n}+\frac{%
G_{A,\,\epsilon }(\epsilon ^{\prime })}{n}+\frac{A^{p}\,\sqrt{\alpha
_{\epsilon }}}{n}+\frac{A^{p}}{\epsilon ^{\prime }\,n^{3/2}}\right) \vspace{%
1mm}
\end{equation*}%
implying for all $n\geq 1$ and $t\geq 0$ that
\begin{equation*}
\sum_{i=1}^{[nt]}\mathrm{E}[\,|\delta (2)_{i}^{n}|\,]\leq Ct\left( \frac{1}{A%
}+G_{A,\,\epsilon }(\epsilon ^{\prime })+A^{p}\,\sqrt{\alpha _{\epsilon }}+%
\frac{A^{p}}{\epsilon ^{\prime }\,n^{1/2}}\right) .\vspace{1mm}
\end{equation*}%
Choosing in this estimate first $A$ sufficiently big, then $\epsilon $
small\thinspace (recall that $\lim_{\epsilon \rightarrow 0}\alpha _{\epsilon
}=0$\thinspace ) and finally $\epsilon ^{\prime }$ small, exploiting that $%
\lim_{\epsilon ^{\prime }\downarrow 0}G_{A,\,\epsilon }(\epsilon ^{\prime
})=0$ for all $A$ and $\epsilon $, we may conclude that
\begin{equation*}
\lim_{n}\,\sum_{i=1}^{[nt]}\mathrm{E}\left[ \,|\delta (2)_{i}^{n}|\,\right]
=0
\end{equation*}%
and thus
\begin{equation*}
\sum_{i=1}^{[nt]}\mathrm{E}\left[ \,\delta (2)_{i}^{n}\,|\,\mathcal{F}_{%
\frac{i-1}{n}}\right] \,\overset{P}{\rightarrow }0.
\end{equation*}
So what remains to be proved is the convergence
\begin{equation*}
\,\sum_{i=1}^{[nt]}\mathrm{E}\,\left[ \delta (3)_{i}^{n}\,|\,\mathcal{F}_{%
\frac{i-1}{n}}\right] \,\,\overset{P}{\rightarrow }0.
\end{equation*}%
As introduced in (\ref{17})
\begin{equation*}
\sqrt{n}\,\triangle _{i}^{n}Y-\beta _{i}^{n}=\sum_{j=1}^{5}\xi
(j)_{i}^{n}=\psi (1)_{i}^{n}+\psi (2)_{i}^{n}
\end{equation*}%
for all $i,n\geq 1$ where
\begin{equation*}
\psi (1)_{i}^{n}=\xi (1)_{i}^{n}+\xi (3)_{i}^{n}+\xi (4)_{i}^{n},
\end{equation*}%
\begin{equation*}
\psi (2)_{i}^{n}=\xi (2)_{i}^{n}+\xi (5)_{i}^{n},
\end{equation*}%
and as
\begin{equation*}
\delta (3)_{i}^{n}=g^{\prime }(\beta _{i}^{n})\cdot (\psi (1)_{i}^{n}+\psi
(2)_{i}^{n})/\sqrt{n}\vspace{1mm}
\end{equation*}%
it suffices to prove
\begin{equation*}
\left( \,\sum_{i=1}^{[nt]}\mathrm{E}\left[ \,g^{\prime }(\beta
_{i}^{n})\cdot \psi (k)_{i}^{n}\,|\,\mathcal{F}_{\frac{i-1}{n}}\right] \,\,/%
\sqrt{n}\,\right) \overset{P}{\rightarrow }0,\ \ \ k=1,2.\vspace{1mm}
\end{equation*}
The case $k=1$ is handled by proving
\begin{equation}
\frac{1}{\sqrt{n}}\,\sum_{i=1}^{[nt]}\mathrm{E}[\,|g^{\prime }(\beta
_{i}^{n})\cdot \xi (j)_{i}^{n}|\,]\rightarrow 0,\ \ \ j=1,3,4.\vspace{1mm}
\label{eqn 11}
\end{equation}%
Using Jensen's inequality it is easily seen that for $j=1,3,4$
\begin{equation*}
\frac{1}{\sqrt{n}}\,\sum_{i=1}^{[nt]}\mathrm{E}[\,|g^{\prime }(\beta
_{i}^{n})\cdot \xi (j)_{i}^{n}|\,]\leq C\,t\cdot \sqrt{\frac{1}{n}%
\,\sum_{i=1}^{[nt]}\mathrm{E}[\,g^{\prime }(\beta _{i}^{n})^{2}]}\,\cdot \,%
\sqrt{\sum_{i=1}^{[nt]}\mathrm{E}[\,(\xi (j)_{i}^{n})^{2}]}
\end{equation*}%
and so using (\ref{12})
\begin{equation*}
\frac{1}{\sqrt{n}}\,\sum_{i=1}^{[nt]}\mathrm{E}[\,|g^{\prime }(\beta
_{i}^{n})\cdot \xi (j)_{i}^{n}|\,]\leq C\,t\cdot \,\sqrt{\sum_{i=1}^{[nt]}%
\mathrm{E}[\,(\xi (j)_{i}^{n})^{2}]}
\end{equation*}%
since almost surely
\begin{equation*}
|g^{\prime }(\beta _{i}^{n})|\leq C\,(1+|\beta _{i}^{n}|^{p})
\end{equation*}%
for all $i,n\geq 1$. From here, (\ref{eqn 11}) is an immediate consequence
of Lemmas \ref{lemma 1st}-\ref{lemma 5th}.\vspace{1mm}
The remaining case $k=2$ is different. The definition of $\psi (2)_{i}^{n}$
implies, using basic stochastic calculus, that $\psi (2)_{i}^{n}/\sqrt{n}$,
for all $i,n\geq 1$, may be written as
\begin{eqnarray*}
&&\int_{(i-1)/n}^{i/n}\left\{ \sigma _{\frac{i-1}{n}}^{\prime }\,\left(
W_{u}-W_{\frac{i-1}{n}}\right) +M(n,i)_{u}\right\} \,\mathrm{d}W_{u} \\
&=&\sigma _{\frac{i-1}{n}}^{\prime }\,\int_{(i-1)/n}^{i/n}\left( W_{u}-W_{%
\frac{i-1}{n}}\right) \,\mathrm{d}W_{u} \\
&&+\triangle _{i}^{n}M(n,i)\cdot \triangle _{i}^{n}W \\
&&+\int_{(i-1)/n}^{i/n}\left( W_{u}-W_{\frac{i-1}{n}}\right) \,\mathrm{d}%
M(n,i)_{u},
\end{eqnarray*}%
where $(M(n,i)_{t})$ is the martingale defined by $M(n,i)_{t}\equiv 0$ for $%
t\leq (i-1)/n$ and
\begin{equation*}
M(n,i)_{t}=v_{\frac{i-1}{n}}^{\ast }\,\left( V_{t}-V_{\frac{i-1}{n}}\right)
+\int_{(i-1)/n}^{t}\int_{E_{n}}\phi \left( \frac{i-1}{n},x\right) (\mu -\nu
)(\mathrm{d}s\,\mathrm{d}x)\vspace{1mm}
\end{equation*}%
otherwise. Thus for fixed $i,n\geq 1$
\begin{equation*}
\mathrm{E}\left[ \,g^{\prime }(\beta _{i}^{n})\cdot \psi (2)_{i}^{n}\,|\,%
\mathcal{F}_{\frac{i-1}{n}}\right] \,\,/\sqrt{n}
\end{equation*}%
is a linear combination of the following three terms
\begin{equation*}
\mathrm{E}\left[ g^{\prime }(\beta _{i}^{n})\cdot \sigma _{\frac{i-1}{n}%
}^{\prime }\,\int_{(i-1)/n}^{i/n}\left( W_{u}-W_{\frac{i-1}{n}}\right) \,%
\mathrm{d}W_{u}\,|\,\mathcal{F}_{\frac{i-1}{n}}\right] \,,
\end{equation*}%
\begin{equation*}
\mathrm{E}\left[ g^{\prime }(\beta _{i}^{n})\cdot \triangle
_{i}^{n}M(n,i)\cdot \triangle _{i}^{n}W\,|\,\mathcal{F}_{\frac{i-1}{n}}%
\right] \,
\end{equation*}%
and%
\begin{equation*}
\mathrm{E}[\,g^{\prime }(\beta _{i}^{n})\cdot \int_{(i-1)/n}^{i/n}W_{u}\,%
\mathrm{d}M(n,i)_{u}\,|\,\mathcal{F}_{\frac{i-1}{n}}\,].
\end{equation*}%
But these three terms are all equal to $0$ as seen by the following
arguments.\vspace{1mm}
The conditional distribution of
\begin{equation*}
\left( W_{t}-W_{\frac{i-1}{n}}\right) _{t\geq \frac{i-1}{n}}|\mathcal{F}_{%
\frac{i-1}{n}}
\end{equation*}%
is clearly not affected by a change of sign. Thus since $g$ being assumed
even and $g^{\prime }$ therefore odd we have
\begin{equation*}
\mathrm{E}\left[ \,g^{\prime }(\beta _{i}^{n})\,\int_{(i-1)/n}^{i/n}\left(
W_{u}-W_{\frac{i-1}{n}}\right) \,\mathrm{d}W_{u}\,|\,\mathcal{F}_{\frac{i-1}{%
n}}\,\right] =0
\end{equation*}%
implying the vanishing of the first term. \vspace{1mm}
Secondly, by assumption, $\left( W_{t}-W_{\frac{i-1}{n}}\right) _{t\geq
\frac{i-1}{n}}$ and $(M(n,i)_{t})_{t\geq \frac{i-1}{n}}$ are independent
given $\mathcal{F}_{\frac{i-1}{n}}$. Therefore, denoting by $\mathcal{F}%
_{i,n}^{\,0}$ the $\sigma $-field generated by
\begin{equation*}
\left( W_{t}-W_{\frac{i-1}{n}}\right) _{\frac{i-1}{n}\leq t\leq i/n}\ \ \
\text{and}\ \ \ \mathcal{F}_{\frac{i-1}{n}},
\end{equation*}%
the martingale property of $(M(n,i)_{t})$ ensures that
\begin{equation*}
\mathrm{E}[\,g^{\prime }(\beta _{i}^{n})\cdot \triangle _{i}^{n}M(n,i)\cdot
\triangle _{i}^{n}W\,|\,\mathcal{F}_{i,n}^{\,0}\,]=0\
\end{equation*}%
and%
\begin{equation*}
\mathrm{E}[\left[ g^{\prime }(\beta _{i}^{n})\cdot
\int_{(i-1)/n}^{i/n}W_{u}\,\mathrm{d}M(n,i)_{u}\,|\,\mathcal{F}_{i,n}^{\,0}%
\right] \,=0.
\end{equation*}%
Using this the vanishing of
\begin{equation*}
\mathrm{E}\left[ \,g^{\prime }(\beta _{i}^{n})\cdot \triangle
_{i}^{n}M(n,i)\cdot \triangle _{i}^{n}W\,|\,\mathcal{F}_{\frac{i-1}{n}}%
\right]
\end{equation*}%
and%
\begin{equation*}
\mathrm{E}\left[ \,g^{\prime }(\beta _{i}^{n})\cdot
\int_{(i-1)/n}^{i/n}W_{u}\,\mathrm{d}M(n,i)_{u}\,|\,\mathcal{F}_{\frac{i-1}{n%
}}\right] \,\vspace{1mm}
\end{equation*}%
is easily obtained by successive conditioning.\vspace{1mm}
The proof of (\ref{13a}) is hereby completed.
\noindent $\square $
\bibliographystyle{chicago}
|
{
"timestamp": "2005-03-30T16:28:41",
"yymm": "0503",
"arxiv_id": "math/0503711",
"language": "en",
"url": "https://arxiv.org/abs/math/0503711"
}
|
\section*{Introduction}
In recent years, the study of singular Yang-Mills fields has been an
extremely active area of research. Considering $\mbox{\rm SU}(2)$ instantons on
four manifolds with codimension two singularities, it was found that
these connections can admit non-trivial holonomy around arbitrarily
small circles linking the embedded singular surface. An analytical theory
for such instantons with holonomy singularity has been developed in
\cite{KM,KM2}. Although we currently have an understanding
of the moduli space for such singular instantons, the literature in this
field has a conspicuous dearth of explicit examples. A singular solution on
$T^{2}\times D^{2}$ is given in the appendix to \cite{KM}, although of
greater interest are solutions on the standard model $S^{4}\setminus S^{2}$.
The first example of a singular instanton was discovered by P.~Forg\'{a}cs,
Z.~Horv\'{a}th, and L.~Palla. In their 1981 paper \cite{FHP1}, they
describe a self-dual Yang-Mills field on $S^{4}\setminus S^{2}$ with
the fractional Chern class $c_{2} = 3/2$. At first their result was not
readily accepted, due to the resistance to the new idea of a
fractional charge.
They later published a second paper
\cite{FHP2} defending their result, and since then over a decade of successful
research into the field has eliminated any initial skepticism.
Nevertheless, their construction itself remains poorly understood.
The goal of this article is to elucidate and extend the work of
Forg\'{a}cs {\em et al.\/}, writing their construction using simpler
notation, explaining the motivation behind their
formulae,
and generalizing to obtain a family of singular instantons with varying
holonomy parameter.
To this work we shall contribute a
mathematical perspective,
exchanging indices and Pauli matrices for more invariant complex,
quaternionic, and spinor notation, and offering geometric interpretations
for the equations involved.
Section 1 is devoted to the construction of instantons on $S^{4}$ employing
the ansatz proposed by the physicists Corrigan, Fairlie, and Wilczek in
1976 and described in \cite{CF, JNR}. Starting with a positive real-valued
function $\r$ on $\mathbb{R}^{4}$, known as the {\em super-potential}, we
consider the Yang-Mills connection
\begin{equation} \label{eq:ansatz}
A = \sum_{\mu,\nu} i\bar{\sigma}_{\mu\nu}\,\partial^{\nu}\log\r\,dx^{\mu}.
\end{equation}
Here the anti-symmetric matrix $\bar{\sigma}_{\mu\nu}$ is defined as%
\SSfootnote{We use the Greek indices $\mu,\nu$ when indexing over 4-space,
while the Roman indices $i,j$ range over 1,2,3.}
\begin{displaymath}
\bar{\sigma}_{\mu\nu} = \left\{ \begin{array}{l}
\bar{\sigma_{ij}} = \frac{1}{4i}
\left[ \sigma_{i},\sigma_j \right] \\
\bar{\sigma_{i0}} =
-\frac{1}{2}\sigma_{i}
\end{array}
\right.
\end{displaymath}
where the $\sigma_{i}$ are the standard Pauli matrices generating the Lie
algebra $\mathfrak{su}(2)$. For such a connection, the self-duality equation
$\ast F_{A} = F_{A}$ is equivalent to the condition $\Delta\rho = 0$.
By reversing orientation, this construction can also be used to generate
anti-self-dual connections from a harmonic super-potential.
This harmonic function ansatz was used by 't Hooft to construct a class
of instantons with $5n$ parameters, corresponding to the centers and
scales of $n$ superimposed basic instantons. Since then, this ansatz
has been shown to be the simplest case of a more general
algebraic-geometric construction involving twistors discussed in \cite{AW}.
More recently, both constructions have been eclipsed by the ADHM
description of instantons given in \cite{ADHM}, which provides a complete construction for all ASD connections on $S^{4}$ up to gauge equivalence.
In Section 1 we recast the harmonic function ansatz in terms of
quaternionic notation. Not only does this greatly simplify the
required calculations, but also it better exhibits the underlying
structure. We then show how these connections arise naturally via
conformal transformations.
In Section 2 we introduce an $\mbox{\rm SO}(3)$-action on $S^{4}$. Taking
advantage of the conformal equivalence
$S^{4}\setminus S^{1} \cong \mathcal{H}^{2}\times S^{2}$, we show that the
symmetric SD and ASD equations over $S^{4}$ are equivalent to the vortex
and anti-vortex equations over hyperbolic space $\mathcal{H}^{2}$. This
technique is known as dimensional reduction. The harmonic function
ansatz for instantons then reduces to a similar ansatz for hyperbolic
vortices, which we also derive using conformal transformations of
hyperbolic space. After computing the vortex equivalents of the
symmetric 't Hooft instantons, we use an equivariant version
of the ADHM construction to provide a classification for all hyperbolic
vortices. Examining gauge transformations, we obtain the surprising
result that if two hyperbolic vortices constructed by the harmonic
function ansatz are gauge equivalent, then they are both completely
determined by the gauge transformation between them.
We return to our primary task of constructing singular instantons in
Section 3. Restricting our attention to $\mbox{\rm SO}(3)$-invariant connections
on $S^{4}$, we can work instead with hyperbolic vortices. Using the unit
disc model of $\mathcal{H}^{2}$, singular instantons correspond to vortices
with a holonomy singularity at the origin. We then proceed to construct
solutions on the cut disc using the harmonic function ansatz, patching them
together with gauge transformations to form global solutions on the
punctured disc. In \S\ref{fhp} we essentially rewrite the paper
\cite{FHP1} in this context, and in the following section we construct
our desired family of singular vortices.
\section{The Harmonic Function ansatz}
\label{instanton-ansatz}
\subsection{Quaternionic Notation}
\label{quaternionic-notation}
For the duration of this section, we adopt the quaternionic notation as used
in \cite{A}. Writing $x\in\H$ in the form $x = x^{0}+ix^{1}+jx^{2}+kx^{3}$,
its conjugate is $\bar{x} = x^{0}-ix^{1}-jx^{2}-kx^{3}$, and the
corresponding differentials are
\begin{align*}
dx = dx^{0} + i\,dx^{1} + j\,dx^{2} + k\,dx^{3} \qquad
d\bar{x} = dx^{0} - i\,dx^{1} - j\,dx^{2} - k\,dx^{3}.
\end{align*}
By analogy with the complex case, we define the partial derivatives
\begin{align*}
\del{x} & = \frac{1}{2} \left(
\del{x^{0}} - i\del{x^{1}} - j\del{x^{2}} - k\del{x^{3}}
\right) \\
\del{\bar{x}} & = \frac{1}{2} \left(
\del{x^{0}} + i\del{x^{1}} + j\del{x^{2}} + k\del{x^{3}}
\right).
\end{align*}
In this notation the Laplacian takes the form
\begin{displaymath}
\Delta = -4\,\del{x}\del{\bar{x}} = -4\,\del{\bar{x}}\del{x}.
\end{displaymath}
As expected, the exterior derivative $d$ may be written as the sum of
$\partial$ and $\bar{\partial}$ components, although there are now two distinct
splittings
\begin{displaymath}
d = dx\,\del{x} + \del{\bar{x}}\,d\bar{x}
= \del{x}\,dx + d\bar{x}\,\del{\bar{x}}
\end{displaymath}
due to the non-abelian nature of the operators involved. Expanding the
2-form $dx\wedge d\bar{x}$ in terms of coordinates as
\begin{equation}\begin{split}
dx\wedge d\bar{x} & = -2 \left[\,
i\left(dx^{0}\wedge dx^{1} + dx^{2}\wedge dx^{3}\right)\,+\,
j\left(dx^{0}\wedge dx^{2} + dx^{3}\wedge dx^{1}\right)
\right. \nonumber \\
& \qquad \left.\mbox{\qquad}\,+\,
k\left(dx^{0}\wedge dx^{3} + dx^{1}\wedge dx^{2}\right)
\,\right], \label{eq:dxdxbar}
\end{split}\end{equation}
we see that $dx\wedge d\bar{x}$ is self-dual and likewise that
$d\bar{x}\wedge dx$ is anti-self-dual.
Rewriting the connection (\ref{eq:ansatz}) in terms of this new
quaternionic notation, the harmonic function ansatz now takes the
surprisingly familiar form
\begin{theorem} \label{theorem-1}
Given a positive real-valued super-potential $\r$ on $\mathbb{R}^{4}$,
the Yang-Mills connection $A^{+}$ defined by
\begin{equation}
\label{eq:anti-self-dual}
A^{+}= - \Im\left(\del{\bar{x}}\log \r\,d\bar{x} \right)
= - \frac{1}{2} \left( \del{\bar{x}}\log \r\,d\bar{x} -
dx\,\del{x}\log\r \right)
\end{equation}
is anti-self-dual and the connection $A^{-}$ defined by the conjugate
expression
\begin{equation}
\label{eq:self-dual}
A^{-} = - \Im \left(\del{x}\log \r\,dx \right)
= - \frac{1}{2} \left( \del{x}\log \r\,dx -
d\bar{x}\,\del{\bar{x}}\log\r \right)
\end{equation}
is self-dual if and only if the super-potential $\r$ is harmonic.
\end{theorem}
Before proceeding with the proof of this theorem the reader may want
to verify that (\ref{eq:ansatz}) and (\ref{eq:self-dual}) both yield the
same self-dual connection. Expanding (\ref{eq:self-dual}) using
coordinates, we obtain the expression
\begin{equation*}\begin{split}
A^{-} & = \frac{1}{2} \left(\,
(+i\partial_{1}+j\partial_{2}+k\partial_{3})\,dx^{0} +
(-i\partial_{0}-k\partial_{2}+j\partial_{3})\,dx^{1}
\right. \nonumber\\
& \qquad \left. \,\mbox{}+
(-j\partial_{0}+k\partial_{1}-i\partial_{3})\,dx^{2} +
(-k\partial_{0}-j\partial_{1}+i\partial_{2})\,dx^{3}
\,\right),
\end{split}\end{equation*}
writing $\partial_i$ as an abbreviation for $\partial_i\log\r$.
\begin{proof}[Proof of Theorem 1]
For the purposes of this proof, we restrict
our attention to the potentially self-dual connection $A^{-}$ given in
(\ref{eq:self-dual}), calling it $A$. Explicitly computing the two
components of the curvature $F_{A} = dA+ A\wedge A$, we obtain
\begin{align*}
A\wedge A & = -\frac{1}{2} \left(
\del{x}\log\r\,dx \wedge d\bar{x}\,\del{\bar{x}}\log\r +
d\bar{x}\,\del{\bar{x}}\log\r \wedge \del{x}\log\r\,dx
\right) \\
dA & = -\frac{1}{2} \left( \del{x}\,dx + d\bar{x}\,\del{\bar{x}} \right)
\left(
\del{x}\log \r\,dx - d\bar{x}\,\del{\bar{x}}\log\r
\right) \\
& = -\frac{1}{2} \left(
- \del{x}\,dx\wedge d\bar{x}\,\del{\bar{x}}\log\r +
d\bar{x}\,\del{\bar{x}}\wedge\del{x}\log\r\,dx
\right) .
\end{align*}
Recalling that the 2-forms $dx\wedge d\bar{x}$ and $d\bar{x}\wedge dx$ are
self-dual and anti-self-dual respectively, the curvature of $A$ then splits
as $F_{A} = F_{A}^{+} + F_{A}^{-}$ with
\begin{align*}
F_{A}^{+} & = \frac{1}{2} \left(
\del{x}\,dx\wedge d\bar{x}\,\del{\bar{x}}\log\r\,-\,
\del{x}\log\r\,dx \wedge d\bar{x}\,\del{\bar{x}}\log\r
\right) \\
F_{A}^{-} & = -\frac{1}{2} \left(
\del{\bar{x}}\del{x}\log\r\,+\,
\del{\bar{x}}\log\r\,\del{x}\log\r
\right) d\bar{x}\wedge dx.
\end{align*}
Noting the identity
\begin{displaymath}
\del{\bar{x}}\del{x}\log\r + \del{\bar{x}}\log\r\,\del{x}\log\r
= \frac{1}{\r}\,\del{\bar{x}}\del{x}\r = -\frac{1}{4\r}\Delta\r,
\end{displaymath}
we see that the self-duality equation $F_{A}^{-} = 0$ is equivalent to
the condition $\Delta\r = 0$ that the super-potential $\r$ be harmonic.
\end{proof}
We now calculate the curvature density $|F_{A}|^{2}$ of the self-dual
connection (\ref{eq:self-dual}), from which we can construct the Yang-Mills
functional $\|F_{A}\|^{2}$ and the Chern class $c_{2}(A)$. Using the
decomposition $F_{A} = F_{A}^{+} + F_{A}^{-}$ given above, we first
compute the anti-self-dual component $|F_{A}^{-}|^{2}$. From equation
(\ref{eq:dxdxbar}) we observe that
$(d\bar{x}\wedge dx)\wedge-(\overline{d\bar{x}\wedge dx}) = 24\,d\mu$,
where $d\mu$ is the volume form, and we immediately obtain
\begin{equation*}
|F_{A}^{-}|^{2} = \frac{3}{8}\left(\frac{1}{\r}\Delta\r\right)^{2},
\end{equation*}
which clearly vanishes if the super-potential $\r$ is harmonic.
\newcommand{\log\r}{\log\r}
\newcommand{\partial_{i}}{\partial_{i}}
\renewcommand{\dj}{\partial_{j}}
On the other hand, the self-dual component $|F_{A}^{-}|^{2}$ of the
curvature density is significantly more difficult to compute. Again
using the expansion (\ref{eq:dxdxbar}) for $dx\wedge d\bar{x}$, we have
\begin{equation*}\begin{split}
|F_{A}^{+}|^{2} & = 2 \left(
\left|
\del{x}\,i\,\del{\bar{x}}\log\r - \left(\del{x}\log\r\right)
i\left(\del{\bar{x}}\log\r\right)
\right|^{2}\right. \\
& \qquad \mbox{}+\left.\left|
\del{x}\,j\,\del{\bar{x}}\log\r - \left(\del{x}\log\r\right)
j\left(\del{\bar{x}}\log\r\right)
\right|^{2}\right. \\
& \qquad \mbox{}+\left.\left|
\del{x}\,k\,\del{\bar{x}}\log\r - \left(\del{x}\log\r\right)
k\left(\del{\bar{x}}\log\r\right)
\right|^{2}\right),
\end{split}\end{equation*}
which when fully expanded in terms of coordinates becomes
\begin{equation*}\begin{split}
|F_{A}^{+}|^{2}
& = \frac{1}{8}\,\sum_{i,j}\left[
4\,(\partial_{i}\dj\log\r)^{2} +
3\,(\partial_{i}\log\r)^{2}(\dj\log\r)^{2} \right. \\*
& \mbox{\qquad} - \left.
8\,(\partial_{i}\dj\log\r)(\partial_{i}\log\r)(\dj\log\r) -
(\partial_{i}^{2}\log\r)(\dj^{2}\log\r) \right. \\
& \rule{0in}{4ex}\mbox{\qquad} + \left.
(\partial_{i}^{2}\log\r)(\dj\log\r)^{2} +
(\partial_{i}\log\r)^{2}(\dj^{2}\log\r)
\right].
\end{split}\end{equation*}
If the super-potential $\r$ is harmonic, then we can take advantage
of the identity $\sum_{i}\partial_{i}^{2}\log\r = -\sum_{i}(\partial_{i}\log\r)^{2}$
to simply this expression for $|F_{A}^{+}|^{2}$ to
\begin{displaymath}
|F_{A}^{+}|^{2} = \frac{1}{2}\,\sum_{i,j}\left[
(\partial_{i}\dj\log\r)^{2} -
2\,(\partial_{i}\dj\log\r)(\partial_{i}\log\r)(\dj\log\r)
\right].
\end{displaymath}
On the other hand, expanding the expression $\Delta\Delta\log\r$, we obtain
\begin{equation*}\begin{split}
\Delta\Delta\log\r
& = \sum_{i,j}\dj^{2}\partial_{i}^{2}\log\r
= -\sum_{i,j}\dj^{2}(\partial_{i}\log\r)^{2} \\
& = -2\,\sum_{i,j}\dj\left[(\partial_{i}\log\r)(\partial_{i}\dj\log\r)\right] \\
& = -2\,\sum_{i,j}\left[
(\partial_{i}\dj\log\r)^{2} + (\partial_{i}\log\r)(\partial_{i}\dj^{2}\log\r)
\right] \\
& = -2\,\sum_{i,j}\left[
(\partial_{i}\dj\log\r)^{2} - (\partial_{i}\log\r)\partial_{i}(\dj\log\r)^{2}
\right] \\
& = -2\,\sum_{i,j}\left[
(\partial_{i}\dj\log\r)^{2} - 2\,(\partial_{i}\log\r)(\dj\log\r)(\partial_{i}\dj\log\r)
\right],
\end{split}\end{equation*}
again assuming that $\r$ is harmonic and using the
identity $\sum_{i}\partial_{i}^{2}\log\r = -\sum_{i}(\partial_{i}\log\r)^{2}$ repeatedly.
Hence if the super-potential $\r$ is harmonic, then the components of
the curvature density $|F_{A}|^{2}$ for the self-dual connection
(\ref{eq:self-dual}) are
\begin{displaymath}
|F_{A}^{+}|^{2} = -\frac{1}{4}\,\Delta\Delta\log\r, \qquad
|F_{A}^{-}|^{2} = 0,
\end{displaymath}
and the Chern class $c_{2}(A)$ and $L^{2}$ norm $\|F_{A}\|^{2}$ are given by
\begin{equation} \label{eq:c2}
c_{2}(A) = \frac{1}{4\pi^{2}}\,\|F_{A}\|^{2}
= -\frac{1}{16\pi^{2}} \int_{\mathbb{R}^{4}}\Delta\Delta\log\r\,d\mu.
\end{equation}
Here we have a factor of $4\pi^{2}$ instead of the customary $8\pi^{2}$
because the function $\xi\mapsto\mbox{Tr}(\xi^{2})$ on the Lie algebras
$\mathfrak{su}(2)$ and $\mathfrak{so}(3)$ corresponds to the map $\xi\mapsto 2\xi^{2}$
in our quaternionic notation. Similarly, if we take the anti-self-dual
connection (\ref{eq:anti-self-dual}) then the two components
$|F_{A}^{+}|^{2}$ and $|F_{A}^{-}|^{2}$ are interchanged and the Chern
class $c_{2}(A)$ switches sign. It is important to note that the scalar
curvature density is gauge invariant. In other words, if two harmonic
super-potentials $\r_{1}$ and $\r_{2}$ yield gauge equivalent connections
via the ansatz of Theorem~\ref{theorem-1}, then they must satisfy the
equation $\Delta\Delta\log\r_{1} = \Delta\Delta\log\r_{2}$.
\subsection{The 't Hooft Construction}
\label{tHooft}
As an example of the harmonic function ansatz, we take for our
super-potential the Green's functions of the Laplacian. Although
these functions have $O(1/r^{2})$ poles, the corresponding
singularities can be removed from the resulting connections by a
gauge transformation. In the simplest case, consider the spherically
symmetric harmonic function
\begin{displaymath}
\r = 1 + \frac{1}{|x|^{2}} = 1 + \frac{1}{x\bar{x}},
\end{displaymath}
the sum of the Green's functions centered at the origin and infinity.
Applying formula~(\ref{eq:anti-self-dual}), this super-potential
generates the anti-self-dual connection
\begin{equation} \label{eq:singular-gauge}
A = \Im\left( \frac{\bar{x}^{-1}\,d\bar{x}}{1 + x\bar{x}} \right)
= -\Im\left( \frac{dx\,x^{-1}}{1 + x\bar{x}} \right),
\end{equation}
which is simply the basic instanton with $c_{2} = 1$ expressed in the
``singular gauge''. Applying the gauge transformation $g = x^{-1}$,
we can remove the $O(1/r)$ pole at the origin to obtain this
instanton's customary form
\begin{equation} \label{eq:standard-gauge}
g(A) = \Im\left(
- x^{-1}\,\frac{dx\,x^{-1}}{1+x\bar{x}}\,x - dx^{-1}\,x
\right)
= \Im\left( \frac{\bar{x}\,dx}{1+x\bar{x}} \right).
\end{equation}
Note that if we switch to coordinates around infinity by putting
$x = y^{-1}$, then we simply interchange these two gauges
(\ref{eq:singular-gauge}) and (\ref{eq:standard-gauge}). This connection
therefore takes the same form about infinity as it does about the origin.
More generally, we can modify the basic
instanton~(\ref{eq:standard-gauge}) by applying a dilation
and translation $x\mapsto\,\lambda^{-1}(x-a)$ with $\lambda>0$ real
and $a\in\H$. The super-potential and associated connection then become
\begin{equation}\label{eq:basic-instanton}
\r = 1 + \frac{\lambda^{2}}{|x-a|^{2}},\qquad
g_{a}(A) = \Im\left( \frac{ (\bar{x} - \bar{a})\,dx }{\lambda^{2} + |x-a|^{2}}
\right).
\end{equation}
Here we have again used a gauge transformation $g_{a} = (x - a)^{-1}$
in order to remove the singularity at the point $x = a$.
One of the interesting features of this ansatz is that it allows us to take
the superposition of several such instantons simply by adding their
super-potentials. For instance, the 't Hooft instantons with $c_{2} = k$
are constructed using the harmonic function
\begin{equation}\label{eq:tHooft}
\r = 1 + \frac{\lambda_{1}^{2}}{|x-a_{1}|^{2}} + \cdots
+ \frac{\lambda_{k}^{2}}{|x-a_{k}|^{2}},
\end{equation}
combining $k$ basic instantons of the form (\ref{eq:basic-instanton})
with scales $\lambda_{1},\ldots,\lambda_{k}$ and distinct centers
$a_{1},\ldots,a_{k}$.
\subsection{Conformal Transformations}
\label{conformal-instanton}
In this section, we discuss a differential geometric interpretation of
the harmonic function ansatz introduced in \S\ref{quaternionic-notation}.
Treating the super-potential as a conformal transformation of flat Euclidean
space, the connections (\ref{eq:anti-self-dual}) and (\ref{eq:self-dual})
arise naturally from the action of the Levi-Civita connection on the
half-spin bundles. We can then express the curvatures of these two
connections in terms of the decomposition of the Riemann curvature into
its scalar, trace-free Ricci, and conformally invariant Weyl curvature
components, thereby providing an alternative proof of Theorem \ref{theorem-1}.
Starting with the flat Euclidean metric $g_{ij} = \delta_{ij}$ on
$\mathbb{R}^{4}$, we consider the conformally equivalent metric $g' = \r^{2}g$,
given a smooth, positive, real-valued super-potential $\r$. The condition
that $\r$ be harmonic enters when calculating the scalar curvature of this
new metric as in the following lemma.
\begin{lemma} \label{lemma-scalar-curvature}
The scalar curvature $R'$ of the conformally Euclidean metric $g'$
given by $g' = \r^{2}\delta_{ij}$ vanishes if and only if the
super-potential $\r$ is harmonic.
\end{lemma}
\begin{proof}
Using the expression for $R'$ computed in \cite[p. 125]{Au},
in dimension $n=4$ we have
\begin{equation*}\begin{split}
R'&=- \r^{-2} \left[
(n-1)\,\sum_{\nu}\partial_{\nu}^{2}\log\r^{2}\,+\,
\frac{(n-1)(n-2)}{4}\,
\sum_{\nu}\left(\partial_{\nu}\log\r^{2}\right)^{2}
\right] \\
&=- 6 \r^{-2}\,\sum_{\nu} \left[ \partial_{\nu}^{2}\log\r +
\left(\partial_{\nu}\log\r\right)^{2}
\right]
= 6 \r^{-3}\Delta\r.\rule{0in}{2.5ex}
\end{split}\end{equation*}
Hence $R' = 0$ if and only if $\Delta\r = 0$.
\end{proof}
Let $\{e_{0},\ldots,e_{3}\}$ be an orthonormal tangent frame for the
original metric $g$. After applying the conformal transformation,
the Levi-Civita connection for the metric $g'$ is given with respect
to this frame by Christoffel's formula
\begin{displaymath}
\Gamma'^{j}_{ik} = \partial_{i}\log\r\:\delta^{j}_{k} +
\partial_{k}\log\r\:\delta^{j}_{i} -
\partial^{j}\log\r\:\delta_{ik}.
\end{displaymath}
In order to express this as an $\mathfrak{so}(4)$ connection, we must rescale the
tangent frame so that it is again orthonormal with respect to the new
metric $g'$. Switching to the frame $e'_{i} = \r^{-1}e_{i}$ introduces
a factor of $-\partial_{i}\log\r\;\delta^{j}_{k}$ into the connection,
cancelling the diagonal term and leaving us with an expression
skew-symmetric in the indices $j$ and $k$.
Taking the double cover $\mbox{\rm Spin}(4)$ of $\mbox{\rm SO}(4)$, we recall that the
Lie algebra isomorphism $\mathfrak{so}(4) \cong \mathfrak{spin}(4)$ associates to a
skew-symmetric matrix $a_{ij}$ the Clifford algebra element%
\SSfootnote{The Clifford algebra $\mbox{Cl}(4)$ is the algebra generated
by $\mathbb{R}^{4}$ subject to the relation $v\cdot w + w\cdot v =
-2(v,w)$, and the Lie algebra $\mathfrak{spin}(4)$ is the subspace
spanned by $\{e_{i}\cdot e_{j}\}_{i\neq j}$.}
$-\frac{1}{4}\sum_{i,j} a_{ij}\,e_{i}\cdot e_{j}$ (see \cite{LM}).
We may thus write the Levi-Civita connection in this $\mathfrak{spin}(4)$
notation as
\begin{displaymath}
A = \frac{1}{2} \sum_{i\neq j}\left( e'_{j}\,\partial^{j}\log\r
\right) \cdot \left( e'_{i}\,dx^{i} \right).
\end{displaymath}
From the decomposition $\mbox{\rm Spin}(4) = \mbox{\rm Sp}(1) \times \mbox{\rm Sp}(1)$, we see
that the complex 4-dimensional spin space splits as the direct sum
$S = S^{+} \oplus S^{-}$ of two half-spin spaces, each of which is
isomorphic to the quaternions $\H$. These spaces $S^{+}$ and $S^{-}$
are called the spaces of self-dual and anti-self-dual spinors
respectively. The two half-spin representations $\gamma^{\pm}$ of the
Lie algebra $\mathfrak{spin}(4)$ on $\H^{\pm}$ are then given by%
\SSfootnote{Here the various signs are determined by the Clifford algebra
relation $\gamma^{\pm}(v\cdot w + w\cdot v) = -2(v,w)$
and also by the convention that
$\gamma^{+}(e'_{0}\cdot e'_{1} - e'_{2}\cdot e'_{3}) = 0$
and
$\gamma^{-}(e'_{0}\cdot e'_{1} + e'_{2}\cdot e'_{3}) = 0$.}
\begin{align*}
\gamma^{+}:v\cdot w \mapsto -\gamma(v)\,\gamma^{\ast}(w) \qquad
\gamma^{-}:v\cdot w \mapsto -\gamma^{\ast}(v)\,\gamma(w),
\end{align*}
where the Clifford action $\gamma(\cdot)$ is simply quaternion
multiplication
\begin{displaymath}
\gamma(e'_{0}) = 1, \qquad \gamma(e'_{1}) = i, \qquad
\gamma(e'_{2}) = j, \qquad \gamma(e'_{3}) = k,
\end{displaymath}
and $\gamma^{\ast}(\cdot)$ is its adjoint. Hence, the Levi-Civita
connection for the conformally transformed metric $g' = \r^{2}g$
splits into the two $\sp(1)$ components
\begin{equation*}
A^{+}=-\Im\left(\del{\bar{x}}\log\r\,d\bar{x}\right) \qquad
A^{-}=-\Im\left(\del{x}\log\r\,dx\right)
\end{equation*}
acting on the positive and negative half-spin spaces respectively.
Note that these two connections agree with the connections $A^{+}$
and $A^{-}$ given in equations (\ref{eq:anti-self-dual}) and
(\ref{eq:self-dual}).
By definition, the Riemann curvature tensor $\mathcal{R}$ is an
$\mathfrak{so}(4)$-valued 2-form. However, using the identification
$\Lambda^{2} \cong \mathfrak{so}(4)$, we may view it as a self-adjoint
linear map $\mathcal{R} : \Lambda^{2} \rightarrow\Lambda^{2}$
given in coordinates by
\begin{displaymath}
\mathcal{R}\left(dx^{i}\wedge dx^{j}\right) =
\frac{1}{2} \sum_{k,l} R_{ijkl}\,dx^{k}\wedge dx^{l}.
\end{displaymath}
Relative to the familar decomposition
$\Lambda^{2} = \Lambda^{2}_{+} \oplus \Lambda^2_{-}$ of the space
of two-forms into its self-dual and anti-self-dual subspaces, the
Riemann curvature can be written in the block matrix form
\begin{displaymath}
\mathcal{R} = \left( \begin{array}{c|c}
\mathcal{W}^{+} - \frac{1}{12}R & R_{0} \\ \hline
R_{0}^{\ast} & \mathcal{W}^{-} - \frac{1}{12}R
\end{array}
\right).
\end{displaymath}
Here $R$ denotes the scalar curvature multiplied by the identity matrix,
while $R_{0} : \Lambda^{2}_{-}\rightarrow \Lambda^{2}_{+}$ is the
trace-free Ricci curvature tensor,
$R_{0}^{\ast} : \Lambda^{2}_{+}\rightarrow \Lambda^{2}_{-}$ is its
adjoint, and $\mathcal{W} = \mathcal{W}^{+} + \mathcal{W}^{-}$ is the conformally invariant Weyl tensor. A standard reference for this material is
\cite{AHS}.
We now consider the Riemann curvature $\mathcal{R}'$ of the metric $g'$
discussed above. Since $g'$ is by definition conformally flat, we see
that the Weyl tensor $\mathcal{W}'$ vanishes. We also recall from
Lemma~\ref{lemma-scalar-curvature} that if our
super-potential $\r$ is harmonic, then the scalar curvature $R'$
vanishes as well. All that remains is the trace-free Ricci tensor $R_{0}'$,
and so the Riemann curvature is simply
\begin{displaymath}
\mathcal{R}' = \left( \begin{array}{c|c}
0 & R_{0}'\\ \hline
R_{0}'^{\ast} & 0
\end{array}
\right).
\end{displaymath}
Note that the splitting $\mathfrak{spin}(4) \cong \sp(1) \oplus \sp(1)$ which we
used to construct the connections $A^{+}$ and $A^{-}$ is isomorphic to
the decomposition $\Lambda^{2} = \Lambda^{2}_{+} \oplus \Lambda^{2}_{-}$.
We can therefore read off the curvatures $F_{A^{+}}$ and $F_{A^{-}}$ of
these connections directly from the block form of the Riemann curvature,
giving us
$$
F_{A^{+}}^{+} = 0 \qquad
F_{A^{+}}^{-} = R_{0}' \qquad
F_{A^{-}}^{+} = R_{0}'^{\ast} \qquad
F_{A^{-}}^{-} = 0.
$$
Hence the connections $A^{+}$ and $A^{-}$ are anti-self-dual and self-dual
respectively as claimed in Theorem \ref{theorem-1}.
\section{Hyperbolic Vortices}
\subsection{Dimensional Reduction}
\label{dimensional-reduction}
In this section, we examine $\mbox{\rm SO}(3)$-invariant instantons, showing that
the SD and ASD equations for Yang-Mills connections over $S^{4}$ with $\mbox{\rm SO}(3)$
symmetry are equivalent to the $\mbox{\rm U}(1)$ vortex equations over the hyperbolic
plane $\mathcal{H}^{2}$. This is an example of dimensional reduction, whereby
the Yang-Mills or (A)SD equations for a symmetric connection reduce to
differential equations for a connection and Higgs fields (sections of the
Lie algebra bundle) over a lower dimensional space.
Viewing $S^{4}$ as the standard conformal compactification
$\mathbb{R}^{4} \cup \{\infty\}$, we let $\mbox{\rm SO}(3)$ act via its fundamental
representation on a three-dimensional subspace of $\mathbb{R}^{4}$.
Expressing this using quaternionic notation, we see that an element
$g\in \mbox{\rm Sp}(1)$ acts on $\H$ according to $g:x\mapsto g x g^{-1}$, fixing
the real part of $x$ and acting by the adjoint representation on its
imaginary part. Regarding $S^{4}$ as the quaternionic projective space
$P(\H^{2})$ with homogeneous coordinates $(x:y)=(x\alpha:y\alpha)$,
the embedding of $\H$ is simply the map $x\mapsto(1:x)$.
The $\mbox{\rm Sp}(1)$-action given by
\[g:(x:y)\mapsto(gx:gy) = (gxg^{-1}:gyg^{-1})\]
then provides an extension of the above action on $\mathbb{R}^{4}$ to all of $S^{4}$.
In order to discuss $\mbox{\rm SO}(3)$-invariant connections, we must lift this
action on $S^{4}$ to an action on the Lie algebra bundle with fibres
$\mathfrak{so}(3)$. There are two possible lifts: either $\mbox{\rm SO}(3)$ acts trivially
on each fibre or it acts via the adjoint representation. For our
purposes, we will consider this second, more interesting, action.
Note that for any $g\in\mbox{\rm SO}(3)$, the adjoint action leaves fixed an
$\mathbb{R} = \u(1)$ subalgebra.
Again adopting quaternionic notation, any $x\in\H$ can be written
in the form $x = t + r Q$, with $t,r$ real, $r \geq 0$, and Q pure
imaginary with $Q^{2} = -1$.
Note that $t,r$ coordinatize the upper half-plane, which we
will later regard as hyperbolic space $\mathcal{H}^{2}$.
If $A$ is an $\mbox{\rm Sp}(1)$-invariant connection, then its connection
one-form satisfies $g A(t,r,Q) g^{-1} = A(t,r,gQg^{-1})$.
The most general connection exhibiting this symmetry is of the form
\begin{displaymath}
A = \frac{1}{2}\,\bigl(Qa + \Phi_{1}\,dQ + \Phi_{2}\,Q\,dQ\bigr), \label{symmetric-connection}
\end{displaymath}
where $a = a_{t}\,dt + a_{r}\,dr$, and the $a_{t}, a_{r}, \Phi_{1}, \Phi_{2}$
are all real functions of $t,r$. The curvature $F_{A}$ of this connection
$A$ is then
\begin{align*}
F_{A}&=\frac{1}{2} \left( Q\,da\,+\,
\frac{1}{2} \left(\Phi_{1}^{2}+\Phi_{2}^{2}+2\Phi_{2}
\right)dQ \wedge dQ\,+\right.\\*
& \qquad\qquad\left.\rule{0in}{3ex}
[ d\Phi_{1} - a ( \Phi_{2} + 1 ) ] \wedge dQ\,+\,
( d\Phi_{2} + a \Phi_{1} ) \wedge Q\,dQ \right).
\end{align*}
Putting $\Phi = \Phi_{1}+ i\,(\Phi_{2}+1)$ and writing $d_{a}\Phi = d\Phi + ia\Phi$,
the curvature can be written much more simply as
\begin{align*}
F_{A}=\frac{1}{2} \left( Q\,da\,-\,
\frac{1}{2}\,\left( 1 - |\Phi|^{2}\right)\,dQ \wedge dQ\,+
\Re(d_{a}\Phi) \wedge dQ \,+\,
\Im(d_{a}\Phi) \wedge Q\,dQ \right).
\end{align*}
Note that multiplication by $Q$ here behaves like multiplication by $i$.
From the above discussion, we see that an $\mbox{\rm SO}(3)$-invariant
connection $A$ on $S^{4}$ gives rise in a natural way to a $\mbox{\rm U}(1)$
connection $ia$ and a complex scalar field $\Phi$ on the upper half-plane.
The next step is to analyze the SD and ASD equations in terms of this
dimensional reduction. To determine the action of the Hodge star operator,
we consider the 2-form $dx\wedge d\bar{x}$ which we already know to be
self-dual.
In coordinates $t,r,Q$, we have
\begin{align*}
dx \wedge d\bar{x} &= (dt+Q\,dr+r\,dQ) \wedge (dt-Q\,dr-r\,dQ) \\*
&= 2Q\,dt\wedge dr\,+\,r^{2}\,dQ\wedge dQ\,+\,
2r\,(dt\wedge dQ\,+\,dr\wedge Q\,dQ),
\end{align*}
and so the Hodge star operator acts according to
\begin{align*}
\ast\,Q\,dt\wedge dr & = \frac{r^{2}}{2}\,dQ\wedge dQ \\
\ast\,dt\wedge dQ & = dr\wedge Q\,dQ \\
\ast\,dr\wedge dQ & = -dt\wedge Q\,dQ.
\end{align*}
Furthermore, using the hyperbolic metric
\begin{displaymath}
h = \frac{1}{r^{2}}\,( dt^{2} + dr^{2} )
\end{displaymath}
on the upper half-plane, the corresponding Hodge star operator
$\ast_{h}$ satisfies $\ast_{h}dt \wedge dr = r^{2}$,
$\ast_{h}dt = dr$, and $\ast_{h}dr = -dt$. Combining this with
the usual Hodge star operator yields
\begin{align*}
\ast\,Q\,da = \frac{1}{2}\,(\ast_{h} da)\,dQ \wedge dQ \qquad
\ast\,\Re\,(d_{a}\Phi) \wedge dQ = \ast_{h} \Re\,(d_{a}\Phi) \wedge Q\,dQ.
\end{align*}
Using complex notation with $z=t+ir$ and noting the identity
$\ast_{h}dz = -i\,dz$, we observe that
$2\,\bar{\partial}_{a}\Phi = d_{a}\Phi - i \ast_{h}d_{a}\Phi$. We therefore
conclude that the $\mbox{\rm SO}(3)$-symmetric self-duality equation
$F_{A} = \ast F_{A}$ on $S^{4}$ is equivalent to the following
two equations on hyperbolic space $\mathcal{H}^{2}$:
\begin{align}
\bar{\partial}_{a} \Phi & = 0 \label{eq:vortex-a} \\
\ast_{h} i F_{a} & = 1 - |\Phi|^{2}, \label{eq:vortex-b}
\end{align}
where $F_{a} = i\,da$ is the curvature of the connection $ia$.
These equations are known as the {\em vortex equations}. Similarly, the
$\mbox{\rm SO}(3)$-symmetric anti-self-dual equation $F_{A} = -\ast F_{A}$
is equivalent to the {\em anti-vortex equations}:
\begin{align}
\partial_{a} \Phi & = 0 \label{eq:anti-vortex-a} \\
\ast_{h} i F_{a} & = |\Phi|^{2} - 1. \label{eq:anti-vortex-b}
\end{align}
The first equation in each pair is simply the condition that $\Phi$ be
holomorphic (or anti-holomorphic) with respect to the holomorphic
structure compatible with the connection $ia$. The second equation
then expresses a form of duality between the connection and Higgs field.
These vortex equations are discussed in great detail in \cite{JT}%
\SSfootnote{After adjusting to the slightly different notation of
\cite{JT}, using $a' = -a$ and $\Phi' = \bar{\Phi}$, the
reader will find that equations (11.5a) and (11.5b) on
p. 99 of \cite{JT} should be switched.}.
Note that if we consider these vortex and anti-vortex equations over
the plane $\mathbb{R}^{2}$ with the flat metric $h = \frac{1}{2}(dx^{2}+dy^{2})$,
then we obtain the Euclidean vortex and anti-vortex equations in their
customary form as given by (1.7) and (1.8) on p. 55 of \cite{JT}.
We now compute the $L^{2}$ norm of the curvature $F_{A}$ using the
standard metric $|\xi|^{2} = \xi\bar{\xi} = -\xi^{2}$ on the Lie
algebra $\sp(1)$ of imaginary quaterions. The Yang-Mills action,
or energy, of this $\mbox{\rm SO}(3)$-invariant connection is thus
\begin{align*}
\|F_{A}\|^{2} & = \int_{S^{4}}
-\,F_{A}\wedge \ast F_{A} \\
& = \frac{1}{8}\int_{S^{4}}
\Bigl( da\wedge \ast_{h} da
\,+\, (1-|\Phi|^{2}) \wedge \ast_{h}(1-|\Phi|^{2}) \,+\,
\\
& \qquad \qquad \rule{0in}{2ex}
2\,\Re d_{a}\Phi \wedge\ast_{h}\Re d_{a}\Phi
\,+\,2\,\Im d_{a}\Phi \wedge\ast_{h}\Im d_{a}\Phi
\Bigr)
\wedge\,\left(\,dQ\wedge Q\,dQ\,\right).
\end{align*}
Note that the left factor of the integrand is independent of the
variable $Q$. Since $Q$ parametrizes the unit 2-sphere with volume
form $\frac{1}{2}(dQ\wedge Q\,dQ)$, we can integrate out a factor of
$\int_{S^{2}}dQ\wedge Q\,dQ = 8\pi$, leaving an integral over
hyperbolic space. The action then becomes
\begin{displaymath}
\|F_{A}\|^{2} = \pi \left( \|F_{a}\|^{2}_{h} \,+\,
2\,\|d_{a}\Phi\|^{2}_{h} \,+\,
\|1-|\Phi|^{2}\|^{2}_{h}
\right),
\end{displaymath}
which we recognize as the $\mbox{\rm U}(1)$ Yang-Mills-Higgs action on hyperbolic
space, at least up to a constant. From this action, we see that a
finite-energy $\mbox{\rm SO}(3)$-invariant connection on $S^{4}$ corresponds to a
pair $(ia,\Phi)$ over $\mathcal{H}^2$ satisfying the boundary conditions
\begin{displaymath}
d_{a}\Phi(x) \rightarrow 0, \qquad |\Phi(x)| \rightarrow 1,
\end{displaymath}
as $|x| \rightarrow \infty$.
Next we examine the relationship between the Chern classes of an
$\mbox{\rm SO}(3)$-invariant connection $A$ on $S^{4}$ and those of the corresponding
connection $ia$ over $\mathcal{H}^2$. Computing $c_{2}(A)$, we first
note that the negative definite form $\xi\mapsto\mbox{Tr}(\xi)^2$
on the Lie algebra $\mathfrak{su}(2)$ corresponds to $\xi\mapsto 2\xi^{2}$
on $\sp(1)$. In this quaternionic notation we therefore have
\begin{align*}
c_{2}(A) & = -\frac{1}{4\pi^{2}}
\int_{S^{4}}F_{A}\wedge F_{A} \\
& = -\frac{1}{16\pi^{2}}
\int_{S^{4}} \left[\,
( 1-|\Phi|^{2} )\,da \,-\,
2\,\Re d_{a}\Phi\wedge\Im d_{a}\Phi
\,\right]
\wedge\,
\left(\,dQ\wedge Q\,dQ\,\right) \\
& = \frac{i}{2\pi} \int_{\mathcal{H}^2} F_{a}
= c_{1}(a).
\end{align*}
Here we again integrate out the $S^{2}$ factor $dQ\wedge Q\,dQ$, and
on the last line we apply Stokes' theorem with the integrand
\begin{align*}
d\,(i\bar{\Phi}\,d\Phi) & = i\,d\bar{\Phi} \wedge d\Phi \,-\,
\Phi\bar{\Phi}\,da \,-\,
(\Phi\,d\bar{\Phi} + \bar{\Phi}d\Phi)\wedge a \\*
& = -|\Phi|^2\,da - 2\,\Re d_{a}\Phi\wedge\Im d_{a}\Phi,
\end{align*}
assuming that $i\bar{\Phi}\,d\Phi$ vanishes at infinity.
\subsection{Another Harmonic Function ansatz}
\label{vortex-ansatz}
We now return to the harmonic function ansatz that we discussed in
Section~\ref{instanton-ansatz}. If we begin with a $\mbox{\rm SO}(3)$-invariant
harmonic super-potential, then the resulting SD or ASD connection will
also exhibit $\mbox{\rm SO}(3)$ symmetry, and from the previous section
we know that such a connection is equivalent to a hyperbolic vortex
or anti-vortex.
In this section, we take advantage of this dimensional reduction
to provide a similar harmonic function ansatz constructing solutions to
the vortex equation over hyperbolic space.
Our first step is to examine the relationship between the Laplacian
on hyperbolic space $\mathcal{H}^{2}$ and the $\mbox{\rm SO}(3)$-symmetric Laplacian
on $\mathbb{R}^{4}$.
\begin{lemma}
An $\mbox{\rm SO}(3)$-invariant function $\r$ on $\mathbb{R}^{4}$ is harmonic if and
only if it can be written as $\r = r^{-1}\phi$, where $\phi$ is
a harmonic function on $\mathcal{H}^2$.
\end{lemma}
\begin{proof}
Using the quaternionic notation $x = t + rQ$ introduced in
\S\ref{dimensional-reduction}, if $\r = \r(r,t)$ is an $\mbox{\rm SO}(3)$-invariant
function on $\mathbb{R}^{4}$, then its Laplacian is
\begin{equation} \label{eq:laplacian}
\Delta \r = -\left( \delsq{t} + \delsq{r} +
\frac{2}{r}\del{r}
\right) \r.
\end{equation}
To cancel the unwanted linear term, we put $\r = r^{-1}\phi$,
where $\phi$ is a function on $\mathcal{H}^{2}$. The Laplacian $\Delta$
then becomes
\begin{displaymath}
\Delta r^{-1}\phi = -\frac{1}{r} \left( \delsq{t} + \delsq{r}
\right) \phi
= r^{-3}\Delta_{h} \phi,
\end{displaymath}
where the Laplacian $\Delta_{h}$ on $\mathcal{H}^{2}$ is
\begin{equation} \label{eq:hyperbolic-laplacian}
\Delta_{h} = -r^{2}\left(\delsq{t} + \delsq{r}\right)
\end{equation}
These two Laplacians are therefore related by%
\SSfootnote{In general, the conformal Laplacian is $L_{g}=\Delta+kR$,
where $R$ is the scalar curvature and $k$ is a constant depending on
the dimension. Taking the metric $g' = e^{2f}g$, it we have
\begin{displaymath}
L_{g'} = e^{-(d+2)f/2}L_{g}\,e^{(d-2)f/2}.
\end{displaymath}
Here the Euclidean metric on $R^{4}$ is $g=dt^{2}+dr^{2}+r^{2}dS^{2}$,
where $dS^{2}$ is the metric on $S^{2}$. Taking the conformally equivalent
metric $g' = r^{-2}(dt^{2}+dr^{2}) + dS^{2}$ on $\mathcal{H}^2 \times S^{2}$,
Lemma \ref{lemma-scalar-curvature} tells us that $R' = 0$ since $r^{-1}$
is harmonic. It follows that $\Delta'=r^{3}\Delta r^{-1}$.}
$\Delta$ = $r^{-3}\Delta_{h}r$ and thus $\r$ is harmonic on
$\mathbb{R}^{4}$ if and only if $\phi$ is harmonic on $\mathcal{H}^{2}$.
\end{proof}
We recall from Theorem \ref{theorem-1} that the connection given by
equation~(\ref{eq:self-dual}),
\begin{displaymath}
A = - \Im\left(\del{x}\log \r\,dx \right),
\end{displaymath}
is self-dual if and only if $\r$ is harmonic. In our current notation,
the quaternionic differential and partial derivative in this expression are
\begin{align*}
dx = dt + Q\,dr + r\,dQ \qquad
\del{x} = \frac{1}{2} \left( \del{t} - Q\,\del{r} - \cdots
\right),
\end{align*}
where we have left out the portions of the partial derivative in the $Q$
directions as these vanish when applied to $\mbox{\rm SO}(3)$-invariant functions.
Taking $\r = r^{-1}\phi$, we see that $\log \r = \log \phi - \log r$.
Expanding equation~(\ref{eq:self-dual}) using these expressions,
our $\mbox{\rm SO}(3)$-invariant self-dual connection becomes
\begin{align*}
A & = \frac{1}{2}\,\left(\,Q \left[\,
\left( \del{r}\log\phi - \frac{1}{r}\right) dt \,-\,
\del{t}\log\phi\,dr\,
\right]\right. \\*
& \qquad \left. \mbox{\qquad}
-\,r\,\del{t}\log\phi\,dQ\,+\,
\left( r\,\del{r}\log\phi - 1 \right) Q\,dQ\,
\right).
\end{align*}
As we did in in \S\ref{dimensional-reduction}, we can extract from
this connection the $\mbox{\rm U}(1)$ connection
\begin{equation} \label{eq:vortex-da}
d_a = d\,+\,i \left[\,
\left( \del{r}\log\phi - \frac{1}{r}\right) dt -
\del{t}\log\phi\,dr\,
\right]
\end{equation}
with curvature
\begin{align*}
F_{a} = -i\left[
\left(\delsq{t} + \delsq{r}\right) \log\phi +
\frac{1}{r^{2}}
\right] dt\wedge dr
= -i \left( 1 - \Delta_{h}\log\phi
\right)\,r^{-2}dt\wedge dr,
\end{align*}
and the complex Higgs field
\begin{equation} \label{eq:vortex-phi}
\Phi = r \left( - \del{t}\log\phi + i\,\del{r}\log\phi \right)
\end{equation}
with norm
\begin{displaymath}
|\Phi|^{2} = r^{2}\left[
\left(\del{t}\log\phi\right)^{2} +
\left(\del{r}\log\phi\right)^{2}
\right] \\*
= \left| \nabla\log\phi \right|_{h}^{2}.
\end{displaymath}
Writing the pair $(a,\phi)$ using complex notation with $z = t + ir$,
we obtain the hyperbolic space analogue of Theorem \ref{theorem-1}.
\begin{theorem} \label{theorem-vortex-ansatz}
Given a positive real-valued super-potential $\phi$ on the hyperbolic
upper half-plane $\mathcal{H}^{2}$, the connection and Higgs field pair
$(a,\Phi)$ defined by
\begin{align}
\bar{\partial}_{a} = \bar{\partial}\,+\,\bar{\partial}\log\phi\,+\,\frac{d\bar{z}}{z-\bar{z}}
\label{eq:vortex-dbar} \qquad
\Phi = i\,(z-\bar{z})\,\del{z}\log\phi,
\end{align}
satisfies the vortex equations (\ref{eq:vortex-a}) and
(\ref{eq:vortex-b}) and the pair $(a',\Phi')$ defined by
\begin{align} \label{eq:anti-vortex-dbar}
\partial_{a}' & = \partial\,+\,\partial\log\phi\,-\,\frac{dz}{z-\bar{z}} \\
\Phi' & = -i\,(z-\bar{z})\,\del{\bar{z}}\log\phi,
\label{eq:anti-vortex-field}
\end{align}
satisfies the anti-vortex equations (\ref{eq:anti-vortex-a})
and (\ref{eq:anti-vortex-b}) if and only if the super-potential
$\phi$ is harmonic.
\end{theorem}
\begin{proof}
Recalling that $\ast_{h} 1 = r^{-2}dt\wedge dr$ with our
hyperbolic metric, we see that the second of the vortex equations
$iF_{a} = \ast_{h}\left( 1 - |\Phi|^{2} \right)$ reduces to
\begin{displaymath}
\frac{1}{\phi}\,\Delta_{h}\phi
= \Delta_{h}\log\phi - \left| \nabla\log\phi \right|_{h}^{2}
= 0.
\end{displaymath}
Using the complex form (\ref{eq:vortex-dbar}),
it is easy to verify that the holomorphicity condition $\bar{\partial}_{a}\Phi = 0$
likewise reduces to $\Delta_{h} \phi = 0$. Hence the pair $(a, \Phi)$ satisfies
the vortex equations if and only if the super-potential $\phi$ is harmonic.
Similarly, the pair $(a',\phi')$ is derived by dimensional reduction from
the ASD connection (\ref{eq:anti-self-dual}), and so it satisfies the
anti-vortex equations if and only if $\phi$ is harmonic.
\end{proof}
Computing the Chern class $c_{1}$ for a pair $(a,\Phi)$ satisfying
the second of the vortex equations (\ref{eq:vortex-b}), we have
\begin{displaymath}
c_{1}(a) = \frac{i}{2\pi}\int_{\mathcal{H}^{2}} F_{a}
= \frac{1}{2\pi}\int_{\mathcal{H}^{2}}
\ast_{h}\left(1-|\Phi|^{2}\right).
\end{displaymath}
For the vortex over the upper half-plane $\mathcal{H}^{2}$ constructed
in (\ref{eq:vortex-da}) and (\ref{eq:vortex-phi}) using a
harmonic super-potential $\phi$, this Chern class takes the form
\begin{align*}
c_{1}(a)=\frac{1}{2\pi}\int_{\mathbb{R}^{2}_{+}}\left(
1 - \Delta_{h}\log\phi
\right)\,r^{-2} dt\wedge dr
=\frac{1}{2\pi}\int_{\mathbb{R}^{2}_{+}}\left(
\frac{1}{r^{2}} - 4\left|\del{z}\log\phi\right|^{2}
\right)\,dt\wedge dr.
\end{align*}
Likewise, if we use the above ansatz to construct the anti-vortex
corresponding to a harmonic super-potential, then the Chern class
switches sign.
As in \S\ref{quaternionic-notation}, we note that the curvature
$F_{a}$ of a vortex is gauge invariant. Therefore, if two harmonic
super-potentials $\phi_{1}$ and $\phi_{2}$ over hyperbolic space
yield gauge equivalent vortices, then they must satisfy the equations
\begin{displaymath}
\left|\del{z}\log\phi_{1}\right| = \left|\del{z}\log\phi_{2}\right|
\end{displaymath}
and $\Delta_{h}\log\phi_{1} = \Delta_{h}\log\phi_{2}$.
Note that it is significantly simpler to calculate $c_{1}$ directly
from the vortex construction on $\mathcal{H}^{2}$ than it is by invoking
dimensional reduction and computing the equivalent Chern class $c_{2}$
for the corresponding $\mbox{\rm SO}(3)$-invariant instanton over $S^{4}$. Indeed,
by comparing the above expression for $c_{1}(a)$ with the expression
(\ref{eq:c2}) for $c_{2}(A)$, we obtain a circuitous proof
of the identity
\begin{displaymath}
\int_{\mathbb{R}^{2}_{+}} \left( 1 - \Delta_{h}\log\phi
\right)\,r^{-2}dt \wedge dr =
- \int_{\mathbb{R}^{2}_{+}} \left(
\frac{1}{2}\,r^{2} \Delta\Delta\log \frac{\phi}{r}
\right) dt \wedge dr
\end{displaymath}
for a harmonic function $\phi$ defined on the upper half-plane, where
$\Delta_{h}$ is the Laplacian on $\mathcal{H}^{2}$ given by
(\ref{eq:hyperbolic-laplacian}) and $\Delta$ is the Laplacian on $\mathbb{R}^{4}$
given by (\ref{eq:laplacian}).
\subsection{Conformal Transformations Revisited}
Instead of relying on dimensional reduction to derive the vortex ansatz
of Theorem~\ref{theorem-vortex-ansatz}, we present here an interpretation
of this construction that is entirely intrinsic to hyperbolic space. As
we did in \S\ref{conformal-instanton}, we can treat the super-potential
as a conformal transformation and then compute the Levi-Civita
connection of the resulting metric. Since we are working on
two-dimensional hyperbolic space, we can take advantage of complex
notation to simplify our task.
Let $\bar{\partial}$ be the standard holomorphic stucture on the complex upper
half-plane. Choosing a holomorphic tangent frame (i.e., a single
holomorphic section) $e$, consider the Hermitian metric $g$ specified
by $(e,e)_{g} = \r^{2}$, where $\r$ is a smooth nonzero real-valued
function. With respect to our holomorphic frame $e$, the unique
connection compatible with both the holomorphic structure $\bar{\partial}$ and
the metric $g$ is specified by the $(1,0)$-form
\begin{displaymath}
a = \r^{-2}(\partial \r^{2}) = 2\,\partial\log\r.
\end{displaymath}
To express this in the form of a unitary connection (in this case given
by a purely imaginary complex 1-form), we must switch to a tangent frame
that is orthonormal with respect to the metric $g$. In terms of the unitary
frame $e' = \r^{-1}e$, the connection $a$ then becomes
\begin{displaymath}
a' = \partial\log\r - \bar{\partial}\log\r
= 2i\,\Im\partial\log\r = -2i\,\Im\bar{\partial}\log\r
\end{displaymath}
and the new holomorphic structure is $\bar{\partial}'=\bar{\partial}-\bar{\partial}\log\r$,
which we observe is compatible with the connection $a'$. In either
frame, the curvature of this connection is given by
\begin{displaymath}
F_{a} = 2\,\bar{\partial}\partial\log\r = -i\,\Delta\log\r\,d\mu,
\end{displaymath}
where the volume form $d\mu$ and the Laplacian $\Delta$ are both
taken here with respect to the Euclidean metric on $\mathbb{R}^{2}$.
When working with $\mathcal{H}^2$, the hyperbolic metric $h$ on the upper
half-plane corresponds to the function $\r = r^{-2}$. Taking a conformal
transformation, we consider the metric $h'$ specified by a function of
the form $\r = \phi^{2}/r^{2}$ with $\phi$ harmonic. The resulting
unitary connection $a$ then splits into the $(0,1)$ and $(1,0)$ components
\begin{equation*}
\bar{\partial}_{a} = \bar{\partial}\,-\,\bar{\partial}\log\phi\,-\,\frac{d \bar{z}}{2ir} \qquad
\partial_{a} = \partial \,+\,\partial\log\phi\,-\,\frac{dz}{2ir},
\end{equation*}
noting that $r = (z - \bar{z})/2i$. The curvature of this connection is then
\begin{displaymath}
i F_{a} = \left( \Delta\log\phi - r^{-2} \right) d\mu
= \left( \Delta_{h}\log\phi - 1 \right) d\mu_{h}
\end{displaymath}
where $d\mu_{h} = r^{-2}d\mu$ is the volume form and $\Delta_{h} =
r^{2}\Delta$ is the Laplacian for the hyperbolic metric $h$.
It is then easy to show that the complex Higgs field $\Phi$ defined by
\begin{displaymath}
\Phi = 2r\,\del{\bar{z}}\log\phi = 2\,\frac{r}{\phi}\,\del{\bar{z}}\phi
\end{displaymath}
satisfies the anti-vortex equations $\partial_{a}\Phi = 0$ and
$\ast_{h} i F_{a} = |\Phi|^{2} - 1$ if the super-potential $\phi$ is
harmonic. We observe that this pair $(a,\Phi)$ agrees with the anti-vortex
(\ref{eq:anti-vortex-dbar}) and (\ref{eq:anti-vortex-field}) constructed
by dimensional reduction of an anti-self-dual connection over the 4-sphere.
Similarly, we can construct the vortex given by (\ref{eq:vortex-dbar})
by reversing orientation, thereby exchanging
the holomorphic and anti-holomorphic structures $\bar{\partial}$ and $\partial$.
\subsection{The Symmetric 't Hooft Construction}
\label{vortex-tHooft}
In \S\ref{tHooft}, as an illustration of the harmonic function ansatz,
we constructed the 't Hooft instantons. These are the instantons formed
by taking the superposition of multiple copies of the basic instanton
with varying scales and distinct centers. For our super-potential, we used
a sum of the Green's functions of the Laplacian, centered at the given
points and weighted according to the corresponding scales. If we impose
$\mbox{\rm SO}(3)$ symmetry on this class of instantons, we see that all of the
centers must lie on a single real line. In fact, as we will demonstrate
in the following section, {\em all\/} $\mbox{\rm SO}(3)$-invariant instantons can be
constructed in this manner---as the superposition of basic instantons on a
line. In this section, we examine the hyperbolic vortices associated
to these symmetric 't Hooft instantons by dimensional reduction.
We begin with the basic instanton with unit scale centered at the origin,
which we recall is given by the $\mathbb{R}^{4}$ super-potential $\r = 1 + |x|^{-2}$.
The corresponding super-potential for hyperbolic space $\mathcal{H}^{2}$ is then
\begin{equation}\begin{split}
\phi = r\r = r + \frac{r}{r^{2}+t^{2}}
= \Im \left( z - \frac{1}{z} \right)
= \frac{ (z-\bar{z})\,(1 + z\bar{z}) }{2i\,z\bar{z} }.
\label{eq:simple-potential}
\end{split}\end{equation}
Taking the complex partial derivatives of its logarithm, we obtain
\begin{align*}
\del{z}\log\phi = + \frac{1}{z-\bar{z}}\,+\,\frac{\bar{z}}{1+z\bar{z}}
\,-\,\frac{1}{z} \qquad
\del{\bar{z}}\log\phi = - \frac{1}{z-\bar{z}}\,+\,\frac{z}{1+z\bar{z}}
\,-\,\frac{1}{\bar{z}}.
\end{align*}
Inserting these expressions into the formula (\ref{eq:vortex-dbar}) gives us the connection and Higgs field
\begin{equation*}
\bar{\partial}_{a} = \bar{\partial}\,-\,\frac{d\bar{z}}{\bar{z}\,(1+z\bar{z})} \qquad
\Phi = i\,\frac{\bar{z}\,(1+z^{2})}{z\,(1+z\bar{z})},
\end{equation*}
satisfying the vortex equations (\ref{eq:vortex-a}) and (\ref{eq:vortex-b}). Note that
as $z$ approaches the real axis, the Higgs field obeys the
boundary condition $|\Phi|\rightarrow 1$.
One of the primary results concerning solutions to the vortex equations
is that they are uniquely specified up to gauge equivalence by the zeros
of the Higgs field (see \cite[Chapter III]{JT}).
In the example above, we see that $\Phi$ vanishes at the point $z = i$.
If we alter the scale of our basic instanton and translate it along the
real axis, the $\mathcal{H}^{2}$ super-potential becomes
\begin{displaymath}
\phi = \Im \left( z - \frac{\lambda}{z-a} \right)
\end{displaymath}
with $\lambda > 0$ and $a$ real. The corresponding vortex is then
\begin{equation*}
\bar{\partial}_{a} = \bar{\partial}\,-\,\frac{\lambda\,d\bar{z}}
{(\bar{z}-\bar{a})\,(1+|z-a|^{2})} \qquad
\Phi = i\,\frac{(\bar{z}-\bar{a})\,(1+(z-a)^{2})}
{(z-a)\,(1+|z-a|^{2})},
\end{equation*}
and we see that $\Phi$ vanishes at the point $z = a + i\sqrt{\lambda}$.
Most generally, given a set of $k$ complex points $\{z_{i}\}$ in the
upper half-plane, the super-potential
\begin{displaymath}
\phi = \Im \left( z - \sum_{i = 1}^{k}
\frac{(\Im z_{i})^2}{z - \Re z_{i}}
\right)
\end{displaymath}
generates the unique hyperbolic vortex with Higgs field vanishing at
the points $\{z_{i}\}$. Hence the centers of the instantons correspond
to the real parts of the complex zeros, while the scales correspond to
their imaginary parts.
\subsection{The Equivariant ADHM Construction}
In this section we shall use an $\mbox{\rm SO}(3)$ equivariant version of the ADHM
construction \cite{ADHM} in order to provide an alternative construction for the
symmetric 't Hooft instantons discussed in the previous section.
In addition, since the ADHM construction actually generates {\em all}
possible anti-self-dual connections on bundles over $S^{4}$, we
will then be able to show that every symmetric instanton must be
gauge equivalent to one constructed using the 't Hooft ansatz. By
dimensional reduction, this gives us a complete classification of
hyperbolic vortices, proving that such vortices are uniquely determined
up to a gauge transformation by the zeros of their Higgs fields. This is
to be contrasted with Euclidean vortices, in which case the classification
theorem may be proved using approximation techniques (see \cite[Chapter
III]{JT}), but no explicit construction for the vortex solutions is known.
\newcommand{W\!\otimes\!\H_{1}}{W\!\otimes\!\H_{1}}
Here we use the quaternionic version of the construction as discussed in
\cite{A}. When dealing with quaternionic vector spaces and linear maps,
we use the convention that scalar multiplication acts on the {\em right}.
Recall from \S\ref{dimensional-reduction} that under our $\mbox{\rm Sp}(1)$-action,
we may view $S^{4}$ as the quaternionic projective space $P(\H_{1}^{2})$,
where $\H_{1}$ is the fundamental representation with $\mbox{\rm Sp}(1)$ acting by
{\em left} quaternion multiplication.
To construct an $\mbox{\rm Sp}(1)$-invariant ASD connection on the bundle
$E\rightarrow S^{4}$ with Chern class $c_{2}(E) = -k$, we introduce
the $k+1$ dimensional {\em quaternionic\/} $\mbox{\rm Sp}(1)$ representation $V$
given by
\begin{equation} \label{eq:V}
V = \mbox{Ker}\,\mathcal{D}_{A}^{\ast} :
\Gamma (S^{4}, E\otimes S^{-}\otimes S^{-})
\rightarrow
\Gamma (S^{4}, E\otimes S^{+}\otimes S^{-})
\end{equation}
and the $k$ dimensional {\em real\/} $\mbox{\rm Sp}(1)$ representation $W$ given by
\begin{equation} \label{eq:W}
W = \left( \mbox{Ker}\,\mathcal{D}_{A}^{\ast} :
\Gamma (S^{4}, E\otimes S^{-})
\rightarrow
\Gamma (S^{4}, E\otimes S^{+})
\right)_{\mathbb{R}}^{\ast},
\end{equation}
where $A$ is an arbitrary connection on $E$ (the spaces $V,W$ are
independent of the connection), $S^{\pm}$ are the two quaternionic
half-spin bundles, and $\mathcal{D}_{A}^{\ast}$ is the adjoint of the Dirac
operator with coefficients in $E\otimes S^{-}$ and $E$ respectively.
Using these spaces $V,W$ the ADHM data consists of the the three maps:
\begin{itemize} \samepage
\item an arbitrary $\mbox{\rm Sp}(1)$ equivariant inclusion
$W\!\otimes\!\H_{1} \hookrightarrow V$
\item an $\mbox{\rm Sp}(1)$ equivariant $\H$-linear map
$B:W\!\otimes\!\H_{1}\rightarrowW\!\otimes\!\H_{1}$ satisfying $B^{\ast} = \bar{B}$
(i.e., $B$ is represented by a symmetric matrix)
\item an $\mbox{\rm Sp}(1)$ equivariant $\H$-linear map
$\Lambda : W\!\otimes\!\H_{1} \rightarrow V\,/\,W\!\otimes\!\H_{1}$.
\end{itemize}
If we fix the inclusion $W\!\otimes\!\H_{1} \hookrightarrow V$, then we say that
two sets of ADHM data $(B,\Lambda)$ and $(B',\Lambda')$ are
{\em equivalent\/} if
\begin{displaymath}
B' = U B U^{-1}, \qquad \Lambda' = v \Lambda U^{-1}
\end{displaymath}
for suitable $U\in O(W)$ and $v\in \mbox{\rm Sp}(V\,/\,W\!\otimes\!\H_{1})$. From the ADHM data,
we construct an $\mbox{\rm Sp}(1)$ equivariant family of $\H$-linear maps
$v(x) : W\!\otimes\!\H_{1} \rightarrow V$ parametrized by $x\in\H$, given by
\begin{displaymath}
v(x) = \left( \begin{array}{c}
\Lambda \\
B - xI
\end{array} \right)
\end{displaymath}
relative to the decomposition $V = (V\,/\,W\!\otimes\!\H_{1}) \oplus (W\!\otimes\!\H_{1})$.
\begin{theorem}[ADHM] \label{ADHM}
There is a one-to-one correspondence between equivalence classes of
ADHM data $(B,\Lambda)$ satisfying the two conditions
\begin{description}
\item[non-degeneracy] $v(x)$ is injective for all $x\in\H$
\item[ADHM condition] $\Lambda^{\ast}\Lambda + B^{\ast}B :
W\!\otimes\!\H_{1} \rightarrow W\!\otimes\!\H_{1}$ is real,
\end{description}
and gauge equivalence classes of $\mbox{\rm Sp}(1)$-invariant ASD connections
on $E$.
\end{theorem}
To construct the connection associated to a set of ADHM data, we
first observe that the non-degeneracy condition implies that
$f : x \mapsto \mbox{Coker}\,v(x)\subset V$ is a smooth map from
$S^{4}$ to the quaternionic projective space $P(V)$ (we map the point
at $\infty$ to the line $V\,/\,W\!\otimes\!\H_{1}$). We then define the corresponding
vector bundle $E$ and connection $A$ to be the pullback of the canonical
quaternionic line bundle over $P(V)$ with its standard connection
(induced by orthogonal projection from the trivial flat connection on V).
The fact that $A$ is ASD follows from the ADHM condition. For a complete
proof of the non-equivariant version of this theorem, see
\cite[\S3.3]{DK} or \cite{A}. The proof of equivariant version then
proceeds with minimal modification.
We now give an even more precise description of $\mbox{\rm Sp}(1)$-invariant
instantons, starting by examining the characters of the representations
$V$ and $W$.
\begin{lemma}
The ADHM representations $V,W$ described in (\ref{eq:V}) and (\ref{eq:W})
corresponding to the bundle $E \rightarrow S^{4}$ with $c_{2}(E) = -k$
are given by
\begin{displaymath}
V = \H_{1}^{k+1}, \qquad W = \mathbb{R}_{0}^{k},
\end{displaymath}
where $\H_{1}$ is the fundamental representation of $\mbox{\rm Sp}(1)$ acting
by left multiplication and $\mathbb{R}_{0}$ is the trivial real representation.
\end{lemma}
\begin{proof}
If $E$ is an $\mbox{\rm Sp}(1)$ equivariant vector bundle, then it is
clearly also equivariant with respect to any one-parameter subgroup $S^{1}$ of
$\mbox{\rm Sp}(1)$. We may therefore apply the results of \cite{Br} and \cite{BA}.
The fixed point set for this $S^{1}$-action on $S^{4}$ is the sphere
$S^{2}$, over which the bundle $E$ splits as $E|_{S^{2}}=L\oplus L^{\ast}$.
Here $L$ is a complex line bundle with an $S^{1}$-action and $L^{\ast}$
is its dual. As in \cite{Br}, such $S^{1}$ equivariant bundles $E$ are
characterized by a pair of constants $(m,k)$, where $m$ is the weight of
the $S^{1}$-action on $L$ and $k = c_{1}(L^{\ast})$. In our case this
$S^{1}$-action is derived from the fundamental representation of $\mbox{\rm Sp}(1)$,
and so its weight is $m = 1/2$. In addition, noting that $2mk = -c_{2}(E)$,
we see that the two definitions of $k$ agree. Using the equivariant index
calculations of \cite{Br}, Braam and Austin compute the representations $V, W$ in
equations (3.4) and (3.5) of \cite{BA}. For $m = 1/2$ we have
$V=(\mathbb{C}_{1/2}\oplus\mathbb{C}_{-1/2})^{k+1}$ and $W=\mathbb{R}_{0}^{k}$ as
representations of $S^{1}$. The corresponding $\mbox{\rm Sp}(1)$ representations
are then $V = \H_{1}^{k+1}$ and $W = \mathbb{R}_{0}^{k}$.
\end{proof}
From this lemma, we see that the ADHM data $(B,\Lambda)$ is a pair of
$\mbox{\rm Sp}(1)$ equivariant maps $B : \H_{1}^{k}\rightarrow\H_{1}^{k}$
and $\Lambda : \H_{1}\rightarrow\H_{1}$. We then have
$gBg^{-1} = B$ and $g\Lambda g^{-1} = \Lambda$ for all $g\in\mbox{\rm Sp}(1)$,
where $g$ acts by quaternion multiplication. It follows that both
$B$ and $\Lambda$ must be {\em real\/} transformations. Recalling that
$B$ is symmetric, we see that $B$ is diagonalizable with real
eigenvalues. Choosing a suitable basis, we can therefore write the map
$v(x) : \H_{1}^{k} \rightarrow \H_{1}^{k+1}$ as a matrix of the form
\begin{displaymath}
v(x) = \left( \begin{array}{ccc}
\lambda_{1} & \cdots & \lambda_{k} \\
b_{1} - x & 0 & 0 \\
0 & \ddots & 0 \\
0 & 0 & b_{k} - x
\end{array}
\right)
\end{displaymath}
with real eigenvalues $\{b_{1},\ldots,b_{k}\}$ and real
scales $\{\lambda_{1},\ldots,\lambda_{k}\}$ with $\lambda_{i} \geq 0$.
The non-degeneracy condition of Theorem \ref{ADHM} implies that
the $b_{i}$ are distinct and the $\lambda_{i}$ are nonzero, while the
ADHM condition is automatically satisfied since $B$ and $\Lambda$ are
real. Computing $\mbox{Coker}\,v(x) = \mbox{Ker}\,v(x)^{\ast}$, we obtain
\begin{theorem}
Given $k$ distinct real centers $\{b_{1},\ldots,b_{k}\}$ and
$k$ positive real scales $\{\lambda_{1},\ldots,\lambda_{k}\}$, let
$f : S^{4} \rightarrow P(\H_{1}^{k+1})$ be the map given by
\begin{displaymath}
f\,:\,x\,\mapsto\,\left( 1\,:\,\frac{\lambda_{1}}{x-b_{1}}\,:\,
\cdots\,:\,\frac{\lambda_{k}}{x-b_{k}} \right).
\end{displaymath}
The $\mbox{\rm Sp}(1)$-invariant connection obtained by taking the pullback
of the standard connection on the canonical bundle over $P(\H_{1}^{k+1})$
is then ASD and has Chern class $c_{2} = -k$. Furthermore,
every $\mbox{\rm Sp}(1)$-invariant connection on a bundle $E\rightarrow S^{4}$
with $c_{2}(E) = -k$ is gauge equivalent to one of this form.
\end{theorem}
We note that the instantons constructed by the above theorem are the
superposition of $k$ basic instantons with distinct centers along the
real axis, which we recognize as the 't Hooft instantons from the previous
section in a different guise. By dimensional reduction, we therefore obtain
a constructive proof of the classification theorem for hyperbolic vortices.
\begin{corollary}
Given a set of $k$ distinct points $\{z_{1},\ldots,z_{k}\}$ in the
complex upper half-plane, there exists a finite action solution
$(a,\Phi)$ to the hyperbolic vortex equations, unique up to gauge
transformation, such that $\{z_{1},\ldots,z_{k}\}$ is the zero set
of the Higgs field $\Phi$. The Chern class of such a vortex is then
$c_{1}(a) = k$, obtained by counting the zeros of the Higgs field.
\end{corollary}
\subsection{Symmetric Gauge Transformations}
\label{gauge-transformation}
Now that we understand the relationship between symmetric instantons and
hyperbolic vortices, we examine how the notion of gauge equivalence
behaves under this dimensional reduction. Starting with a symmetric
$\mbox{\rm SU}(2)$ gauge transformation on $S^{4}$, we compute the resulting
$\mbox{\rm U}(1)$ gauge transformation on hyperbolic space $\mathcal{H}^{2}$. Then,
continuing to work in the simpler $\mathcal{H}^{2}$ picture, we examine
the conditions under which two hyperbolic vortices given by the
harmonic function ansatz of \S\ref{vortex-ansatz} are gauge
equivalent.
Using the quaternionic notation $x = t + rQ$, the most general
$\mbox{\rm Sp}(1)$-invariant gauge transformation on $S^{4}$ has the form
\begin{displaymath}
g = e^{Q\,\chi(t,r)} = \cos\chi(t,r) + Q\sin\chi(t,r),
\end{displaymath}
where $\chi$ is a real-valued function on $\mathcal{H}^{2}$. Computing
its differential, we have
\begin{equation*}
dg\,g^{-1} = Q\,d\chi\,+\,\sin\chi\:dQ\:e^{-Q\chi}
= Q\,d\chi\,-\,\frac{1}{2}\,Q\,(e^{2Q\chi}-1)\,dQ.
\end{equation*}
Applying this gauge transformation to the general $\mbox{\rm SO}(3)$-invariant
connection (\ref{symmetric-connection}), we obtain
\begin{align*}
g(A)&= \frac{1}{2} \, e^{Q\chi}\,\left[\,
Qa\,+\,(\Phi_{1}+Q\Phi_{2})\,dQ
\,\right]\,e^{-Q\chi}
\,-\,dg\,g^{-1} \\
&=\frac{1}{2}\left[\,
Q\,(a - 2\,d\chi) \,+\,
\left( e^{2Q\chi}\,[\,\Phi_{1}+Q\,(\Phi_{2}+1)\,] - Q \right)dQ
\,\right],
\end{align*}
noting that $g\,Q = Q\,g$ while $g\,dQ = dQ\,g^{-1}$.
The associated connection $ia$ and Higgs field $\Phi = \Phi_{1} +
i\,(\Phi_{2}+1)$ over $\mathcal{H}^{2}$ then transform according to
\begin{displaymath}
g(a) = a - 2\,d\chi, \qquad
g(\Phi) = e^{2i\chi}\Phi.
\end{displaymath}
Hence, the corresponding gauge transformation over $\mathcal{H}^{2}$
is simply $g = e^{2i\chi}$.
Suppose that we have two gauge equivalent hyperbolic vortices
$(a_{+},\Phi_{+})$ and $(a_{-},\Phi_{-})$ constructed by the
harmonic function ansatz of \S\ref{vortex-ansatz}, using
equation (\ref{eq:vortex-dbar})
with the super-potentials $\phi_{+}$ and $\phi_{-}$ respectively.
If these two vortices satisfy $a_{-} = g(a_{+})$ and
$\Phi_{-} = g(\Phi_{+})$ with a gauge transformation of the form
$g = e^{2i\chi}$ as discussed above, then we obtain the system
of differential equations
\begin{align}
\label{eq:equivalent-connections}
\del{\bar{z}} \log\phi_{-} & = \del{\bar{z}} \log\phi_{+}\,-\,
\del{\bar{z}}\,2i\chi \\
\label{eq:equivalent-fields}
\del{z} \log\phi_{-} & = e^{2i\chi}\,\del{z} \log\phi_{+}.
\end{align}
Note that equation (\ref{eq:equivalent-connections}) implies that
$\chi$ is harmonic,
\begin{displaymath}
\del{z}\del{\bar{z}}\,\chi = 0,
\end{displaymath}
as it is the imaginary part of a holomorphic function. Taking the
conjugate of (\ref{eq:equivalent-fields}) and inserting it into
(\ref{eq:equivalent-connections}), we can eliminate either
$\log\phi_{+}$ or $\log\phi_{-}$ from these equations to obtain
\begin{align*}
\del{\bar{z}}\log\phi_{+} =
\frac{e^{2i\chi}}{e^{2i\chi}-1}\,\del{\bar{z}}\,2i\chi
& \qquad
\del{z}\log\phi_{+} =
\frac{1}{e^{2i\chi}-1}\,\del{z}\,2i\chi \\
\del{\bar{z}}\log\phi_{-} =
\frac{1}{e^{2i\chi}-1}\,\del{\bar{z}}\,2i\chi
& \qquad
\del{\bar{z}}\log\phi_{-} =
\frac{e^{2i\chi}}{e^{2i\chi}-1}\,\del{z}\,2i\chi.
\end{align*}
These equations may also be written in either of the two simpler forms
\begin{align*}
\del{\bar{z}}\log\phi_{\pm} =
\frac{e^{\pm i\chi}}{\sin\chi}\,\del{\bar{z}}\,\chi
& \qquad
\del{z}\log\phi_{\pm} =
\frac{e^{\mp i\chi}}{\sin\chi}\,\del{z}\,\chi
\end{align*}
or
\begin{align}
\label{eq:dzbar}
\del{\bar{z}}\log\phi_{\pm} & =
\del{\bar{z}}\log\left( e^{\pm 2i\chi}-1 \right) \qquad
\del{z}\log\phi_{\pm} =
\del{z}\log\left( e^{\mp 2i\chi}-1 \right).
\end{align}
Substituting these formulae for the partial derivatives into
(\ref{eq:vortex-dbar}), we can
express the two gauge equivalent hyperbolic vortices completely
in terms of the gauge transformation without reference to their
super-potentials.
Furthermore, computing the Laplacian of the super-potentials
$\phi_{1},\phi_{2}$ in terms of the function $\chi$, we have
\begin{equation*}
\frac{1}{\phi_{\pm}}\,\del{\bar{z}}\del{z}\,\phi_{\pm}
=
\del{\bar{z}}\log\phi_{\pm}\,\del{z}\log\phi_{\pm} \,+\,
\del{\bar{z}}\del{z}\log\phi_{\pm}
=
\frac{e^{\pm i\chi}}{\sin\chi}\,\del{z}\del{\bar{z}}\,\chi.
\end{equation*}
Hence, the requirements that $\phi_{1},\phi_{2}$ be harmonic reduce
simply to the condition that $\chi$ be harmonic, which we have already
established as a corollary to equation~(\ref{eq:equivalent-connections}).
We have thus proved
\begin{theorem} \label{vortex-construction}
Let $\chi$ be a real-valued harmonic function on the hyperbolic
upper half-plane $\mathcal{H}^2$. The two pairs $(a_{+},\Phi_{+})$
and $(a_{-},\Phi_{-})$ given by
\begin{align*}
\bar{\partial}_{a_{\pm}} & = \bar{\partial}\,+\,
\bar{\partial}\log\left( e^{\pm 2i\chi}-1 \right)
\,+\, \frac{d\bar{z}}{z-\bar{z}} \\
\Phi_{\pm} & = i\,(z-\bar{z})\,
\del{z}\log\left( e^{\mp 2i\chi}-1 \right)
\end{align*}
then satisfy the hyperbolic vortex equations (\ref{eq:vortex-a})
and (\ref{eq:vortex-b}) and are related by the gauge transformation
$g = e^{2i\chi}$. Conversely, any two gauge equivalent hyperbolic
vortices constructed via the harmonic function ansatz of
Theorem~\ref{theorem-vortex-ansatz} can be expressed in this form.
\end{theorem}
\subsection{The Unit Disc Model}
\label{unit-disc}
Until now, we have always used the upper half-plane model for hyperbolic
space $\mathcal{H}^2$. In some circumstances, it will be more convenient to
use the unit disc model. While the upper half-plane arises naturally by
the dimensional reduction technique discussed in the previous sections,
the calculations in the following section become much simpler and
exhibit significantly more symmetry if we can work on the unit disk.
Here we make the transition beween the two coordinate systems, showing
how the formulae of the previous sections behave under the transformation.
Letting $z$ be the complex coordinate for the upper half-plane
and $w$ the coordinate for the unit disc, these two models are related
by the conformal transformation
\begin{equation} \label{eq:coordinate-transform}
w = \frac{i-z}{i+z}, \qquad
z = i\,\frac{1-w}{1+w}.
\end{equation}
This map takes the upper half-plane to the interior of the unit disc,
mapping the real axis to the unit circle. In particular, the point
$z = i$ maps to the origin, while the origin maps to $w = 1$ and the
point at infinity maps to $w = -1$. The positive imaginary axis in
$z$-coordinates maps to the interval $(-1,1)$ on the real axis in
$w$-coordinates.
The hyperbolic metric on the unit disk is given by
\begin{displaymath}
h = \frac{4}{\left(1-|w|^2\right)^2}\,dw\,d\bar{w}.
\end{displaymath}
The vortex equations remain fixed under this change of coordinates;
equation (\ref{eq:vortex-a}) is preserved because the transformation
is holomorphic, while equation (\ref{eq:vortex-b}) is a relationship
between coordinate-invariant scalar quantities. To compute the new
connection and Higgs fields generated by the harmonic function ansatz,
we first note that
\begin{displaymath}
z - \bar{z} = 2i\,\frac{1-|w|^2}{|1+w|^2},
\end{displaymath}
and that the partial derivatives transform according to
\begin{displaymath}
\del{w} = -\frac{2i}{(1+w)^2}\,\del{z}, \qquad
\del{\bar{w}} = \frac{2i}{(1+\bar{w})^2}\,\del{\bar{z}}.
\end{displaymath}
Converting the formula (\ref{eq:vortex-dbar})
for the hyperbolic vortex associated to a harmonic super-potential $\phi$
to $w$-coordinates on the unit disc, Theorem~\ref{theorem-vortex-ansatz}
then becomes
\begin{theorem} \label{w-theorem-vortex-ansatz}
Given a positive real-valued super-potential $\phi$ over the hyperbolic
disc $\mathcal{H}^{2}$, the $\mbox{\rm U}(1)$ connection and Higgs field pair
$(a, \Phi)$ defined by
\begin{align} \label{eq:w-dbar}
\bar{\partial}_{a} = \bar{\partial} \,+\, \bar{\partial}\log\phi \,+\,
\frac{d\bar{w}}{1-|w|^2}\,\frac{1+w}{1+\bar{w}} \qquad
\Phi =-i\,(1-|w|^2)\,\frac{1+w}{1+\bar{w}}\,\del{w}\log\phi,
\end{align}
satisfies the vortex equations (\ref{eq:vortex-a}) and (\ref{eq:vortex-b})
if and only if the super-potential $\phi$ is harmonic.
\end{theorem}
The Chern class $c_{1}(a)$ of this connection is given in these
coordinates by
\begin{equation}\begin{split}
c_{1}(a) =\frac{1}{2\pi}\int_{\mathcal{H}^{2}}
\ast_{h}\left(1-|\Phi|^{2}\right)
=\frac{1}{2\pi}\int_{D^{2}}\left(
\frac{4}{\left(1-|w|^{2}\right)^{2}} -
4\,\left|\del{w}\log\phi\right|^{2}
\right) d\mu, \label{eq:c1-w}
\end{split}\end{equation}
where $d\mu$ is the volume element on the disc.
We now return to the simple hyperbolic vortex constructed
in \S\ref{vortex-tHooft}, which we obtained by a dimensional
reduction of the basic $c_{2}=1$ instanton. In $w$-coordinates on the
unit disc, the super-potential (\ref{eq:simple-potential}) becomes
\begin{equation} \label{eq:w-simple-potential}
\phi = 2\,\frac{1-|w|^{4}}{\left|1-w^{2}\right|^{2}}.
\end{equation}
Computing the partial derivatives of its logarithm, we obtain
\begin{equation*}
\del{w}\log\phi = \frac{2w}{1-|w|^{4}}\,
\frac{1-\bar{w}^{2}}{1-w^{2}} \qquad
\del{\bar{w}}\log\phi = \frac{2\bar{w}}{1-|w|^{4}}\,
\frac{1-w^{2}}{1-\bar{w}^{2}},
\end{equation*}
which when inserted into formula (\ref{eq:w-dbar}) above give the vortex
\begin{equation*}
\bar{\partial}_{a} = \bar{\partial} + \frac{d\bar{w}}{1+|w|^{2}}\,
\frac{1+w}{1-\bar{w}}\, \qquad
\Phi = -\frac{2iw}{1+|w|^{2}}\,\frac{1-\bar{w}}{1-w}.
\end{equation*}
Note that here the Higgs field $\Phi$ vanishes only at the origin,
where it has a simple zero, and so the Chern class of this vortex
should be $c_{1}(a) = 1$.
Using (\ref{eq:c1-w}) to explicitly calculate this Chern class,
we obtain the integral
\begin{displaymath}
c_{1}(a) = \frac{1}{2\pi} \int_{D^{2}} \left(
\frac{4}{\left(1-|w|^{2}\right)^{2}} -
\frac{16\,|w|^{2}}{\left(1-|w|^{4}\right)^{2}}
\right) d\mu.
\end{displaymath}
Taking polar coordinate $r,\theta$ on the unit disc, this
integral becomes
\begin{equation*}
c_{1}(a) = \int_{0}^{1} \left(
\frac{4\,r}{\left(1-r^2\right)^2} -
\frac{16\,r^3}{\left(1-r^4\right)^2}
\right) dr
= \left[ \frac{2}{1-r^2} - \frac{4}{1-r^4}
\right]_{r=0}^{r=1}.
\end{equation*}
To evaluate this expression at $r=1$, we substitute $r=1-x$
and expand it about $x=0$, giving us
\begin{align*}
\frac{2}{1-(1-x)^{2}} - \frac{4}{1-(1-x)^4}
&= \frac{2}{2x-x^2+O(x^3)} - \frac{4}{4x-6x^2+O(x^3)} \\*
&= \left(\frac{1}{x} + \frac{1}{2} + O(x)\right) -
\left(\frac{1}{x} + \frac{3}{2} + O(x)\right) \\*
&= -1 + O(x) \rule{0in}{3.5ex}.
\end{align*}
We therefore see that the Chern class of this vortex is
indeed $c_{1}(a) = 1$ as we predicted by counting the zeros
of the Higgs field.
The formulae from \S\ref{gauge-transformation} giving the
two super-potentials in terms of the gauge transformation remain
unchanged, except for replacing all the $z$'s with $w$'s. In
particular, if the vortices determined by the super-potentials
$\phi_{+}$ and $\phi_{-}$ are gauge equivalent by a transformation
of the form $g = e^{2i\chi}$, then the differential equations
(\ref{eq:equivalent-connections}) and (\ref{eq:equivalent-fields}) become
\begin{align}
\label{eq:w-equivalent-connections}
\del{\bar{w}} \log\phi_{-} & = \del{\bar{w}} \log\phi_{+}\,-\,
\del{\bar{w}}\,2i\chi \\
\label{eq:w-equivalent-fields}
\del{w} \log\phi_{-} & = e^{2i\chi}\,\del{w} \log\phi_{+}.
\end{align}
The unit disc version of Theorem \ref{vortex-construction} is then
\begin{theorem} \label{w-vortex-construction}
Let $\chi$ be a real-valued harmonic function on the hyperbolic
unit disc $\mathcal{H}^2$. The two pairs $(a_{+},\Phi_{+})$
and $(a_{-},\Phi_{-})$ given by
\begin{align*}
\bar{\partial}_{a_{\pm}} & = \bar{\partial}\,+\,
\bar{\partial}\log\left( e^{\pm 2i\chi}-1 \right)
\,+\,
\frac{d\bar{w}}{1-|w|^2}\,\frac{1+w}{1+\bar{w}} \\
\Phi_{\pm} & = -i\,(1-|w|^2)\,\frac{1+w}{1+\bar{w}}\,
\del{w}\log\left( e^{\mp 2i\chi}-1 \right)
\end{align*}
then satisfy the hyperbolic vortex equations (\ref{eq:vortex-a})
and (\ref{eq:vortex-b}) and are related by the gauge transformation
$g = e^{2i\chi}$. Conversely, any two gauge equivalent hyperbolic
vortices constructed via the harmonic function ansatz of
Theorem~\ref{w-theorem-vortex-ansatz} can be expressed in this form.
\end{theorem}
\section{Holonomy Singularity}
\subsection{The Forg\'{a}cs, Horv\'{a}th, Palla Instanton}
\label{fhp}
In this section, we construct the singular instanton described by
P. Forg\'{a}cs, Z. Horv\'{a}th, and L. Palla in \cite{FHP1}.
In order to obtain a connection on $S^4\setminus S^2$ with a holonomy
singularity, Forg\'{a}cs {\em et al.\/} patch together two non-singular
connections on overlapping simply connected regions using a gauge
transformation. (This process is not unlike the clutching construction,
which creates ``twisted'' vector bundles given their local trivializations
and transition functions.) These two non-singular solutions are generated
by the harmonic function ansatz of Section~\ref{instanton-ansatz},
and since the super-potentials they use are $\mbox{\rm SO}(3)$-invariant,
the resulting connection can be analyzed in terms of the dimensional
reduction to hyperbolic space $\mathcal{H}^2$ discussed in Section 2.
Using the quaternionic notation $x = t + rQ$ with $t, r$ real, $r > 0$
and $Q$ pure imaginary satisfying $Q^{2} = -1$, we want to construct an
$\mbox{\rm SO}(3)$-invariant self-dual connection singular along the 2-sphere
$t=0, r=1$. By dimensional reduction, this translates into a vortex
over the hyperbolic upper half-plane with non-trivial holonomy around
the point $z = i$. If instead we work using the unit disc model of
hyperbolic space, our task takes the more symmetric form of finding a
hyperbolic vortex on the punctured disc with a holonomy singularity at
the origin.
We therefore set out to construct two gauge equivalent hyperbolic
vortices on the punctured disc, using the harmonic function ansatz
of \S\ref{vortex-ansatz}. For the two simply connected
regions, we let $P_{1}$ be the disc with a cut along the positive
real axis, and let $P_{2}$ be the disc with a cut along the negative
real axis. The areas of overlap are then the upper and lower half-discs,
excluding the real axis. Let $(a_{1},\Phi_{1})$ and $(a_{2},\Phi_{2})$
be the hyperbolic vortices corresponding to the super-potentials
$\phi_{1}$ and $\phi_{2}$ on the regions $P_{1}$ and $P_{2}$ respectively.
In light of Theorem~\ref{vortex-construction}, we begin by examining
the gauge transformation between these two vortices, rather than focusing
on the super-potentials. Here our gauge transformation $g$ is specified
by a real-valued harmonic function $\chi$, and we take
\begin{displaymath}
g = \left\{ \begin{array}{ll}
e^{+2i\chi} & \mbox{for $\Im w > 0$} \\
e^{-2i\chi} & \mbox{for $\Im w < 0$}.
\end{array}
\right.
\end{displaymath}
In other words, on the lower half-disc we use the inverse of the gauge
transformation that we use on the upper half-disc. In the notation
of \S\ref{gauge-transformation}, switching the sign of $2i\chi$
simply interchanges the resulting gauge equivalent super-potentials
$\phi_{+}$ and $\phi_{-}$ determined by equation (\ref{eq:dzbar}). Note that although the resulting $g$ is undefined
along the real axis, this does not pose a problem for our construction.
Indeed, we use $g$ directly only on regions excluding the real axis,
and we will find that the two hyperbolic vortices $(a_{1},\Phi_{1})$
and $(a_{2},\Phi_{2})$ constructed from $g$ are continuous across the
negative and positive axes respectively.
In \cite{FHP1}, Forg\'{a}cs {\em et al.\/} use the gauge transformation
specified by
\begin{equation} \label{eq:imlog}
2\chi = \left(\,
\frac{\pi}{2}\,+\,2 \arctan \frac{T_{2}}{1-T_{1}}
\,+\,2 \arctan \frac{T_{1}}{1-T_{2}}
\,\right).
\end{equation}
We will define the $T_{i}$ below, but before doing so we first
study the behavior of this gauge transformation for general values
of $T_{i}$. In particular, we would like to coerce $T_{1}$ and
$T_{2}$ into being the real and imaginary parts of a holomorphic
(or anti-holomorphic) function $f(w)$. Then where $|f(w)| = 1$,
the arguments of the arctans resemble the half-angle formula for
$\tan(\theta)$. In such circumstances we have
\begin{equation*}
\arctan \frac{T_{2}}{1-T_{1}} = \Im\log\,(1-\bar{f}) \qquad
\arctan \frac{T_{1}}{1-T_{2}} = \Im\log\,(1+if),
\end{equation*}
and we can write $\chi$ in terms of $f$ using
\begin{displaymath}
2\chi = \Im\log\left( i\,\frac{(1+if)^{2}}{(1-f)^{2}}
\right).
\end{displaymath}
We then see that $\chi$ is indeed harmonic as it is the imaginary part
of a holomorphic (or anti-holomorphic) function. Exponentiating, we
obtain
\begin{equation} \label{eq:exp-2ichi}
e^{2i\chi} = i\,\frac{(1-\bar{f})\,(1+if)}{(1-f)\,(1-i\bar{f})},
\end{equation}
and the expressions $e^{2i\chi}-1$ and $e^{-2i\chi}-1$ take the form
\begin{equation*}
e^{+2i\chi} - 1 = -\frac{(1-i)\,(1-f\bar{f})}{(1-f)\,(1-i\bar{f})}, \qquad
e^{-2i\chi} - 1 = -\frac{(1+i)\,(1-f\bar{f})}{(1-\bar{f})\,(1+if)},
\end{equation*}
which we shall use in equation (\ref{eq:dzbar}).
Note that if $|f| = 1$ then $\bar{f} = f^{-1}$, and we observe that
$e^{\pm 2i\chi} = 1$.
In equation (\ref{eq:imlog}) above, the $T_{i}$ are defined by
\begin{align*}
T_{1}&=\frac{1}{2\,S^{5}}\,\sqrt{(S+S_{-})^{2}-4}\,\left\{
\frac{1}{4} \left[4 - (S-S_{-})^2 \right]^{2}\,+\,
S^2 S_{-}^2\,-\,3\,(z+\bar{z})^{2}
\right\} \\
T_{2}&=\frac{1}{2\,S^{5}}\,\sqrt{4-(S-S_{-})^{2}}\,\left\{
\frac{1}{4} \left[4 - (S+S_{-})^2 \right]^{2}\,+\,
S^2 S_{-}^2\,-\,3\,(z+\bar{z})^{2}
\right\}
\end{align*}
with
\begin{displaymath}
S = |z + i|, \qquad S_{-} = |z - i|,
\end{displaymath}
using the complex coordinate $z$ on the upper half-plane. With formulae
such as these, it is not surprising that the mathematical community was
incredulous. Changing to the complex coordinate $w$ on the unit disc
via the conformal transformation (\ref{eq:coordinate-transform}), we have
\begin{displaymath}
S = \frac{2}{|w+1|}, \qquad
S_{-} = 2\,\frac{|w|}{|w+1|} = |w| S.
\end{displaymath}
Calculating the various components of the $T_{i}$, we obtain
\begin{align*}
\sqrt{(S+S_{-})^{2}-4} & =
2\,\frac{|\sqrt{w}-\sqrt{\bar{w}}|}{|w+1|} =
\left|\sqrt{w}-\sqrt{\bar{w}}\right|S \\
\sqrt{4-(S-S_{-})^{2}} & =
2\,\frac{|\sqrt{w}+\sqrt{\bar{w}}|}{|w+1|} =
\left|\sqrt{w}+\sqrt{\bar{w}}\right|S \\
\end{align*}
and
\begin{displaymath}
z + \bar{z} = -2i\,\frac{w-\bar{w}}{|w+1|^{2}}
= -\frac{i}{2}\,(w-\bar{w})\,S^{2}.
\end{displaymath}
Putting these pieces together, the $T_{i}$ are given much more simply by
\begin{align*}
T_{1} & = \frac{1}{2} \left|\sqrt{w}-\sqrt{\bar{w}}\right|
\left[\,
\frac{1}{4}\left(\sqrt{w}+\sqrt{\bar{w}}\right)^{4} \,+\,
w\bar{w} \,+\, \frac{3}{4}\,(w-\bar{w})^{2}
\,\right] \\*
& = \frac{1}{2} \left|w^{1/2}-\bar{w}^{1/2}\right|
\left(
w^2 + w^{3/2}\bar{w}^{1/2} + w\bar{w} +
w^{1/2}\bar{w}^{3/2} + \bar{w}^{2}
\right) \\*
& = \mp\frac{i}{2} \left(w^{5/2}-\bar{w}^{5/2}\right)
= \left(\Im w^{5/2}\right)
\left(\mbox{sign}\:\Im w^{1/2}\right),
\end{align*}
and
\begin{align*}
T_{2} & = \frac{1}{2} \left|\sqrt{w}+\sqrt{\bar{w}}\right|
\left[\,
\frac{1}{4}\left(\sqrt{w}-\sqrt{\bar{w}}\right)^{4} \,+\,
w\bar{w} \,+\, \frac{3}{4}\,(w-\bar{w})^{2}
\,\right] \\*
& = \frac{1}{2} \left|w^{1/2}+\bar{w}^{1/2}\right|
\left(
w^2 - w^{3/2}\bar{w}^{1/2} + w\bar{w} -
w^{1/2}\bar{w}^{3/2} + \bar{w}^{2}
\right) \\*
& = \pm\frac{1}{2} \left(w^{5/2}+\bar{w}^{5/2}\right)
= \left(\Re w^{5/2}\right)
\left(\mbox{sign}\:\Re w^{1/2}\right).
\end{align*}
Hence, we see that $T_{1}$ and $T_{2}$ are indeed the imaginary and real
parts of the function $f(w)$ defined by
\begin{displaymath}
f(w) = \left\{ \begin{array}{rcrcl}
w^{5/2} && 0 & \le\;\;\arg w\;\;\le & \pi \\
-\bar{w}^{5/2} && \pi & \le\;\;\arg w\;\;\le & 2\pi
\end{array}
\right.
\end{displaymath}
on the region $P_{1}$ or equivalently
\begin{displaymath}
f(w) = \left\{ \begin{array}{rcrcl}
\bar{w}^{5/2} && -\pi & \le\;\;\arg w\;\;\le & 0 \\
w^{5/2} && 0 & \le\;\;\arg w\;\;\le & \pi \\
\end{array}
\right.
\end{displaymath}
on the region $P_{2}$. This function $f(w)$ is holomorphic on the upper
half-disc and anti-holomorphic on the lower half-disc, and we note that
$f(w)$ is well defined and continuous over the whole unit disc.
We now have all that we need to calculate the gauge equivalent
connections and Higgs fields $a_{i},\Phi_{i}$ satisfying
$a_{2} = g(a_{1})$ and $\Phi_{2} = g(\Phi_{1})$.
We first consider
the super-potential $\phi_{1}$ defined on the region
$0 < \arg w < 2\pi$. Using the notation of
\S\ref{gauge-transformation}, on the upper half-disc
we have $\phi_{1} = \phi_{+}$ and $f = w^{5/2}$, giving us
\begin{displaymath}
\del{\bar{w}}\log\phi_{1}
= \del{\bar{w}}\log\frac{1-|w|^{5}}
{(1-w^{5/2})\,(1-i\,\bar{w}^{5/2})}
=\frac{5}{2}\,\frac{i\,\bar{w}^{3/2}}{1-|w|^5}\,\frac{1+i\,w^{5/2}}{1-i\,\bar{w}^{5/2}}.
\end{displaymath}
On the lower half-disc, we obtain the same expression
\begin{displaymath}
\del{\bar{w}}\log\phi_{1}
= \del{\bar{w}}\log\frac{1-|w|^{5}}
{(1+w^{5/2})\,(1-i\,\bar{w}^{5/2})}
= \frac{5}{2}\,\frac{i\,\bar{w}^{3/2}}{1-|w|^5}\,\frac{1+i\,w^{5/2}}{1-i\,\bar{w}^{5/2}},
\end{displaymath}
although this time we take $\phi_{1}=\phi_{-}$ and $f=-\bar{w}^{5/2}$.
Similarly, considering the super-potential $\phi_{2}$ defined
for $-\pi < \arg w < \pi$, on the upper half-disc with
$\phi_{2} = \phi_{-}$ and $f = w^{5/2}$ we have
\begin{displaymath}
\del{\bar{w}}\log\phi_{2}
= \del{\bar{w}}\log\frac{1-|w|^{5/2}}
{(1-\bar{w}^{5/2})\,(1+i\,w^{5/2})}
= \frac{5}{2}\,\frac{\bar{w}^{3/2}}{1-|w|^5}\,\frac{1-w^{5/2}}{1-\bar{w}^{5/2}},
\end{displaymath}
while on the lower half-disc, we again get the same expression
\begin{displaymath}
\del{\bar{w}}\log\phi_{2}
= \del{\bar{w}}\log\frac{1-|w|^{5/2}}
{(1-\bar{w}^{5/2})\,(1-i\,w^{5/2})}
= \frac{5}{2}\,\frac{\bar{w}^{3/2}}{1-|w|^5}\,\frac{1-w^{5/2}}{1-\bar{w}^{5/2}},
\end{displaymath}
taking $\phi_{2} = \phi_{+}$ and $f = \bar{w}^{5/2}$. Hence, as we
predicted, the vortices determined by $\phi_{1}$ and $\phi_{2}$ extend
continuously across the negative and positive real axes respectively,
even though the gauge transformation $g$ between them does not.
For the purposes of our vortex construction, we need not know the
super-potentials explicitly; rather, all we require are the complex partial
derivatives of their logarithms which we computed above. Nevertheless,
here we present the super-potentials as given in \cite{FHP1}. There the two
super-potentials take center stage, defined on their own instead of
being constructed from the gauge transformation as we have done.
Forg\'{a}cs {\em et al.\/} define $\phi_{1}$, $\phi_{2}$ by%
\SSfootnote{Actually, \cite{FHP1} uses $+2\,S^{5}\,T_{1}$ and $+2\,S^{5}\,T_{2}$
in the denominators of $\phi_{1}$ and $\phi_{2}$ respectively.
The resulting super-potentials are still harmonic, and they do
indeed generate gauge equivalent hyperbolic vortices. However,
this choice of sign is not consistent with the gauge
transformation they use, given here by (\ref{eq:imlog}).}
\begin{displaymath}
\phi_{1}=\frac{S^{5}-S_{-}^{5}}{S^{5}+S_{-}^{5}-2\,S^{5}\,T_{1}},
\qquad
\phi_{2}=\frac{S^{5}-S_{-}^{5}}{S^{5}+S_{-}^{5}-2\,S^{5}\,T_{2}}.
\end{displaymath}
Simplifying these expressions and writing them using the unit disc model
of hyperbolic space, we obtain
\begin{align} \label{eq:w-phi1}
\phi_{1}&=\frac{1-|w|^{5}}{1\,-\,2\,\Im w^{5/2}\,+\,|w|^5}
=\frac{1-|w|^{5}}{\left|1+i\,w^{5/2}\right|^{2}} \\
\label{eq:w-phi2}
\phi_{2}&=\frac{1-|w|^{5}}{1\,-\,2\,\Re w^{5/2}\,+\,|w|^5}
=\frac{1-|w|^{5}}{\left|1-w^{5/2}\right|^{2}}.
\end{align}
The reader may want to verify that the partial derivatives
$\del{\bar{w}}\log\phi_{1}$ and $\del{\bar{w}}\log\phi_{2}$
agree with those calculated on the previous page.
We now compute the Chern class $c_{1}$ of the vortex patched together from
the two super-potentials $\phi_{1}$ and $\phi_{2}$. Using equation
(\ref{eq:c1-w}), we have
\begin{displaymath}
c_{1}(a) = \frac{1}{2\pi}\int_{D^{2}}\left(
\frac{4}{(1-|w|^{2})^{2}} -
\frac{25\,|w|^{3}}{(1-|w|^{5})^{2}}
\right) d\mu.
\end{displaymath}
Taking polar coordinates $r,\theta$ on the disc $D^{2}$, this integral
becomes
\begin{equation*}
c_{1}(a)=\int_{0}^{1}\left(
\frac{4\,r}{(1-r^{2})^{2}} -
\frac{25\,r^{4}}{(1-r^{5})^{2}}
\right) dr
=\left[\frac{2}{1-r^{2}} - \frac{5}{1-r^{5}}
\right]_{r=0}^{r=1},
\end{equation*}
and evaluating this expression by the method used at the end of
\S\ref{unit-disc} shows that $c_{1}(a) = 3/2$. We can also arrive
at this same result by naively counting the zeros (with multiplicity)
of the Higgs field, as both $\del{w}\log\phi_{1}$ and
$\del{w}\log\phi_{2}$ vanish to order $3/2$ at the origin
but are otherwise nonzero.
\subsection{A Family of Singular Vortices}
In this section we will generalize the construction of \cite{FHP1} to
produce a family of hyperbolic vortices with varying Chern class
$c_{1}$. This family includes both the standard $c_{1} = 1$
vortex of \S\ref{unit-disc} and the fractionally charged
vortex of the previous section, as well as vortices with arbitrary
real $c_{1}$. We will continue to work using the unit disc model of
hyperbolic space.
\newcommand{\epsilon}{\epsilon}
\newcommand{\bar{\epsilon}}{\bar{\epsilon}}
Observing the resemblance between the two super-potentials
(\ref{eq:w-simple-potential}) and (\ref{eq:w-phi2}), we consider
a more general super-potential of the form
\begin{equation} \label{eq:phi-c}
\phi = \frac{1-|w|^{2c}}{\left|1-w^{c}\right|^{2}},
\end{equation}
where $c$ is a nonzero real constant. For non-integral $c$, this
$\phi$ is not well defined over the whole unit disc; rather, we
must restrict it to a simply-connected cut disc. If we loop around
the origin once in the positive direction, crossing our cut in
the unit disc, then this super-potential becomes
\begin{equation} \label{eq:phi-c-epsilon}
\phi' = \frac{1-|w|^{2c}}{\left|1-\epsilon w^{c}\right|^{2}},
\end{equation}
introducing a factor of $\epsilon = e^{2\pi ic}$ in the denominator.
In both of these cases, we note that $\phi$ and $\phi'$ vanish on
on the unit circle, except at the roots of $w^{a} = 1$
(or $w^{a} = \bar{\epsilon}$) where they have simple poles.
Taking the logarithmic derivatives of the super-potential $\phi$,
we obtain
\begin{equation*}
\del{w}\log\phi = \frac{c\,w^{c-1}}{1-|w|^{2c}}\,
\frac{1-\bar{w}^{c}}{1-w^{c}} \qquad
\del{\bar{w}}\log\phi = \frac{c\,\bar{w}^{c-1}}{1-|w|^{2c}}\,
\frac{1-w^{c}}{1-\bar{w}^{c}},
\end{equation*}
which we can use in equation (\ref{eq:w-dbar})
to construct a hyperbolic vortex $(a,\Phi)$. After looping around the
origin, the logarithmic derivatives become
\begin{equation*}
\del{w}\log\phi' = \frac{c\,\epsilon\,w^{c-1}}{1-|w|^{2c}}\,
\frac{1-\bar{\epsilon}\,\bar{w}^{c}}{1-\epsilon\,w^{c}} \qquad
\del{\bar{w}}\log\phi' = \frac{c\,\bar{\epsilon}\,\bar{w}^{c-1}}{1-|w|^{2c}}\,
\frac{1-\epsilon\,w^{c}}{1-\bar{\epsilon}\,\bar{w}^{c}},
\end{equation*}
and we let $(a',\Phi')$ be the corresponding hyperbolic vortex.
In order to construct a single hyperbolic vortex over the whole of
the unit disc, we would like to find a gauge transformation $g$
taking the vortex $(a,\Phi)$ to the vortex $(a',\Phi')$. Using equation
(\ref{eq:w-equivalent-fields}) for gauge equivalent Higgs fields,
we have
\begin{displaymath}
\del{w}\log\phi' = g\,\del{w}\log\phi,
\end{displaymath}
and so our gauge transformation must be
\begin{equation} \label{eq:g}
g = \epsilon\,\frac{1-\bar{\epsilon}\,\bar{w}^{c}}{1-\epsilon\,w^{c}}\,
\frac{1-w^{c}}{1-\bar{w}^{c}}.
\end{equation}
We must also verify that this gauge transformation $g$ satisfies equation
(\ref{eq:w-equivalent-connections}) for gauge equivalent connections
\begin{displaymath}
\del{\bar{w}}\log\phi' = \del{\bar{w}}\log\phi - \del{\bar{w}}\log g,
\end{displaymath}
which we leave to the reader. Hence when we loop around the origin, we
obtain a vortex that is gauge equivalent to our original one, and so
this vortex is well defined over the punctured disc.
Suppose that instead of defining $\epsilon = e^{2\pi ic}$, we use
an arbitrary constant $\epsilon$ with $|\epsilon| = 1$ in equations
(\ref{eq:phi-c-epsilon}) and (\ref{eq:g}). In this case, the gauge
transformation $g$ still maps between the two vortices corresponding
to the super-potentials $\phi$ and $\phi'$. Indeed, if we take
$c = 5/2$ and $\epsilon = -i$, then the resulting $\phi$, $\phi'$,
and $g$ correspond to the super-potentials (\ref{eq:w-phi2}) and
(\ref{eq:w-phi1}) and the gauge transformation (\ref{eq:exp-2ichi})
we used in \S\ref{fhp}.
Writing out the connection $a$ and Higgs field $\Phi$ of this vortex
explicitly using formula (\ref{eq:w-dbar}),
we obtain
\begin{align*}
\bar{\partial}_{a}&= \bar{\partial} + \left(
\frac{c\,\bar{w}^{c-1}}{1-|w|^{2c}}\,
\frac{1-w^{c}}{1-\bar{w}^{c}} +
\frac{1}{1-|w|^{2}}\,\frac{1+w}{1+\bar{w}}
\right) d\bar{w} \\
\Phi &= -ic\,w^{c-1}\,\frac{1-|w|^{2}}{1-|w|^{2c}}\,
\frac{1+w}{1+\bar{w}}\,
\frac{1-\bar{w}^{c}}{1-w^{c}}.
\end{align*}
The Chern class $c_{1}(a)$ of this vortex is given by equation
(\ref{eq:c1-w}), yielding the integral
\begin{align*}
c_{1}(a) & = \frac{1}{2\pi}\int_{D^{2}}\left(
\frac{4}{\left(1-|w|^2\right)^{2}} -
\frac{4c^{2}|w|^{2c-2}}{\left(1-|w|^{2c}\right)^{2}}
\right) d\mu \\
& = \int_{0}^{1}\left(
\frac{4r}{(1-r^{2})^{2}} -
\frac{4c^{2}r^{2c-1}}{(1-r^{2c})^{2}}
\right) dr
= \left[
\frac{2}{1-r^{2}} - \frac{2c}{1-r^{2c}}
\right]_{r=0}^{r=1},
\end{align*}
where we have used polar coordinates $r,\theta$ on the unit disc.
Evaluating this final expression by the method used at the end of
\S\ref{unit-disc}, we have $c_{1}(a) = c - 1$.
For $c = 2$, this construction yields the standard $c_{1} = 1$
hyperbolic vortex which we discussed in \S\ref{unit-disc}.
If we take $c = 5/2$ then we obtain the $c_{1} = 3/2$ vortex
given by the super-potential (\ref{eq:w-phi2}) on the cut
disc $P_{2}$. Note that with our construction, it is no longer
necessary to cover the disc with two overlapping regions as
we did in \S\ref{fhp}. Rather, it is sufficient to take
a single vortex on the cut disc and then study how it behaves
across that cut. For integer values of $c$, the vortex is
continuous across the cut, while for other values the vortex
changes by a gauge transformation. The flat $c=1$ vortex in
this family is
\begin{equation*}
\bar{\partial}_{a} = \bar{\partial} + \frac{2\,d\bar{w}}{1-\bar{w}^{2}} \qquad
\Phi = -i\,\frac{1+w}{1+\bar{w}}\,\frac{1-\bar{w}}{1-w},
\end{equation*}
which we readily see has $|\Phi| = 1$ and $F_{a} = 0$, and we
therefore have $c_{1} = 0$ as expected. We observe that this
vortex is equivalent to the standard flat vortex $(a=0, \Phi=1)$
using the gauge transformation (\ref{eq:g}) with $\epsilon = -1$.
To compute the holonomy around loops circling the origin, we
introduce polar coordinates $r, \theta$ on the unit disc. In
these coordinates, the complex differentials $dw$ and $d\bar{w}$
are
\begin{displaymath}
dw = \frac{w}{|w|}\,dr + iw\,d\theta, \qquad
d\bar{w} = \frac{\bar{w}}{|\bar{w}|} \, dr - i\bar{w}\,d\theta.
\end{displaymath}
In a small neighborhood of the origin, our connection is approximated by
\begin{equation*}
a \approx c \left( \bar{w}^{c-1}\,d\bar{w} - w^{c-1}\,dw \right)
= -ic \left( w^{c} + \bar{w}^{c} \right) d\theta + \cdots
\end{equation*}
(only the $d\theta$ term is needed for calculating the holonomy).
The contribution to the holonomy around the circle $|w| = r$ due to
the connection $a$ is then given by the loop integral
\begin{align*}
\oint_{|w|=r}a &= \int_{0}^{2\pi}\!-ic r^{c}
\left( e^{ic\theta} + e^{-ic\theta}
\right) d\theta \\
&=r^{c} \left( -e^{2\pi ic} + e^{-2\pi ic} \right)
=-2 r^{c} \sin 2\pi ic.
\end{align*}
For $c > 0$, this expression vanishes as $r \rightarrow 0$, and thus
the limit holonomy around the origin comes entirely from the gauge
transformation $g$ given by (\ref{eq:g}). Near the origin, we have
$g\approx\epsilon=e^{2\pi ic}$. Hence, the limit of the holonomy around
small loops centered at the origin is $e^{2\pi ic}$.
By the dimensional reduction technique of Section 2, our family of
singular hyperbolic vortices corresponds to a similar family of
$\mbox{\rm SO}(3)$-invariant self-dual connections over $S^{4}\setminus S^{2}$.
Using the results and quaternionic notation of \S\ref{gauge-transformation},
we see that the holonomy around small loops linking the singular surface
$S^{2}$ is $e^{\pi Qc}$. Viewing these connections as $\mbox{\rm SU}(2)$ connections
on a bundle $E$, we then obtain a splitting $E=L\oplus L^{\ast}$ on a
neighborhood of $S^{2}$ with respect to which the holonomy takes the
standard form
\begin{displaymath}
\left( \begin{array}{cc}
e^{2\pi i\alpha} & 0 \\
0 & e^{-2\pi i\alpha}
\end{array} \right)
\end{displaymath}
for a constant $\alpha$ in the range $[0,1/2)$. In the $\mbox{\rm SO}(3)$-symmetric
case, the complex line bundle $L$ on $S^{2}$ has Chern class $c_{1}(L)=-1$.
For our family of singular solutions, the holonomy parameter is
$\alpha=(c-\lfloor c\rfloor)/2$, where $\lfloor c\rfloor$ is the greatest
integer less than or equal to $c$.
|
{
"timestamp": "2005-03-26T00:10:29",
"yymm": "0503",
"arxiv_id": "math/0503611",
"language": "en",
"url": "https://arxiv.org/abs/math/0503611"
}
|
\section{Introduction}
Entanglement, the nonclassical correlations between spatially separated particles, is typically a signature of interactions in the past or
emergence from a common source. However, it can also arise as the interference of identical particles \cite{Yur92}.
By postselecting experimental data based on the ``click'' of detectors \cite{Shi88,Ou88}, photons scattered at a beam splitter have violated a Bell
inequality, even if they originated from independent sources \cite{Pit03,Fat04}. In reverse, triggered by an interferometric
Bell-state measurement, entanglement has been swapped \cite{Zuk93} to initially uncorrelated photons of different Bell pairs \cite{Pan98a,Pan98b,Jen02}.
The observation of these nonclassical interference effects is an important step on the road towards an optical approach of quantum information
processing \cite{Kni01,Fra02}.
Being furnished by interference, the ability of a beam splitter to entangle the polarizations of two independent photons depends on their
indistinguishability \cite{Fey69}. One of the incident photons is horizontally polarized in state $|{\rm H};\psi\rangle$, the other
vertically polarized in $|{\rm V};\phi\rangle$. The photons are partially distinguishable by their temporal degrees of freedom captured
in the kets $|\psi\rangle$ and $|\phi\rangle$. Besides temporal which-path information inherited from incident photons, a scattered
two-photon state possibly holds polarization which-path information. We make no assumptions about the scattering amplitudes
connecting polarizations at the beam splitter, except that they constitute a unitary scattering matrix.
Translated to a polarization-conserving beam splitter, this corresponds to incident photons in states $|\sigma;\psi\rangle$ and
$|\sigma';\phi\rangle$ where $\sigma$, $\sigma'$ are arbitrary superpositions of ${\rm H}$, ${\rm V}$.
Our analysis generalizes existing work on a polarization-conserving beam splitter where $\sigma={\rm H}$ and $\sigma'={\rm V}$ \cite{Bos02,Fat04}.
The polarization-state $\rho$ of a scattered photon pair is established from the scattering amplitudes of the beam splitter, the shape and timing of
photonic wavepackets ($|\psi\rangle$, $|\phi\rangle$) and the time-window of coincidence detection. If not erased by
ultra-coincidence detection, an amount of temporal distinguishability of $(1-|\langle \psi|\phi \rangle|^{2})$ pertains corresponding to a mixed state
$\rho$. We calculate both its concurrence and the Bell-CHSH parameter.
The ability of the latter to witness entanglement can disappear in the presence of a Mandel dip.
In terms of a polarization-conserving beam splitter, this corresponds to a deviation of $\sigma$, $\sigma'$ from
$\sigma={\rm H}$ and $\sigma'={\rm V}$.
\section{Formulation of the problem}
In a second-quantized notation, the incident two-photon state $|{\rm H};\psi\rangle_{\rm L}|{\rm V};\phi\rangle_{\rm R}$ takes the form
\begin{equation}
|{\rm \Psi}_{\rm in}\rangle=
{\rm \Psi}_{\rm H,L}^{\dagger}{\rm \Phi}_{\rm V,R}^{\dagger}|0\rangle,
\label{Psiin}
\end{equation}
with field creation operators given by (see Fig. \ref{beamsplitter})
\begin{equation}
{\rm \Psi}_{\rm H,L}^{\dagger}=\int d\omega\, a_{\rm H}^{\dagger}(\omega)\psi^{*}(\omega), \quad
{\rm \Phi}_{\rm V,R}^{\dagger}=\int d\omega\, b_{\rm V}^{\dagger}(\omega)\phi^{*}(\omega).
\end{equation}
(The subscripts R,L indicate the two sides of the beam splitter.)
The operators $a_{i}(\omega)$ with $i={\rm H},{\rm V}$ satisfy commutation rules
\begin{equation}
[a_{i}(\omega),a_{j}(\omega')]=0, \quad [a_{i}(\omega),a_{j}^{\dagger}(\omega')]=\delta_{ij}\delta(\omega-\omega').
\end{equation}
The same commutation rules hold for the operators $b_{i}(\omega)$, with commutation among
$a$ and $b$.
The outgoing operators $c_{i}(\omega)$, $d_{i}(\omega)$ are related to the incoming ones
$a_{i}(\omega)$, $b_{i}(\omega)$ by a $4 \times 4$ unitary scattering matrix $S$, decomposed in
$2 \times 2$ reflection and transmission matrices $r$,$t$,$t'$,$r'$:
\begin{equation}
\left(\begin{array}{c} c(\omega) \\ d(\omega) \end{array} \right) =
\left( \begin{array}{ll}
r & t' \\
t & r' \end{array} \right)
\left(\begin{array}{c} a(\omega) \\ b(\omega) \end{array} \right), \quad
a(\omega) \equiv \left(\begin{array}{c} a_{\rm H}(\omega) \\ a_{\rm V}(\omega) \end{array} \right),
\label{inout}
\end{equation}
and vectors $b(\omega)$, $c(\omega)$, $d(\omega)$ defined similarly. The scattering amplitudes are
frequency-independent.
\begin{widetext}
The outgoing state $|{\rm \Psi}_{\rm out}\rangle$ can be conveniently written in a matrix notation
\begin{equation}
|{\rm \Psi}_{\rm out}\rangle=
\int d\omega \int d\omega' \, \psi^{*}(\omega)\phi^{*}(\omega')
\left(\begin{array}{c} c^{\dagger}(\omega) \\ d^{\dagger}(\omega) \end{array} \right)^{\rm T}
\left( \begin{array}{ll}
r\sigma_{\rm in}t'^{\rm T} & r\sigma_{\rm in}r'^{\rm T} \\
t\sigma_{\rm in}t'^{\rm T} & t\sigma_{\rm in}r'^{\rm T} \end{array} \right)
\left(\begin{array}{c} c^{\dagger}(\omega') \\ d^{\dagger}(\omega') \end{array} \right)|0\rangle.
\label{psimatout}
\end{equation}
\end{widetext}
Here we used the unitarity of $S$ and $\sigma_{\rm in}=(\sigma_{x}+i\sigma_{y})/2$, with $\sigma_{x}$ and $\sigma_{y}$ Pauli matrices,
corresponds to the polarizations of the incoming photons cf. Eq. (\ref{Psiin}).
The matrix $\sigma_{\rm in}$ has rank 1 reflecting the fact that
polarizations are not entangled prior to scattering. Since we make no assumptions about the scattering amplitudes (apart from the unitarity of $S$),
the choice of $\sigma_{\rm in}$ is without loss of generality (see Appendix \ref{stostate}).
\begin{figure}
\includegraphics[width=8cm]{beamsplitter}
\caption{
Schematic drawing of generation and detection of polarization-entanglement at a beam splitter.
The independent sources SL and SR each create a photon in modes $\{a\}$ and $\{b\}$ cf. Eq. (\ref{Psiin}).
The beam splitter with unitary $4 \times 4$ scattering matrix $S$ couples the polarization of incoming modes to the
polarization of outgoing modes $\{c\}$ and $\{d\}$. Polarizations are mixed by $R_{\rm L},R_{\rm R}$ and absorbed by photodetectors
DL,DR. A coincidence circuit C registers simultaneous detection of photons.
\label{beamsplitter}
}
\end{figure}
The joint probability per unit $\mbox{(time)}^{2}$ of absorbing a photon with polarization $i$ at detector DL and a photon with
polarization $j$ at detector DR at times $t$ and $t'$ respectively is given by \cite{Gla63}
\begin{equation}
w_{ij}(t,t') \propto \langle {\rm \Psi}_{\rm out}|E_{i{\rm L}}^{(-)}(t)E_{j{\rm R}}^{(-)}(t')E_{j{\rm R}}^{(+)}(t')E_{i{\rm L}}^{(+)}(t)
|{\rm \Psi}_{\rm out}\rangle,
\end{equation}
where $E_{i{\rm L}}^{(+)}(t)$ and $E_{i{\rm R}}^{(+)}(t)$ are the positive frequency field operators of polarization $i$ at detectors DL and DR.
The probability $C_{ij}(t)$ of a coincidence event within time-windows $\tau$ around $t$ is given by
\begin{equation}
C_{ij}(t)=\int_{t-\frac{\tau}{2}}^{t+\frac{\tau}{2}} dt'\int_{t-\frac{\tau}{2}}^{t+\frac{\tau}{2}} dt''w_{ij}(t',t'').
\label{Corstart}
\end{equation}
Experimentally, the time-window $\tau$ has typically a lower bound determined by the random rise time of an avalanche of charge
carriers in response to a photon absorption event.
The polarization-entanglement is detected by violation of the Bell-CHSH inequality \cite{Cla69}. This requires two local polarization mixers
$R_{\rm L}$ and $R_{\rm R}$. The Bell-CHSH parameter ${\cal E}$ is
\begin{equation}
{\cal E}=|E(R_{\rm L},R_{\rm R})+E(R'_{\rm L},R_{\rm R})+E(R_{\rm L},R'_{\rm R})-E(R'_{\rm L},R'_{\rm R})|,
\label{Bellpar}
\end{equation}
where $E(R_{\rm L},R_{\rm R})$ is related to the correlators $C_{ij}(R_{\rm L},R_{\rm R})$ by
\begin{equation}
E=\frac{C_{\rm HH}+C_{\rm VV}-C_{\rm HV}-C_{\rm VH}}{C_{\rm HH}+C_{\rm VV}+C_{\rm HV}+C_{\rm VH}}.
\label{Edef}
\end{equation}
Substituting the correlators of Eq. (\ref{Corstart}) into Eq. (\ref{Edef}), we see that
\begin{equation}
E(R_{\rm L},R_{\rm R})={\rm Tr}\, \rho \, (R_{\rm L}^{\dagger}\sigma_{z}R_{\rm L}) \otimes (R_{\rm R}^{\dagger}\sigma_{z}R_{\rm R}),
\label{Erho}
\end{equation}
where $\sigma_{z}$ is a Pauli matrix and $\rho$ a $4 \times 4$ polarization density matrix with elements
\begin{widetext}
\begin{equation}
\rho_{ij,mn} = \frac{1}{\mathcal{N}} \left( (1 + |\alpha|^{2})(\gamma_{1})_{ij}(\gamma_{1})^{*}_{mn}+(1 - |\alpha|^{2})
(\gamma_{2})_{ij}(\gamma_{2})^{*}_{mn} \right).
\end{equation}
The parameter $\alpha$ is given by
\begin{equation}
\alpha = \frac{\int_{t-\frac{\tau}{2}}^{t+\frac{\tau}{2}} dt' \int d\omega \int d\omega'
\phi(\omega)\psi^{*}(\omega')e^{i(\omega-\omega')t'}}
{\sqrt{\left(\int_{t-\frac{\tau}{2}}^{t+\frac{\tau}{2}} dt' \int d\omega \int d\omega' \phi(\omega)\phi^{*}(\omega')
e^{i(\omega-\omega')t'}\right)
\left(\int_{t-\frac{\tau}{2}}^{t+\frac{\tau}{2}} dt' \int d\omega \int d\omega' \psi(\omega)\psi^{*}(\omega')e^{i(\omega-\omega')t'}\right)}}
\end{equation}
\end{widetext}
and $\gamma_{1}$,$\gamma_{2}$ are $2 \times 2$ matrices related to the scattering amplitudes by
\begin{equation}
\gamma_{1} = r\sigma_{\rm in}r'^{\rm T}+t'\sigma_{\rm in}^{\rm T}t^{\rm T}, \quad
\gamma_{2} = r\sigma_{\rm in}r'^{\rm T}-t'\sigma_{\rm in}^{\rm T}t^{\rm T}.
\label{gammas}
\end{equation}
The normalization factor $\mathcal{N}$ takes the form
\begin{equation}
\mathcal{N}=(1 + |\alpha|^{2}){\rm Tr}\,\gamma_{1}^{\dagger}\gamma_{1}^{\vphantom{\dagger}}+
(1 - |\alpha|^{2}){\rm Tr}\,\gamma_{2}^{\dagger}\gamma_{2}^{\vphantom{\dagger}}.
\label{norm}
\end{equation}
The parameter $1-|\alpha|^{2} \in (0,1)$ represents the amount of temporal which-path information.
Generally, the time-window $\tau$ is much larger than the coherence times or temporal difference of the wavepackets. We may then take the
limit $\tau \rightarrow \infty$ and $\alpha$ reduces to the overlap of wavepackets
\begin{equation}
\alpha=\int d\omega \phi(\omega)\psi^{*}(\omega).
\label{overlap}
\end{equation}
In the opposite limit of ultra-coincidence detection where $\tau \rightarrow 0$,
temporal which-path information is completely erased corresponding to $|\alpha|^{2}=1$.
\section{Entanglement of formation}
\label{EOF}
The entanglement of formation of the mixed state $\rho$ is quantified by the concurrence $\mathcal{C}$ \cite{Woo98} given by
\begin{equation}
\mathcal{C}={\rm max}\left(0,\sqrt{\lambda_{1}}-\sqrt{\lambda_{2}}-\sqrt{\lambda_{3}}-\sqrt{\lambda_{4}}\right).
\end{equation}
The $\lambda_{i}$'s are the eigenvalues of the matrix product $\rho\tilde{\rho}$, where
$\tilde{\rho}=(\sigma_{y}\otimes\sigma_{y})\rho^{*}(\sigma_{y}\otimes\sigma_{y})$,
in the order $\lambda_{1} \ge \lambda_{2} \ge \lambda_{3} \ge \lambda_{4}$. The concurrence ranges from 0 (no entanglement) to
1 (maximal entanglement).
For simplicity of notation it is convenient to define $(\widehat{xy})_{ij,mn} \equiv x_{ij}y^{*}_{mn}$. The matrix $\tilde{\rho}$ can be
written as
\begin{equation}
\tilde{\rho} = \frac{1}{\mathcal{N}} \left( (1 + |\alpha|^{2})\widehat{\tilde{\gamma}_{1}\tilde{\gamma}_{1}}+
(1 - |\alpha|^{2})\widehat{\tilde{\gamma}_{2}\tilde{\gamma}_{2}} \right),
\end{equation}
with $\tilde{\gamma} \equiv \sigma_{y}\gamma^{*}\sigma_{y}$.
The product $\rho\tilde{\rho}$ takes the simple form
\begin{equation}
\rho\tilde{\rho}=\frac{{\rm Tr}\,\gamma_{1}^{\dagger}\tilde{\gamma}_{1}^{\vphantom{\dagger}}}{\mathcal{N}^{2}}
\left((1 + |\alpha|^{2})^{2}\widehat{\gamma_{1}\tilde{\gamma}_{1}}-
(1 - |\alpha|^{2})^{2}\widehat{\gamma_{2}\tilde{\gamma}_{2}} \right),
\end{equation}
where we have used the multiplication rule $\widehat{xy}\widehat{vw}=({\rm Tr}\,y^{\dagger}v) \widehat{xw}$ and
\begin{equation}
{\rm Tr}\,\gamma_{1}^{\dagger}\tilde{\gamma}_{1}^{\vphantom{\dagger}}=-{\rm Tr}\,\gamma_{2}^{\dagger}
\tilde{\gamma}_{2}^{\vphantom{\dagger}}, \quad
{\rm Tr}\,\gamma_{1}^{\dagger}\tilde{\gamma}_{2}^{\vphantom{\dagger}}={\rm Tr}\,\gamma_{2}^{\dagger}
\tilde{\gamma}_{1}^{\vphantom{\dagger}}=0.
\label{traces1}
\end{equation}
The results for the tilde inner products of Eq. (\ref{traces1}) hold since the photons are not polarization-entangled
prior to scattering (${\rm Det}\,\sigma_{\rm in}=0$).
\begin{widetext}
The non-Hermitian matrix $\rho\tilde{\rho}$ has eigenvalue-eigenvector decomposition
\begin{equation}
\rho\tilde{\rho}=\frac{|{\rm Tr}\,\gamma_{1}^{\dagger}\tilde{\gamma}_{1}^{\vphantom{\dagger}}|^{2}}{\mathcal{N}^{2}}
\left(\sum_{i=1,2} \widehat{\gamma_{i}s_{i}}\right)
\left( (1 + |\alpha|^{2})^{2}\widehat{s_{1}s_{1}}+
(1 - |\alpha|^{2})^{2}\widehat{s_{2}s_{2}} \right)
\left(\sum_{i=1,2} \widehat{\gamma_{i}s_{i}}\right)^{-1},
\label{simtrans}
\end{equation}
\end{widetext}
where we have defined orthonormal states
$s_{1}=(1/2)(\openone+\sigma_{z})$ and $s_{2}=(1/2)(\sigma_{x}+i\sigma_{y})$.
The pseudo-inverse is easily seen to be
\begin{equation}
\left(\sum_{i=1,2} \widehat{\gamma_{i}s_{i}}\right)^{-1}=
\frac{1}{({\rm Tr}\,\gamma_{1}^{\dagger}\tilde{\gamma}_{1}^{\vphantom{\dagger}})^{*}}
\left(\widehat{s_{1}\tilde{\gamma}_{1}}-\widehat{s_{2}\tilde{\gamma}_{2}}\right).
\end{equation}
It follows that
\begin{equation}
\mathcal{C}=\frac{2|\alpha|^{2}|{\rm Tr}\,\gamma_{1}^{\dagger}\tilde{\gamma}_{1}^{\vphantom{\dagger}}|}{\mathcal{N}}.
\label{Ctemp}
\end{equation}
The trace that appears in the numerator of Eq. (\ref{Ctemp}) is given by
\begin{equation}
|{\rm Tr}\,\gamma_{1}^{\dagger}\tilde{\gamma}_{1}^{\vphantom{\dagger}}| =
2\sqrt{{\rm Det}\,X^{\dagger}X\,{\rm Det}(\openone-X^{\dagger}X)},
\label{trace2}
\end{equation}
where we have defined a ``hybrid'' $2 \times 2$ matrix $X$ as
\begin{equation}
X=\left(\begin{array}{cc} r_{\rm HH} & t'_{\rm HV} \\ r_{\rm VH} & t'_{\rm VV} \end{array}\right).
\end{equation}
The normalization factor $\mathcal{N}$ given by Eq. (\ref{norm}) can be expressed in terms of $X$ using
\begin{equation}
{\rm Tr}\,\gamma_{1}^{\dagger}\gamma_{1}^{\vphantom{\dagger}} = {\rm Tr}\,X^{\dagger}X-2\,{\rm Per}\,X^{\dagger}X,
\label{trace3}
\end{equation}
\begin{equation}
{\rm Tr}\,\gamma_{2}^{\dagger}\gamma_{2}^{\vphantom{\dagger}} = {\rm Tr}\,X^{\dagger}X-2\,{\rm Det}\,X^{\dagger}X.
\label{trace4}
\end{equation}
(``Per'' denotes the permanent of a matrix.) In the derivation of Eqs. (\ref{trace2},\ref{trace3},\ref{trace4}) we have made
use of the unitarity of $S$.
The concurrence becomes
\begin{equation}
\mathcal{C}=\frac{2|\alpha|^{2}\sqrt{{\rm Det}\,X^{\dagger}X\,{\rm Det}(\openone-X^{\dagger}X)}}{
{\rm Tr}\, X^{\dagger}X-(1 + |\alpha|^{2}){\rm Per}\, X^{\dagger}X-(1 - |\alpha|^{2}){\rm Det}\, X^{\dagger}X}.
\label{Cfinal}
\end{equation}
Entanglement depends on the amount of temporal indistinguishability $|\alpha|^{2}$ and the Hermitian matrix
\begin{equation}
X^{\dagger}X=\left( \begin{array}{cc}
|{\mathbf r}_{\rm H}|^{2} & {\mathbf r}_{\rm H}\cdot {\mathbf t}'_{\rm V} \\
({\mathbf r}_{\rm H}\cdot {\mathbf t}'_{\rm V})^{*} & |{\mathbf t}'_{\rm V}|^{2} \end{array} \right),
\end{equation}
containing the states ${\mathbf r}_{\rm H}=(r_{\rm HH},r_{\rm VH})$ and ${\mathbf t}'_{\rm V}=(t'_{\rm HV},t'_{\rm VV})$
of a reflected and transmitted photon to the left of the beam splitter.
The determinant of $X^{\dagger}X$ measures the size of the span of ${\mathbf r}_{\rm H}$ and ${\mathbf t}'_{\rm V}$ as
\begin{equation}
{\rm Det}\, X^{\dagger}X = |{\mathbf r}_{\rm H}|^{2} |{\mathbf t}'_{\rm V}|^{2}\left(1-\frac{|{\mathbf r}_{\rm H}\cdot{\mathbf t}'_{\rm V}|^{2}}{
|{\mathbf r}_{\rm H}|^{2} |{\mathbf t}'_{\rm V}|^{2}} \right).
\end{equation}
If ${\mathbf r}_{\rm H}$ and ${\mathbf t}'_{\rm V}$ are parallel (${\rm Det}X^{\dagger}X=0$), a scattered photon to the left of the
beam splitter is in a definite state, giving rise to an unentangled two-photon state ($\mathcal{C}=0$).
Similarly,
\begin{equation}
{\rm Det}(\openone-X^{\dagger}X)=
|{\mathbf t}_{\rm H}|^{2} |{\mathbf r}'_{\rm V}|^{2}\left(1-\frac{|{\mathbf t}_{\rm H}\cdot {\mathbf r}'_{\rm V}|^{2}}{
|{\mathbf t}_{\rm H}|^{2} |{\mathbf r}'_{\rm V}|^{2}} \right)
\end{equation}
involves scattered states ${\mathbf t}_{\rm H}=(t_{\rm HH},t_{\rm VH})$ and ${\mathbf r}'_{\rm V}=(r'_{\rm HV},r'_{\rm VV})$ to the right
of the beam splitter.
The denominator of Eq. (\ref{Cfinal}) is the probability of finding a scattered state with one photon on either
side of the beam splitter. It deviates from its classical value
$(X^{\dagger}X)_{\rm HH}+(X^{\dagger}X)_{\rm VV}-2(X^{\dagger}X)_{\rm HH}(X^{\dagger}X)_{\rm VV}$
by an amount $-2|\alpha|^{2}|(X^{\dagger}X)_{\rm HV}|^{2}$ due to photon bunching.
This reduction of coincidence count probability is the Mandel dip \cite{Hon87}. It measures the indistinguishability of a reflected and transmitted
photon as the product of temporal indistinguishability $|\alpha|^{2}$ and polarization indistinguishability $|(X^{\dagger}X)_{\rm HV}|^{2}$.
\section{Violation of the Bell-CHSH inequality}
\label{BellCHSH}
The maximal value ${\cal E}_{\rm max}$ of the Bell-CHSH parameter (\ref{Bellpar}) for an arbitrary mixed state
was analyzed in Refs. \cite{Hor95,Ver02}. For a pure state with concurrence $\mathcal{C}$ one has simply
${\cal E}_{\rm max}=2\sqrt{1+\mathcal{C}^{2}}$ \cite{Gis91}. For a mixed state there is no one-to-one
relation between $\mathcal{C}$ and ${\cal E}_{\rm max}$. Depending on the density matrix, ${\cal E}_{\rm max}$
can take on values between $2\mathcal{C}\sqrt{2}$ and $2\sqrt{1+\mathcal{C}^{2}}$.
The dependence of ${\cal E}_{\rm max}$ on $\rho$ involves the two largest eigenvalues
of the real symmetric $3 \times 3$ matrix $R^{\rm T}R$ constructed from $R_{kl}={\rm Tr}\rho\, \sigma_{k}
\otimes \sigma_{l}$, where $\sigma_{1}=\sigma_{x}$,$\sigma_{2}=\sigma_{y}$ and $\sigma_{3}=\sigma_{z}$.
In terms of $\gamma_{1}$ and $\gamma_{2}$, the elements $R_{kl}$ take the form
\begin{equation}
R_{kl}=\frac{(1+|\alpha|^{2})}{\mathcal{N}}{\rm Tr}\,\gamma_{1}^{\dagger}\sigma_{k}\gamma_{1}^{\vphantom{\dagger}}
\sigma_{l}^{\rm T}+
\frac{(1-|\alpha|^{2})}{\mathcal{N}}{\rm Tr}\,\gamma_{2}^{\dagger}\sigma_{k}\gamma_{2}^{\vphantom{\dagger}}
\sigma_{l}^{\rm T}.
\label{Rdef}
\end{equation}
The matrix $\gamma_{2}$ has a polar decomposition $\gamma_{2}=U\sqrt{\xi}V$ where $U$ and $V$ are unitary
matrices and $\xi$ is a diagonal matrix holding the eigenvalues of
$\gamma_{2}^{\dagger}\gamma_{2}^{\vphantom{\dagger}}$.
The real positive $\xi_{i}$'s are determined by
\begin{equation}
\xi_{1}+\xi_{2}={\rm Tr}\,\gamma_{2}^{\dagger}\gamma_{2}^{\vphantom{\dagger}}, \quad
2\sqrt{\xi_{1}\xi_{2}}=|{\rm Tr}\,\gamma_{2}^{\dagger}\tilde{\gamma}_{2}^{\vphantom{\dagger}}|.
\label{xi}
\end{equation}
The matrix $\gamma_{1}$ can be conveniently expressed as (see Appendix \ref{semipol})
\begin{equation}
\gamma_{1}=UQ\sqrt{\xi}V, \quad \mbox{where} \quad Q=\left( \begin{array}{cc} c_{1} & c_{2} \\
c_{3} & -c_{1} \end{array}\right).
\label{gam1}
\end{equation}
The parameters $c_{1}$,$c_{2}$,$c_{3}$ are real numbers. The matrix $Q$ is traceless due to the orthogonality of $\gamma_{1}$ and
$\tilde{\gamma}_{2}$. The number $c_{1} \in (-1,1)$ on the diagonal is related to the inner product of
$\gamma_{1}$ and $\gamma_{2}$ and takes the form
\begin{equation}
c_{1}=\frac{{\rm Tr}\,\gamma_{1}^{\dagger}\gamma_{2}^{\vphantom{\dagger}}}{\xi_{1}-\xi_{2}}, \quad \mbox{with} \quad
{\rm Tr}\,\gamma_{1}^{\dagger}\gamma_{2}^{\vphantom{\dagger}}={\rm Tr}\,\sigma_{z}X^{\dagger}X.
\label{c1}
\end{equation}
The numbers $c_{2}$,$c_{3}$ are determined by the norm
and tilde inner product of $\gamma_{1}$ and satisfy the relations
\begin{equation}
c_{1}^{2}+c_{2}c_{3}=1, \quad c_{1}^{2}(\xi_{1}+\xi_{2})+c_{2}^{2}\xi_{2}+c_{3}^{2}\xi_{1}=
{\rm Tr}\, \gamma_{1}^{\dagger}\gamma_{1}^{\vphantom{\dagger}}.
\label{c2&c3}
\end{equation}
We substitute $\gamma_{1}$ of Eq. (\ref{gam1}) and the polar decomposition of $\gamma_{2}$ in Eq. (\ref{Rdef}) and parameterize
\begin{equation}
U^{\dagger}\sigma_{k} U = \sum_{i=1}^{3} N_{ki}\sigma_{i}, \quad
V\sigma_{k}^{\rm T} V^{\dagger} = \sum_{i=1}^{3} M_{ki} \sigma_{i}^{\rm T},
\end{equation}
in terms of two $3 \times 3$ orthogonal matrices $N$ and $M$.
The matrix $R$ takes the form
\begin{equation}
R=N R' M^{\rm T},
\end{equation}
where $R'$ is given by Eq. (\ref{Rdef}) with substitutions $R \rightarrow R'$,
$\gamma_{2} \rightarrow \sqrt{\xi}$ and $\gamma_{1} \rightarrow Q\sqrt{\xi}$.
With the help of Eqs. (\ref{xi},\ref{c1},\ref{c2&c3}), the eigenvalues $u_{i}$ of
$R^{\rm T}R$ can now be expressed as (see Appendix \ref{RtReig})
\begin{equation}
u_{1}=\frac{1}{2\mathcal{N}^{2}}\left(\mathcal{T}+\sqrt{\mathcal{T}^{2}-4\mathcal{D}}\right),
\label{u1}
\end{equation}
\begin{equation}
u_{2}=\frac{1}{2\mathcal{N}^{2}}\left(\mathcal{T}-\sqrt{\mathcal{T}^{2}-4\mathcal{D}}\right),
\label{u2}
\end{equation}
\begin{equation}
u_{3}=4\frac{|\alpha|^{4}|{\rm Tr}\,\gamma_{1}^{\dagger}\tilde{\gamma}_{1}^{\vphantom{\dagger}}|^{2}}{\mathcal{N}^{2}},
\label{u3}
\end{equation}
where
\begin{widetext}
\begin{equation}
\mathcal{T}=\mathcal{N}^{2}+4|{\rm Tr}\,\gamma_{1}^{\dagger}\tilde{\gamma}_{1}^{\vphantom{\dagger}}|^{2}
-4(1-|\alpha|^{4})\left({\rm Tr}\,\gamma_{1}^{\dagger}\gamma_{1}^{\vphantom{\dagger}}
{\rm Tr}\,\gamma_{2}^{\dagger}\gamma_{2}^{\vphantom{\dagger}}-
{\rm Tr}^{2}\,\gamma_{1}^{\dagger}\gamma_{2}^{\vphantom{\dagger}}\right),
\label{mathT}
\end{equation}
\begin{equation}
\mathcal{D}=4|{\rm Tr}\,\gamma_{1}^{\dagger}\tilde{\gamma}_{1}^{\vphantom{\dagger}}|^{2}\left(\mathcal{N}^{2}-4(1-|\alpha|^{4})
{\rm Tr}\,\gamma_{1}^{\dagger}\gamma_{1}^{\vphantom{\dagger}}
{\rm Tr}\,\gamma_{2}^{\dagger}\gamma_{2}^{\vphantom{\dagger}}\right).
\label{mathD}
\end{equation}
\end{widetext}
We can relate the $u_{i}$'s to $X^{\dagger}X$ and $|\alpha|^{2}$ using Eqs.
(\ref{norm},\ref{trace2},\ref{trace3},\ref{trace4},\ref{c1}).
The parameter ${\cal E}_{\rm max}$ depends on the two largest eigenvalues of $R^{\rm T}R$ as
\begin{equation}
{\cal E}_{\rm max}=2\sqrt{u_{1}+{\rm max}(u_{2},u_{3})}.
\label{Emax}
\end{equation}
Generically, the expression for ${\cal E}_{\rm max}$ takes a complicated form where ordering of $u_{2}$ and
$u_{3}$ depends on $X^{\dagger}X$ and $|\alpha|^{2}$.
\section{Discussion}
\label{Dis}
The objective of the discussion is to reveal the role played by the Mandel dip $-2|\alpha|^{2}|(X^{\dagger}X)_{\rm HV}|^{2}$
in the connection between $\mathcal{C}$ and ${\cal E}_{\rm max}$.
We first consider the case $|(X^{\dagger}X)_{\rm HV}|^{2}=0$. The concurrence of Eq. (\ref{Cfinal}) reduces to
\begin{equation}
\mathcal{C}=\frac{2|\alpha|^{2}\prod_{i={\rm H,V}}\sqrt{(X^{\dagger}X)_{ii}(1-(X^{\dagger}X)_{ii})}}{(X^{\dagger}X)_{\rm HH}+(X^{\dagger}X)_{\rm VV}
-2(X^{\dagger}X)_{\rm HH}(X^{\dagger}X)_{\rm VV}}.
\end{equation}
The maximal value of the Bell-CHSH parameter takes the form
\begin{equation}
{\cal E}_{\rm max}=2\sqrt{1+\mathcal{C}^{2}}
\label{Enomix}
\end{equation}
and $\mathcal{C} > 0$ implies ${\cal E}_{\rm max} > 2$.
\begin{figure}
\includegraphics[width=8cm]{regions}
\caption{
Parameter space of a beam splitter with $(X^{\dagger}X)_{ii}=1/2$ spanned by $|\alpha|^{2} \in (0,1)$ and $|(X^{\dagger}X)_{\rm HV}|^{2} \in (0,1/4)$.
All points correspond to a non-vanishing polarization-entanglement ($\mathcal{C} > 0$) except the line segments $|\alpha|^{2}=0$ and
$|(X^{\dagger}X)_{\rm HV}|^{2}=1/4$ where entanglement vanishes ($\mathcal{C}=0$). Only in the shaded region, the Bell-CHSH parameter is
able to detect entanglement (${\cal E}_{\rm max} > 2$). The lines correspond to the functions $f$, $g$ of Eqs. (\ref{f},\ref{g})
respectively.
\label{regions}
}
\end{figure}
In the presence of a Mandel dip ($|\alpha|^{2}|(X^{\dagger}X)_{\rm HV}|^{2} > 0$), the ability of ${\cal E}$ to witness entanglement can
disappear. We consider the special case $(X^{\dagger}X)_{ii}=1/2$.
The concurrence of Eq. (\ref{Cfinal}) reduces to
\begin{equation}
\mathcal{C}=\frac{|\alpha|^{2}\left(1-4|(X^{\dagger}X)_{\rm HV}|^{2}\right)}{1-4|\alpha|^{2}|(X^{\dagger}X)_{\rm HV}|^{2}}.
\end{equation}
To find ${\cal E}_{\rm max}$ we have to consider the ordering of $u_{2}$ and $u_{3}$ which depends on $|(X^{\dagger}X)_{\rm HV}|^{2}$ and $|\alpha|^{2}$.
The function
\begin{equation}
f(|\alpha|^{2})=\frac{|\alpha|^{2}}{2(1+|\alpha|^{2})}
\label{f}
\end{equation}
divides parameter space in the region $|(X^{\dagger}X)_{\rm HV}|^{2} \le f$ where
${\cal E}_{\rm max}=2\sqrt{u_{1}+u_{3}}$ and the region
$|(X^{\dagger}X)_{\rm HV}|^{2} > f$ where ${\cal E}_{\rm max}=2\sqrt{u_{1}+u_{2}}$. The equation ${\cal E}_{\rm max}=2$ has a solution
$g(|\alpha|^{2})$ for $|(X^{\dagger}X)_{\rm HV}|^{2}$ that lies in the region $|(X^{\dagger}X)_{\rm HV}|^{2} \le f$. The function $g$ takes the form
\begin{equation}
g(|\alpha|^{2})=\frac{1}{4}\left(1-|\alpha|^{2}+|\alpha|^{4}-(1-|\alpha|^{2})\sqrt{1+|\alpha|^{4}}\right)
\label{g}
\end{equation}
and breaks parameter space in two fundamental regions: a region $|(X^{\dagger}X)_{\rm HV}|^{2}<g$ where ${\cal E}_{\rm max} > 2$
and a region $|(X^{\dagger}X)_{\rm HV}|^{2}>g$ where ${\cal E}_{\rm max} < 2$.
We have drawn these regions in Fig. \ref{regions}.
The maximal value of the Bell-CHSH parameter is given by
\begin{equation}
{\cal E}_{\rm max}=2\mathcal{C}|\alpha|^{-2}\sqrt{1+|\alpha|^{4}}
\end{equation}
in the region $|(X^{\dagger}X)_{\rm HV}|^{2} \le f$.
\section{Conclusions}
In summary, we have calculated the amount of polarization-entanglement (concurrence $\mathcal{C}$) and its witness
(maximal value of the Bell-CHSH parameter ${\cal E}$) induced by two-photon interference at a lossless beam splitter.
The ability of ${\cal E}$ to witness entanglement (${\cal E}_{\rm max} > 2$) depends on the Mandel dip $-2|\alpha|^{2}|(X^{\dagger}X)_{\rm HV}|^{2}$.
In the absence of a Mandel dip, $\mathcal{C} > 0$ implies ${\cal E}_{\rm max} > 2$ cf. Eq. (\ref{Enomix}), whereas in its presence this is not
necessarily true. In the latter case, as we have demonstrated in Sec. \ref{Dis} with $(X^{\dagger}X)_{ii}=1/2$, the witnessing ability of ${\cal E}$
depends on the individual contributions of temporal ($|\alpha|^{2}$) and polarization indistinguishability
($|(X^{\dagger}X)_{\rm HV}|^{2}$).
Our results can be applied to interference of other kinds of particles, getting entangled in some $2 \otimes 2$ Hilbert space and being ``marked''
by an additional degree of freedom. However, determining the indistinguishability parameter $|\alpha|^{2}$ requires careful analysis of the detection scheme.
In case of fermions, the matrices $\gamma_{1}$ and $\gamma_{2}$ of Eq. (\ref{gammas}) are to be interchanged.
Systems without a time-reversal symmetry are captured by
the analysis, as we did not make use of the symmetry of the scattering matrix.
\acknowledgments
I am grateful to C. W. J. Beenakker for discussions and advice.
This work was supported by the Dutch Science Foundation NWO/FOM and by the U.S. Army Research Office (Grant No. DAAD 19-02-0086).
|
{
"timestamp": "2005-03-25T18:46:05",
"yymm": "0503",
"arxiv_id": "quant-ph/0503204",
"language": "en",
"url": "https://arxiv.org/abs/quant-ph/0503204"
}
|
\section{Introduction}
Numerical calculations of lattice QCD predict a transition from
ordinary hadronic matter to a deconfined state of quarks and gluons when
the temperature of the
system is of the order of $T_{crit}\approx$ 0.17 GeV~\cite{latt}.
The existence of such a phase transition manifests itself clearly
in the QCD equation-of-state (EoS) on the lattice by a sharp jump of
the (Stefan-Boltzmann) scaled energy density, $\varepsilon(T)/T^4$,
at the critical temperature, reminiscent of a first-order phase
change\footnote{The order of the phase transition itself is not
exactly known: the pure SU(3) gauge theory is first-order whereas
introduction of 2+1 flavours makes it of a fast cross-over
type~\cite{latt}.}. The search for evidences of this deconfined
plasma of quarks and gluons (QGP) is the main driving
force behind the study of relativistic nuclear collisions
at different experimental facilities in the last 20 years. Whereas several
experimental results have been found consistent with the formation
of the QGP both at CERN-SPS~\cite{sps_qgp} and BNL-RHIC~\cite{rhic_qgp}
energies, it is fair to acknowledge that there is no incontrovertible proof
yet of bulk deconfinement in the present nucleus-nucleus data. In this paper,
we present a detailed study of the only experimental signature, thermal photons,
that can likely provide direct information on the {\it thermodynamical}
properties (and, thus, on the equation-of-state) of the underlying QCD
matter produced in high-energy heavy-ion collisions.
Electromagnetic radiation (real and virtual photons) emitted in the course
of a heavy-ion reaction, has long~\cite{feinberg,shuryak_photons} been considered
a privileged probe of the space-time evolution of the colliding
system\footnote{Excellent reviews on photon production in relativistic nuclear collisions
have been published recently~\cite{peitz_thoma_physrep,yellow_rep,gale_rep}.},
inasmuch as photons are not distorted by final-state interactions due
to their weak interaction with the surrounding medium. Direct photons,
defined as real photons not originating from the decay of final hadrons, are
emitted at various stages of the reaction with several contributing processes.
Two generic mechanisms are usually considered:
(i) {\it prompt} (pre-equilibrium or pQCD) photon emission from perturbative
parton-parton scatterings in the first tenths of fm/$c$ of the collision process,
(ii) subsequent $\gamma$ emission from the {\it thermalized} partonic (QGP) and hadronic
(hadron resonance gas, HRG) phases of the reaction.\\
\begin{sloppypar}
Experimentally, direct $\gamma$ have been indeed measured in Pb+Pb
collisions at CERN-SPS ($\sqrt{s_{\mbox{\tiny{\it{NN}}}}}$ = 17.3
GeV)~\cite{wa98_photons}. However, the relative contributions to
the total spectrum of the pQCD, QGP and HRG components have not
been determined conclusively. Different hydrodynamics
calculations~\cite{srivastava_sps_rhic,alam_sps_rhic,peressou,steffen_sps_rhic_lhc,finnish_hydro}
require ``non-conventional'' conditions: high initial temperatures
($T_{0}^{max}>$ $T_{crit}$), strong partonic and/or hadronic
transverse velocity flows, or in-medium modifications of hadron
masses, in order to reproduce the observed photon spectrum.
However, no final conclusion can be drawn from these results due
mainly to the uncertainties in the exact amount of radiation
coming from primary parton-parton collisions. In a situation akin
to that affecting the interpretation of high $p_T$ hadron data at
SPS~\cite{dde_sps}, the absence of a concurrent baseline
experimental measurement of prompt photon production in p+p
collisions at the same $\sqrt{s}$ and $p_T$ range as the
nucleus-nucleus data, makes it difficult
to have any reliable empirical estimate of the actual
thermal $\gamma$ excess in the Pb+Pb spectrum. In the theoretical
side, the situation at SPS is not fully under control either: (i)
next-to-leading-order (NLO) perturbative calculations are known to
underpredict the experimental reference nucleon-nucleon $\gamma$
differential cross-sections below $\sqrt{s}\approx$
30~GeV~\cite{aurenche} (a substantial amount of parton
intrinsic transverse momentum $k_T$~\cite{wong}, approximating the
effects of parton Fermi motion and soft gluon radiation, is
required~\cite{apanasevich}), (ii) the implementation of the extra nuclear $k_T$
broadening observed in the nuclear data (``Cronin
enhancement''~\cite{cronin} resulting from multiple soft and
semi-hard interactions of the colliding partons on their way
in/out the traversed nucleus) is
model-dependent~\cite{dumitru,ina,levai} and introduces an additional
uncertainty to the computation of the yields, and (iii)
hydrodynamical calculations usually assume initial conditions
(longitudinal boost invariance, short thermalization times, zero
baryochemical potential) too idealistic for SPS energies. The
situation at RHIC (and LHC) collider energies is undoubtedly far
more advantageous. Firstly, the photon spectra for different
centralities in Au+Au~\cite{ppg042} and in (baseline) p+p~\cite{ppg049}
collisions at $\sqrt{s}$ = 200 GeV are already experimentally available.
Secondly, the p+p baseline reference is well under control
theoretically (NLO calculations do not require extra
non-perturbative effects to reproduce the hard spectra at
RHIC~\cite{ppg049,ppg024}). Thirdly, the amount of nuclear
Cronin enhancement experimentally observed is very modest
(high $p_T$ $\pi^0$ are barely enhanced in d+Au collisions at
$\sqrt{s_{NN}}$ = 200 GeV~\cite{dAu_phnx}), and one expects even
less enhancement for $\gamma$ which, once produced, do not gain
any extra $k_T$ in their way out through the nucleus.
Last but not least, the produced system at midrapidity in heavy-ion
reactions at RHIC top energies is much closer to the zero net baryon
density and longitudinally boost-invariant conditions customarily
presupposed in the determination of the parametrized photon rates
and in the hydrodynamical implementations of the reaction evolution.
In addition, the thermalization times usually assumed in the hydrodynamical
models ($\tau_{\mbox{\tiny{\it{therm}}}}\lesssim$ 1 fm/$c$) are, for the
first time at RHIC, above the lower limit imposed by the transit time of
the two colliding nuclei ($\tau_0 = 2R/\gamma\approx$ 0.15 fm/$c$ for Au+Au at 200 GeV).
As a matter of fact, it is for the first time at RHIC that hydrodynamics
predictions agree {\it quantitatively} with most of the differential observables
of bulk (``soft'') hadronic production below $p_T\approx$ 1.5 GeV/$c$ in
Au+Au reactions~\cite{kolb_heinz_rep,teaney_hydro,hirano}.\\
\end{sloppypar}
\begin{sloppypar}
In this context, the purpose of this paper is three-fold.
First of all, we present a relativistic Bjorken hydrodynamics model that
reproduces well the identified hadron spectra measured at all centralities
in Au+Au collisions at $\sqrt{s_{NN}}$ = 200 GeV (and, thus, the
centrality dependence of the total charged hadron multiplicity).
Secondly, using such a model complemented with the most up-to-date
parametrizations of the QGP and HRG photon emission rates, we determine the
expected thermal photon yields in Au+Au reactions and compare them to the
prompt photon yields computed in NLO perturbative QCD. The combined inclusive
(thermal+pQCD) photon spectrum is successfully confronted to recent results
from the PHENIX collaboration as well as to other available predictions.
Thirdly,
after discussing in which $p_T$ range the thermal photon signal
can be potentially identified experimentally, we address the issue
of how to have access to the thermodynamical properties (temperature,
entropy density) of the radiating matter. We propose the correlation of two
experimentally measurable quantities: the thermal photon slope and
the multiplicity of charged hadrons produced in the reaction,
as a direct method to determine the underlying degrees of freedom
and the equation of state, $s(T)/T^3$, of the dense and hot QCD
medium produced in Au+Au collisions at RHIC energies.
\end{sloppypar}
\section{Hydrodynamical model}
\subsection{Implementation}
\begin{sloppypar}
Hydrodynamical approaches of particle production in heavy-ion
collisions assume {\it local} conservation of energy and momentum
in the hot and dense strongly interacting matter produced in the course
of the reaction and describe its evolution using the equations of motion of
perfect (non-viscous) relativistic hydrodynamics. These equations are nothing
but the conservation of:
\begin{description}
\item (i) the energy-momentum tensor: $\partial_{\mu}
T^{\mu\nu} = 0$ with $T^{\mu\nu} = (\varepsilon +
p)u^{\mu}u^{\nu}-p\,g^{\mu\nu}$ [where $\varepsilon$, $p$, and
$u^{\nu}=(\gamma,\gamma$v) are resp. the energy density,
pressure, and collective flow 4-velocity fields, and
$g^{\mu\nu}$=diag(1,-1,-1,-1) the metric tensor], and
\item (ii) the conserved currents in strong interactions: $\partial_\mu
J^{\mu}_{i} = 0$, with $J^{\mu}_{i}=n_{i}u^{\mu}$ [where $n_i$ is
the number density of the net baryon, electric charge, net strangeness,
etc. currents].
\end{description}
\end{sloppypar}
These equations complemented with three input ingredients:
(i) the initial conditions ($\varepsilon_0$
at time $\tau_0$),
(ii) the equation-of-state of the system, $p(\varepsilon,n_{i})$,
relating the local thermodynamical quantities,
and (ii) the freeze-out conditions, describing the transition from
the hydrodynamics regime to the free streaming final particles,
are able to reproduce most of the bulk hadronic observables
measured in heavy-ion reactions at RHIC~\cite{kolb_heinz_rep,teaney_hydro,hirano}.\\
\begin{sloppypar}
The particular hydrodynamics implementation used in this work is
discussed in detail in~\cite{peressou}.
We assume cylindrical symmetry in the transverse direction ($r$)
and longitudinal ($z$) boost-invariant (Bjorken) expansion~\cite{bjorken}
which reduces the equations of motion to a one-dimensional problem
but results in a loss of the dependence of
the observables on longitudinal degrees of freedom. Our results,
thus, are only relevant for particle production within a finite range
around midrapidity\footnote{The experimental $\pi^\pm$ and $K^\pm$
$dN/dy$ distributions at RHIC are Gaussians~\cite{brahms_hadrons},
as expected from perturbative QCD initial conditions~\cite{eskola_hydro}.
Thus, although there is no Bjorken rapidity plateau, the widths of the distributions
are quite broad and within $|y|\lesssim$ 2, deviations from boost
invariance are not very large~\cite{eskola_hydro}.}.
The equation-of-state used here describes a first order phase
transition from a QGP to a HRG at $T_{crit}$ = 165 MeV with latent
heat\footnote{Although the lattice results seem to indicate that
the transition is of a fast cross-over type, the predicted change
of $\Delta\varepsilon \approx$ 0.8 GeV/fm$^3$ in a narrow temperature
interval of $\Delta T\approx$ 20 MeV~\cite{latt} can be interpreted as the latent
heat of the transition.} $\Delta\varepsilon\approx$ 1.4 GeV/fm$^3$,
very similar to that used in other works~\cite{kolb_heinz_rep}.
The QGP is modeled as an ideal gas of massless quarks ($N_f$
= 2.5 flavours) and gluons with total degeneracy
$g_{\mbox{\tiny{\it{QGP}}}} = (g_{\mbox{\tiny{\it{gluons}}}}+7/8\,
g_{\mbox{\tiny{\it{quarks}}}})$ = 42.25. The corresponding EoS,
$p=1/3\varepsilon-4/3B$ ($B$ being the bag constant), has sound
velocity $c_s^2=\partial p/\partial \varepsilon = 1/3$.
The hadronic phase is modeled as a non-interacting gas of
$\sim$400 known hadrons and hadronic resonances with masses below
2.5 GeV/$c^2$. The inclusion of heavy hadrons leads to an equation
of state significantly different than that of an ideal gas of
massless pions: the velocity of sound in the HRG phase is
$c_s^2\approx$ 0.15, resulting in a relatively soft hadronic EoS
as suggested by lattice calculations~\cite{mohanty_cs}; and the
effective number of degrees of freedom at $T_{c}$ is
$g_{\mbox{\tiny{\it{HRG}}}}\approx$ 12 (as given by $g_{\ensuremath{\it eff}} =
45\,s/(2\pi^2\,T^3)$, see later). Both phases are
connected via the standard Gibbs' condition of phase equilibrium,
$p_{\mbox{\tiny{\it{QGP}}}}(T_{c}) =
p_{\mbox{\tiny{\it{HRG}}}}(T_{c})$, during the mixed phase.
The external bag pressure, calculated to fulfill this condition at
$T_c$, is $B\approx$ 0.38 GeV/fm$^3$. The system of equations is solved
with the MacCormack two-step (predictor-corrector) numerical scheme~\cite{maccormack}
with time and radius steps: $\delta t$ = 0.02 fm/$c$ and $\delta r$ = 0.1 fm
respectively.\\
\end{sloppypar}
Statistical model analyses of particle production in nucleus-nucleus
reactions~\cite{pbm_thermal} provide a very good description of
the measured particle ratios at RHIC assuming that all hadrons are
emitted from a thermalized system reaching chemical equilibrium at
a temperature $T_{chem}$ with baryonic, strange and isospin chemical
potentials $\mu_{i}$. In agreement with those observations, our specific
hydrodynamical evolution reaches chemical freeze-out at $T_{chem}=150$ MeV
with $\mu_{B}=25$ MeV (as given by the latest statistical fits to hadron
ratios~\cite{andronic05}), and has $\mu_{S}=\mu_{I}=0$. For temperatures above $T_{chem}$
we conserve baryonic, strange and charge currents, but not particle numbers,
while for temperatures below $T_{chem}$ we explicitly conserve particle
numbers by introducing individual (temperature-dependent)
chemical potentials for each hadron.
The final differential hadron $dN/dp_T$ spectra are produced via a standard
Cooper-Frye ansatz~\cite{cooper_frye} at the kinetic freeze-out
temperature ($T_{\ensuremath{\it fo}}=120$ MeV) when the hydrodynamical
equations lose their validity, i.e. when the microscopic length (the
hadrons mean free path) is no longer small compared to the size of the system.
Unstable resonances are then allowed to decay with their appropriate branching
ratios~\cite{PDG}. Table I summarizes the most important
parameters describing our hydrodynamic evolution.
The only free parameters are the initial energy density $\varepsilon_0$
in the center of the reaction zone for head-on (impact parameter $b$ = 0 fm)
Au+Au collisions at the starting time $\tau_0$, and the temperature at
freeze-out time, $T_{\ensuremath{\it fo}}$.
\subsection{Initialization}
\begin{sloppypar}
We distribute the initial energy density within the reaction volume
according to the geometrical Glauber\footnote{The density of participant
and colliding nucleons are obtained from the nuclear overlap function
$T_{AA}(b)$ computed with a Glauber Monte Carlo code which parametrizes
the Au nuclei with Woods-Saxon functions with radius $R$ = 6.38~fm and
diffusivity $a$ = 0.54~fm~\cite{hahn}.}
prescription proposed by Kolb {\it et al.}~\cite{kolb_heinz_finnishgroup}.
Such an ansatz ascribes 75\% of the initial entropy production in a given
centrality bin, $s_0(b)$, to soft processes (scaling with the transverse
density of participant nucleons $N_{part}(b)$) and the remaining 25\% to
hard processes (scaling with the density of point-like collisions,
$N_{coll}(b)$, proportional to the nuclear overlap function $T_{AA}(b)$):
\begin{equation}
s(b) = C\cdot(0.25\cdot N_{part}(b) + 0.75 \cdot N_{coll}(b)),
\end{equation}
where $C$ is a normalization coefficient chosen so that we produce the correct
particle multiplicity at $b$ = 0 fm.
For each impact parameter, we construct an azimuthally symmetric hydrodynamical
source from the (azimuthally deformed) initial Glauber entropy distribution,
by defining a coordinate origin in the middle point between the centers of
the two colliding nuclei and averaging the entropy density over all azimuthal
directions. We then transform $\varepsilon_0(b)\propto s_0(b)^{4/3}$.
This method provides a very good description of the measured
centrality dependence of the final charged hadron rapidity
densities $dN_{ch}/d\eta$ measured at RHIC as can be seen in Figure~\ref{fig:dNch}.
Note that in our implementation of this prescription, we explicitly added
the contribution of the particle multiplicity coming from hard processes
(i.e. from hadrons having $p_T>$ 1 GeV/$c$) obtained from the scaled pQCD
calculations (see later). Such a ``perturbative'' component accounts for a
roughly constant $\sim$7\% factor of the total hadron multiplicity for all centralities.
The good reproduction of the measured charged hadron integrated yields is an
important result for our later use of $dN_{ch}/d\eta|_{\eta=0}$ as an empirical
measure of the initial entropy density in different Au+Au centrality classes
(see Section~\ref{sec:eos}).\\
\end{sloppypar}
\begin{figure}[htbp]
\begin{center}
\psfig{figure=dNchdeta_vs_Npart.eps,width=9.cm}
\end{center}
\caption{Charged hadron multiplicity at midrapidity (normalized by the
number of participant nucleon pairs) as a function of centrality (given by
the number of participants, $N_{part}$) measured in Au+Au
at $\sqrt{s_{\mbox{\tiny{\it{NN}}}}}$ = 200 GeV by PHENIX~\protect\cite{ppg019}
(circles), STAR~\protect\cite{star_Nch} (stars), PHOBOS~\protect\cite{phobos_Nch}
(squares) and BRAHMS~\protect\cite{brahms_Nch} (crosses), compared to our
hydrodynamics calculations (dashed line), our scaled pQCD ($p_T>$ 1 GeV/$c$)
p+p yields~\protect\cite{vogel_hadrons} (dashed-dotted line), and to the sum
hydro+pQCD (solid line).}
\label{fig:dNch}
\end{figure}
\begin{table*}[htb]
\caption{Summary of the thermodynamical parameters characterizing
our hydrodynamical model evolution for central ($b = 0$ fm) Au+Au
collisions at $\sqrt{s_{\mbox{\tiny{\it{NN}}}}}$ = 200 GeV. Input
parameters are the (maximum) initial energy density $\varepsilon_0$
(with corresponding ideal-gas entropy densities $s_0$ and
temperature $T_0$)
at time $\tau_0$, the baryochemical potential $\mu_{B}$, and the chemical and
kinetic freeze-out temperatures $T_{chem}$ and $T_{\ensuremath{\it fo}}$ (or
energy density $\varepsilon_{\ensuremath{\it fo}}$). The energy densities at the end
of the pure QGP ($\varepsilon_{\mbox{\tiny{\it{QGP}}}}^{\mbox{\tiny{\it{min}}}}$),
and at the beginning of the pure hadron gas phase
($\varepsilon_{\mbox{\tiny{\it{HRG}}}}^{\mbox{\tiny{\it{max}}}}$) are
also given for indication, as well as the average (over total volume) values of the
initial energy density $\langle\varepsilon_0\rangle$, entropy density $\langle s_0\rangle$,
and temperature $\langle T_0\rangle$.}
\begin{center}
\begin{tabular}{c|c|c|c|c|c|c|c|c|c}
\hline\hline $\tau_0$ & $\varepsilon_0$ ($\langle\varepsilon_0\rangle)$
& $s_0$ ($\langle s_0\rangle$)
& $T_0$ ($\langle T_0\rangle$) &
$\varepsilon_{\mbox{\tiny{\it{QGP}}}}^{\mbox{\tiny{\it{min}}}}$ &
$\varepsilon_{\mbox{\tiny{\it{HRG}}}}^{\mbox{\tiny{\it{max}}}}$ &
$\mu_{B}$ &
$T_{chem}$ & $T_{\ensuremath{\it fo}}$ & $\varepsilon_{\ensuremath{\it fo}}=\varepsilon_{\mbox{\tiny{\it{HRG}}}}^{\mbox{\tiny{\it{min}}}}$ \\
(fm/$c$) & (GeV/fm$^3$) & (fm$^{-3}$) &
(MeV) & (GeV/fm$^3$) & (GeV/fm$^3$) & (MeV) & (MeV) & (MeV) & (GeV/fm$^3$)
\\\hline
0.15 & 220 (72) & 498 (190) & 590 (378) & 1.7 & 0.35 & 25. & 150 & 120 & 0.10 \\
\hline\hline
\end{tabular}
\label{tab:hydro_parameters}
\end{center}
\end{table*}
For the initial conditions (Table~\ref{tab:hydro_parameters}),
we choose $\varepsilon_0$ = 220 GeV/fm$^3$ (maximum energy density at
$b$ = 0 fm, corresponding to an {\it average} energy density over
the total volume for head-on collisions of
$\langle\varepsilon_0\rangle$ = 72 GeV/fm$^3$) at a
time $\tau_0 = 2R/\gamma\approx$ 0.15 fm/$c$ equal to the transit time
of the two Au nuclei at $\sqrt{s_{\mbox{\tiny{\it{NN}}}}}$ = 200 GeV.
The choice of this relatively short value of $\tau_0$, -- otherwise typically
considered in other hydrodynamical studies of thermal photon
production at RHIC~\cite{srivastava_sps_rhic,finnish_hydro,frankfurt_rhic_lhc}
--, rather than the ``standard'' thermalization time of $\tau_{therm}$
= 0.6 fm/$c$~\cite{kolb_heinz_rep,teaney_hydro,hirano}, is driven by our will to
consistently take into account within our space-time evolution the
emission of photons from secondary ``cascading'' parton-parton
collisions~\cite{bass,bass2} taking place in the {\it thermalizing} phase
between prompt pQCD emission (at $\tau\sim 1/p_T\lesssim$0.15 fm/$c$) and full
equilibration (see Sect.~\ref{sec:extra_gamma}). Though it may be questionable
to identify such photons from second-chance parton-parton collisions as
genuine {\it thermal} $\gamma$, it is clear that their spectrum reflects the
momentum distribution of the partons during this thermalizing phase\footnote{Note
also that it is precisely those secondary partonic interactions that are actually
driving the system towards (local) thermal equilibrium.}. Additionally, recent
theoretical works~\cite{berges04,arnold04} do seem to support the application
of hydrodynamics equations in such ``pre-thermalization'' conditions.
Our consequent space-time evolution leads to a value of the energy density
of $\varepsilon\approx$ 30 GeV/fm$^3$ at $\tau_{therm}$ = 0.6 fm/$c$,
in perfect agreement with other 2D+1 hydrodynamic calculations which do not
invoke azimuthal symmetry~\cite{kolb_heinz_rep,teaney_hydro}
as well as more numerically involved 3D+1 approaches~\cite{hirano}.
Thus, our calculations reproduce the final hadron spectra as well,
at least, as those other works do. As a matter of fact, by using
$\tau_0$ = 0.15 fm/$c$ (rather than 0.6 fm/$c$), the system has a
few more tenths of fm/$c$ to develop some extra transverse collective
flow and there is no need to consider in our initial conditions a
supplemental input radial flow velocity parameter, $v_{r_0}$, as done
in other works~\cite{peressou,kolb_rapp_flow} in order to reproduce the hadron
spectra.
\subsection{Comparison to hadron data}
Figure~\ref{fig:hadron_spectra_AuAu200GeV} shows the pion, kaon,
and proton\footnote{For a suitable comparison to the (feed-down
corrected) PHENIX~\cite{ppg026}, PHOBOS~\cite{phobos_lowpt_had}
and BRAHMS~\cite{brahms_hadrons} yields, the STAR proton
spectra~\cite{star_hadrons} have been appropriately corrected for
a $\sim$40\% ($p_T$-independent) contribution from weak
decays~\cite{star_hadrons2}.} transverse spectra measured by
PHENIX~\cite{ppg026}, STAR~\cite{star_hadrons,star_hadrons2},
PHOBOS~\cite{phobos_lowpt_had} and
BRAHMS~\cite{brahms_hadrons} in central (0--10\% corresponding
to $\mean{b}$ = 2.3 fm) and peripheral (60--70\% corresponding to $\mean{b}$ = 11.9 fm)
Au+Au collisions at $\sqrt{s_{\mbox{\tiny{\it{NN}}}}}$ = 200 GeV,
compared to our hydrodynamical predictions (dashed lines)
and to properly scaled p+p NLO pQCD expectations~\cite{vogel_hadrons}
(dotted lines). At low transverse momentum, the agreement data--hydro
is excellent
starting from the very low $p_T$ PHOBOS data ($p_T<$ 100 MeV)
up to at least $p_T\approx$ 1.5 GeV/$c$.
Above this value, contributions from perturbative
processes (parton fragmentation products) start to dominate
over bulk hydrodynamic production.
Indeed, particles with transverse momenta $p_T\gtrsim$ 2 GeV/$c$
are mostly produced in primary parton-parton collisions at times
of order $\tau\sim 1/p_T\lesssim$ 0.15 fm/$c$ (i.e. during the
interpenetration of the colliding nuclei and {\it before}
any sensible time estimate for equilibration),
and as such, they are {\it not} in thermal equilibrium with the
bulk particle production. Therefore, one does not expect
hydrodynamics to reproduce the spectral shapes beyond $p_T\approx$ 2 GeV/$c$.
The dotted lines of Fig.~\ref{fig:hadron_spectra_AuAu200GeV} show
NLO predictions for $\pi$, $K$ and $p$ production in p+p
collisions at $\sqrt{s}$ = 200 GeV~\cite{vogel_hadrons} scaled by
the number of point-like collisions ($N_{coll}\propto T_{AA}$) times
an empirical quenching factor, $R_{AA}$ = 0.2 (0.7) for 0-10\% central
(60-70\% peripheral) Au+Au, to account for the observed constant suppression
factor of hadron yields at high $p_T$~\cite{phenix_hiptpi0_200,star_hipt_200}
(such a suppression is not actually observed in the $p,\bar{p}$ spectra
at intermediate $p_T\approx$ 3 -- 5 GeV/$c$, see discussion below).\\
\begin{figure*}[htbp]
\psfig{figure=hadron_spec_AuAu200GeV_cent.eps,width=8cm,height=9cm
\psfig{figure=hadron_spec_AuAu200GeV_periph.eps,width=8cm,height=9cm
\caption{Transverse momentum spectra for $\pi^{\pm,0}$,$K^{\pm,0}$, and protons
measured in the range $p_T$ = 0 -- 5.5 GeV/$c$ by PHENIX~\protect\cite{ppg026},
STAR ($K^0_s$ are preliminary)~\protect\cite{star_hadrons,star_hadrons2},
PHOBOS~\protect\cite{phobos_lowpt_had} and
BRAHMS~\protect\cite{brahms_hadrons} in central (0-10\% centrality, left)
and peripheral (60-70\%, right) Au+Au collisions at
$\sqrt{s_{\mbox{\tiny{\it{NN}}}}}$ = 200 GeV, compared to our hydrodynamics
calculations (dashed lines), to the scaled pQCD p+p rates~\protect\cite{vogel_hadrons}
(dotted lines), and to the sum hydro+pQCD (solid lines).}
\label{fig:hadron_spectra_AuAu200GeV}
\end{figure*}
Fig.~\ref{fig:ratios_hadron_spectra_theory} shows more clearly (in linear rather than
log scale as the previous figure) the relative agreement between the experimental hadron
transverse spectra and the hydrodynamical plus (quenched) pQCD yields presented in this work.
The data-over-theory ratio plotted in the figure is obtained by taking the quotient of
the pion, kaon and proton data measured in central Au+Au reactions (shown in the left plot
of Fig.~\ref{fig:hadron_spectra_AuAu200GeV}) over the corresponding sum of hydrodynamical
plus perturbative results (solid lines in Fig.~\ref{fig:hadron_spectra_AuAu200GeV}).
In the low $p_T$ range dominated by hydrodynamical production, there exist some local
$p_T$-dependent deviations between the measurements and the calculations. However,
the same is true within the independent data sets themselves and, thus, those differences
are indicative of the amount of systematic uncertainties associated with the different
measurements. High $p_T$ hadro-production, dominated by perturbative processes, agrees
also well within the $\sim$20\% errors associated with the standard scale uncertainties
for pQCD calculations at this center-of-mass energy.
It is, thus, clear from Figs.~\ref{fig:hadron_spectra_AuAu200GeV}
and~\ref{fig:ratios_hadron_spectra_theory} that
identified particle production at $y$ = 0 in nucleus-nucleus
collisions at RHIC can be fully described in their whole $p_T$ range
and for all centralities by a combination of hydrodynamical
(thermal+collective boosted) emission plus (quenched) prompt
perturbative production. An exception to this rule are, however,
the (anti)protons~\cite{phnx_ppbar}. Although due to their higher masses,
they get an extra push from the hydrodynamic flow up to
$p_T\sim 3 $ GeV/$c$, for even higher transverse momenta the
combination of hydro plus (quenched) pQCD still clearly undershoots the
experimental proton spectra. This observation has lent support to the
existence of an additional mechanism for baryon production at intermediate
$p_T$ values ($p_T\approx$ 3 -- 5 GeV/$c$)
based on quark recombination~\cite{recomb}. This mechanism will not,
however, be further considered in this paper since it has no practical
implication for photon production and/or for the overall hydrodynamical
evolution of the reaction. The overall good theoretical reproduction of the
differential $\pi,K,p$ experimental spectra for all centralities is obviously
consistent with the previous observation that our calculated total integrated hadron
multiplicities agree very well with the experimental data measured at
mid-rapidity by the four different RHIC experiments (Fig.~\ref{fig:dNch}).
\begin{figure*}[htbp]
\psfig{figure=ratio_hadron_data_hydro+pQCD_AuAu_cent.eps,height=7.cm}
\caption{Ratio of $\pi^{\pm,0}$,$K^{\pm,0}$, and protons yields
measured in the range $p_T$ = 0 -- 5.5 GeV/$c$ by PHENIX~\protect\cite{ppg026},
STAR (note that $K^0_s$ are preliminary)~\protect\cite{star_hadrons,star_hadrons2},
PHOBOS~\protect\cite{phobos_lowpt_had} and BRAHMS~\protect\cite{brahms_hadrons}
in 0-10\% most central Au+Au collisions at $\sqrt{s_{\mbox{\tiny{\it{NN}}}}}$ = 200 GeV,
over the sum hydro ($\mean{b}$ = 2.3 fm) plus (quenched) pQCD. Theoretical calculations
above $p_T\approx$ 2 GeV/$c$ have an overall $\pm$20\% uncertainty (not shown) dominated
by pQCD scale uncertainties.}
\label{fig:ratios_hadron_spectra_theory}
\end{figure*}
\section{Direct photon production}
As in the case of hadron production, the total direct photon
spectrum in a given Au+Au collision at impact parameter $b$ is
obtained by adding the primary production from perturbative
parton-parton scatterings to the thermal emission rates integrated
over the whole space-time volume of the produced fireball. Three
sources of direct photons are considered
corresponding to each one of the phases of the reaction: prompt
production, partonic gas emission, and hadronic gas radiation.
\subsection{Prompt photons}
\begin{sloppypar}
For the prompt $\gamma$ production we use the NLO pQCD predictions of
W.~Vogelsang~\cite{vogel_gamma} scaled by the corresponding
Glauber nuclear overlap function at $b$, $T_{AA}(b)$, as expected
for hard processes in A+A collisions unaffected by final-state effects
(as empirically confirmed for photon production in Au+Au~\cite{ppg042}).
This pQCD photon spectrum is obtained with CTEQ6M~\cite{cteq6} parton
distribution function (PDF), GRV~\cite{grv_photons} parametrization of
the $q,g\rightarrow\gamma$ fragmentation function (FF), and
renormalization-factorization scales set equal to the transverse
momentum of the photon ($\mu = p_T$). Such NLO calculations
provide an excellent reproduction of the inclusive direct
$\gamma$~\cite{ppg049} and large-$p_T$ $\pi^0$~\cite{ppg024}
spectra measured by PHENIX in p+p
collisions at $\sqrt{s}$ = 200 GeV without any additional
parameter (in particular, at variance with results at lower
energies~\cite{wong}, no primordial $k_T$ is needed to describe
the data). We do {\it not} consider any modification of the prompt
photon yields in Au+Au collisions due to partially counteracting
initial-state (IS) effects such as: (i) nuclear modifications
(``shadowing'') of the Au PDF
($<20$\%, in the relevant ($x,Q^2$) kinematical range considered
here~\cite{frankfurt_rhic_lhc,ina,jamal}), and (ii) extra nuclear
$k_T$ broadening (Cronin enhancement) as described e.g. in~\cite{dumitru}.
Both IS effects are small and/or approximately cancel each other at mid-rapidity
at RHIC as evidenced experimentally by the barely modified nuclear modification
factor, $R_{dAu}\lesssim$1.1, for $\gamma$ and $\pi^0$ measured in d+Au collisions
at $\sqrt{s_{\mbox{\tiny{\it{NN}}}}}$ = 200 GeV~\cite{qm05}.
Likewise, we do {\it not} take into account any
possible final-state (FS) {\it photon} suppression due to energy
loss of the jet-fragmentation (aka. ``anomalous'') component of the
prompt photon cross-section~\cite{dumitru,jamal,arleo04,dde_hq04}, which,
if effectively present (see~\cite{zakharov04} and discussion in Sect.~\ref{sec:extra_gamma}),
can be in principle experimentally determined by detailed measurements of the
isolated and non-isolated direct photon baseline spectra in p+p collisions
at $\sqrt{s}$ = 200 GeV~\cite{dde_hq04}.
\end{sloppypar}
\subsection{Thermal photon rates}
\begin{sloppypar}
For the QGP phase we use the most recent full leading order (in $\alpha_{em}$
and $\alpha_s$ couplings) emission rates from Arnold {\it et al.}~\cite{arnold}.
These calculations include hard thermal loop diagrams to all orders and
Landau-Migdal-Pomeranchuk (LPM) medium interference effects.
The parametrization given in~\cite{arnold}
assumes zero net baryon density (i.e. null quark chemical
potential, $\mu_q$ = 0), and {\it chemical} together with thermal
equilibrium. Corrections of the QGP photon rates due to net quark
densities are $\mathcal{O}[\mu_q^2/(\pi T)^2]$~\cite{traxler} i.e.
marginal at RHIC energies where the baryochemical potential is
close to zero at midrapidity ($\mu_B=3\,\mu_q\sim$ 25 MeV) and
neglected here. Similarly, although the early partonic phase is
certainly not chemically equilibrated (the first instants of the
reaction are strongly gluon-dominated) the two main effects from
chemical non-equilibrium composition of the QGP: reduction of
quark number and increase of the temperature, nearly cancel in the
photon spectrum~\cite{yellow_rep,gelis} and have not been
considered either.
For the HRG phase, we use the latest improved parametrization
from Turbide {\it et al.}~\cite{turbide} which includes hadronic
emission processes
not accounted for before in the literature. In all calculations,
we use a temperature-dependent
parametrization of the strong coupling\footnote{According to this parametrization,
$\alpha_S(T)$ = 0.3 -- 0.6 in the range of temperatures of interest here ($T\approx$ 600 -- 150 MeV).},
$\alpha_s(T) = 2.095/\{\frac{11}{2\pi}\ln{(Q/\Lambda_{\overline{MS}})} +
\frac{51}{22\pi}\ln{[2\ln(Q/\Lambda_{\overline{MS}})]}\}$ with $Q = 2\pi T$,
obtained from recent lattice results~\cite{karsch_alphaS}.
\end{sloppypar}
\subsection{Extra photon contributions}
\label{sec:extra_gamma}
Apart from the aforementioned (prompt and thermal) photon production mechanisms,
S.~Bass {\it et al.}~\cite{bass,bass2} have recently evaluated within
the Parton Cascade Model (PCM), the contribution to the total
Au+Au photon spectrum from secondary (cascading) parton-parton
collisions taking place before the attainment of thermalization
(i.e. between the transit time of the two nuclei, $\tau \approx$ 0.15 fm/$c$,
and the standard $\tau_{therm}$ = 0.6 fm/$c$ considered at RHIC).
Since such cascading light emission is due to second-chance
partonic collisions which are, simultaneously, driving the system
towards equilibrium, we consider not only ``valid'' (see the discussion of
refs.~\cite{berges04,arnold04}) but more self-consistent within our framework
to account for this contribution
with our hydrodynamical evolution alone. We achieve this by starting
hydrodynamics (whose photon rates also include the expected LPM reduction of
the secondary rates~\cite{bass2}) at $\tau_{0}$ = 0.15 fm/$c$. By doing that,
at the same time that we account for this second-chance emission, our initial
plasma temperature and associated thermal photon production can be
considered to be at their {\it maximum values} for RHIC energies.\\
Likewise, we do not consider the conjectured extra $\gamma$ emission
due to the passage of quark jets (Compton-scattering and annihilating) through
the dense medium~\cite{fries_rhic_lhc,fries_gammajet2,zakharov04} since such contribution
is likely partially compensated by: (i) the concurrent non-Abelian
energy loss of the parent quarks going through the system~\cite{turbide05},
plus (ii) a possible {\it photon} suppression due to energy loss of the
``anomalous'' component of the prompt photon cross-section~\cite{dumitru,jamal,dde_hq04,arleo04}.
As a matter of fact, some approximate cancellation of all those effects must exist
since the experimental Au+Au photon spectra above $p_T\approx$ 4 GeV/$c$ turn out to be
well reproduced by primary (pQCD) hard processes alone for all centralities,
as can be seen in the comparison of pQCD NLO predictions with PHENIX data~\cite{ppg042}
(Fig.~\ref{fig:photon_spec_AuAu_cent_periph}). The apparent agreement between the
experimental spectra above $p_T\approx$ 4 GeV/$c$ and the NLO calculations does not
seem to leave much room for extra radiation contributions. A definite conclusion on the
existence or not of FS effects on photon production will require in any case
precision $\gamma$ data in Au+Au, d+Au and p+p collisions. The more critical
issue of the role of the jet bremsstrahlung component needs to be estimated, for
example, via measurements of isolated and non-isolated direct photon baseline
p+p spectra as discussed in~\cite{dde_hq04}. Additional IS effects not considered so far
due, for example, to isospin corrections\footnote{Direct photon cross-sections
depend on the light quark electric charges and are thus disfavoured in a nucleus target
less rich in up quarks than the standard proton reference~\cite{arleo05}.} will require a
careful analysis and comparison of Au+Au to reference d+Au photon cross-sections too.\\
\begin{figure*}[htbp]
\psfig{figure=photon_spec_AuAu200GeV_cent.eps,height=9.cm}
\psfig{figure=photon_spec_AuAu200GeV_periph.eps,height=9.cm}
\caption{Photon spectra for central (0--10\%, left) and peripheral
(60--70\%, right) Au+Au reactions at $\sqrt{s_{\mbox{\tiny{\it{NN}}}}}$ = 200 GeV
as computed with our hydrodynamical model [with the contributions
for the QGP and hadron resonance gas (HRG) given separately]
compared to the expected NLO pQCD p+p yields for the prompt
$\gamma$~\protect\cite{vogel_gamma} (scaled by the corresponding
nuclear overlap function), and to the experimental photon
yields measured by the PHENIX collaboration~\protect\cite{ppg042}.}
\label{fig:photon_spec_AuAu_cent_periph}
\end{figure*}
\subsection{Total direct photon spectra}
Figure~\ref{fig:photon_spec_AuAu_cent_periph} shows our computed
total direct photon spectra for central (left) and peripheral (right)
Au+Au collisions at $\sqrt{s_{\mbox{\tiny{\it{NN}}}}}$ = 200 GeV,
with the pQCD, QGP, and HRG components differentiated\footnote{We
split the mixed phase contribution onto QGP and HRG components
calculating the relative proportion of QGP (HRG) matter in it.}.
In central reactions, thermal photon production (mainly of QGP origin)
outshines the prompt pQCD emission below $p_T\approx$ 3 GeV/$c$.
Within $p_T\approx$ 1 -- 4 GeV/$c$, thermal photons account for
roughly 90\% -- 50\% of the total photon yield in central Au+Au,
as can be better seen in the ratio total-$\gamma$/pQCD-$\gamma$
shown in Fig.~\ref{fig:RAA_photon}. Photon production in peripheral
collisions is, however, clearly dominated by the primary parton-parton radiation.
In both cases, hadronic gas emission prevails only for lower $p_T$ values.
In Figure~\ref{fig:photon_spec_AuAu_cent_periph} we also compare our
computed spectra to the inclusive Au+Au photon spectra published
recently by the PHENIX collaboration~\cite{ppg042}.
The total theoretical (pQCD+hydro) differential cross-sections are in
good agreement with the experimental yields, though for central reactions
our calculations tend to ``saturate'' the upper limits of the data
in the range below $p_T\approx$ 4 GeV/$c$ where thermal photons dominate.
New preliminary PHENIX Au+Au direct-$\gamma^*$ results~\cite{qm05,qm05_akiba}
are also systematically above (tough still consistent with) these published
spectra in the range $p_T\approx$ 1 -- 4 GeV/$c$ and, if confirmed, will
bring our results to an even better agreement with the data.\\
To better distinguish the relative amount of thermal radiation in the theoretical
and experimental {\it total} direct photon spectra in central Au+Au collisions,
we present in Figure~\ref{fig:RAA_photon} the nuclear modification factor
$R_{AA}^\gamma$ defined as the ratio of the total over prompt (i.e. $T_{AA}$-scaled
p+p pQCD predictions) photon yields:
\begin{equation}
R_{AA}^{\gamma}(p_T)\;=\;\frac{dN_{AuAu}^{total\;\gamma}/dp_{T}}{T_{AA}\cdot d\sigma_{pp}^{\gamma\;pQCD}/dp_{T}}.
\label{eq:RAA}
\end{equation}
\begin{figure}[htbp]
\psfig{figure=photon_RAA_AuAu200GeV_cent.eps,height=6.cm}
\caption{Direct photon ``nuclear modification factor'', $R_{AA}^\gamma$ (Eq.~\ref{eq:RAA}),
obtained as the ratio of the total over the prompt $\gamma$ spectra for
0--10\% most central Au+Au reactions at $\sqrt{s_{\mbox{\tiny{\it{NN}}}}}$ = 200 GeV.
The solid line is the ratio resulting from our hydro+pQCD model.
The points show the PHENIX data~\protect\cite{ppg042} over the same NLO yields and
the dashed-dotted curves indicate the theoretical uncertainty of the NLO calculations
(see text).}
\label{fig:RAA_photon}
\end{figure}
A value $R_{AA}^\gamma \approx$ 1 would indicate that all the photon yield can be
accounted for by the prompt production alone.
Of course, since our total direct-$\gamma$ result for central Au+Au includes thermal
emission from the QGP and HRG phases, we theoretically obtain $R_{AA}^\gamma \approx$ 10 -- 1
in the $p_T\approx$ 1 -- 4 GeV/$c$ region where the thermal component is significant
(Fig.~\ref{fig:RAA_photon}). In this very same $p_T$ range, although the available PHENIX
results have still large uncertainties\footnote{Technically, the PHENIX data points below
$p_T$ = 4 GeV/$c$ have ``lower errors that extend to zero'', i.e. a non-zero direct-$\gamma$ signal is
indeed observed in the data but the associated errors are larger than the signal itself~\cite{ppg042}.},
the central value of most of the data points is clearly consistent with the existence of
a significant excess over the NLO pQCD expectations. A note of caution is worth here, however,
regarding the $R_{AA}^\gamma\gg$ 1 value observed for both the theoretical and experimental
spectra below $p_T\approx$ 4 GeV/$c$ since it is not yet clear to what extent the
NLO predictions, entering in the denominator of Eq.~(\ref{eq:RAA}), are realistic in
this thermal-photon ``region of interest''.
Indeed, in this comparatively low $p_T$ range the theoretical prompt yields are dominated
by the jet bremsstrahlung contribution~\cite{dde_hq04} which is intrinsically non-perturbative
(i.e. not computable) and determined solely from the parametrized parton-to-photon
GRV~\cite{grv_photons} FF which is relatively poorly known in this kinematic range.
The standard scale uncertainties in the NLO pQCD calculations are $\pm$20\% above
$p_T\approx$ 4 GeV/$c$ but we have assigned a much more pessimistic $_{+50}^{-200}$\%
uncertainty to these calculations in the range $p_T\approx$ 1 -- 4 GeV/$c$ (dashed-dotted
lines in Fig.~\ref{fig:RAA_photon}). Precise measurements of the direct-$\gamma$ baseline
spectrum in p+p collisions at $\sqrt{s}$ = 200 GeV above $p_T$ = 1 GeV/$c$ are mandatory
before any definite conclusion can be drawn on the existence or not of a thermal excess
from the Au+Au experimental data.\\
\begin{figure}[htbp]
\begin{center}
\psfig{figure=photon_spec_AuAu200GeV_cent_thermal_comparison.eps,height=8.5cm}
\end{center}
\caption{Thermal photon predictions for central Au+Au reactions at $\sqrt{s_{_{NN}}}$ = 200 GeV
as computed with different hydrodynamical~\protect\cite{srivastava_sps_rhic,alam_sps_rhic,finnish_hydro}
or ``dynamical fireball''~\protect\cite{turbide} models, compared to (i) our hydro calculations
(dashed curve), (ii) the expected perturbative $\gamma$ yields ($T_{AA}$-scaled
NLO p+p calculations~\protect\cite{vogel_gamma}), and (iii) the experimental total direct photon spectrum
measured by PHENIX~\protect\cite{ppg042}.}
\label{fig:compare}
\end{figure}
As a final cross-check of our computed hydrodynamical photon yields, we have compared
them to previously published predictions for thermal photon production in Au+Au collisions
at top RHIC energy: D.~K.~Srivastava {\it et al.}~\cite{srivastava_sps_rhic}
(with initial conditions $\tau_0\approx$ 0.2 fm/$c$ and $T_0\approx$ 450 -- 660 MeV),
Jan-e Alam {\it et al.}\footnote{Alam {\it et al.} have recently~\cite{alam05} recomputed their
hydrodynamical yields using higher initial temperatures ($T_0$ = 400 MeV
at $\tau_0$ = 0.2 fm/$c$) and getting a better agreement with the data.}~\cite{alam_sps_rhic}
($\tau_0$ = 0.5 fm/$c$ and $T_0$ = 300 MeV),
F.~D.~Steffen and M.~H.~Thoma~\cite{steffen_sps_rhic_lhc} ($\tau_0$ = 0.5 fm/$c$ and
$T_0$ = 300 MeV), S.~S.~Rasanen {\it et al.}~\cite{finnish_hydro}
($\tau_0$ = 0.17 fm/$c$ and $T_0$ = 580 MeV), N.~Hammon {\it et al.}~\cite{frankfurt_rhic_lhc}
($\tau_0$ = 0.12 fm/$c$ and $T_0$ = 533 MeV), and Turbide {\it et al.}\footnote{Note that
{\it stricto senso} Turbide's spectra are not obtained with a pure hydrodynamical computation but
using a simpler ``dynamical fireball'' model which assumes constant acceleration in longitudinal
and transverse directions.}~\cite{turbide} ($\tau_0$ = 0.33 fm/$c$ and $T_0$ = 370 MeV).
For similar initial conditions, the computed total thermal yields in those works are compatible
within a factor of $\sim$2 with those presented here. Some of those predictions are shown
in Figure~\ref{fig:compare} confronted to our calculations. Our yields are, in general,
above all other predictions since, as aforementioned, both our initial thermalization time
and energy densities (temperatures) have the most ``extreme'' values possible consistent with
the RHIC charged hadron multiplicities. They agree specially well with the hydrodynamical
calculations of the Jyv\"askyl\"a group~\cite{finnish_hydro} which have been computed
with the same up-to-date QGP rates used here.
Given the current (large) uncertainties of the available published data, all thermal photon
predictions are consistent with the experimental results. However, as aforementioned, newer
(preliminary) PHENIX direct-$\gamma^\star$ measurements
have been reported very recently~\cite{qm05,qm05_akiba} and indicate a clear excess of direct photons
over NLO pQCD for Au +Au at $\sqrt{s_{_{NN}}}$ = 200 GeV in this $p_T$ range in excellent agreement
with our thermal photon calculations.
\section{Thermal photons and the QCD equation-of-state}
\label{sec:eos}
In order to experimentally isolate the thermal photon spectrum one needs
to subtract from the total direct $\gamma$ spectrum the non-equilibrated
``background'' of prompt photons. The prompt $\gamma$ contribution emitted
in a given Au+Au centrality can be measured separately in
reference p+p (or d+Au) collisions at the same $\sqrt{s}$, scaled by the
corresponding nuclear overlap function $T_{AA}(b)$, and subtracted from the
total Au+Au $\gamma$ spectrum~\cite{dde_hq04}. The simpler expectation is
that the remaining photon spectrum for a given impact parameter $b$
\begin{equation}
\frac{dN_{AuAu}^{thermal\;\gamma}(b)}{dp_T} \; = \; \frac{dN_{AuAu}^{total\;\gamma}(b)}{dp_T}
- T_{AA}(b)\cdot\frac{d\sigma_{pp}^{\gamma}}{dp_T}\;,
\label{eq:thermal_spec}
\end{equation}
will be just that due to thermal emission from the partonic and hadronic phases
of the reaction. Such a subtraction procedure can be effectively applied to all the
$\gamma$ spectra measured in different centralities as long as both the total Au+Au
and baseline p+p photon spectra are experimentally measured with reasonable
($\lesssim$15\%) point-to-point (systematical and statistical) uncertainties~\cite{dde_hq04}.
The subtracted spectra~(\ref{eq:thermal_spec}) can be therefore subject to
scrutiny in terms of the thermodynamical properties of the radiating medium.
\subsection{Determination of the initial temperature}
\begin{sloppypar}
Due to their weak electromagnetic interaction with the surrounding medium,
photons produced in the reaction escape freely the interaction region immediately
after their production. Thus, even when emitted from an equilibrated source, they are
not reabsorbed by the medium and do not have a black-body spectrum at the source temperature.
Nonetheless, since all the theoretical thermal $\gamma$ rates~\cite{arnold,turbide}
have a general functional dependence of the form\footnote{The $T^2$ factor is
just an overall normalization factor in this case (since its temporal variation is
small compared to the short emission times) and does not significantly alter the
exponential shape of the spectra.} $E_\gamma\,dR_{\gamma}/d^3p\propto T^2\cdot \exp{(-E_{\gamma}/T)}$,
one would expect the final spectrum to be locally exponential with an inverse slope
parameter strongly correlated with the (local) temperature $T$ of the radiating medium.
Obviously, such a general assumption is complicated by several facts.
On the one hand,
the final thermal photon spectrum is a sum of exponentials with different temperatures
resulting from emissions at different time-scales and/or from different
regions of the fireball which has strong temperature gradients (the core being much hotter
than the ``periphery''). On the other hand, collective flow effects (stronger for
increasingly central collisions) superimpose on top of the purely thermal emission
leading to an effectively larger inverse slope parameter ($T_{\ensuremath{\it
eff}}\approx\sqrt{(1+\beta)/(1-\beta)}\,T$)~\cite{peressou}.
One of the main results of this paper is to show that, based
upon a realistic hydrodynamical model, such effects do not
destroy completely the correlation between the apparent photon temperature
and the maximal temperature actually reached at the beginning of
the collision process. We will show that such a correlation indeed
exists and that the local inverse slope parameter obtained by fitting
to an exponential, at high enough $p_T$, the thermal photon spectrum obtained
via the expression (\ref{eq:thermal_spec}), indeed provides a good proxy of the
initial temperature of the system without much
distortion due to collective flow (and other) effects.\\
\end{sloppypar}
To determine to what extent the thermal slopes are indicative of the original
temperature of the system, we have fitted the thermal spectra obtained from our
hydrodynamical calculations in different Au+Au centralities to an exponential
distribution in different $p_T$ ranges. Since, -- according to our Glauber prescription
for the impact-parameter dependence of the hydrodynamical initial conditions --,
different centralities result in different initial energy densities, we can
in this way explore the dependence of the apparent thermal photon temperature on
the maximal initial temperatures $T_{0}$ (at the core) of the system. The upper plot
of Figure~\ref{fig:thermal_slopes} shows the obtained local slope parameter,
$T_{\ensuremath{\it eff}}$, as a function of the initial
$T_{0}$ for our default QGP+HRG hydrodynamical evolution (Table~\ref{tab:hydro_parameters}).
We find that although all the aforementioned effects smear the correlation between
the apparent and original temperatures, they do not destroy it completely.
The photon slopes are indeed approximately proportional to the initial temperature
of the medium, $T_{0}$. There is also an obvious anti-correlation between the $p_T$
of the radiated photons and their time of emission.
At high enough $p_T$ the hardest photons issuing from the
hottest zone of the system swamp completely any other softer contributions
emitted either at later stages and/or from outside the core region of the fireball.
Thus, the higher the $p_T$ range, the closer is $T_{\ensuremath{\it eff}}$
to the original $T_{0}$ at the center of the system. According to our calculations,
empirical thermal slopes measured above $p_T\approx$ 4 GeV/$c$ in central Au+Au collisions
are above $\sim$400 MeV i.e. only $\sim$30\% lower than the ``true'' maximal (local)
temperature of the quark-gluon phase. On the other hand, local $\gamma$ slopes
in the range below $p_T\approx$ 1 GeV/$c$ have almost constant value
$T_{\ensuremath{\it eff}}\sim$ 200 MeV (numerically close to $T_{crit}$)
for all centralities and are almost insensitive to the initial temperature of the
hydrodynamical system but mainly specified by the exponential prefactors in the
hadronic emission rates, plus collective boost effects.\\
\begin{figure}[htbp]
\centerline{\psfig{figure=Teff_vs_T0_qgp+hrg.eps,height=6.cm,width=9.5cm}}
\centerline{\psfig{figure=Teff_vs_T0_hrg.eps,height=5.cm,width=9.5cm}}
\caption{Local photon slope parameters $T_{\ensuremath{\it eff}}$
(obtained from exponential fits of the thermal photon
spectrum in different $p_T$ ranges) plotted versus the initial
(maximum) temperature $T_{0}$ of the fireball produced at different
centralities in Au+Au collisions at $\sqrt{s}=200$ GeV. Upper
plot - hydrodynamical calculations with QGP+HRG EoS
(Table~\ref{tab:hydro_parameters}), bottom - HRG EoS (with initial
conditions: $\varepsilon_0 = 30$ GeV/fm$^3$ at
$\tau_0=0.6$~fm/$c$).}
\label{fig:thermal_slopes}
\end{figure}
To assess the dependence of the thermal photon spectra on the underlying EoS,
we have rerun our hydro evolution
with just the EoS of a hadron resonance gas. We choose now as initial conditions:
$\varepsilon_0 = 30$ GeV/fm$^3$ at $\tau_0=0.6$~fm/$c$,
which can still reasonably describe the experimental hadron spectra.
Obviously, any description in terms of hadronic degrees of freedom
at such high energy densities is unrealistic but we are interested
in assessing the effect on the thermal photon slopes of a non ideal-gas EoS
as e.g. that of a HRG-like system with a large number of heavy resonances
(or more generally, of any EoS with exponentially rising number of mass states).
The photon slopes for the pure HRG gas EoS (Fig.~\ref{fig:thermal_slopes}, bottom)
are lower ($T_{\ensuremath{\it eff}}^{\ensuremath{\it max}}\approx$ 220 MeV)
than in the default QGP+HRG evolution, not only because the input HRG $\varepsilon_0$
is smaller (the evolution starts at a later $\tau_0$) but, specially because for the
same initial $\varepsilon_0$ the effective number of degrees of freedom in a system
with a HRG EoS is higher than that in a QGP\footnote{Note that $g(T)\propto \varepsilon/T^4$
increases exponentially with $T$ for a HRG-like EoS, and at high enough temperatures will
clearly overshoot the QGP constant number of degrees of freedom.}
and therefore the initial temperatures are lower. A second difference is that, for all
$p_T$ ranges, we find almost the same exact correlation between the local $\gamma$ slope
and $T_{0}$ indicating a single underlying (hadronic) radiation mechanism dominating
the transverse spectra at all $p_T$.\\
Two overall conclusions can be obtained from the study of the hydrodynamical photon
slopes. First, the observation in the data, via Eq.~(\ref{eq:thermal_spec}), of a thermal
photon excess above $p_T\approx$ 2.5 GeV/$c$ with exponential slope
$T_{\ensuremath{\it eff}}\gtrsim$ 250 MeV is an unequivocal proof of the formation
of a system with maximum temperatures above $T_{crit}$ since no realistic collective
flow mechanism can generate such a strong boost of the photon slopes, while simultaneously
reproducing the hadron spectra.
Secondly, pronounced $p_T$ dependences of the local thermal slopes seem to be characteristic
of space-time evolutions of the reaction that include an ideal-gas QGP radiating phase.
\subsection{Determination of the QCD Equation of State (EoS)}
As we demonstrated in the previous section, $T_{\ensuremath{\it eff}}$ is
approximately proportional to the maximum temperature reached in a nucleus-nucleus reaction.
One can go one step further beyond the mere analysis of the thermal photon slopes
and try to get a direct handle on the equation of state of the radiating medium by looking
at the correlation of $T_{\ensuremath{\it eff}}$ with experimental observables
related to the initial energy or entropy densities of the system.
For example, assuming an isentropic expansion (which is implicit in our perfect
fluid hydrodynamical equations with zero viscosity) one can estimate the
{\it initial} entropy density $s$ at the time of photon emission
from the total {\it final} particle multiplicity $dN/dy$ measured in the reaction.
Varying the centrality of the collision, one can then explore the form of the dependence $s\,=\,s(T)$
at the first instants of the reaction, extract the underlying equation of state
of the radiating system and trace any signal of a possible phase transition.
Indeed, the two most clear evidences of QGP formation from QCD calculations on
the lattice are: (i) the sharp rise of $\varepsilon(T)/T^4$, or equivalently
of $s(T)/T^3$, at temperatures around $T_{crit}$, and (ii) the flattening
of the same curve above $T_{crit}$. The sharp jump is of course due to the
sudden release of a large number of (partonic) degrees of freedom at $T_{crit}$.
The subsequent plateau is due to the full formation of a QGP with a
{\it fixed} (constant) number of degrees of freedom.\\
We propose here to use $T_{\ensuremath{\it eff}}$ as a proxy for the initial
temperature of the system, and directly study the evolution, versus $T_{\ensuremath{\it eff}}$,
of the effective number of degrees of freedom defined as\footnote{Units are in GeV and fm.
$\zeta(4)$ = $\pi^4/90$, where $\zeta(n)$ is the Riemann zeta function.}
\begin{equation}
g(s,\,T)\,=\,\frac{\pi^2}{4\,\zeta(4)}\frac{s}{T^3}\,(\hbar
c)^3=\,\frac{45}{2 \pi^2}\,\frac{s}{T^3}(\hbar c)^3,
\label{eq:ndf}
\end{equation}
which coincides with the degeneracy of a weakly interacting gas of massless particles.
[In a similar avenue, B.~Muller and K.~Rajagopal~\cite{muller_rajagopal05}
have recently proposed a method to estimate the number of thermodynamic
degrees of freedom via $g_{\ensuremath{\it eff}}\propto s^4/\epsilon^3$, where $s$ is
also determined from the final hadron multiplicities].
The dashed line in Fig.~\ref{fig:EoS} (top) shows the evolution of the {\it true}
number of degrees of freedom $g_{\ensuremath{\it hydro}}(s_0,T_0)$ computed via
Eq.~(\ref{eq:ndf}), as a function of the (maximal) temperatures and entropies
directly obtained from the initial conditions of our hydrodynamical model in different
Au+Au centralities\footnote{In the most peripheral reactions, the bag entropy has been
subtracted to make more apparent the drop near $T_c$.}.
The first thing worth to note is that $g(s,T)$ remains constant at the expected
degeneracy $g_{\ensuremath{\it hydro}}$ = 42.25 of an ideal gas of
$N_f$ = 2.5 quarks and gluons for basically {\it all} the
maximum temperatures accessible in the different centralities of Au+Au at
$\sqrt{s_{\mbox{\tiny{\it{NN}}}}}$ = 200 GeV. This indicates that
at top RHIC energies and for most of the impact parameters, $T_0$ is (well)
above $T_{crit}$ and the hottest parts of the initial fireball are in the QGP phase.
The expected drop in $g_{\ensuremath{\it hydro}}$ related to the transition
to the hadronic phase is only seen, if at all, for the very most peripheral
reactions (with $T_0\approx T_c$).
Thus, direct evidence of the QGP-HRG phase change itself via the study of the centrality
dependence of any experimentally accessible observable would only be potentially feasible
at RHIC in Au+Au reactions at {\it lower} center-of-mass energies~\cite{dde_dima}.\\
\begin{figure}[htbp]
\begin{center}
\psfig{figure=ndf_dNchdeta_vs_Teff_qgp+hrg.eps,height=6.cm,width=9.cm}
\includegraphic
[height=4.5cm,width=9.cm,clip=true,viewport=0 0 567 310]{ndf_dNchdeta_vs_Teff_hrg.eps}
\end{center}
\caption{Effective initial number of degrees of freedom obtained from our
hydrodynamical calculations with a QGP+HRG EoS (upper plot), and with a
pure HRG EoS (bottom), plotted as a function of the temperature ($T_{0}$) or
thermal photon slope ($T_{\ensuremath{\it eff}}$) in different Au+Au centrality
classes at $\sqrt{s_{_{NN}}}$ = 200 GeV. The number of degrees of freedom are
computed respectively: (i) From our initial thermodynamical conditions
$(s_0,\,T_0)$ via Eq.~(\protect\ref{eq:ndf}) (dashed line),
(ii) from the obtained charged hadron multiplicity
$dN_{ch}/d\eta$ and the {\it true} initial temperature $T_0$
via Eq.~(\protect\ref{eq:geff}) (dotted-dashed line); and
(iii) from $dN_{ch}/d\eta$ and the thermal photon slopes
$T_{\ensuremath{\it eff}}$ measured in different $p_T$ ranges
via Eq.~(\protect\ref{eq:geff}) (solid lines). For illustrative purposes,
the open squares indicate the approximate position of the different Au+Au
centrality classes (in 10\% percentiles) for the values of $g_{\ensuremath{\it eff}}$
obtained using the thermal photon slopes measured above $p_T$ = 4 GeV/$c$.}
\label{fig:EoS}
\end{figure}
As aforementioned, we can empirically trace the QCD EoS shown in Fig.~\ref{fig:EoS}
(and eventually determine the temperature-evolution of the thermodynamic degrees of freedom
of the produced medium) using the estimate of the initial temperature given by the
thermal photon slopes, $T_{\ensuremath{\it eff}}$, and a second observable closely
related to the initial entropy of the system such as the final-state hadron multiplicity, $dN/dy$.
Although one could have also considered to obtain $g_{\ensuremath{\it eff}}$ via
$\varepsilon/T^4 \propto (dE_{T}/dy)/T_{\ensuremath{\it eff}}^4$,
using the transverse energy per unit rapidity $dE_{T}/dy$ measured in different Au+Au
centralities~\cite{ppg019}, we prefer to use the expression (\ref{eq:ndf}) which contains
the entropy-, rather than the energy-, density for two reasons:
\begin{description}
\item (i) the experimentally accessible values of $dN/dy$ remain constant in an isentropic expansion
(i.e. $dN/dy \propto s_0$) whereas, due to longitudinal work, the measured final
$dE_{T}/dy$ provides only a {\it lower limit} on the initial $\varepsilon$
($dE_{T}/dy \lesssim \varepsilon_0$); and
\item (ii) $g_{\ensuremath{\it eff}}\propto s/T_{\ensuremath{\it eff}}^3$ is less sensitive to
experimental uncertainties associated to the measurement of $T_{\ensuremath{\it eff}}$ than
$g_{\ensuremath{\it eff}}\propto \varepsilon/T_{\ensuremath{\it eff}}^4$ is.
\end{description}
\begin{sloppypar}
Again, in the absence of dissipative effects, the space-time evolution of the
produced system in a nucleus-nucleus reaction is isentropic and the entropy density
(per unit rapidity) at the thermalization time $\tau_{0}$ can be directly connected
(via $s \,\approx\, 4 \,\rho$~\cite{wong_book}) to the final charged hadron
pseudo-rapidity density\footnote{This formula uses $N_{tot}/N_{ch}$ = 3/2,
and the Jacobian $|d\eta/dy|=E/p\approx$ 1.2.}:
\begin{equation}
s\,\approx\,4\cdot\frac{dN}{dV}\,\approx\,
\frac{7.2}{\mean{A_\perp}\cdot\tau_0}\cdot\frac{dN_{ch}}{d\eta}
\label{eq:entropy}
\end{equation}
where we have written the volume of the system, $dV=\mean{A_\perp}\tau_0\,d\eta$,
as the product of the (purely geometrical) average transverse overlap area for
each centrality times the starting proper time of our hydro evolution ($\tau_0=0.15$ fm/$c$),
and where $dN_{ch}/d\eta$ is the {\it charged} hadron multiplicity customarily
measured experimentally at mid-rapidity\footnote{Note again that both the
photon slopes and the charged hadron multiplicities are proxies of the thermodynamical
conditions of the system {\it at the same time} $\tau_{0}$.}.
By combining, Eqs.~(\ref{eq:ndf}) and (\ref{eq:entropy}),
we obtain an estimate for the number of degrees of freedom of the system
produced in a given A+A collision at impact parameter $b$:
\begin{equation}
g_{\ensuremath{\it eff}}\left(\frac{dN_{ch}(b)}{d\eta},\,T_{\ensuremath{\it eff}}(b)\right)\,\approx
\,\frac{150}{\pi^2}\cdot\frac{(\hbar c)^3}{\mean{A_\perp(b)}\cdot\tau_0\cdot T^{3}_{\ensuremath{\it eff}}(b)}\cdot
\frac{dN_{ch}(b)}{d\eta}\;,
\label{eq:geff}
\end{equation}
which can be entirely determined with two experimental observables: $dN_{ch}/d\eta$ and
$T_{\ensuremath{\it eff}}$.\\
\end{sloppypar}
Let us first assess to what extent the ansatz~(\ref{eq:geff}) is affected by the assumption that
Eq.~(\ref{eq:entropy}) indeed provides a good experimental measure of the initial entropy
density $s$. The dotted-dashed curve in Fig.~\ref{fig:EoS} has been obtained via
Eq.~(\ref{eq:geff}) using the $(dN_{ch}/d\eta)/\mean{A_\perp}$ values obtained from
our hydrodynamical model, and the {\it true} (input) initial temperature of the system $T_0$,
and thus it is only sensitive to the way we estimate the entropy density. The resulting curve
is a factor of $\sim$3 below the expected ``true'' $g_{\ensuremath{\it hydro}}$ curve, i.e.
$g_{\ensuremath{\it eff}}\left(dN_{ch}/d\eta,T_{0}\right)\approx 3\cdot g_{\ensuremath{\it hydro}}(s_0,T_0)$,
indicating that Eq.~(\protect\ref{eq:geff}) underestimates by the same amount the maximal entropy of the
original medium. This is so because our estimate $(dN_{ch}/d\eta)/\mean{A_\perp}$ specifies the entropy
density averaged over the {\it whole} Glauber transverse area $\mean{A_\perp}$, whereas the maximal
entropy area in the {\it core} of the system (from where the hardest thermal photons are emitted)
is $\sim$3 times {\it smaller}. Although one could think of a method to correct for this difference,
this would introduce an extra model-dependence that we want to avoid at this point.
We prefer to maintain the simple (geometrical overlap) expression of the transverse area
$\mean{A_\perp(b)}$ in Eq.~(\protect\ref{eq:geff}), and exploit the fact that, although such an
equation does not provide the true {\it absolute} number of degrees of freedom, it does provide a
very reliable indication of the dependence of $g_{\ensuremath{\it eff}}$ on the temperature
of the system and, therefore, of the exact {\it form} of the underlying EoS.\\
Finally, let us consider the last case where we use Eq.~(\ref{eq:geff}) with
the values of $dN_{ch}/d\eta$ {\it and} $T_{\ensuremath{\it eff}}$ that can be actually
experimentally measured. The different solid curves in the upper plot of Fig.~\ref{fig:EoS}
show the effective degeneracy $g_{\ensuremath{\it eff}}$, computed using
Eq.~(\ref{eq:geff}) and the local photon slopes $T_{\ensuremath{\it eff}}$
measured in different $p_T$ ranges for our default QGP+HRG evolution.
As one could expect from Fig.~\ref{fig:thermal_slopes}, the best reproduction
of the shape of the underlying EoS
is obtained with the effective temperatures measured in higher $p_T$ bins.
For those $T_{\ensuremath{\it eff}}$, the computed $g_{\ensuremath{\it eff}}$'s
show a relatively constant value in a wide range of centralities as expected for
a weakly interacting QGP. Deviations from this ideal-gas plateau appear
for more central collisions, due to an increasing difference between the
(high) initial temperatures, $T_0$, and the apparent
temperature given by the photon slopes (Fig.~\ref{fig:thermal_slopes}).
Such deviations do not spoil, however, the usefulness of our estimate
since, a non-QGP EoS would result in a considerably different
dependence of $g_{\ensuremath{\it eff}}$ on the reaction centrality. Indeed,
the different curves in the bottom plot of Fig.~\ref{fig:EoS} obtained with
a pure hadron resonance gas EoS clearly indicate\footnote{Accidentally,
$g_{\ensuremath{\it eff}}\gtrsim g_{\ensuremath{\it hydro}}$ in the case of a HRG EoS,
because the underestimation of the apparent temperature (raised to the cube)
compensates for the aforementioned area averaging of the entropy.}
that a HRG EoS, or in general any EoS with exponentially increasing number of mass states,
would bring about a much more dramatic rise of $g_{\ensuremath{\it eff}}$ with $T_{\ensuremath{\it eff}}$.\\
In summary, the estimate~(\ref{eq:geff}) indeed provides a direct experimental handle
on the {\it form} of the EoS of the strongly interacting medium produced in the first
instants of high-energy nuclear collisions. More quantitative conclusions on the possibility
to extract the exact shape of the underlying EoS and/or the absolute number of degrees of freedom
of the produced medium require more detailed theoretical studies (e.g. with varying
lattice-inspired EoS's~\cite{dde_dima} and/or using more numerically involved 3D+1 hydrodynamical approaches).
In any case, we are confident that by experimentally measuring the thermal photon slopes
in different Au+Au centralities and correlating them with the associated charged
hadron multiplicities as in Eq.~(\ref{eq:geff}), one can approximately observe the
expected ``plateau'' in the number of degrees of freedom indicative of QGP formation
above a critical value of $T$.
\section{Conclusions}
We have studied thermal photon production in Au+Au reactions at
$\sqrt{s_{\mbox{\tiny{\it{NN}}}}}$ = 200 GeV using a Bjorken hydrodynamic
model with longitudinal boost invariance. We choose the initial conditions
of the hydrodynamical evolution so as to efficiently reproduce the
observed particle multiplicity in central Au+Au collisions at RHIC
and use a simple Glauber prescription to obtain the corresponding initial
conditions for all other centralities. With such a model we can perfectly
reproduce the identified soft pion, kaon and proton $p_T$-differential spectra measured at RHIC.
Complementing our model with the most up-to-date parametrizations of the QGP and
HRG thermal photon emission rates plus a NLO pQCD calculation of the
prompt $\gamma$ contribution, we obtain direct photon spectra which
are in very good agreement with the Au+Au direct photon (upper limit) yields
measured by the PHENIX experiment. In central collisions, a thermal
photon signal should be identifiable as a factor of $\sim$8 -- 1 excess
over the pQCD $\gamma$ component within $p_T\approx$ 1 -- 4 GeV/$c$,
whereas pure prompt emission clearly dominates the photon spectra
at all $p_T$ in peripheral reactions. The local inverse slope parameter of the
thermal photon spectrum is found to be directly correlated to the maximum temperature
attained in the course of the collision. The experimental measurement of local thermal photon
slopes above $p_T\approx$ 2.5 GeV/$c$, with values $T_{\ensuremath{\it eff}}\gtrsim$ 250 MeV
and with pronounced $p_T$ dependences can only be reproduced
by space-time evolutions of the reaction that include a QGP phase.\\
Finally, we have proposed and tested within our framework, an empirical method to
determine the effective thermodynamical number of degrees of freedom of the produced medium,
$g(s,T)\propto s(T)/T^3$, by correlating the thermal photon slopes with the final-state charged
hadron multiplicity measured in different centrality classes. We found that one can
clearly distinguish between the equation of state of a weakly interacting
quark-gluon plasma and that of a system with rapidly rising number of mass
states with $T$. Stronger quantitative conclusions on the exact shape of the
underlying EoS and/or the absolute number of degrees of freedom
of the produced medium require more detailed theoretical studies as well as high
precision photon data in Au+Au and baseline p+p, d+Au collisions.
In any case, the requirement for hydrodynamical models of
concurrently describing the experimental bulk hadron and thermal photon spectra for
different Au+Au centralities at $\sqrt{s_{\mbox{\tiny{\it{NN}}}}}$ = 200 GeV, imposes
very strict constraints on the form of the equation of state of the underlying
expanding QCD matter produced in these reactions.
\section{Acknowledgments}
We would like to thank Werner Vogelsang for providing us with his NLO
pQCD calculations for photon production in p+p collisions at
$\sqrt{s}$ = 200 GeV; Sami Rasanen for valuable comments on hydrodynamical
photon production; and Helen Caines and Olga Barannikova for useful discussions
on (preliminary) STAR hadron data. D.P. acknowledges support from
MPN of Russian Federation under grant NS-1885.2003.2.
|
{
"timestamp": "2006-02-09T13:01:30",
"yymm": "0503",
"arxiv_id": "nucl-th/0503054",
"language": "en",
"url": "https://arxiv.org/abs/nucl-th/0503054"
}
|
\subsection*{Notation}
We use the standard notation for the coproduct-insertion maps:
we say that an ordered set is a pair of a finite set $S$ and a
bijection $\{1,\ldots,|S|\} \to S$.
For $I_1,\dots,I_m$ disjoint ordered subsets of $\{1,\dots,n\}$,
$(U,\Delta)$ a Hopf algebra and $a \in U^{\otimes m}$,
we define
$$a^{I_1,\dots,I_n}= \sigma_{I_1,\ldots,I_m} \circ
(\Delta^{|I_1|}\otimes \cdots \otimes \Delta^{|I_n|})(a),
$$
with $\Delta^{(1)}=\on{id}$, $\Delta^{(2)}=\Delta$,
$\Delta^{(n+1)}=({\on{id}}^{\otimes n-1} \otimes \Delta)\circ \Delta^{(n)}$,
and $\sigma_{I_1,\ldots,I_m} : U^{\otimes \sum_i |I_i|} \to
U^{\otimes n}$ is the
morphism corresponding to the map $\{1,\ldots,\sum_i |I_i|\}
\to \{1,\ldots,n\}$ taking $(1,\ldots,|I_1|)$ to $I_1$,
$(|I_1| + 1,\ldots,|I_1| + |I_2|)$ to $I_2$, etc.
When $U$ is cocommutative, this definition depends only on
the sets underlying $I_1,\ldots,I_m$.
\subsection*{Acknowledgements}
We would like to thank V. Dolgushev, P. Etingof and L.-C. Li for
discussions.
\section{Solutions of the functional twist equations}
If ${\mathfrak{g}}$ is a Lie algebra, we denote by
${\mathcal O}_{{\mathfrak{g}}^*} = \wh S({\mathfrak{g}})$ the formal series ring of functions on the
formal neighborhood of $0$ in ${\mathfrak{g}}^*$. We define by ${\mathfrak{m}}_{{\mathfrak{g}}^*} \subset
{\mathcal O}_{{\mathfrak{g}}^*}$ the maximal ideal of this ring. If $k$ is an integer $\geq 1$,
we denote by ${\mathcal O}_{({\mathfrak{g}}^*)^k} = \wh S({\mathfrak{g}})^{\wh\otimes k}$ the
\footnote{$\wh\otimes$ is
the completed tensor product, defined by
$V_0[[x_1,\ldots,x_n]] \wh\otimes W_0[[y_1,\ldots,y_n]] :=
V_0\otimes W_0[[x_1,\ldots,y_n]]$, where $V_0,W_0$ are vector spaces.}
ring of formal functions functions on $({\mathfrak{g}}^*)^k$, by ${\mathfrak{m}}_{({\mathfrak{g}}^*)^k}$
its maximal ideal and by ${\mathfrak{m}}_{({\mathfrak{g}}^*)^k}^i$ the $i$th power of this ideal.
If $f,g\in {\mathfrak{m}}_{({\mathfrak{g}}^*)^k}^2$, then the series
$f \star g = f + g + {1\over 2} \{f,g\} +
\cdots + B_n(f,g) + \cdots$ is convergent, where $\sum_{i\geq 1}
B_i(x,y)$ is the Baker-Campbell-Hausdorff series specialized to the
Poisson bracket of ${\mathfrak{m}}_{({\mathfrak{g}}^*)^k}^2$. The product $\star$ defines a
group structure on ${\mathfrak{m}}_{({\mathfrak{g}}^*)^k}^2$.
If $f\in {\mathcal O}_{{\mathfrak{g}}^*}^{\wh\otimes n}$
and $P_1,\dots, P_m$ are disjoint subsets of
$\{1,\dots,m\}$, one defines $f^{P_1,\dots,P_n}$ as
in the Introduction using the cocommutative coproduct of
${\mathcal O}_{{\mathfrak{g}}^*}$ (dual to the addition of ${\mathfrak{g}}^*$).
\medskip
Let ${\mathfrak{g}}$ be a Lie algebra and $Z\in \wedge^3({\mathfrak{g}})^{\mathfrak{g}}$.
\begin{proposition}
\label{lift1}
There exists $\varphi\in ({\mathfrak{m}}_{{\mathfrak{g}}^*}^{\wh\otimes 3})^{\mathfrak{g}}
(\subset {\mathfrak{m}}_{({\mathfrak{g}}^*)^3}^2)$, the image of
which under the map
${\mathfrak{m}}_{{\mathfrak{g}}^*}^{\wh\otimes 3} \to
({\mathfrak{m}}_{{\mathfrak{g}}^*} / {\mathfrak{m}}_{{\mathfrak{g}}^*}^2)^{\otimes 3} = {\mathfrak{g}}^{\otimes 3}
\stackrel{\on{Alt}}{\to} \wedge^3({\mathfrak{g}})$
equals $Z$ (here $\on{Alt}$ is the total antisymmetrization map)
and satisfying the functional pentagon equation
$$
\varphi^{1,2,34} \star \varphi^{12,3,4} = \varphi^{2,3,4} \star
\varphi^{1,23,4} \star \varphi^{1,2,3}.
$$
Such a $\varphi$ (we call it a lift of $Z$) is unique up to the action
of an element of $({\mathfrak{m}}_{{\mathfrak{g}}^*}^{\wh\otimes 2})^{\mathfrak{g}}$
by $\sigma \cdot \varphi = \sigma^{2,3} \star \sigma^{1,23} \star
\varphi \star (-\sigma)^{12,3} \star (-\sigma)^{1,2}$.
\end{proposition}
{\em Proof.} In \cite{Dr:QH}, Proposition 3.10, Drinfeld constructed a solution
$\Phi\in U({\mathfrak{g}})^{\otimes 3}[[\hbar]]$ of the pentagon equation
\begin{equation} \label{pent}
\Phi^{1,2,34} \Phi^{12,3,4} = \Phi^{2,3,4} \Phi^{1,23,4} \Phi^{1,2,3}
\end{equation}
such that $\varepsilon^{(2)}(\Phi)=1$ and $\Phi = 1^{\otimes 3} + O(\hbar)$
(here $\varepsilon^{(2)} = \id\otimes \varepsilon\otimes\id$; applying $\varepsilon$ to the
first and third factors of (\ref{pent}), we also get $\varepsilon^{(1)}(\Phi) =
\varepsilon^{(3)}(\Phi)=1$).
In \cite{EH2}, we stated that $\Phi$ can be transformed into an admissible
solution $\Phi'$ of the same equations, using an invariant twist. In Appendix
\ref{app:A}, we explain why the proof given in \cite{EH2} is wrong and
we give a correct proof.
The classical limit of $\hbar\log(\Phi')$ then satisfies the functional
pentagon equation.
This gives the existence of $\varphi$. One can also construct $\varphi$
directly using cohomological methods, as it will be done for $\rho$ later.
Let us prove uniqueness: let $\varphi$ and $\varphi'$ be two lifts of
$Z$. The classes of $\varphi$ and $\varphi'$ are the same in
${\mathfrak{m}}_{{\mathfrak{g}}^*}^{\wh\otimes 3}/({\mathfrak{m}}_{{\mathfrak{g}}^*}^{\wh\otimes 3}\cap {\mathfrak{m}}_{({\mathfrak{g}}^*)^3} ^3)$,
as this space is $0$.
Let $N$ be an integer $\geq 3$; assume that we have found $\sigma_N
\in ({\mathfrak{m}}_{{\mathfrak{g}}^*}^{\wh\otimes 2})^{\mathfrak{g}}$ such that $\sigma_N \cdot \varphi$
and $\varphi'$ are equal modulo ${\mathfrak{m}}_{{\mathfrak{g}}^*}^{\wh\otimes 3} \cap
{\mathfrak{m}}_{({\mathfrak{g}}^*)^3}^N$. Write $\varphi'=\sigma_N \cdot \varphi+\psi$,
with $\psi \in ({\mathfrak{m}}_{{\mathfrak{g}}^*}^{\wh\otimes 3} \cap {\mathfrak{m}}_{({\mathfrak{g}}^*)^3}^N)^{\mathfrak{g}}$.
We will use the following lemma (see \cite{EGH}, p. 2477):
\begin{lemma}
\label{lemmetech}
For any $k\geq 1$ and $n \geq 2$,
$f,h \in {\mathfrak{m}}_{({\mathfrak{g}}^*)^k}^2$ and
$g \in {\mathfrak{m}}_{({\mathfrak{g}}^*)^k}^n$, one has
$$
f \star (h + g) = f \star h + g, \quad
(f+g)\star h = f \star h +g
\hbox{\ modulo\ }{\mathfrak{m}}_{({\mathfrak{g}}^*)^k}^{n+1}.
$$
\end{lemma}
Let $\overline\psi$ be the class of $\psi$ in
$({\mathfrak{m}}_{{\mathfrak{g}}^*}^{\wh\otimes 3}\cap{\mathfrak{m}}_{({\mathfrak{g}}^*)^3}^N)^{\mathfrak{g}} /
({\mathfrak{m}}_{{\mathfrak{g}}^*}^{\wh\otimes 3}\cap {\mathfrak{m}}_{({\mathfrak{g}}^*)^3}^{N+1})^{\mathfrak{g}} =
(S^{>0}({\mathfrak{g}})^{\otimes 3})^{\mathfrak{g}}_N$. Then
$\overline\psi^{1,2,34} +\overline\psi^{12,3,4} =
\overline\psi^{2,3,4} + \overline\psi^{1,23,4} + \overline\psi^{1,2,3}$,
which means that $\overline\psi$ is a cocycle in the subcomplex
$((S^{>0}({\mathfrak{g}})^{\otimes \cdot})^{\mathfrak{g}},d)$ of the
co-Hochschild\footnote{We
denote by $S({\mathfrak{g}})$ the symmetric algebra of ${\mathfrak{g}}$, by $S^{>0}({\mathfrak{g}})$
is positive degree part; the index $N$ means the part of total degree $N$.}
complex $(S({\mathfrak{g}})^{\otimes \cdot},d)$.
Using \cite{Dr:QH}, Proposition 3.11, one can prove that the $k$th
cohomology group of this complex is $\wedge^k({\mathfrak{g}})^{\mathfrak{g}}$ and that the
antisymmetrization map coincides with the canonical map from the space of
cocycles to the cohomology.
For $N=3$, the hypothesis implies that $\on{Alt}(\overline\psi) = 0$,
so $\overline\psi$ is a coboundary of an element $\overline\tau_3 \in
(S^{>0}({\mathfrak{g}})^{\otimes 2})^{\mathfrak{g}}_3$. For $N >3$, $\overline\psi$ is the
a coboundary of an element $\overline\tau_N\in (S^{>0}({\mathfrak{g}})^{\otimes 2})^{\mathfrak{g}}_N$,
since the degree $N$ part of the relevant cohomology group
vanishes.
We then set $\sigma_{N+1} = \sigma_N + \tau_N$, where $\tau_N
\in ({\mathfrak{m}}_{{\mathfrak{g}}^*}^{\wh\otimes 2} \cap {\mathfrak{m}}_{({\mathfrak{g}}^*)^2}^N)^{\mathfrak{g}}$ is a lift of
$\overline\tau_N$. Then
$\sigma_{N+1} \cdot \varphi$ and $\varphi'$ are equal modulo
${\mathfrak{m}}_{{\mathfrak{g}}^*}^{\wh\otimes 3} \cap {\mathfrak{m}}_{({\mathfrak{g}}^*)^3}^{N+1}$. The sequence
$(\sigma_N)_{N\geq 3}$ has a limit $\sigma$. Then $\sigma \cdot \varphi =
\varphi'$.
\hfill \qed \medskip
We now construct a lift of $r$:
\begin{theorem}
\label{lift2}
There exists $\rho\in {\mathfrak{m}}_{{\mathfrak{g}}^*}^{\wh\otimes 2}$, the image of which in
${\mathfrak{g}}^{\otimes 2}$ under the square of
the projection ${\mathfrak{m}}_{{\mathfrak{g}}^*} \to {\mathfrak{m}}_{{\mathfrak{g}}^*} / {\mathfrak{m}}_{{\mathfrak{g}}^*}^2 = {\mathfrak{g}}$
equals $r$, and such that
\begin{equation}
\label{twistequation}
\rho^{1,2}\star \rho^{12,3} = \rho^{2,3} \star \rho^{1,23} \star \varphi.
\end{equation}
Such a $\rho$ (we call it a lift of $r$) is unique up to the action of
${\mathfrak{m}}_{{\mathfrak{g}}^*}$
by $\lambda \cdot \rho = \lambda^{1} \star \lambda^{2} \star \rho
\star (-\lambda)^{12}$. We call equation (\ref{twistequation}) the functional
cocycle equation.
\end{theorem}
{\em Proof.}
Let us construct $\rho$ by induction: we will construct a convergent sequence
$\rho_N \in {\mathfrak{m}}_{{\mathfrak{g}}^*}^{\wh\otimes 2}$ ($N\geq 2$)
satisfying (\ref{twistequation}) in ${\mathfrak{m}}_{{\mathfrak{g}}^*}^{\wh\otimes 3} /
({\mathfrak{m}}_{{\mathfrak{g}}^*}^{\wh\otimes 3} \cap {\mathfrak{m}}_{({\mathfrak{g}}^*)^3}^N)$. When $N = 3$, we take for
$\rho_2$ any lift of $r$ to ${\mathfrak{m}}_{{\mathfrak{g}}^*}^{\wh\otimes 2}$; then
equation (\ref{twistequation}) is automatically satisfied.
Let $N$ be an integer $\geq 3$;
assume that we have constructed $\rho_N$ in ${\mathfrak{m}}_{{\mathfrak{g}}^*}^{\wh\otimes 2}$
satisfying equation (\ref{twistequation}) in ${\mathfrak{m}}_{{\mathfrak{g}}^*}^{\wh\otimes 3}/
({\mathfrak{m}}_{{\mathfrak{g}}^*}^{\wh\otimes 3}\cap{\mathfrak{m}}_{({\mathfrak{g}}^*)^3} ^N)$.
Set $\alpha_N :=
\rho_N^{1,2}\star \rho_N^{12,3} -\rho_N^{2,3} \star \rho_N^{1,23}
\star \varphi$. Then $\alpha_N$ belongs to
${\mathfrak{m}}_{{\mathfrak{g}}^*}^{\wh\otimes 3}\cap{\mathfrak{m}}_{({\mathfrak{g}}^*)^3}^N$, and the following
equalities hold in ${\mathfrak{m}}_{{\mathfrak{g}}^*}^{\wh\otimes 4}/({\mathfrak{m}}_{{\mathfrak{g}}^*}^{\wh\otimes 4} \cap
{\mathfrak{m}}_{({\mathfrak{g}}^*)^4}^{N+1})$:
\begin{align*}
\alpha_N^{12,3,4} = &~
\rho_N^{1,2} \star \alpha_N^{12,3,4}=
\rho_N^{1,2} \star \rho_N^{12,3}\star \rho_N^{123,4} -
\rho_N^{1,2} \star \rho_N^{3,4} \star \rho_N^{12,34} \star \varphi^{12,3,4}
\\ & ~(\hbox{using Lemma \ref{lemmetech}})
\\
=&~(\alpha_N^{1,2,3}+ \rho_N^{2,3} \star \rho_N^{1,23}\star \varphi^{1,2,3})
\star \rho_N^{123,4}-\rho_N^{3,4}\star \rho_N^{1,2}\star \rho_N^{12,34}
\star \varphi^{12,3,4} \\
=&~\alpha_N^{1,2,3} + \rho_N^{2,3} \star \rho_N^{1,23}
\star \rho_N^{123,4}\star \varphi^{1,2,3}
-\rho_N^{3,4}\star (\rho_N^{2,34}\star \rho_N^{1,234} \star \varphi^{1,2,34}
+\alpha_N^{1,2,34}) \star \varphi^{12,3,4} \cr
&~(\hbox{using Lemma \ref{lemmetech}, the invariance of }\varphi\hbox{ and
the definition of }\alpha_N^{1,2,34})\cr
=&~\alpha_N^{1,2,3} + \rho_N^{2,3} \star (\alpha_N^{1,23,4}+
\rho_N^{23,4} \star \rho_N^{1,234} \star \varphi^{1,23,4})\star
\varphi^{1,2,3}\cr
&- \alpha_N^{1,2,34}-\rho_N^{3,4}\star \rho_N^{2,34}\star \rho_N^{1,234}
\star \varphi^{1,2,34} \star \varphi^{12,3,4} \cr
&~(\hbox{using the definition of }\alpha_N^{1,23,4}
\hbox{ and Lemma \ref{lemmetech}})\cr
=&~\alpha_N^{1,2,3} + \alpha_N^{1,23,4}+ (\rho_N^{3,4} \star
\rho_N^{2,34}\star \varphi^{2,3,4}+ \alpha_N^{2,3,4} )\star \rho_N^{1,234}
\star \varphi^{1,23,4}\star \varphi^{1,2,3}\cr
&- \alpha_N^{1,2,34}-\rho_N^{3,4}\star \rho_N^{2,34}\star \rho_N^{1,234}
\star \varphi^{1,2,34} \star \varphi^{12,3,4} \cr
&~ (\hbox{using the definition of }\alpha_N^{2,3,4} \hbox{ and Lemma
\ref{lemmetech}}) \cr
&~ = \alpha_N^{1,2,3} + \alpha_N^{1,23,4} - \alpha_N^{1,2,34} + \alpha_N^{2,3,4}
\end{align*}
(using Lemma \ref{lemmetech}, the invariance of $\varphi$ and the
fact that $\varphi$ satisfies the functional pentagon equation).
Let us denote by $\overline\alpha_N$ the image of $\alpha_N$ in
$({\mathfrak{m}}_{{\mathfrak{g}}^*}^{\wh\otimes 3}\cap{\mathfrak{m}}_{({\mathfrak{g}}^*)^3}^N) /
({\mathfrak{m}}_{{\mathfrak{g}}^*}^{\wh\otimes 3}\cap{\mathfrak{m}}_{({\mathfrak{g}}^*)^3}^{N+1})
= (S^{>0}({\mathfrak{g}})^{\otimes 3})_N$, then we get
$$
\overline\alpha_N^{12,3,4} + \overline\alpha_N^{1,2,34}
=\overline\alpha_N^{1,2,3} + \overline\alpha_N^{1,23,4}
+\overline\alpha_N^{2,3,4}.
$$
This means that $\alpha$ is a cocycle for the subcomplex
$(S^{>0}({\mathfrak{g}})^{\otimes\cdot},d)$ of the co-Hochschild complex.
Using \cite{Dr:QH}, Proposition 3.11, one proves that the $k$th
cohomology group of this subcomplex is $\wedge^k({\mathfrak{g}})$, and that
antisymmetrization coincides with the canonical projection from the space of
cocycles to the cohomology group.
For $N=3$, the equation $\on{CYB}(r)=Z$ implies
$\on{Alt}(\overline\alpha_3)=0$, hence $\overline\alpha_3$ is the coboundary
of an element $\overline\beta_3\in (S^{>0}({\mathfrak{g}})^{\otimes 2})_3$.
For $N>3$, $\overline\alpha_N$ is the coboundary of an element
$\overline\beta_N \in (S^{>0}({\mathfrak{g}})^{\otimes 2})_N$, since the degree $N$
part of the cohomology vanishes. We then set $\rho_{N+1} := \rho_N + \beta_N$,
where $\beta_N\in{\mathfrak{m}}_{{\mathfrak{g}}^*}^{\wh\otimes 2} \cap {\mathfrak{m}}_{({\mathfrak{g}}^*)^2}^N$ is a
representative of $\overline\beta_N$. Then $\rho_{N+1}$ satisfies
(\ref{twistequation}) in ${\mathfrak{m}}_{{\mathfrak{g}}^*}^{\wh\otimes 3}
/({\mathfrak{m}}_{{\mathfrak{g}}^*}^{\wh\otimes 3} \cap {\mathfrak{m}}_{({\mathfrak{g}}^*)^3}^{N+1})$.
The sequence $(\rho_N)_{N\geq 2}$ has a limit $\rho$, which then satisfies
(\ref{twistequation}).
The second part of the theorem can be proved either by analyzing the
choices for $\overline\beta_N$ in the above proof, or following the proof
of the previous proposition.
\hfill \qed \medskip
\begin{remark} \label{rem:sigma}
If $\varphi$ is replaced by $\varphi' = \sigma \star \varphi$, then a
solution of (\ref{twistequation}) is $\rho' = \rho \star (-\sigma)$.
\end{remark}
\section{Isomorphism of formal Poisson manifolds ${\mathfrak{g}}^* \simeq G^*$}
Let us assume that ${\mathfrak{g}}$ is a finite dimensional coboundary Lie bialgebra.
the following result was proved in \cite{EEM} when ${\mathfrak{g}}$ is quasitriangular;
the result of \cite{EEM} is itself a generalization of the formal version
of the Ginzburg-Weinstein isomorphism (\cite{GW,A,Bo}).
\begin{corollary}
\label{coroprinc}
There exists an isomorphism of formal Poisson manifolds ${\mathfrak{g}}^* \simeq G^*$.
\end{corollary}
{\em Proof.} Let $P : \wedge^2({\mathcal O}_{{\mathfrak{g}}^*}) \to {\mathcal O}_{{\mathfrak{g}}^*}$ be the Poisson
bracket on ${\mathcal O}_{{\mathfrak{g}}^*}$ corresponding to the
Lie-Poisson\footnote{or Kostant-Kirillov-Souriau, or linear} Poisson
structure on ${\mathfrak{g}}^*$.
Then $(\O_{{\mathfrak{g}}^*},m_0,P,\Delta_0)$ is a Poisson formal series Hopf (PFSH)
algebra; it corresponds to the formal Poisson-Lie group $({\mathfrak{g}}^*,+)$
equipped with its Lie-Poisson structure.
Set ${}^{\rho}\Delta(f) = \rho\star\Delta_0(f)\star (-\rho)$ for any
$f\in{\mathcal O}_{{\mathfrak{g}}^*}$. It follows from the fact that $\rho$ satisfies the
functional cocycle equation that $({\mathcal O}_{{\mathfrak{g}}^*},m_0,P,{}^{\rho}\Delta_0)$
is a PFSH algebra.
Let us denote by ${\bf PFSHA}$ and ${\bf LBA}$ the categories of PSFH algebras
and Lie bialgebras. We have a category equivalence
$c : {\bf PFSHA} \to {\bf LBA}$, taking $({\mathcal O},m,P,\Delta)$ to the Lie bialgebra
$(\c,\mu,\delta)$, where $\c := {\mathfrak{m}}/{\mathfrak{m}}^2$ (${\mathfrak{m}}\subset{\mathcal O}$ is the maximal ideal),
the Lie cobracket of $\c$ is induced by $\Delta - \Delta^{2,1} : {\mathfrak{m}}\to
\wedge^2({\mathfrak{m}})$, and the Lie bracket of $\c$ is induced by the Poisson
bracket $P : \wedge^2({\mathfrak{m}}) \to {\mathfrak{m}}$. The inverse of the functor $c$
takes $(\c,\mu,\delta)$ to ${\mathcal O} = \wh S(\c)$ equipped with its usual product;
$\Delta$ depends only on $\delta$ and $P$ depends on $(\mu,\delta)$.
Then $c$ restricts to a category equivalence $c_{\on{fd}} :
{\bf PFSHA}_{\on{fd}} \to {\bf LBA}_{\on{fd}}$ of subcategories of
finite-dimensional objects (in the case of ${\bf PFSH}$, we say that ${\mathcal O}$ is
finite-dimensional iff ${\mathfrak{m}}/{\mathfrak{m}}^2$ is).
Let $\on{dual} : {\bf LBA}_{\on{fd}} \to {\bf LBA}_{\on{fd}}$ be the duality
functor. It is a category antiequivalence; we have $\on{dual}({\mathfrak{g}},\mu,\delta) =
({\mathfrak{g}}^*,\delta^t,\mu^t)$. Then $\on{dual} \circ c_{\on{fd}} :
{\bf PFSHA}_{\on{fd}} \to {\bf LBA}_{\on{fd}}$ is a category antiequivalence.
Its inverse it the usual functor ${\mathfrak{g}}\mapsto U({\mathfrak{g}})^*$. If $G$ is the formal
Poisson-Lie group with Lie bialgebra ${\mathfrak{g}}$, one sets ${\mathcal O}_G = U({\mathfrak{g}})^*$.
Let us apply the functor $c$ to $({\mathcal O}_{{\mathfrak{g}}^*},m_0,P,{}^\rho\Delta_0)$.
We obtain $\c = {\mathfrak{m}}/{\mathfrak{m}}^2 = {\mathfrak{g}}$; the Lie bracket is unchanged w.r.t.
the case $\rho=0$, so it is the Lie bracket of ${\mathfrak{g}}$; the Lie cobracket
is given by $\delta(x) = [r,x\otimes 1 + 1\otimes x]$ since the reduction of
$\rho$ modulo $({\mathfrak{m}}_{{\mathfrak{g}}^*})^2\wh\otimes {\mathfrak{m}}_{{\mathfrak{g}}^*} + {\mathfrak{m}}_{{\mathfrak{g}}^*} \wh\otimes
({\mathfrak{m}}_{{\mathfrak{g}}^*})^2$ is equal to $r$.
Then applying $\on{dual} \circ c_{\on{fd}}$ to
$({\mathcal O}_{{\mathfrak{g}}^*},m_0,P,{}^\rho\Delta_0)$, we obtain the Lie bialgebra
${\mathfrak{g}}^*$. So this PFSH algebra is isomorphic to the PFSH algebra of the formal
Poisson-Lie group $G^*$. In particular, the Poisson algebras
${\mathcal O}_{{\mathfrak{g}}^*}$ and ${\mathcal O}_{G^*}$ are isomorphic. It is easy to check that
the map ${\mathfrak{g}} = {\mathfrak{m}}_{{\mathfrak{g}}^*} / {\mathfrak{m}}_{{\mathfrak{g}}^*}^2 \to {\mathfrak{m}}_{G^*}/{\mathfrak{m}}_{G^*}^2 = {\mathfrak{g}}$
induced by this isomorphism is the identity (here ${\mathfrak{m}}_{G^*} \subset {\mathcal O}_{G^*}$
is the maximal ideal).
\hfill \qed \medskip
\begin{remark} When ${\mathfrak{g}}$ is infinite dimensional, one can define
${\mathcal O}_{G^*}$ as the image of ${\mathfrak{g}}$ under ${\bf LBA} \to {\bf PFSHA}$
and then show that the Poisson algebras ${\mathcal O}_{G^*}$ and ${\mathcal O}_{{\mathfrak{g}}^*} =
(\wh S({\mathfrak{g}}),$ linear Poisson structure)
are isomorphic.
\end{remark}
\section{The morphism $S({\mathfrak{g}}^*)^{\mathfrak{g}} \hookrightarrow U({\mathfrak{g}}^*)$}
In this section, ${\mathfrak{g}}$ is a finite dimensional coboundary Lie bialgebras.
The following fact is well-known (\cite{STS2}):
\begin{lemma} \label{lemma:O}
${\mathcal O}_G^{\mathfrak{g}} \subset {\mathcal O}_G$ is a Poisson commutative subalgebra.
\end{lemma}
Here the action of ${\mathfrak{g}}$ on ${\mathcal O}_G$ corresponds to adjoint
action of $G$. We recall the proof: if $f,g\in {\mathcal O}_G$, then
$\{f,g\} = m(({\bf L} - {\bf R})(r)(f\otimes g))$, where
${\bf L}, {\bf R}$ are the infinitesimal left and right actions
and $m$ is the product map.
If $\varphi \in {\mathcal O}_G^{\mathfrak{g}}$, then ${\bf L}(a)(\varphi) = {\bf R}(a)(\varphi)$
for any $a\in{\mathfrak{g}}$,
therefore if $f,g\in{\mathcal O}_G^{\mathfrak{g}}$, then $({\bf L} - {\bf R})(r)(f\otimes g) = 0$,
hence $\{f,g\}=0$.
The inclusion ${\mathcal O}_G^{\mathfrak{g}} \subset {\mathcal O}_G$ is a morphism of Poisson
algebras with a decreasing filtration. By passing to the associated
graded, we obtain:
\begin{lemma} \label{lemma:poisson}
$S({\mathfrak{g}}^*)^{\mathfrak{g}} \subset S({\mathfrak{g}}^*)$ is a Poisson commutative subalgebra.
\end{lemma}
{\em Another proof.} If $\alpha,\beta\in{\mathfrak{g}}^*$, then $[\alpha,\beta] =
\on{ad}^*(R(\beta))(\alpha) - \on{ad}^*(R(\alpha))(\beta)$,
where $R : {\mathfrak{g}}^* \mapsto {\mathfrak{g}}$ is given by $R(\xi) = (\on{id}\otimes\xi)(r)$.
Let $f,g \in S({\mathfrak{g}}^*)^{\mathfrak{g}}$ be of degrees $k$ and $\ell$.
Write $f = \sum_\alpha a_1^\alpha \cdots a_k^\alpha$,
$g = \sum_\beta b_1^\beta \cdots b_\ell^\beta$. Then
$$
\{f,g\} =
\sum_\beta \sum_{j=1}^\ell
b_1^\beta \cdots \check b_j^\beta\cdots b_\ell^\beta
\on{ad}^*(R(b_j^\beta))(f)
- \sum_\alpha \sum_{i=1}^k
a_1^\alpha \cdots \check a_i^\alpha\cdots a_k^\alpha
\on{ad}^*(R(a_i^\alpha))(g).
$$
When $f$ and $g$ are both invariant, this bracket vanishes.
\hfill \qed \medskip
We now prove that $S({\mathfrak{g}}^*)^{\mathfrak{g}} \subset S({\mathfrak{g}}^*)$ is also the
associated graded of an inclusion of noncommutative algebras
with an increasing filtration:
\begin{theorem}
\label{theoprinc}
There exists a morphism of filtered algebras:
$$\theta : S({\mathfrak{g}}^*)^{\mathfrak{g}} \to U({\mathfrak{g}}^*),$$
the associated graded morphism of which is the canonical inclusion
$S({\mathfrak{g}}^*)^{\mathfrak{g}} \subset S({\mathfrak{g}}^*)$.
\end{theorem}
{\em Proof.} Let us denote by ${\bf FSHA}$ the category of
formal series Hopf (FSH) algebras and by ${\bf FilAlg}$ the category
of filtered
algebras. There is a contravariant functor (restricted duality)
${\bf FSHA} \to {\bf FilAlg}$, defined by ${\mathcal O}\mapsto {\mathcal O}^\circ$,
where ${\mathcal O}^\circ = \{\ell\in {\mathcal O}^* | \exists n\geq 0, \ell({\mathfrak{m}}^n) = 0\}
\subset {\mathcal O}^*$; here ${\mathfrak{m}}\subset{\mathcal O}$ is the maximal ideal of ${\mathcal O}$.
The algebra structure of ${\mathcal O}^\circ$ is defined by $(\ell_1 \cdot \ell_2)(f)
= (\ell_1 \otimes \ell_2)(\Delta(f))$; its filtration is defined
by $({\mathcal O}^\circ)_{\leq n} = \{\ell\in{\mathcal O}^* | \ell({\mathfrak{m}}^{n+1}) = 0\}$.
Note that we have a category equivalence ${\bf FSHA} \to {\bf LCA}$,
where ${\bf LCA}$ is the category of Lie coalgebras, taking
${\mathcal O}$ to ${\mathfrak{m}}/{\mathfrak{m}}^2$, equipped with the cobracket induced by
$\Delta - \Delta^{2,1}$. Then the composed functor
${\bf LCA} \to {\bf FSHA} \to {\bf FilAlg}$ is $\c\mapsto U(\c^*)$
(recall that $\c^*$ is a Lie algebra).
$({\mathcal O}_{{\mathfrak{g}}^*},\Delta_0) = (\wh S({\mathfrak{g}}),\Delta_0)$ is a graded FSH algebra.
Its restricted dual is the graded algebra $S({\mathfrak{g}}^*)$. Recall that
${\mathcal O}_{{\mathfrak{g}}^*}$ is also a Poisson algebra. We define the set of
Poisson traces on ${\mathcal O}_{{\mathfrak{g}}^*}$ as the
subspace of all $\ell\in{\mathcal O}_{{\mathfrak{g}}^*}^\circ$, such that $\ell(\{u,v\}) = 0$
for any $u,v\in{\mathcal O}_{{\mathfrak{g}}^*}$. Then $\{$Poisson traces on ${\mathcal O}_{{\mathfrak{g}}^*}\}
\subset {\mathcal O}_{{\mathfrak{g}}^*}^\circ$ identifies with $S({\mathfrak{g}}^*)^{\mathfrak{g}} \subset S({\mathfrak{g}}^*)$;
this is a graded subalgebra of ${\mathcal O}_{{\mathfrak{g}}^*}^\circ$. This defines a graded
algebra structure on $\{$Poisson traces on ${\mathcal O}_{{\mathfrak{g}}^*}\}$.
Consider the FSH algebra $({\mathcal O}_{{\mathfrak{g}}^*},{}^\rho\Delta_0)$. It is
isomorphic (as a filtered vector space) to $({\mathcal O}_{{\mathfrak{g}}^*},\Delta_0)$,
and this isomorphism induces an algebra isomorphism between their associated
graded FSH algebras. It follows that we have an isomorphism
of filtered vector spaces
between the filtered algebra $({\mathcal O}_{{\mathfrak{g}}^*},{}^\rho\Delta_0)^\circ$
and $S({\mathfrak{g}}^*)$, and the associated graded of this morphism is
an algebra isomorphism $\on{gr} (({\mathcal O}_{{\mathfrak{g}}^*},{}^\rho\Delta_0)^\circ) \to
S({\mathfrak{g}}^*)$.
Recall that the vector spaces underlying $({\mathcal O}_{{\mathfrak{g}}^*},\Delta_0)^\circ$ and
$({\mathcal O}_{{\mathfrak{g}}^*},{}^\rho\Delta_0)^\circ$ are the same (i.e., ${\mathcal O}_{{\mathfrak{g}}^*}^\circ$).
We claim that the canonical inclusion $\{$Poisson traces on ${\mathcal O}_{{\mathfrak{g}}^*}\}
\subset ({\mathcal O}_{{\mathfrak{g}}^*},{}^\rho\Delta_0)^\circ$ is a morphism of filtered
algebras.
Indeed, let us denote by $\cdot_\rho$ (resp., $\cdot$) the product of
$({\mathcal O}_{{\mathfrak{g}}^*},{}^\rho\Delta_0)^\circ$ (resp., $({\mathcal O}_{{\mathfrak{g}}^*},\Delta_0)^\circ$).
Let $\ell_1,\ell_2$ be Poisson
traces on ${\mathcal O}_{{\mathfrak{g}}^*}$. Then for any $x\in{\mathcal O}_{{\mathfrak{g}}^*}$, we have
$(\ell_1 \cdot_\rho \ell_2)(x) = (\ell_1 \otimes \ell_2)
(\rho \star \Delta_0(f) \star (-\rho))$. Now Leibniz's rule implies that
$(\ell_1\otimes \ell_2)(\{u,v\})=0$ for any $u,v\in{\mathcal O}_{{\mathfrak{g}}^*}^{\wh\otimes 2}$,
therefore
$(\ell_1 \cdot_\rho \ell_2)(x) = (\ell_1\otimes \ell_2)(\Delta_0(x))
= (\ell_1\cdot \ell_2)(x)$. So $\{$Poisson traces on ${\mathcal O}_{{\mathfrak{g}}^*}\}
\subset ({\mathcal O}_{{\mathfrak{g}}^*},{}^\rho\Delta_0)^\circ$ is an algebra morphism.
Since the filtrations on the vector spaces underlying
$({\mathcal O}_{{\mathfrak{g}}^*},\Delta_0)^\circ$ and $({\mathcal O}_{{\mathfrak{g}}^*},{}^\rho\Delta_0)^\circ$
are the same, and since the filtration on $\{$Poisson traces on ${\mathcal O}_{{\mathfrak{g}}^*}\}$
is induced by that of $({\mathcal O}_{{\mathfrak{g}}^*},\Delta_0)^\circ$, this morphism is filtered,
and its associated graded is the canonical inclusion
$S({\mathfrak{g}}^*)^{\mathfrak{g}} \subset S({\mathfrak{g}}^*)$.
Now the FSH algebra isomorphism ${\mathcal O}_{G^*} \simeq
({\mathcal O}_{{\mathfrak{g}}^*},{}^\rho\Delta_0)$ (Corollary \ref{coroprinc})
induces a filtered algebra isomorphism
$({\mathcal O}_{{\mathfrak{g}}^*},{}^\rho\Delta_0)^\circ \to {\mathcal O}_{G^*}^\circ = U({\mathfrak{g}}^*)$.
The fact that the associated graded of this morphism is the canonical
isomorphism $S({\mathfrak{g}}^*) \to \on{gr}(U({\mathfrak{g}}^*))$ follows from the fact that the
completed graded of the FSH algebras ${\mathcal O}_{G^*}$ and
$({\mathcal O}_{{\mathfrak{g}}^*},{}^\rho\Delta_0)$ are both $({\mathcal O}_{{\mathfrak{g}}^*},\Delta_0)$.
We now compose the filtered algebra morphism
$\{$Poisson traces on ${\mathcal O}_{{\mathfrak{g}}^*}\}
\subset ({\mathcal O}_{{\mathfrak{g}}^*},{}^\rho\Delta_0)^\circ$
with the filtered algebra isomorphism
$({\mathcal O}_{{\mathfrak{g}}^*},{}^\rho\Delta_0)^\circ \to {\mathcal O}_{G^*}^\circ$
and obtain a filtered algebra morphism $S({\mathfrak{g}}^*)^{\mathfrak{g}} \to U({\mathfrak{g}}^*)$,
whose associated graded is the canonical inclusion $S({\mathfrak{g}}^*)^{\mathfrak{g}}
\subset S({\mathfrak{g}}^*)$.
The situation may be summarized as follows:
$$
\begin{matrix}
& & ({\mathcal O}_{{\mathfrak{g}}^*},\Delta_0)^\circ = S({\mathfrak{g}}^*) & & \\
& \scriptstyle{(a)} \nearrow & & & & \\
S({\mathfrak{g}}^*)^{\mathfrak{g}} = \{\hbox{Poisson\ traces\ on\ }{\mathcal O}_{{\mathfrak{g}}^*}\} & &
\scriptstyle{(c)}\uparrow & & & \\
& \scriptstyle{(b)}\searrow & & & \\
& & ({\mathcal O}_{{\mathfrak{g}}^*},{}^\rho\Delta_0)^\circ & \stackrel{(d)}{\to} &
{\mathcal O}_{G^*}^\circ = U({\mathfrak{g}}^*)
\end{matrix}
$$
Here $S({\mathfrak{g}}^*)^{\mathfrak{g}}$ and $S({\mathfrak{g}}^*)$ are graded algebras,
$({\mathcal O}_{{\mathfrak{g}}^*},{}^\rho\Delta_0)^\circ$ and ${\mathcal O}_{G^*}^\circ$
are filtered algebras; $(a)$ is a morphism of graded algebras, $(c)$ is
an isomorphism of filtered vector spaces, $(b)$ and $(d)$ are morphisms
of filtered algebras ($(d)$ is an isomorphism). The associated graded
of $(c)$ is an isomorphism of graded algebras.
\hfill \qed \medskip
\begin{remark} The restricted dual of the isomorphism ${\mathcal O}_{{\mathfrak{g}}^*} \to
{\mathcal O}_{G^*}$ appearing in the above proof is an
isomorphism of filtered vector spaces $\sigma : S({\mathfrak{g}}^*) \to U({\mathfrak{g}}^*)$, whose
associated graded is the canonical isomorphism
$S({\mathfrak{g}}^*) \to \on{gr}(U({\mathfrak{g}}^*))$. These properties are also satisfied by
the symmetrization map $\on{Sym}$, however $\sigma$ depends on $\rho$,
so in general $\on{Sym}$ and $\sigma$ are different.
\end{remark}
\begin{remark} One can check that the morphism $\theta$ is independent
on the choice of $(\rho,\varphi)$ (these choices are described in Remark
\ref{rem:sigma} and in Theorem \ref{lift2}).
\end{remark}
\section{Duality of QUE and QFSH algebras} \label{sect:duality}
In this section, we recall some facts from \cite{Dr:QG} (proofs can be found in
\cite{Gav}). Let us denote by ${\bf QUE}$ the category of quantized universal
enveloping (QUE) algebras and by ${\bf QFSH}$ the category of quantized formal
series Hopf (QFSH) algebras. We denote by ${\bf QUE}_{\on{fd}}$ and
${\bf QFSH}_{\on{fd}}$ the subcategories corresponding to finite dimensional
Lie bialgebras.
We have contravariant functors ${\bf QUE}_{\on{fd}} \to {\bf QFSH}_{\on{fd}}$,
$U\mapsto U^*$ and ${\bf QFSH}_{\on{fd}} \to {\bf QUE}_{\on{fd}}$,
${\mathcal O}\mapsto {\mathcal O}^\circ$. These functors are inverse to each other.
$U^*$ is the full topological dual of $U$, i.e., the space of all
continuous (for the $\hbar$-adic topology) ${\mathbb{K}}[[\hbar]]$-linear maps
$U \to {\mathbb{K}}[[\hbar]]$.
${\mathcal O}^\circ$ the space of continuous ${\mathbb{K}}[[\hbar]]$-linear forms
${\mathcal O}\to {\mathbb{K}}[[\hbar]]$,
where ${\mathcal O}$ is equipped with the ${\mathfrak{m}}$-adic topology (here ${\mathfrak{m}}\subset {\mathcal O}$
is the maximal ideal).
We also have covariant functors ${\bf QUE} \to {\bf QFSH}$, $U\mapsto U'$
and ${\bf QFSH} \to {\bf QUE}$, ${\mathcal O}\mapsto {\mathcal O}^\vee$. There functors are
also inverse to each other. $U'$ is a subalgebra of $U$, while ${\mathcal O}^\vee$
is the $\hbar$-adic completion of $\sum_{k\geq 0} \hbar^{-k} {\mathfrak{m}}^k \subset
{\mathcal O}[1/\hbar]$.
We also have canonical isomorphisms $(U')^\circ \simeq (U^*)^\vee$
and $({\mathcal O}^\vee)^* \simeq ({\mathcal O}^\circ)'$.
If $\a$ is a finite dimensional Lie bialgebra and $U = U_\hbar(\a)$
is a QUE algebra quantizing $\a$, then $U^* = {\mathcal O}_{A,\hbar}$ is a
QFSH algebra quantizing the Poisson-Lie group $A$ (with Lie bialgebra $\a$),
and $U' = {\mathcal O}_{A^*,\hbar}$ is a QFSH algebra quantizing the Poisson-Lie
group $A^*$ (with Lie bialgebra $\a^*$). If now ${\mathcal O} = {\mathcal O}_{A,\hbar}$
is a QFSH algebra quantizing $A$, then ${\mathcal O}^\circ = U_\hbar(\a)$ is a
QUE algebra quantizing $\a$ and ${\mathcal O}^\vee = U_\hbar(\a^*)$ is a QFSH algebra
quantizing $\a^*$.
We now compute these functors explicitly in the case of cocommutative
QUE and commutative QFSH algebras. If $U = U(\a)[[\hbar]]$ with
cocommutative coproduct
(where $\a$ is a Lie algebra), then $U'$ is a completion of
$U(\hbar \a[[\hbar]])$; this is a flat deformation of $\wh S(\a)$
equipped with its linear Lie-Poisson structure. If $G$ is a formal group
with function ring ${\mathcal O}_G$, then ${\mathcal O} := {\mathcal O}_G[[\hbar]]$ is a QFSH algebra,
and ${\mathcal O}^\vee$ is a commutative QUE algebra; it is a quantization of
$(S({\mathfrak{g}}^*)$, commutative product, cocommutative coproduct, co-Poisson
structure induced by the Lie bracket of ${\mathfrak{g}})$.
\section{Relation between twist quantization and its functional version}
\label{sect:rel}
Let us define a twist quantization of
the coboundary Lie bialgebra $({\mathfrak{g}},r,Z)$ as
a pair $(J,\Phi)$, $J\in U({\mathfrak{g}})^{\otimes 2}[[\hbar]]$,
$\Phi \in U({\mathfrak{g}})^{\otimes 3}[[\hbar]]$, such that $\Phi$
is invariant, and $(J,\Phi)$ satisfies the
twisted cocycle relation
\begin{equation}
\label{eq1}
J^{1,2}J^{12,3}=J^{2,3}J^{1,23}\Phi,
\end{equation}
$(\varepsilon\otimes\id)(J)=(\id\otimes\varepsilon)(J)=1$,
$J = 1^{\otimes 2} + O(\hbar)$, $\Phi = 1^{\otimes 3} + O(\hbar)$,
$\on{Alt}((J-1^{\otimes 2})/\hbar) = r + O(\hbar)$,
$\on{Alt}((\Phi-1^{\otimes 3})/\hbar^2)
= Z + O(\hbar)$.
These conditions imply that $\Phi$ satisfies the pentagon relation, as well as
$\varepsilon^{(i)}(\Phi) = 1^{\otimes 2}$, $i=1,2,3$.
(We know that such a twist quantization always exists
when ${\mathfrak{g}}$ is triangular or quasi-triangular.)
Our purpose is to relate twist quantization with its functional version.
The first step is to show that $(J,\Phi)$ can be transformed into an
admissible pair, in a sense which we now precise.
\begin{definition}
\label{admissible}
{\it An element $x$ in a QUE algebra
$U$ is admissible if $x\in 1 + \hbar U$, and
if $\hbar \log x$ is in $U' \subset U$.}
\end{definition}
We will use the isomorphism $U({\mathfrak{g}})^{\otimes k}[[\hbar]] \simeq
U({\mathfrak{g}}^{\oplus k})[[\hbar]]$ to view $U({\mathfrak{g}})^{\otimes k}[[\hbar]]$
as a QUE algebra.
\begin{proposition}
\label{propadmi}
Any twist quantization $(J,\Phi)$ of a coboundary Lie bialgebra $({\mathfrak{g}},r,Z)$
is gauge equivalent to an admissible twist quantization $(J',\Phi')$
(i.e., such that $J'$ and $\Phi'$ are admissible).
\end{proposition}
{\em Proof.} Let us set $U = U({\mathfrak{g}})[[\hbar]]$.
According to Proposition \ref{prop:assoc}, one can find
an invariant $F \in U^{\wh\otimes 2}$, such that
$F \in 1^{\otimes 2} + \hbar U_0^{\wh\otimes 2}$ and
$\Phi' := {}^F\Phi = F^{2,3} F^{1,23} \Phi (F^{1,2}F^{12,3})^{-1}$
is admissible. In particular, $\Phi' \in 1^{\otimes 3}
+ \hbar^2 U_0^{\wh\otimes 3}$.
Then if we set $J_0 := JF$, we have
$J_0^{1,2} J_0^{12,3} = J_0^{2,3} J_0^{1,23} \Phi'$,
and $J_0 \in 1^{\otimes 2} + \hbar U_0^{\wh\otimes 2}$.
For any $u\in 1^{\otimes 3} + \hbar U_0$,
${}^uJ_0 := u^1 u^2 J_0 (u^{12})^{-1}$
is such that $({}^uJ_0,\Phi')$ is a twist quantization of
$({\mathfrak{g}},r,Z)$. It remains to find $u$ such that $J' := {}^u J_0$
is admissible.
We will construct $u$ as a product $\cdots u_2 u_1$, where
$u_n\in 1 + \hbar^n U_0$, in such a way that if
$J_n := {}^{u_n\cdots u_1}J_0$, then
$\hbar\log(J_n) \in U_0^{\prime\wh\otimes 2} + \hbar^{n+2} U_0^{\wh\otimes 2}$.
We have already $\hbar\log(J_0) \in \hbar^2 U_0^{\wh\otimes 2}$.
Expand $J_0 = 1^{\otimes 2} + \hbar j_1 + \cdots$, then $\on{Alt}(j_1) = r$.
Moreover, the coefficient of $\hbar$ in $J_0^{1,2}J_0^{12,3} =
J_0^{2,3}J_0^{1,23}\Phi$ yields $d(j_1) = 0$, where $d : U({\mathfrak{g}})_0^{\otimes 2}
\to U({\mathfrak{g}})_0^{\otimes 3}$ is the co-Hochschild differential.
It follows that for some $a_1\in U({\mathfrak{g}})_0$, we have $j_1 = r+d(a_1)$.
Then if we set $u_1 := \exp(\hbar a_1)$ and $J_1 = {}^{u_1}J_0$,
we get $J_1 \in 1^{\otimes 2} + \hbar r + \hbar^2 U_0^{\wh\otimes 2}$.
Then $\hbar\log(J_1) \in \hbar^2 r + \hbar^3 U_0^{\wh\otimes 2}
\subset U_0^{\prime\wh\otimes 2} + \hbar^3 U_0^{\wh\otimes 3}$.
Assume that for $n\geq 2$, we have constructed $u_1,\ldots,u_{n-1}$ such that
$\alpha_{n-1} := \hbar\log(J_{n-1}) \in U_0^{\prime\wh\otimes 2}
+ \hbar^{n+1} U_0^{\wh\otimes 2}$.
Let us denote by $\bar\alpha$ the image of the class of
$\alpha_{n-1}$ in $U({\mathfrak{g}})_0^{\otimes 2} / (U({\mathfrak{g}})_0^{\otimes 2})_{\leq n+1}$
under the isomorphism of this space with $(U_0^{\prime\wh\otimes 2}
+ \hbar^{n+1} U_0^{\wh\otimes 2}) / (U_0^{\prime\wh\otimes 2}
+ \hbar^{n+2} U_0^{\wh\otimes 2})$ (see Lemma \ref{lemma:quot}).
Let $\alpha\in U({\mathfrak{g}})_0^{\otimes 2}$ be a representative of $\bar\alpha$, then
$\alpha_{n-1} = \alpha' + \hbar^{n+1}\alpha$, where $\alpha'\in
U_0^{\prime\wh\otimes 2} + \hbar^{n+2} U_0^{\wh\otimes 2}$. Let us set
$\varphi' := \hbar\log(\Phi')$, then the twist equation gives
$$
(-\alpha'-\hbar^{n+1}\alpha)^{1,23} \star_\hbar
(-\alpha'-\hbar^{n+1}\alpha)^{2,3} \star_\hbar
(\alpha'+\hbar^{n+1}\alpha)^{1,2} \star_\hbar
(\alpha'+\hbar^{n+1}\alpha)^{12,3} = \varphi',
$$
where $\star_\hbar$ is defined as in Appendix \ref{app:A}.
According to Lemma \ref{lemma:approx}, the image of this equality
in $(U^{\wh\otimes 3} + \hbar^{n+1} U^{\prime\wh\otimes 3}) /
(U^{\wh\otimes 3} + \hbar^{n+2} U^{\prime\wh\otimes 3})
\simeq U({\mathfrak{g}})^{\otimes 3} / (U({\mathfrak{g}})^{\otimes 3})_{\leq n+1}$
is $d(\bar\alpha)$, where $d$ is the co-Hochschild differential on
$U({\mathfrak{g}})_0^{\otimes \cdot} / (U({\mathfrak{g}})_0^{\otimes\cdot})_{\leq n+1}$.
Since $n\geq 2$, the relevant cohomology group vanishes, so
$\bar\alpha = d(\bar\beta)$, where $\bar\beta\in U({\mathfrak{g}})_0
/(U({\mathfrak{g}})_0)_{\leq n+1}$. Let $\beta\in U({\mathfrak{g}})_0$ be a representative
of $\bar\beta$ and set $u_n := \exp(\hbar^n\beta)$, $J_n := {}^{u_n}J_{n-1}$,
$\alpha_n := \hbar\log(J_n)$. Then
$$
\alpha_n = (\hbar^{n+1}\beta)^1 \star_\hbar (\hbar^{n+1}\beta)^2
\star_\hbar \alpha_{n-1} \star_\hbar (-\hbar^{n+1}\beta)^{12}.
$$
According to Lemma \ref{lemma:approx}, the image of $\alpha_n$
in
$$
(U_0^{\wh\otimes 2} + \hbar^{n+1} U_0^{\prime\wh\otimes 2})
/ (U_0^{\wh\otimes 2} + \hbar^{n+2} U_0^{\prime\wh\otimes 2})
\simeq U({\mathfrak{g}})_0^{\otimes 2}/(U({\mathfrak{g}})^{\otimes 2}_0)_{\leq n+1}
$$
is $\bar\alpha - d(\bar\beta)=0$. So $\alpha_n$ belongs to
$U_0^{\wh\otimes 2} + \hbar^{n+2} U_0^{\prime\wh\otimes 2}$, as
required. This proves the induction step.
\hfill \qed \medskip
If now $(J',\Phi')$ is an admissible twist quantization, then
$\rho := \hbar \log(J')_{|\hbar=0}$ and $\varphi :=
\hbar \log(\Phi')_{|\hbar=0}$
are formal functions on ${\mathfrak{m}}_{{\mathfrak{g}}^*}^{\wh\otimes 2}$ and
${\mathfrak{m}}_{{\mathfrak{g}}^*}^{\wh\otimes 3}$, solutions of the functional twist equation.
\section{Quantization of ${\mathcal O}_G^{\mathfrak{g}} \subset {\mathcal O}_G$} \label{sect:O}
Using a (non necessarily admissible) twist quantization, we construct
a formal noncommutative deformation of the inclusion of algebras
of Lemma \ref{lemma:O}:
\begin{proposition}
We have an injective algebra morphism
${\mathcal O}_G^{\mathfrak{g}}[[\hbar]] \hookrightarrow {\mathcal O}_{G,\hbar}$
deforming ${\mathcal O}_G^{\mathfrak{g}}\subset {\mathcal O}_G$, where
${\mathcal O}_{G,\hbar}$ is a quantization of the PFSH algebra
${\mathcal O}_G$ and ${\mathcal O}_G^{\mathfrak{g}}[[\hbar]]$ is the trivial deformation of
the commutative algebra ${\mathcal O}_G^{\mathfrak{g}}$ (it is also commutative).
\end{proposition}
{\em Proof.} Let us first construct the QFSH algebra ${\mathcal O}_{G,\hbar}$.
For $x\in U({\mathfrak{g}})[[\hbar]]$, set
${}^J\Delta_0(x) = J\Delta_0(x) J^{-1}$,
where $\Delta_0$ is the usual cocommutative coproduct.
Then
$U_\hbar({\mathfrak{g}}) = (U({\mathfrak{g}})[[\hbar]],m_0,{}^J\Delta_0)$ is a quantization
of the Lie bialgebra ${\mathfrak{g}}$ (here $m_0$ is the product on $U({\mathfrak{g}})$).
The dual ${\mathcal O}_{G,\hbar} := U_\hbar({\mathfrak{g}})^*$ of this QUE algebra
is a QFSH algebra quantizing the PFSH algebra ${\mathcal O}_G$.
The product in this QFSH algebra is defined by
$(f \star g)(x) = (f \otimes g)(J\Delta_0(x)J^{-1})$
for $f,g\in U_\hbar({\mathfrak{g}})^*$ and $x\in U_\hbar({\mathfrak{g}})$.
On the other hand, the FSH algebra ${\mathcal O}_G$ is equal to $U({\mathfrak{g}})^*$,
and its product is defined by $(fg)(x) = (f\otimes g)(\Delta_0(x))$
for $f,g\in U({\mathfrak{g}})^*$ and $x\in U({\mathfrak{g}})$.
We say that $f\in U({\mathfrak{g}})^*$ is a trace iff $f(xy) = f(yx)$
for any $x,y\in U({\mathfrak{g}})$. Then the inclusion
$\{$traces on $U({\mathfrak{g}})\} \subset U({\mathfrak{g}})^*$
identifies with ${\mathcal O}_G^{\mathfrak{g}} \subset {\mathcal O}_G$.
In the same way, we define $\{$traces on $U({\mathfrak{g}})[[\hbar]]\}$;
this is a subalgebra of $U({\mathfrak{g}})[[\hbar]]^* \simeq {\mathcal O}_G[[\hbar]]$,
which identifies with ${\mathcal O}_G^{\mathfrak{g}}[[\hbar]]$.
The canonical map $\{$traces on $U({\mathfrak{g}})[[\hbar]]\} \to U_\hbar({\mathfrak{g}})^*$
is an algebra morphism. Indeed, if $f_1,f_2$ are traces on $U({\mathfrak{g}})[[\hbar]]$,
then $f_1\otimes f_2$ is a trace on $U({\mathfrak{g}})^{\otimes 2}[[\hbar]]$, so
$(f_1 \star f_2)(x) = (f_1\otimes f_2)(J\Delta_0(x) J^{-1}) =
(f_1\otimes f_2)(\Delta_0(x)) = (f_1 f_2)(x)$ for any $x\in U({\mathfrak{g}})[[\hbar]]$,
so $f_1\star f_2 = f_1 f_2$. So we have obtained an algebra morphism
${\mathcal O}_G^{\mathfrak{g}}[[\hbar]] \to U_\hbar({\mathfrak{g}})^* = {\mathcal O}_{G,\hbar}$. It is clearly a
deformation of the canonical inclusion ${\mathcal O}_G^{\mathfrak{g}}\subset {\mathcal O}_G$.
\hfill \qed \medskip
\section{Quantization of $S({\mathfrak{g}}^*)^{\mathfrak{g}} \hookrightarrow U({\mathfrak{g}}^*)$}
\label{sect:SG}
Assume now that $(J,\Phi)$ is an admissible twist quantization.
We will construct a formal deformation of the inclusion of
algebras of Theorem \ref{theoprinc}.
\begin{theorem}
There is an injective algebra morphism:
$$
\theta_\hbar : S({\mathfrak{g}}^*)^{\mathfrak{g}}[[\hbar]] \hookrightarrow U_\hbar({\mathfrak{g}}^*),
$$
where $U_\hbar({\mathfrak{g}}^*)$ is a quantization of ${\mathfrak{g}}^*$.
Its reduction modulo $\hbar$ coincides with the morphism
$S({\mathfrak{g}}^*)^{\mathfrak{g}} \hookrightarrow U({\mathfrak{g}})$ from Theorem \ref{theoprinc}.
\end{theorem}
{\em Proof.} Recall that $U({\mathfrak{g}})[[\hbar]]'$ is a cocommutative QFSH
algebra; we denote by $m_0$, $\Delta_0$ its product and
coproduct.
Since $(\varepsilon\otimes\id)(J) = (\id\otimes\varepsilon)(J)=1$, we have
$\hbar\log(J) \in {\mathfrak{m}}_0^{\wh\otimes 2}$, where ${\mathfrak{m}}_0\subset
U({\mathfrak{g}})[[\hbar]]'$ is the kernel of the counit. According to \cite{EH1},
Proposition 3.1, this implies that the inner automorphism
$z\mapsto JzJ^{-1}$ of $U({\mathfrak{g}})^{\otimes 2}[[\hbar]]$ restricts to an
automorphism of $U({\mathfrak{g}})^{\otimes 2}[[\hbar]]'$.
We then equip $U({\mathfrak{g}})[[\hbar]]'$ with the coproduct
${}^J\Delta : x\mapsto J \Delta_0(x) J^{-1}$.
Then $(U({\mathfrak{g}})[[\hbar]]',m_0,{}^J\Delta_0)$ is a QFSH
algebra. Its classical limit is the PFSH algebra
$({\mathcal O}_{{\mathfrak{g}}^*},m_0,P,{}^\rho\Delta_0)$. We have seen that
this PSFH algebra is isomorphic to ${\mathcal O}_{G^*}$, hence
$(U({\mathfrak{g}})[[\hbar]]',m_0,{}^J\Delta_0)$ is a quantization of ${\mathcal O}_{G^*}$.
It now follows from Section \ref{sect:duality} that
$(U({\mathfrak{g}})[[\hbar]]',m_0,{}^J\Delta_0)^\circ$ is a
quantization of $U({\mathfrak{g}}^*)$, which we denote by $U_\hbar({\mathfrak{g}}^*)$.
Let us say that $\varphi\in (U({\mathfrak{g}})[[\hbar]]')^\circ$ is a trace if
$\varphi(xy) = \varphi(yx)$ for any $x,y\in U({\mathfrak{g}})[[\hbar]]'$.
Then $\{$traces on $U({\mathfrak{g}})[[\hbar]]'\} \subset (U({\mathfrak{g}})[[\hbar]]')^\circ$
is a subalgebra. Indeed, if $\ell_1,\ell_2$ are traces then
$\ell_1\otimes \ell_2$ is also a trace, so for
$x,y\in U({\mathfrak{g}})[[\hbar]]'$, we have $(\ell_1\ell_2)(xy) =
(\ell_1\otimes \ell_2)(\Delta(x)\Delta(y)) =
(\ell_1\ell_2)(yx)$. This inclusion
identifies with the inclusion $({\mathcal O}_G[[\hbar]]^\vee)^{\mathfrak{g}} \subset
{\mathcal O}_G[[\hbar]]^\vee$. Indeed, the Drinfeld
functors have the property that $(U')^\circ = (U^*)^\vee$ for any QUE algebra
$U$.
Now we show that the map
$\{$traces on $U({\mathfrak{g}})[[\hbar]]'\} \subset (U({\mathfrak{g}})[[\hbar]]',{}^J\Delta_0)^\circ$
is also an algebra morphism. Indeed, let $\cdot_J$ be the product of the latter
algebra. If $\ell_1,\ell_2$ are traces
and $x,y\in U({\mathfrak{g}})[[\hbar]]$, then
$(\ell_1 \cdot_J \ell_2)(x) = (\ell_1\otimes \ell_2)(J\Delta_0(x)J^{-1})
= (\ell_1\otimes \ell_2)(\Delta_0(x)) = (\ell_1\ell_2)(x)$, so
$\ell_1 \cdot_J \ell_2 = \ell_1 \ell_2$. So we have constructed an
algebra morphism $({\mathcal O}_G[[\hbar]]^\vee)^{\mathfrak{g}} \to U_\hbar({\mathfrak{g}})$. It is clearly a
deformation of the morphism constructed in Theorem \ref{theoprinc}.
Recall that ${\mathcal O}_G[[\hbar]]^\vee$ is the $\hbar$-adic completion of
$\sum_{k\geq 0} \hbar^{-k} {\mathfrak{m}}_G^k \subset
{\mathcal O}_G((\hbar))$.\footnote{${\mathcal O}_G[[\hbar]]^\vee$ may be also be viewed
as the formal Rees algebra associated to the decreasing filtration
${\mathcal O}_G \supset {\mathfrak{m}}_G \supset {\mathfrak{m}}_G^2 \cdots$.}
Then ${\mathcal O}_G[[\hbar]]^\vee$ is a topologically
free ${\mathbb{K}}[[\hbar]]$-commutative algebra; its specialization at $\hbar=0$ is
${\mathcal O}_G[[\hbar]]^\vee / \hbar {\mathcal O}_G[[\hbar]]^\vee \simeq S({\mathfrak{g}}^*)$.
The action of ${\mathfrak{g}}$ on ${\mathcal O}_G$ induces an action of ${\mathfrak{g}}$ on
${\mathcal O}_G[[\hbar]]^\vee$.
Then $({\mathcal O}_G[[\hbar]]^\vee)^{\mathfrak{g}}$ is the $\hbar$-adic completion of
$\sum_{k\geq 0} \hbar^{-k} ({\mathfrak{m}}_G^k)^{\mathfrak{g}}$.
We have an inclusion of topologically free ${\mathbb{K}}[[\hbar]]$-algebras
$({\mathcal O}_G[[\hbar]]^\vee)^{\mathfrak{g}} \subset {\mathcal O}_G[[\hbar]]^\vee$.
Now the dual of the symmetrization map induces an algebra isomorphism
$\wh S({\mathfrak{g}}^*) = {\mathcal O}_{\mathfrak{g}}\simeq {\mathcal O}_G$ (dual to the exponential map ${\mathfrak{g}}\to G$).
This isomorphism induces a ${\mathfrak{g}}$-equivariant isomorphism of
${\mathcal O}_G[[\hbar]]^\vee$ with the $\hbar$-adic completion of
$\sum_{k\geq 0} \hbar^{-k} {\mathfrak{m}}_{\mathfrak{g}}^k \subset {\mathcal O}_{\mathfrak{g}}((\hbar))$.
So we have an algebra isomorphism ${\mathcal O}_G[[\hbar]]^\vee \simeq S({\mathfrak{g}}^*)[[\hbar]]$.
It restricts to an isomorphism $({\mathcal O}_G[[\hbar]]^\vee)^{\mathfrak{g}} \simeq S({\mathfrak{g}}^*)^{\mathfrak{g}}[[\hbar]]$.
Composing its inverse with the morphism $({\mathcal O}_G[[\hbar]]^\vee)^{\mathfrak{g}} \to
U_\hbar({\mathfrak{g}}^*)$, we get the announced morphism
$S({\mathfrak{g}}^*)^{\mathfrak{g}}[[\hbar]] \to U_\hbar({\mathfrak{g}}^*)$.
\hfill \qed
\medskip
\section{The quasitriangular case}
\label{sect:Sem}
A quasitriangular Lie bialgebra (QTLBA) is a pair $({\mathfrak{g}},r')$, where
${\mathfrak{g}}$ is a Lie algebra and $r'\in {\mathfrak{g}}^{\otimes 2}$ is such that
$\on{CYB}(r')=0$ and $t:= r'+r^{\prime 2,1} \in S^2({\mathfrak{g}})^{\mathfrak{g}}$. Any QTLBA
gives rise to a coboundary Lie bialgebra $({\mathfrak{g}},r,Z)$, where
$r=(r'-r^{\prime 2,1})/2$ and $Z = [t^{1,2},t^{2,3}]/4$.
We call a QTLBA {\it nondegenerate} if ${\mathfrak{g}}$ is finite dimensional and
$t$ is nondegenerate.
Let $D : {\mathfrak{g}}^* \to {\mathfrak{g}}^*$ be the composition of
the Lie cobracket $\delta : {\mathfrak{g}}^* \to \wedge^2({\mathfrak{g}}^*)$ with the Lie bracket
of ${\mathfrak{g}}^*$. It is a derivation and a coderivation, and it induces a
derivation of $U({\mathfrak{g}}^*)$, which we also denote by $D$ (or
sometimes $D_{{\mathfrak{g}}^*}$).
\begin{proposition}
For any scalar $s$, $C_s := \on{Ker}(\delta - s (D\otimes \id)\circ \Delta_0)$
is a commutative subalgebra of $U({\mathfrak{g}}^*)$.
\end{proposition}
{\em Proof.} The condition $\ell\in C_s$ means that
for any $u,v\in {\mathcal O}_{G^*}$, we have $\ell(\{u,v\} - s D^*(u)v) = 0$
(here $D^*$ is the derivation of ${\mathcal O}_{G^*}$ dual to the coderivation
$D$).
Let $\ell_1,\ell_2$ belong to $C_s$. Then for any $u,v\in {\mathcal O}_{G^*}$,
\begin{align*}
(\ell_1\ell_2)(\{u,v\} - s D^*(u)v) &
= (\ell_1\otimes \ell_2)(\{\Delta(u),\Delta(v)\}
- s \Delta(D^*(u)) \Delta(v))
\\ & = (\ell_1\otimes \ell_2)(\{u^{(1)},v^{(1)}\} \otimes u^{(2)}v^{(2)}
+ u^{(1)}v^{(1)} \otimes \{u^{(2)},v^{(2)}\}
\\ & - s D^*(u^{(1)})v^{(1)} \otimes u^{(2)}v^{(2)} - u^{(1)}v^{(1)}
\otimes sD^*(u^{(2)})v^{(2)}) = 0,
\end{align*} hence $\ell_1\ell_2\in C_s$.
Here $\Delta$ is the coproduct of ${\mathcal O}_{G^*}$.
Moreover, we constructed in \cite{EGH} an element $\varrho\in
{\mathfrak{m}}_{G^*}^{\wh\otimes 2}$, such that $\Delta'(u) = \varrho \star \Delta(u) \star
(-\varrho)$ for any $u\in{\mathcal O}_{G^*}$; if $(U_\hbar({\mathfrak{g}}),{\mathcal R})$ is
any quantization of $({\mathfrak{g}},r')$, then $\hbar\log({\mathcal R}) \in
{\mathfrak{m}}_\hbar^{\wh\otimes 2}$, where ${\mathfrak{m}}_\hbar \subset U_\hbar({\mathfrak{g}})'$
is the augmentation ideal, and the reduction of $\hbar\log({\mathcal R})$
mod $\hbar$ equals $\varrho$. Then it follows from $(S^2\otimes S^2)
({\mathcal R}) = {\mathcal R}$ that $(S_{\mathcal O}^2\otimes S_{\mathcal O}^2)(\hbar\log{\mathcal R}) = \hbar\log{\mathcal R}$,
where $S$ is the antipode of $U_\hbar({\mathfrak{g}})$ and
$S_{\mathcal O} = S_{|U_\hbar({\mathfrak{g}})'}$ is the antipode of
$U_\hbar({\mathfrak{g}})'\subset U_\hbar({\mathfrak{g}})$;
since the specialization for $\hbar=0$ of $\hbar^{-1}(S_{\mathcal O}^2 -\id)$
is $D^*$, we get
$(D^* \otimes \id + \id \otimes D^*)(\varrho) = 0$.
Then if $\ell_1,\ell_2\in C_s$, then
$(\ell_2\ell_1)(u) = (\ell_1\otimes \ell_2)(\Delta'(u))
= (\ell_1\otimes \ell_2)(\varrho \star \Delta(u) \star (-\varrho))
= (\ell_1\otimes \ell_2)(\Delta(u)) + \sum_{n\geq 1} (1/n!)
(\ell_1\otimes \ell_2)(\{\varrho,\{\varrho,\ldots,\{\varrho,\Delta(u)\}\})$.
Now if $f\in {\mathcal O}_{G^*}^{\wh\otimes 2}$, then $(\ell_1\otimes \ell_2)
(\{\varrho,f\}) = s(\ell_1\otimes \ell_2)((D^*\otimes \id +
\id \otimes D^*)(\varrho)f) = 0$.
It follows that $\ell_2\ell_1 = \ell_1 \ell_2$.
\hfill \qed \medskip
\begin{remark}
If $A$ is a quasitriangular Hopf algebra with antipode $S$,
set $C_{s,A} := \{\ell\in A^* | \forall a,b\in A,
\ell(ab) = \ell(bS^{-2s}(a))\}$ for any $s\in{\mathbb{Z}}$. Then it follows from
\cite{Dr:coco} that $C_{s,A}$ is a commutative algebra, and that we have
isomorphisms $C_s \simeq C_{s+2}$ for any $s\in{\mathbb{Z}}$. The isomorphism
takes $\ell\in C_s$ to $\overline\ell\in C_{s+2}$ defined by
$\overline\ell(x) = \ell(xu^{-1}S(u))$, where $u = m\circ
(\id\otimes S)(R)$ ($m,R$ are the product and $R$-matrix of $A$).
The definition of
$C_{s,A}$ can be generalized to $s\in{\mathbb{K}}$ when $A = (U_\hbar({\mathfrak{g}}),{\mathcal R})$
is a quasitriangular QUE Hopf algebra. Define $U_\hbar({\mathfrak{g}}^*)$
as $(U_\hbar({\mathfrak{g}})')^\circ = (U_\hbar({\mathfrak{g}})^*)^\vee \supset U_\hbar({\mathfrak{g}})^*$.
Then $C_{s,\hbar} := \{\ell\in (U_\hbar({\mathfrak{g}})')^\circ | \forall
a,b\in U_\hbar({\mathfrak{g}})', \ell(ab) = \ell(b (S^2)^{-s}(a)) \}$
is a commutative subalgebra of $U_\hbar({\mathfrak{g}}^*)$, and its reduction
modulo $\hbar$ is contained in $C_s$. In this case, $u^{-1}S(u)$ does
not necessarily belong to $U_\hbar({\mathfrak{g}})'$, therefore $C_{s,\hbar}$
and $C_{s+2,\hbar}$ are not necessarily isomorphic.
\end{remark}
\begin{remark}
If $({\mathfrak{g}},r,Z)$ is a coboundary Lie bialgebra, then $r$ is $D$-invariant iff
$(\mu\otimes \id)(Z)$ is symmetric (where $\mu$ is the Lie bracket of ${\mathfrak{g}}$).
Otherwise, if we set $\varrho := \rho^{2,1} \star (-\rho)$, then
$(D^*\otimes \id + \id \otimes D^*)(\varrho) \neq 0$, so unless $s=0$,
one cannot prove that $C_s$ is commutative.
\end{remark}
For each nondegenerate QTLBA $({\mathfrak{g}},r')$, Semenov-Tian-Shansky defined
an algebra morphism $\Theta : Z(U({\mathfrak{g}})) \to U({\mathfrak{g}}^*)$, where $Z(A)$
denotes the center of an algebra $A$ (\cite{STS1}).
Let us recall the construction of $\Theta$.
There are unique Lie algebra morphisms $L,R : {\mathfrak{g}}^* \to {\mathfrak{g}}$,
defined by $L(\ell) = (\ell\otimes \id)(r')$, $R(\ell) =
-(\id\otimes \ell)(r')$
for any $\ell\in{\mathfrak{g}}^*$. We denote by $\alpha : U({\mathfrak{g}}^*) \to U({\mathfrak{g}})$
the composed map $U({\mathfrak{g}}^*) \stackrel{\Delta_0}{\to} U({\mathfrak{g}}^*)^{\otimes 2}
\stackrel{L \otimes (S_0\circ R)}{\to}
U({\mathfrak{g}})^{\otimes 2} \stackrel{m_0}{\to} U({\mathfrak{g}})$. Here $m_0,\Delta_0$
are the standard product and coproduct maps, we still denote by $L,R$
the algebra morphisms induced by $L,R$, and $S_0$ denotes the antipode of
$U({\mathfrak{g}})$. The associated graded of the map $\alpha$ is the isomorphism
$S({\mathfrak{g}}^*) \to S({\mathfrak{g}})$ induced by $t$, hence $\alpha$ is an isomorphism.
Then $\Theta : Z(U({\mathfrak{g}})) \to U({\mathfrak{g}}^*)$ is defined as the restriction of
$\alpha^{-1}$ to $Z(U({\mathfrak{g}}))$; one can prove that it is an algebra morphism.
We will show, together with Proposition \ref{8:10}:
\begin{proposition} \label{prop:sem} \label{8:4}
$\on{Im}(\Theta) = C_1 \subset U({\mathfrak{g}}^*)$. The associated graded of
$C_1$ (for the degree filtration of $U({\mathfrak{g}}^*)$) is $S({\mathfrak{g}}^*)^{\mathfrak{g}}$.
\end{proposition}
\begin{remark} Let $\theta$ be as in Theorem \ref{theoprinc}.
The image of $\theta : S({\mathfrak{g}}^*)^{\mathfrak{g}} \to U({\mathfrak{g}}^*)$
is $\{$Poisson traces on ${\mathcal O}_{G^*}\}$, i.e.,
this is $\on{Ker}(\delta)$, where $\delta :
U({\mathfrak{g}}^*) \to \wedge^2 U({\mathfrak{g}}^*)$ is the co-Poisson map of $U({\mathfrak{g}}^*)$.
So the images of $\Theta$ and $\theta$ do not coincide.
\hfill \qed \medskip
\end{remark}
Let us now construct a deformation $\Theta_\hbar$ of
$\Theta$. The following lemma is proved in \cite{Dr:coco}.
\begin{lemma} \label{8:9}
Let $(A,\Delta,R)$ be a quasitriangular Hopf algebra
with antipode $S$.
Define a linear map $\alpha_A : A^* \to A$ by $\alpha_A(\ell) =
(\ell\otimes \id)(R^{21}R)$. Then $\alpha_A$ induces an
algebra morphism $C_{1,A} \to Z(A)$.
\end{lemma}
\begin{lemma} \label{lemma:p7}
Assume moreover that $A$ is finite dimensional and $R^{2,1}R$
is nondegenerate. Then the map $C_1 \to Z(A)$ is a linear
isomorphism. Its inverse induces an algebra morphism $\Theta_A :
Z(A) \to A^*$.
\end{lemma}
{\em Proof.} We have to check that if $\ell\in A^*$ is such that
$\alpha_A(\ell) \in Z(A)$, then $\ell$ is a trace. The condition
$\alpha_A(\ell) \in Z(A)$ means that
for any $a\in A$, we have $(\ell \otimes \id)([R^{2,1}R,1\otimes a])=0$.
It follows that for any $a\in A$, we have
$S^{-1}(a^{(4)})
(\ell\otimes \id)([R^{2,1}R, a^{(2)}S^{-1}(a^{(1)}) \otimes a^{(3)}]) =0$.
Since $R^{2,1}R$ commutes with the image of $\Delta_A$,
$(\ell\otimes \id)((a^{(2)} \otimes S^{-1}(a^{(4)}) a^{(3)})
[R^{2,1}R, S^{-1}(a^{(1)}) \otimes 1])
=0$. Therefore
$(\ell\otimes \id)((a^{(2)} \otimes 1) R^{2,1}R (S^{-1}(a^{(1)}) \otimes \id))
= \varepsilon(a) (\ell\otimes \id)(R^{2,1} R)$.
Since $R^{2,1}R$ is nondegenerate, this means that for any $b\in A$,
we have $\ell(a^{(2)} b S^{-1}(a^{(1)})) = \varepsilon(a)\ell(b)$. Replacing
$a\otimes b$ by $a^{(1)} \otimes S(a^{(2)}) b$, we get
$\ell(bS^{-1}(a)) = \ell(S(a)b)$, so that $\ell\in C_1$.
\hfill \qed \medskip
The QUE algebra version of these lemmas is 1), 2) of the following
proposition.
Let $({\mathfrak{g}},r')$ be a QTLBA and let $(U_\hbar({\mathfrak{g}}),\Delta,{\mathcal R})$ be a
quantization of $({\mathfrak{g}},r')$.
\begin{proposition} \label{8:10}
1) The linear map $U_\hbar({\mathfrak{g}})^* \to U_\hbar({\mathfrak{g}})$, $\ell\mapsto
(\ell\otimes \id)({\mathcal R}\cR^{2,1})$ extends to a map
$\alpha_\hbar : U_\hbar({\mathfrak{g}}^*) \to U_\hbar({\mathfrak{g}})$.
2) If $({\mathfrak{g}},r')$ is nondegenerate, then $\alpha_\hbar$ is a linear
isomorphism, and it restricts to an algebra isomorphism $C_{1,\hbar} \to
Z(U_\hbar({\mathfrak{g}}))$.
3) Proposition \ref{8:4} is true.
\end{proposition}
{\em Proof.} Let us prove 1).
Define $L_\hbar, R'_\hbar : U_\hbar({\mathfrak{g}})^* \to U_\hbar({\mathfrak{g}})$
by $L_\hbar(\xi) = (\xi\otimes \id)({\mathcal R})$, $R'_\hbar(\xi) =
(\id\otimes \xi)({\mathcal R})$. According to \cite{EH1}, $\hbar\log({\mathcal R})
\subset (U_\hbar({\mathfrak{g}})'_0)^{\wh\otimes 2} \subset U_\hbar({\mathfrak{g}})'_0
\wh\otimes \hbar U_\hbar({\mathfrak{g}})_0$, so that $\log({\mathcal R}) \in
U_\hbar({\mathfrak{g}})'_0 \wh\otimes U_\hbar({\mathfrak{g}})_0$. According to \cite{EGH}, appendix,
the image of $\log({\mathcal R})$
in $({\mathfrak{m}}_{G^*}/{\mathfrak{m}}_{G^*}^2) \wh\otimes U({\mathfrak{g}})_0$ (by reduction mod $\hbar$
followed by projection) is $r'$. It follows that ${\mathcal R}\in U_\hbar({\mathfrak{g}})'
\wh\otimes U_\hbar({\mathfrak{g}})$, therefore $L_\hbar$ extends to a map
$U_\hbar({\mathfrak{g}}^*) \to U_\hbar({\mathfrak{g}})$; this map is necessarily a QUE algebra
morphism. The quasitriangularity identities imply
that the image of ${\mathcal R}$ in ${\mathcal O}_{G^*} \wh\otimes
U({\mathfrak{g}})$ has the form $\on{exp}(\rho)$, where $\rho\in {\mathfrak{m}}_{G^*}\otimes {\mathfrak{g}}$
is a lift of $r$. It follows that the reduction mod $\hbar$ of $L_\hbar$
is the morphism induced by ${\mathfrak{g}}^*\to{\mathfrak{g}}$, $\ell \mapsto (\ell\otimes\id)(r)$.
In the same way, $R'_\hbar$ extends to a (anti)morphism $U_\hbar({\mathfrak{g}}^*) \to
U_\hbar({\mathfrak{g}})$.
Define $\alpha_\hbar : U_\hbar({\mathfrak{g}}^*) \to U_\hbar({\mathfrak{g}})$, by
$x\mapsto m \circ (L_\hbar\otimes R'_\hbar) \circ \Delta$.
Then $\alpha_\hbar$ extends $\ell\mapsto (\ell\otimes \id)({\mathcal R}\cR^{2,1})$.
Let us prove 2). The reduction mod $\hbar$ of $\alpha_\hbar$ is $\alpha$,
which is a linear isomorphism; hence $\alpha_\hbar$ is a linear
isomorphism. The second part is proved as Lemma \ref{8:9}.
Let us prove Proposition \ref{8:4}. Assume that $U_\hbar({\mathfrak{g}})$ is as in
\cite{EK}, hence $U_\hbar({\mathfrak{g}}) \simeq U({\mathfrak{g}})[[\hbar]]$ as algebras. Then
$Z(U_\hbar({\mathfrak{g}})) \simeq Z(U({\mathfrak{g}}))[[\hbar]]$. 2) implies that $\alpha$ induces an
isomorphism $(\on{mod\ }\hbar)(C_{1,\hbar}) \to Z(U({\mathfrak{g}}))$; here $(\text{mod\
}\hbar)$ is the reduction modulo $\hbar$. On the other hand,
$(\text{mod\ }\hbar)(C_{1,\hbar})\subset C_1$, therefore $\Theta(Z(U({\mathfrak{g}})))
\subset C_1$.
The map $\delta - (D\otimes \id)\circ \Delta_0 : U({\mathfrak{g}}^*) \to
U({\mathfrak{g}}^*)^{\otimes 2}$ is filtered, and its associated graded is the dual
$\delta : S({\mathfrak{g}}^*) \to \wedge^2(S({\mathfrak{g}}^*))$ of the Poisson bracket of
$S({\mathfrak{g}})$. We have a surjective morphism $S({\mathfrak{g}})_{\mathfrak{g}} = S({\mathfrak{g}}) / \{{\mathfrak{g}},S({\mathfrak{g}})\}
\twoheadrightarrow S({\mathfrak{g}})/\{S({\mathfrak{g}}),S({\mathfrak{g}})\}$ to the cokernel of this Poisson
bracket, hence $\on{Ker}(\delta) \hookrightarrow (S({\mathfrak{g}})_{\mathfrak{g}})^* = S({\mathfrak{g}}^*)^{\mathfrak{g}}$. We
have $\on{gr}(C_1) \subset \on{Ker}(\delta)$, hence $\on{gr}(C_1) \subset
S({\mathfrak{g}}^*)^{\mathfrak{g}}$. Now since $\Theta$ is filtered and its associated graded takes
$\on{gr}(Z(U({\mathfrak{g}}))) \simeq S({\mathfrak{g}})^{\mathfrak{g}}$ to $S({\mathfrak{g}}^*)^{\mathfrak{g}}$, we get $\on{gr}(C_1)
= S({\mathfrak{g}}^*)^{\mathfrak{g}}$ and $\Theta(Z(U({\mathfrak{g}}))) = C_1$.
\hfill \qed \medskip
We denote by $\Theta_\hbar :
Z(U_\hbar({\mathfrak{g}})) \to U_\hbar({\mathfrak{g}}^*)$ the algebra morphism
inverse to $\alpha_\hbar$. $\Theta_\hbar$
is the QUE algebra version of $\Theta_A$ defined above.
The image of $\Theta_\hbar$ is $C_{1,\hbar}$.
When the quantization is an in \cite{EK}, $U_\hbar({\mathfrak{g}}) \simeq
U({\mathfrak{g}})[[\hbar]]$, so this image is not the same as that of $\theta_\hbar$,
which is $\{$traces on $U_\hbar({\mathfrak{g}})'\} = C_{0,\hbar}$.
Therefore in this case,
the images of $\theta_\hbar$ and $\Theta_\hbar$ do not coincide.
\section{On the canonical derivation of ${\mathcal O}_{G^*}$} \label{sect:D}
Let $(\a,\mu_\a,\delta_\a)$ be a finite dimensional Lie bialgebra.
Then ${\mathcal O}_A$ is a Poisson-Lie group, dual to $U(\a)$.
Set $D_\a := \mu_\a \circ \delta_\a$, then $D_\a$ is a
derivation of $U(\a)$, such that if $U_\hbar(\a)$
is any quantization of $U(\a)$ with antipode $S$, then
$D_\a = \hbar^{-1}(S^2-\id)_{|\hbar=0}$ (see \cite{Dr:coco}).
It follows that the dual derivation $D_\a^*$ of ${\mathcal O}_A$ has
the same property.
When $\a = ({\mathfrak{g}},r')$ is a quasitriangular Lie bialgebra,
$D_\a$ is inner, given by $D_\a(x) = -[\mu(r'),x]$ for any $x\in U({\mathfrak{g}})$;
here $\mu$ is the Lie bracket of ${\mathfrak{g}}$ (see \cite{Dr:coco}).
\begin{proposition} \label{prop:inner}
If ${\mathfrak{g}}$ is a nondegenerate quasitriangular Lie bialgebra, then
the derivation $D_{{\mathfrak{g}}^*}^*$ of ${\mathcal O}_{G^*}$ is inner, i.e.,
there exists a function $h\in {\mathcal O}_{G^*}$ such that
$D_{{\mathfrak{g}}^*}^*(f) = \{h,f\}$ for any $f\in {\mathcal O}_{G^*}$.
\end{proposition}
{\em Proof.} We assume that ${\mathfrak{g}}$ is the double $\a_+\oplus \a_-$
of a Lie bialgebra $\a_+$ (here $\a_- = \a_+^*$);
the general case is similar. Then ${\mathfrak{g}}^*$ is (as a Lie algebra)
the direct sum $\a_+\oplus \a_-$. Let $A_\pm$ be the formal groups
corresponding to $\a_\pm$. The morphism $\alpha : U({\mathfrak{g}}^*) \to U({\mathfrak{g}})$
is now $U(\a_+) \otimes U(\a_-) \to U({\mathfrak{g}})$, $x_+ \otimes x_- \mapsto
x_- S(x_+)$. The dual morphism $\alpha^* : {\mathcal O}_{G} \to {\mathcal O}_{G^*}$
takes $F\in {\mathcal O}_G$ to $f\in {\mathcal O}_{G^*}$ given by $f(g_+,g_-) :=
F(g_- g_+^{-1})$.
\begin{lemma}
Let $D^*_{\mathfrak{g}}$, $D^*_{{\mathfrak{g}}^*}$ be the canonical derivations of
${\mathcal O}_G$ and ${\mathcal O}_{G^*}$. Then $\alpha^* \circ D^*_{{\mathfrak{g}}} =
D^*_{{\mathfrak{g}}^*} \circ \alpha^*$. Moreover, $D^*_{{\mathfrak{g}}} =
{\bold L}_{\mu(r')} - {\bold R}_{\mu(r')}$, where $\mu$ is the Lie bracket of
${\mathfrak{g}}$ and ${\bold L}_a f(g) = (d/d\varepsilon)_{|\varepsilon=0} F(e^{\varepsilon a}g)$,
${\bold R}_a f(g) = (d/d\varepsilon)_{|\varepsilon=0} F(ge^{\varepsilon a})$.
\end{lemma}
{\em Proof of Lemma.} $D_{{\mathfrak{g}}^*}$ is a coderivation, so $\Delta_0 :
U({\mathfrak{g}}^*) \to U({\mathfrak{g}}^*)^{\otimes 2}$ intertwines $D_{{\mathfrak{g}}^*}$ and
$D_{{\mathfrak{g}}^*} \otimes \id + \id\otimes D_{{\mathfrak{g}}^*}$; $L$ and $R$ are
Lie bialgebra morphisms, so they intertwine $D_{{\mathfrak{g}}^*}$ and $D_{{\mathfrak{g}}}$;
$S$ commutes with $D_{{\mathfrak{g}}}$; and $D_{\mathfrak{g}}$ is a derivation, so $m_0$ intertwines
$D_{{\mathfrak{g}}} \otimes \id + \id \otimes D_{\mathfrak{g}}$ with $D_{\mathfrak{g}}$.
Hence $\alpha \circ D_{{\mathfrak{g}}^*} = D_{\mathfrak{g}} \circ \alpha$. The first part follows.
According to \cite{Dr:coco}, $D_{\mathfrak{g}}(x) = -[\mu(r'),x]$, which implies the
second part. \hfill \qed \medskip
In \cite{STS2}, the image of the Poisson bracket on $G^*$ under the
formal isomorphism $\alpha : G^*\to G$ dual to $\alpha^*$ was
computed. Let $f,h\in {\mathcal O}_{G^*}$ and
$F = (\alpha^*)^{-1}(f)$, $H = (\alpha^*)^{-1}(h)$, then
\begin{align} \label{PB:sem}
(\alpha^*)^{-1}(\{f,h\})(g) & =
\langle (d_{{\bold R}} - d_{\bold L}) F(g) \otimes d_{\bold L} H(g), r' \rangle
+ \langle (d_{{\bold R}} - d_{\bold L}) F(g) \otimes d_{\bold R} H(g), (r')^{2,1} \rangle
\\ & \nonumber
= \langle (d_{\bold L} - d_{\bold R}) F(g), L(d_{\bold R} H(g)) - R(d_{\bold L} H(g))
\rangle
\end{align}
where $g\in G$, $d_{\bold L} F(g), d_{\bold R} F(g) \in {\mathfrak{g}}^*$ are the left and right
differentials defined by
$\langle d_{\bold L} F(g) , a\rangle = ({\bold L}_a F)(g)$,
$\langle d_{\bold R} F(g) , a\rangle = ({\bold R}_a F)(g)$
for any $a\in {\mathfrak{g}}$.
\begin{lemma} \label{9:3}
There exists a function $H(g)\in {\mathcal O}_G$ such that
$L(d_{{\bold R}} H(g)) - R(d_{\bold L} H(g)) = \mu(r')$.
\end{lemma}
{\em Proof of lemma.} We prove this when ${\mathfrak{g}}$ is the double
$\a_+ \oplus \a_-$ of a Lie bialgebra $\a_+$. Then set $a = (a_+,a_-)$
where $a_\pm\in \a_\pm$. We have ${\mathfrak{g}}^* = \a_+ \oplus \a_-$, and we
should solve: $d_{{\bold R}} H_a(g) = \mu(r')_- + u_+(g)$,
$d_{{\bold L}} H_a(g) = \mu(r')_+ + u_-(g)$,
where $u_\pm(g)$ are functions $G \to \a_\pm$.
Now $d_{\bold L} H(g) = \operatorname{Ad}(g) (d_{\bold R} H(g))$, hence
$\mu(r')_+ + u_-(g) = \operatorname{Ad}(g)(\mu(r')_- + u_+(g))$.
Let us decompose
$g = g_- g_+^{-1}$, where $g_\pm \in A_\pm = \exp(\a_\pm)$, we get
$\operatorname{Ad}(g_+^{-1})(u_+(g)) - \operatorname{Ad}(g_-^{-1})(u_-(g)) = \operatorname{Ad}(g_-^{-1})( \mu(r')_+)
- \operatorname{Ad}(g_+^{-1})(\mu(r')_-)$.
Therefore
$$
u_+(g) = \operatorname{Ad}(g_+) \big( \operatorname{Ad}(g_-^{-1})(\mu(r')_+)
- \operatorname{Ad}(g_+^{-1})(\mu(r')_-)\big)_+
$$
and the condition is
$$
d_{\bold R} H(g) = \operatorname{Ad}(g_+) \big( \operatorname{Ad}(g_+^{-1})(\mu(r')_-)\big)_-
+ \operatorname{Ad}(g_+) \big( \operatorname{Ad}(g_-^{-1})(\mu(r')_+) \big)_+ ,
$$
i.e.,
\begin{equation} \label{flatness}
{\bold R}_\alpha H_a(g) =
\langle \mu(r')_-, \operatorname{Ad}(g_+)\big((\operatorname{Ad}(g_+^{-1})(\alpha))_+\big) \rangle
+ \langle \mu(r')_+, \operatorname{Ad}(g_-) \big( (\operatorname{Ad}(g_+^{-1})(\alpha))_- \big) \rangle
\end{equation}
for any $\alpha\in {\mathfrak{g}}$.
Let us denote by $A_\alpha(g)$ the r.h.s. of (\ref{flatness}).
Let us compute ${\bold R}_\alpha A_\beta - {\bold R}_\beta A_\alpha$,
for $\alpha,\beta\in{\mathfrak{g}}$. Recall that $g = g_- g_+^{-1}$, then
we have ${\bold R}_\alpha(g) = g \alpha$, so ${\bold R}_\alpha(g_\pm^{-1}) =
\pm (\operatorname{Ad}(g_\pm^{-1})(\alpha))_\pm g_\pm^{-1}$. After computations, we find:
$$
{\bold R}_\alpha A_\beta - {\bold R}_\beta A_\alpha = A_{[\beta,\alpha]}
+ B_{\alpha,\beta},
$$
where
\begin{align*}
& B_{\alpha,\beta}(g) = - \langle [(\operatorname{Ad}^*(g_+^{-1})(\beta))_+,
(\operatorname{Ad}^*(g_+^{-1})(\alpha))_+],
(\operatorname{Ad}(g_+^{-1})(\mu(r')_-))_-\rangle
\\ & + \langle [(\operatorname{Ad}^*(g_+^{-1})(\beta))_-,(\operatorname{Ad}^*(g_+^{-1})(\alpha))_-],
(\operatorname{Ad}(g_-^{-1})(\mu(r')_+))_+\rangle .
\end{align*}
Now for $u,v\in\a_+$, we have
\begin{align*}
& \langle [u,v], (\operatorname{Ad}(g_+^{-1})(\mu(r')_-))_- \rangle
= \langle [u,v], \operatorname{Ad}(g_+^{-1})(\mu(r')_-) \rangle
\\ &
= \langle [\operatorname{Ad}^*(g_+)(u), \operatorname{Ad}^*(g_+)(v)], \mu(r')_- \rangle
= \langle [\operatorname{Ad}^*(g_+)(u), \operatorname{Ad}^*(g_+)(v)], \mu(r') \rangle
\\ &
= \langle \operatorname{Ad}^*(g_+)(u) \otimes \operatorname{Ad}^*(g_+)(v),
\delta(\mu(r')) \rangle =0,
\end{align*}
since $\delta(\mu(r')) = 0$ (see \cite{Dr:coco}). In the same way, the second
term of $B_{\alpha,\beta}(g)$ vanishes. Hence the system
(\ref{flatness}) has a solution
(it is unique if we impose that $H$ vanishes at the origin).
\hfill \qed \medskip
{\em End of proof of Proposition \ref{prop:inner}.}
Now if $h = -\alpha^*(H)$
with $H$ as in Lemma \ref{9:3} and for any $f\in{\mathcal O}_{G^*}$, we have
\begin{align*}
& D_{{\mathfrak{g}}^*}(f) = \alpha^*(D_{\mathfrak{g}}^*(F)) =
\alpha^*(({\bold L}_{\mu(r')} - {\bold R}_{\mu(r')})(F))
\\ & = \alpha^*(\langle (d_{\bold L} - d_{\bold R})(F)(g), R(d_{\bold L} H(g)) - L(d_{\bold R} H(g))
\rangle) = \{h,f\}.
\end{align*}
\hfill \qed \medskip
|
{
"timestamp": "2005-03-25T17:52:25",
"yymm": "0503",
"arxiv_id": "math/0503608",
"language": "en",
"url": "https://arxiv.org/abs/math/0503608"
}
|
\section{Introduction}
In the search for a suitable system for quantum information
processing, certain requirements have to be met \cite{04}, such as
scalability of the physical system, the capability of initializing
and reading out the qubits, and the possibility of having a set of
universal logic gates. Neutral atoms are one of the most promising
candidates for storing and processing quantum information. A qubit
can be encoded in the internal or motional state of an atom, and
several qubits can be entangled using atom-light interactions or
atom-atom interactions. Schemes for quantum gates for neutral
atoms have been theoretically proposed, that rely on dipole-dipole
interactions \cite{02,qg1,qg2,qg3} or controlled collisions
\cite{03,jaksch,10,12}. Such schemes can be implemented in optical
lattices with a controlled filling factor, as shown in ref.
\cite{bloch} where multi-particle entanglement via controlled
collisions was demonstrated.
Presently a major challenge
is to combine controlled collisions with the loading and the
addressing of individually trapped atoms. Recently techniques to
confine single atoms in micron-sized \cite{07,RS,01} or larger
\cite{meschede} dipole traps have been experimentally
demonstrated. A set of qubits can be obtained by creating an
array of such dipole traps, each one storing a single atom
\cite{register}. Gate operations require the addressability of
individual trapping sites and reconfigurability of the array.
Arrays of dipole traps, each containing many atoms, were obtained
using either arrays of micro-lenses \cite{09} or holograms
\cite{salomon}.
Actually, holographic techniques allow one to realize arrays of
very small dipole traps \cite{grier}, which can trap single atoms.
Holographic optical tweezers use a computer designed diffractive
optical element to split a single collimated beam into several
beams, which are then focused by a high numerical aperture lens
into an array of tweezers. Recently holographic optical tweezers
for individual Rubidium atoms have been implemented by using
computer-driven liquid crystal Spatial Light Modulators (SLM)
\cite{Bergamini2004}. The advantage of these systems is that the
holograms corresponding to various arrays of traps can be
designed, calculated and optimized on a computer. As a
consequence, the trap array can be (slowly) controlled and
reconfigured by writing these holograms on the SLM in real-time.
Here we want to combine such an holographic array with a fast
moving tweezer, in order to implement quantum gates based on a
state selective collision between two atoms, by using a Feschbach
resonance. Optimizing the control of the atoms motion is then of
crucial importance, and is the subject of the present paper.
\section{Quantum register with holographic dipole traps}
The present approach for neutral atoms quantum gates is related to
several schemes which have been proposed for trapped ions
\cite{wineland,zoller}, and it uses a quantum register made of
individual atoms stored in an array of holographic dipole traps.
The atoms encoding the qubit will be stored in this register,
which can be slowly reconfigured to move the atoms around, but
does not allow fast precise motion, which is required to implement
a controlled collision between two atoms. As a consequence, the
register has to be combined with one (or several) fast tweezers,
which can rapidly move an atom from one place to an other. There
are then several options~: either there is an atom in the moving
tweezer, which can be entangled and disentangled with the atoms in
the register (``moving head" scheme, similar to the one proposed
in ref. \cite{zoller}). One can also consider a configuration with
two tweezers, which catch two atoms in the register and bring them
to interaction.
The fast tweezer (or tweezers) consist of a laser beam passing
through an acousto-optical modulator (AOM), which allow to control
simultaneously the deflection and the intensity of the beam with
high accuracy. In the present paper, we will consider only two
such tweezers, each containing one atom, and we will show that a
quantum gate can be implemented with high fidelity by using
optimal control techniques. The parameters of the calculations
will be inspired by the experiment described in ref.
\cite{01,07,RS,Bergamini2004}, but the scheme may work as well in
a large range of parameter values. Typically, the size of the beam
waist for the tweezer will be less than a micron, resulting in
oscillation frequencies of 130 kHz in the radial directions, and
about 30 kHz in the axial direction. In addition, we will assume
that a standing wave is added along the propagation axis. This has
two important consequences~: first, the axial oscillation
frequency is increased up to a value which is typically close or
above the radial oscillation frequency; second, it will confine
the two atoms within the same ``pancake", therefore maximizing the
non-linear phase shift acquired during a controlled cold
collision. In the following, we will also assume that the two
atoms have been prepared in the ground state of the tweezer.
Though this was not implemented yet, it can in principle be done,
by using either side band cooling, or evaporative cooling down to
the single atom level \cite{RS}.
\section{Atom transport in a time-dependent double-well potential}
The transport mechanism is discussed in~\cite{calarco2004} for
atoms in a time dependent, optical super lattice which has the
form of a periodic array of double well potentials. Here,
however, we consider a one-dimensional system with a single double
well potential of the form
\begin{equation}
V(x,t) = -A(t) \; e^{-x^2/2w^2} - B(t) \; e^{-(x+d(t))^2/2w^2}.
\label{potential}
\end{equation}
\begin{figure}[h!]
\begin{center}
\includegraphics[]{potential.eps}
\end{center}
\caption{
The double well potential at (a) the initial time $t=0$, (b) at an
intermediate time $t=T/2$ and (c) at the end of the transport
process $t=T$. The position is given in units of
$a\equiv\sqrt{\hbar/m\omega}$ and the energy in units of
$\varepsilon\equiv{\hbar^2/2ma^2}$ as described in the text. The
horizontal, dashed lines indicate the eigenenergies of the system.
The solid lines are the (real) eigenfunctions of $H(t)$
corresponding to the moving and the register atom, i.e.
$\psi_2(x,t)$ and $\psi_0(x,t)$, respectively.
The shown potentials correspond to experimental parameters as
described in \cite{01,07,RS}. \label{fig:sequence}}
\end{figure}
In the geometry described above, it is sufficient to consider the
one-dimensional case, where the position coordinate $x$
corresponds to the distance between the two tweezers. We will thus
assume that the motional state along the two other axis does not
change during the transport process to be described (this point is
further discussed later in this section).
The location and the depth of the minima of the
potential~(\ref{potential}) is determined by the time dependent
control parameters $A(t),B(t)$ and $d(t)$. The time evolution of
the motional degrees of freedom of a single particle in the trap
is governed by the time dependent Schr\"odinger equation
\begin{equation}
\mathrm{i} \hbar \frac{d}{dt}\psi(x,t) = H(t)\psi(x,t) \label{eq:SG}
\end{equation}
with
\begin{equation}
H(t) = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V(x,t).
\end{equation}
In the following, distances are measured in units of a harmonic
oscillator length $a\equiv\sqrt{\hbar/m\omega}$ and energies in
units of $\varepsilon\equiv{\hbar^2/2ma^2}$. In case of
${}^{87}$Rb and $\omega=2\pi\times 100$kHz this defines a length
scale of $a = 34\,\mathrm{nm}$ and an energy scale of $\varepsilon
= 2\pi\hbar\times50\,\mathrm{kHz}$.
As in~\cite{calarco2004}, we assume that there is initially one
atom in the ground state of each well and that the barrier is
sufficiently high to prevent tunnelling between the wells. This
allows to raise rapidly the left potential well, such that at time
$t=0$ the situation depicted in figure~\ref{fig:sequence}(a) can
be created: The lowest motional state of the left atom corresponds
to the second excited state of the double well system while the
right atom is in the ground state. The moving atom can then be
adiabatically transported to the right well by lowering the left
well and the barrier simultaneously (see
figure~\ref{fig:sequence}(b)), and raising the barrier again while
the left well is further lowered leading to the final
configuration at $t=T$ shown in figure~\ref{fig:sequence}(c).
During the whole process the moving atom stays always in the
second excited instantaneous eigenstate of the system which
corresponds eventually to the first excited state of the right
well while the register atom remains in the ground state.
\begin{figure}[t!]
\begin{center}
\includegraphics[]{pulses.eps}
\end{center}
\caption{%
(a) The lowest instantaneous eigenenergies of the double-well system
during the transport process (the ground state energy is set to
zero). The transported atom corresponds to the solid line and the
static atom to the zero line. (b)(c) The pulse functions $A(t)$,
$B(t)$ and $d(t)$ control the depth and the distance of the two
wells. In (a)-(c), the parameters for $t=0$, $t=T/2$ and $t=T$ are
the same as in figure~\ref{fig:sequence}.
\label{fig:pulses}}
\end{figure}
The adiabatic transport is possible since by choosing appropriate
pulse functions $A(t),B(t)$ and $d(t)$, level crossings of the
eigenenergies of the system during the process can be avoided. An
example for such pulse functions, as well as the corresponding
instantaneous eigenenergies of $H(t)$, are shown in
figure~\ref{fig:pulses}. In figure~\ref{fig:pulses}(a) the time
dependent energy of the moving atom is given by the bold line (the
ground state energy is set to zero). Back transport of the moving
atom to its original position is obtained by time inversion of the
pulses.
In order to study the dynamics of the transport process we
introduce the occupation probabilities
\begin{equation}
P_n^A(t) = \left\vert \int dx\, {\psi^A}^*(x,t)
\psi_\mathrm{n}(x,t) \right\vert^2,
\end{equation}
where $\psi_\mathrm{n}(x,t)$ with $n=0,1,2,...$ is the $n$th
instantaneous eigenfunction of the double well potential. The
superscript $A\in\{M,R\}$ indicates the wavefunction of the atom
to be transported (moving atom) and the atom which is supposed to
stay located at its well (register atom), i.e. $\psi^M(x,t)$
[$\psi^R(x,t)$] is the solution of the time dependent single
particle Schr\"odinger equation (\ref{eq:SG}) with initial
condition $\psi^M(x,0)=\psi_\mathrm{2}(x,0)$
[$\psi^R(x,0)=\psi_\mathrm{0}(x,0)$] as shown in
figure~\ref{fig:sequence}(a). The fidelities of the processes are
then given by $F^M \equiv P^M_2(T)$ and $F^R \equiv P^R_0(T)$.
\begin{figure}[t!]
\begin{center}
\includegraphics[]{fidelities.eps}
\end{center}
\caption{
(a) Fidelity $F^M(t)$ corresponding to the moving atom during the
transport for $T=250\hbar/\varepsilon$ (i),
$T=350\hbar/\varepsilon$ (ii), $T=400\hbar/\varepsilon$ (iii),
$T=500\hbar/\varepsilon$ (iv), $T=1000\hbar/\varepsilon$ (v),
$T=5000\hbar/\varepsilon$ (vi). (b) Fidelity $F^R(t)$
corresponding to the register atom during the transport for
$T=500\hbar/\varepsilon$. (c) Probabilities $P_0^M(t)$ (solid
line), $P_1^M(t)$ (dash-dotted line), $P_3^M(t)$ (dashed line) and
$P_4^M(t)$ (dotted line) of finding the moving atom in the
respective eigenstates during its transport.
\label{fig:fidelity}}
\end{figure}
For the example shown in Fig.~\ref{fig:pulses} we get $F^M =
99.7\%$ for propagating the moving atom wavefunction and $F^R =
99.9\%$ for propagating the register atom wavefunction from $t=0$
to $t=T=500\hbar/\varepsilon$. In the case of Rubidium this would
correspond to a time $T=1.6\,\mathrm{ms}$.
Figure~\ref{fig:fidelity} shows the probability $P^M_2(t)$ that
the moving atom remains in the second instantaneous eigenstate
during the transport for various operation times $T$. For
$T=500\hbar/\varepsilon$ the fidelity is always greater than
$98.7\%$ and the corresponding occupation probability $P^R_0(t)$
of the register atom (shown in figure \ref{fig:fidelity}(b)) is
always larger than $99.9\%$. The corresponding probabilities of
finding the moving atom in the ground, the first excited, the
third excited and the fourth excited instantaneous eigenstates are
displayed in figure~\ref{fig:fidelity}(c). The occupation
probabilities of higher excited states are smaller than
$4\times10^{-5}$ and are not shown. An excitation energy of
$100$kHz, which would be required for the excitation of radial
motional states, would correspond at least roughly to the eighth
excited state along the axis of motion. The occupation probability
of this (and higher) state(s) is found to be smaller than
$5\times10^{-9}$ which justifies the one-dimensional model used in
this paper.
In the following we will discuss the influence of experimental
imperfections, especially variations in the laser intensities,
which are proportional to the pulse functions $A(t)$ and $B(t)$,
and variations in the distance of the lasers creating the double
well potential, which affect the pulse function $d(t)$. Motivated
by the experimental conditions we assume variations of the pulse
functions of the form
\begin{eqnarray}
\tilde d(t) &=& d(t) + \delta d \sin(\Omega t) \\
\tilde A(t) &=& A(t) + \delta A \sin(\Omega t) \\
\tilde B(t) &=& B(t) + \delta B \sin(\Omega t)
\end{eqnarray}
with $\Omega = 2\pi/1$ms. Assuming a drift $\delta d = 1nm$ while
$\delta A=\delta B=0$ leads for $T=500\hbar/\varepsilon$ to a
slight reduction of the fidelity to $F^M = 99.4\%$ and a higher fidelity $F^R$. A variation of $\delta A = 0.1\varepsilon$ while $\delta B =
0$ results in $F^M = 99.5\%$ and $F^R = 99.9 \%$.
If $A(t)$ and $B(t)$ undergo the {\em same} perturbation, i.e.
$\delta A = \delta B$, the shape of the
potential~(\ref{potential}) does not change significantly if the
variation is not too large (except for an approximately constant
shift of the potential). Therefore, the level spacing as shown in
figure~\ref{fig:pulses}(a) remains roughly the same and it is
expected that the transport can be done as fast as without
fluctuations. This behavior is confirmed by numerical simulations:
Assuming $\delta A = \delta B = 10\varepsilon$, which corresponds
to variations of the laser intensity of approximately $1\%$, we
get the fidelities $F^M = 99.7\%$ and $F^R = 99.9\%$.
This analysis shows that the current transport scheme is
relatively insensitive to noise which affects both parameters,
$A(t)$ and $B(t)$, in the same way, while it is more sensitive to
different perturbations in these parameters. In this case, level
crossings in the energy diagram~\ref{fig:pulses} can appear,
leading to significant leakage into higher excited states, which
would require a more sophisticated engineering to be controlled
and will be a subject of future investigations.
\section{Quantum gates by optimal control of molecular interactions}
Performing gate operations requires a strong molecular interaction
between atoms. They can be coupled to molecular states either by
means of Feshbach resonances \cite{Feshbach} or through Raman
photo-association laser pulses \cite{photoassociation}. For the
sake of concreteness, we focus here on Feshbach resonances --
however, all of our arguments can be adapted, e.g., to Raman
photo-association. We consider $^{87}$Rb atoms.
Feshbach resonances occur when a bound molecular state $|n\rangle$
crosses the dissociation threshold for a state having the same
quantum numbers \cite{Feshbach} while changing an external
magnetic field $B$. Close to resonance, the scattering length
varies as
\begin{equation}
A(B)=A_{bg}\left( 1-\frac{\Delta_n}{B-B_0}\right) ,
\end{equation}%
where $A_{bg}$ is a non resonant background scattering length,
$B_0$ is the resonant magnetic field, and $\Delta_n$ is the width
of the resonance. The resonance energy varies almost linearly with
the field
\begin{equation}
\varepsilon_n(B)=s_n(B-B_0), \label{resenergy}
\end{equation}%
with a slope $s_n$. We are interested in the dynamics of such a
system in a confined geometry. Following \cite{Mies00}, we shall
model it by the effective Hamiltonian
\begin{equation}
H_{\rm res}=\varepsilon_n(B)|n\rangle\langle
n|+\sum_v(v\hbar\nu|v\rangle\langle v|+V_v|v\rangle\langle n|+{\rm
h.c.}),
\end{equation}
where the $|v\rangle$'s are the trapped relative-motion atomic
eigenstates of an isotropic harmonic oscillator trap having
frequency $\nu$. The couplings to the resonance are
\begin{equation}
\label{couplings} V_v=2\hbar\nu
\sqrt{\sqrt{4v+3}\;a_{bg}\delta_n/\pi}
\end{equation}
with $a_{bg}\equiv A_{bg}\sqrt{m\nu/\hbar}$,
$\delta_n\equiv\Delta_n s_n/(\hbar\nu)$. In a different geometry,
for instance in an elongated trap characterized by a ratio
$\gamma$ between the ground level spacings in the transverse and
in the longitudinal potential, the couplings can be calculated by
projection on the corresponding eigenstates \cite{calarco2004}.
Accurate values for the resonance parameters $\Delta_n$ and $B_0$,
as well as for $A_{bg}$, are now available from both theoretical
calculations and recent measurements \cite{Verhaar02}.
The possibility of controlling the resonance energy via an
external magnetic field, as described by Eq.~(\ref{resenergy}),
provides a straightforward way to steer the interaction between
the atoms. Indeed, the coupling to a specific resonant state
$|n\rangle$ is only effective for a particular entrance channel,
{\em i.e.} a specific combinations of atomic hyperfine states
(that is, of logical qubit states in our case), while in general
all other channels will be unaffected by the resonance. Thus the
resonance-induced energy shift will cause a two-particle phase to
appear only for that particular two-qubit computational basis
state.
\begin{figure}[t!]
\begin{center}
\includegraphics[]{gate.eps}
\end{center}
\caption{Two-qubit gate operation via optimal magnetic field
control: optimized field time dependence (top left); overlap between
initial and evolved state (bottom left); accumulated two-particle
phase $\varphi$ (top right); decrease of the infidelity with
increasing iterations (bottom right).
\label{fig:gate}}
\end{figure}
We will identify our qubit logical states with the
clock-transition states
\begin{equation}
\left\vert 0 \right\rangle \equiv \left\vert
F=1,m_{F}=0\right\rangle,\quad \left\vert 1 \right\rangle \equiv
\left\vert F=2,m_{F}=0\right\rangle.
\end{equation}%
The main advantage of this choice is that the qubit states are not
sensitive to the magnetic field, and hence not subject to
decoherence due to its fluctuations. We will use the resonance for
the channel $|00\rangle$ occurring around $B_0=685$ G, having a
width $\Delta_n=16$ mG. For obtaining a two-qubit gate, the
magnetic field is ramped across $B_0$, and eventually tuned out of
the Feshbach resonance again, getting the following truth table
for the operation:
\begin{eqnarray}
\ket{00}&\rightarrow &e^{\mathrm{i} \varphi}\ket{00}, \nonumber\\
\ket{01}&\rightarrow &\ket{01}, \nonumber\\
\ket{10}&\rightarrow &\ket{10}, \nonumber\\
\ket{11}&\rightarrow &\ket{11},
\end{eqnarray}
where we included the phase $\varphi$ accumulated by state
$|00\rangle$ during the ramping process due to the interaction
energy shift, whose value can be adjusted by controlling the
magnetic field. If $\varphi=\pi$, a C-phase gate between the two
atoms is obtained. Note that laser addressing of single qubits is
never required throughout the procedure.
The magnetic ramping process can be even performed
non-adiabatically, provided that all population is finally
returned to the trapped atomic ground state. This can be
accomplished via a quantum optimal control technique in analogy
with the above discussion for the transport process. The control
parameter in this case is the external magnetic field $B$. Care
has to be taken in optimizing not only the absolute value of the
overlap of the final state onto the goal state, but also its phase
$\varphi$. Fig.~\ref{fig:gate} shows the optimization results for
a trap with transverse frequencies of 100 kHz and a longitudinal
frequency of about 25 kHz, corresponding to the right well of
Fig.~\ref{fig:sequence}. The final infidelity is about
$2\times10^{-5}$ in this case.
\section{Outlook}
We have described a scheme using moving tweezers and
state-dependent controlled collisions, which is able to implement
a quantum gate between two individual atoms with a high fidelity.
The sensitivity of the scheme to intensity or position
fluctuations has been examined, and the controlled motion is
found to be very tolerant to ``common-mode" noise between the two
tweezers. It is even relatively tolerant to differential noise,
because the overall process is close to adiabatic, and designed in
such a way as to avoid unwanted level crossings.
The magnetic field which is used here to obtain the Feshbach
resonance would be ultimately very advantageously replaced by an
optical field \cite{schlyap,grimm}, which can be switched on and
off with high speed and precision, and which will not perturb the
neighboring atoms stored in the holographic array. Though the
overall scheme is clearly not easy to implement, optimal control
techniques as used here certainly help to make it closer to
realistic.
\ack
This research was supported by a Marie Curie Intra-European Fellowship
within the 6th European Community Framework Programme,
the RTN ``CONQUEST'',
and by the IST/FET/QIPC projects ``QGATES'' and ``ACQP''. The Institut
d'Optique group acknowledges partial support from ARDA/NSA.
\vspace{0.5truecm}
|
{
"timestamp": "2005-03-22T12:40:11",
"yymm": "0503",
"arxiv_id": "quant-ph/0503180",
"language": "en",
"url": "https://arxiv.org/abs/quant-ph/0503180"
}
|
\section{Introduction}
One of the most remarkable lattices in Euclidean space is the
Leech lattice, the unique even unimodular lattice
$\Gamma _1\subset ({\mathbb{R}} ^{24}, (,)) $ of dimension 24 that
does not contain vectors of square length 2.
Here a lattice $\Lambda \subset ({\mathbb{R}}^n,(,))$ is called
{\em unimodular}, if $\Lambda $ equals its {\em dual lattice}
$$\Lambda ^{\#} := \{ x\in {\mathbb{R}} ^n \mid (x,\lambda ) \in {\mathbb{Z}} \mbox{ for}\mbox{ all } \lambda \in \Lambda \} $$
and {\em even}, if the quadratic form $x\mapsto (x,x) $ takes only even values
on $\Lambda $.
\cite{SieMod} studies spaces of Siegel modular forms generated by the Siegel
theta-series of the 24 isometry classes of lattices in the genus of
$\Gamma _1$.
The present paper extends this investigation to further genera of lattices,
closely related to $\Gamma _1$.
A unified construction is given in \cite{RS98s}:
Consider the Matthieu group
$M_{23} \leq \mbox{\rm Aut} (\Gamma _1)$, where the {\em automorphism group} of
a lattice $\Lambda \subset ({\mathbb{R}} ^n,(,)) $ is
$\mbox{\rm Aut}(\Lambda ):= \{ g\in O(n) \mid \Lambda g = \Lambda \} .$
Let $g\in M_{23}$ be an element of square-free order
$l:=|\langle g \rangle |$. Then
$$l\in \{ 1,2,3,5,6,7,11,14,15,23 \} =: {\cal N} = \{ n\in {\mathbb{N}} \mid
\sigma_1(n):=\sum _{d \mid n} d \mbox{ divides } 24 \} $$
and for each $l\in {\cal N}$, there is an up to conjugacy unique
cyclic subgroup $\langle g \rangle \leq M_{23}$ of order $l$.
Let $\Gamma _l := \{ \lambda \in \Gamma _1 \mid \lambda g = \lambda \}$
denote the fixed lattice of $g$.
Then $\Gamma _l$ is an extremal strongly modular lattice of level $l$ and
of dimension $2k_l$, where $$k_l:=12 \sigma_0(l)/\sigma _1(l)$$
and $\sigma_0(l)$ denotes the number of divisors of $l$.
In particular $\Gamma _1$ is the Leech lattice,
$\Gamma _2$ the 16-dimensional Barnes-Wall lattice and
$\Gamma _3$ the Coxeter-Todd lattice of dimension 12.
Let $\Lambda $ be an even lattice.
The minimal $l\in {\mathbb{N}} $ for which $\sqrt{l} \Lambda ^{\# }$ is even,
is called the {\em level} of $\Lambda $.
Then $l\Lambda ^{\#} \subset \Lambda $.
For an exact divisor $d$ of $l$ let
$$\Lambda ^{\#,d} := \Lambda ^{\#} \cap \frac{1}{d} \Lambda $$
denote the {\em $d$-partial dual} of $\Lambda $.
A lattice $\Lambda $ is called {\em strongly $l$-modular}, if
$\Lambda $ is isometric to $\sqrt{d} \Lambda ^{\# ,d}$ for all
exact divisors $d$ of the level $l$ of $\Lambda $.
If $l$ is a prime, this coincides with the notion of {\em modular}
lattices, which just means that the lattice is similar to its dual
lattice.
The Siegel theta-series
$$\Theta ^{(m)} _{\Lambda } (Z) := \sum _{(\lambda _1,\ldots , \lambda _m) \in
\Lambda ^m }\exp ( i \pi \mbox{\rm trace} ((\lambda _i,\lambda _j)_{i,j} Z)) $$
(which is a holomorphic function on
the Siegel halfspace ${\cal H}^{(m)} = $ $ \{ Z\in \mbox{\rm Sym} _m({\mathbb{C}})
\mid $ $ \Im (Z) $ positive definite $ \} $)
of a strongly $l$-modular lattice is a modular form for the
$l$-th congruence subgroup
$\Gamma _0^{(m)}(l)$ of $ \mbox{\rm Sp}_{2m}({\mathbb{Z}} )$ (to a certain character)
invariant under
all Atkin-Lehner-involutions (cf. \cite{Andrianov}).
In particular for $m=1$ and $l\in {\cal N}$
the relevant ring of modular forms is a polynomial
ring in 2 generators as shown in \cite{Quebmodular}, \cite{Quebstmodular}.
Explicit generators of this ring allow to bound the
minimum of an $n$-dimensional strongly
$l$-modular lattice $\Lambda $ with $l\in {\cal N}$,
$$
\min (\Lambda ) := \min _{0\neq \lambda \in \Lambda }(\lambda, \lambda )
\leq 2 + 2 \lfloor \frac{n}{2k_l} \rfloor .$$
Lattices $\Lambda $ achieving this bound
are called {\em extremal}.
For all $l \in {\cal N}$ there is a unique
extremal strongly $l$-modular lattices of dimension $2k_l$
and this is
the lattice $\Gamma _l$ described above.
All the genera are presented in the nice survey article
\cite{SchaSchuPi}.
In this paper we investigate the spaces of Siegel modular forms
generated by the Siegel theta-series of the lattices
in the genus ${\cal G} (\Gamma _l)$
for $l\in {\cal N}$ using similar methods as for
the case $l=1$ which is treated in \cite{SieMod}.
The vector space ${\cal V} := {\cal V}({\cal G})$
of all complex formal linear combinations of the
isometry classes of lattices in
any genus ${\cal G}$ forms a finite dimensional
commutative
${\mathbb{C}} $-algebra with positive definite Hermitian scalar product.
Taking theta-series defines linear operators
$\Theta ^{(m)}$ from ${\cal V}$ into a certain space of
modular forms and hence a filtration of ${\cal V}$ by
the kernels of these operators.
This filtration behaves nicely under the multiplication
and is invariant under all Hecke-operators.
With the Kneser neighbouring process we construct a
family of commuting self-adjoint linear operators on ${\cal V}$.
Their common eigenvectors provide
explicit examples of Siegel cusp forms.
The genera ${\cal G} (\Gamma _l) $ ($l\in {\cal N}$)
share the following properties:
\begin{kor}
Let $l\in {\cal N}$ and
let $p$ be the smallest prime not dividing $l$.
The mapping $\Theta ^{(k_l)}$ is injective on ${\cal V}({\cal G}(\Gamma _l))$.
For $l\neq 7$, the construction described in \cite{BFW} (see Paragraph \ref{BFW})
gives a non-zero cusp form $\mbox{\rm BFW} (\Gamma _l,p) =
\Theta ^{(k_l)}(\mbox{\rm Per} (\Gamma _l,p))$.
The eigenvalue of the Kneser operator $K_2$ at
the eigenvector $\mbox{\rm Per} (\Gamma _l,p) $
is the negative of the number of pairs of
minimal vectors in $\Gamma _l$ which is also the
minimal eigenvalue of $K_2$.
\end{kor}
\begin{remark}
In Section \ref{results} we also list the eigenvalues of some of the
operators $T(q)$ defined in Subsection \ref{Hecke}.
These eigenvalues suggest that for even values of $k_l$, the
cusp form $\mbox{\rm BFW} (\Gamma _l,p)$ is a generalized Duke-Imamoglu-Ikeda lift
(see \cite{Ikeda}) of the elliptic cusp form of minimal weight
$k_l$.
\end{remark}
{\bf Acknowledgement.}
We thank R. Schulze-Pillot for helpful comments, suggestions and
references.
\section{Methods}
The general method has already been explained in \cite{SieMod}
(see also \cite{SchuPi0},\cite{SchuPi1},
\cite{SchuPi2} and \cite{Birch} for similar strategies).
\subsection{The algebra ${\cal V} = {\cal V}({\cal G})$}
Let ${\cal G}$ be a genus of lattices in the Euclidean space
$({\mathbb{R}} ^{2k},(,))$.
Then ${\cal G}$ is the disjoint union of finitely many isometry classes
$${\cal G} = [\Lambda _1] \cup \ldots \cup [\Lambda _h ] .$$
Let ${\cal V} := {\cal V}({\cal G})\cong {\mathbb{C}} ^h$ be the complex vector space with basis
($[\Lambda _1], \ldots , [\Lambda _h])$.
Let ${\cal V}_{{\mathbb{Q}} } = \langle
[\Lambda _1], \ldots , [\Lambda _h] \rangle _{{\mathbb{Q}} } \cong {\mathbb{Q}} ^h$
be the rational span of the basis.
The space ${\cal V}$ can be identified with the algebra ${\cal A} $
of complex
functions on the double cosets
$G({\mathbb{Q}}) \backslash G({\mathbb{A}}) / \mbox{\rm Stab} _{G({\mathbb{A}} )} (\Lambda _{{\mathbb{A}} } )
= \cup _{i=1}^h G({\mathbb{Q}}) x_i \mbox{\rm Stab} _{G({\mathbb{A}} )} (\Lambda _{{\mathbb{A}} }) $
where $G$ is the integral form of the real orthogonal group $G({\mathbb{R}} )= O_{2k}$
defined by $\Lambda _1$, ${\mathbb{A}} $ denotes the ring
of rational ad\`eles and $\Lambda _{{\mathbb{A}}} $ the ad\'elic completion of $\Lambda _1$.
If $\chi _{i}$ denotes the characteristic function
mapping
$G({\mathbb{Q}}) x_j \mbox{\rm Stab} _{G({\mathbb{A}} )} (\Lambda _{{\mathbb{A}} }) $ to $\delta _{ij}$ and
$\Lambda _i = x_i \Lambda _1$ ($i=1,\ldots , h$) then
the isomorphism maps $[\Lambda _i ]$
to $|\mbox{\rm Aut} (\Lambda _i)| \chi _i $.
The usual Petersson scalar product then translates into the
Hermitian scalar product on ${\cal V}$ defined by
$$< [\Lambda _i], [\Lambda _j] > := \delta _{ij} |\mbox{\rm Aut} (\Lambda _i)| $$
and the multiplication of ${\cal A}$ defines
a commutative and associative multiplication $\circ $ on ${\cal V}$ with
$$[ \Lambda _i ] \circ [ \Lambda_j ] := \# (\mbox{\rm Aut}(\Lambda _i )) \delta_{i,j}
[ \Lambda _i ]$$
(see for instance \cite[Section 1.1]{Boe}).
Note that the Hermitian form
$\langle , \rangle $ is associative, i.e.
$$\langle v_1 \circ v_2 , v_3 \rangle = \langle v_1 , v_2 \circ v_3 \rangle
\mbox{ for}\mbox{ all } v_1,v_2,v_3\in {\cal V} . $$
\subsection{The two basic filtrations of ${\cal V}$}
For simplicity we now assume that ${\cal G}$ consists of
even lattices.
Let $l$ be the level of the lattices in ${\cal G}$.
Taking the degree-$n$ Siegel theta-series
$\Theta _{\Lambda _i }^{(n)} $ ($n=0,1,2,\ldots $)
of the lattices $\Lambda _i$
($i=1,\ldots , h $) then defines a linear map
$$
\Theta ^{(n)} : {\cal V}\rightarrow M_{n,k}(l) \mbox{ by }
\Theta ^{(n)}(\sum _{i=1 }^h
c_i [ \Lambda _i] ):=
\sum _{i=1}^h c_{i } \Theta _{\Lambda _i }^{(n)} $$
with values in a space of modular forms
of degree $n$ and weight $k$ for the group $\Gamma _{0}^{(n)}(l)$
(see \cite{Andrianov}).
For $n=0,\ldots, 2k $ let
${\cal V}_{n}:= \ker (\Theta ^{(n)})$ be the kernel of this linear map.
Then we get the filtration
$${\cal V}=:{\cal V}_{-1} \supseteq {\cal V}_0 \supseteq {\cal V}_1 \supseteq \ldots \supseteq {\cal V}_{2k} = \{ 0 \}$$ where
${\cal V}_0 = \{ v =\sum _{i=1}^h c_{i } [\Lambda _i ] \mid
\sum _{i=1}^h c_{i } = 0 \}$ is of codimension 1 in ${\cal V}$.
Clearly $\Theta ^{(n)}({\cal V}_{n-1})$ is the kernel of the
Siegel $\Phi $-operator mapping $\Theta ^{(n)}({\cal V})$
onto $\Theta ^{(n-1)} ({\cal V})$.
For square-free level one even has
\begin{theorem} (see \cite[Theorem 8.1]{BoeSchuPi}) {\label{cusp}}
If $l$ is square-free, then
$\Theta ^{(n)} ({\cal V}_{n-1}) $ is the space of cusp forms in
$\Theta ^{(n)} ({\cal V})$.
\end{theorem}
Let ${\cal W}_n := {\cal V}_n^{\perp }$ be the orthogonal complement of ${\cal V}_n$.
We then have the ascending filtration
$$0={\cal W}_{-1} \subseteq {\cal W}_0 \subseteq {\cal W}_1 \subseteq \ldots \subseteq {\cal W}_{2k} = {\cal V}.$$
By \cite[Proposition 2.3, Corollary 2.4]{SieMod}
one has the following lemma:
\begin{lemma}\label{mult}
$${\cal W}_n \circ {\cal W}_m \subset {\cal W}_{n+m} \mbox{ for}\mbox{ all } m,n \in \{ -1,\ldots , 2k \} $$
and
$${\cal W}_n \circ {\cal V}_m \subset {\cal V}_{m-n} \mbox{ for}\mbox{ all } m>n \in \{ -1,\ldots , 2k \} .$$
\end{lemma}
Since theta-series have rational coefficients,
both filtrations are rational, i.e.
${\cal V}_n = {\mathbb{C}} \otimes ({\cal V}_n \cap {\cal V}_{{\mathbb{Q}} })$
and ${\cal W}_n = {\mathbb{C}} \otimes ({\cal W}_n \cap {\cal V}_{{\mathbb{Q}} })$, hence the same statements
hold when ${\cal V}$ is replaced by ${\cal V}_{{\mathbb{Q}} }$.
\subsection{The Borcherds-Freitag-Weissauer cusp form}{\label{BFW}}
The article \cite{BFW} gives a quite general construction of a
cusp form of degree $k$.
Let $\Lambda $ be a $2k$-dimensional even lattice and
choose some prime $p$ such that the quadratic space
$(\Lambda / p\Lambda , Q_p) $ (where $Q_p(x) := \frac{1}{2} (x,x) + p{\mathbb{Z}}$)
is isometric to
the sum of $k$ hyperbolic planes.
Fix a totally isotropic subspace
$F$ of $\Lambda / p\Lambda $ of dimension $k$.
For $\lambda := (\lambda _1,\ldots, \lambda _k) \in \Lambda ^k$
we put
$E(\lambda ):=\langle \lambda _1,\ldots, \lambda _k \rangle + p\Lambda $
and
$S(\lambda ) := \frac{1}{p} ((\lambda _i, \lambda _j )_{i,j}) \in
\mbox{\rm Sym} _k ({\mathbb{R}} )$.
Define $ \epsilon (E(\lambda) ) = \epsilon (\lambda ):= (-1) ^{\dim (F\cap E(\lambda ))} $
if $E(\lambda )$ is a $k$-dimensional totally isotropic subspace of
$\Lambda /p \Lambda $ and
$\epsilon (E(\lambda) ) = \epsilon (\lambda ) := 0$ otherwise.
\begin{defi}
$\mbox{\rm BFW}(\Lambda ,p) (Z) := \sum _{\lambda \in \Lambda ^k} \epsilon(\lambda )
\exp (i \pi \mbox{\rm trace} (S(\lambda )Z )) $.
\end{defi}
By \cite{BFW} the form
$\mbox{\rm BFW}(\Lambda, p)$ is a linear combination of Siegel theta-series
of lattices in the genus of $\Lambda $:
For any $k$-dimensional totally isotropic subspace $E$ of
$\Lambda / p\Lambda $ let $\Gamma (E) := \langle E , p\Lambda \rangle $
be the full preimage of $E$.
Dividing the scalar product by $p$, one
obtains a lattice $\ ^{1/p} \Gamma (E) := (\Gamma (E) , \frac{1}{p} (,) ) \in {\cal G}$.
Then we define
$$\mbox{\rm Per} (\Lambda ,p):= \sum _{E} \epsilon(E) [ \ ^{1/p} \Gamma (E) ] \in {\cal V}$$
where the sum runs over all $k$-dimensional totally isotropic
subspaces of $\Lambda / p\Lambda $.
As $\epsilon $ is only defined up to a sign, also $\mbox{\rm Per} (\Lambda ,p) $
is only well defined up to a factor $\pm 1 $.
It is shown in \cite[Theorem 2]{BFW} that
$$\Theta ^{(k)} (\mbox{\rm Per} (\Lambda ,p)) = \mbox{\rm BFW} (\Lambda , p ).$$
In analogy to the notation in \cite{KV} we call
$\mbox{\rm Per} (\Lambda , p)$ the {\em perestroika} of $\Lambda $.
Clearly $\mbox{\rm BFW} (\Lambda ,p )$ is in the kernel of the $\Phi $-operator
and hence a cusp form, if the level of $\Lambda $ is
square-free by Theorem \ref{cusp}.
\subsection{Hecke-actions}\label{Hecke}
Strongly related to the Borcherds-Freitag-Weissauer construction
are the Hecke operators $T(p)$ which define self-adjoint linear
operators on ${\cal V}$ and whose action on theta series
coincides with the one of $T(p)$ in
\cite[Theorem IV.5.10]{Freitag} and \cite[Proposition 1.9]{Yoshida}
up to a scalar factor (depending on the degree of the theta series).
Assume that the genus ${\cal G}$ consists of even $2k$-dimensional
lattices of level $l$.
For primes $p$ not dividing $l$
we define $T(p) : {\cal V} \to {\cal V}$ by
$$T(p) ([\Lambda ]):= \sum _{E} [ \ ^{1/p} \Gamma (E) ] $$
where the sum runs over all $k$-dimensional totally isotropic
subspaces of $(\Lambda /p\Lambda , Q_p)$.
Note that $T(p)$ is 0 if
$(\Lambda /p\Lambda , Q_p)$ is not isomorphic to the
sum of $k$ hyperbolic planes.
The following operators commute with the $T(p)$
and are usually easier to calculate using the
Kneser neighbouring-method (see \cite{Kneser}):
For a prime $p$
define the linear operator $K_p$ by
$$K_p ([ \Lambda ] ) := \sum _{\Gamma } [ \Gamma ], \mbox{ for}\mbox{ all } \Lambda \in {\cal G}$$
where the sum runs over all lattices $\Gamma $ in ${\cal G}$
such that
the intersection $ \Lambda \cap \Gamma$ has index $p$ in
$\Lambda $ and in $\Gamma $.
If $p$ does not divide the level $l$
\cite[Proposition 1.10]{Yoshida} shows that the operators
$K_p$ are essentially the Hecke operators $T^{(m-1)}(p^2)$
(up to a summand, which is a multiple of the identity and a
scalar factor).
Also if $p$ divides $l$, the operators $K_p$ are
self-adjoint:
For $\Lambda $ and $\Gamma $ in ${\cal G}$, the
number $n(\Gamma,[\Lambda ])$
of neighbours of $\Gamma $ that are isometric to
$\Lambda $
equals the number of rational matrices $X\in \mbox{\rm GL} _{2k}({\mathbb{Z}} ) \mbox{\rm diag} (
p^{-1},1^{2k-1},p) \mbox{\rm GL}_{2k }({\mathbb{Z}} )$ solving
$$I(\Gamma , \Lambda ): \ \
X F_{\Gamma } X^{tr} = F_{\Lambda }$$ (where $F_{\Gamma }$
and $F_{\Lambda }$ denote fixed Gram matrices of $\Gamma $ respectively
$\Lambda $) divided by the order of the automorphism group of
$\Lambda $ (since one only counts lattices, $X$ and $gX$
have to be identified for all $g\in \mbox{\rm GL} _{2k}({\mathbb{Z}})$ with
$g F_{\Lambda } g^{tr} = F_{\Lambda }$).
Mapping $X$ to $X^{-1}$ gives a bijection between the
set of solutions of $I(\Gamma , \Lambda )$ and
$I(\Lambda , \Gamma )$.
Therefore
$$n(\Gamma , [\Lambda ]) |\mbox{\rm Aut} (\Lambda )| =
n(\Lambda , [\Gamma ]) |\mbox{\rm Aut} (\Gamma )| .$$
Hence the linear operators $K_p$ and $T(p)$ generate a commutative subalgebra
$${\cal H}:= \langle T(q), K_p \mid q,p \mbox{ primes }, q \teiltnicht l \rangle \leq \mbox{\rm End}^s ({\cal V}) $$
of the space of self-adjoint endomorphisms of ${\cal V}$
and ${\cal V}$ has an orthogonal basis
$(d_1,\ldots , d_h)$,
consisting of common eigenvectors of ${\cal H}$.
For each $1\leq i \leq h$ we define $v(i)\in \{ -1,\ldots, 2k-1 \}$ by
$d_i \in {\cal V}_{v(i)},\ \ d_i \not\in {\cal V}_{v(i)+1} .$
\\
Analogously let $w(i) \in \{ 0,\ldots ,2k \}$ be defined by
$d_i \in {\cal W}_{w(i)},\ \ d_i \not\in {\cal W}_{w(i)-1} .$
\begin{lemma}(\cite[Lemma 2.5]{SieMod})\label{mult1}
Let $1\leq i \leq h$ and assume that $d_i$ generates a full
eigenspace of ${\cal H}$.
Then
$w(i) = v(i)+1$.
\end{lemma}
If the genus ${\cal G}$ is strongly modular of level $l$, by which we mean that
$\sqrt{d} \Lambda ^{\# , d} \in {\cal G}$ for all $\Lambda \in {\cal G}$
and all exact divisors $d$ of $l$, then
the Atkin-Lehner involutions
$$ W_d: [\Lambda ] \mapsto [\sqrt{d} \Lambda ^{\# ,d } ] $$
for exact divisors $d$ of $l$
define further self-adjoint linear operators on ${\cal V}$.
In this case let
$$\hat{{\cal H}}:= \langle {\cal H}, W_{d} \mid d \mbox{ exact divisor of } l \rangle . $$
If all lattices in ${\cal G}$ are strongly modular, then $W_d = 1 $ for all
$d$ and
$\hat{\cal H} = {\cal H}$ is commutative.
Again, the Hecke action is rational on ${\cal V}_{{\mathbb{Q}} }$ hence
the ${\mathbb{Q}}$-algebras ${\cal H}_{{\mathbb{Q}} }$ and $\hat{\cal H}_{{\mathbb{Q}}}$
spanned by the $K_p$ respectively the $K_p$ and $W_d$
act on ${\cal V}_{{\mathbb{Q}} }$.
\begin{remark}
For $v\in \{ -1,0,\ldots , 2k-1 \} $ let
$${\cal D}_v := \langle d_i \mid v(i) = v \rangle .$$
If all eigenspaces of ${\cal H}$ are 1-dimensional, the
decomposition ${\cal V} = \oplus _{v=-1}^{2k-1} {\cal D}_v$
is preserved under any semi-simple algebra ${\cal A}$
with ${\cal H} \leq {\cal A} \leq \mbox{\rm End} (V) $ that respects the
filtration.
\end{remark}
\section{Results}\label{results}
The explicit calculations are performed in MAGMA (\cite{MAGMA}).
Fix $l\in {\cal N}$, let ${\cal G}:= {\cal G}(\Gamma _l)$, ${\cal V} = {\cal V}({\cal G})$ and
denote by $\Lambda _1 := \Gamma _l, \Lambda _2,\ldots, \Lambda _h$
representatives of the isometry classes of lattices in ${\cal G}$.
We find that in all cases ${\cal H} = \langle K_2,K_3 \rangle \cong {\mathbb{C}} ^h$
is a maximal commutative subalgebra of $\mbox{\rm End} ({\cal V}) $.
Therefore the common eigenspaces are of dimension one
and it is straightforward to calculate an explicit
orthogonal basis
$(d_1,\ldots , d_h)$ of ${\cal V}$ consisting of eigenvectors of ${\cal H}$.
In particular $v(i) = w(i) -1 $ for all $i=1,\ldots, h$
by Lemma \ref{mult1}.
Here we choose
$d_1 := \sum _{i=1} ^h | Aut(\Lambda _i)|^{-1} [\Lambda _i ]
\in {\cal V}_0 \setminus {\cal V}_1$ to be the unit element of ${\cal V}$
and (for $l\neq 7$) $d_h = \mbox{\rm Per} (\Gamma _l , p) \in {\cal V}_{k_l-1}$,
where $p$ is
the smallest prime not dividing $l$.
We then determine some Fourier-coefficients of the
series $\Theta ^{(n)} (d_i)$ $(n=0,1,\ldots, k_l)$ to get upper
bounds on $v(i)$.
In all cases the degree-$k_l$ Siegel theta-series of the lattices are
linearly independent hence ${\cal V}_{k_l} = \{ 0 \}$.
Moreover ${\cal V}_{{k_l}-1} = \langle d_h \rangle $ if $l\neq 7$.
We also know that $w(1)=0$ and we may choose $d_2$ such that
$w(2) =1 $.
By
Lemma \ref{mult} and \ref{mult1}
the product
$d_j \circ d_i $ lies in $ {\cal W} _{w(i)+w(j)}$.
If the coefficient of $d_h$ in the product is
non-zero, this yields lower bounds on the sum $w(i)+w(j)$
which often yield sharp lower bound for $w(i)$ and $w(j)$.
The method is illustrated in \cite[Section 3.2]{SieMod} and
an example is given in Paragraph \ref{BW}.
\subsection{The genus of the Barnes-Wall lattice in dimension 16.}\label{BW}
The lattices in this genus are given in \cite{BW}.
The class number is $h=24$ and we find
$$\langle K_2 , K_3 \rangle = {\cal H}_{{\mathbb{Q}} } \cong {\mathbb{Q}} ^{13} \oplus F_1 \oplus F_2 \oplus F_3 $$
where the totally real number fields $F_i \cong {\mathbb{Q}}[x] / (f_i(x)) $ are given by
$$\begin{array}{ll}
f_1 = & x^3 - 11496x^2 + 41722560x - 47249837568 \\
f_2 = & x^3 - 1704x^2 + 400320x + 173836800 \\
f_3 = & x^5 - 11544x^4 + 42868800x^3 - 53956108800x^2 + 1813238784000x
\\ & + 20094119608320000
\end{array}
$$
and $\langle K_2 , K_3, W_2 \rangle =
\hat{{\cal H}}_{{\mathbb{Q}} } \cong {\mathbb{Q}}^{13} \oplus \mbox{\rm Mat} _3 ( {\mathbb{Q}}) \oplus \mbox{\rm Mat} _3({\mathbb{Q}} ) \oplus \mbox{\rm Mat} _5 ({\mathbb{Q}} ).$
Let $\alpha _i$, $\beta _i$ and $\gamma _j$ ($ i=1,\ldots 3, j=1,\ldots, 5$)
denote the complex roots of
the polynomials $f_1$, $f_2$ respectively $f_3$.
Let $\epsilon _i$ ($i=1,\ldots , 3$) denote the primitive idempotents of
${\cal H}_{{\mathbb{Q}} }$ with ${\cal H}_{{\mathbb{Q}}} \epsilon _i \cong F_i$.
Since the image of ${\cal V}_{{\mathbb{Q}} }$ under $\Theta ^{(n)}$ has rational
Fourier-coefficients, the functions $v$ and $w$ are constant on the
eigenspaces $E_i = {\cal V} \epsilon _i $ ($i=1,2,3$).
We therefore give their values in one line in the following tabular:
\begin{theorem}
The functions $v$ and and the eigenvalues of $ev_2$ and $ev_3$ of
$K_2$ respectively $K_3$ on $(d_1,\ldots , d_{24})$
are as follows: \\
\begin{center}
\begin{tabular}{|l|c|r|r|c|l|c|r|r|}
\hline
$i$ & $v(i)$ & $ev_2$ & $ev_3$ & &
$i$ & $v(i)$ & $ev_2$ & $ev_3$ \\
\hline
$1$ & $-1$ & $34560$ & $7176640$ & & $15$ & $3,4$ & $1320$ & $8640$ \\
$2$ & $0$ & $ 16200$ & $2389440$ & & $E_2$ & $4$ & $\beta _j$ & $31680$ \\
$3$ & $1$ & $8760$ & $792000$ & & $19$ & $3,4,5$ & $1080$ & $-45120$ \\
$4$ & $1$ & $7128$ & $804288$ & & $20$ & $3,4,5$ & $312$ & $4032$ \\
$E_1$ & $2$ & $\alpha _j$ & $266688$ & & $21$ & $5$ & $-216$ & $8640$ \\
$8$ & $3$ & $2664$ & $90048$ & & $22$ & $5$ & $-216$ & $20928$ \\
$9$ & $3$ & $1320$ & $77760$ & & $23$ & $6$ & $-936$ & $13248$ \\
$E_3$ & $3$ & $\gamma _j$ &$ 100800$ & & $24$ & $7$ & $-2160$ & $39360$ \\
\hline
\end{tabular}
\end{center}
For the dimensions of ${\cal D}_v$ one finds
$$
\begin{array}{|l|ccccccccc|}
\hline
v & -1 & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\
\hline
\dim ({\cal D}_v) & 1 & 1 & 2 & 3 & 7\mbox{-}10 & 3\mbox{-}5 & 2\mbox{-}4 & 1 & 1 \\
\hline
\end{array}
$$
\end{theorem}
\underline{Proof.}
By explicit calculations of the Fourier-coefficients the values
given in the table are upper bounds for the $v(i)$.
By Lemma \ref{mult1} they also
provide upper bounds on the $w(i) = v(i) + 1 $.
We see that
$$d_i \circ d_j = A_{ij} d_{24} + \sum _{m=1}^{23} b_{ij}^m d_m $$
with a nonzero coefficient $A_{ij}$ for the following
pairs $(i,j)$:
$$(23,2),\ (22,3),\ (21,4),\ (E_1,E_2),\ (E_3,E_3),\ (8,8), \ (9,9) $$
(where $(E_1,E_2)$ means that there is some vector in $E_1$ and
some in $E_2$ such that this coefficient is non-zero, similarly $(E_3,E_3)$).
Since $d_m \in {\cal W}_7$ for all $m\leq 23$ and
$d_j\circ d_i \in {\cal W}_{w(i)+w(j)}$ the inequality
$w(i)+w(j) \leq 7$ together with $A_{ij} \neq 0$
implies that $d_{24} \in {\cal W}_7$ which is a contradiction.
Hence $w(i) + w(j) \geq 8$ for all pairs $(i,j)$ above.
This yields equality for all values $v(i)$ and $v(j)$
for these pairs.
Similarly we get $3\leq v(i) $ for $i=15,19,20$ since $A_{i,i} \neq 0$
for these $i$.
\hfill{q.e.d.}
\begin{Conjecture}
$v(19) = 5$ and $v(20) = 5 $.
\end{Conjecture}
Since $d_{15}\circ d_2 = \sum _{m=1}^{18} c_m d_m + A_1 d_{19} + A_2 d_{20} $ with $A_1 \neq 0 \neq A_2$, we get
$w(15) + 1 \geq \max (w(19),w(20))$.
\begin{remark}
If the conjecture is true, then
$v(15) = 4$ and
$\dim ({\cal D}_3 )= 7$,
$\dim ({\cal D}_4 )= 4$, and
$\dim ({\cal D}_5 )= 4$.
\end{remark}
Using the formula in \cite[Korollar 3]{Krieg} (resp. \cite[Proposition 1.9]{Yoshida}) we may calculate the eigenvalues of $T^{(m-1)}(3^2)$ from the
one of $K_3$ and compare them with the ones given in \cite[formula (7)]{BK}.
The result suggests that
$\Theta ^{(2)}(d_4)$,
$\Theta ^{(4)}(v)$ (for some $v\in E_3$),
$\Theta ^{(6)}(d_{19})$ and
$\Theta ^{(8)}(d_{24})$ are
generalized Duke-Imamoglu-Ikeda-lifts (cf. \cite{Ikeda}) of the elliptic cusp forms
$\delta _8 \theta _{D_4}^i$ (i=3,2,1,0) where
$\delta _8 = \frac{1}{96} (\theta _{D_4}^4-\theta _{\Gamma _3})$
is the cusp form of $\Gamma _0(2)$ of
weight 8 and $\theta _{D_4}$ the theta series of the 4-dimensional
2-modular root lattice
$D_4$.
This would imply that $v(19)=5$ and, with Lemma \ref{mult}, $v(15)=4$.
\subsection{The genus of the Coxeter-Todd lattice in dimension 12.}
For $l=3$ one has $h=10$,
all lattices in this genus are modular, and
${\cal H} _{{\mathbb{Q}} } = \langle K_2 \rangle \cong {\mathbb{Q}} ^{10} = \hat{\cal H} _{{\mathbb{Q}} }$
\begin{theorem}
There is some $a\in \{ 0,1 \}$ such that
the function $v$ and the eigenvalues $ev_2$ of $K_2$ and $e_2$ of $T(2)$
are as follows:
\begin{center}
\begin{tabular}{|c|c|r|r|c|c|c|r|r|}
\hline
$i$ & $v(i)$ & $ev_2$ & $e_2$ & & $i$ & $v(i)$ & $ev_2 $ & $e_2$ \\
\hline
$1$ & $-1$ & $2079$ & $151470 $ & &
$6$ & $3-a$ & $234$ & $7560 $ \\
$2$ & $0$ & $1026$ & $-27540 $ & &
$7$ & $3$ & $126$ & $2376 $ \\
$3$ & $1$ & $594$ & $17820 $ & &
$8$ & $3$ & $-36$ & $432 $ \\
$4$ & $1$ & $432$ & $3240 $ & &
$9$ & $4$ & $-144$ & $-864 $ \\
$5$ & $2$ & $288$ & $-5400 $ & &
$10$ & $5$ & $-378$ & $1944 $ \\
\hline
\end{tabular}
\end{center}
For the dimensions of ${\cal D}_v$ one finds
$$
\begin{array}{|l|ccccccc|}
\hline
v & -1 & 0 & 1 & 2 & 3 & 4 & 5 \\
\hline
\dim ({\cal D}_v) & 1 & 1 & 2 & 1+a & 3-a & 1 & 1 \\
\hline
\end{array}
$$
\end{theorem}
We conjecture that $a=0$ but cannot prove this using
Lemma \ref{mult}.
The eigenvalues of $T(2)$ suggest that $\Theta ^{(2)}(d_3)$,
$\Theta ^{(4)}(d_6)$ and $\Theta ^{(6)}(d_{10})$ are
generalized Duke-Imamoglu-Ikeda-lifts (cf. \cite{Ikeda}) of the elliptic cusp forms
$\delta _6 \theta _{A_2}^2$,
$\delta _6 \theta _{A_2}$, respectively
$\delta _6 $, where
$\delta _6 = \frac{1}{36} (\theta _{A_2}^6-\theta _{\Gamma _3})$
is the cusp form of $\Gamma _0(3)$ of
weight 6 and $\theta _{A_2}$ the theta series of the hexagonal lattice
$A_2$.
This would imply $v(3) = 1$, $v(6)=3$ and $v(10)=5$ and hence $a=0$.
\subsection{The genus of the 5-modular lattices in dimension 8.}
The class number of this genus is $h=5$,
all lattices in this genus are modular, and
${\cal H} _{{\mathbb{Q}} } = \langle K_2 \rangle \cong {\mathbb{Q}} ^5 = \hat{\cal H} _{{\mathbb{Q}} }$
\begin{theorem}
For $l=5$ one has
$\dim ({\cal D}_{v} ) = 1 $ for $v=-1,0,1,2,3$.
The function $v$ and the eigenvalues $ev_2$ of $K_2$ and
$e_p$ of $T(p)$ ($p=2,3$)
are given in the following table:
\begin{center}
\begin{tabular}{|c|c|r|r|r|c|c|c|r|r|r|}
\hline
$i$ & $v(i)$ & $ev_2$ & $e_2$ & $ e_3$ & & $i$ & $v(i)$ & $ev_2 $ & $e_2$ & $ e_3$ \\
\hline
$1$ & $-1$ & $135$ & $270$ & $2240 $ & &
$4$ & $2$ & $-8$ & $-16 $ & $56 $ \\
$2$ & $0$ & $70$ & $-120 $ & $ 160 $ & &
$5$ & $3$ & $-60$ & $10$ & $ 420 $ \\
$3$ & $1$ & $42$ & $84 $ & $ 256 $ & & & & & & \\
\hline
\end{tabular}
\end{center}
\end{theorem}
\subsection{The genus of the strongly 6-modular lattices in dimension 8.}
The class number of ${\cal G}(\Gamma _6)$ is
$h=8$, the Hecke-algebras are
$\hat{{\cal H}} _{{\mathbb{Q}} } = \langle K_2 , W_2 \rangle \cong {\mathbb{Q}}^5 \oplus \mbox{\rm Mat} _3({\mathbb{Q}} ) $
and ${\cal H}_{{\mathbb{Q}} }= \langle K_2 \rangle \cong {\mathbb{Q}}^5 + {\mathbb{Q}}[x]/(f(x))$
where $$f(x) = x^3 - 66x^2 - 216x + 31104.$$
Let $\delta _i \in {\mathbb{R}} $ ($i=1,2,3$) denote the roots of $f$.
\begin{theorem}
Then the function $v$
and the eigenvalues $ev_2$ of $K_2$ and $e_5$ of $T(5)$
are given in the following table:
\begin{center}
\begin{tabular}{|c|c|r|r|c|c|c|r|r|}
\hline
$i$ & $v(i)$ & $ev_2$ & $ e_5 $ & & $i$ & $v(i)$ & $ev_2$ & $ e_5 $ \\
\hline
$1$ & $-1$ & $144$ & $39312 $ & & $E$ & $1$ & $\delta _j $ & $ 1872 $ \\
$2$ & $0$ & $54$ & $1872 $ &
& $7$ & $2$ & $-6$ & $432 $ \\
$3$ & $1$ & $ 18 $ & $ 1008 $ & &
$8$ & $3$ & $-36$ & $ 4752 $ \\
\hline
\end{tabular}
\end{center}
Hence
$\dim ({\cal D}_v) = 1$ for $v=-1,0,2,3$ and $\dim({\cal D}_1) = 4$.
\end{theorem}
\subsection{The genus of the 7-modular lattices in dimension 6.}
The class number is $h=3$, all lattices are modular, and
$\hat{\cal H} _{{\mathbb{Q}} } = {\cal H} _{{\mathbb{Q}} } = \langle K_2 \rangle \cong {\mathbb{Q}}^3$.
In contrast to the other genera,
the perestroika $\mbox{\rm Per} (\Gamma _7,2) $ and hence also $\mbox{\rm BFW} (\Gamma _7,2)$
vanishes due to the fact that the image of $\mbox{\rm Aut} (\Gamma _7)$
in $GO_6^+(2)$ is not contained in the derived subgroup $O_6^+(2)$.
In fact, $\Theta ^{(2)} $ is already injective.
Since the discriminant of the space is not a square modulo $3$ and $5$,
the Hecke operators $T(3)$ and $T(5)$ vanish.
\begin{theorem}
We have $v(i) = i-2$ for $i=1,2,3$ and hence
$\dim ({\cal D}_v) = 1 $ for $v=-1,0,1 $.
The eigenvalues of $K_2$ are $35$, $19$, and $5$,
the ones of $T(2)$ are $30$, $-18$, and $10$, and
$T(11)$ has eigenvalues
$2928$, $ -144$, and $ 248$.
\end{theorem}
\subsection{The genus of the strongly $l$-modular lattices in dimension 4
for $l=11,14,15$.}
For $l=11,14,15$ the genus ${\cal G}(\Gamma _l)$ consists of 3 isometry classes
and ${\cal H} _{{\mathbb{Q}} } = \langle K_2 \rangle \cong {\mathbb{Q}} ^3 = \hat{\cal H} _{{\mathbb{Q}} }$ since all lattices in the genus
are strongly modular.
\begin{theorem}
For $l=11,14,15$ one has
$\dim ({\cal D}_{v}) = 1 $ for $v=-1,0,1 $.
The eigenvalues $ev_2$ of $K_2$ and $e_p$ of $T(p)$ for primes $p\leq 7$ not dividing
$l$
are given in the following table:
\begin{center}
\begin{tabular}{|c|c||r|r|r|r|r||r|r|r||r|r|r|}
\hline
& & \multicolumn{5}{|c||}{$l=11$} &
\multicolumn{3}{|c||}{$l=14$} &
\multicolumn{3}{|c|}{$l=15$} \\
\hline
$i$ & $v(i)$ & $ev_2$ & $e_2$ & $e_3$ & $e_5$ & $e_7$ &
$ev_2$ & $e_3$ & $e_5$ &
$ev_2$ & $e_2$ & $e_7$ \\
\hline
$1$ & $-1$ & $9$ & $6$ & $8$ & $12$ & $16$ &
$8$ & $8$ & $12$ &
$9$ & $6$ & $16$ \\
$2$ & $0$ & $4$ & $-4$ & $-2 $ & $2 $ & $-4 $ &
$2$ & $-4$ & $0$ &
$1$ & $ -2 $ & $ 0 $ \\
$3$ & $1$ & $ -6 $ & $1$ & $3 $ & $ 7 $ & $ 6 $ &
$ -4 $ & $2$ & $6$ &
$ -3 $ & $2$ & $8 $ \\
\hline
\end{tabular}
\end{center}
\end{theorem}
\subsection{The genus of the $23$-modular lattices in dimension 2.}
In the smallest possible dimension $2$ the genus ${\cal G}(\Gamma _{23})$ consists of only
2 isometry classes and ${\cal H} _{{\mathbb{Q}} } = \langle K_2 \rangle \cong {\mathbb{Q}} ^2 =
\hat{\cal H} _{{\mathbb{Q}} }$ for the same argument that all lattices in the genus are
modular.
\begin{theorem}
For $l=23$ one has
$\dim ({\cal D}_{v}) = 1 $ for $v=-1,0$.
One has $v(1) = -1$, $v(2)= 0$,
$d_1 K_2 = 2 d_1 $ and
$d_2 K_2 = - d_2 $.
For the $T(p)$ for primes $p<23$ we find
$T(2)=T(3)=T(13) = K_2$ and $T(5)=T(7)=T(11)=T(17)=T(19)=0$.
\end{theorem}
|
{
"timestamp": "2005-03-22T08:59:06",
"yymm": "0503",
"arxiv_id": "math/0503447",
"language": "en",
"url": "https://arxiv.org/abs/math/0503447"
}
|
\section{Introduction}
Einstein-Podolsky-Rosen (EPR) \cite{einstein} suggested that
quantum mechanics was incomplete and that hidden variables may be
needed for completion. This was the subject of an extensive debate
with Bohr \cite{bohr} who denied the existence of such hidden
variables using his well known reasoning that is at the basis of
the Copenhagen interpretation of quantum mechanics. A mathematical
non-existence proof for these hidden variables was presented by
von Neumann within the framework of quantum mechanics. Bell
\cite{bellbook}, however, showed that von Neumann's proof assumed
the simultaneous measurability of certain quantities that could
not possibly be simultaneously measured. Subsequently Bell himself
presented a non-existence proof \cite{bell} in form of
inequalities that are derived by using more complex assumptions
that did not necessarily include simultaneous measurability.
The inequalities of Bell \cite{bell} are derived by the use of
probability theory, in essence by the use of very elementary facts
about random variables within the framework pioneered by
Kolmogorov. The derivation of the inequalities does not involve
physics or quantum mechanics, yet the inequalities have assumed an
important role for the foundations of quantum mechanics. This role
is a consequence of the assumptions for the random variables (and
other possible variables) that are used to derive the Bell
inequalities. It is commonly believed that these assumptions are
needed and moreover are equivalent to the basic postulates of an
``objective local" parameter space \cite{leggett}. These
postulates or conditions encompass Einstein-locality and the
existence of elements of reality in the sense of Mach that
co-determine the outcome of measurements. The fact that some
results of quantum mechanics violate the Bell inequalities has
therefore led to the belief that no objective local parameter
space can exist that explains all the results of quantum
mechanics.
Subsequent to these discussions experiments were proposed
\cite{ba}, \cite{bohm} and later realized \cite{eprex} that
confirmed the theoretical result of quantum mechanics. This seemed
to leave only difficult options for theoretical physics such as
(i) to deny that the microscopic entities of physics have
objective reality (non-existence of objective local parameters) or
(ii) to assert that an influence can be propagated faster than the
speed of light \cite{moore}. Additional options involving the
validity of counterfactual reasoning have also been suggested and
will be discussed below.
We show in this paper that because of the historical sequence of
events, viz. the development of Bell's theory before the
performance of the actual experiments, some very important facts
of probability theory have not been considered and/or
misinterpreted . These facts are connected with the concept of a
probability space that is important to link probability theory and
mathematical statistics to the evaluation of actual experiments.
We reconsider here these concepts and show that violations of the
Bell inequalities have a purely mathematical reason, in particular
that the Bell inequalities represent a special case of theorems
given earlier by Bass \cite{bass}, Vorob'ev \cite{vorob1},
\cite{vorob} and Schell \cite{schell}. These theorems permit us to
deduce that, for all the possible Bell inequalities to be valid,
it is a necessary and sufficient condition that the random
variables involved in their proof are defined on one common
probability space. Beyond this we show that the requirement of the
use of one common probability space does not follow from the
requirements of objective local spaces and vice versa. In fact, we
show that there exist objective local random variables that can
not be defined on a common probability space and therefore do not
need to obey the Bell inequalities. As mentioned, Bell has
dismissed von Neumann's proof that assumed from the start the
simultaneous measurability of the observables that correspond to
our random variables and that can not be measured simultaneously.
Our contribution here is that we also dismiss non-existence proofs
of certain systems of random variables that need to be defined on
one common probability space when it is clear from the outset that
these systems of random variables can not be defined on any common
probability space. We believe that our results give additional
options of explanation for Aspect-type experiments without
violating relativity or denying objective reality.
\section{Mathematical model of a singlet spin state EPR
experiment}
As outlined above, probability theory provides a probability space
and random variables and the link to the statistical treatment of
the data from an actual experiment. Bell \cite{bell} considered an
experimental situation advocated by Bohm and Aharonov \cite{ba},
and by Bohm and Hiley \cite {bohm}. This proposal was transformed
into an actual experiment by Aspect et al. \cite{eprex} including
the suggestion of Bell and others that a rapid change of the
settings needed to be implemented to accomplish a delayed choice
situation \cite{eprex}. We develop now an idealized experiment and
a probability model for this actual experiment by Aspect et al
within the framework of Kolmogorov. We note that our procedure
also applies to other related experiments.
We first recall the concepts of a (discrete) probability space and
of a random variable defined on it. As Feller states
\cite{feller}``If we want to speak about experiments or
observations in a theoretical way and without ambiguity, we first
must agree on the simple events representing the thinkable
outcomes; {\it they define the idealized experiment}...... By
definition {\it every indecomposable result of the (idealized)
experiment is represented by one, and only one, sample point}. The
aggregate of all sample points will be called the {\it sample
space}." In our case of the idealized EPR experiment, the simple
event can be chosen, for example, as the event of sending out one
(and only one) correlated pair. With this event we associate an
element $\omega$. In order to avoid mathematical technicalities
that are not needed for the purpose of our paper we assume that
the sample space $\Omega$ is at most countable. Each simple event
$\omega \in \Omega$ is assigned the probability $P(\omega)$ that
$\omega$ occurs. $P$ is a set function, defined for all subsets of
$\Omega$, that satisfies the usual axioms, such as countable
additivity and it assigns to $\Omega$ the value $P(\Omega) = 1$.
The pair $(\Omega, P)$ is called a probability space. A random
variable is a real-valued function on $\Omega$, but if needed it
also can assume values in high-dimensional space.
We now turn to the specifics of an idealized EPR-experiment. A
correlated spin pair in the singlet state is sent out from a
source in opposite directions toward measurement stations. These
stations are characterized by certain randomly and rapidly
switched settings which we denote by three dimensional unit
vectors ${\bf a}, {\bf b}, {\bf c}, ...$. The measurements in the
stations are mathematically represented by random variables $A =
\pm 1, B = \pm 1, C = \pm 1, ...$ that may in turn be functions of
other random variables e.g. a source parameter $\Lambda$ that
characterizes all the properties of the particles sent out from
the source. $A$ indicates that the measurements that correspond to
the outcomes of random variable $A$ have been performed using the
setting $\bf a$ and similarly for $B$ and $C$.
We perform three categories of experiments, each with a different
pair of setting vectors. The first category is characterized by
the vectors $\bf a$ in station $S_1$ and $\bf b$ in station $S_2$.
According to our notational convention we denote the pair of
measurements $(A, B)$ and the joint probability density of $A$ and
$B$ by $f_1$. Thus $f_1$ is given by
\begin{equation}
f_1 (+1, +1) = P(A=+1, B=+1) \text{ , }f_1(-1, +1) = P(A=-1,
B=+1) \nonumber
\end{equation}
\begin{equation}
f_1 (+1, -1) = P(A=+1, B=+-1) \text{ , }f_1(-1, -1) = P(A=-1,
B=-1) \label{bvc1}
\end{equation}
The second category of experiments will be characterized by the
vectors $\bf a$ in $S_1$ and $\bf c$ in $S_2$ with the resulting
pair of measurements $(A, C)$ having density $f_2$, and the third
category by the vectors $\bf b$ in $S_1$ and $\bf c$ in $S_2$
resulting in a pair of measurements $(B, C)$ with density $f_3$.
The measurement outcomes on both sides need to be completely
random and $\pm 1$ with equal probability, i.e. all three
distributions have identical marginals. This is dictated by the
rules of quantum mechanics and verified by experiment. From this
it follows that the $f_i, i =1, 2, 3$ have the center of gravity
for their point masses at the origin $(0, 0)$ and
\begin{equation}
f_i(+1,-1) + f_i(+1,+1) = \frac {1} {2} = f_i(+1,+1) +
f_i(-1,+1)\text{ for } i = 1,2,3 \label{bv1}
\end{equation}
The idealized mathematical model with exactly the properties
described above and used within the framework of Kolmogorov is the
basis for all our further considerations and we call it the Ma-EPR
model.
\section{Ma-EPR and the theorems of Bass and Vorob'ev}
We start with an example that illustrates the theorems of Bass
\cite{bass} and Vorob'ev \cite{vorob} for the special case of the
Ma-EPR model. The essential point of these theorems is that, in
general, it is not possible to find three random variables $A, B$
and $C$, defined on a common probability space such that the three
pairs $(A,B)$, $(A,C)$, and $(B,C)$ of random variables have their
joint densities equal to $f_1, f_2$ and $f_3$, respectively. Hence
the notation that is commonly used and that we also introduced in
section 2 above is misleading in the sense that it suggests there
exist three random variables $A, B$ and $C$ that can reproduce the
joint densities $f_1$, $f_2$ and $f_3$, when in fact they can not.
Here is a modification of an example of Vorob'ev \cite{vorob}.
\begin{table}[ht]
\centering
\begin{tabular}{|l||r|r|r|r|}\hline
& $(+1,+1)$ & $(+1,-1)$ & $(-1,+1)$ & $(-1,-1)$
\\ \hline
$f_1(.,.)$ & $3/8$ & $1/8$
& $1/8$ & $3/8$\\ \hline
$f_2(.,.)$ & $3/8$ & $1/8$
& $1/8$ & $3/8$\\ \hline
$f_3(.,.)$ & $1/8$ & $3/8$
& $3/8$ & $1/8$\\ \hline
\end{tabular}
\caption{Vorob'ev-type example \cite{vorob}.}\label{TA:ma}
\end{table}
Clearly Eq(\ref{bv1}) holds. Suppose now that three such random
variables $A, B$ and $C$ exist and are defined on one common
probability space. Then the first two rows would imply that $P(A =
B) = \frac {3} {4} = P(A = C)$, and so $P(B \neq C) \leq \frac {1}
{2}$ , contradicting the fact that according to the third row $P(B
= C) = \frac {1} {4}$. Another easy way to see that three such
random variables cannot be defined on a common probability space
follows from the fact that, for instance, it is not possible to
assign a probability to the event $(A = 1, B = 1, C = 1)$.
According to the first entry of the third row this probability
could not exceed $\frac {1} {8}$. Subtracting this value from the
first entry of the first row we obtain that P(A = 1, B = 1, C =
-1) would have to be at least $\frac {1} {4}$. But this is in
conflict with the second entry of the second row. The reason for
this phenomenon is that, picturesquely speaking, the three pair
distributions form a closed loop. The joint densities of $(A, B)$
and of $(A, C)$ already contain some information about the joint
density of $(B, C)$. Hence we do not have complete freedom to
choose the latter one. This was shown for three general pair
distributions by Jean Bass \cite{bass} and independently by Schell
\cite{schell} who also investigated the connection with certain
problems in economics. Vorob'ev \cite{vorob1}, \cite{vorob}
established necessary and sufficient conditions that any complex
of distributions must possess so that these distributions can be
realized as marginal distributions of a set of random variables
defined on a common probability space.
\begin{table}[ht]
\centering
\begin{tabular}{|l||r|r|r|r|}\hline
& $(+1,+1)$ & $(+1,-1)$ & $(-1,+1)$ & $(-1,-1)$
\\ \hline
$f_1(.,.)$ & ${\frac {1} {4}}(1 + \sigma_1)$ & ${\frac {1} {4}}(1 - \sigma_1)$
& ${\frac {1} {4}}(1 - \sigma_1)$ & ${\frac {1} {4}}(1 + \sigma_1)$\\ \hline
$f_2(.,.)$ & ${\frac {1} {4}}(1 + \sigma_2)$ & ${\frac {1} {4}}(1 - \sigma_2)$
& ${\frac {1} {4}}(1 - \sigma_2)$ & ${\frac {1} {4}}(1 + \sigma_2)$\\ \hline
$f_3(.,.)$ & ${\frac {1} {4}}(1 + \sigma_3)$ & ${\frac {1} {4}}(1 - \sigma_3)$
& ${\frac {1} {4}}(1 - \sigma_3)$ & ${\frac {1} {4}}(1 + \sigma_3)$\\ \hline
\end{tabular}
\caption{Pair densities in terms of covariances}\label{TA:ob}
\end{table}
It is easy to show that under the assumption of Eq(\ref{bv1}) the
joint pair densities can be expressed in terms of the covariances
$\sigma_i, i=1,2,3$ defined by these pair densities. The pair
densities are then given by Table \ref{TA:ob} (see also the Lemma
below). Note that the covariances $\sigma_i$ do not exceed $1$ in
absolute value. Suppose now that there exist three random
variables $A, B, C$ defined on one common probability space that
reproduce the densities $f_1, f_2, f_3$ in Table \ref{TA:ob}. Then
$\sigma_1 = E(AB)$, $\sigma_2 = E(AC)$, and $\sigma_3 = E(BC)$
where $E$ denotes the expectation value. Expressing the entries of
Table \ref{TA:ob} in terms of the eight unknown probabilities $P(A
= \pm 1, B = \pm 1, C = \pm1)$ will result in a system of twelve
linear equations in these eight unknowns that can be solved in an
elementary way. In particular, solving this system shows that
these eight probabilities can be expressed in terms of the three
covariances $\sigma_i, i = 1,2,3$. It turns out that five of these
twelve linear equations are redundant. Thus this system has
infinitely many solutions. Taking into account that the solutions
of this system represent probabilities $P \geq 0$ we obtain in a
straightforward way that the following four inequalities are
necessary and sufficient conditions for the solvability of the
consistency problem for the three pair distributions given in
Table \ref{TA:ob}:
\begin{equation}
1 + \sigma_1 + \sigma_2 + \sigma_3 \geq 0 \label{bvcc1}
\end{equation}
\begin{equation}
1 + \sigma_1 - \sigma_2 - \sigma_3 \geq 0 \label{bvcc2}
\end{equation}
\begin{equation}
1 - \sigma_1 + \sigma_2 - \sigma_3 \geq 0 \label{bvcc3}
\end{equation}
\begin{equation}
1 - \sigma_1 - \sigma_2 + \sigma_3 \geq 0 \label{bvcc4}
\end{equation}
Of course, the necessity part of this conclusion can be shown
directly and trivially by modifying the standard proofs of the
Bell inequality along the lines shown in \cite{bell}.
\section{Bell's inequalities as a special case of Bass-Vorob'ev}
Replacing the covariances $\sigma$ by the corresponding
expectation values, one obtains from
Eqs.(\ref{bvcc2}-\ref{bvcc3}):
\begin{equation}
E(AB) - E(AC) \leq 1 - E(BC) \label{bv6}
\end{equation}
and
\begin{equation}
-E(AB) + E(AC) \leq 1 - E(BC) \label{bv7}
\end{equation}
Eqs.(\ref{bv6}) and (\ref{bv7}) give
\begin{equation}
|E(AB) - E(AC)| \leq 1 - E(BC) \label{bv8}
\end{equation}
This is, of course, one of the celebrated Bell inequalities. Five
more can be obtained in analogous fashion from
Eqs.(\ref{bvcc1}-\ref{bvcc4}) giving a total of 6 (4 choose 2).
These can also be obtained by cyclic permutation in Eq.(\ref{bv8})
and replacing both minus signs by plus signs.
Bass \cite{bass} proved that for three general pair distributions
the consistency problem can be solved if and only if the triple
$(\sigma_1, \sigma_2, \sigma_3)$ considered as a point in $R^3$
belongs to a certain domain. In the special case we have been
considering this domain reduces to the tetrahedron defined by the
inequalities of Eqs.(\ref{bvcc1}-\ref{bvcc4}). We shall call it
the covariance tetrahedron. It is diplayed in Fig.
\ref{fig:Tetrahedra}.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.30\textwidth]{Tetrahedra.eps}
\caption{Covariance Tetrahedron. The solid point represents a
choice of values that violates the Bell inequalities.}
\label{fig:Tetrahedra}
\end{figure}
We formulate now our findings for the Ma-EPR experiment as a
theorem. We first collect a few facts of section 3 above in form
of a lemma.
Lemma: Let $f$ be a density supported on the four vertices $(\pm
1, \pm 1)$ of a square. Suppose that
\begin{equation}
f(+1, +1) + f(+1, -1) = f(+1, +1) + f(-1, +1) = \frac {1} {2}
\label{bvcc5}
\end{equation}
Then
\begin{equation}
\sum x f(x, y) = \sum y f(x, y) = 0 \label{bvcc6}
\end{equation}
where the sums are extended over the four points $(x, y) = (\pm 1,
\pm 1)$. Conversely, if $f$ satisfies Eq.(\ref{bvcc6}) then $f$
also satisfies Eq.(\ref{bvcc5}).
Set
\begin{equation}
\sigma := \sum x y f(x, y) \label{bvcc7}
\end{equation}
with the same proviso for the sum. Then $f$ can be expressed in
terms of $\sigma$ by the equations
\begin{equation}
f(+1, +1) = f(-1, -1) = {\frac {1} {4}}(1 + \sigma) \label{bvcc8}
\end{equation}
\begin{equation}
f(-1, +1) = f(+1, -1) = {\frac {1} {4}}(1 - \sigma) \label{bvcc9}
\end{equation}
Theorem1: Let $f_1, f_2, f_3$ be three probability densities
satisfying the hypotheses of the Lemma with corresponding
covariances $\sigma_1, \sigma_2, \sigma_3$. Then the following
statements are equivalent
\begin{enumerate}
\item[(I)] The point $(\sigma_1, \sigma_2, \sigma_3) \in R^3$
satisfies the system of inequalities Eqs.(\ref{bvcc1}-\ref{bvcc4})
and therefore belongs to the covariance tetrahedron.
\item[(II)] The point $(\sigma_1, \sigma_2, \sigma_3)$ satisfies
the following six Bell-type inequalities
\begin{equation}
|\sigma_1 - \sigma_2| \leq 1 - \sigma_3 \text{ , }|\sigma_1 +
\sigma_2| \leq 1 + \sigma_3 \nonumber
\end{equation}
\begin{equation}
|\sigma_1 - \sigma_3| \leq 1 - \sigma_2 \text{ , }|\sigma_1 +
\sigma_3| \leq 1 + \sigma_2 \nonumber
\end{equation}
\begin{equation}
|\sigma_2 - \sigma_3| \leq 1 - \sigma_1 \text{ , }|\sigma_2 +
\sigma_3| \leq 1 + \sigma_1 \label{bvcc10}
\end{equation}
\item[(III)] There exist three random variables $A, B, C$ defined
on a single common probability space with the following
properties. The joint probability densities of $(A, B), (A, C)$
and $(B, C)$ are $f_1, f_2$ and $f_3$ respectively. In particular,
the expectation values equal
\begin{equation}
E(A) = E(B) = E(C) = 0 \label{bvcc11}
\end{equation}
the covariances equal
\begin{equation}
E(AB) = \sigma_1 \text{ , }E(AC) = \sigma_2 \text{ , }E(BC)
= \sigma_3 \label{bvcc12}
\end{equation}
Using Eqs.(\ref{bvcc10}) and (\ref{bvcc12}) one obtains the six
Bell inequalities for the expectation values $E(AB), E(AC),
E(BC)$.
\end{enumerate}
Proofs: The proof of the Lemma is straightforward. The proof that
conditions (I) and (II) of Theorem1 are equivalent can be done by
inspection. The proof that (III) implies (II) or the six Bell
inequalities obtained from Eq.(\ref{bvcc10}) and Eq.(\ref{bvcc12})
can be carried out by a simple modification of the standard proof
of the Bell inequalities \cite{bell}. Finally, the proof that (I)
implies (III) was outlined at the end of section 3.
We have shown therefore the following. The inequalities of Bell
are a special case of the theorems of Bass and Vorob'ev for the
Ma-EPR experiment. If the 6 Bell inequalities are valid then it is
possible to find three random variables $A, B$ and $C$, defined on
one common probability space that reproduce the three joint pair
densities $f_i$ of Table \ref{TA:ob} and their covariances
$\sigma_1 = E(AB), \sigma_2 = E(AC)$ and $\sigma_3 = E(BC)$. These
covariances satisfy the Bell inequalities. Therefore, if quantum
mechanics predicts that, for a given idealized experiment
involving random variables $A, B$ and $C$ and $E(AB)$, $E(AC)$,
and $E(BC)$, one of the six Bell inequalities in Eq.(\ref{bvcc10})
will be violated or equivalently if the point with coordinates
$(E(AB), E(AC), E(BC)) \in R^3$ does not belong to the covariance
tetrahedron of Fig. \ref{fig:Tetrahedra}, then the random
variables $A, B$ and $C$, that are supposed to form the basis for
the model of this idealized experiment, can not be defined on one
common probability space. We note that the work of Fine
\cite{fine} has already shown the importance of a joint density
and therefore of a common probability space in the derivations of
Bell-type inequalities. The importance of a common probability
space was also stressed more recently in \cite{entrop1} and other
publications.
In summary, we have shown that the definability of $A, B$ and $C$
on one common probability space (OCPS) is a necessary and
sufficient condition for the validity of Bell's inequalities and
that this condition is of a purely mathematical nature and has
nothing to do with the questions of non-locality or counterfactual
reasoning that usually surround discussions of the Bell
inequalities. The condition is, however, related to some of the
physics of EPR experiments in a variety of ways that will be
discussed in section 6.
We add here that other inequalities of similar type such as the
Clauser-Horne-Holt-Shimony (CHHS) \cite{chhs} inequalities can be
treated similarly, although with greater algebraic exertion (16
linear equations in 16 unknowns). Their validity is again a
necessary and sufficient reason that all involved random variables
are defined on one common probability space. In fact, a theorem
analogous to Theorem1 above holds, with the covariance tetrahedron
replaced by a four-dimensional polytope. This polytope equals the
intersection of the four-dimensional parallelepiped, defined by
the four CHHS inequalities, and the four-dimensional cube with
vertices $(\pm 1, \pm 1,\pm 1,\pm1)$. The details will be
published elsewhere.
\section{Bell-type proofs and Bass-Vorob'ev}
In view of the OCPS condition and the theorems of Bass and
Vorob'ev, the proofs for the Bell inequalities as given by Bell
and others become obvious and at the same time lacking physical
justification.
Consider Bell's original proof \cite{bell}. Here Bell assumes that
all random variables $A, B, C$ are in turn functions of a single
random variable $\Lambda$. Then it is clear that $A, B, C$ are
defined on one common probability space and therefore the
inequalities can not be violated by the pair expectation values as
explained above. It is clear that no $\Lambda$ can exist that
leads to a violation of the inequalities for purely mathematical
reasons as already found by Bass much earlier. Bell's physical
justification is wanting because he attempts to show that the
inequalities follow from the fact that $\Lambda$ does not depend
on the settings ${\bf a}, {\bf b},...$. In fact, it does not
matter on what $\Lambda$ depends as long as the resulting $A, B$
and $C$ are random variables defined on one probability space. We
will discuss this in more detail below.
Other well known proofs \cite{leggett} invoke ``counterfactual"
reasoning of the following kind: If, for example, $A$ is measured
given a certain information that we denote by $\lambda$ (a value
that $\Lambda$ assumes in a given experiment) and that is carried
by the correlated spin pair, then one could have measured with
another setting, say $\bf b$ and the same $\lambda$. As we have
explained in more detail previously \cite{hpnp}, it is permissible
to ask the question of what would have been obtained if the
measurement had been performed with a different setting. It is
also permissible to hypothesize the existence of an element of
reality related to that different setting if that different
setting had been chosen. However, to assume then, as is always
done in Bell type proofs, that all these possible different
measurement results are actually contained in the data set of
actual outcomes of the idealized experiment is arbitrary and
against all the rules of modelling and simulation especially for
the particular case of the Aspect-type experiment and all other
known EPR experiments \cite{hpnp}. Naturally, we do not have to
pay for all items on a restaurant's menu just because we could
have chosen them. We call this latter assumption the extended
counterfactual assumption (ECA). ECA is equivalent to the
assumption that $A, B$ and $C$ only depend on one random variable
$\Lambda$ and is therefore an assumption, not a proof. As a
consequence, ECA implies that $A, B$ and $C$ are defined on one
common probability space. In view of the Bass-Vorob'ev theorem it
leads to a contradiction from the outset irrespective and
independent of any physical considerations.
\section{EPR-physics and probability spaces}
A number of physical conditions have been given in the past that
have been thought to be necessary and sufficient for the Bell
inequalities to be valid. Most prominently among these conditions
ranks the definition of an objective local parameter space
\cite{peres}, \cite{leggett}. This definition involves several
conditions that are automatically fulfilled in our Kolmogorovian
model as has been outlined before \cite{hpnp}; it further implies
the existence of elements of reality that contain information
related to the spin (represented by the random variable $\Lambda$)
and, most importantly Einstein locality. Armed with the knowledge
that the validity of the Bell inequalities as described above is
equivalent to the assumption that $A, B, C$ can be defined on one
common probability space, we must now ask the question how this
fact can be related to the condition of an objective local
parameter space i.e. essentially to Einstein locality and the
existence of elements of reality.
We first deal with the question of the relation between the
elements of reality that are ``carried" by the correlated spin
pair and the elements $\omega$ of a probability space. Part of the
work around the Bell theorem concentrates on the question whether
elements of reality that determine (or at least co-determine) the
outcome of the spin measurement can exist. Is not $\omega$ such an
element of reality and do we not assume then its existence to
start with? The answer is that the $\omega$'s represent only a
necessary tool to count and average all measurements correctly.
Whether or not the outcome of a single measurement is the causal
consequence of an element of reality is, at this point, not
discussed. The symbol $\omega$ represents just the choice of the
goddess Tyche (Fortuna) for the given experiment. Of course, if an
element of reality exists, $\omega$ can just represent this
element. The question of whether such elements of reality can
exist in nature and do explain the EPR experiments was, of course,
a subject of the Einstein-Bohr debate and is also subject of our
discussion here. To explore this question using the Bell
inequalities we need to explore whether there exist physical
reasons that demand the definition of $A, B, C$ on one probability
space.
\subsection{Physical reasons for definition on one probability space
for source parameters only}
A very important and broadly applicable physical reason for the
definition of $A, B, C$ on one common probability space arises for
the case in which all random variables are characterized only by
the information emanating from a common source. If in addition
this information is stochastically independent of the settings
(delayed choice arguments), then in line with our notational
convention $A, B, C$ are completely determined by one random
variable $\Lambda$ corresponding to the elements of reality
$\lambda$. These elements of reality can be viewed as the value
the random variable $\Lambda$ assumes for the experiment that we
denoted by $\omega$ i.e. we have the relation $\Lambda(\omega) =
\lambda$. The settings may, of course, also be treated as random
variables and may be defined on a separate probability space.
However, because $\Lambda$ and the settings are stochastically
independent, all random variables can be defined on one common
probability space namely the product space. We have discussed
details of these facts in \cite{hpnp}. Under these conditions the
Bell inequalities will hold and the mathematical model obeying
these conditions is in contradiction to quantum mechanics. We will
show in the next section how this contradiction can be resolved by
still using a classical space-time framework and just adding time
and setting dependent equipment random variables in addition to
the source random variable $\Lambda$. We would like to emphasize,
however, that even though the system consisting of source
parameters only correctly can be ruled out, this fact does not
necessarily have anything to do with Einstein locality. For
example, we can introduce a source parameter represented by a
random variable $\Lambda_1$ that operates only if the settings
${\bf a}, {\bf b}$ and $\bf c$ are employed and $\Lambda_1$
``knows" of these settings by action at a distance. Similarly we
admit a completely different source parameter $\Lambda_2$ that
operates and operates only if the three different settings ${\bf
d}, {\bf e}$ and $\bf f$ are going to be chosen. Again $\Lambda_2$
``knows" of these settings ${\bf d}, {\bf e}$, $\bf f$ by action
at a distance. As long as $\Lambda_1$ is a random variable defined
on some probability space and $\Lambda_2$ is a random variable
defined on some possibly different probability space, the Bell
inequalities formed as before for the settings ${\bf a}, {\bf b}$,
$\bf c$ respectively for ${\bf d}, {\bf e}$,$\bf f$ are valid in
spite of the assumption of action at a distance.
Thus a contradiction exists between the results of quantum
mechanics and the physical assumptions that have just been
described and that appear, on the surface, to be very general.
This contradiction has therefore been explained by some authors
invoking violations of Einstein locality \cite{bellbook}. Others
have given more reasonable, albeit noncommittal, explanations by
postulating that (i) the elements of reality simply do not exist
and/or (ii) there exists a ``contextuality" as discussed in
\cite{peres}. Different contexts of measurements provide then
different probability spaces. There were also other choices to
explain the difficult situation such as (iii) counterfactual
reasoning was held responsible for the difficulties \cite{peres}.
As we have shown, no counterfactual reasoning is necessary to
derive the inequalities and the extended counterfactual reasoning
(ECA) described above is flawed from the viewpoint of mathematical
modelling. We will show in the next section that explanations (i)
and (ii) can, in principle, be reformulated in such a way as to
have a natural explanation in the space-time of relativity. We
note that (i) and (ii) contain in essence Bohr's interpretation:
the spin is determined in the moment of measurement and, with
respect to measurements in any of the two wings of the experiment,
there is essentially the question of ``an influence on the very
conditions which define the possible types of prediction..."
\cite{bohr}.
\subsection{A space-time interpretation of Ma-EPR that agrees
with Bohr in essence}
As the basis for our reasoning in this section, we will assume or
postulate certain properties for the parameters and random
variables of the probability theory that are in harmony with
special relativity. We define with each basic experiment that
corresponds to an element $\omega$ of the probability space a pair
of light-cones corresponding to locations and time at which the
experiments are performed as shown in Fig \ref{fig:LightCones}.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.30\textwidth]{LightCones.eps}
\caption{Light cone figure}
\label{fig:LightCones}
\end{figure}
The elements of reality and the corresponding random variables of
the mathematical model are permitted to be functions of the
space-time coordinates of the respective light-cones. As parameter
random variables we admit not only source parameters but also
equipment parameters for each measurement station. What we
introduce below is a dependence of the equipment parameters of a
given station on the setting vector in the light-cone of that
station and an additional dependence on the time of measurement of
a clock in the inertial frame of the equipment.
All we need to achieve is to derive a model for Ma-EPR within the
space-time of relativity that is not refuted by Bell-type
inequalities and agrees with (i) and (ii) in spirit (if not the
letter). For this it is only necessary to find an Einstein local
model with random variables $A, B, C$ that can not be defined on
one common probability space. To show that this is possible we
revert to the standard notation using the settings as subscripts
and denoting the functions in the two experimental wings by
$A_{\bf a}, A_{\bf b}$ on one side and $B_{\bf b}, B_{\bf c}$ on
the other. We continue to permit all functions to be functions of
a source parameter $\Lambda$ which may have a time dependence e.g.
$\Lambda$ may depend on the time of emission of the correlated
pair. However, we also add equipment random variables. Of course,
equipment parameters have been discussed before in many research
articles. But none of them considered the role of time
dependencies of these equipment parameters except our work (see
\cite{hpnp}). We permit that the probability densities of these
additional random variables depend on the time of measurement
$t_m$ as shown by a local clock and also to depend on the local
setting. We indicate this latter fact by denoting the additional
random variable e.g. for setting $\bf a$ by $\Lambda_{\bf a}(t_m)$
we then have $A_{\bf a} = A_{\bf a}(\Lambda, \Lambda_{\bf
a}(t_m))$ and similar for the other settings and the $B$'s on the
other side e.g. $B_{\bf b} = B_{\bf b}(\Lambda, \Lambda_{\bf
b}(t_m))$. Notice that all light-cones for different measurement
times may contain different $\Lambda_{\bf a}(t_m)$ even though the
setting is the same. No matter how a probabilistic model is
conceived, different light-cones can certainly support different
probability distributions for the elements of reality. Assume now
that as in the Aspect-type experiment the settings on each side
are sequentially changed. Because according to relativity this
change of settings to take place requires a time interval of
length bounded away from 0 by a positive constant $c_0$, all
light-cone pairs of a sequence of measurements are different and
each such experiment may be on a different probability space with
a different density of the involved random variables. Furthermore,
let $\eta$ be an element of a probability space that determines
the random times of measurement i.e. $t_m(\eta)$ is the actual
measurement time of a given experiment. We now show that physical
reasons, derived from the framework of relativity only,
necessitate the involvement of different probability spaces if one
postulates the existence of time and setting dependent Einstein
local equipment parameters.
Theorem2: Assume that there exist a source parameter $\Lambda$ and
equipment parameters $\Lambda_{\bf a}, \Lambda_{\bf b}$ and
$\Lambda_{\bf c}$ such that $\Lambda_{\bf a}, \Lambda_{\bf b}$ and
$\Lambda_{\bf c}$ not only depend on the setting vectors ${\bf a},
\bf b$ and $\bf c$, respectively, but also on the time $t_m$ of a
given measurement. Here we consider $t_m$ to be a random variable
$t_m = t_m(\eta)$. Thus
\begin{equation}
\Lambda_{\bf a} = \Lambda_{\bf a}(t_m(\eta)) \text{ ,
}\Lambda_{\bf b} = \Lambda_{\bf b}(t_m(\eta)) \text{ ,
}\Lambda_{\bf c} = \Lambda_{\bf c}(t_m(\eta)) \label{ccc1}
\end{equation}
The source parameter $\Lambda = \Lambda(\omega)$ is permitted to
depend on emission time. We assume further that the random
variables corresponding to the measurements of spin $A_{\bf a},
A_{\bf b}, B_{\bf b}$ and $B_{\bf c}$ all equal to $\pm 1$ are
functions of the source parameter and of the equipment parameters
$\Lambda_{\bf a}, \Lambda_{\bf b}$ and $\Lambda_{\bf c}$. Then,
under the assumption that the velocity of light in vacuo is an
upper limit for the velocities by which the settings can be
changed, there is no probability space on which all of
\begin{equation}
A_{\bf a} = A_{\bf a}(\Lambda(\omega), \Lambda_{\bf a}(t_m(\eta))
\nonumber
\end{equation}
\begin{equation}
A_{\bf b} = A_{\bf b}(\Lambda(\omega), \Lambda_{\bf b}(t_m(\eta))
\nonumber
\end{equation}
\begin{equation}
B_{\bf b} = B_{\bf b}(\Lambda(\omega), \Lambda_{\bf b}(t_m(\eta))
\nonumber
\end{equation}
\begin{equation}
B_{\bf c} = B_{\bf c}(\Lambda(\omega), \Lambda_{\bf c}(t_m(\eta))
\label{ccc2}
\end{equation}
can be consistently defined.
Proof: Let $I$ be any time interval of length $|I| \leq \frac {1}
{2} c_0$. Let $M$ be a measurable set in the range of $\Lambda$
and let $F, G$ and $H$ be sets in the ranges of $\Lambda_{\bf a}$,
$\Lambda_{\bf b}$ and $\Lambda_{\bf c}$ respectively. Then
\begin{equation}
[(\omega, \eta): t_m(\eta) \in I, \Lambda(\omega) \in M,
\Lambda_{\bf a}(t_m(\eta)) \in F, \Lambda_{\bf b}(t_m(\eta)) \in
G, \Lambda_{\bf c}(t_m(\eta)) \in H] \label{bvcc13}
\end{equation}
is the impossible event and therefore has probability 0. Recall
that $\omega$ signifies the sending out of a particular particle
pair from the source. This result simply reflects the
impossibility in the space-time of relativity to accomplish two
different settings on both sides within the same short time
interval and all for the same $\omega$. Hence for each time
interval $I$ each of the sixteen probabilities
\begin{equation}
P[(\omega, \eta): t_m(\eta) \in I, A_{\bf a}(\cdot) = \pm 1,
A_{\bf b}(\cdot) = \pm 1, B_{\bf b}(\cdot) = \pm 1, B_{\bf
c}(\cdot) = \pm 1] = 0 \label{bvcc15}
\end{equation}
must vanish. Here $(\cdot)$ denotes the dependence on source and
equipment parameters that in turn depend on $\omega$ and $\eta$
respectively just as in Eq.(\ref{ccc2}). Now let $J$ be a finite
but arbitrarily long time interval. Then $J$ can be split up into
a large but finite number $N$ of intervals $I_i, i = 1,2,...,N$
with length $|I_i| \leq \frac {1} {2} c_0$. Then the probability
in Eq.(\ref{bvcc15}) with $I$ replaced by $J$ also must vanish
because of finite additivity and thus $A_{\bf a}, A_{\bf b},
B_{\bf b}$ and $B_{\bf c}$ cannot be defined on a common
probability space as claimed.
In other words, not only must we have different probability spaces
involved in the Aspect-type experiment for mathematical reasons,
we must have different probability spaces for physical reasons,
the requirements of relativity. We emphasize that none of the
assumptions in the above proof imply any synchronization of the
measurement times with certain settings. Both settings and
measurement times can be chosen randomly, only the measurement
times in $S_1$ and $S_2$ are correlated for any given photon pair.
Note that the essence of Bohr's discussion is not violated by the
above. We just need to view both spin and measurement equipment in
the sense of information theory: the measurement outcome is really
not the single consequence of the source information $\lambda$
that characterizes particle properties but also that of the
measurement equipment and the corresponding $\lambda_{\bf a}(t_m)$
etc.. These equipment parameters correspond to the use of decoding
machines in information theory \cite{shannon}. Both the source
information content together with that of the decoding machines or
equipment parameters (that involve different probability spaces)
determine the measurement outcomes i.e. the values $\pm 1$ that
the functions $A_{\bf a}$ etc. assume. In a larger sense this
fulfills the spirit of Bohr. The spin does not really exist before
the measurement, but only in the very moment of measurement is the
outcome determined (decoded) and can not be separated from the
equipment and act of measurement. The contextuality is implicitly
contained in the dependence of the probability densities of the
various variables on measurement time. For example, it is now
incorrect to say that it makes no difference if one measures with
setting $\bf b$ or setting $\bf c$ on the other side. It does make
a difference because one necessarily makes these different
measurements during different time intervals. The measurements in
both wings are also performed at the same clock-time or at least
at correlated clock-times which opens the possibility of
correlations between the two wings even though the settings are
randomly chosen.
What is the meaning then of the Aspect et al. \cite{eprex}
experiment in view of the above discussions? If one assumes that
this experiment is free from any problems related to non-ideal
experimental conditions and if one assumes that a space-time
explanation must be possible then the Aspect et al. experiment has
proven the existence of setting and time dependent equipment
parameters.
\section{Conclusion}
We have shown that the inequalities of Bell can be derived as
special cases of a more general theorem found by Bass ten years
earlier. We have further shown that the Bell inequalities are
valid if and only if the three random variables involved can
actually be defined on a common probability space. As a
consequence the Bell theorem is correct at least for the following
systems of hidden variables, in the sense that these systems can
be ruled out:
\begin{enumerate}
\item{} Source parameter $\Lambda$ only,
\item{} Source parameter $\Lambda$ and equipment parameters $\Lambda_{\bf a}$,
$\Lambda_{\bf b}$ and $\Lambda_{\bf c}$ that depend only on the
respective settings.
\end{enumerate}
On the other hand, equipment parameters that depend on the
measurement times as well as on the respective instrument settings
can not be ruled out. A space-time explanation of the Aspect et
al. experiment is therefore not ruled out by Bell's inequalities.
Any such space-time explanation can not rely on source parameters
only but must involve a certain type of time and setting dependent
equipment parameter random variables. Thus, the validity of the
Bell inequalities for objective local parameter spaces has not
been proven by any of the proofs reported in the literature
\cite{bellbook}, \cite{leggett}, \cite{peres}.
\section{Acknowledgement}
The authors would like to thank M. Aschwanden for creating the
figures of the manuscript and helpful suggestions. Support of the
Office of Naval Research (N00014-98-1-0604) is gratefully
acknowledged.
|
{
"timestamp": "2005-03-03T22:01:05",
"yymm": "0503",
"arxiv_id": "quant-ph/0503044",
"language": "en",
"url": "https://arxiv.org/abs/quant-ph/0503044"
}
|
\section{Introduction}
To obtain information about a system, a measurement has to be made.
Based on the results of this measurement we assign to the
system our state of knowledge. For a classical system this state
takes the form of a probability distribution $P(x',t)$, while for a
quantum system we have a state matrix $\rho(t)$. \footnote{Here we are not
concerned with where the division between classical systems and
quantum systems occurs. Instead we recognize that both
descriptions are valid and the system dynamics determine which is
appropriate.} In this paper we are concerned with efficient
simulation techniques for {\em partly} observed systems; that is,
systems for which the observer cannot obtain enough information to
assign the system a pure state, $P(x',t)=\delta[x'-x(t)]$ or
$\rho(t)=\ket{\psi(t)}\bra{\psi(t)}$.
The chief motivation for wishing to know the conditional state of a system
is for the purpose of feedback control \cite{Jac93,Bel87,WisMil94,DohJac99}. That is because
for cost functions that are additive in time, the optimal basis for controlling the
system is the observer's state of knowledge about the system. Even if such a control strategy
is too difficult to implement in practice, it plays the important role of bounding the
performance of any strategy, which helps in seeking the best practical strategy.
It is well known that the quantum state of an open quantum system,
given continuous-in-time measurements of the bath, follows a
stochastic trajectory through time \cite{BelSta92}. In the quantum
optics community this is referred to as a quantum trajectory
\cite{Car93a,GarParZol92,MolCasDal93,WisMil93a,GoeGra93,GoeGra94,Wis96,GamWis01,WisDio01,GarZol00}.
The form of this trajectory can be either jump-like in nature or
diffusive depending on how we choose to measure the system; that
is, the arrangement of the measuring apparatus. In this paper we
review quantum trajectory theory for partially observed systems by
presenting a simple model: A three level atom that emits into two
separate environments, only one of which is accessible to our detectors. Such partially
observed systems cannot be described by a stochastic Schr\"odinger~~equation (SSE)
\cite{Car93a, GarParZol92,MolCasDal93}, but rather requires a more general form
of a quantum trajectory that has been called a stochastic master equation (SME)
\cite{WisMil93a}. This is an instance of the fact that the most general
form of quantum measurement theory requires the full Kraus
representation of operations \cite{Kra83,BraKha92}, rather than just
measurement operators
\cite{BraKha92}.
It is also well known that if we have a classical system and we
make measurements on it with a measurement apparatus that has
associated with it a Gaussian noise, then the evolution of this
classical state in the continuous-in-time limit obeys a
Kushner-Stratonovich equation (KSE) \cite{Mcg74}. To review these
dynamics for partially observed classical systems we present the
KSE for a system that experiences an `internal' unobservable white
noise process. That is, the evolution in the absence of the
measurements is given by a Fokker-plank equation \cite{Gar85}. This is the
classical analogue to the quantum master equation.
The new work in this paper is a simple numerical technique that
allows us to reduce the numerical resources required to calculate
the continuous-in-time trajectories. This method relies on the
implementation of linear or `ostensible' \cite{Wis96} measurement theory,
classical \cite{Mcg74} and quantum
\cite{GoeGra93,GoeGra94,Wis96,GamWis01}. For the classical case
our method reduces the problem from solving the KSE for the
probability distribution to simulating the ensemble average of two
coupled stochastic differential equations (SDE). For the quantum
case our method reduces the problem from solving a conditional
SME to simulating the ensemble average of a
SSE plus a c-number SDE. Thus in both
the classical and the quantum case, our method reduces the size of
the problem by a factor of $N,$ the number of basis states
required to represent the system.
Recently Brun and Goan \cite{Brun} have used a similar
idea to investigate a partially observed quantum system.
However, since they did not use measurement theory with ostensible
probabilities, their claim that they can generate a typical trajectory
conditioned on some partial record ${\bf R}$ is not valid.
This is demonstrated in detail in \ref{AppendixQuant}. (In their
method, the record ${\bf R}$ can only be generated randomly, and
can be found only by doing the ensemble average over the fictitious noise,
but that is not the issue of concern here.)
Finally, we combine these theories to consider the following case:
a quantum system is monitored continuously in time by a
classical system but we only have access to the results of
non-ideal measurements performed on the classical system. Note
that such joint systems have recently been studied by Warszawski
{\em et al} \cite{WarWisMab01,WarWis03a,WarGamWis04} and
Oxtoby {\em et al} \cite{Neil}. Warszawski {\em et al} considered
continuous-in-time monitoring of a quantum optical system with
realistic photodections while Oxtoby {\em et al} considered
continuous-in-time monitoring of a quantum solid-state system with
a quantum point contact. We show that our ostensible
numerical technique can be applied to these types of systems,
greatly simplifying the simulations.
The format of this paper is as follows. In Secs.~\ref{measQ} and
\ref{measC} we review quantum and classical measurement theory respectively.
This is essential as it allows us to define both the notations and
the physical insight that will be used throughout this paper. In
Secs.~\ref{quantum}, \ref{class}, and \ref{both} we investigate
the above mentioned quantum, classical and joint systems respectively,
and present our ostensible numerical technique for each specific
case. Finally in Sec.~\ref{con} we conclude with a discussion.
\section{Quantum Measurement theory (QMT)}\label{measQ}
\subsection{General theory}
In quantum mechanics the most general way we can represent the
state of the system is via a state matrix $\rho(t)$. This is a
positive semi-definite operator that acts in the system Hilbert
space ${\cal H}_{\rm s}$. In this paper we take the view that this
represents our state of knowledge of the system. Taking this view
allows us to simply interpret the ``collapse of the
wavefunction'', upon measurement, as an update in the observer's
knowledge of the system \cite{CavFucSch02,Fuc02}. If we now assume
that we have a measurement apparatus that allows us to measure
observable $R$ of the system, then the conditional state $\rho_{r}(t')$ of the
system given result $r$ is determined by
\cite{Kra83}
\begin{equation}\label{QuantumUpdate}
\rho_{r}(t')=\frac{\hat{\cal O}_r(t',t)\rho(t)}
{P(r,t')},
\end{equation} where $P(r,t')$ is the probability of getting result
$r$ at time $t'=t+T$, where $T$ is the measurement duration time. Here
$\hat{\cal O}_r(t',t)$ is known as the operation of the
measurement and is a completely positive superoperator and for efficient measurements
can be defined by
\begin{equation}\label{OperationDef}
\hat{\cal O}_r(t',t)\rho(t)=\hat{\cal
J}[\hat{M}_{r}(T)]\rho(t)=\hat{M}_{r}(T)\rho(t)\hat{M}_{r}^\dagger(T),
\end{equation} where $\hat{M}_r(T)$ is called a measurement operator.
The probability of getting result $r$ is given by
\begin{eqnarray}\label{QuantumProb}
P(r,t')={\rm Tr}[\hat{\cal O}_r(t',t)\rho(t)]={\rm
Tr}[\hat{F}_r(T)\rho(t)],
\end{eqnarray} where the set $\{\hat{F}_r(T)=\hat{M}_r^\dagger(T)\hat{M}_r(T)\}$ is the positive
operator measure (POM) for observable $R$. By completeness, the
sum of all the POM elements satisfies
\begin{equation}\label{QuantumMeas}
\sum_r \hat{F}_r(T)=\hat{1}.
\end{equation}
So far we have only considered efficient, or purity-preserving measurements.
That is if $\rho(t)$ was initially $\ket{\psi(t)}\bra{\psi(t)}$
then the state after the measurement would also be of this form.
In a more general theory we must dispense with the measurement
operator $\hat{M}_r(T)$ and define the Kraus operator
$\hat{K}_{r,f}(T)$ \cite{Kra83}. This has the effect of changing
the definition of the operation of the measurement $\hat{\cal
O}_r(t',t)$ [\erf{OperationDef}] to
\begin{equation}\label{OperationDefComplete}
\hat{\cal O}_r(t',t)=\sum_f\hat{\cal
J}[\hat{K}_{r,f}(T)],
\end{equation} and the POM elements for this measurement are now given by
\begin{equation} \label{KrausEffec}
\hat{F}_{r}(T)=\sum_f\hat{K}_{r,f}^\dagger(T)\hat{K}_{r,f}(T).
\end{equation} Note $\hat{F}_{r}(T)$ still satisfies the completeness condition
[\erf{QuantumMeas}]. We can think of $f$ as labelling results of fictitious measurement.
If one is only interested in the average evolution of the system,
this can be found via
\begin{equation} \label{AverageO}
\rho(t')=\sum_r P(r)\rho_r(t')=\hat{\cal O}(t',t)\rho(t),
\end{equation}
where $\hat{{\cal O}}(t',t)=\sum_r\hat{\cal O}_r(t',t)$ is the
non-selective operation.
\subsection{Quantum trajectory theory}
Quantum trajectory theory is simply quantum measurement theory
applied to a continuous in-time monitored system
\cite{Car93a,GarParZol92,WisMil93a,MolCasDal93,GoeGra93,GoeGra94,Wis96,GamWis01,WisDio01,GarZol00}.
In continuous monitoring, repeated measurements of duration $T=d
t$ are performed on the system. This results in the state being
conditioned on a record ${\bf R}}%_{(0,t]}$, which is a string containing the
results $r_k$ of each measurement from time 0 to $t$ but not
including time 0. Here the subscript $k$ refers to a measurement
completed at time $t_{k}=k d t$. From the record ${\bf R}}%_{(0,t]}$, the conditioned
state at time $t$ can be written as
\begin{equation}
\rho_{\bf R}(t)=\frac{\tilde\rho_{\bf R}(t)}{{P}({\bf R}}%_{(0,t]})},
\end{equation}
where $\tilde\rho_{\bf R}(t)$ is an unnormalized state defined by
\begin{equation}
\tilde\rho_{\bf R}(t)=\hat{\cal
O}_{r_{k}}(t_k,t_{k-1})\ldots\hat{\cal O}_{r_2}(t_2,t_1)\hat{\cal
O}_{r_1}(t_1,0)\rho(0).
\end{equation}
The probability of observing the record ${\bf R}}%_{(0,t]}$ is
\begin{equation}
P({\bf R}}%_{(0,t]})={\rm Tr}[\tilde\rho_{{\bf R}}%_{(0,t]}}(t)].
\end{equation}
If we now assume that the coupling between the apparatus (bath)
and the system is Markovian then the average state
\begin{eqnarray}
\rho(t)=\hat{\cal O}(t_k,t_{k-1})\ldots\hat{\cal
O}(t_2,t_1)\hat{\cal O}(t_1,0)\rho(0)
\end{eqnarray} is equivalent to the reduced state
\begin{equation}
\rho_{\rm red}(t)={\rm Tr}_{\rm bath}[\ket{\Psi(t)}\bra{\Psi(t)}],
\end{equation} which itself obeys the Master equation \cite{Lin76}
\begin{equation}\label{QuantumMaster}
\dot{\rho}(t)=\hat{\cal L}\rho(t)=-i[\hat{H},\rho(t)]+\sum_j\gamma_j\hat{\cal
D}[\hat{L}_j]\rho(t).
\end{equation} Here $\hat{\cal
D}[\hat{A}]$ is the superoperator defined by
\begin{equation}\label{DampSuper}
\hat{\cal
D}[\hat{A}]\rho=\hat{A}\rho\hat{A}^\dagger-\hat{A}^\dagger\hat{A}\rho/2-\rho\hat{A}^\dagger\hat{A}/2,
\end{equation} and represents dissipation of information about the system into the
baths.
\subsection{Fictitious quantum trajectories: the ostensible
numerical technique} \label{OstensibleQ}
If the system is only partly observed ($f$
in \erf{OperationDefComplete} represents the unobservable processes)
this state will be mixed. This is not a problem for simple systems but for a
large system a numerical simulation for $\rho_{{\bf R}}(t)$ would be impractical. This brings
us to the goal of this section which is to demonstrate that $\rho_{{\bf R}}(t)$ can be
numerically simulated by using SSEs, requiring less space to
store on a computer.
To do this we assume that a fictitious measurement with record
${\bf F}}%_{(0,t]}$ is actually made on the unobservable process. Then we can
expand $\rho_{{\bf R}}(t)$ to
\begin{equation}\label{eq29}
\rho_{{\bf R}}(t)=\sum_{{\bf F}}%_{(0,t]}}\rho_{{\bf R, F}}(t){ P}({\bf F}}%_{(0,t]}|{\bf R}}%_{(0,t]}),
\end{equation}
where
\begin{equation}\label{eq30}
\rho_{{\bf R, F}}(t)=\ket{{\psi}_{{\bf R, F}}(t)}\bra{{\psi}_{{\bf R, F}}(t)}.
\end{equation} Here $\ket{{\psi}_{{\bf R, F}}(t)}$ is a normalised state conditioned on both ${\bf F}}%_{(0,t]}$ and
${\bf R}}%_{(0,t]}$. In quantum trajectory theory this is defined as
\begin{equation}\label{eq31}
\ket{{\psi}_{{\bf R, F}}(t)}=\frac{\ket{\tilde{\psi}_{{\bf R, F}}(t)}}{\rt{{P}({\bf F}}%_{(0,t]},{\bf R}}%_{(0,t]})}},
\end{equation} where
\begin{equation}\label{eq32}
\ket{\tilde{\psi}_{{\bf R, F}}(t)}=\hat{M}_{r_{k},f_{k}}(dt)....\hat{M}_{r_{1},f_{1}}(dt)\ket{\psi(0)}.
\end{equation} Here $r_{k}$ and $f_{k}$ are the results of the measurement
operator
\begin{equation}\label{eq33}
\hat{M}_{r_k,f_k}(dt)=\bra{r_k}\bra{f_k}\hat{U}(t_k,t_{k-1})\ket{0}\ket{0},
\end{equation} where $\ket{0}\ket{0}$ is the initial bath state.
This indicates that given that we have a real
record ${\bf R}}%_{(0,t]}$ we can calculate $\rho_{{\bf R}}(t)$ from averaging over an
ensemble of pure states $\ket{{\psi}_{{\bf R, F}}(t)}$. But as shown in
\ref{AppendixQuant} the fact that future real results are not
necessarily independent from the current fictitious results means that
we cannot generate single trajectories without knowing the full
solution. However by using quantum measurement theory with
ostensible distributions we can get around this problem.
Under ostensible quantum trajectory theory
\cite{GoeGra94,Wis96,GamWis01} we can define a state, $\ket{\bar{\psi}_{{\bf R, F}}(t)}$ as,
\begin{equation}\label{eq38}
\ket{\bar{\psi}_{{\bf R, F}}(t)}=\frac{\ket{\tilde{\psi}_{{\bf R, F}}(t)}}{\rt{\Lambda({\bf F}}%_{(0,t]}, {\bf R}}%_{(0,t]})}},
\end{equation} where $\Lambda({\bf F}}%_{(0,t]}, {\bf R}}%_{(0,t]})$ is an ostensible probability distribution.
This is simply a guessed distribution that only has the requirement that
it be a probability distribution and be non-zero when $P({\bf F}}%_{(0,t]},
{\bf R}}%_{(0,t]})$ is non-zero. Note this state is no longer normalized to one
and this is why we signify it with the bar. The true probability can be
related to the ostensible probability by
\begin{equation}\label{eq40}
{P}({\bf R}}%_{(0,t]},{\bf F}}%_{(0,t]})=\bra{\bar{\psi}_{{\bf R, F}}(t)}{{\bar{\psi}_{{\bf R, F
}}(t)}}\rangle\Lambda({\bf F}}%_{(0,t]}, {\bf R}}%_{(0,t]}),
\end{equation} which is a generalized Girsanov transformation
\cite{BelSta92,GoeGra94,Wis96,GamWis01,GatGis91}.
Going back to \erf{eq29} and using the above equations we can
write $\rho_{{\bf R}}(t)$ as
\begin{equation}\label{eq42}
\rho_{{\bf R}}(t)=\frac{\sum_{{\bf F}}%_{(0,t]}}\ket{\bar{\psi}_{{\bf R, F}}(t)}\bra{\bar{\psi}_{{\bf R, F}}(t)}\Lambda({\bf F}}%_{(0,t]}, {\bf R}}%_{(0,t]})}{{
P}({\bf R}}%_{(0,t]})},
\end{equation}
where
\begin{equation}\label{eq43}
{P}({\bf R}}%_{(0,t]})=\sum_{{\bf F}}%_{(0,t]}}\langle\bar{\psi}_{{\bf R}, {\bf F}}(t) \ket{\bar{\psi}_{{\bf R, F}}(t)} \Lambda({\bf F}}%_{(0,t]},{\bf R}}%_{(0,t]}).
\end{equation}
Note that the sum containing $\Lambda({\bf F}}%_{(0,t]}, {\bf R}}%_{(0,t]})$ in the above equations
simply represents the ensemble average over all possible
fictitious records. Thus we can rewrite \erf{eq42} as
\begin{equation}\label{MainEquation}
\rho_{{\bf R}}(t)=\frac{{\rm E}_{\bf F} \Big{[}\ket{\bar{\psi}_{{\bf R, F}}(t)}\bra{\bar{\psi}_{{\bf R, F}}(t)}\Big{]}}
{{\rm E}_{\bf F}\Big{[}\langle\bar{\psi}_{{\bf R}, {\bf F}}(t) \ket{\bar{\psi}_{{\bf R, F}}(t)}\Big{]}}.
\end{equation}
\section{Classical Measurement theory (CMT)} \label{measC}
\subsection{General theory}
In this paper when considering what we call a classical system,
we are referring to a system that can be described by the
probability distribution $P(x,t)$ (i.e a vector of probabilities)
rather than a state matrix. That is, with respect to a fixed basis
$x$ the coherences (off diagonal elements) are always zero. If
we now measure observable $R$ of the system, then after a
measurement which yielded result $r$, the state of the system is
given by \cite{Bayes}
\begin{equation}\label{BayesTheorem}
P_r(x,t)=\frac{P(r,t|x,t)P(x,t)}{P(r,t)},
\end{equation}
where
\begin{equation}\label{Pr}
P(r,t)=\int dx P(r,t|x,t)P(x,t).
\end{equation} This is known as Bayes' theorem. Here $ P_r(x,t)\equiv P(x,t|r,t)$ is called
a conditional state and represents our new state of knowledge
given that we observed result $r$. Here we have only considered
minimally disturbing classical measurements. That is, there is no
back action acting on the system in the measurement process. To
generalize Bayes' theorem to deal with measurements which incur
back action we mathematically split the measurement into a two
stage process. The first is the Bayesian update, followed by a
second stage described by $ B_r(x',t'|x,t)$, the probability for
the measurement to cause the system to make a transition from $x$
at time $t$ to $x'$ at time $t'=t+T$, given the result $r$. Thus
for all $x'$, $x$ and $r$
\begin{eqnarray}\label{Bpropeties1}
B_r(x',t'|x,t)&\geq& 0,\\
\int dx' B_r(x',t'|x,t)&=& 1\label{Bpropeties}.
\end{eqnarray} Now by defining the operation
\begin{eqnarray}
{\cal O}_r(x',t'|x,t)&=& B_r(x',t'|x,t)P(r,t|x,t)
\end{eqnarray}
the conditional system state after the measurement becomes
\begin{equation}\label{GeneralBayessTheorem}
P_r(x',t')=\frac{ \int dx {\cal O}_r(x',t'|x,t)P(x,t)}{P(r,t')},
\end{equation} where
\begin{eqnarray}\label{Pr2}
P(r,t')&=&\int dx'\int dx {\cal O}_r(x',t'|x,t) P(x,t).
\end{eqnarray} Using \erf{Bpropeties} this can be rewritten as
\begin{eqnarray}\label{Pr3}
P(r,t')&=&\int dx F_r(x,t) P(x,t),
\end{eqnarray}where $F_r(x,t)=P(r,t|x,t)$,
which by definition satisfies
\begin{equation}
\sum_r F_r(x,t)=1,
\end{equation} is the classical analogue of the POM element.
The average evolution of the system is given by
\begin{eqnarray}\label{Ave}
P(x',t')&=&\sum_{r}P_r(x',t')P(r,t')\nonumber\\&=&\int dx {\cal
O}(x',t'|x,t)P(x,t),
\end{eqnarray}
where ${\cal O}(x',t'|x,t)=\sum_r{\cal O}_r(x',t'|x,t)$ is the non-selective operation.
Note that for any $B_r(x',t'|x,t)$ that satisfies
\erfs{Bpropeties1}{Bpropeties} we can rewrite it as
\begin{equation}\label{Bansatz}
B_r(x',t'|x,t)=\sum_f \delta [x'-x_{r,f}(t')]P(f,t'|x,t;r,t),
\end{equation} where $x_{r,f}(t')$ is the new system configuration $x'$ at time $t'$
given the measurement result $r$ and
extra noise $f$ (the stochastic part of the back action). The parameter $f$ is
analogous to the fictitious measurement results in the quantum case. Thus the
operation for the measurement can be written as
\begin{eqnarray}\label{orf}
{\cal O}_r(x',t'|x,t)&=& \sum_f \delta [x'-x_{r,f}(t')]P(f,t';r,t|x,t),\nonumber\\ &=&\sum_f
{\cal J}_{r,f}(x',t'|x,t).
\end{eqnarray} This is the classical equivalent of \erf{OperationDefComplete}.
In the above we have purposely structured QMT and CMT so that the
theories appear to be similar and as a general rule we will push
this point of view throughout the rest of this paper. However, it
is important to point out the key differences between these
theories. In the quantum case we can always write the measurement
operator (or Kraus operator) as
$\hat{M}_r=\hat{U}_r\rt{\hat{F}_r}$ where $U_r$ is a unitary
operator. That is we can always interpret
a measurement as a two stage process, where $\rt{\hat{F}_r}$ is
responsible for the wavefunction collapse and the gain in
information by the observer and $\hat{U}_r$ is some extra
evolution that entails no information gain (as the entropy of
the system is not changed by this evolution). It simply adds
surplus back action to the system. In the classical case we can
also write the measurement as a two stage process. However, the
first process by definition has no back action; it is simply the
update in the observer's knowledge of the system. Furthermore the
second stage is not necessary unitary evolution (and as such can
change the entropy of the system). Thus back action in the quantum
and classical case are physically different processes and one can
not separate all the back action in the quantum case from the
observer's information gain. Mathematically speaking, the
difference arises from the fact that a quantum state is
represented by a positive {\em matrix}, the state matrix, while a
classical state is represented by a positive {\em vector}, the vector of
probabilities.
\subsection{Classical trajectory theory}
To achieve continuous-in-time measurements theory for a classical
system we simply let the measurement time tend to $dt$ and extend
the number of consecutive measurements to $t/dt$. Then the state
of the classical system given the measurement record ${\bf R}}%_{(0,t]}$ is
\begin{equation}
P_{{\bf R}}%_{(0,t]}}(x,t)=\frac{\tilde{P}_{{\bf R}}%_{(0,t]}}(x,t)}{{P}({\bf R}}%_{(0,t]})},
\end{equation}
where $\tilde{P}_{{\bf R}}%_{(0,t]}}(x,t)$ is an unnormalized state defined by
\begin{eqnarray}
&&\hspace{-.8cm} \tilde{P}_{{\bf R}}%_{(0,t]}}(x,t)=\int dx_{k-1}...\int
dx_1\int dx_0 \nn \\ &&\times{\cal O}_{r_{k}}(x,t |x_{k-1},t_{k-1})
\ldots {\cal O}_{r_{2}}(x_2,t_2 |x_1,t_1) \nn \\ &&\times{\cal
O}_{r_{1}}(x_1,t_1|x_0,0)P(x_0,0).
\end{eqnarray}
The probability of observing the record ${\bf R}}%_{(0,t]}$ is
\begin{equation}
P({\bf R}}%_{(0,t]})=\int dx \tilde{P}_{{\bf R}}%_{(0,t]}}(x,t).
\end{equation}
If we now assume that the noise added by the measurement apparatus
is white, and the form of the back action is independent of the
results ${\bf R}$, then the unconditional state
\begin{eqnarray}
&&\hspace{-.8cm} {P}(x,t)= \int dx_{k-1}...\int dx_1\int dx_0
\nn \\ &&\times{\cal O}(x,t |x_{k-1},t_{k-1})
\ldots {\cal O}(x_2,t_2 |x_1,t_1) \nn \\ &&\times{\cal
O}(x_1,t_1|x_0,0)P(x_0,0)
\end{eqnarray}
is the solution of the Fokker Plank Equation
\cite{Gar85}
\begin{equation}\label{Eq.FPE2}
\partial_t
P(x,t)=-\partial_{x}[A(x,t)
P(x,t)]+\smallfrac{1}{2}\partial^2_{x}[
D^2(x,t)P(x,t)],
\end{equation} where $A(x,t)$ determines the amount of drift and $D(x,t)$ determines the amount of
diffusion.
\subsection{Fictitious classical trajectories: The ostensible numerical
technique}\label{sec.fitcla}
The basic principle behind this technique is that we assume that
the unobservable process, ${\bf F}}%_{(0,t]}$, that generates the back action part of the measurement is
fictitiously simulated. To be more
specific we can define
\begin{eqnarray}\label{ptilde}
&&\hspace{-.8cm} \tilde{P}_{{\bf R, F}}%_{(0,t]}}(x,t)=\int dx_{k-1}...\int
dx_1\int dx_0 \nn \\ &&\times{\cal J}_{r_{k},f_{k}}(x,t
|x_{k-1},t_{k-1})
\ldots {\cal J}_{r_{2},f_{2}}(x_2,t_2 |x_1,t_1) \nn \\ &&\times{\cal
J}_{r_{1},f_{1}}(x_1,t_1|x_0,0)P(x_0,0).
\end{eqnarray}
where ${\cal J}_{r,f}(x',t'|x,t)$ is defined implicitly in
\erf{orf}. From this the conditional state, $P_{{\bf R}}%_{(0,t]}}(x,t)$, is
given by
\begin{equation}\label{conclass}
{P}_{{\bf R}}%_{(0,t]}}(x,t)=
\sum_{{\bf F}}%_{(0,t]}}{P}_{{\bf R, F}}%_{(0,t]}}(x,t){{P}({{\bf F}}%_{(0,t]}|{\bf R}}%_{(0,t]}})},
\end{equation} where
\begin{equation}
{P}_{{\bf R, F}}%_{(0,t]}}(x,t)=\frac{\tilde{P}_{{\bf R, F}}%_{(0,t]}}(x,t)}{{P}({{\bf R}}%_{(0,t]},{\bf F}}%_{(0,t]}})}.
\end{equation} But as in the quantum case this cannot be directly
calculated and as a result we must use an ostensible theory. We
define the ostensible state by
\begin{equation}\label{pbar}
\bar{P}_{{\bf R, F}}%_{(0,t]}}(x,t)=\frac{\tilde{P}_{{\bf R, F}}%_{(0,t]}}(x,t)}{{\Lambda}({{\bf R}}%_{(0,t]},{\bf F}}%_{(0,t]}})},
\end{equation} and the true probability can be related to the
ostensible by
\begin{equation}
{{P}({{\bf R}}%_{(0,t]},{\bf F}}%_{(0,t]}})}=\int dx
\bar{P}_{{\bf R, F}}%_{(0,t]}}(x,t){{\Lambda}({{\bf R}}%_{(0,t]},{\bf F}}%_{(0,t]}})},
\end{equation} the classical Girsanov transformation.
Using the above we can rewrite \erf{conclass} as
\begin{equation}\label{Pxgrvialin}
{P}_{{\bf R}}%_{(0,t]}}(x,t)=
\frac{\sum_{{\bf F}}%_{(0,t]}}\bar{P}_{{\bf R, F}}%_{(0,t]}}(x,t){{\Lambda}({{\bf F}}%_{(0,t]},{\bf R}}%_{(0,t]}})}}{P({\bf R}}%_{(0,t]})},
\end{equation} where
\begin{equation}
{P({\bf R}}%_{(0,t]})}=\sum_{{\bf F}}%_{(0,t]}}\int dx
\bar{P}_{{\bf R, F}}%_{(0,t]}}(x,t){{\Lambda}({{\bf F}}%_{(0,t]},{\bf R}}%_{(0,t]}})}.
\end{equation} As in the quantum case we can rewrite \erf{Pxgrvialin} as
\begin{equation}\label{EPxgrvialin}
{P}_{{\bf R}}%_{(0,t]}}(x,t)=
\frac{{\rm E}_{\bf F}\Big{[}\bar{P}_{{\bf R, F}}%_{(0,t]}}(x,t)\Big{]}}{{\rm E}_{\bf F}\Big{[}\int dx\bar{P}_{{\bf R, F}}%_{(0,t]}}(x,t)\Big{]}
},
\end{equation}
where $\bar{P}_{{\bf R}}%_{(0,t]},{\bf F}}%_{(0,t]}}(x,t)$ is an unnormalized pure classical
state. That is, it is of the form
$\bar{P}(x,t)=p_{{\bf R, F}}%_{(0,t]}}\delta[x-x_{{\bf R, F}}%_{(0,t]}}(t)]$, where $p_{{\bf R, F}}%_{(0,t]}}$
is the norm of the ostensible state. To show this we consider a
system initially in the state $\bar{P}(x,0)=p\delta(x-x_0)$ then
by using \erfs{pbar}{ptilde} with ${\cal
J}_{r_{1},f_{1}}(x',t_1|x,0)$ defined implicitly in \erf{orf} we can rewrite $\bar{P}_{r_1,f_1}(x',t_1)$ as
\begin{equation}
\bar{P}_{r_1,f_1}(x',t_1)=p_{r_1,f_1}(t_1)\delta
[x'-x_{f_1,r_1}(t_1)],
\end{equation} which is still of the $\delta$-function form. Here
$p_{r_1,f_1}(t_1)$ is given by
\begin{equation}\label{pdifff}
p_{r_1,f_1}(t_1) = P(f_1,t_1;r_1,0|x_0,0)p(0)/\Lambda({r_1,f_1}),
\end{equation} and $x_{f_1,r_1}(t_1)$ is determined by the underlying
dynamics. That is, we can simulate the distribution by solving
the two coupled SDEs, $\dot{x}_{\bf R, F}(t)$ and $\dot{{p}}_{\bf
R, F}(t)$.
\section{A Quantum system with an unobserved
process}\label{quantum}
To illustrate a quantum system where a complete measurement can
not be performed, due to some physical constraint, the system in
Fig.~\ref{fig1} was considered. This system is a three level atom
with lowering operators $\hat{L}_1=\ket{1}\bra{3}$ and
$\hat{L}_2=\ket{2}\bra{3}$, and decay rates $\gamma_{1}$ and
$\gamma_{2}$ respectively.
\begin{figure}\begin{center}
\includegraphics[width=0.25\textwidth]{PartlyObservedEvolutionFig01}
\caption{\label{fig1} A simple system (a three level atom) which
has two outputs due to the to lowering operators $\hat{L}_1$ and
$\hat{L}_2$.}\end{center}
\end{figure}
\subsection{Master equation}
With no external driving [$\hat{H}=0$ in \erf{QuantumMaster}], the
solution of the master equation can be determined analytically. To
illustrate a non-trivial solution we calculated this solution for
the initial condition
$\ket{\psi(0)}=0.4123\ket{1}+0.1\ket{2}+(0.9+0.1i)\ket{3}$ and
coupling constants $\gamma_{1}=0.5$ and $\gamma_{2}=1$. This is
shown in Fig.~\ref{fig2}. In this figure it is observed that as
time goes on, the state becomes mixed. This is seen as the purity
$p(t)={\rm Tr}[\rho^2(t)]$ of the state decays (although not monotonically)
as time increases. This figure also shows that the state becomes a mixture of the two
ground states, with the ground state associated with the larger
coupling constant being weighted more heavily, even though it
started with less weight.
\begin{figure}[t]\begin{center}
\includegraphics[width=0.45\textwidth]{PartlyObservedEvolutionFig02}
\vspace{0.2cm} \caption{\label{fig2} The solution to the master
equation. The first subplot shows $\rho_{33}(t)$ (solid line),
$\rho_{22}(t)$ (dashed line) and $\rho_{11}(t)$ (dotted line). The
second and third subplot show the real and imaginary parts
respectively of $\rho_{12}(t)$ (solid line), $\rho_{31}(t)$
(dashed line) and $\rho_{32}(t)$ (dotted line). The fourth subplot
illustrates the purity of this state. This is all for the initial
condition
$\ket{\psi(0)}=0.4123\ket{1}+0.1\ket{2}+(0.9+0.1i)\ket{3}$ and
$\gamma_{1}=0.5$ and $\gamma_{2}=1$.}\end{center}
\end{figure}
\subsection{Conditional evolution: The quantum trajectory} \label{Incomplete}
In this section we consider the trajectory $\rho_{{\bf R}}%_{(0,t]}}(t)$ which
occurs when output $\hat{L}_1$ is monitored using homodyne-$x$
detection and output $\hat{L}_2$ is un-monitored. A schematic of
this measurement process is shown in Fig.~\ref{fig3}. Because this
arrangement is an inefficient measurement we have to use the
operation defined in \erf{OperationDefComplete}. To determine the
Kraus operators we need to present the underlying dynamics in more
detail. For the interaction of this system with a Markovian bath
(and under the rotating wave approximation and in the interaction
frame) the total Hamiltonian is
\begin{eqnarray}\label{eq7}
H(t)&=&i\hbar\rt{\gamma_{1}}\int
\delta(t-t')[\hat{L}_1\hat{b}_r^\dagger(t')-\hat{L}_1^\dagger\hat{b}_r(t')]dt'
\nn \\ &&+i\hbar\rt{\gamma_{2}}\int
\delta(t-t')[\hat{L}_2\hat{b}_f^\dagger(t')-\hat{L}_2^\dagger\hat{b}_f(t')]dt'.\nonumber\\
\end{eqnarray} Here $\hat{b}_r(t)$ and $\hat{b}_f(t)$ are the
temporal-mode annihilation operators for the detected
($\hat{b}_r$) and non-detected ($\hat{b}_f$) fields (baths). Since
these fields are Markovian there will be a commutator relationship
for the field of the following form
\begin{equation}\label{eq8}
[{\hat{b}_{i}(t),\hat{b}_{j}^\dagger(s)}]=\delta(t-s)\delta_{i,j},
\end{equation} where $i$, $j$ denotes either of the two baths.
This indicates that the field operators are gaussian white noise
operators. Thus they obey It\^o~ calculus and the infinitesimal
evolution operator is \cite{GarParZol92,GarZol00}
\begin{eqnarray}\label{U}
\hat{U}(t+dt,t)&=&\exp\Big{\{}\rt{\gamma_{1}}[\hat{L}_1d\hat{B}_r^\dagger(t)-\hat{L}_1^\dagger
d\hat{B}_r(t)]\nn \\ && +\rt{\gamma_{2}}
[\hat{L}_2d\hat{B}_f^\dagger(t)-\hat{L}_2^\dagger d\hat{B}_f(t)]\Big{\}},
\end{eqnarray} where $d\hat{B}_i$ satisfies the commutator
relation
\begin{equation}
[d\hat{B}(t)_{i},d\hat{B}_{j}^\dagger(t)]=dt\delta_{i,j}.
\end{equation} Thus $\hat{U}(t+dt,t)$ is an operator acting in the Hilbert
space ${\cal H}_{s}\otimes {\cal H}_{r}\otimes {\cal H}_{f}$,
where ${\cal H}_{s}$, ${\cal H}_{r}$ and ${\cal H}_{f}$ are the
Hilbert spaces for the system, detected field and non detected
field respectively.
\begin{figure}\begin{center}
\includegraphics[width=0.4\textwidth]{PartlyObservedEvolutionFig03}
\vspace{0.2cm} \caption{\label{fig3} A schematic representing
homodyne measurement of one of the outputs of the three level
atom. In an ordinary homodyne measurement the signal is coupled to
a classical local oscillator (LO) via a low reflective beam
splitter and then detected using a photoreceiver.}\end{center}
\end{figure}
Now, given that a projective measurement is made on bath field
$\hat{b}_r(t)$ and bath field $\hat{b}_f(t)$ is completely
unobserved the state of the system after this measurement (time
$dt$ later) is given by \erfs{QuantumUpdate}{OperationDefComplete}
with $T=dt$, and the Kraus operator is
\begin{equation}\label{Kraus2}
\hat{K}_{r,f}(dt)=\bra{f}_f\bra{r}_r\hat{U}(t+dt,t)\ket{0}_{r}\ket{0}_f.
\end{equation} Here $\{\ket{r}_r\}$ is the set of orthogonal states the bath
is projected into, while $\{\ket{f}_f\}$ is any arbitrary
orthogonal basis set. For a homodyne-$x$ measurement of bath
$\hat{b}_r(t)$ the set $\{\ket{r}_r\}$ corresponds to the eigenset
of the operator $d\hat{B}_{r}(t)+d\hat{B}_{r}^\dagger(t)$
\cite{GoeGra94} and the results $r$ are the corresponding
eigenvalues. Note we have assumed that initially the baths, for
all the temporal-modes, are in the vacuum state.
After some simple rearrangement and using $(rdt)^2=dt$, the POM
elements for this measurement are of the form
\begin{equation}
\hat{F}_{r}(dt)=|\bra{r}{0}\rangle_{r}|^2[1+\rt{\gamma_{1}}r(t+dt)dt
\hat{x}_1],
\end{equation} where $\hat{x}_1=\hat{L}_1+\hat{L}_1^\dagger$. Thus
\begin{eqnarray}\label{eq20}
{ P}(r,t+dt)&=&|\bra{r}{0}\rangle|^2[1+rdt\rt{\gamma_{1}}\langle \hat{x}_1\rangle_t],
\end{eqnarray} where $\langle \hat{x}_1\rangle_t={\rm Tr}[\hat{x}_1\rho(t)]$.
Using the fact that $\ket{r}$ is a temporal-quadrature state,
\begin{equation}\label{eq22}
|\bra{r}{0}\rangle_{r}|^2=\rt{\frac{dt}{2\pi}}\exp\Big{(}-\frac{r^{2}}{2/dt}\Big{)},
\end{equation} we can rearrange this to
\begin{equation}\label{eq23}
{ P}(r,t+dt)=\rt{\frac{dt}{2\pi}}\exp\Big{[}-\frac{[r-\rt{\gamma_{1}}\langle \hat{x}_1\rangle_t]^{2}}{2/dt}\Big{]}.
\end{equation} This implies that the random variable associated with this distribution, $r(t+dt)dt$, is a gaussian
random variable (GRV) of mean $\rt{\gamma_{1}}\langle
\hat{x}_1\rangle_t dt$ and variance $dt$. That is,
\begin{equation}\label{eq24}
r(t+dt) dt=dW(t)+dt\rt{\gamma_{1}}\langle \hat{x}_1\rangle_t,
\end{equation} where $dW(t)$ is a Wiener increment \cite{Gar85}.
Using the above and \erfs{QuantumUpdate}{OperationDefComplete} the
stochastic master equation for this system is
\begin{eqnarray}\label{eq16}
d\rho_{\bf R}(t+dt)&&=dt\Big{(}\gamma_{2}{\cal D}[\hat{L}_2]
+\gamma_{1}{\cal D}[\hat{L}_1]
\nn \\ &&+dW(t)\rt{\gamma_{1}}{\cal
H}[\hat{L}_1]/dt\Big{)}\rho_{\bf R}(t),\nonumber\\
\end{eqnarray} where ${\cal H}[\hat{A}]$ is the superoperator
\begin{equation}\label{eq17}
{\cal H}[\hat{A}]\rho=\hat{A}\rho+\rho\hat{A}^\dagger -{\rm
Tr}[\hat{A}\rho+\rho\hat{A}^\dagger]\rho.
\end{equation}
To illustrate an example quantum trajectory, \erf{eq16} was solved
for a randomly chosen record ${\bf R}}%_{(0,t]}$ and the same parameters used
in Fig. $\ref{fig2}$. This is shown in Fig. \ref{fig4}. It is
observed that this state evolution is stochastic in time and
becomes mixed (but not as mixed as the average evolution). It is
interesting to note that by performing this measurement the
coherence $\rho_{12, {\bf R}}(t)$, which was a constant of motion
for the average state, becomes comparable to the other coherence
and does not decay with time.
\begin{figure}[t]\begin{center}
\includegraphics[width=0.45\textwidth]{PartlyObservedEvolutionFig04}
\vspace{0.2cm} \caption{\label{fig4} The solution to $\rho_{{\bf R}}$
written in matrix elements. The first subplot shows $\rho_{33,{\bf
R}}(t)$ (solid line), $\rho_{22,{\bf R}}(t)$ (dashed line),
$\rho_{11,{\bf R}}(t)$ (dotted line). The second and third subplot
show the real and imaginary parts respectively of $\rho_{12,{\bf
R}}(t)$ (solid line), $\rho_{31,{\bf R}}(t)$ (dashed line) and
$\rho_{32,{\bf R}}(t)$ (dotted line). The fourth subplot
illustrates the purity. We have used the same parameters as in
Fig. \ref{fig2}}\end{center}
\end{figure}
\subsection{The ostensible numerical technique}
In Sec.~\ref{OstensibleQ} we observed that the conditional
evolution of a partly monitored system could be simulated by
assuming that fictitious measurements are made on the unobservable
process.
For this system we assume that a fictitious homodyne-$x$ measurement is made on
output $\hat{L}_2$. Note we could have chosen any unraveling for ${\bf F}}%_{(0,t]}$.
To determine the SSE for the ostensible state $\ket{\bar{\psi}_{{\bf R, F}}(t)}$ [the state
which we substitute into \erf{MainEquation} to determine the
actual conditional evolution] we have to derive the measurement
operator for the combined real and fictitious measurements, as well
as make a convenient choice for $\Lambda({\bf F}}%_{(0,t]}, {\bf R}}%_{(0,t]})$. Using
\erf{eq33} and the fact that we are performing homodyne-$x$
measurements the measurement operator is
\begin{eqnarray}\label{eq34}
\hat{M}_{r,f}(dt)&=&\bra{f}0\rangle\bra{r}0\rangle\Big{(} 1+\rt{\gamma_{1}}rdt\hat{L}_1+\rt{\gamma_{2}}fdt\hat{L}_2\nn \\ &&\hspace{-1cm}-\gamma_{1}dt
\hat{L}_1^\dagger\hat{L}_1/2-\gamma_{2}dt
\hat{L}_2^\dagger\hat{L}_2/2\Big{)},
\end{eqnarray} where the bath states $\ket{f}$ and $\ket{r}$ are
temporal quadrature states acting in Hilbert spaces ${\cal H}_f$
and ${\cal H}_r$ respectively. To derive this we have expanded \erf{U} to first order in $dt$ and used the fact that
$(fdt)^2=(rdt)^2=dt$. Since the real distribution is Gaussian
(with a variance $1/dt$) a convenient choice for $\Lambda({\bf F}}%_{(0,t]},
{\bf R}}%_{(0,t]})$ is $\Lambda({\bf F}}%_{(0,t]})\Lambda({\bf R}}%_{(0,t]})$ where
$\Lambda({\bf F}}%_{(0,t]})=\Lambda(f_k)\ldots\Lambda(f_1)$ and
$\Lambda({\bf R}}%_{(0,t]})=\Lambda(r_k)\ldots\Lambda(r_1)$ with
\begin{eqnarray}\label{o1}
\Lambda(r) &=& \rt{\frac{dt}{2\pi}}\exp\Big{[}-\frac{(r-\lambda)^{2}}{2/dt}\Big{]} \\
\Lambda(f) &=&
\rt{\frac{dt}{2\pi}}\exp\Big{[}-\frac{(f-\mu)^2}{2/dt}\Big{]}\label{ostenFit}.
\end{eqnarray} Here $\lambda$ and $\mu$ are arbitrary parameters.
With these ostensible distributions, \erf{eq34}, and \erf{eq38},
the ostensible SSE is
\begin{eqnarray}\label{eq50}
d\ket{\bar{\psi}_{\bf R, F}(t)}&=&dt
\Big{(}[r-\lambda](\rt{\gamma_{1}}\hat{L}_1-\lambda/2)+ [f-\mu]\nn \\ &&\times(\rt{\gamma_{1}}\hat{L}_2-\mu/2)
-\smallfrac{1}{2} [\gamma_{1}\hat{L}_1^\dagger\hat{L}_1+\gamma_{2}\hat{L}_2^\dagger\hat{L}_2
\nn \\ &&-\rt{\gamma_{1}}\lambda\hat{L}_1-\rt{\gamma_{2}}\mu\hat{L}_2+\lambda^2/4+\mu^2/4]
\Big{)}
\nn \\ &&\times
\ket{\bar{\psi}_{\bf R, F}(t)}.
\end{eqnarray}
Now since we are interested in calculating $\rho_{{\bf R}}(t)$ based on an
assumed known real record ${\bf R}}%_{(0,t]}$, we can rewrite \erf{eq50} as
\begin{eqnarray}\label{eq52}
dc_{1}&=&
c_{3}[\rt{\gamma_{1}}(r-\lambda)dt+dt\lambda/2]-c_1[\rt{\gamma_2}
d{\cal
W}\mu\nn \\ &&+\rt{\gamma_1}(r-\lambda)dt\lambda+dt\lambda^2/4+dt\mu^2/4]/2,\\
dc_{2}&=&c_{3}[\rt{\gamma_{2}}d{\cal
W}+dt\mu/2]-c_2[\rt{\gamma_2} d{\cal
W}\mu\nn \\ &&+\rt{\gamma_1}(r-\lambda)dt\lambda+dt\lambda^2/4+dt\mu^2/4]/2\\
dc_{3}&=&c_3[-\gamma dt+\rt{\gamma_2} d{\cal
W}\mu+\rt{\gamma_1}(r-\lambda)dt\lambda\nn \\ &&+dt\lambda^2/4+dt\mu^2/4]/2,
\end{eqnarray} where $\gamma=\gamma_1+\gamma_2$. Here we have used the identity
\begin{equation}\label{eq47}
\ket{\bar{\psi}(t)}=c_{1}\ket{1}+c_{2}\ket{2}+c_{3}\ket{3},
\end{equation} and replaced
$fdt$ with $d{\cal W}(t)+\mu dt$, where $d{\cal W}(t)$
is a Wiener increment.
To illustrate the convergence of our method the ensemble average
of the above ostensible SSE for $\lambda=\mu=0$ was calculated for
$n=10$ and $n=1000$. To quantify how closely the ensemble method
reproduces $\rho_{{\bf R}}%_{(0,t]}}(t)$ we used the fidelity measure, which
for two different quantum states is defined as
\begin{equation}\label{felquant}
F^{\rm (Q)}(t)={\rm Tr}[\rt{\rt{\rho_1(t)}\rho_2(t)\rt{\rho_1(t)}}].
\end{equation} Note this measure ranges from 0 to 1 with 0
indicating two orthogonal states and 1 indicating the same state.
The result of this measure for the actual $\rho_{{\bf R}}%_{(0,t]}}(t)$ and the
ensemble version are shown in part A of figure \ref{felquantf}.
Here we see that for larger ensemble size the fidelity is closer
to one, indicating that as we increase the ensemble size our
ostensible method approaches the actual $\rho_{{\bf R}}%_{(0,t]}}(t)$.
To illustrate the effect of choosing different ostensible
distributions we considered the case when $\lambda=0$ and
\begin{equation}\label{eq422}
\mu=\rt{\gamma_2}\frac{\bra{\bar{\psi}_{\bf R,
F}(t)}\hat{L}_2+\hat{L}_2^\dagger \ket{\bar{\psi}_{\bf R,
F}(t)}}{\bra{\bar{\psi}_{\bf R, F}(t)}\bar{\psi}_{\bf R,
F}(t)\rangle}.
\end{equation} That is, the ostensible probability for the $k^{th}$ fictitious
results is the true probability we would expect based on the past
real and fictitious results up to, but not including the time
$kdt$. The motivation for this choice is that with $\mu=0$, the improbable
trajectories, ones that tend towards being inconsistent with the full real
record, will have norms that are very small and as such have little
contribution to the ensemble average. By contrast, using \erf{eq422}, the improbable
trajectories are less likely to be generated, so avoiding useless simulations.
With this ostensible distribution
the fidelity measure was calculated for $n=10$ and $n=1000$. These
results are shown in part $B$ of figure \ref{felquantf}. Here we
see that for the smaller ensemble size the fidelity is closer to
one than that observed using the first ostensible case. This
indicates that the rate of convergence for this case is greater
than the $\lambda=\mu=0$ case.
\begin{figure}[t]\begin{center}
\includegraphics[width=0.45\textwidth]{PartlyObservedEvolutionFig05}
\vspace{0.2cm} \caption{\label{felquantf} This figure shows the
fidelity between the actual $\rho_{{\bf R}}%_{(0,t]}}(t)$ and our ensemble
method for ensembles sizes 10 (dotted) and 1000 (solid). Part $A$
corresponds to a linear ostensible distribution while part $B$
refers to the non-linear ostensible distribution. The same
parameters were used as in Fig. \ref{fig2}.}\end{center}
\end{figure}
\section{A Classical system with an internal unobserved process}\label{class}
In this section we consider continuous-in-time measurements with
Gaussian precision of a classical system driven by an unobservable
noise process. This for example could correspond to a measurement
of the voltage across a resistor that is driven by a noisy classical
current.
\subsection{The average evolution}
We restrict ourselves to unconditional state evolution
described by the Fokker
Plank equation \erf{Eq.FPE2}. This equation has as its solution a
distribution that diffuses and drifts though time. Using Eq.
(\ref{Ave}) and only considering one interval in time we can write
\begin{equation}
P(x',t+dt)=\int dx {\cal O}(x',t+dt|x,t) P(x,t),
\end{equation} which when compared to \erf{Eq.FPE2} implies that
RHS of the above equation equals
\begin{eqnarray}\label{Eq.FPE3}
&&\hspace{-.8cm} \int dx [1 -dt\partial_{x'}A(x,t)
+dt\partial^2_{{x'}}D^2(x,t)/2]\delta(x'-x) \nn \\ &&\times P(x,t).
\end{eqnarray} By introducing an arbitrary Gaussian distribution $P(f,t+dt)$ with mean
$m(t)$ and variance $1/dt$, that is
\begin{equation}\label{RealFclass}
P(f,t+dt)=
\rt{\frac{dt}{2\pi}}\exp\Big{[}-\frac{[f-m(t)]^{2}}{2/dt}\Big{]},
\end{equation} \erf{Eq.FPE3} can be rewritten as
\begin{eqnarray}\label{Eq.FPE4}
&&\hspace{-.8cm} \int df P(f,t+dt) \int dx [1 -dt\partial_{x'}A(x,t)
-dt [f-m(t)]\nn \\ &&\times \partial_{x'}D_f(x,t)+dt\partial^2_{{x'}}D^2(x,t)/2]\delta(x'-x)
P(x,t).\nonumber\\
\end{eqnarray} By using It\^o~ calculus and a Taylor expansion this can be
rewritten as
\begin{eqnarray}\label{Eq.FPE6}
&&\hspace{-.8cm} \int dx {\rm E}_f \Big{\{}\delta[x'-x-dt
A(x,t)- dt[f(t+dt)-m(t)] \nn \\ &&\times D(x,t)]\Big{\}}P(x,t).
\end{eqnarray}
where $f(t+dt)dt=m(t)dt +d {\cal W}(t)$. Thus
\begin{eqnarray}\label{OO}
{\cal O}(x',t+dt|x,t)={\rm E}_f
\Big{\{}\delta[x'-x_f(t+dt)]\Big{\}},
\end{eqnarray} where $x_f(t+dt)$ is determined by the following SDE
\begin{equation} \label{Eq.Linear1}
d x_{\bf F}(t)=dt
A[x_{\bf F}(t),t]+ dt[f(t+dt)-m(t)]D[x_{\bf F}(t),t].
\end{equation} Note here we have written the SDE for the complete
record ${\bf F}$.
\subsection{Conditional evolution: The Kushner-Stratonovich
equation}\label{KSEsec}
To derive the KSE we start by deriving ${\cal O}_r(x',t'|x,t)$ and
$P(r,t+dt)$. For the case when the classical measurement has a
back action that is independent of the result $r(t+dt)$, the
operation for the measurement is given by
\begin{equation} {\cal
O}_r(x',t'|x,t)={\cal
O}(x',t'|x,t)P(r,t|x,t),
\end{equation} where $ {\cal O}(x',t'|x,t)=B(x',t'|x,t)$.
Thus to derive ${\cal O}_r(x',t'|x,t)$ we need only $P(r,t|x,t)$.
For a measurement that has a precision limited by Gaussian white
noise it follows that
\begin{equation}\label{Eq.PIgivenx}
P(r,t|x,t)=F_r(x,t)=\frac{\sqrt{dt}}{\sqrt{2\pi \beta}}\exp[-(r-x)^2
dt/2\beta],
\end{equation} where $\beta$ is a constant characterizing the classical measurement strength.
To find $P(r,t)$ we substitute \erf{Eq.PIgivenx} into \erf{Pr3}.
This gives
\begin{equation}\label{Eq.PI}
P(r,t+dt)=\int dx\frac{\sqrt{dt}}{\sqrt{2\pi
\beta }}\exp[-(r-x)^2 dt/2\beta]P(x,t).
\end{equation} After some simple stochastic algebra and using $r^2=\beta/dt$ this
can be simplified to \cite{WarWis03a}
\begin{equation}\label{Eq.PI5}
P(r,t+dt)
=\frac{\sqrt{dt}}{\sqrt{2\pi
\beta}} \exp[-(r-\langle x\rangle_t)^2dt/2\beta ],
\end{equation} where for the classical system $\langle x\rangle_t=\int
x P(x,t)dx$. From \erf{Eq.PI5} the stochastic representation of
$r(t+dt)$ is a Gaussian random variable with mean $\langle
x\rangle_t$ and variance $\beta dt$. That is,
\begin{equation}\label{Eq.I}
r(t+dt)=\langle x\rangle_t+\sqrt{\beta}dW(t)/dt.
\end{equation}
With all the above information and \erf{GeneralBayessTheorem} the
conditional state at time $t'=t+dt$ is
\begin{eqnarray}
&& \hspace{-0.8cm} P_r(x',t+dt)=\int dx {\rm E}_f \Big{\{}\delta[x'-x_f(t+dt)]\Big{\}}
\Big{\{}1\nn \\ &&+[x-\langle x\rangle_t]
[r-\langle x\rangle_t]dt/\beta\Big{\}}P(x,t).
\end{eqnarray}
Here we have expanded the exponentials in \erf{Eq.PI5} and
\erf{Eq.PIgivenx} to second order in $dt$ and used $r^2=\beta/dt$.
Taylor expanding the delta function and averaging over the
$f(t+dt)$ [using \erf{RealFclass}] for each step in time gives the
KSE
\begin{eqnarray} \label{KS}
P_{\bf R}(x,t+dt)&=&P_{\bf R}(x,t)+dt[x-\langle
x\rangle_t][r(t+dt)-\langle x\rangle_t]\nn \\ &&\times P_{\bf
R}(x,t)/\beta -dt\partial_x [{A}({x},t)P_{\bf
R}(x,t)]\nn \\ &&+\smallfrac{1}{2} dt\partial^2_x [D^2({x},t)P_{\bf
R}(x,t)]
\end{eqnarray} and $\langle x\rangle_t$ becomes $\int
x P_{\bf R}(x,t)dx$. In general to
solve this equation we need to solve for all $x$. For some
${A}({x},t)$ and $D(x,t)$ this can be a rather lengthy numerical
problem. In the following section we will present our ostensible
technique which allows us to reformulate the problem to solving
two coupled SDEs, at the cost of performing an ensemble average.
\subsection{The ostensible numerical
technique}\label{sec.linearclass}
As shown in Sec. \ref{sec.fitcla}, if we consider the
unobservable process ${\bf F}$ as actually occurring then we can
simulate the KSE by using \erf{EPxgrvialin}, and
$\bar{P}_{{\bf R}}%_{(0,t]},{\bf F}}%_{(0,t]}}(x,t)$ is determined by solving two coupled
SDEs. For the case when the classical measurement has Gaussian
precision and the back action only depends on the white noise
process $f(t)$, we can rewrite $P(f,t';r,t|x,t)$ in \erf{pdifff} as
$P(f,t')P(r,t|x,t)$ where $P(f,t')$ is given by \erf{RealFclass}
and $P(r,t|x,t)$ is given by \erf{Eq.PIgivenx}. Thus $\dot{x}_{\bf
R,F}(t)$ becomes $\dot{x}_{\bf F}(t)$ and is given by
\erf{Eq.Linear1}. To find the differential equation for
$\dot{p}_{\bf R, F}(t)$ we need to assume a form for the
ostensible distribution
$\Lambda(f,r)$. We use $\Lambda(f,r)=\Lambda(f)\Lambda(r)$, where
\begin{eqnarray}\label{ostenRit}
\Lambda(r) &=& \rt{\frac{dt}{2\pi}}\exp\Big{[}-\frac{(r-\lambda)^{2}}{2\beta/dt}\Big{]}
\end{eqnarray} and $\Lambda(f)$ is given by \erf{ostenFit}. Extending \erf{pdifff} to continuous measurements gives
\begin{eqnarray}\label{Eq.Linear2}
d p_{\bf R, F}(t)&=&
dt[m(t)-\mu][f(t+dt)-\mu]p_{\bf R, F}(t)\nn \\ &&+dt[x(t)-\lambda][r(t+dt)-\lambda]p_{\bf R,
F}(t)/\beta.\nonumber\\
\end{eqnarray}
Thus to determine $\bar{P}_{\bf R, F}(x,t)$ we only need to
simulate \erfs{Eq.Linear1}{Eq.Linear2} with ${\bf R}$ assumed
known and $f(t+dt)$ given by \erf{ostenFit}. $P_{\bf R}(x,t)$ is
then determined by \erf{Pxgrvialin}. Since the theory
requires $\bar{P}_{\bf R, F}(x,t)$ to be a delta function, one might
conclude that this method is only valid for initial conditions of
the form ${P}(x,0)\rho(0)=\delta(x-x_0)$. Infact, we are not limited to this case.
To consider other initial conditions we simply choose the initial
value $x_0$ in \erf{Eq.Linear1} from the distribution ${P}(x,0)$.
\subsection{A simple example}
To illustrate the classical theory we consider a Gaussian
measurement of a classical system that is driven by an an
unobservable white noise process with $m(t)=0$ and drift and
diffusion functions given by \begin{eqnarray} \label{cond1}
A({x},t) &=& -kx+l, \\
D({x},t) &=& b.\label{cond2}
\end{eqnarray} If this is the case then
$P_{{\bf R}}%_{(0,t]}}(x,t)$ has a Gaussian solution with a mean $\langle
x_{\bf R}\rangle_t$ and variance $\nu_{\bf R}(t)$ given by
\begin{eqnarray}
d\langle x_{\bf R}\rangle_{t}&=&dt\{\nu_{\bf R}(t)[r(t+dt)-\langle x_{\bf R}\rangle_t
]/\beta- k \langle x_{\bf R}\rangle_t\nn \\ &&+l\},\label{Eq.bKS22}\\
\label{Eq.vKS22}
d\nu_{\bf R}(t)&=&dt[-\nu_{\bf R}^2(t)/\beta-2k\nu_{\bf R}(t)+b^2],
\end{eqnarray} and $r(t+dt)=\langle x_{\bf R}\rangle_t+dW(t)$. That is,
as time increases the measurement has
the effect of reducing the variance but the diffusive coefficient
$b$ causes this variance to increase. The mean, however contains
both the deterministic evolution and a random term due the
measurement. To illustrate this solution we have
simulated \erfs{Eq.bKS22}{Eq.vKS22} for the case when $ A(x,t) = 1-x$,
$D=1$ and $\beta=1$. The results of this simulation are shown in
Fig. \ref{Fig.PxGivenIEquations} as a solid line. Here we see that
the mean follows some stochastic path conditioned on the record
${\bf R}}%_{(0,t]}$, while the variance follows a smooth function.
\begin{figure}\begin{center}
\includegraphics[width=0.45\textwidth]{PartlyObservedEvolutionFig06}
\caption{\label{Fig.PxGivenIEquations} The mean and variance of
${P}_{{\bf R}}%_{(0,t]}}(x,t)$ when $\beta= 1$, $A=1-x$ and $D=1$ for both
${P}_{{\bf R}}%_{(0,t]}}(x,t)$ calculated exactly (solid) and via the linear
method for an ensemble size of 10 000 (dotted). }\end{center}
\end{figure}
To illustrate our ostensible method we use the above record and
solve numerically \erfs{Eq.Linear1}{Eq.Linear2} with
$\lambda=\mu=0$. The mean
and variance is then found via
\begin{eqnarray}
\langle x_{\bf R}\rangle_{t}&=&\frac{{\rm E}_{\bf F}\Big{[}{x}_{\bf
F}(t)p_{\bf R, F}(t)\Big{]}}{
{\rm E}_{\bf F}\Big{[}p_{\bf R, F}(t)\Big{]}} \\
\nu_{\bf R}(t)&=&\frac{{\rm E}_{\bf F}\Big{[}x_{\bf F}^2(t)p_{\bf R,
F}(t)\Big{]}}{
{\rm E}_{\bf F}\Big{[}p_{\bf R, F}(t)\Big{]}}-\langle x_{\bf R}\rangle^2_{t}\nonumber\\
\end{eqnarray}
where ${\rm E}_{\bf F}$ denotes an ensemble average over all
possible fictitious records. The numerical values for the mean and
variance are shown in Fig. \ref{Fig.PxGivenIEquations} (dotted)
for an ensemble size of 10 000. To get an indication of the
numerical error in the solution from our method, the difference from
the exact solution is shown in Fig. \ref{Fig.EPxGivenIF1000}. The
dotted line corresponds to an ensemble of 100 and the solid to one
of 10 000. Here we see that the ostensible method solution agrees well with
the exact solution and as we increase the ensemble size the
difference between these solutions decreases. To get a better
indication of how well our method reproduces the actual
${P}_{{\bf R}}%_{(0,t]}}(x,t)$, we also calculated the classical fidelity, which is defined by
\begin{equation} \label{felclass}
F^{(C)}(t)=\int dx \rt{P_1(x,t)}\rt{P_2(x,t)}.
\end{equation} This was calculated under the assumption that
the state calculated via the ostensible
method was also Gaussian. This is illustrated in Fig.
\ref{Fig.EPxGivenIF1000}, where we see that for the larger
ensemble the fidelity is very close to one, implying that the
distributions are almost identical.
\begin{figure}\begin{center}
\includegraphics[width=0.45\textwidth]{PartlyObservedEvolutionFig07}
\caption{\label{Fig.EPxGivenIF1000} The first and second plot show the difference between the
mean and variance of ${P}_{{\bf R}}%_{(0,t]}}(x,t)$ calculated by the linear
method and the know result for ensemble sizes 100 (dotted) and 10
000 (solid). The third plot shows the Fidelity between ${P}_{{\bf R}}%_{(0,t]}}(x,t)$ calculated by the linear
method and the know result for ensemble sizes 100 (dotted) and 10
000 (solid). The parameters are the same as in Fig.
\ref{Fig.PxGivenIEquations}.}\end{center}
\end{figure}
\section{An unobservable quantum system driving a Classical
system}\label{both}
In this section we consider the following situation: a quantum
system is monitored continuously in time by a classical system. This
in turn is measured with Gaussian precision, and these are the only results
to which we have
access. This for example occurs when the signal
from the quantum system enters a detector with a bandwidth $B$,
resulting in the state of the detector being related to ${\bf F}}%_{(0,t]}$ by
\cite{WarWis03a}
\begin{equation}\label{pollysyst}
x(t)=\int_{-\infty}^{t} ds B\exp[-B(t-s)]{f}(s).
\end{equation} Thus in a measurement that reveals $x(t)$ with perfect
precession [eg $F_r(x)=P(r,t|x,t)=\delta(r-x)$] we could determine ${\bf F}}%_{(0,t]}$
(the quantum signal) by inverting the convolution in \erf{pollysyst}. But if this
measurement has Gaussian precision [\erf{Eq.PIgivenx}] then we
must treat the state of the detector as a classical probability
distribution and use a mixture of CMT and QMT to describe the
conditional state of the supersystem (classical and quantum
system). To denote the supersystem we use the
notation $\rho(x,t)$, where $x$ refers to the classical
configuration space and $\rho$ denotes an object acting on a
Hilbert space. This has the interpretation whereby $P(x,t)={\rm Tr}[\rho(x,t)]$ is the
(marginal) classical state and $\rho(t)=\int \rho(x,t)dx$ is the (reduced) quantum state.
For uncorrelated quantum and classical states, $\rho(x,t)=P(x,t)\rho(t)$.
\subsection{Conditional evolution} \label{ConSec}
We denote the state of the supersystem conditioned on the
classical result $r$ at time $t+dt$ as $\rho_{r}(x,t+dt)$. Assuming that the quantum
system is not affected by the classical system, this
can be expanded as
\begin{equation}\label{s}
\rho_{r}(x,t+dt)=\sum_{f}P_r(f,t+dt)P_{r,f}(x,t+dt)\rho_f(t+dt),
\end{equation} where $\rho_f(t+dt)$ is the state that an observer who had access to all the
quantum information would ascribe to the quantum system. That is,
$f(t+dt)$ can be regarded as really existing (with the collapse of the wavefunction
occurring at this level); it is just that the real observer does not
have access to this information. The state of knowledge of this real observer
is different from, but consistent with, that of the hypothetical
observer who has access to {\bf F}.
In terms of the operation of
the measurement, the conditional state can be written as
\begin{eqnarray}\label{ss}
\rho_{r}(x',t+dt)&=&\frac{\tilde{\rho}_{r}(x',t+dt)}{P(r,t+dt)},
\end{eqnarray} where
\begin{eqnarray}\label{sss}
\tilde{\rho}_{r}(x',t+dt)&=&\int dx
\sum_{f} {\cal J}_{r,f}(x',t+dt|x,t) \nn \\ &&\times\hat{\cal O}_f(t+dt,t)
\rho(x,t)/P(f,t+dt) \hspace{.8cm}
\end{eqnarray} and
\begin{eqnarray}\label{ssss}
P(r,t+dt)&=&\int dx' {\rm Tr}\Big{[}\tilde{\rho}_{r}(x',t+dt) \Big{]}.
\end{eqnarray} The quantum part of the operation of measurement in defined by
\erf{OperationDefComplete} and the classical part is defined in
\erf{orf} with the replacement of $P(f,t';r,t|x,t)\rightarrow P(f,t')P(r,t|x,t)$ because in
this system the quantum signal does not depend on the classical state.
To illustrate the above we consider the case when we are
monitoring with Gaussian precision the classical system defined by
\erf{pollysyst} which is in turn monitoring the $x$
quadrature flux coming from a classically driven two level atom (TLA).
This is the same as the system considered in Ref \cite{WarWis03a}
and as such we will simply list the important equations. The
quantum part of operation is given by $\hat{\cal
O}_f(t+dt,t)=\hat{\cal J}[\hat{M}_f(dt)]$ where
\begin{equation}\label{m}
\hat{M}_f(dt)=\bra{f}0\rangle[1-dt(i\hat{H}-\rt{\gamma}f\hat{\sigma}+\gamma
\hat{\sigma}^\dagger\hat{\sigma}/2)].
\end{equation} The fictitious quantum signal statistic obeys
\begin{equation}
P(f,t+dt)=\int dx {\rm Tr}[\hat{\cal O}_f(t+dt,t)\rho(x,t)],
\end{equation} which for a homodyne-$x$ measurement
can be shown to be of the form displayed
in \erf{RealFclass} with $m(t)=\rt{\gamma}{\rm Tr} [(\hat
\sigma+\hat\sigma^\dagger)\rho(t)]$. Here $\hat{\sigma}$ is the
lowering operator for the TLA and $\gamma$ is the decay rate. Note
here we have assumed all the quantum signal is fed into the
classical system, if we wanted to simulate some inefficiency we
would simply use the Kraus represention, and for the case
where this inefficiency is a constant, $\eta$, we simply replace
$\sigma$ in the above equations by $\rt{\eta}\sigma$.
As shown in Sec. \ref{class} for a classical measurement with
Gaussian precision and a back action that does not depend on the
results of the measurement, the classical part of the operation is
\begin{equation}\label{p}
{\cal
J}_{r,f}(x',t+dt|x,t)=\delta[x'-x_f(t+dt)]P(f,t+dt)P(r,t|x,t),
\end{equation} where $P(r,t|x,t)$ is defined in \erf{Eq.PIgivenx}
and $x_f(t+dt)$ is given by \erf{Eq.Linear1}. For the system we
are considering, to find $A(x,t)$ and $D(x,t)$ we simply
differentiate \erf{pollysyst} and equate this with
\erf{Eq.Linear1}. Doing this gives
\begin{eqnarray}\label{Ad}
A(x,t)&=&-B x +B m(t),\\
D(x,t)&=&B. \label{Dd}
\end{eqnarray}
Combining the quantum and classical parts of the operation and
using the same techniques as in Sec. \ref{KSEsec} allows us to
rewrite \erf{ss} for continuous-in-time measurements as
\begin{eqnarray}\label{supertrajextory}
d\rho_{\bf R}(x,t)&=&dt \Big{(} B\partial_{x}x
+\smallfrac{1}{2}B^2\partial^2_{x}
+\hat{\cal
L}\Big{)}\rho_{\bf R}(x,t) \nn \\ &&+dt
\Big{(}\frac{[x-\langle x_{\bf R}\rangle_t][r(t+dt)-\langle x_{\bf R}\rangle_t]}{\beta}\Big{)}\rho_{\bf R}(x,t)
\nn \\ &&-dt\rt{\gamma}\partial_x B [\hat\sigma\rho_{\bf R}(x,t)+\rho_{\bf R}(x,t)\hat\sigma^\dagger],
\end{eqnarray} where $\langle x_{\bf R}\rangle_t=\int x{\rm Tr}[ \rho_{\bf R}(x,t)]dx $ and
\begin{equation}\label{rboth}
r(t+dt)dt= \langle x_{\bf R}\rangle_tdt+\rt{\beta}dW(t).
\end{equation} This equation (\ref{supertrajextory}) has been labeled the
Superoperator-Kushner-Stratonovich equation \cite{WarWis03a} and
represents the evolution of the combined supersystem. The first
line contains the free evolution for both the quantum and the
classical systems. For this quantum system
\begin{equation}
\hat{\cal L}[\hat\sigma]\rho=\frac{-i \Omega}{2}[\hat{\sigma}_x,\rho]
+\gamma\hat{\cal D}[\hat\sigma]\rho,
\end{equation} where $\Omega$ is the Rabi frequency and $\hat{\cal D}$ is the
damping superoperator and is
defined in \erf{DampSuper}. The second line of Eq.~(\ref{supertrajextory}) describes the gaining of
knowledge about the state of classical system via Gaussian
measurements. Lastly the third line describes the coupling of the
quantum and classical system.
For a TLA we can write
the state of the supersystem as
\begin{equation}\label{superrho}
\rho(x,t)=\smallfrac{1}{2}[P(x,t)\hat{1}+X(x,t)\hat{\sigma}_x+Y(x,t)\hat{\sigma}_y+Z(x,t)\hat{\sigma}_z].
\end{equation} Note that $P(x,t)$ is the marginal state of knowledge for the classical
system (found via tracing out the
quantum degrees of freedom).
Substituting this into \erf{supertrajextory} gives the following
four coupled partial differential equations
\begin{eqnarray}\label{classical}
\dot{P}_{\bf R}&=&[x-\langle x_{\bf R}\rangle_t][r-\langle
x_{\bf R}\rangle_t]P_{\bf R}/\beta+B\partial_x[x P_{\bf R}\nn \\ &&-\rt{\gamma}X_{\bf R}]
+\smallfrac{1}{2}B^2\partial_x^2P_{\bf R}\\
\dot{X}_{\bf R}&=&[x-\langle x_{\bf R}\rangle_t][r-\langle
x_{\bf R}\rangle_t]X_{\bf R}/\beta+\smallfrac{1}{2}B^2\partial_x^2X_{\bf R}\nn \\ &&+B\partial_x[x
X_{\bf R}-\rt{\gamma}P_{\bf R}-\rt{\gamma}Z_{\bf R}]
-\smallfrac{1}{2}\gamma X_{\bf R}\\
\dot{Y}_{\bf R}&=&[x-\langle x_{\bf R}\rangle_t][r-\langle
x_{\bf R}\rangle_t]Y_{\bf R}/\beta+B\partial_x[x Y_{\bf R}]\nn \\ &&+\smallfrac{1}{2}
B^2\partial_x^2Y_{\bf R}-\Omega Z_{\bf R}-\smallfrac{1}{2}\gamma Y_{\bf R}\\
\dot{Z}_{\bf R}&=&[x-\langle x_{\bf R}\rangle_t][r-\langle
x_{\bf R}\rangle_t]Z_{\bf R}/\beta+B\partial_x[x Z_{\bf R}\nn \\ &&+\rt{\gamma}X_{\bf R}
]+\smallfrac{1}{2}B^2\partial_x^2Z_{\bf R}+\Omega Y_{\bf R}-\gamma(P_{\bf R}+Z_{\bf R}).\nonumber
\\ \label{classical4}
\end{eqnarray} To determine the state of knowledge for the quantum
system we simply integrate out the classical degrees of freedom.
To illustrate a trajectory for this supersystem the following
parameters were used; $\beta=0.5$, $B=2$, $\gamma=1$ and $\Omega=5$.
The results are shown in Fig. \ref{figsuperI} (solid line) for
a randomly chosen record ${\bf R}}%_{(0,t]}$. This figure displays the mean and
the variance of the classical trajectory found via tracing over
the quantum degrees of freedom as well as the quantum state in
Bloch representation after we have integrated out the classical
degrees of freedom.
\begin{figure}\begin{center}
\includegraphics[width=0.45\textwidth]{PartlyObservedEvolutionFig08}
\caption{\label{figsuperI} ${\rho}_{{\bf R}}%_{(0,t]}}(x,t)$ calculated via
numerical integration (solid) and via the ostensible method for an
ensemble size of 10 000 (dotted). The parameters are $\beta=0.5$,
$B=2$, $\gamma=1$ and $\Omega=5$ and initial conditions
${\rho}(x,0)=P(x)\ket{g}\bra{g}$ where $P(x)$ is a Gaussian with
mean zero and variance 0.1. }\end{center}
\end{figure}
\subsection{Fictitious trajectories: The ostensible numerical technique}
In the above section we observed that to be able to calculate the
supersystem trajectory we needed to solve four coupled partial differential
functions (each involving derivatives with respect to a classical configuration coordinate $x$).
This is a rather lengthy calculation which for higher dimensional
($d$) quantum systems will require $d^2$ partial differential equations. Here we
present our linear method that allows us to reduce the problem to
$d+2$ couple differential equations. The
expense, again, is that an ensemble average must be performed.
To do this we simply note that we can define the following quantum
and classical states
\begin{equation}\label{linearquantum}
\bar\rho_f(t+dt)=\frac{\hat{\cal O}_f(t+dt,t)\rho(t)}{\bar{\Lambda}(f)}
\end{equation} and
\begin{equation}\label{linearclassical}
\bar P_{r,f}(x',t+dt)=\frac{\int dx \bar{\cal O}_{r,f}(x',t+dt|x,t)
P(x,t)}{\Lambda(f)\Lambda(r)},
\end{equation} where
\begin{equation}
\bar{\cal O}_{r,f}(x',t+dt|x,t)=\delta[x'-x_{f}(t+dt)]P(r,t|x,t)\bar{\Lambda}(f).
\end{equation} Note the bar above $\bar{\Lambda}(f)$ means that the ostensible distribution used to
scale the quantum state does not have to be the same as that used
to scale the classical state. Here for simplicity we
consider only the case when they are the same (as no numerical
advantage is gain by different choices). Using the above equations
we can rewrite \erfs{ss}{ssss} as
\begin{eqnarray}\label{li}
\rho_{r}(x',t+dt)&=&\frac{\sum_{f}\Lambda(f)\Lambda(r)\bar{P}_{r,f}(x',t+dt)\bar\rho_f(t+dt)}{P(r,t+dt)},\nonumber\\\\
P(r,t+dt)&=&\int dx \sum_f{\rm Tr}
[\Lambda(f)\Lambda(r)\bar{P}_{r,f}(x,t+dt)\nn \\ &&\times\bar\rho_f(t+dt)].
\end{eqnarray} Thus to simulate $\rho_{\bf R}(x,t)$ we need only to calculate
$\bar{P}_{\bf R, F}(x,t)$ and $\bar\rho_{\bf F}(t)$ for a specific
record ${\bf R}}%_{(0,t]}$.
For the above TLA-classical detector system with $\Lambda(r)$ and
$\Lambda(f)$ defined by \erfs{ostenRit}{ostenFit} respectively,
$\bar P_{\bf R, F}(x',t)$ has a solution of the form $p_{\bf
R}(t)\delta[x'-x_{\bf R, F}(t)]$ where $x_{\bf F}(t)$ is given by
\begin{equation} \label{Eq.Linear1a}
d{x}_{\bf F}(t)=dt [-B x_{\bf F}(t) +B f(t+dt)]
\end{equation} and $p_{\bf R, F}(t)$ is given by
\begin{equation}\label{Eq.Linear3}
dp_{\bf R, F}(t)=dt[x_{\bf F}(t)-\lambda][r(t+dt)-\lambda]p_{\bf R, F}(t)/\beta,\hspace{1cm}.
\end{equation}
Thus we can rewrite \erf{li} as
\begin{equation}\label{liMain}
\rho_{\bf R}(x,t)=\frac{{\rm E}_{\bf F}\Big{[}\delta[x-x_{\bf F}(t)]p_{\bf R,F}(t)\bar\rho_{\bf F}(t)\Big{]}}
{{\rm E}_{\bf F}\Big{[}p_{\bf R,F}(t)\check{p}_{\bf F}(t) \Big{]}},
\end{equation} where $\check{p}_{\bf F}(t) ={\rm Tr}[\bar\rho_{\bf F}(t)]$.
To determine the evolution of the ostensible quantum state we
simply substitute the measurement operator defined in \erf{m} with
$\hat{H}=\Omega\hat{\sigma}_x/2$ and the ostensible distribution
${\Lambda}(f)$ into \erf{linearquantum}. Doing this gives
\begin{eqnarray}\label{linearquantumtraj}
d\bar\rho_{\bf F}(t)&=&dt\frac{-i\Omega}{2}[\hat{\sigma}_x,\rho_{\bf F}(t)]+dt\gamma\hat{\cal
D}[\hat{\sigma}]\rho_{\bf F}(t)+\nn \\ && dt [f(t+dt)-{\mu}][\rt{\gamma}\hat{\sigma}\rho_{\bf F}(t)
\nn \\ &&+\rt{\gamma}\rho_{\bf F}(t)\hat{\sigma}^\dagger-{\mu}\rho_{\bf F}(t)].
\end{eqnarray} However since we have assumed that all the quantum
signal is fed into the detector the evolution of the ostensible quantum state can be
written as an ostensible SSE. That is,
\begin{eqnarray}
d\ket{\bar{\psi}_{\bf F}(t)}&=&dt
\Big{(}-\frac{i\Omega}{2}\hat{\sigma}_x+[f(t+dt)-\mu](\rt{\gamma}\hat{\sigma}-\mu/2)\nn \\ &&
-\smallfrac{1}{2} [\gamma\hat{\sigma}^\dagger\hat{\sigma}
-\rt{\gamma}\mu\hat{\sigma}+\mu^2/4]
\Big{)}
\ket{\bar{\psi}_{\bf F}(t)}.\hspace{.8cm}
\end{eqnarray}
Thus to determine $\rho_{\bf R}(x,t)$ all we need to do is solve
the above SSE and \erfs{Eq.Linear1a}{Eq.Linear3} for ${\bf R}$
assumed known and $f(t+dt)dt=d{\cal W} +dt \mu$ where $d{\cal W}$
is a Wiener increment. Once solved the quantum state conditioned
on ${\bf R}}%_{(0,t]}$ is given by
\begin{eqnarray}
{\chi}_{\bf R}(t)&=&\frac{{\rm E}_{\bf F}\Big{[}p_{\bf R,
F}(t)\check{\chi}_{\bf F}(t)\Big{]}}{ {\rm E}_{\bf F}\Big{[}p_{\bf
R, F}(t)\check{p}_{\bf F}(t)\Big{]}},
\end{eqnarray} where ${\chi}_i =\{\check{x}_i,\check{y}_i,\check{z}_i\}$ are the Bloch vectors of the quantum state.
The moments of the
classical state are given by
\begin{eqnarray}
\langle x^m_{\bf R} \rangle_{t}&=&\frac{{\rm E}_{\bf
F}\Big{[}{x}^m_{\bf F}(t)p_{\bf R, F}(t)\check{p}_{\bf
F}(t)\Big{]}}{ {\rm E}_{\bf F}\Big{[}p_{\bf R, F}(t)\check{p}_{\bf
F}(t)\Big{]}} . \hspace{.5cm}
\end{eqnarray}
To illustrate this method we considered two choices for the
ostensible distributions. The first is $\lambda=\mu=0$; that is,
all the ostensible distributions are Gaussian distributions of
mean zero and variance $dt$. The second case corresponds to the
situation when $\lambda=0$ and $\mu=\rt{\gamma}{\rm Tr} [(\hat
\sigma+\hat\sigma^\dagger)\rho_{\bf F}(t)]$; that is, the fictitious
distribution is treated as the real unobservable distribution.
Both cases were simulated to show the robustness of our numerical
technique and to demonstrate that while any ostensible
distributions can be chosen a more realistic choice will result in
a faster convergence. To demonstrate this we numerically solved \erf{supertrajextory}
and used this as our reference solution. Then we compared the mean
and variance of the classical marginal states
and the fidelity for quantum reduced states
(using \erf{felquant} once the
classical space has been removed) for both ostensible cases and with ensemble sizes
of 100 and 10 000.
These results are shown Fig. \ref{figsuper2} where it is observed
that for the larger ensemble size the difference in the classical marginal state
is small and the quantum
fidelity is close to one, indicating that our ostensible method
has reproduced the known result and is converging. Furthermore it
is observed that for the second case for the same ensemble size
this difference is smaller thereby indicating that the second method
convergence is faster.
\begin{figure}\begin{center}
\includegraphics*[width=0.45\textwidth]{PartlyObservedEvolutionFig09}
\caption{\label{figsuper2} This figure shows the quantum and
classical fidelity between the actual solution and our ostensible
solution for ${\rho}_{{\bf R}}%_{(0,t]}}(x,t)$. Part A corresponds to the
$\lambda=\mu=0$ case while part B represents the $\lambda=0$ and
$\mu=\rt{\gamma}{\rm Tr} [(\hat \sigma+\hat\sigma^\dagger)\rho_{\bf
F}(t)]$ case. In both case an ensemble size of $n=100$ (dotted)
and $n=10 000$ (solid) was used. The system parameters are the
same as in Fig. \ref{figsuperI}.}\end{center}
\end{figure}
\section{Discussion and Conclusion}\label{con}
The central topic of this paper was to investigate the conditional
dynamics of partially observed systems (classical and quantum).
Due to the fact that the information obtained is incomplete we have
to assign a mixed state to the system. For a quantum system this
means the state of knowledge given result $r$ is given by the
state matrix $\rho_{r}(t)$ and for a classical system a
probability distribution $P_{r}(x,t)$ has to be used. If we
consider a joint system (for example a classical detector is used
to monitor a quantum system) the conditional state is given by
$\rho_{r}(x,t)$.
Even when we consider continuous-in-time monitoring we can still
have incomplete information because of unobserved processes. For
this case the conditional state trajectories obey either a
stochastic master equation (for a quantum system), a Kushner
Stratonovich equation (for a classical system) or a superoperator
Kushner-Stratonovich equation (for the joint system). That is, to
simulate the conditional state we have to solve a rather
numerically expensive equation. In this paper we showed that by
introducing a fictitious record ${\bf F}}%_{(0,t]}$ for the unobserved
processes and ostensible measurement theory we can reduce this
problem to solving pure states (stochastic Schr\"odinger~~ equations for the
quantum system or stochastic differential equations for the
classical system) conditioned on both ${\bf R}}%_{(0,t]}$ and ${\bf F}}%_{(0,t]}$. Then by
averaging over all possible ${\bf F}}%_{(0,t]}$ we get the require conditional
state. That is the numerical memory requirements are decreased by
a factor of $N$, the number of basis states for the system.
However, this is at the cost of an ensemble average.
In summary, our ostensible method will be useful for investigating
realistic situations where the dimensions of the systems are
large. It is also much easier to implement numerically than the standard
technique, so we expect
it to find immediate applications.
\acknowledgments
We would like to acknowledge the interest shown and help provided
by K. Jacobs and N. Oxtoby. This work was supported by the
Australian Research Council (ARC) and the State of Queensland.
|
{
"timestamp": "2007-04-06T23:01:15",
"yymm": "0503",
"arxiv_id": "quant-ph/0503241",
"language": "en",
"url": "https://arxiv.org/abs/quant-ph/0503241"
}
|
\section{Introduction}
\hspace{3ex}
It is interesting to consider the elastic rod of a large mass $M$, the
left end of which is joined with mass $m << M$
and body of mass $m$ is fixed to the right end of the rod. Then,
it is interesting to study the consequences of the application of the
the force of the delta-function form to the left side of the rod.
The delta-function is chosen for simplicity. This
function can be replaced by the different functions. We show that the internal
motion of the elastic rod medium is controlled by the wave equation. We derive
the mathematical form of the mechanical motion of the considered string or rod.
Our problem represents
the missing problem in the Newton ``Principia mathematica'' [1]
and in any textbook on mechanics. The relation of our theory to the quark-string model of mesons is evident.
\section{Classical theory of interaction of particle with an impulsive force}
We will first show that use of the impulsive force of the delta-function form
is physically meaningful in a classical mechanics of a point particle.
We idealize the impulsive force by the Dirac $\delta$-function.
Newton's second law in the one-dimensional form for the interaction
of a massive particle with mass $m$ with force $F$
$$ma = F\eqno(1)$$
with $F$ being an impulsive force $P\delta(t)$ is as follows:
$$m\frac {d^{2}x}{dt^{2}} = P\delta(\alpha t),\eqno(2)$$
where $P$ and $\alpha$ are some constants, with MKSA dimensionality [$P$]
= ${\rm kg.m.s}^{-2}$, [$\alpha$] = ${\rm s}^{-1}$. We put $|\alpha| = 1$.
Using the Laplace transform [2] in the last equation, with
$$\int_{0}^{\infty}e^{-st}x(t)dt \stackrel{d}{=} X(s), \eqno(3)$$
$$\int_{0}^{\infty}e^{-st}{\ddot x}(t)dt = s^{2}X(s) - sx(0) - {\dot x}(0),
\eqno(4)$$
$$\int_{0}^{\infty}e^{-st}\delta(\alpha t)dt = \frac{1}{\alpha}, \eqno(5)$$
we obtain:
$$ms^{2}X(s) - msx(0) - m\dot x(0) = P/\alpha.\eqno(6)$$
For a particle starting from the rest with $\dot x(0) = 0, x(0) = 0$, we get
$$X(s) = \frac{P}{ms^{2}\alpha}.\eqno(7)$$
Using the inverse Laplace transform, we obtain
$$x(t) = \frac{P}{m\alpha}t\eqno(8)$$
and
$$\dot x(t) = \frac{P}{m\alpha}.\eqno(9)$$
In case of the harmonic oscillator with the damping force and under
influence of the general force $F(t)$, the Newton law is as follows:
$$m\frac {d^{2}x(t)}{dt^{2}} + b{\dot x}(t)
+ kx(t) = F(t).\eqno(10)$$
After application of the Laplace transform (3) and with regard to the same
initial conditions as in the preceding situation,
$\dot x(0) = 0, x(0) = 0$, we get the following algebraic equation:
$$ms^{2}X(s) +bsX(s) + kX(s) = F(s),\eqno(11)$$
or,
$$X(s) = \frac{F(s)}{m\omega_{1}}
\frac {\omega_{1}}{(s + b/2m)^{2} + \omega_{1}^{2}} \eqno(12)$$
with $\omega_{1}^{2} = k/m - b^{2}/4m^{2}$.
Using inverse Laplace transform denoted by symbol ${\cal L}^{-1}$ applied
to multiplication of functions $f_{1}(s)f_{2}(s)$,
$${\cal L}^{-1}(f_{1}(s)f_{2}(s)) = \int_{0}^{t} d\tau
F_{1}(t-\tau)F_{2}(\tau),
\eqno(13)$$
we obtain with $f_{1}(s) = F(s)/m\omega_{1}, \quad f_{2}(s) = \omega_{1}/
((s + b/2m)^{2} + \omega_{1}^{2}), \quad F_{1}(t) = F(t)/m\omega_{1},\\
F_{2}(t) = \exp{(-bt/2m)}\sin\omega_{1}t$.
$$x(t) = \frac {1}{m\omega_{1}}\int_{0}^{t}F(t-\tau)e^{-\frac {b}{2m}\tau}
\sin(\omega_{1}\tau)d\tau.\eqno(14)$$
For impulsive force $F(t) = P\delta(\alpha t)$, we have from the last formula
$$x(t) = \frac {(P/\alpha)}{m\omega_{1}}e^{-\frac {b}{2m}t}\sin\omega_{1}t.
\eqno(15)$$
\section{The pulse propagating in a rod}
\hspace{3ex}
In this section we will solve the motion of a string or rod with the massive
ends (the body with mass $m$ is fixed to the every end of the string)
on the assumption that the tension in the string is linear and the
applied force is of the Dirac delta-function.
First, we will derive the Euler wave equation from the Hook law of tension
and then we will give the
rigorous mathematical formulation of the problem. Linearity of the
wave equation enables to solve this problem by the Laplace transform method.
We follow [3] and the author preprint [4] where this method was used
to solve the Gassendi model of gravity. Although Gassendi [5] is
known in physics as the founder of the modern atomic theory of matter,
his string model of gravity was not accepted. The Newton reaction to this
model was empirical. He said: ``Hypotheses non fingo''. It seems that
Gassendi ideas was applied later by Faraday in his theory of
electromagnetism. We know also that Gassendi was independent thinker
and he was persecuted. Every
independent thinker is persecuted in any society.
The present problem can be also defined as a central collision of two
bodies (balls). While in the basic mechanics the central collision is
considered as a contact collision of the two balls, here,
the collision is mediated by the string, or, rod.
To our knowledge, the present problem is not involved
in the textbooks of mathematical physics or in the mathematical journals.
This problem was not possible to define and solve in the Newton
period, because the method of solution is based on the
Euler partial wave equation,
the Laplace transform, The Riemann-Mellin transform, the Bromwich
integral and Bromwich contour and other ingredients of the operator calculus
which was elaborated after the Newton period. So, this is why the
problem is not involved in the Newton ``Principia Mathematica'' [1].
Now, let us consider the rod (or string) of the length $L$,
the left end of which is joined with mass $m$ and
the right end is joined with mass $m$. The force
of the delta-function form is applied to the left end
and the initial state of the rod is the sate of equilibrium.
The deflection of the rod element $dx$ at point $x$ and time $t$ let
be $u(x,t)$ where $x\in(0,L)$.
The differential equation of motion of string elements can be derived
by the following way [3]. We suppose that the force acting on
the element $dx$ of the string is given by the law:
$$T(x,t) = ES\left(\frac {\partial \*u}{\partial x}\right), \eqno(16)$$
where $E$ is the modulus of elasticity, $S$ is the cross section of
the string. We easily derive that
$$T(x+dx)-T(x) = ESu_{xx}dx. \eqno(17)$$
The mass $dm$ of the element $dx$ is $\varrho ESdx$,
where $\varrho = const$ is the mass
density of the string matter and the dynamical equilibrium gives
$$\varrho\*Sdx u_{tt} = ESu_{xx}dx. \eqno(18)$$
So, we get
$$\frac {1}{c^2}u_{tt} - u_{xx} = 0; \quad
c = \left(\frac {E}{\varrho}\right)^{1/2}. \eqno(19)$$
Now, we get the problem of the mathematical physics in the form:
$$u_{tt} = c^{2}u_{xx}\eqno(20)$$
with the initial conditions
$$u(x,0) = 0; \quad u_{t}(x,0) = 0\eqno(21)$$
and with the boundary conditions
$$mu_{tt}(0,t) = au_{x}(0,t) + P\delta(\alpha t);\quad
mu_{tt}(L,t) = au_{x}(L,t),\eqno(22)$$
where we have put
$$a = - ES;\quad P = {\rm some\; constant}.\eqno(23)$$
The delta-function can be approximatively realized by the strike of
the hammer to the left end of the rod.
The equation (20) with the initial and boundary conditions (21) and (22)
represents one of the standard problems of the mathematical physics and can
be easily solved using the Laplace transform [2]:
$$\hat L u(x,t) \stackrel{d}{=}\int_{0}^{\infty}e^{-pt}
u(x,t)dt \stackrel{d}{=} \varphi(x,p). \eqno(24)$$
Using (24) and (20) we get:
$$\hat Lu _{tt}(x,t) = p^{2}\varphi(x,p) - pu(x,0) - u_{t}(x,0) =
p^{2}\varphi(x,p),\eqno(25)$$
$$\hat L u_{xx}(x,t) = \varphi_{xx}(x,p);\quad
\hat L \delta(\alpha, t) = 1/\alpha .
\eqno(26)$$
After elementary mathematical operations we get the differential
equation for $\varphi$ in the form
$$\varphi_{xx}(x,p) - k^{2}\varphi(x,p) = 0; \quad
k = p/c. \eqno(27)$$
with the boundary condition in eq. (22).
We are looking for the the solution of eq. (27) in the form
$$\varphi(x,p) =c_{1}\cosh k x + c_{2}\sinh k x .\eqno(28)$$
We get from the boundary conditions in eq. (22)
$$c_{1} = \frac {1}{p}\;\frac {ac(P/\alpha) \cosh (pL/c) -
(P/\alpha)mpc^2\sinh (pL/c)}
{\sinh(pL/c)(a^2 - m^2 p^2 c^2)},\eqno(29)$$
$$c_{2} = -\frac {(P/\alpha)c}{ap} + \frac {(P/\alpha)mac^2\cosh(pL/c) -
(P/\alpha)pm^2c^3\sinh (pL/c)}{a\sinh (pL/c)
(a^2 - m^2 c^2 p^2)}.\eqno(30)$$
The corresponding $\varphi(x,p)$ is of the form:
$$\varphi(x,p)= \frac {1}{p}\;\frac {ac(P/\alpha)\cosh (pL/c) -
(P/\alpha)mpc^2\sinh(pL/c)}
{\sinh (pL/c)(a^2 - m^2 p^2 c^2)}\cosh (px/c)\quad + $$
$$\left[-\frac {(P/\alpha)c}{ap} +
\frac {a(P/\alpha)mc^2\cosh(pL/c) -
bpm^2c^3\sinh (pL/c)}{a\sinh (pL/c)(a^2 - m^2 c^2 p^2)}\right]\sinh(px/c).
\eqno(31)$$
The corresponding function $u(x,t)$ follows from the theory of the Laplace
transform as the mathematical formula (res is residuum)[2]:
$$u(x,t) = \frac {1}{2\pi i}\oint e^{pt}\varphi(x,p) dp =
\sum_{p=p_{n}}{\rm res}\;e^{pt}\varphi(x,p) = $$
$$\sum _{p=p_{n}}{\rm res}\;e^{pt}
\frac {1}{p}\;\frac {ac(P/\alpha) \cosh (pL/c)}
{\sinh (pL/c)(a^2 - m^2 p^2 c^2)}\cosh (px/c) \quad -$$
$$\sum _{p=p_{n}}{\rm res}\;e^{pt}
\frac {(P/\alpha)mc^2}
{(a^2 - m^2 p^2 c^2)}\cosh (px/c)\quad - $$
$$\sum _{p=p_{n}}{\rm res}\;e^{pt}
\left[\frac {(P/\alpha)c}{ap}\right]\sinh (px/c)\quad + $$
$$\sum _{p=p_{n}}{\rm res}\;e^{pt}
\left[\frac {m(P/\alpha)c^2\cosh (pL/c)}{\sinh (pL/c)(a^2 - m^2
p^2 c^2)}\right]\sinh( px/c)\quad - $$
$$\sum _{p=p_{n}}{\rm res}\;e^{pt}
\left[\frac{(P/\alpha)pm^2c^3}{a}\;\frac {1}{(a^2 - m^2 c^2p^2)}\right]
\sinh (px/c)\quad =$$
$$u_{1} - u_{2} - u_{3} + u_{4} - u_{5},
\eqno(32)$$
where
$$u_{j} = \sum {\rm res}\; e^{pt}\frac{A_{j}}{B_{j}}; \quad j = 1, 2, 3, 4, 5
\eqno(33)$$
and
$$A_{1} = {ac(P/\alpha) \cosh (pL/c)}\cosh (px/c);\quad
B_{1} = p\sinh (pL/c)(a^2 - m^2 p^2 c^2)\eqno(34)$$
$$A_{2} = (P/\alpha)mc^2\cosh (px/c); \quad
B_{2} = (a^2 - m^2 p^2 c^2)\eqno(35)$$
$$\quad A_{3} = (P/\alpha)c\sinh(px/c); \quad B_{3} = ap \eqno(36)$$
$$A_{4} = (P/\alpha)mc^2\cosh (pL/c)\sinh (px/c); \quad
B_{4} = \sinh (pL/c)(a^2 - m^2 p^2 c^2)\eqno(37)$$
$$A_{5} = (P/\alpha)pm^2c^3\sinh (px/c); \quad
B_{5} = a(a^2 - m^2 p^2 c^2).\eqno(38)$$
We know from the theory of the complex functions that if the pole of
some function $f(z)/g(z)$ is simple and it is at point $a$,
then the residuum is as follows [2]:
$${\rm residuum} = \frac{f(a)}{g'(a)}.\eqno(39)$$
If the pole at point $a$ of the function $f(z)$ is multiply of the
order $m$, then the residuum is defined as follows:
$${\rm residuum} = \frac{1}{(m-1)!} \lim_{z\to a}\frac{d^{m-1}}{dz^{m-1}}
\left[(z-a)^m f(z) \right].\eqno(40)$$
Let us first determined the function
$$u_{1} = \sum {\rm res}\; e^{pt}\frac{A_{1}}{B_{1}}.\eqno(41)$$
Poles of $B_{1}$ are
at points $p=0$, this is pole of the order 2, $p = + a/mc, p = -a/mc$ and
$p_{n} = + i\pi nc/L, p_{n} = -i\pi nc/L, n = 1, 2, 3, ...$. So, the
function $u_{1}$ is as follows:
$$u_{1} = \frac{(P/\alpha)c^2}{La}t -
\frac{(P/\alpha)c}{a}\cosh\left(\frac{aL}{mc^2}\right)
\cosh\left(\frac{ax}{mc^2}\right)\sinh\left(\frac{at}{mc}\right)
\quad + $$
$$\sum_{n=1}^{n= \infty} \frac{2a(P/\alpha)c}{\pi n}
\frac{L^2}{a^2 L^2 + m^2\pi^2
n^2 c^4}\cos\left(\frac{\pi nx}{L}\right)
\sin\left(\frac{\pi n c t}{L}\right).
\eqno(42)$$.
For the function $u_{2}$ we get:
$$u_{2} =
\left(-\frac{(P/\alpha)c}{a}\right)\sinh\left(\frac{at}{mc}\right)\cosh
\left(\frac{ax}{mc^2}\right).\eqno(43)$$
$$u_{3} = 0 .\eqno(44)$$
For $u_{4}$ and $u_{5}$ we get:
$$u_{4} =
\left(-\frac{(P/\alpha)c}{a}\right)
\coth\left(\frac{aL}{mc^2}\right)\sinh\left(\frac{ax}{mc^2}\right)
\sinh\left(\frac{at}{mc}\right)\eqno(45)$$
$$u_{5} =
\left(-\frac{(P/\alpha)c}{a}\right)
\sinh\left(\frac{ax}{mc^2}\right)
\sinh\left(\frac{at}{mc}\right)\eqno(46)$$
The dimensionality of $u$ is [$u$] = m and $u(x,0) = 0$. The momentum
of a left particle $p = mu(0,t)$, or right particle $p = mu(L,t)$ is
not conserved. Only the total momentum of a system is conserved.
\section{Discussion}
Our problem is the modification of some problems involved in the
textbooks on mathematical physics. However, our approach is
pedagogically original in
the sense that we use the initial force of a delta-function form
to show the internal motion of the string, or, rod. The delta-function
form of electromagnetic pulse was used by author in [6] and [7] to
discuss the quantum motion of an electron in the laser pulse.
We have considered here the real strings and rods in the real
space and we do not use extra-dimensions and unrealistic strings.
The M-dimensional geometrical object cannot be realized in
N-dimensional space for M $>$ N [8].
The mathematical theory of unrealistic strings is well known
as the string theory in particle physics.
Our problem with the real strings and rods
can be generalized for the two-dimensional and three-dimensional
situation. It can be also generalized to the
situation with the dissipation of waves in the strings and rods. In
this case it is necessary to write the wave equation with the
dissipative term and then to solve this problem ``ab initio''.
While we have solved the problem for the situation where the pulse was
generated by the force of the delta-form, we give some general
ideas following from the wave equation. It is well known that
the solution of this equation is in general in the form [9]:
$$f\left(t - \frac{x}{c}\right); \quad g\left(t + \frac{x}{c}\right)
\eqno(47)$$
where functions $f, g$ are general. It means it involves also the
function of he delta-form. For the wave propagating from the left side
to the right side, we take function $f$. The corresponding tension in the
rod is
$$T = ESu_{x}(x, t) = ESf'\left(t - \frac{x}{c}\right)
\left(\frac{-1}{c}\right).\eqno(48)$$
We easily see that $T(x = 0, t= 0) = T(x = L,t = L/c )$, and it means
that when the pulse force is created at the left end of the rod then
it propagates in the rod and after time $L/c$
it is localized in the right end of the rod.
However, we have seen that the pulse force generated in the
system with the massive ends
of strings or rods develops in time according to laws of the mathematical
physics of strings and rods and cannot be intuitively predicted.
Only rigorous solution of the dynamics of the system can give the
answer on the real motion of tension in the string.
There is no information in the Newton ``Principia mathematica '' [1] and
in any textbook on mechanics on the central collision of two particles
where the force is mediated by string or rod.
Similarly, there is not the
solution of our problem in the famous monograph by Pars [10].
So, This is the missing problem in the textbooks on mechanics.
The propagation of a pulse in one direction was confirmed
experimentally by author using
the heavy elastic rod (the segment of a rail). The delta-form force (tension) was
generated approximately by the strike of hammer.
The experiment was performed as the table experiment and
it can be repeated by any theorist.
The proposed model with the string with massive ends
can be also related in the modified form to the problem of
the radial motion of quarks bound by a strings, and
used to calculate the excited
states of such system. The resent analysis of such problem
was performed by Lambiase and Nesterenko [11] and Nesterenko and
Pirozhenko [12], and others. So, can we hope that our approach
and their approach will be unified to generate the new revolution
of the string theory of matter and space-time? Why not?
\vspace{5mm}
|
{
"timestamp": "2005-03-01T23:12:12",
"yymm": "0503",
"arxiv_id": "math-ph/0503003",
"language": "en",
"url": "https://arxiv.org/abs/math-ph/0503003"
}
|
\section{Introduction}
It is a well-known consequence of the simultaneous uniformisation
theorem of Bers \cite{Bers} that given two abstractly isomorphic
Fuchsian groups $G_1\subset PSL_2(\mbox{$\mathbb R$})$ and $G_2\subset PSL_2(\mbox{$\mathbb R$})$,
acting on the upper and lower complex half-planes respectively,
each having limit set $\hat{\mathbb R}={\mathbb R} \cup \infty$,
and such that the action of $G_1$ on $\hat{\mathbb R}$ is
topologically conjugate to that of $G_2$, the actions of $G_1$ and
$G_2$ can be {\it mated} to obtain a quasifuchsian Kleinian group
$G \subset PSL_2(\mbox{$\mathbb C$})$. This {\it mating} is a group which is
abstractly isomorphic to both $G_1$ and $G_2$, it has limit set
$\Lambda(G)$ a simple closed (fractal) curve, and the actions of
$G$ on the two components of $\Omega={\hat{\mathbb C}}-\Lambda$
are conformally conjugate to those of $G_1$ on ${\mathcal U}$ and
$G_2$ on ${\mathcal L}$.
\medskip
It is also well-known that given two polynomial maps $P$ and $Q$
of the same degree $n$, in appropriate circumstances one can find
a rational map $R$ which realises a {\it mating} between the
actions of $P$ and $Q$ on their filled Julia sets, in a precise
sense as defined for example in \cite{HT}. A necessary condition
for a mating between two quadratic polynomials $P:z \to z^2+c$ and
$Q:z \to z^2+c'$ to exist is that $c$ and $c'$ should not belong
to conjugate limbs of the connectivity locus (the Mandelbrot Set)
in parameter space: this was first shown also to be a sufficient
condition in the case that $P$ and $Q$ are {\it postcritically
finite} \cite{ree,T}, and subsequently for much more general
classes of $P$ and $Q$ \cite{HT}.
\medskip
In \cite{BP} the first examples of holomorphic correspondences
realising {\it matings} between Fuchsian groups and polynomials
were presented. {\it Holomorphic correspondences} on the Riemann
sphere are multi-valued maps $f:z\to w$ defined by polynomial
equations $p(z,w)=0$. Examples of holomorphic correspondences are
those defined by a union of the graphs of some finite set of
M\"obius transformations, or by the graph of a rational map (or
its inverse). We say that such a correspondence has {\it bidegree}
$(m:n)$ if a generic point $z$ has $n$ images $w$ and a generic
point $w$ has $m$ inverse images $z$.
\medskip
{\bf Definition} {\it Let $q_c:z \to z^2+c$ be a quadratic map
with connected filled Julia set $K(q_c)$. A holomorphic
correspondence $f:z \to w$ of bidegree $(2:2)$ is called a {\it
mating} between $q_c$ and the modular group $PSL_2(\mbox{$\mathbb Z$})$ if:
\medskip
(a) there exists a completely invariant open simply-connected
region $\Omega \subset \hat{\mbox{$\mathbb C$}}$ and a conformal bijection $h$
from $\Omega$ to the upper half-plane conjugating the two branches
of $f\vert_\Omega$ to the pair of generators $z \to z+1,\ z \to
z/(z+1)$ of $PSL_2(\mbox{$\mathbb Z$})$;
\medskip
(b) the complement of $\Omega$ is the union of two closed sets
$\Lambda_-$ and $\Lambda_+$, which intersect in a single point and
are equipped with homeomorphisms $h_\pm: \Lambda_\pm \to K(q_c)$,
conformal on interiors, respectively conjugating $f$ restricted to
$\Lambda_-$ as domain and codomain to $q_c$ on $K(q_c)$, and
conjugating $f$ restricted to $\Lambda_+$ as domain and codomain
to $q_c^{-1}$ on $K(q_c)$.}
\medskip
In \cite{BP} the one parameter family of correspondences
$$\left(\frac{az+1}{z+1}\right)^2+\left(\frac{az+1}{z+1}\right)
\left(\frac{aw-1}{w-1}\right)+\left(\frac{aw-1}{w-1}\right)^2=3
\leqno(1) $$
was shown to contain examples of matings between quadratic maps
and the modular group. The following conjecture is implicit in the
discussion in Sections 1 and 6 of that paper.
\medskip
{\bf Conjecture 1} {\it The family $(1)$ of $(2:2)$
correspondences contains matings between $PSL_2(\mbox{$\mathbb Z$})$ and {\it
every} quadratic polynomial having a connected Julia set, that is
to say every $z \to z^2+c$ with $c \in {\mathcal M}$, the {\it
Mandelbrot set}.}
\medskip
Supporting evidence was provided by proofs for particular examples
and numerical experiments suggesting the resemblance between the
space of matings and the Mandelbrot set. However difficulties in
adapting the theory of {\it polynomial-like maps} \cite{DH} to the
setting of {\it pinched polynomial-like maps} prevented a proof.
\medskip
A different question turned out to be easier to answer. The
modular group may be considered as a representation of the free
product $C_2*C_3$ of cyclic groups, of orders two and three, in
$PSL_2(\mbox{$\mathbb C$})$. Up to conjugacy there is a one parameter family of
such representations and in the parameter space there is a set
${\mathcal D}$, homeomorphic to a once-punctured closed disc, for
which the representation is discrete and faithful. The modular
group corresponds to a particular {\it boundary} point of
$\mathcal D$. Let $r$ be any representation of $C_2*C_3$
corresponding to a parameter value in the {\it interior}
${\mathcal D}^\circ$ of $\mathcal D$. The ordinary set $\Omega(r)$
of the Kleinian group defined by such a representation $r$ is
connected and the limit set $\Lambda(r)$ is a Cantor set. In
\cite{BH} the notion of a mating between such a representation $r$
of $C_2*C_3$ and a quadratic polynomial $q_c: z \to z^2+c$ was
introduced: $\Lambda_-$ and $\Lambda_+$ are now disjoint, and
their complement $\Omega$ is canonically associated to $\Omega(r)$
(see Section 2.2). By the application of
polynomial-like mapping theory the following analogue of
Conjecture 1 was proved in \cite{BH}.
\medskip
\begin{thm}\label{mating} For every quadratic map $q_c:z \to
z^2 + c$ with $c \in {\mathcal M}$ and every faithful discrete
representation $r$ of $C_2*C_3$ in $PSL_2(\mbox{$\mathbb C$})$ having connected
ordinary set, there exists a polynomial relation $p(z,w)=0$
defining a $(2:2)$ correspondence which is a mating between $q_c$
and $r$.
\end{thm}
An outline of the proof of Theorem \ref{mating} is presented in
Section 2.2, as a prelude to applying pinching techniques to
the matings it shows to exist.
\medskip
We describe an involution $J$ on $\hat{\mathbb C}$ as {\it
compatible} with a mating $f$ if $(J\circ f) \cup I_{\hat{\mathbb
C}}$ is an equivalence relation, where $I_{\hat{\mathbb C}}$
denotes the identity map on $\hat{\mathbb C}$ and $(J\circ f) \cup
I_{\hat{\mathbb C}}$ denotes the $3:3$ correspondence defined by
the algebraic curve
$$p(z,J(w))(z-w)=0$$ (Here $p(z,w)=0$ is the curve defining $f$.)
\begin{prop}\label{compatible} Every mating with a compatible
involution is conjugate to a correspondence in the following two
parameter family (also considered in \cite{BP}):
$$\left(\frac{az+1}{z+1}\right)^2+\left(\frac{az+1}{z+1}\right)
\left(\frac{aw-1}{w-1}\right)+\left(\frac{aw-1}{w-1}\right)^2=3k
\leqno(2) $$
\end{prop}
\medskip
As we shall see, the matings constructed in \cite{BH} have
compatible involutions, so they have representatives in the family
(2), a fact observed in \cite{BH} but for which Proposition
\ref{compatible} (proved in Section 2) provides a more conceptual
setting.
\medskip
The basic idea of pinching can be seen in the process by which the
modular group can be obtained from any chosen standard
representation $r_*$ of $C_2*C_3$ lying in the interior of
$\mathcal D$, that is to say a faithful discrete representation
with connected ordinary set $\Omega(r_*)$ (and therefore limit set
a Cantor set). We first recall that each Kleinian representation
of $C_2*C_3$ comes equipped with a canonical involution $\chi$
which conjugates the generators $\sigma\in C_2$ and $\rho\in C_3$
to their inverses (see Section 2.1); we let $G$ denote the group
$<\chi,\sigma,\rho>$. For each rational number $p/q$ there is an
arc $\delta_{p/q}$ on the orbit space $\Sigma=\Omega(r_*)/G$ which
lifts to simple closed geodesic $\tilde{\delta}_{p/q}$ of winding
number $p/q$ on a certain torus $\tilde{\Sigma}$ double-covering
$\Sigma$ (see Lemma \ref{arcs-exist} in Section 3.1 for details).
The arc $\delta_{p/q}$ lifts to an arc $\alpha_{p/q}$ on
$\Omega(r_*)$ together with its translates under $G$. This arc
$\alpha_{p/q}$ is {\it precisely $<<g>>$-invariant} for any
loxodromic element $g \in G$ which stabilises it. (Here $<<g>>$
denotes the maximal elementary subgroup of $G$ containing $g$, and
saying that an arc $\alpha$ is {\it precisely $<<g>>$-invariant}
means that $<<g>>\alpha=\alpha$ and $h(\alpha)\cap \alpha =
\emptyset$ for all $h\in G$ not in $<<g>>$). In this situation
Maskit's Theorem \cite{M} states that the representation of $G$ in
$PSL_2(\mbox{$\mathbb C$})$ can be deformed to one in which $\alpha_{p/q}$ and its
translates under $G$ are pinched to points and $g$ becomes
parabolic. We deduce that we may pinch $\delta_0$, and hence its
lift $\alpha_0$, to a point, thereby deforming the representation
$r_*$ of $C_2*C_3$ to the representation $PSL_2(\mbox{$\mathbb Z$})$, which lies
on the boundary of the deformation space $\mathcal D$. Similarly
for $p/q \ne 0$ we may pinch $\delta_{p/q}$ to a point and so
deform the representation $r_*$ to a faithful discrete
representation which we denote $r_{p/2q}$. This has ordinary set a
disjoint union of a countable infinity of open round discs, and
limit set a circle-packing. The representation $r_{p/2q}$ depends
only on the value of $p/2q$ mod $2$: pinching $\delta_{(2nq+p)/q}$
in place of $\delta_{p/q}$ amounts to approaching the same limit
representation $r_{p/2q}$ but by a non-isotopic path in ${\mathcal
D}$. We remark that by a deep result of McMullen \cite{mc2} the
representations $r_{p/2q}$ are dense in the boundary of ${\mathcal
D}$.
\medskip
Recently, Ha\"{\i}ssinsky \cite{H2}, Cui \cite{cui} and
Ha\"{\i}ssinsky and Tan \cite{HT} proved analogous results to
Maskit's in the context of rational maps, showing that, under
appropriate hypotheses, given a rational map $R$ and an
$R$-invariant union of arcs joining attracting to repelling
cycles, one can continuously deform the map in such a way that the
arcs, and their pre-images, are pinched to points and the cycles
become parabolic.
\medskip
In this paper we adapt the techniques of \cite{H2} and \cite{HT}
to apply them to the holomorphic correspondences constructed in
\cite{BH}. In Section 3, for any correspondence $p_0(z,w)=0$ which
is a mating between $r_*$ and $q_c$, and for any rational number
$p/q$, we identify an arc $\gamma_{p/q}$ such that the grand orbit
of $\gamma_{p/q}$ under the correspondence is a union of
infinitely many disjoint copies of $\gamma_{p/q}$ (or of copies of
a quotient of $\gamma_{p/q}$ by an involution), and such that
pinching each connected component of this union to a point
corresponds to deforming the representation $r_*$ to $r_{p/2q}$.
We describe the pinching process formally as follows.
\medskip
{\bf Definition} {\it A convergent pinching deformation for
$\gamma_{p/q}$ is a family of quasiconformal maps $(\varphi_t)_{0\le t <1}$ of
the Riemann sphere such that the conjugate correspondences $p_t$
defined by
$$p_t(z,w)= p_0(\varphi_t^{-1}(z),\varphi_t^{-1}(w))$$
are holomorphic and satisfy the following\,:
\begin{itemize}
\item $(p_t,\varphi_t)$ are uniformly convergent to a pair
$(p_1,\varphi_1)$ as $t$ tends to $1$ , \item the non-trivial fibres
of $\varphi_1$ are exactly the closure of the connected components of the orbit of
$\gamma_{p/q}$. \end{itemize}}
\medskip
There are two technical conditions that we require the quadratic
map $q_c$ to satisfy in order to apply the techniques of \cite{HT}
to $\gamma_0$:
\medskip
(i) if the critical point $0$ of $q_c$ is recurrent, the
$\beta$-fixed point of $q_c$ is not in the $\omega$-limit set of
$0$;
\medskip
(ii) $q_c$ is weakly hyperbolic, that is, there are constants
$r>0$ and $\delta<\infty$ such that, for all $z\in J_q\smallsetminus
\{\text{preparabolic points}\}$, there is a subsequence of
iterates $(q^{n_k})_k$ such that
$$\mbox{deg}(W_k(z)\stackrel{q^{n_k}}{\longrightarrow} D(q^{n_k}(z),r) )\le
\delta\,$$
where $W_k(z)$ is the connected component of
$q^{-n_k}(D(q^{n_k}(z),r) )$ containing $z$.
\medskip
In Section 4 we prove:
\medskip
\begin{thm}\label{simple}
Let $p_0(z,w)$ be a mating between the representation $r_*$ and
$q_c$, where $q_c$ satisfies conditions (i) and (ii) above. Then
there exists a pinching deformation of $p_0$ such that $(p_t)_{0
\le t <1}$ converges uniformly to a mating $p_1$ between
$PSL_2(\mbox{$\mathbb Z$})$ and $q_c$.\end{thm}
\begin{figure}
\begin{center}
\includegraphics{pinchfig1.eps}
\caption{A mating of a representation of $C_2*C_3$ with a Douady
rabbit (and zoom). The arc $\gamma_0$ and its images are shown.
Pinching these gives a mating of $PSL_2(\mbox{$\mathbb Z$})$ with the rabbit, by
Theorem 2.}
\end{center}
\end{figure}
\medskip
{\bf Corollary} {\it Conjecture 1 is true for all quadratic maps
$q_c$ which satisfy conditions (i) and (ii).}
\medskip
The class of {\it weakly hyperbolic} quadratic maps is quite
large: for example it contains all quadratic maps which satisfy
the {\it Collet-Eckmann} condition \cite{pr}, and those which
contain parabolic points.
\medskip
We next investigate pinching $\gamma_{p/q}$, for $p/q \ne 0$. In
Section 3.2, we define the notion of a mating between the
circle-packing representation $r_{p/2q}$ of $C_2*C_3$ and $q_c$.
This generalises our earlier definition of a {\it mating between
$PSL_2(\mbox{$\mathbb Z$})$ and $q_c$}, replacing $K(q_c)$ by a certain
identification space $K(q_c)/\sim_{p/q}$ and replacing the
condition that $\Lambda_+\cap\Lambda_-$ be a point by the
condition that it consist of $q$ points (the $p/q$ {\it Sturmian
orbit} on the boundary of $K(q_c)$). We show that a mating between
$r_{p/2q}$ and $q_c$ depends only on $p/q$ mod $1$. To avoid
technical difficulties we restrict attention to the special case
that the quadratic map is $q_0:z \to z^2$. Using the techniques of
\cite{H2}, we prove the following:
\begin{thm}\label{rat} Let $p_0(z,w)$ be a mating between the
representation $r_*$ and $q_0$, and let $p/q$ be any rational.
Then there exists a pinching deformation of $p_0$ such that
$(p_t)_{0 \le t <1}$ converges uniformly to a mating $p_1$ between
the circle-packing representation $r_{p/2q}$ of $C_2*C_3$ in
$PSL_2(\mbox{$\mathbb C$})$ and $q_0$.
\end{thm}
The following is the natural generalisation of Conjecture 1.
\medskip
{\bf Conjecture 2} {\it For every $0\le p/q<1$, the family $(2)$
of $(2:2)$ correspondences contains matings between the
circle-packing representation $r_{p/2q}$ and every quadratic
polynomial $q_c$ which has $c \in {\mathcal M}\setminus{\mathcal
M}_{1-p/q}$, where ${\mathcal M}_{1-p/q}$ denotes the
$(1-p/q)$-limb of the Mandelbrot set ${\mathcal M}$.}
\medskip
The condition that $c$ does not lie in ${\mathcal M}_{1-p/q}$ is
necessary for elementary topological reasons. One might hope to
generalise the techniques of the present paper to prove Conjecture
2 in the case that $q_c$ satisfies conditions (i) and (ii) of the
hypotheses of Theorem \ref{simple}, but the technical details could be
formidable.
\medskip
{\bf Warning} As will already be apparent, certain of the
constructions and results in this article depend on $p/q \in
{\mathbb Q}$, certain depend only on $p/q$ mod $1$ (the class of
$p/q$ in ${\mathbb Q}/{\mathbb Z}$), and certain on $p/q$ mod $2$.
We shall try to make the dependence clear in each case, but
briefly the situation may be summed up as follows. A
circle-packing representation $r_{p/2q}$ of $C_2*C_3$ depends on
$p/q$ mod $2$ but the route to it (in the moduli space ${\mathcal
D}$) given by pinching $\delta_{p/q}$ depends on $p/q \in {\mathbb
Q}$. A mating between $r_{p/2q}$ of $C_2*C_3$ and $q_c$ depends
only on $p/q$ mod $1$, but again the route to it (in mating space)
given by pinching $\gamma_{p/q}$ depends on $p/q \in {\mathbb Q}$.
\section{Matings between quadratic maps and representations of $C_2*C_3$}
We define what we mean by {\it matings} between quadratic maps and
representations of $C_2*C_3$ in $PSL_2(\mbox{$\mathbb C$})$ which lie in
${\mathcal D}^o$, we recall the main ideas of the proof \cite{BH}
of Theorem \ref{mating}, we prove Proposition \ref{compatible}, and we
present a group-theoretic description of the `ordinary set'
$\Omega(f)$ of a mating.
\subsection{Faithful discrete
representations with connected ordinary sets}
Up to conjugacy each representation $r$ of $C_2*C_3$ in
$PSL_2(\mbox{$\mathbb C$})$ is determined by a single complex parameter, the
cross-ratio between the fixed points on $\hat{\mathbb C}$ of the
action of the generator $\sigma$ of $C_2$ and those of the
generator $\rho$ of $C_3$. Such a representation comes equipped
with a (unique) involution $\chi$ which exchanges the two fixed
points of $\sigma$ and also those of $\rho$, so that
$\chi\sigma=\sigma\chi$ and $\chi\rho=\rho^{-1}\chi$. On the
Poincar\'e $3$-ball $\chi$ is simply rotation through $\pi$ around
the common perpendicular to the axes of $\sigma$ and $\rho$. Write
$G$ for the group $<\sigma,\rho,\chi>$, and note that it has
ordinary set $\Omega(G)$ the same as that of $<\sigma,\rho>$.
\medskip The faithful discrete actions $r:C_2*C_3 \subset
PSL_2(\mbox{$\mathbb C$})$ with connected ordinary set $\Omega(G)$ form a single
quasiconformal conjugacy class, the class of representations for
which one can find simply-connected fundamental domains for
$\sigma$ and $\rho$ with interiors together covering the whole
Riemann sphere (the conditions of the simplest form of the Klein
Combination Theorem are satisfied) \cite{mas}. Such fundamental
domains may be constructed as illustrated in figure 2.
\begin{figure}
\begin{center}
\input{pinchfig2.pstex_t}
\caption{A fundamental domain $D_G$ for the group
$G=<\sigma,\rho,\chi>$.} \label{fund}
\end{center}
\end{figure}
\medskip
Here $P$ and $P'$ are the fixed points of $\rho$, $Q$ and $Q'$ are
the fixed points of $\sigma$, $R$ is a fixed point of (the
involution) $\chi\rho$ and $S$ and $S'$ are the fixed points of
$\chi\sigma$. The lines $l,m$ and $n$, joining $R$ to $S$, $Q$ to
$S$ and $R$ to $P$, are chosen such that they are smooth and
remain non-intersecting in the quotient orbifold $\Omega(G)/G$.
The region bounded by $n,\rho n, \chi n$ and $\chi\rho n$ is a
fundamental domain for $\rho$, and the region exterior to the loop
made up of $m,\sigma m,\chi m$ and $\chi\sigma m$ is a fundamental
domain for $\sigma$. The intersection of these two regions is a
fundamental domain for the (faithful) action of $C_2*C_3$ on
$\Omega(G)$, and the half $D_G$ of this intersection bounded by
$n,l,m,\sigma m, \chi l$ and $\rho n$ is a fundamental domain for
the action of $G$. The union of all translates of $D_G$ under
elements of $C_2*C_3$ is a topological disc $D$ which is a
fundamental domain for the action of $\chi$ on $\Omega(G)$. The
complement $\Lambda(G)$ of $\Omega(G)= D \cup \chi(D)$ in
$\hat{\mathbb C}$ is a Cantor set.
\medskip
The orbifold $\Omega(G)/G$ is a sphere $\Sigma$, which has a
complex structure with four cone points, which we may also label
$P,Q,R,S$, where $P$ has angle $2\pi/3$ and $Q,R,S$ each have
angle $\pi$. For a given representation of $C_2*C_3$, a set of
lines $l,m,n$ as in figure 2 descend to an isotopy class of
non-intersecting paths joining the corresponding cone points in
$\Sigma$. By considering the choices we may make of the various
labels and lines in figure 2 we can obtain a description of
$\widetilde{\mathcal D}^0$, the universal cover of the moduli
space ${\mathcal D}^0$.
\begin{lemma}\label{Kleinian-markings}
There is a homeomorphism $\Phi$ between ${\mathcal D}^o$ and the
space $\mathcal S$ of spheres $\Sigma$ having a complex structure
with four marked cone points $P,Q,R,S$ where $P$ has angle
$2\pi/3$ and $Q,R,S$ each have angle $\pi$. This homeomorphism
$\Phi$ lifts to a homeomorphism $\tilde{\Phi}$ between
$\widetilde{\mathcal D}^o$ and the space $\tilde{\mathcal S}$ of
spheres $\Sigma \in {\mathcal S}$ marked with an isotopy class of
non-intersecting paths $PR$, $RS$ and $SQ$.
\end{lemma}
{\bf Proof.} For $r\in {\mathcal D}^o$ define $\Phi(r)$ to be the
orbifold $\Omega(G)/G$, where $G=<\sigma,\rho,\chi>$ is the
subgroup of $PSL_2(\mbox{$\mathbb C$})$ corresponding to the representation $r$.
Clearly $\Phi$ is continuous as ${\mathcal D}^o$ is endowed with
the topology induced by its parametrisation by the cross-ratio
$(Q,Q';P,P')$. To define an inverse to $\Phi$, observe that given
any $\Sigma \in {\mathcal S}$, we may obtain a representation $r$
by regarding $\Sigma$ as a quasiconformal deformation of the
orbifold corresponding to $r_*$, lifting the corresponding ellipse
field to $\hat{\mbox{$\mathbb C$}}$, and applying the Measurable Riemann Mapping
Theorem.
\medskip
To lift $\Phi$ to a homeomorphism $\tilde{\Phi}$ we have to
consider markings. Note that given a representation of $C_2*C_3$
which lies in ${\mathcal D}^o$, there is only one choice for which
of the pair $P,P'$ (in figure 2) to label $P$, namely the fixed
point of $\rho$ around which the rotation is {\it anticlockwise}.
There is also just one choice (up to isotopy) for the arc $n$. The
labels $Q$ and $Q'$ are interchangeable (provided that we also
interchange the labels $S$ and $S'$), but once a choice has been
made for $Q$ the arc $m$ is determined, and even if the labels $Q$
and $Q'$ are exchanged the arc $QS$ in the orbifold $\Sigma$ is
unchanged up to isotopy. This just leaves us a choice of the arc
$l$ in figure 2. We can alter $l$ to wind an extra $n$ times
around the central `hole' for any integer $n$, or $n+1/2$ times if
we switch the labels $Q$ and $Q'$. Changing the winding number of
$l$ corresponds to choosing a different isotopy class of paths
between the points labelled $R$ and $S$ in the orbifold $\Sigma$.
$\square$
\medskip
Let $t_\alpha$ denote the automorphism of $\widetilde{\mathcal
D}^o$ corresponding to turning the internal boundary of figure 2
through an angle $2\pi\alpha$. Note that $t_{1/4}$ moves the pair
of points labelled $Q,Q'$ to the pair labelled $S,S'$ and vice
versa. Let $\iota:{\mathcal D}^o \to {\mathcal D}^o$ denote the
involution obtained by replacing the generating pair
$\{\sigma,\rho\}$ of $C_2*C_3$ by $\{\sigma',\rho \}$, where
$\sigma'=\chi\sigma$. This corresponds to composing the
representation with an outer automorphism of $C_2*C_3$. The
following result is now self-evident.
\begin{lemma}\label{quarter-twist} The automorphism
$t_{1/4}:\widetilde{\mathcal D}^o \to \widetilde{\mathcal D}^o$
covers $\iota:{\mathcal D}^o \to {\mathcal D}^o$, and $t_{1/2}$
generates the group of covering transformations of
$\widetilde{\mathcal D}^o \to {\mathcal D}^o$. $\square$
\end{lemma}
\subsection{Matings between $q_c$ and $r \in {\mathcal D}^o$}
As in the previous subsection, $G$ denotes the group
$<\sigma,\rho,\chi>$.
\medskip
{\bf Definition} {\it A $(2:2)$ holomorphic correspondence $f:z
\to w$ is called a {\it mating} between a faithful discrete
representation $r$ of $C_2*C_3$ in $PSL_2(\mbox{$\mathbb C$})$ having connected
ordinary set $\Omega(G)$ and a polynomial $q_c:z \to z^2+c$ having
connected filled Julia set $K(q_c)$, if the Riemann sphere
$\hat{\mathbb C}$ is the disjoint union of a connected open set
$\Omega(f)$ and a closed set $\Lambda(f)$ made up of two
components, $\Lambda_+(f)$ and $\Lambda_-(f)$ such that each of
$\Omega(f)$ and $\Lambda(f)$ is completely invariant under $f$
and:
\medskip
(a) the action of $f$ on $\Omega(f)$ is discontinuous and there is
a conformal bijection between the grand orbit space $\Omega(f)/f$
and $\Omega(G)/G$;
\medskip
(b) there is a quasiconformal homeomorphism defined from a neighbourhood of
$\Lambda_-(f)$ onto a neighbourhood of $K(q_c)$ in $\mbox{$\mathbb C$}$, which realises a
hybrid equivalence, conjugating $f$ to $q_c$. Similarly there is a
hybrid equivalence between $(f^{-1},\Lambda_+(f))$ and
$(q_c,K(q_c))$, this time conjugating $f^{-1}$ to $q_c$.}
\medskip
(See \cite{DH} for the definition of the term `hybrid
equivalence'.)
\bigskip
The construction of a holomorphic correspondence which realises a
mating between given $q_c$ and $r$ proceeds as follows (see
\cite{BH} for more details).
\medskip
We first associate an annulus $A$ to $q_c:z \to z^2+c$. There is a
holomorphic conjugacy (the B\"ottcher coordinate) from $z \to z^2$
to $q_c$ on a neighbourhood of $\infty$, fixing the point $\infty$
and tangent to the identity map there \cite{DH1}. An {\it
equipotential} for $q_c$ is the image of a circle $\{Re^{2\pi it}:
0 \le t <1\}$ under this conjugacy. It is a smooth Jordan curve
parameterized by {\it external angle} $t$. The region bounded by
such an equipotential is a simply-connected domain $V$, mapped
$2:1$ by $q_c$ onto a larger domain $U \supset \overline{V}$ which
also has boundary an equipotential parametrised by external angle.
Let $A$ denote the annulus $U-V$, and denote its inner and outer
boundaries by $\partial_1 A$ and $\partial_2 A$ respectively. The
map $q_c$ sends $\partial_1 A$ two-to-one onto $\partial_2 A$.
There is an involution $i:z \to -z$ on $V$ sending each $z \in V$
to the other point which has the same image in $U$ under $q_c$,
and there are many choices possible of an orientation-reversing
smooth involution $j$ on $\partial_2 A$, a canonical choice being
given by $t \to 1-t$ on external angles.
\begin{figure}
\begin{center}
\input{pinchfig3.pstex_t}
\caption{The set $D_G \cup \rho(D_G) \cup
\rho^{-1}(D_G)$ and its quotient the annulus $B$.}
\label{ann}
\end{center}
\end{figure}
\medskip
The next ingredient is an annulus $B$ associated to $r$. Recall
the fundamental domain $D_G$ constructed above for the group
$G=<\sigma,\rho,\chi>$ and illustrated in figure 2. Let $B$ denote
the annulus consisting of the three copies $D_G \cup \rho D_G \cup
\rho^{-1}D_G$ of $D_G$, with the boundary identifications (induced
by $\chi$) indicated in figure 3. The rotations $\rho$ and
$\rho^{-1}$ mapping $D_G \cup \rho D_G \cup \rho^{-1} D_G$ to
itself descend to a $2:2$ correspondence $g$ on $B$, mapping each
$z \in B$ to the pair $\{\rho z, \rho^{-1}z\}$ (or rather to their
equivalence classes under the action of $\chi$). The set $D_G$
descends to a `fundamental domain' for the action of $g$ on $B$.
The boundary of $B$ is divided into three segments (two inner and
one outer, figure 3), each of which is mapped to the other two by
$g$. Thus when its domain is restricted to the inner boundary
$\partial_1 B$, and its range is restricted to the outer boundary
$\partial_2 B$, the correspondence $g$ defines a two-to-one map.
When restricted to a correspondence from the inner boundary to
itself, $g$ defines a (fixed point free) bijection. Moreover the
involution $\sigma$ descends to an involution (which we also
denote $\sigma$) on the outer boundary $\partial_2 B$ of $B$,
having fixed points $Q$ and $S$.
\medskip
\begin{lemma}\label{AB} There exists a quasiconformal homeomorphism
$h$ from $A$ to $B$ which restricts to a smooth homeomorphism from
$\partial A$ to $\partial B$ conjugating the boundary maps
$(q_c:\partial_1 A \to
\partial_2 A,\ j:\partial_2 A \to \partial_2 A)$ to the
boundary maps $(\sigma \circ g:\partial_1 B \to
\partial_2 B,\ \sigma: \partial_2 B \to \partial_2 B)$.
\end{lemma}
\medskip This lemma is proved \cite{BH} by applying standard
techniques of Ahlfors and Bers. Now to construct a mating between
$q_c$ and $r$ first glue together $U$ and a second copy $U'$ of
$U$, via the boundary involution $j$, to obtain a sphere $U \cup
U'$, equipped with an involution (also denoted $j$) exchanging $U$
with $U'$ and restricting to the original $j$ on the common
boundary. Inside $U'$ is a simply-connected subdomain $V'$
corresponding to $V \subset U$. Let $q_c'=j\circ q_c \circ j:V'
\to U'$ denote the quadratic map corresponding to $q_c:V \to U$
and $A'$ denote the annulus $U'-V'$. To define a $2:2$ topological
correspondence $f$ on $U \cup U'$ we fit together:
$\bullet$ $q_c:V \to U$ (a $2:1$ correspondence);
$\bullet$ $(q_c')^{-1}=j\circ q_c^{-1} \circ j:U' \to V'$ (a $1:2$
correspondence);
$\bullet$ $j\circ i:V \to V'$ (a $1:1$ correspondence), and
$\bullet$ $j\circ g:A \to A'$ (a $2:2$ correspondence),
where $g:A \to A$ is the $2:2$ correspondence constructed earlier.
Now define an ellipse field on $A$ by using Lemma \ref{AB} to
transport the standard complex structure from the annulus $B$.
Using $j$ extend this ellipse field to $A'$ and pulling back via
$q_c^{-1}$ and $q_c'^{-1}$ extend it to an ellipse field on the
whole of $\hat{\mathbb C}-(K(q_c) \cup K(q_c'))$, which transforms
equivariantly under the action of the $2:2$ correspondence $f$.
Extend this ellipse field to the whole of $\hat{\mathbb C}$ by
using the standard complex structure on $K(q_c) \cup K(q_c')$. By
applying the Measurable Riemann Mapping Theorem we obtain a
complex structure respected by $f$, completing our outline proof
of Theorem \ref{mating}.
\medskip
For any mating $f$ constructed by the method of the proof above,
the $3:3$ correspondence $(j\circ f)\cup I_{\hat{\mathbb C}}$
sends each $z\in V$ to the triple of points $\{z,i(z),jq_c(z)\}$,
each $z\in A$ to the triple $\{z,g(z)\}$ (recall that $g$ is $2:2$
so $g(z)$ contains two points), and each $z \in U'$ to the triple
$\{z,q_c^{-1}j(z)\}$. It is easily checked that each of these
triples is the grand orbit under $(j\circ f)\cup I_{\hat{\mathbb
C}}$ of any one of its elements, in other words the $3:3$
correspondence is an equivalence relation. The involution $j$ is
therefore {\it compatible} with the mating $f$ in the sense
defined in Section 1. To show that $f$ is conjugate to a
correspondence in the family $(2)$ it now only remains to prove
Proposition \ref{compatible}. But a holomorphic correspondence
which is an equivalence relation is necessarily the covering
correspondence of a rational map, and so there is a rational map
$Q$ of degree three such that $(J\circ f)\cup I_{\hat{\mathbb
C}}=Cov^Q$ where
$$Cov^Q: z \to w \quad \Leftrightarrow \quad Q(w)-Q(z)=0.$$
We deduce that $f=J\circ Cov^Q_0$, where
$$Cov^Q_0: z \to w \quad \Leftrightarrow \quad
\frac{Q(w)-Q(z)}{w-z}=0.$$ Counting singular points of $f$ now
tells us that $Q$ has one double and two single critical points,
and that therefore up to pre- and post-compositions by M\"obius
transformations $Q$ is the polynomial $Q(z)=z^3-3z$. It follows
that up to conjugacy we may write $f$ in the form
$$z \to w \quad \Leftrightarrow \quad (Jw)^2+(Jw)z+z^2=3.$$
It is easy to see that if we apply a further conjugacy to
transform $J$ to the involution $J(z)=-z$, the equation defining
the correspondence $f$ becomes a member of the family $(2)$. This
completes the proof of Proposition \ref{compatible}.
\subsection{A group-theoretic description of $\Omega$ for a mating}
We shall be pinching unions of arcs in $\Omega(f)$ which are lifts
of simple closed curves in the grand orbit space $\Omega(f)/f$,
where $f$ is one of the matings provided by Theorem 1. With a view
to describing these arcs we examine the structure of $\Omega(f)$
and its relationship with $\Omega(G)$. Our first step will be to
find a {\it Fuchsian} uniformisation for $\Omega(G)/G$.
\medskip
Let $\Gamma$ denote the abstract group $<\sigma,\rho,\tau:
\sigma^2=\rho^3=\tau^2=(\sigma\rho\tau)^2=1>$.
\medskip
Let ${\mathcal F}$ denote the moduli space of conjugacy classes of
faithful discrete co-compact representations of $\Gamma$ in
$PSL_2(\mbox{$\mathbb R$})$ (recall that a Fuchsian group is said to be {\it
co-compact} if the quotient of the Poincar\'e disc by its action
is compact). An example of a representation of $\Gamma$ which lies
in ${\mathcal F}$ is illustrated in figure 4. Let ${\mathcal F}_0$
denote the path-component of ${\mathcal F}$ containing the
representation illustrated. Thus a faithful discrete
representation of $\Gamma$ lies in ${\mathcal F}_0$ if and only if
there is a fundamental domain $D_\Gamma$ for $\Gamma$ isotopic to
that illustrated in figure 4, with boundary passing through the
fixed points of the corresponding elements of $\Gamma$, in the
same order but with the intervening boundary segments no longer
necessarily geodesic.
\begin{figure}
\begin{center}
\input{pinchfig4.pstex_t}
\caption{A Fuchsian representation of $\Gamma$. Here $P,Q,R$ and
$S$ are the fixed points of $\rho,\sigma,\tau$ and
$\sigma\rho\tau$. The heavy lines indicate the boundary of
$D_{\Gamma_1}$.} \label{Gamma1}
\end{center}
\end{figure}
\medskip
Let $\sigma'=\rho\tau\sigma$. Then $\sigma'$, $\rho$ and $\tau$
together generate $\Gamma=<\sigma,\rho,\tau>$, and satisfy the
same relations. Changing to the new generating set amounts to
applying an (outer) automorphism, which we denote $\beta$, to
$\Gamma$. Let $\psi$ be the automorphism of ${\mathcal F}_0$
induced by composing the representation $\gamma \to PSL_2(\mbox{$\mathbb R$})$
with $\beta$ and replacing the boundary of $D_\Gamma$ in figure 3
with that given by moving $S$ up to $Q$ (the fixed point of
$\sigma'\rho\tau$), and $Q$ up to $\sigma(S)$ (the fixed point of
$\sigma'$), but keeping $P$ and $R$ unchanged.
\medskip
Recall our description in Section 2.1 of the universal cover
$\widetilde{{\mathcal D}^o}$ of the space ${\mathcal D}^o$ of
conjugacy classes of faithful discrete representations of
$C_2*C_3$ in $PSL_2(\mbox{$\mathbb C$})$ having connected ordinary set.
\begin{prop}\label{Kleinian-Fuchsian} There is a
homeomorphism $\Psi:\widetilde{{\mathcal D}^o} \to {\mathcal
F}_0$, which carries $t_{1/4}$ to $\psi$ and hence induces
homeomorphisms:
(i) ${\mathcal D}^o \to {\mathcal F}_0/<\psi^2>$;
(ii) ${\mathcal D}^o/\iota \to {\mathcal F}_0/<\psi>$.
\end{prop}
{\bf Proof.} By Lemma \ref{Kleinian-markings}, Section 2.1, a
point of $\widetilde{\mathcal D}^o$ corresponds to an element of
$\tilde{\mathcal S}$, that is to say a sphere equipped with a
complex structure having cone points $P$ of angle $2\pi/3$, and
$Q,R$ and $S$ all of angle $\pi$, together with an isotopy class
of paths $PR$, $RS$ and $SQ$. Obviously it suffices to define a
homeomorphism between $\tilde{\mathcal S}$ and ${\mathcal F}_0$.
\medskip
To do this we uniformise each marked orbifold $\Sigma\in
\tilde{\mathcal S}$ as a quotient of the Poincar\'e disc $\Delta$
by isometries. The marked arcs on $\Sigma$ lift to a union of
arcs, tiling $\Delta$ by translates of a polygon isotopic to that
labelled $D_\Gamma$ in figure 4. The group of covering
transformations of the projection from $\Delta$ to $\Sigma$ is
isomorphic to $\Gamma$ by Poincar\'e's polygon theorem \cite{B}.
Conversely, given a faithful discrete representation of $\Gamma$
lying in ${\mathcal F}_0$, its quotient orbifold $\Sigma$ is an
element of $\tilde{\mathcal S}$. Thus we have a bijection
$\tilde{\mathcal S} \to {\mathcal F}_0$ which, by construction, is
continuous and has a continuous inverse. Since $t_{1/4}$ and
$\psi$ have identical effects on $\Sigma$, our composite
homeomorphism $\Psi:\widetilde{{\mathcal D}^o} \to {\mathcal F}_0$
carries $t_{1/4}$ to $\psi$, and the assertions (i) and (ii) are
immediate corollaries. $\square$
\medskip {\bf Remark.}
The question of finding explicit formulae for bijections between
moduli spaces of representations of Kleinian groups and Fuchsian
groups, such as the bijection provided by Proposition
\ref{Kleinian-Fuchsian}, is in general highly non-trivial, a
classical example being to relate each Schottky group to a
Fuchsian group representing the same surface.
\bigskip Now let $\Gamma_1 \subset \Gamma$ be the subgroup
generated by $\rho\tau$ (which has infinite order), the involution
$\rho^{-1}\tau\rho$, and all involutions of the form
$W\rho^{-1}\tau\rho W^{-1}$, where $W$ runs through those words in
$\sigma$ and $\rho$ which have rightmost letter $\sigma$. Then
$\Gamma_1$ has as fundamental domain the region $D_{\Gamma_1}$
bounded by heavy lines in figure 4. Note that
$D_{\Gamma_1}/\Gamma_1$ is a topological cylinder, the top edge of
the region $D_{\Gamma_1}$ in figure 4 being identified with the
bottom edge, each of the arcs on the left hand edge being folded
in onto an interval, and each of the arcs on the right hand edge
also being folded in onto an interval.
\medskip Suppose $f$ is a $2:2$ holomorphic correspondence which is
a mating, constructed as in Theorem 1, between a faithful discrete
representation of $C_2*C_3$ in $PSL_2(\mbox{$\mathbb C$})$ having connected
ordinary set and a quadratic map $z \to z^2+c$ having connected
Julia set. Let $\Gamma\subset PSL_2(\mbox{$\mathbb R$})$ be the Fuchsian
representation associated to it by Lemma \ref{Kleinian-Fuchsian},
and let $\Gamma_1$ be the subgroup of $\Gamma$ defined above.
\begin{prop}\label{groupify} There is a bi-analytic homeomorphism
$$D_{\Gamma_1}/{\Gamma_1}\cong \Delta/{\Gamma_1}
\to \Omega(f)$$ carrying the action of the pair $\{\sigma \rho,
\sigma\rho^{-1}\}$ on $D_{\Gamma_1}/{\Gamma_1}$ to that of the
correspondence $f$ on $\Omega(f)$. \end{prop}
{\bf Proof.} From the construction of the mating $f$ in our
outline proof of Theorem \ref{mating} (in Section 2.2), it is
apparent that $(\Delta,{\Gamma_1})$ uniformises $\Omega(f)$: the
set $D_{\Gamma_1} \cup \rho D_{\Gamma_1} \cup \rho^{-1}
D_{\Gamma_1}$ in figure 4, when quotiented by the boundary
identifications induced by $\Gamma_1$, becomes the annulus $B$ of
figure 3, and the maps $\sigma\rho$ and $\sigma\rho^{-1}$ become
the two `branches' of the correspondence $f$ on $\Omega(f)$.
$\square$
\begin{cor}\label{iota} A mating between $q_c$ and $r\in {\mathcal D}^o$
constructed by the method of Theorem \ref{mating} is canonically
isomorphic to a mating between $q_c$ and $\iota(r)$.
\end{cor}
{\bf Proof.} The outer automorphism $\beta$ defined by replacing
the generator $\sigma$ of $\Gamma$ by $\sigma'=\rho\tau\sigma$
stabilises $\Gamma_1$, and the correspondence induced by
$\{\sigma\rho,\sigma\rho^{-1}\}$ on $\Delta/\Gamma_1$ is the same
as that induced by $\{\sigma'\rho,\sigma'\rho^{-1}\}$, since
$\sigma'\rho=\rho\tau\sigma\rho$ and
$\sigma'\rho^{-1}=\rho\tau\sigma\rho^{-1}$. $\square$
\medskip
{\bf Remarks}
\medskip 1. The idea of regarding $\Omega(f)$ as a quotient of
$\Delta$ by an infinitely generated Fuchsian group is originally
due to Chris Penrose.
\medskip
2. We can recover the action of the Kleinian group
$G=<\sigma,\rho,\chi>$ on $\Omega(G)$ from the action of the
corresponding Fuchsian group $\Gamma=<\sigma,\rho,\tau>$ on
$\Delta$, as follows. Take the polygon
$D_{\Gamma_2}=D_{\Gamma_1}\cup \rho\tau(D_{\Gamma_1})$ formed by
two copies of $D_{\Gamma_1}$, one above the other, identify the
top and bottom edges of this polygon to form a cylinder, then fold
and glue the left-hand edge together and fold and glue the right
hand edge together, to form a sphere. The quotient
$D_{\Gamma_2}/\sim$, which can also be described as an orbit space
$\Delta/\Gamma_2$ for an appropriate infinitely generated subgroup
$\Gamma_2 \subset \Gamma$, is conformally equivalent to
$\Omega(G)$. Indeed $\Gamma_2\cong \pi_1(\Omega(G))$, and the
projection $\Delta \to \Delta/\Gamma_2$ is the universal cover for
$\Omega(G)$. Under the bijection from $D_{\Gamma_2}/\sim$ to
$\Omega(G)$ the ends of $D_{\Gamma_2}$ (the cusps) become the
points of the limit set $\Lambda(G)$ of the action of the Kleinian
group $G$ on $\hat{\mbox{$\mathbb C$}}$.
\section{The pinching deformation}
\subsection{The arcs to be pinched}
To describe the arcs that we shall pinch later, we first fix a
standard faithful discrete representation $r_*$ of $C_2*C_3$
having connected ordinary set, and a path $l$ from a fixed point
$R$ of $\chi\rho$ to a fixed point $S$ of $\chi\sigma$ (so $R$ and
$S$ are as illustrated in figure 2). For convenience we may choose
$r_*$ and $l$ so that the corresponding group $\Gamma$ has the
reflection symmetry in the horizontal axis apparent in figure 4.
Now consider the double cover $\tilde{\Sigma}$ of the orbifold
$\Sigma$ ramified at all four cone points. This is a torus, with a
single cone point $P$ of angle $4\pi/3$, represented by the
central hexagon $D_\Gamma \cup \sigma D_\Gamma$ illustrated in
figure 4, with the top edge identified with the bottom edge, and
the left-hand edge identified with the right-hand edge. While
$\tilde{\Sigma}$ is not itself a quotient of the unit disc
$\Delta$ by a subgroup of $PSL_2(\mbox{$\mathbb C$})$ (since the cone point is not
of angle $2\pi/n$), nevertheless we may equip ${\tilde{\Sigma}}$
with the metric induced by the restriction of the hyperbolic
metric on $\Delta$ to the hexagon $D_\Gamma \cup \sigma D_\Gamma$.
The involution $\sigma$ (on $\Delta$) induces an involution
$\tilde\sigma$ on $\tilde{\Sigma}$ such that ${\tilde{\Sigma}}/
\tilde{\sigma}={\Sigma}$.
\begin{lemma}\label{arcs-exist} For each rational number $p/q$
there is a geodesic arc $\delta_{p/q}$ in ${\Sigma}$ which has end
points two of the three cone points of angle $\pi$, which misses
the other cone point of angle $\pi$ and the cone point of angle
$2\pi/3$, and which has lift $\tilde{\delta}_{p/q}$ to
$\tilde{\Sigma}$ a simple closed geodesic of winding number $p/q$.
\end{lemma}
{\bf Proof.} For each such $p/q$ (in lowest terms), there is a
simple closed curve of winding number $p/q$ on the torus
${\tilde{\Sigma}}$, passing through (i) the cone points $Q$ and
$S$ if $q$ is even, (ii) the cone points $Q$ and $R$ if $p$ and
$q$ are both odd, and (iii) the cone points $R$ and $S$ if $p$ is
even. Examples are illustrated in figure 5 for typical cases of
each type.
\begin{figure}
\begin{center}
\input{pinchfig5.pstex_t}
\caption{The arcs $\tilde{\delta}_{p/q}$ for $p/q=1/2$, $1/3$,
$2/3$ and $4/3$ respectively.} \label{winding-nunmbers}
\end{center}
\end{figure}
\medskip
Note that when we add an even integer to $p/q$ the new
$\delta_{p/q}$ is an arc between the same two cone points on
${\Sigma}$. But when we add an odd integer the roles of $Q$ and
$S$ are interchanged.
\medskip In every case the simple closed curve on
$\tilde{\Sigma}$ can be chosen to be invariant under
$\tilde\sigma$. Since it passes through the lifts of two cone
points, it descends to an arc on ${\Sigma}$ joining these two
points. We define $\delta_{p/q}$ to be a representative of
shortest length in the isotopy class of this arc, relative to its
end points and the other two cone points on ${\Sigma}$. Note that
there must exist such a minimal length example, as arcs which pass
through one or both of the other cone points have lengths which
are local maxima (since all the cone points have cone angle less
that $2 \pi$). $\square$
\medskip
Let $A_{p/q}$ denote the lift of $\delta_{p/q}$ to the cylinder
$(D_\Gamma \cup \sigma D_\Gamma)/\Gamma_1$ constructed by
identifying the top and bottom of the hexagon. Thus $A_{p/q}$
consists of $q$ arcs each running from one boundary circle of this
cylinder to the other. Consider the union $\Gamma A_{p/q}$ of all
lifts of $\delta_{p/q}$. Recall that $D_{\Gamma_1}/{\Gamma_1}$ is
a cylinder, with ends corresponding to $\partial \Lambda_-$ and
$\partial \Lambda_+$ (by Proposition \ref{groupify}), that the
correspondence $f$ acts on $\partial \Lambda_-$ as a quotient of
the doubling map, and that $f^{-1}$ acts on $\partial \Lambda_+$
as a quotient of the doubling map. For simplicity of description
assume that $\partial \Lambda_-$ is a topological circle and the
action of $f$ on it is that of the doubling map (this is the case
when the quadratic map in the mating corresponds to a value of $c$
in the interior of the main cardioid of the Mandelbrot set):
obvious adaptations are possible for the cases where $\partial
\Lambda_-$ is a proper quotient of the circle.
\begin{figure}
\begin{center}
\input{pinchfig6.pstex_t}
\caption{The three arcs linking $\Lambda_-$ to $\Lambda_+$ in the case
$p/q=1/3$ (the other images of these arcs under $\Gamma$ are not
shown).}
\label{arcs-on-omega}
\end{center}
\end{figure}
\medskip
If we label the ends of $\partial D_{\Gamma_1}$ by binary
sequences as indicated in figure 6 then the folding
identifications induced by $\Gamma_1$ impose the usual quotient
from the space of binary sequences to the unit circle, carrying
the shift to the doubling map. Thus, under our assumption that
$\partial \Lambda_-$ is the circle, points of $\partial \Lambda_-$
are labelled (figure 6) in such a way that $f^{-1}:\partial
\Lambda_- \to \partial \Lambda_-$ (a $1:2$ correspondence) is
defined by ``right shift and insert $0$ or $1$'' according as the
branch of $f^{-1}$ is $\rho\sigma$ or $\rho^{-1}\sigma$
respectively, and points of $\partial \Lambda_+$ are labelled in
such a way that $f:\partial \Lambda_+ \to \partial \Lambda_+$
(also a $1:2$ correspondence) is defined by ``right shift and
insert $0$ or $1$'' according as the branch of $f$ is $\sigma\rho$
or $\sigma\rho^{-1}$ respectively. We adopt the usual notational
convention that a bar over a symbol (or group of symbols)
indicates the infinite repetition of that symbol (or group of
symbols).
\medskip {\bf Definition} {\it An infinite sequence of $0$'s and
$1$'s is known as {\it Sturmian} if the binary number it
represents on the circle has orbit under the doubling map a
sequence of points arranged in the same order around the circle as
for a rigid rotation.}
\medskip One may assign a rotation number to each Sturmian sequence
$s$, namely the limit as $n$ tends to infinity of the proportion
of the first $n$ digits of $s$ which are $1$'s, or equivalently
the rotation number of the rigid rotation having orbit points in
the same order as those of $s$. Note that such a rotation number
is only defined mod $1$. For each rational $p/q$ (mod $1$) there
is a unique periodic Sturmian orbit of rotation number $p/q$ (this
was observed by Morse and Hedlund, who introduced the notion of
Sturmian sequences). We remark that the points of each periodic
Sturmian orbit ${\mathcal O}$ must be contained in an interval of
length less than $1/2$ on the circle $\mbox{$\mathbb R$} /\mbox{$\mathbb Z$}$, as the doubling map
must preserve the cyclic order of ${\mathcal O}$ (see \cite{BS}
for more about this and other properties of Sturmian sequences).
\medskip
{\bf Examples}
\medskip
The infinite sequences $\overline{01}$, $\overline{001}$ and
$\overline{00101}$ are Sturmian, of rotation numbers $1/2,1/3$ and
$2/5$ respectively.
\begin{prop}\label{landing-points} $(\Gamma A_{p/q}\cap
D_{\Gamma_1})/\Gamma_1$ contains exactly $q$ arcs which join
$\Lambda_-$ to $\Lambda_+$. These land on $\partial\Lambda_-$ at
points of the unique Sturmian orbit of rotation number $p/q$ (mod
$1$) of the $2:1$ map $f:\partial\Lambda_- \to
\partial\Lambda_-$ and at the other end they land on $\partial
\Lambda_+$ at points of the unique Sturmian orbit of $f^{-1}$ of
rotation number $p/q$ (mod $1$).
\end{prop}
{\bf Proof.} The fact that there are exactly $q$ arcs joining
$\Lambda_-$ to $\Lambda_+$ follows at once from the fact that
exactly $q$ arcs in $(\Gamma A_{p/q}\cap D_{\Gamma_1})/\Gamma_1$
cross the equator circle of the central cylinder $(D_\Gamma \cup
\sigma D_\Gamma)/\Gamma_1$ (the vertical line in the central
hexagon in $D_{\Gamma_1}$). The action of the correspondence
$f^{-1}=\{\rho^{-1}\sigma, \rho\sigma\}$ on these arcs is to map
the $j$th arc to the $(j+p)$th arc for each $j$, where the arcs
are counted modulo $q$, from the bottom of the central hexagon
upwards. Thus the action of $f^{-1}$ on the landing point of the
$j$th arc on $\Lambda_+$ is to send it to the landing point of the
$(j+p)$th arc, for each $j$. Similarly $f$ sends the $j$th landing
point on $\Lambda_-$ to the $(j+p)$th. $\square$
\medskip
{\bf Definition of the arc $\gamma_{p/q}$.} {\it For each $p/q$ we
pick as $\gamma_{p/q}$ one of the $q$ components of $(\Gamma
A_{p/q}\cap D_{\Gamma_1})/\Gamma_1$ which cross the equator circle
of the central cylinder and therefore join $\Lambda_-$ to
$\Lambda_+$. For definiteness, when $q$ is odd we take
$\gamma_{p/q}$ to be the component which passes through $R$ (the
fixed point of $\tau$) and when $q$ is even we take it to be the
component which passes through $S$ (the fixed point of
$\sigma\rho\tau$). We remark that in the case $p/q=0$ there is
just one component crossing the vertical symmetry line of the
central hexagon, and it passes through both of these points.}
\medskip
In figure 6 we illustrate $\gamma_{1/3}$, which joins
$\overline{010}\in \Lambda_-$ to $\overline{100}\in \Lambda_+$,
and its two images which also join $\Lambda_-$ to $\Lambda_+$.
These join $\overline{100}\in \Lambda_-$ to $\overline{010}\in
\Lambda_+$, and $\overline{001}\in \Lambda_-$ to
$\overline{001}\in \Lambda_+$ respectively. Arcs
$\gamma_{(3n+1)/3}$ for values of $n$ other than $0$, and their
images, join the same pairs of points in $\Lambda_-$ and
$\Lambda_+$, but wind a different number of times around the
cylinder $D_{\Gamma_1}/\Gamma_1$.
\medskip
For general rational $p/q$ we have the following:
\medskip
{\bf Algorithm} {\it Each point in $\Lambda_-$ represented by a
Sturmian $p/q$ word $\overline{u_1\ldots u_q}$ is joined (by
$\gamma_{p/q}$ or one of its images) to the point in $\Lambda_+$
represented by the Sturmian $p/q$ word
$\overline{u_{q-1}u_{q-2}\ldots u_1u_q}$.}
\medskip
{\bf Proof.} Both $\sigma \rho$ and $\sigma\rho^{-1}$ map the
fixed point $P$ of $\rho$ to $\sigma P$. It follows that $f$ maps
the pair of geodesics landing on $\Lambda_-$ either side of $\bar
1$ to the pair of geodesics landing on $\Lambda_+$ either side of
$\bar 1$ (figure 6). The pair of landing points either side of
$\bar 1$ are represented by the maximum and minimum Sturmian $p/q$
words, $M_{p/q}$ and $m_{p/q}$ respectively, so the arcs landing
at these points of $\Lambda_-$ have their opposite ends at the
points of $\Lambda_+$ represented by $s(m_{p/q})$ and $s(M_{p/q})$
respectively, where $s$ denotes left shift (i.e. `forget the first
digit'). Since it is easily proved from the {\it staircase
algorithm} for Sturmian words \cite{BS} that the minimum word
$m_{p/q}=\overline{v_q\ldots v_1}$ is the reverse of the maximum
word $M_{p/q}=\overline{v_1 \ldots v_q}$, the result follows.
Indeed we may regard the $q$ arcs joining $\Lambda_-$ to
$\Lambda_+$ as indexed by a marked digit in a bi-infinite Sturmian
word, and the action of $f$ and $f^{-1}$ on these arcs as moving
the marker left and right. $\square$
\bigskip {\bf Remarks.}
\medskip
1. Which two of the three cone points on ${\Sigma}$ of cone angle
$\pi$ are the end points of the arc $\delta_{p/q}$ is determined
by the reflection symmetries of the bi-infinite periodic Sturmian
word of rotation number $p/q$ mod $1$. Each such word has
reflection symmetries of exactly two of four possible types:
reflection at a $0$, or at a $1$, or between two adjacent $0$'s or
$1$'s. Which two types occur depends on whether (after reduction
of $p/q$ mod $1$) $p$ is even, $q$ is even, or $p$ and $q$ are
both odd. For example the bi-infinite word generated by
$\overline{00101}$, a case where $p$ is even, has reflection
points between the first two $0$'s and at the third $0$. The
stabiliser of any lift of $\delta_{p/q}$ to $\Delta$ is an
infinite dihedral group, generated by a pair of involutions fixing
adjacent lifts of cone points on the arc, and indeed isomorphic to
the group of symmetries of the bi-infinite periodic Sturmian word.
\medskip
2. The same construction of geodesic arcs crossing the central
hexagon can be followed through for {\it irrational} slope $\nu$
in place of $p/q$. One then obtains a lamination on
$D_{\Gamma_1}/{\Gamma_1}$, with singular leaves passing through
the fixed point of $\rho$ and its translates. In this case the
leaves crossing the hexagon join a Cantor set in $\partial
\Lambda_-$, the unique closed invariant Sturmian set of rotation
number $\nu$ mod $1$, to the analogous Cantor set in $\partial
\Lambda_+$. The algorithm above also applies in this case to tell
us which points are joined to which; we omit details here.
\bigskip It remains to describe the grand orbit of $\gamma_{p/q}$
under the correspondence $f$.
\medskip We start with the special case $p/q=0$. The arc $\gamma_0$
is the lower boundary component of the region $D_{\Gamma_1}$ in
figure 4. Under $f$ this component maps to itself and to the
boundary component of $D_{\Gamma_1}$ which passes through the
point $\sigma(T)$. The grand orbit of $\gamma_0$ under $f$ is the
union of all the boundary components of $D_{\Gamma_1}$, and
quotienting by $f$, or equivalently by $\Gamma_1$, folds all these
components (except the original one) into ``spikes''.
\medskip
We now turn to general $p/q$. From the explicit construction of
matings in Section 2.2 it follows that the branch of $f$ mapping
$\Lambda_-$ to $\Lambda_+$ is defined as follows: given a word $W$
in $0$'s and $1$'s representing a point in $\partial\Lambda_-$ the
$f$-image in $\partial\Lambda_+$ of that point is represented by
the word $\phi(W)$ obtained by changing the parity of the first
digit of $W$. It is now a straightforward computation that when
$q$ is even the set of $q$ arcs joining $\Lambda_-$ to $\Lambda_+$
is mapped two to one by this branch to a set of $q/2$
``concentric'' arcs connecting pairwise the $q$ points of
$\Lambda_+$ obtained by applying the operation $\phi$ to the
Sturmian $p/q$ orbit (i.e. the points of the circle {\it opposite}
to points of the Sturmian orbit). When $q$ is odd, the set of $q$
arcs joining $\Lambda_-$ to $\Lambda_+$ is mapped by this branch
of $f$ to a set of $(q-1)/2$ concentric arcs together with an
innermost spike (figure 7) which lands on $\Lambda_+$ at a single
point, the point opposite to the middle point of the Sturmian
$p/q$ orbit. This spike arises from the fact that for $q$ odd the
geodesic $\gamma_{p/q}$ passes through the fixed point of the
involution $\tau$. Hence its image under the branch of $f$ we are
considering passes through the fixed point of an involution in the
group $\Gamma_1$. This fixed point is on the boundary of
$D_{\Gamma_1}$ (indeed in figure 4 it is the point $\sigma(T)$),
and becomes the end point of a spike in the quotient
$D_{\Gamma_1}/\Gamma_1\cong \Omega(f)$.
\begin{figure}
\begin{center}
\input{pinchfig7.pstex_t}
\caption{The Sturmian orbits of rotation number $2/5$ on
$\Lambda_-$ and $\Lambda_+$, the five arcs joining them, and the
first images of these under the correspondence and its inverse
(subsequent images are not shown).}
\label{images_of_arcs}
\end{center}
\end{figure}
\medskip
Applying $f$ again arbitrarily may times to our ``concentric'' set
of $q/2$ arcs (or $(q-1)/2$ arcs plus a spike, if $q$ is odd), we
obtain smaller and smaller copies around $\partial\Lambda_+$, and
applying $\sigma$ to these copies we obtain similar copies around
$\partial\Lambda_-$, together making up the grand orbit under $f$
of our original set of $q$ arcs.
\subsection{Matings between $q_0$ and circle-packing representations of $C_2*C_3$}
We can now define precisely what we mean by the {\it mating}
between $q_0$ and $r_{p/2q}$ referred to in the statement of Theorem
\ref{rat}. After the arcs which make up the grand orbit of
$\gamma_{p/q}$ have been pinched, the intersection $\Lambda_+\cap
\Lambda_-$ is no longer empty, but consists of the $p/q$ Sturmian
orbit of the correspondence on $\partial\Lambda_+$, identified
with the same orbit (in the opposite direction) on
$\partial\Lambda_-$. The set $\Omega$ for the pinched
correspondence has $q$ components whose boundaries meet this
orbit. These form what we call the {\it principal cycle} of
components of $\Omega$. Together with $\Lambda_-\cap\Lambda_+$
itself, they separate the Riemann sphere into two parts, one
containing $\Lambda_-\setminus(\Lambda_+\cap\Lambda_-)$ and the
other containing $\Lambda_+\setminus (\Lambda_+\cap\Lambda_-)$.
The stabilizer (under the iterated pinched correspondence) of each
of the components of the principal cycle is a group, since these
components do not contain ``fold'' points. Moreover it is not hard
to see that this group is isomorphic to $C_2*C_3$.
\medskip
{\bf Definition.} {\it A holomorphic correspondence is said to be
a mating between $r_{p/2q}$ and $q_0$ if it is topologically
conjugate to a correspondence obtained by pinching to a point each
component of the grand orbit of $\gamma_{p/q}$ for a mating
between $r_*$ and $q_0$, and if moreover the action of the
stabiliser of each component of the principal cycle of the
correspondence is conformally conjugate to the action of
$PSL_2(\mbox{$\mathbb Z$})$ on the upper half-plane.}
\medskip In a mating between $q_0$ and $r_{p/2q}$, the sets
$\Lambda_+$ and $\Lambda_-$ are no longer copies of $K(q_0)$ (the
unit disc) but are now each homeomorphic to a quotient
$K(q_0)_{p/q}$ of $K(q_0)$ by an equivalence relation $\sim_{p/q}$
on $\partial K(q_0)$ (the unit circle) which may be described as
follows. Let $\omega'_{p/q}$ denote the points of the circle
opposite to points of the Sturmian $p/q$ orbit $\omega_{p/q}$, so
$\omega_{p/q}$ and $\omega'_{p/q}$ are contained in disjoint
intervals. To define the relation $\sim_{p/q}$ we identify the
`outermost' pair of points of $\omega'_{p/q}$, and similarly we
identify the next pair of points from the outside, and so on,
folding the points of $\omega'_{p/q}$ together in pairs. Similarly
we identify in pairs the corresponding inverse images of points of
$\omega'_{p/q}$ under the doubling map, and repeat so that the
relation $\sim_{p/q}$ becomes invariant under this inverse.
\bigskip {\bf Remarks.}
\medskip 1. The justification for describing the construction in
the definition as ``a mating between $q_0$ and $r_{p/2q}$'' is
two-fold. Firstly, both the construction and $r_{p/2q}$ are
obtained by pinching the same simple closed curve $\delta_{p/q}$
on the same orbifold $\Sigma$, and secondly the definition agrees
with our earlier definition for a mating between $q_0$ and the
modular group. However when $p/q\notin {\mathbb Z}$ the most
direct relationship we know of between $\Omega(r_{p/2q})$ and
$\Omega(f)$ for the correspondence pinched along $\gamma_{p/q}$ is
that given by pinching $\delta_{p/q}$ in the Fuchsian picture of
$\Omega(r_*)$, described in Remark 2 following Corollary
\ref{iota} (in Section 2.4).
\medskip
2. Corollary \ref{iota} implies that a mating between $q_0$ and
$r_{p/2q}$ is isomorphic to a mating between $q_0$ and
$r_{(p+q)/2q}$. For example a mating between $q_0$ and $r_{1/2}$
is isomorphic to one between $q_0$ and the modular group. This
example is easily understood directly, since $r_{1/2}$ is the
faithful discrete representation of $C_2*C_3$ for which the limit
set is a single round circle, like $PSL_2(\mbox{$\mathbb Z$})$, but for which the
generator $\sigma$ of $C_2$ acts by interchanging the two
components of the complement. We remark that $r_{p/2q}$ and
$r_{(p+q)/2q}$ always have the same limit set, since the second
representation is obtained from the first by composing with an
(outer) automorphism of $C_2*C_3$.
\subsection{Invariant collar neighbourhoods of arcs}
For the proofs of Theorem \ref{simple} and Theorem \ref{rat} we shall need
well-behaved neighbourhoods of our arcs on which to support the
pinching deformations. We define an {\it invariant collar
neighbourhood} of an arc $A$ joining $\Lambda_-$ to $\Lambda_+$ to
be a closed set ${\mathcal N}(A)$ containing $A$, bounded by a
pair of arcs joining the end points of $A$, such that under the
action of $f$ the set ${\mathcal N}(A)$ has stabiliser isomorphic
to the infinite dihedral group, and ${\mathcal N}(A)$ is {\it
precisely invariant} under the action of this stabiliser.
(Strictly speaking, ${\mathcal N}(A)$ is not a topological
neighbourhood of $A$, since the end points of $A$ are on the
boundary of ${\mathcal N}(A)$.)
\begin{lemma}\label{collars_exist}
The arc $\gamma_{p/q}$ has an invariant collar neighbourhood.
\end{lemma}
{\bf Proof.} A collar neighbourhood of each of the $q$ arcs which
join $\Lambda_-$ to $\Lambda_+$ is obtained by lifting any collar
neighbourhood of the $p/q$ geodesic $\delta_{p/q}$ on the orbifold
$\Sigma$. It is immediate from the action of $\sigma\rho$ and
$\sigma\rho^{-1}$ on the lift of such a neighbourhood that its
stabiliser under the action of $f$ is an infinite dihedral group,
generated by the appropriate branch of $f^q$ and by $\sigma$
(which is a branch of $f^{-1}ff^{-1}$) composed with a branch of
whichever $f^r$ maps the $\sigma$ image of the arc back to the
arc. This lifted collar neighbourhood is precisely invariant under
the action of the stabiliser. $\square$
\medskip The small copies of the $q$ arcs have collar
neighbourhoods that are the images of the original collar
neighbourhoods under appropriate branches of forward or backward
iterates of $f$. These images are each either a bijective copy, or
(in the case of a ``spike'') a quotient by an involution, of one
of the original collar neighbourhoods. In the case of the arc
$\gamma_0$, joining the fixed points of the doubling map on
$\partial \Lambda_-$ and $\partial \Lambda_+$, {\it all} the
images are such quotients.
\subsection{A pinching deformation}
Let us consider a correspondence $p$ which represents the mating
of a quadratic polynomial $q$ with a faithful and discrete
representation of $C_2 * C_3$ with connected ordinary set, and let
$f:\Lambda_-\to\Lambda_-$ be the $2:1$-branch of $p$. We fix the curve of
rotation number $p/q$ and consider its lifts ${\cal R}$ (for red) to
$\overline{\mbox{$\mathbb C$}}$. Thus $\gamma=\gamma_{p/q}$ is one of the connected components of
${\cal R}$ which joins $\Lambda_-$ to $\Lambda_+$. Let us denote its collar
neighbourhood defined above by ${\cal N}(\gamma)$. Then $\mbox{Stab}\,_p({\cal N}(A))$
is isomorphic to the infinite dihedral group. Let $B_-$ and $B_+$
be both components of ${\cal N}(A)\setminus \gamma$.
\medskip
We will first define an appropriate quasiconformal deformation on a model
strip and then implement it on the dynamical plane \cite{HT}.
\medskip
Our model space will be a closed horizontal strip on the upper
half-plane. Choose a collection of numbers $0< L_y < L_r$ (the
indices $y,r$ are colours yellow and red respectively), and then
an increasing $C^1$-function $\tau:[0,1[\to [L_r,+\infty[$. Let
$M\subset \mbox{$\mathbb R$}^2$ be the closed subset bounded by
$$([0,1]\times\{0\})\cup (\{0\}\times [0,L_r])\cup (\{1\}\times [0,+\infty[
) \cup ( \{(t,\tau(t)),t\in [0,1[\})\ .$$ Choose $v_t(y)$ so that
$v_t(y)=y$ for $0\le y\le L_y$ and that $(t,y)\mapsto (t,v_t(y))$
is a $C^1$-diffeomorphism from $[0,1]\times [0,L_r]\smallsetminus
\{(1,L_r) \}\to M$.
\begin{figure}
\begin{center}
\input{deform.pstex_t}
\caption{The diffeomorphism $(t,y)\mapsto (t, v_t(y))$.}
\label{deform}
\end{center}
\end{figure}
\medskip We also make the following technical assumption: for any
$L'<L_r$, there is $t(L')\in ]0,1[$ with $t(L')\to 1$ as ${L'\to
L_r}$, such that for any $(s,y) \in\, ]t(L'),1] \times [0,L']$,
we have $v_s(y)=v_{t(L')}(y)$. Now on the straight strip $\{0\le x
\le L_r\}$, and for every $t\in [0,1]$ , set
$$\widetilde{P}_t(x+iy)=x+ i\cdot v_t(y)\ .$$ This map satisfies
the following properties\,:
\begin{enumerate}
\item It commutes with the translation by $1$ (and by any other
real number).
\item It is the identity on the sub-strip $\{0\le y\le L_y\}$.
\item The coefficient of the Beltrami form
$$\left.\frac{\partial \widetilde{P}_t /\partial \bar{z}}{\partial
\widetilde{P}_t /\partial z}\right|_{x+iy}=
\frac{1-\frac{\partial}{\partial
y}v_t(y)}{1+\frac{\partial}{\partial y}v_t(y)}$$ is continuous on
$(t,x+iy)\in [0,1]\times \{0\le y\le L_r\}$, whose norm is locally
uniformly bounded from $1$ if $(t,y)\ne (1,L_r)$ and tends to $1$
as $(t,y)\to (1,L_r)$.
\end{enumerate}
Define conformal maps $\psi_\pm :B_\pm\to \mbox{$\mathbb R$}\times (0,L_r)$ which
map $\gamma$ to $\mbox{$\mathbb R$}\times\{L_r\}$. For $t\in [0,1[$, set
$\sigma'_t=(\widetilde{P}_t\circ \psi_\pm)^*(\sigma_0)$ to be the
pull-back of the standard complex structure on $B_\pm$. Since the
action is properly discontinuous on $\mbox{$\Omega$} (f)$, we may spread
$\sigma_t'$ to the whole orbit of ${\cal N}(\gamma)$ under the correspondence
$p$. We let $\sigma_t$ be the extension of this almost complex
structure to the whole Riemann sphere by setting $\sigma_t=\sigma_0$ on
the complement. It is a $p$-invariant complex structure. We let
${\cal Y}$ (for yellow) be the set of points $z$ such that $\sigma_t(z)$
is not the standard conformal structure for some $t$.
\medskip
The family of $p$-invariant complex structures $(\sigma_t)_{t\in
[0,1)}$ defines a pinching deformation supported on ${\cal R}$. We let
$h_t$ be the quasiconformal map given by the Measurable Riemann Mapping
Theorem applied to $\sigma_t$ normalised so that $h_t$ fixes both
critical points of $f|_{\Lambda_-}$ and $f^{-1}|_{\Lambda_+}$ and the
point at infinity as well. The correspondence $p_t$ defined by
$p_t(z,w)=p(h_t^{-1}(z),h_t^{-1}(w))$ is holomorphic by
construction, and the family of pairs $(p_t,h_t)_{t\in [0,1)}$
defines a marked pinching deformation.
\section{Convergence of the pinching deformation}
The proofs of both Theorem \ref{simple} and Theorem \ref{rat} follow
essentially the same lines. We must prove that the pinching
deformation defined in the previous section converges uniformly in
each case, and we must prove that in each case the limit
correspondence has as stabiliser of each of the components of the
principal cycle of $\Omega$ a group conformally equivalent to
$PSL_2(\mbox{$\mathbb Z$})$. The strategy for proving uniform convergence is
inspired by \cite{H2,HT} where analogous statements are proved for
rational maps and where detailed proofs can be found.
\medskip
We proceed to prove both theorems simultaneously as far as
possible. We refer to \cite{H2} and \cite{HT} when we can, instead
of repeating the detailed arguments presented in these papers. The
parts of the proofs which differ for the two theorems are
postponed to $4.1$ and $4.2$. In particular we delay the proof of
the key Lemma \ref{nbhd} (stated below). The first step in the
proof of the theorems is to prove that the path of quasiconformal homeomorphisms
$(h_t)$ is equicontinuous. We will apply the following criterion
the proof of which is elementary (cf. Lemma 2.5 in \cite{HT}).
\begin{lemma}{\em\bf (Equicontinuity criterion at a point)}
Let ${\cal H}=\{h:\mbox{$\mathbb D$}\to \mbox{$\mathbb C$}\}$ be a family of continuous injective
maps such that $\cup_{h \in{\cal H}}h(\mbox{$\mathbb D$})$ avoids at least 2 points
in $\mbox{$\mathbb C$}$. Let $(U_{n})_{n\ge 0}$ be a nested sequence of disc-like
neighbourhoods of the origin in the unit disc $\mbox{$\mathbb D$}$ such that
$A_{n}=\mbox{$\mathbb D$}\smallsetminus \overline{U_{n}}$ is an annulus. If there
exists a sequence $\eta_n \nearrow +\infty$ such that
$$\forall\,h\in{\cal H},\ \forall\,n\ge 0,\ \mbox{\em mod\,}h(A_{n})\,\ge\,
\eta_n,$$
then ${\cal H}$ is equicontinuous at the origin.
\end{lemma}
This means that we need to get infinitely many annuli with
controlled moduli. The assumption on the fixed point $\beta$ will
give us information on the support of the deformation\,: this will
enable us to prove the following lemma in the respective cases.
\begin{lemma}\label{nbhd}{\em\bf (One good annulus around each Julia point)}
Fix $r>0$.
\begin{itemize}\item[(i)]
For any $x\in \partial\Lambda_-\cup\partial\Lambda_+\smallsetminus{\cal R}$,
there are two open neighbourhoods $N'(x)$ and $N(x)$ of $x$ in $
D(x,\frac{r}{4})$ and $m>0$ such that ${\rm mod}\,
h_t(N(x)\smallsetminus \overline{N'(x)})\ge m$ for all $t$.
\item[(ii)] For any $x=\beta_\gamma \in {\cal R}\cap (
\partial\Lambda_-\cup\partial\Lambda_+)$, with $\gamma$ an
${\cal R}$-component, there is a sequence $(t_n)$ in $[0,1)$ tending
to $1$, a nested sequence of annuli $(A_n)_n$ surrounding $\gamma$,
and a constant $m > 0$ such that ${\rm mod}\, h_t(A_n)\ge m/n$
for $t\ge t_n$.
\end{itemize}\end{lemma}
Then the weak hyperbolicity condition is used to spread these
annuli at every point and at every scale and therefore to imply
the equicontinuity of $(h_t)$ (cf. the proof of the Proposition
2.3 in \cite{HT} or \S 3 in \cite{H2}). The estimates of the
conformal moduli also enable us to analyse the structure of the
fibres of any limit map and to conclude that its fibres are
exactly the closures of the connected components of ${\cal R}$.
\medskip
Any limit $h_1$ satisfies the conclusion of the theorem and we may
also extract a convergent sequence $(p_{t_n})$ of the
correspondences to a correspondence $p_1$ (cf. Appendix A in
\cite{HT}).
\medskip
Since the fibre structure is well understood, it follows that if
there are other limits $(\widehat{h},\widehat{p})$, then
$\widehat{h}\circ h_1^{-1}$ defines a conjugacy which is conformal
off $h_1(\partial\Lambda_-\cup\partial\Lambda_+)$ (cf. Lemma A.2 in
\cite{HT}).
\medskip
Now it can be shown as in \cite{HT} that all the limit
correspondences satisfy the ``weak hyperbolicity'' condition on
the image of $\partial\Lambda_-\cup\partial\Lambda_+$. Since
$\partial\Lambda_-\cup\partial\Lambda_+$ has no interior, a standard
argument of Sullivan implies that the Lebesgue measure of
$h_1(\partial\Lambda_-\cup\partial\Lambda_+)$ is zero (cf. Theorem 4.1
\cite{H1}). Furthermore, the weak hyperbolicity condition on $p_1$
implies that the following rigidity statement holds.
\begin{prop}\label{conjugacy-conformal-off-limits}
Let $p_0$ and $p_1$ be two correspondences which are matings of
weakly hyperbolic polynomials with discrete representations of
$C_2*C_3$. If $p_0$ and $p_1$ are conjugate by a topological
homeomorphism which is conformal off the limit sets, then the
conjugacy is a M\"obius transformation.\end{prop}
The proof of this proposition follows the same lines as
Proposition 6.3 and Theorem 0.2 in \cite{H1}. $\square$
\medskip Thus $\widehat{h}\circ h_1^{-1}$ is a M\"obius
transformation, whence the uniqueness of the limits $(p_t,h_t)$ as
$t$ tends to $1$.
\medskip To complete the proofs of Theorem \ref{simple} and Theorem
\ref{rat} it now remains only to prove Lemma \ref{nbhd} in both
cases, and to prove that in each case the limit of the family of
pinching deformations corresponds to the mating we are looking
for.
\subsection{The simple case (winding number zero)}
We shall make use of the statements proved in \cite{HT} for
simple pinchings of rational maps, so we have to show how to get
to that setting.
\medskip
Using McMullen's gluing lemma (Proposition 5.5 in \cite{mc1}), we
may construct a rational map $R$ of degree $2$ which induces a
partition of the sphere $\overline{\mbox{$\mathbb C$}}= K\sqcup {\cal F}$ where $K$ is the
filled-in Julia set of a quadratic-like map induced by a
restriction of $R$ hybrid-equivalent to $q$, and ${\cal F}$ is the
basin of attraction of a fixed point at infinity of multiplier
$1/2$. For the domains of the quadratic-like map, we first choose
a linearising disc $D$ for the point at infinity which contains
the critical value, and set $V=\overline{\mbox{$\mathbb C$}}\setminus \overline{D}$. If
$V'=R^{-1}(V)$, then $R:V'\to V$ is quadratic-like. Furthermore,
we may find a forward-invariant Jordan arc $\kappa$ in ${\cal F}$
joining the point at infinity with the corresponding $\beta$-fixed
point which only cuts $\partial V$ once, and then transversally.
Let $\widehat{{\cal R}}$ be the grand orbit of $\kappa$ for $R$. It
follows that $(\widehat{{\cal R}}\setminus\kappa)\cap \partial
V=\emptyset$.
\begin{prop}\label{comp} There is a quasiconformal $\Phi:\overline{\mbox{$\mathbb C$}}\to\overline{\mbox{$\mathbb C$}}$ such that
\begin{itemize}
\item $\Phi(\Lambda_-)=K$ and $\Phi({\cal R})=\widehat{{\cal R}}$, \item
$\Phi\circ f= R\circ\Phi$ in a neighbourhood of $\Lambda_-$, \item
$\overline{\partial}\Phi =0$ a.e. on $\Lambda_-$.
\end{itemize}\end{prop}
{\bf Proof.} We already know that there is a quasiconformal map
$\phi:\overline{\mbox{$\mathbb C$}}\to\overline{\mbox{$\mathbb C$}}$ which fulfills the conclusions of the
Proposition except for the condition on the curves. We let
$U'\subset \subset U$ be simply connected domains such that the
extension $f:U'\to U$ of the branch of the correspondence
$f:\Lambda_-\to\Lambda_-$ is a quadratic-like map hybrid-equivalent to
$q$. It follows from the construction of $f$ that we may assume
that $U$ is a fundamental domain for the involution $J$.
Furthermore, we may also assume that $\phi(U)=V$.
\medskip
We let $\phi_0:U\to V$ be a quasiconformal homeomorphism isotopic to $\phi$ rel.
$\Lambda_-$ through an isotopy which maps $\partial U$ to
$\partial V$ throughout, and such that $$\phi_0(\gamma_0\cap
(\overline{U}\setminus U'))= \kappa\cap (\overline{V}\setminus
V')\mbox{ and } R\circ \phi_0|_{\partial U'}=\phi_0\circ
f|_{\partial U'}.$$ This is possible since both sets $U\setminus
\Lambda_-$ and $V\setminus K$ are annuli and since the action of the
maps $f$ and $R$ are 2:1 coverings. Define $(\phi_n)$ inductively
so that $\phi_{n+1}\circ f=R\circ\phi_n$ so that
$\phi_n|_{\Lambda_-}=\phi|_{\Lambda_-}$ and
$\phi_n|_{\overline{U}\setminus U'}=\phi_0|_{\overline{U}\setminus
U'}$. This sequence is a normal family quasiconformal mappings which admits
at least one limit $\Phi:U\to V$. This map satisfies the
conclusion of the proposition. $\square$
\medskip
We now provide a proof of Lemma \ref{nbhd} under the
assumptions of Theorem \ref{simple}.
\medskip
{\bf Proof of Lemma \ref{nbhd}.} We first assume that $q$ is not
conjugate to $z\mapsto z^2 +1/4$. Then by Lemma 2.7 in \cite{HT}
we have the result we seek but for $\widehat{{\cal R}}$ and the
rational map $R$ in place of ${\cal R}$ and the correspondence. By
Proposition \ref{comp} this is all we need, except for the case of
the only ${\cal R}$-component, $\gamma_0$, which is not contained in
the neighbourhood $U$ of $\Lambda_-$. But $\gamma_0$ is a double
cover of any other component $\gamma$ of ${\cal R}$ by a branch of the
correspondence, and $\gamma_0$ has a neighbourhood which is a
double cover of a disc neighbourhood of $\gamma$, by the same
branch.
\medskip
We now deal with $q(z)=z^2+1/4$. Let us denote by $p$ the mating
of $ q$ with $C_2*C_3$ and let us define $q_0(z)=z^2$, $p_0$ and
$R_0$ the corresponding mating and rational map. We let
$(p_t,\widehat{h}_t)$ be the simple pinching of $p_0$ considered
above, and $\Phi_0:\Lambda_-(p_0)\to \overline{\mbox{$\mathbb D$}}$ be given by
Proposition \ref{comp}. It follows from Corollary 3.10 in
\cite{HT} that there is a $\mu$-homeomorphism , in the sense of David,
$\phi:\mbox{$\mathbb C$}\to\mbox{$\mathbb C$}$, conjugating $p_0|_{\mbox{$\Omega$} (p_0)}$ conformally to
$p|_{\mbox{$\Omega$} (p)}$. Furthermore, a constant $K_0\ge 1$ exists such that
the set of points $z\in \mbox{$\mathbb C$}$ for which the dilatation ratio
$K_\phi(z)$ is at least $K_0$ is contained in the disjoint union
of the orbit of an invariant sector $S\subset int(\Lambda(p_0))$ with
vertex $\beta$ (see Lemma 2.1 \cite{H} for details).
\medskip
We claim that the image under $\phi$ of the controlled annuli for
$p_0$ have also controlled moduli. For points outside the red set,
this is because the set where $K_\phi$ is large is contained in
the union of sectors so that the Key lemma in \cite{HT}, which
implies the bounds on the moduli, also holds for these domains.
\medskip
For points in the red set, we must be more precise and use
intermediate results which are established for the proof of Lemma
2.7 in \cite{HT}. We refer to \S 2.5 in \cite{HT} for the
details. We let $Y$ be the connected component of ${\cal Y}(p_0)$ which contains
$\gamma_0$. In the proof of the equicontinuity at those points, it is
shown that there is a sequence $\psi_n: A_n \to
(-C-(n+1),C+(n+1))^2\setminus[-C-n,C+n]^2$ of homeomorphisms,
where $C$ is a fixed positive real number, such that, for $t\ge
t_n$, $\psi_n\circ \widehat{h}_t^{-1}$ is uniformly quasiconformal
off ${\cal Y}\setminus Y$. Moreover, $\psi_n$ maps $\Phi_0(S)\cap A_n$
onto a rectangle $Q_n=[-C-(n+1),-C-n]\times [C_1,C_2]$ for fixed
constants $C_1$ and $C_2$.
\medskip
The bound on the moduli for the cauliflower map $z\mapsto z^2+1/4$
comes from a length-area argument provided by metrics $(\rho_n^t)$
defined as follows. Let $t\ge t_n$; on
$\widehat{h}_t({\cal Y}(p_0)\setminus Y)$, we let $\rho_n^t=0$ and on
its complement we define
$$\rho_n^t=\frac{1}{|\partial_z \widehat{h}_t\circ\psi_n^{-1}|-|\partial_
{\bar z}\widehat{h}_t\circ\psi_n^{-1}|} \circ (\psi_n\circ
\widehat{h}_t^{-1})\,.$$ This kind of metric is used to prove the
quasi-invariance of moduli of annuli for quasiconformal maps. This metric
yields the bound $\mod \widehat{h}_t(A_n)\ge m/n$ where $m>0$ is
independent of $n$.
\medskip
Similarly, we let $\widehat{\rho}^t_n=0$ for points in
$h_t\circ\phi({\cal Y}(p_0)\setminus Y)$ and on the complement, we let
$$\widehat{\rho}^t_n=\frac{\rho_n^t}{|\partial_z \phi_t|-|\partial_{\bar
z} \phi_t|}\circ\phi_t^{-1}\,,$$ where $\phi_t= h_t\circ \phi\circ
\chi_t^{-1}$. It follows from the construction of $\phi$ that
$K_{\phi}\asymp n$ on $Q_n $ (see Lemma 2.1 in \cite{H}), so that
the area of $h_t(\phi(Q_n))$ is at most a multiple of $n$, as the
area of $h_t(\phi(A_n\setminus Q_n))$, for the metric
$\widehat{\rho}^t_n$. Thus, we get $\mod h_t(\phi_0(A_n))\ge c/n$.
Whence we obtain the estimates of the moduli for these points
also. $\square$
\medskip The following proposition now completes the proof of Theorem
\ref{simple}.
\begin{prop}\label{simple-mating} Under the assumptions of Theorem \ref{simple},
the limit $p_1$ of $(p_t)$ is a mating of $q$ with $PSL_2(\mbox{$\mathbb Z$})$.
\end{prop}
{\bf Proof.} The limiting correspondence $p_1$ inherits a
compatible involution $J$ from $p_0$, so by Proposition
{\ref{compatible} (Section 1) this correspondence is conjugate to
some member of the family (2), or equivalently to $J \circ
Cov_0^Q$ for $Q(z)=z^3-3z$ and $J$ some (M\"obius) involution. The
proof of the Proposition now follows the same steps as the proof
of Theorem 7.1 in \cite{bf}, which states an analogous result for
the degree $4$ Chebyshev polynomial in place of $Q$. We summarise
the steps but refer the reader to \cite{bf} for technical details.
The topological dynamics of $p_1$ ensure that there exist a
transversal $D_Q$ for $Q$ and a fundamental domain $D_J$ for $J$
such that the complement of the union of the interiors of $D_Q$
and $D_J$ consists precisely of the fixed point $\Lambda_+ \cap
\Lambda_-$. This fixed point is parabolic for $f$ and it follows
from local anaysis that in a neighbourhood the boundaries of $D_Q$
and $D_J$ may be chosen to be smooth curves, tangent to one
another at the fixed point. The set $D_Q\cap D_J$ is a fundamental
domain for the action of $f|_\Omega$, and since $f|_\Omega$ and
$f^{-1}|_\Omega$ have no critical points (only double points) we
know that $f|_\Omega$ is conformally conjugate to
$\{\sigma\rho,\sigma\rho^{-1}\}$ for some Fuchsian representation
of $C_2*C_3$ acting on the open upper half of the complex plane.
To show that this action is indeed that of $PSL(2,\mbox{$\mathbb Z$})$ it suffices
to show that in the upper half-plane the images of $\partial D_Q$
and $\partial D_J$ converge to the same point on the real axis.
This can be shown to follow from the fact that $\partial D_Q$ and
$\partial D_J$ are smooth curves which meet tangentially (see
\cite{bf}). $\square$
\subsection{Pinching arcs of non-zero rational winding number}
Let $p$ be a correspondence which is a mating between $z\mapsto
z^2$ and a faithful discrete representation of $C_2*C_3$ in
$PSL_2(\mbox{$\mathbb C$})$ with connected ordinary set. In this section we prove
Lemma \ref{nbhd} for curves in $\mbox{$\Omega$} (p)$ with non-zero rational
rotation number. The fact that the Julia set is a quasicircle will
be crucial in the proof, which closely follows the argument in \S
3 of \cite{H2}.
\medskip
The first step is to straighten the limit set and the support of
the pinching. Figure \ref{chi} illustrates an example.
\begin{lemma} There is a quasiconformal map $\chi:\overline{\mbox{$\mathbb C$}}\to\overline{\mbox{$\mathbb C$}}$ such that
$\chi(\partial \Lambda_-)=\mbox{$\mathbb S$}^1$, which satisfies the following
properties\,:
\begin{itemize}
\item $\chi$ is conformal on the interior of $\Lambda_-$\,; \item
$\chi$ conjugates $f$ to $z\mapsto z^2$ in a neighbourhood of the
interior of $\Lambda_-$\,; \item components of ${\cal Y}$ which are
attached at two points $x$ and $y$ to $\Lambda_-$ are mapped into
rectangles in (log)-polar coordinates with base
$[\chi(x),\chi(y)]$; \item components $Y$ of ${\cal Y}$ which are
attached at a single point $x$ to $\Lambda_-$ are mapped into sectors
based at $\chi(x)$;
\end{itemize}\end{lemma}
{\bf Proof.} The restriction of $\chi$ to $\Lambda_-$ is given by the
B\"ottcher coordinates of $f$. The extension of $\chi$ to the
outside makes use of a pull-back argument (see pp.\,14-15 in
\cite{H2}). $\square$
\medskip
\begin{figure}
\begin{center}
\input{chi.pstex_t}
\caption{Image under $\chi$ of the collars of the first two
generations of the orbit of $\gamma_{p/q}$, in the case $p/q=2/5$
(cf. fig. 7).} \label{chi}
\end{center}
\end{figure}
\medskip
The next step of the proof is to control the moduli of many
annuli. We place ourselves in the coordinates given by $\chi$. As
in \cite{H2}, we may define annuli bounded by rectangles in the
log-polar coordinates which avoid the image of ${\cal Y}$ under
$\chi$.
\medskip
As in the case of simple pinchings, there is no problem with the
curves which link both components of $\Lambda$, because they cover
other components which do not. This enables us to prove Lemma
\ref{nbhd} (cf. Proposition 3.3 and 3.4 in \cite{H2}).
\medskip
Finally, the following proposition completes the proof of Theorem
\ref{rat}.
\begin{prop}\label{rational-mating} Under the assumptions of
Theorem \ref{rat}, the limit $p_1$ of $(p_t)$ is a mating of $z\mapsto
z^2$ with the circle-packing representation $r_{p/2q}$ of
$C_2*C_3$.
\end{prop}
{\bf Proof.} As in the proof of Proposition \ref{simple-mating}
the limiting correspondence $p_1$ is necessarily conjugate to some
member of the family (2), or equivalently to $J \circ Cov_0^Q$ for
$Q(z)=z^3-3z$ and $J$ some (M\"obius) involution. Once again we
can now follow the same steps as in the proof of Theorem 7.1 in
\cite{bf}. Transversals $D_Q$ and $D_J$ can be chosen this time
such that the complement of the union of their interiors consists
precisely of the period $q$ parabolic orbit $\Lambda_+ \cap
\Lambda_-$, and such that in a neighbourhood of any point of this
orbit the boundaries of these transversals are smooth curves,
tangent to one another at the orbit point. From the fact that
$\Omega$ is now a countable union of topological discs and our
knowledge of the topological dynamics of $f$ (using convergence of
the pinching deformation) we know that $f|_\Omega$ and
$f^{-1}|_\Omega$ have no critical points (only double points) and
that for any component of $\Omega$ which meets the period $q$
orbit $\Lambda_+\cap\Lambda_-$ the iterated branches of $f$ which
stabilise the component are conformally conjugate to the elements
of the group generated by $\{\sigma\rho,\sigma\rho^{-1}\}$ for
some Fuchsian representation of $C_2*C_3$ acting on the open upper
half of the complex plane. As in the proof of Proposition
\ref{simple-mating} the properties of the boundaries of $D_Q$ and
$D_J$ again ensure that this representation is indeed conformally
conjugate to $PSL_2(\mbox{$\mathbb Z$})$. $\square$
|
{
"timestamp": "2005-03-30T15:10:07",
"yymm": "0503",
"arxiv_id": "math/0503706",
"language": "en",
"url": "https://arxiv.org/abs/math/0503706"
}
|
\section*{References}
\noindent\REF{[1]}S. Sachdev, {\it Quantum Phase Transitions} (Cambridge
Univ. Press, Cambridge, 1999).
\REF{[2]} J. A. Hertz, Phys. Rev. B {\bf 14}, 1165 (1976).
\REF{[3]} F. lachello and A. Arima, {\it The Interacting Boson Model}
(Cambridge University
Press, Cambridge, 1987).
\REF{[4]} A. Borh and B. R. Mottelson, {\it Nuclear Structure} Vol. I
(Benjamin, New York, 1969); Vol. II (Benjamin, New York, 1975).
\REF{[5]} F. lachello, AIP Conf. Proc. {\bf 726},
111 (2004).
\REF{[6]} A. E. L. Dieperink, O. Scholten, and F. lachello, Phys.
Rev. Lett. {\bf 44}, 1747 (1980).
\REF{[7]} D. H. Feng, R. Gilmore, and S. R. Deans, Phys. Rev. {\bf
C23}, 1254 (1981).
\REF{[8]} O. S. Van Roosmalen, {\it Algebraic Description of Nuclear
and Molecular
Rotation-Vibration Spectra}, Ph.D. Thesis, University of Groningen,
The Netherlands, 1982.
\REF{[9]} R. Gilmore and D. H. Feng, Nucl. Phys. {\bf A301}, 189 (1978);
R. Gilmore, J. Math. Phys. {\bf 20}, 89 (1979).
\REF{[10]} R. F. Casten, in {\it Interacting Bose-Fermi System},
ed. F. Iachello (Plenum, 1981).
\REF{[11]} F. Iachello, Phys. Rev. Lett. {\bf 85}, 3580 (2000).
\REF{[12]} R. M. Clark, M. Cromaz, M. A. Deleplanque, M. Descovich,
R. M. Diamond, P. Fallon, I. Y. Lee, A. O. Macchiavelli,
H. Mahmud, E. Rodriguez-Vieitez, F. S. Stephens, and D. Ward,
Phys. Rev. {\bf C69}, 064322 (2004).
\REF{[13]} J. M. Arias, J. Dukelsky, and J. E. Garcia-Ramos,
Phys. Rev. Lett. {\bf 91}, 162502 (2003).
\REF{[14]} D. J. Rowe, Phys. Rev. Lett. {\bf 93}, 122502 (2004).
\REF{[15]} F. Iachello, N. V. Zamfir, Phys. Rev. Lett. {\bf 92}, 212501 (2004).
\REF{[16]} P. Cejnar, S. Heinze, and J. Dobe{\v{s}},
Phys. Rev. C {\bf 71}, 011304 (2005).
\REF{[17]} J. M. Arias, C. E. Alonso, A. Vitturi, J. E. Garcia-Ramos,
J. Dukelsky, A. Frank, Phys. Rev. {\bf C68}, 041302 (2003).
\end{document}
\begin{figure}
\begin{center}
\epsfig{file=fig6.eps,width=3.8cm}~~~
\epsfig{file=fig7.eps,width=3.8cm}
\epsfig{file=fig8.eps,width=3.8cm}
\epsfig{file=fig9.eps,width=3.8cm}
\end{center}
{\scriptsize Fig. 2. Overlaps of the ground state wavefunction, where
the full line shows the overlap $|\langle 0_g;x|0_g;x=0\rangle|$, and
the dotted line shows the overlap $|\langle 0_g;x|x_g;x=1\rangle|$.}
\end{figure}
|
{
"timestamp": "2005-03-29T15:04:57",
"yymm": "0503",
"arxiv_id": "nucl-th/0503071",
"language": "en",
"url": "https://arxiv.org/abs/nucl-th/0503071"
}
|
\section*{Introduction}
Myosin V is a motor protein involved in different forms of intracellular
transport \cite{Reck-Peterson.Mercer2000,Vale2003b}. Because it was the first
discovered processive motor from the myosin superfamily and due to its unique
features, including a very long step size, it has drawn a lot of attention in
recent years and now belongs to the best studied motor proteins. The
experiments have characterized it mechanically
\cite{Mehta.Cheney1999,Rock.Spudich2000,Rief.Spudich2000,Veigel.Molloy2002,Purcell.Sweeney2002},
biochemically
\cite{De_La_Cruz.Sweeney1999,De_La_Cruz.Ostap2000a,De_La_Cruz.Ostap2000b,Yengo.Sweeney2002,Purcell.Sweeney2002},
optically \cite{Ali.Ishiwata2002,Forkey.Goldman2003,Yildiz.Selvin2003} and
structurally
\cite{Walker.Knight2000,Burgess.Trinick2002,Wang.Sellers2003,Coureux.Houdusse2003}.
These studies have shown that myosin V walks along actin filaments in a
hand-over-hand fashion \cite{Yildiz.Selvin2003} with an average step size of
about 35 nm, roughly corresponding to the periodicity of actin filaments
\cite{Mehta.Cheney1999,Rief.Spudich2000,Veigel.Molloy2002,Ali.Ishiwata2002}, a
stall force of around 2 pN \cite{Rief.Spudich2000} and a run length of a few
microns \cite{Rief.Spudich2000,Sakamoto.Sellers2003,Baker.Warshaw2004}. Under
physiological conditions, ADP release was shown to be the time limiting step in
the duty cycle \cite{De_La_Cruz.Sweeney1999,Rief.Spudich2000}. Two stages of
the power stroke have been resolved: one about 20nm, possibly connected with
the release of phosphate, and another one of 5nm, probably occurring upon
release of ADP \cite{Veigel.Molloy2002}. Despite all this progress, the
definite answer to the questions how the mechanical and the chemical cycle are
coupled and how the heads communicate with each other to coordinate their
activity has not yet been found.
Theoretical models for processive molecular motors can follow different goals.
What most models have in common is that they identify a few long-living states
in the mechanochemical cycle and assume stochastic (Markovian) transitions
between them. The differences between models start in the way these states are
chosen. An approach that has been applied to myosin V
\cite{Kolomeisky.Fisher2003}, kinesin
\cite{peskin95,schief2001,Thomas.Tawada2002}, as well as to other biological
mechanisms of force generation, including actin polymerization \cite{peskin93}
and RNA polymerase \cite{wang-oster98}, models the motors as stochastic
steppers. These models describe the whole motor as an object that can go
through a certain number of conformations (typically a few) with different
positions along the track. After the completion of one cycle (which is, in
models for myosin V and kinesin, tightly coupled to the hydrolysis of one ATP
molecule), the motor advances by one step. All steps are reversible and at
loads above the stall the motor is supposed to walk backwards and thereby
regenerate ATP. The approach has been particularly useful for interpreting the
measured force-velocity relations and relating them to the kinetic parameters
and positions of substeps
\cite{schief2001,Fisher.Kolomeisky2001,Kolomeisky.Fisher2003}. A limitation of
such models is that they assume coordinated activity of both heads rather than
explaining it. They also assume that the motor strictly follows the regular
cycle and there is no place for events like steps of variable length and
dissociation from the track, although the latter can be incorporated into the
models by proposing a different dissociation rate for each state in the cycle.
In this Article we present a physical model for the processive motility of
myosin V. The basic building block of our model is an individual head, which
we model in a similar way as the models for conventional myosins do
\cite{hill74}, albeit with different rate constants. The head is connected to
the lever arm, which we model as an elastic rod, whose geometry we infer from
electron microscopy studies \cite{Walker.Knight2000,Burgess.Trinick2002}. The
two lever arms are connected through a flexible joint and this is the exclusive
way of communication between them. We will derive the properties of the dimer
from those of the individual head.
\section*{The Model}
To describe each myosin V head we use a model based on the 4-state cycle as
postulated by \citet{lymn71} and used in many quantitative muscle models
\cite{hill74} (Fig.~\ref{fig:2}A). We restrict ourselves to
the long-living states in the cycle: detached with ADP.Pi, bound with ADP.Pi,
bound with ADP, detached with ADP and bound without a nucleotide. The bound
state with ATP and the free state with ATP have both been found to be very
short-lived \cite{De_La_Cruz.Sweeney1999} and we therefore omit them in our
description, i.e., we assume that binding of ATP to a bound head leads to
immediate detachment and ATP hydrolysis. The detached state without a
nucleotide is very unlikely to be occupied because of the low transition rates
leading to it and we omit it from our scheme as well.
One question that has not yet been definitely answered, is whether Pi release
occurs before or during the power stroke, i.e., whether a head which is
mechanically restrained form conducting its power stroke can release Pi or not.
The 4-state model assumes a tight linkage between the Pi release and the power
stroke.
While the 4-state model has been successfully applied to myosin II (e.g.,
\citet{duke99,vilfan2003b}), recent experimental evidence suggests that the lead head can
release Pi before the power stroke \cite{Rosenfeld.Sweeney2004}. We
therefore also discuss an alternative 5-state model. In the 5-state model we
introduce an additional state ADP$'$ in which the phosphate is already
released, but the lever-arm is still in the pre-powerstroke state. The next
transition, ADP release, however, is still linked to the completion of the full
power-stroke. This is necessary in order to explain head coordination and also
in agreement with experiments that show a strain-dependence in the ADP release
rate in single-headed molecules \cite{Veigel.Molloy2002}. The extended duty
cycle of a head is shown in Fig.~\ref{fig:2}B.
\begin{figure}[htbp]
\figurecontents{
\includegraphics{Figure1}
}
\mycaption{The myosin V dimer is modeled as two heads, each connected to a
lever arm which leaves the head at a certain angle $\phi$, depending on the
state of the head. The two lever arms, modeled as elastic beams, are
connected with a flexible joint, which is also connected to the external
load.}
\label{fig:1}
\end{figure}
\begin{figure}
\figurecontents{
A)\includegraphics{Figure2a}\\
B)\includegraphics{Figure2b}
}
\mycaption{A) The mechanochemical cycle of each individual head. The head
attaches to actin in the state with ADP and Pi bound on it, undergoes a
large conformational change upon Pi release, another smaller conformational
change upon ADP release, then binds ATP and enters the very weakly bound
state, which dissociates quickly. B) The mechanochemical cycle in the
5-state model. In this scenario, the phosphate release and the
power stroke are two separate transitions.}
\label{fig:2}
\end{figure}
\begin{table}
\mycaption{Geometric parameters of a myosin V head (see also
Fig.~\ref{fig:1} for their definition).}
\label{tab:I}
\begin{center}
\begin{tabular}{lll}
\hline
Lever arm length & $L$ & $26\,{\rm nm}$\\
Lever arm start & $R$ & $8\,{\rm nm}$\\
Lever arm start & $\delta_{\rm ADP.Pi}$ & $0\,{\rm nm}$\\
Lever arm start & $\delta_{\rm ADP,apo}$ & $3.5\,{\rm nm}$\\
Angle ADP.Pi & $\phi_{\rm ADP.Pi}$ & $115^{\circ}$\\
Angle ADP & $\phi_{\rm ADP}$ & $50^{\circ}$\\
Angle apo & $\phi_{\rm apo}$ & $40^{\circ}$\\
\hline
\end{tabular}
\end{center}
\end{table}
A head always binds to an actin subunit in the same relative position. In each
state, the proximal end of the lever arm leaves the head in a fixed direction
in space, determined by the polar angle $\phi$ towards the filament plus end
and the azimuthal angle $\theta=\theta_0 i$ of the actin subunit $i$ to which
the head is bound. The geometry of the molecule and the angles were inferred
from images obtained with electron microscopy
\cite{Walker.Knight2000,Burgess.Trinick2002}. They are summarized in Table
\ref{tab:I}. In our calculations we assume a 13/6 periodicity of the
actin helix (6 rotations per 13 subunits), which means $\theta_0=2\pi \times
6/13$.
We assume that the lever arm has the properties of a linear, uniform and
isotropic elastic rod, described with the bending modulus $EI$. Then the local
curvature $\kappa$ is determined from $M=EI {\mathbf\kappa}$, where $M$ is the
local bending moment (torque). The lever arms from both heads are joined
together (and to the tail) with a flexible joint which allows free rotation in
all directions. For a certain configuration of chemical states, binding
sites of both heads and a given external force, the three-dimensional shape and
the bending energy of both lever arms can be calculated numerically as
described in the Appendix. Some of the calculated shapes are shown in
Fig.~\ref{fig:3}.
\begin{figure*}[htbp]
\figurecontents{
\includegraphics{Figure3}
}
\mycaption{Calculated shapes and bending energies of dimers, bound $i$
subunits apart ($i=-2,2,\ldots,15$) and in different states: first in post-,
second in the pre-powerstroke state (upper row), both in the post-powerstroke
state (middle row) and both in the pre-powerstroke state (bottom row). Each
configuration is shown in side and front view. If both heads are in the same
state (bottom two rows) there is a significant cost in elastic energy needed
to buckle one of the lever arms. Binding of the lead head before the trail
head undergoes the power stroke is therefore unlikely. }
\label{fig:3}
\end{figure*}
We calculate the free energy of a dimer state as
\begin{equation}
G=G_1 + G_2 + U_1 + U_2 + F x\;,
\label{eq:1}
\end{equation}
where $G_1$ and $G_2$ are the intrinsic free energies of both heads (which
depend on the chemical state of the head and the concentrations of
nucleotides), $U_1$ and $U_2$ are the energies stored in the elastic
deformation of each lever arm, and $F x$ is the work done against the external
load ($x$ denotes the coordinate of the flexible joint along the filament axis
with positive values towards the plus end, while positive values of $F$ denote
a force pulling towards the minus end, against the direction of motion of an
unloaded motor).
\subsection*{Transition rates}
There are two exact statements we can make about the kinetic rates of the duty
cycle that follow from the principle of detailed balance. The first statement
relates the forward and the backward rate of any reaction to the free energy
difference between the initial and the final state. For any transition the
principle of detailed balance states that
\begin{equation}
\label{eq:2}
\frac{k_{+i}}{k_{-i}} =
\frac{k^0_{+i}}{k^0_{-i}} e^{-\frac{\Delta U + F \Delta x}{k_B T}}
\end{equation}
where $\Delta U$ denotes the change in elastic energy of the dimer
and $F\Delta x$ the work performed against the external load.
The second exact statement can be derived by multiplying together the detailed
balance conditions for a monomer in the absence of any external force along a
closed pathway in Fig.~\ref{fig:2}. After one cycle the free energy of the
bound monomeric head returns to its initial value, while the total free energy
change in the system equals the amount gained from the hydrolysis of one ATP
molecule. The resulting relation reads
\begin{multline}
\label{eq:3}
\frac{k^0_{\rm +A} k^0_{\rm -Pi} k^0_{\rm -ADP} k_{\rm +ATP} [{\rm ATP}]}{k_{\rm -A} k^0_{\rm +Pi} [{\rm Pi}] k^0_{\rm +ADP}
[{\rm ADP}] k^0_{\rm -ATP}} \\
=e^{\frac{\Delta G_{ATP}}{k_B T}}=e^{\frac{\Delta G^0}{k_B T}}
\frac{[{\rm ATP}]}{[{\rm ADP}][{\rm Pi}]}
\end{multline}
and provides an important constraint on the kinetic rates of the model.
In the 5-state model, we obtain an equivalent equation,
\begin{multline}
\label{eq:4}
\frac{k^0_{\rm +A} k^0_{\rm -Pi} k^0_{\rm +PS} k^0_{\rm -ADP} k_{\rm +ATP}
[{\rm ATP}]}{k_{\rm -A} k^0_{\rm +Pi} [{\rm Pi}] k^0_{\rm -PS} k^0_{\rm
+ADP}
[{\rm ADP}] k^0_{\rm -ATP}} \\
=e^{\frac{\Delta G^0}{k_B T}} \frac{[{\rm ATP}]}{[{\rm ADP}][{\rm Pi}]}\;.
\end{multline}
A similar statement also holds for the rates along the inner loop in the
reaction scheme, which involves attachment, power stroke and detachment, all in
the ADP state. Because we assume that the detachment rate in the
pre-powerstroke and the post-powerstroke state are both the same ($k'_{\rm
-A}$), the relation reads
\begin{equation}
\label{eq:5}
\frac{ k^{0\prime\prime}_{\rm +A} k^0_{\rm +PS} } { k^{0\prime}_{\rm +A} k^0_{\rm -PS} }=1 \;.
\end{equation}
When it comes to the actual force dependence of transition rates we have to
rely on approximations. An approach that is most widely used when modeling
motor proteins, but also other conformational changes, like the gating of ion
channels, involves the Arrhenius theory of reaction rates \cite{hill74}. It
proposes that the protein has to reach an activation point ($x_a$) somewhere
between the initial ($x_i$) and the final state ($x_f$) by thermal diffusion,
but completes the reaction rapidly after that. Therefore, the force dependence
of the forward rate can be modeled as
\begin{equation}
\label{eq:6}
k_{+i}=k_{+i}^0 e^{-\frac {U(x_a)-U(x_i)}{k_B T}} \qquad
k_{-i}=k_{-i}^0 e^{-\frac {U(x_a)-U(x_f)}{k_B T}}
\end{equation}
where $U(x)$ means the total potential (bending of both lever-arms and work
done against the external load) which a head has to overcome to bring the lever
arm angle into a given state. We use the variable $\epsilon$ to denote the
relative position of the activation point between the initial and the final
state, so that $x_a=(1-\epsilon) x_i +\epsilon x_f$. Unless otherwise noted,
we will assume $\epsilon=0.5$. Not precisely identical, but useful for
practical purposes is also the approximation
$U(x_a)=(1-\epsilon)U(x_i)+\epsilon U(x_f)$. Therefore we get the following
expression for the force-dependence of the transition rate:
\begin{equation}
\label{eq:7}
k_{+i}=k_{+i}^0 e^{\frac{\epsilon \Delta U}{k_B T}}
\end{equation}
For reactions that involve the binding and unbinding of a head, Eq.~\ref{eq:2}
is valid, but one expects the activation point to be much closer to the bound
state. The strain-dependence of the detachment rate for heads in the ADP and
ATP.Pi state has not yet been measured and we therefore neglect it, assuming
that the detachment rate is force-independent, $k_{\rm -A}\equiv k_{\rm -A}^0$.
The attachment rate then relates to the potential difference as
\begin{equation}
k_{\rm +A}=k_{\rm +A}^0 e^{-\frac{\Delta U}{k_B T}}\;.
\end{equation}
\subsection*{Choice of kinetic parameters}
\begin{table*}[htbp]
\mycaption{Kinetic parameters of the model}
\label{tab:II}
\figurecontents{
\begin{tabular}{lp{5.5cm}llp{6cm}}
\multicolumn{2}{c}{Parameter} & \multicolumn{2}{c}{Value}& Source\\
&&4-state&5-state&\\
\hline\\
$k^0_{\rm +A}$ & actin binding with ADP.Pi & $5000\,{\rm s^{-1}}$&
$5000\,{\rm s^{-1}}$&est.~from run length \\
%
$k_{\rm -A}$ & actin release with ADP.Pi & $1\,{\rm s^{-1}}$& $50\,{\rm
s^{-1}}$&est.~from run length \\
%
$k^{0\prime}_{\rm +A}$ & actin binding with ADP & $5000\,{\rm s^{-1}}$& $5000\,{\rm s^{-1}}$&
$\approx k^0_{\rm +A}$ \cite{De_La_Cruz.Sweeney1999} \\
%
$k^{\prime}_{\rm -A}$ & actin release with ADP & $0.1\,{\rm s^{-1}}$ & $0.1\,{\rm s^{-1}}$ & $0.032\,{\rm
s^{-1}}$ \cite{De_La_Cruz.Sweeney1999}, $1.1 \,{\rm s^{-1}}$ \cite{Baker.Warshaw2004}\\
%
$k^0_{\rm -Pi}$ & Pi release & $200\,{\rm s}^{-1}$ & $200\,{\rm s}^{-1}$
& $>250\,{\rm s}^{-1}$ \cite{De_La_Cruz.Sweeney1999}, $110\,{\rm s}^{-1}$
\cite{Yengo.Sweeney2004}, $228\,{\rm s}^{-1}$
\cite{Rosenfeld.Sweeney2004}\\
$\epsilon_{\rm -Pi}$ & activation point & $0.3$ &--&F-v relation at high loads\\
%
$k^0_{\rm +Pi}$ & Pi binding & $10^{-4}\,{\mu \rm M}^{-1}{\rm s}^{-1}$ &
$10^{-2}\,{\mu \rm M}^{-1}{\rm s}^{-1}$ &
guess \\
%
$k^0_{\rm +PS}$ & power stroke & -- &
$10^{4}\,{\rm s}^{-1}$ &
guess \\
%
$k^0_{\rm -PS}$ & reverse stroke & -- &
$0.05\,{\rm s}^{-1}$ &
$k^0_{\rm +PS}/k^0_{\rm -PS}$ from the stall force \\
%
$k^0_{\rm -ADP}$ & ADP release & $20\,{\rm s}^{-1}$ & $20\,{\rm s}^{-1}$ &
$k_{\rm -ADP}=13\,{\rm s}^{-1}$ for dimers \cite{Rief.Spudich2000}\\
%
$k^0_{\rm +ADP}$ & ADP binding & $12\,{\mu \rm M}^{-1}{\rm s}^{-1}$ & $12\,{\mu \rm M}^{-1}{\rm s}^{-1}$ &
$12.6\,{\mu \rm M}^{-1}{\rm s}^{-1}$ \cite{De_La_Cruz.Sweeney1999},
$14\,{\mu \rm M}^{-1}{\rm s}^{-1}$ \cite{Wang.Sellers2000}\\
%
$k_{\rm +ATP}$ & ATP binding, actin release & $0.7 \,{\mu \rm M}^{-1}{\rm
s}^{-1}$ & $0.7 \,{\mu \rm M}^{-1}{\rm s}^{-1}$ &
$0.9 \,{\mu \rm M}^{-1}{\rm s}^{-1}$
\cite{De_La_Cruz.Sweeney1999,Rief.Spudich2000}, $0.6-1.5 \,{\mu \rm
M}^{-1}{\rm s}^{-1}$ \cite{Veigel.Molloy2002}\\
%
$k^0_{\rm -ATP}$ & actin binding with ATP release & $0.07\,{\rm s}^{-1}$
& $1.2\,{\rm s}^{-1}$ &
Eq.~\ref{eq:3}, Eq.~\ref{eq:4} \\
\end{tabular}
}
\end{table*}
Some of the transition rates in the cycle are well known from the literature.
$k_{\rm -ADP}$ is the limiting rate both for running myosin V molecules and for
single-headed constructs at low ATP concentrations. The measured values are
$13\,{\rm s}^{-1}$ \cite{Rief.Spudich2000} for dimers and $12\,{\rm s}^{-1}$
\cite{De_La_Cruz.Sweeney1999}, $13$--$22\,{\rm s^{-1}}$
\cite{Trybus.Freyzon1999}, and $4.5$--$7\,{\rm s}^{-1}$
\cite{Molloy.Veigel2003} for monomers. Because the actual rate in a dimer is
slowed down as compared to the monomer, we use the value $k^0_{\rm
-ADP}=20\,{\rm s}^{-1}$. The reverse rate, $k_{\rm +ADP}$ can be determined
from the inhibitory effect of ADP on the velocity and has been estimated as
$12.6\,{\rm \mu M}^{-1}{\rm s}^{-1}$ \cite{De_La_Cruz.Sweeney1999}, $4.5\,{\rm
\mu M}^{-1}{\rm s}^{-1}$ \cite{Rief.Spudich2000}, $14\,{\rm \mu M}^{-1}{\rm
s}^{-1}$ \cite{Wang.Sellers2000}.
Equally well known is the rate for ATP binding, $k_{\rm +ATP}$, which has been
measured as $0.9\,{\rm \mu M}^{-1}{\rm s}^{-1}$
\cite{De_La_Cruz.Sweeney1999,Rief.Spudich2000}, $0.6$--$1.5\,{\rm \mu
M}^{-1}{\rm s}^{-1}$ \cite{Veigel.Molloy2002}. For the Pi release rate the
estimates range from $k_{\rm -Pi}>250\,{\rm s}^{-1}$
\cite{De_La_Cruz.Sweeney1999} to $110\,{\rm s}^{-1}$ \cite{Yengo.Sweeney2004}.
We therefore use the value $k_{\rm -Pi}=200\,{\rm s}^{-1}$.
There is some more discrepancy between the current values for the release rate
from actin in the ADP state. While direct measurements gave $k'_{-\rm
A}=0.032\,{\rm s}^{-1}$ \cite{De_La_Cruz.Sweeney1999} and $0.08\,{\rm
s}^{-1}$ \cite{Yengo.Sweeney2004}, a recent estimate from the run length led
to a higher value of $1.1\,{\rm s}^{-1}$ \cite{Baker.Warshaw2004}. We use an
intermediate value of $k'_{-\rm A}=0.1\,{\rm s}^{-1}$. For the attachment rate
in the ADP state, we set $k^{0 \prime}_{+\rm A}\approx k^0_{+\rm A}$, based on
kinetic measurements \cite{De_La_Cruz.Sweeney1999}.
This leaves us with a total of 4 unknown kinetic rates, of which 3 need to be
estimated from the measured stepping behavior and run length data, while one
can be determined from Eq.~\ref{eq:3}.
\section*{Results}
\subsection*{Choice of the value for the bending modulus}
There are two ways to estimate the bending stiffness of the myosin V lever arm
- one from its structure and analogy with similar molecules and the other one
from the observed behavior of the dimeric molecule. The lever arm consists of 6
IQ motifs, forming an $\alpha$-helix, surrounded by 6 calmodulin or other light
chains \cite{Wang.Sellers2003,Terrak.Dominguez2003}. One possible estimate for
the stiffness of the lever arm can be obtained by approximating it with a
coiled-coil domain, as has been done by \citet{Howard.Spudich1996}. Generally,
the stiffness of a semiflexible molecule is related to its persistence length
$\ell_p$ as $EI=\ell_p k_B T$. Howard and Spudich estimated the persistence
length of a coiled-coil domain as $100\,{\rm nm}$, which yields $EI\approx
400\,{\rm pN\,nm}^2$. Other researchers report values of $\ell_p=130\,{\rm nm}$
for myosin \cite{Hvidt.Ferry1982} and $\ell_p=150\,{\rm nm}$ for tropomyosin
\cite{Swenson.Stellwagen1989,Phillips.Chacko1996}.
On the other hand, we can estimate the stiffness from the force a lever arm has
to bear under conditions close to stall. We do this by calculating the
distribution of binding probabilities to different sites at $F=1.8\,{\rm pN}$,
which is close to stall force. We assume that the binding rate to each site is
proportional to its Boltzmann weight, $\exp(- G / k_B T)$, which is equivalent
to assuming that the activation point of the binding process is close to the
final state and that the reverse reaction (detachment in the state with ADP.Pi)
has no force-dependence in its rate. The expectation value of the binding
position of the lead head relative to the trail head is shown in
Fig.~\ref{fig:4}. It shows that a stiffness of $EI\gtrsim 1000\,{\rm pN\, nm^2
}$ is necessary to allow stepping at loads of this magnitude.
\begin{figure}[htbp]
\figurecontents{
\includegraphics{Figure4}
}
\mycaption{The average step size under a load of $F=1.8\,{\rm pN}$ as a
function of the lever arm elasticity $EI$. The step size was calculated from
attachment probabilities of the lead head (ADP.Pi state) relative to the
bound trail head (ADP state).}
\label{fig:4}
\end{figure}
For these reasons, we use the value $EI=1500\,{\rm pN\,nm^2}$. This
corresponds to an elastic constant (measured at the joint) of
\begin{equation}
\label{eq:9}
k=3 EI/L^3=0.25\,{\rm pN/nm}\;.
\end{equation}
The elastic constant for longitudinal forces (with respect to the lever arm) is
much higher. If we approximate the lever arm with a homogeneous cylinder of
radius $r=1\,{\rm nm}$, we can estimate it as $k_L=4 EI /(r^2 L)=230\,{\rm
pN/nm}$. We therefore neglect the longitudinal extensibility of the lever
arm in all calculations.
A similar value ($EI=1300\,{\rm pN\,nm^2}$) has also been obtained by analyzing
data from optical trap experiments on single-headed myosin V molecules with
different lever arm lengths \cite{Moore.Warshaw2004}. Even though it is
somewhat larger (about 3 times) than the values estimated for myosin II
\citep{Howard.Spudich1996}, there is no solid evidence that the structures with
different light chains have the same bending stiffness. On the other hand,
there could have been some evolutionary pressure to increase the lever arm
stiffness, as it is directly related to the stall force of myosin V. While we
are not able to give a definite answer to the question whether the lever arm
behaves like a uniform elastic rod or whether there is a pliant region close to
the head, we favor the first hypothesis because the estimated lever arm
elasticity already is more than sufficient to explain the mechanical properties
of the dimeric molecule.
\subsection*{Step size distribution}
Figure \ref{fig:3} shows the energies stored in the elastic distortions of the
lever arms of both heads in the pre-powerstroke or the post-powerstroke state.
For example, if the first head is in the ADP.Pi state and the second head binds
before the first one undergoes a power stroke, this is connected with an energy
cost of $6.6\,k_B T$. The attachment rate of the lead head before the power
stroke in the trail head is therefore more than 100 times slower than after the
power stroke.
Because the lead head normally attaches to actin while the trail head is in the
ADP state, we can determine the probability that the lead head binds to an
actin site $i$ subunits in front of the trail head from the Boltzmann factors
formed from the bending energy in the final configuration, $P_i \propto \exp (-
(U_1+U_2)/k_B T)$. Here $U_1+U_2$ denotes the sum of elastic energies stored
in both lever arms if the trail head is in the ADP state and the lead head in
the ADP.Pi state, bound $i$ sites in front of the trail head. The resulting
distributions for different lever arm lengths are shown in Fig.~\ref{fig:5}.
For the lever arm consisting of 6 IQ motifs, the result is a mixture of 11 and
13 subunit steps, whereby 13 subunits dominate. Azimuthal distortion plays a
major role in the bending energy, therefore binding is only likely to sites 2,
11, 13 and 15, on which the azimuthal angles of both heads differ by not more
than $27^\circ$.
\begin{figure}[htbp]
\figurecontents{
\includegraphics{Figure5}
}
\mycaption{Step size distribution for 4 different lever arm lengths L: 10nm
(2IQ), 18nm (4IQ), 26nm (6IQ) and 34nm (8IQ) and no external load. The
histograms show the probability that a lead head (ADP.Pi state) will bind
$i$ sites in front of the trail head in the post-powerstroke ADP state. The
probabilities were determined from the Boltzmann factors, resulting from
the elastic distortion energy of the configuration. Azimuthal distortion
plays a crucial role role in determining the step size, which is the reason
why the binding is always concentrated on sites 2, 11, 13 and 15. Taking
into account the fluctuations in the actin would lead to a broader
distribution, in better agreement with experiments
\citep{Walker.Knight2000}.}
\label{fig:5}
\end{figure}
\subsection*{The gated step in the cycle}
\label{sec:}
A question that has been a subject of intense discussion is which step in the
cycle is deciding for the coordination of the two heads. A currently often
favored hypothesis proposes that the lead head undergoes its power stroke
immediately after binding, thereby storing energy into elastic deformation of
its lever arm and releasing it after the unbinding of the trail head. An
alternative hypothesis proposes that the release of the rear head is necessary
for the power stroke in the front head. As we will show below, our model
favors this picture. In the 4-state scenario, this implies that the lead head
is waiting in the ADP.Pi. In the 5-state scenario it is in the ADP$'$ state
(the pre-powerstroke ADP state). The trail head spends most of its cycle in
the ADP state in both scenarios at saturating ADP concentrations.
Because this model challenges the currently prevailing view, we should first
critically review the arguments supporting it. One argument includes the
direct observation of telemark-shaped molecules, with the leading head leaning
forward and then the lever arm tilted strongly backwards
\cite{Walker.Knight2000}. A more detailed image analysis, however, showed that
the converter of the leading head is in the pre-powerstroke state
\cite{Burgess.Trinick2002}. Another piece of evidence comes from experiments
by \citet{Forkey.Goldman2003} which show a fraction of tags on the lever arm
(30-50\%) that do not tilt while moving, but again the data provide no
conclusive proof because the method does not allow detection of tilts symmetric
with respect to the vertical axis. To conclude, one cannot say that the
present experimental evidence excludes any of the two hypotheses about the
moment of phosphate release and of the power stroke.
From the theoretical side, we will argue that in a model with linear elasticity
the mechanism with immediate power stroke in the lead head cannot work under
loads for which the motor is known to be operational. It is known that the
monomeric constructs of myosin V undergo a normal duty cycle
\cite{De_La_Cruz.Sweeney1999,Yengo.Sweeney2002}, which means that no step in
the cycle requires mechanical work from the outside for its completion (which
would be, for example, the case if the head needed to be pulled away from actin
to complete the cycle). This excludes the possibility that the free energy
gain connected with binding and the power stroke exceeds $\Delta G_{\rm
ATP}=100\,{\rm pN nm}$, the total available energy for one cycle. Because
this and other transitions in the cycle need to be forward-running, we use the
still conservative estimate that the free energy gain from binding and the
power stroke cannot exceed $80\,{\rm pN nm}$. On the other hand, we can
estimate the free energy that would be necessary for a head to bind to a site
13 units ahead and then undergo a conformational change. The amount of energy
needed to bring the dimer into the hypothetical state with both heads in the
post-powerstroke state and a strong distortion, especially of the leading lever
arm, is plotted in Fig.~\ref{fig:6}. The calculation shows that
the binding of the front head with the subsequent power stroke before the rear
head detaches (for a load of $F=1.8\,{\rm pN}$) is only possible for values of
$EI\lesssim 450\,{\rm pN\,nm^2}$, which is inconsistent with the lower estimate
based on the observed step size (Fig.~\ref{fig:4}). Of course, we
cannot rule out that there is some additional state in the middle of the power
stroke which is occupied immediately while the lead head waits for the trail
head to detach. But within the scope of the geometrical model with a single
power stroke connected with the Pi release, we consider the scenario where the
lead head instantaneously undergoes the power stroke without waiting for the
detachment of the trail head unrealistic.
\begin{figure}[htbp]
\figurecontents{
\includegraphics{Figure6}
}
\mycaption{The amount of energy needed for the binding of the lead head and
the subsequent power-stroke, plotted against the lever arm elasticity. The
load pulling on the tail is $F=1.8\,{\rm pN}$. The lower curve shows the
energy needed to pull the external load and distort the lever arms in order
to bind the new lead head 13 sites in front of the trailing head. Note
that most of this work will be performed by Brownian motion, but the
potential well in the bound state still has to be strong enough to
stabilize the bound state. The middle curve shows the energy needed mainly
for the distortion of the lever arms when the lead head undergoes a
power-stroke before the trailing head detaches. Since the sum of both
cannot be higher than $80\,{\rm pN\,nm}$, we estimate that this
hypothetical scenario would only be possible if the lever arm stiffness was
$EI \lesssim 450\,{\rm pN\,nm^2}$. This is inconsistent with other
requirements of the model, so we rule this scenario out.}
\label{fig:6}
\end{figure}
\subsection*{Hidden power strokes in the dimer configuration}
\begin{figure}[htbp]
\figurecontents{
\includegraphics{Figure7}
}
\mycaption{For a single head, the $x$-component of the power-stroke upon ADP
release equals 3.3nm (for zero load). In the dimer with both heads bound,
only 0.07nm of that power stroke reach the load. As a consequence, the
load-dependence of transition rates between states with both heads bound is
negligible.}
\label{fig:7}
\end{figure}
An immediate consequence of the elastic lever arm model is that the tail
position is mainly determined by the geometry of the triangle and less by the
conformations of individual heads. For a monomeric head or a dimer bound by a
single head, the power-stroke upon ADP release has an $x$-component (in the
direction of the actin filament) of about $3.3\,{\rm nm}$ (Fig.~\ref{fig:7}).
If the lead head is attached, however, the power stroke as measured on the tail
is reduced by about a factor of 50. The tail movement is also closely related
to the force-dependence of transition rates, which means that transitions
between states with both heads bound do not show any significant load
dependence. In the kinetic scheme we use here this implies that the rates of
ADP release and ATP binding (the two rate limiting steps at low or forward
loads) are both constant, in agreement with the flat F-v curve measured by
\citet{Mehta.Cheney1999}.
\begin{figure*}[htbp]
\figurecontents{
\begin{tabular}{ll}
\hspace*{-0.5cm}\includegraphics{Figure8a}& \includegraphics{Figure8b}
\\ \hspace*{-0.5cm}A)&B)
\end{tabular}
}
\mycaption{ Most probable kinetic pathways for a dimer in the 4-state (A) and
in the 5-state model (B). The thick arrows denote the regular pathway and
the thin arrows side branches that can result in dissociation from actin.
Note that the simulation was not restricted to the pathways shown here, but
included all possible combinations of transitions between monomer
states. }
\label{fig:8}
\end{figure*}
\begin{figure}[htbp]
\figurecontents{
A) \includegraphics{Figure9a}\\
B) \includegraphics{Figure9b}\\
}
\mycaption{A)
Force-velocity curves in the 4-state model, obtained from a stochastic
simulation. The solid curve shows the values for $1000\mu \rm M$ ATP and
the dashed curve for $1 \mu \rm M$ ATP. Both curves are compared with the
prediction of the simplified analytical expression (Eq.~\ref{eq:13}),
dotted lines. The minor deviation is mainly due to cycles taking other
pathways, neglected force-dependence of the ADP release rate and variation
in the step size. Note that the velocities above $\sim 2.5\,{\rm pN}$ are
not well defined because the dissociation time becomes comparable with the
step time. B) Inhibition by ADP and Pi. The force-velocity relation with
1mM ATP is shown by the continuous line. The dashed line shows the same
relation with additional $10\mu{\rm M}$ ADP and the dotted line with 1mM
phosphate. The velocity reduction through ADP occurs at low or negative
loads, while the inhibition by Pi only becomes significant close to stall
conditions. }
\label{fig:9}
\end{figure}
\begin{figure}[htbp]
\figurecontents{
\includegraphics{Figure10}
}
\mycaption{Force-velocity relation of the 5-state model with 1mM ATP
(solid), 1mM ATP+ 10$\mu$M ADP (dashed) and 1$\mu$M ATP (dotted). Note the
sharper drop at high loads as compared to the 4-state coupled model
(Fig.~\ref{fig:9}).}
\label{fig:10}
\end{figure}
\subsection*{Force-velocity and run length curves}
The bending energies, calculated for each possible dimer configuration, and the
transition rates were fed into a kinetic simulation to determine the average
velocity of a dimeric motor and its dissociation rate from actin. The most
probable kinetic pathway of the dimer is indicated by thick arrows in
Fig.~\ref{fig:8}, while the thin arrows indicate some of the possible side
branches that can lead to dissociation. Figure \ref{fig:9} shows the resulting
force-velocity curves and Fig.~\ref{fig:11} the dissociation rates.
An analytical solution of the 4-state model would, in theory, require solving
the occupation probabilities for a system with about $6+8\times 3 \times 3=78$
states (6 states with one head bound, plus configurations with both heads
bound, where each head can occupy 3 different states and the relative positions
of both heads can have 8 different values). Such a system could easily be
solved numerically, but would be too complex for obtaining an insightful
analytical expression. However, we will show that a simplified pathway can
already lead to expressions that agree reasonably well with simulation data and
are therefore useful for fitting model parameters to experimental data.
In the following, we give approximate expressions for the most significant
steps in the mechanochemical cycle. The average time it takes for a head in
the state 0 to bind an ATP molecule can be estimated as
\begin{equation}
\left< t_{\rm +ATP} \right> = \frac{1}{k_{\rm +ATP} [{\rm ATP}] }
\left(1+ \frac{k_{\rm +ADP} [{\rm ADP}]}{k_{-\rm ADP}} \right)
\end{equation}
where the second term takes into account a reduction of the forward rate due to
ADP rebinding. The second rate limiting process (especially at hight loads) is
the release of phosphate. The average dwell time in the state with one head
free and the other one in the ADP.Pi state is
\begin{equation}
\label{eq:11}
\left< t_{\rm -Pi} \right> = \frac {1}{k_{\rm -Pi}}
\end{equation}
The third rate limiting step is the ADP release, with the time constant
\begin{equation}
\left< t_{\rm -ADP} \right> = \frac 1 {k_{\rm -ADP}}\;.
\end{equation}
With these three average dwell times, the motor velocity can be calculated as
\begin{equation}
\label{eq:13}
v=\frac{\left< d \right>}{\left< t_{\rm -Pi} \right>+ \left< t_{\rm -ADP} \right> +
\left< t_{\rm +ATP} \right> }\;,
\end{equation}
where $\left< d \right>$ denotes the average step size, which is about
$35\,{\rm nm}$. The individual rates that appear in this expression can be
estimated as follows: $k_{\rm -Pi}\approx k_{\rm -Pi}^0 \exp(-F \epsilon_{\rm
-Pi}d_{\rm PS}/k_B T)$ with $d_{\rm PS}=L (\cos \phi_{\rm ADP}-\cos \phi_{\rm
ADP.Pi})+\delta$ and $k_{\rm -ADP}\approx k_{\rm -ADP}^0 \exp(-\Delta U_{\rm
-ADP}/2k_B T) \approx 0.65 k_{\rm -ADP}^0 $. The results for two different
ATP concentrations are shown in Fig.~\ref{fig:9}A and compared with a
simulation result. The analytical expression reproduces the simulation result
well, with a small deviation being mainly the result of alternative pathways,
neglected force-dependence of the ADP release rate and variation in the step
size. The experimentally measured force-velocity curves
\citep{Mehta.Cheney1999,Uemura.Ishiwata2004} are also well reproduced, although
the experiments show a more abrupt drop in velocity at high loads, with no
measurable effect up to about 1pN.
In the 5-state model the power-stroke can be fast and reversible, in which case
the pre- and the post-powerstroke state can reach an equilibrium and the
limiting rate is proportional to the probability of the post-powerstroke state
$1/(1+ \exp(F d_{\rm PS} /k_B T))$ - a significantly sharper load dependence
than the 4-state model (Fig.~\ref{fig:10}).
\subsection*{Inhibition by ADP and phosphate}
It is a well established observation that ADP can slow down myosin V by binding
to heads in the state with no nucleotide and thereby preventing them from
accepting an ATP molecule. The rate of ADP rebinding is already taken into
account in the kinetic constants and the model naturally reproduces the
observed behavior, as shown in Figs.~\ref{fig:9} and \ref{fig:12} for the
4-state model and in Fig.~\ref{fig:10} for the 5-state model. Not yet
experimentally investigated has been the inhibition by phosphate. Its
intensity depends on the reverse power-stroke rate, which is one of the open
parameters of our model. In the 4-state model, Pi re-binding is necessary for
the reverse power stroke and therefore some inhibition effect can be expected
at high loads. The simulation shows clearly that the phosphate concentration
has no effect on zero-load velocity, but it does slow down the motor close to
stall (Fig.~\ref{fig:9}B). A similar effect of Pi on isometric force has also
been observed in muscle \cite{Cooke.Pate1985}. In the 5-state model Pi
rebinding is not mechanically sensitive and its effect is roughly
force-independent. However, with the parameters chosen here, it is negligible.
\subsection*{Three dissociation pathways}
As we can see from the kinetic scheme (Fig.~\ref{fig:8}), there are three
significant pathways in the cycle that can lead to the dissociation of the
myosin V dimer from an actin filament. The first pathway leaves the cycle if a
dimer bound with one head in the ADP.Pi state detaches before the second head
can attach. The second pathway runs through a state in which the bound head
releases ADP and binds a new ATP molecule before the free head can bind. With
the third pathway we denote all processes that involve the detachment of a head
in the ADP state. This is the pathway favored by recent results of
\citet{Baker.Warshaw2004}. Figure \ref{fig:11} shows the dissociation rate,
separated by contributions of the three pathways. They have the following
characteristics:
\emph{Pathway 1:} With this pathway we denote the dissociation of a head in the
ADP.Pi state. Because this state is long-lived at high loads in the 4-state,
but short-lived in the 5-state model, the resulting force-dependence of the
dissociation rate differs significantly in both scenarios. In the 4-state
model, the contribution to the dissociation probability per step shows a strong
load-dependence, but no significant dependence on the ATP concentration. It can
be estimated as
\begin{equation}
\label{eq:14}
P_{\rm diss}\approx \frac {k_{\rm -A}}{k_{\rm -Pi}}\approx \frac{k_{\rm -A}}{k^0_{\rm -Pi}}
e^{\frac{F \epsilon_{\rm -Pi} d_{\rm PS}}{k_B T}}
\end{equation}
with $d_{\rm PS}=L (\cos \phi_{\rm ADP}-\cos \phi_{\rm ADP.Pi})+\delta$. The
dissociation rate is higher for positive loads. From the estimated run length
at $1\,{\rm pN}$ load and saturating ATP concentration of about 15 steps
\cite{Clemen.Rief2003}, we can estimate the unbinding rate as $k_{\rm
-A}\approx 1\,{\rm s}^{-1}$. In order to account for reported run lengths of
over 50 steps at low loads, we tentatively assign $k_{\rm +A}^0\approx
5000\,{\rm s}^{-1}$.
In the 5-state model, the situation is reversed. There the dissociation
process on path 1 takes place if the trail head releases ADP before the lead
head releases Pi, which can happen in two different ways: on one the rate is
approximately force-independent, on the other it grows with negative (forward)
loads. In order to obtain a significant contribution to the detachment rate on
this pathway, we choose a higher detachment rate $k_{-A}$ than in the 4-state
model ($50\,{\rm s}^{-1}$ instead of $1\,{\rm s}^{-1}$).
\emph{Pathway 2:} Because the process of unbinding requires an ATP molecule,
the per-step dissociation rate grows with the ATP concentration. In addition,
it is proportional to the ratio of the $ADP$ dissociation rate and the actin
binding rate, $k_{\rm -ADP}/k_{\rm +A}$, which is higher for negative (forward)
loads. This holds in both the 4- and the 5-state scenario.
\emph{Pathway 3:} The dissociation probability on pathway 3 is proportional to
the detachment rate in the ADP state, $k'_{\rm -A}$. Of all three pathways,
this one shows the weakest load-dependence, although it is higher for forward
loads.
We expect that systematic data on mean run length as a function of load and nucleotide
concentrations will be helpful to determine the remaining model
parameters.
\begin{figure}
\figurecontents{
\includegraphics{Figure11a}~\\
\includegraphics{Figure11b}~
}
\mycaption{Dissociation rate of myosin V dimers from actin under a high (top)
and a low (bottom) ATP concentration (4-state model). The continuous line
shows the total dissociation rate, the dashed line the dissociation via
pathway 1, the dot-dashed line via pathway 2 and the dotted line via
pathway 3.}
\label{fig:11}
\end{figure}
\begin{figure}
\figurecontents{ \includegraphics{Figure12} }
\mycaption{Velocity (continuous, left scale) and mean run length (dashed,
right scale) as a function of ADP concentration in the 4-state model for
zero load and 1mM ATP.}
\label{fig:12}
\end{figure}
\begin{figure}[htbp]
\figurecontents{\includegraphics{Figure13}}
\mycaption{Force-dependence of the dissociation rate in the 5-state model.
The load dependence for positive loads is much weaker than in the 4-state
model (Fig.~\ref{fig:11})}
\label{fig:13}
\end{figure}
\subsection*{ Reverse stepping in the 5-state model }
As a consequence of both the reversibility of the power stroke and the slower
dissociation rate at high loads, the motor can step backwards under loads
exceeding the stall force (Fig.~\ref{fig:14}). Note that these steps are not
the simple reversal of forward steps (which would involve ATP synthesis), but
rather indicate a different pathway in the kinetic scheme, in which both heads
stay in the ADP state and alternately release actin at the leading position and
rebind at the trailing. The time scale of reverse stepping is determined by
the dissociation rate of a head in the ADP state, $k'_{\rm -A}$, which we chose
as $0.1\,{\rm s}^{-1}$. With a higher value of $k'_{\rm -A}$, especially for
the pre-powerstroke state (so far we assumed that the rate is equal in both ADP
states), faster stepping would also be possible, although there is an upper
limit on $k'_{\rm -A}$, imposed by the dissociation rate on pathway 3.
\begin{figure}[htbp]
\figurecontents{
\includegraphics{Figure14}
}
\mycaption{Reverse stepping in the 5-state model under a high load (4.5pN),
$10\,{\mu \rm M}$ ATP and $1\,{\mu \rm M}$ ADP. There is also some
creeping motion between the steps, which results from the attachment and
detachment of the two heads on neighboring sites, and only takes place if
myosin V is allowed to follow a helical path on actin. If binding is
constrained to one side of the actin filament (like on a coverslip), then
only regular reverse steps with the periodicity of the helix are observed
(not shown).}
\label{fig:14}
\end{figure}
\section*{Discussion}
We used the geometrical data of the myosin V molecule as obtained from EM
images to calculate the conformations and elastic energies in all dimer
configurations. These data were first used in a model with a four-state cycle
and subsequently in a five-state model.
The first result, which follows directly from the bending potentials and is
independent of the underlying cycle is that the elastic lever arm model
explains two key components of the coordination between heads: why the lead
head does not bind to actin before the power stroke in the trail head and why
it does not undergo its power stroke before the trail head detaches. It also
allows us to calculate the distribution of step sizes. The results for
different lever-arm lengths (Fig.~\ref{fig:5}) give realistic values, in
agreement with step size and helicity measurements
\cite{Purcell.Sweeney2002,Ali.Ishiwata2002}, even though they have a slight
tendency towards underestimation and also show a narrower distribution than
direct electron microscopy observations \cite{Walker.Knight2000}. A possible
explanation for the broader distribution than predicted by the model lies in
the fact that in reality the actin structure does not follow the perfect helix,
as assumed in our model, but has angular deviations of up to $10^{\circ}$ per
subunit \cite{Egelman.DeRosier1982}. Taking these fluctuations into account
would clearly broaden the distribution of our step sizes, but alone it cannot
explain the tendency towards longer steps. The most straightforward
explanation for the longer steps is that the power stroke has an additional
right-handed azimuthal component. Then the configuration with the lowest
energy is reached if the lead head is twisted to the right relatively to the
trail head, which is the case if it is bound further away along the helix. The
observation that the actin repeat is often somewhat longer than 13 subunits
(some results suggest a structure closer to a 28/13 helix
\citep{Egelman.DeRosier1982}) could also partially explain the deviation.
An issue that has been much discussed is the contribution of Brownian motion
and the power stroke to the total step size. With the geometric data used in
this study, the power stroke, i.e., the distance of the lever arm tip movement
between the states ADP.Pi and ADP, is about 31nm, or 5nm less than the average
step size. Note that the second, smaller power stroke connected with ADP
release does not contribute to the step size because it is normally followed by
the detachment of the same head. Its function could be suppressing premature
dissociation before the lead head binds and thus improving the processivity.
The remaining 5nm can be overcome by Brownian motion before the lead head
binds. However, at low loads, the binding of the lead head does not move the
load, but rather stores the energy into bent lever arms. This energy gets
released when the rear head detaches, which leads to an elastic power stroke
immediately preceding the power stroke upon Pi release. At higher loads the
situation is different, because the 5nm load movement occurs when the lead head
binds. In neither case we expect the 5nm power stroke to be resolvable under
normal conditions because it always immediately precedes or follows the large
power stroke. However, it is possible that the substeps become observable in
the presence of chemicals that slow down the power stroke
\citep{Uemura.Ishiwata2004}.
In order to fully reproduce the substeps as reported by
\citet{Uemura.Ishiwata2004}, some modifications would be necessary to the
model. First, part of the power-stroke would have to occur immediately upon Pi
release, resulting in a lever arm move of about $12\,\rm nm$ (first substep).
This step would need a very strong force-dependence in its transition rate
(activation point near the final state). The subsequent longer power stroke
(ADP'$\to$ADP) would then need a slower rate ($\sim 200\,\rm s^{-1}$) with less
force dependence (activation point close to the initial state). However, the
finding that the substep position is independent of force remains difficult to
explain, because the substep involves transition between a stiff configuration,
bound on both heads, and a more compliant state, bound on a single head.
The main value of both models (4- and 5-state) is that they provide a
quantitative explanation of the coordinated head-over-head motility of the
dimeric molecule while using only the properties of a single head as input.
Both models also explain the observed force-velocity curves at high and low ATP
concentration and the effect of additional ADP, but these features already
reveal some testable differences between the two scenarios. One of them is the
shape of the force-velocity curve. In the 4-state scenario the reverse
power-stroke needs the rebinding of a phosphate molecule. This makes the
cutoff behavior at high loads dependent on the Pi concentration: the velocity
drop is more gradual at low, but might become sharper at high Pi concentrations
(Fig.~\ref{fig:9}B). In the 5-state scenario the velocity decline
is more abrupt regardless of the Pi concentration. This is the first
suggestion how experiments with improved precision and a wider range of
chemical conditions could help distinguishing between the two scenarios.
The main difference between the two scenarios is the predicted shape of the run
length. Because the dissociation can take place on three different pathways,
its rate depends on a number of parameters, of which a few cannot yet be
determined by other methods. In the 4-state model the dissociation
rate at high loads is dominated by detachment of a head in the ADP.Pi state and
it therefore depends on the ratio $k^0_{\rm -A}/ k^0_{\rm -Pi}$
(Eq.~\ref{eq:14}). A strong increase with the load is characteristic
for the 4-state model, because the load slows down the phosphate
release and prolongs the dwell time in the state that is most vulnerable to
dissociation. Dissociation at negative (forward) loads is dominated by
pathways 2 (ATP mediated actin release in one head before the other head has
bound) and 3 (dissociation of a head with ADP). In the 5-state model
all three pathways can contribute towards the dissociation rate, but there is
no significant increase for positive loads - in fact, the dissociation rate can
even decrease.
The run length shortens with an increasing ADP concentration in both scenarios.
The decrease in run length is weaker than the decrease in the velocity
(Fig.~\ref{fig:12}), which is consistent with recent observations
\cite{Baker.Warshaw2004}. However, we cannot reproduce the reported complete
saturation of run length at hight ADP concentrations.
\citet{Baker.Warshaw2004} explain this saturation with a big difference
(50-fold) between the attachment rates of the lead head depending whether the
trail head is in the ADP or apo state, which we currently cannot reproduce with
the relatively small power stroke ($10^{\circ}$) upon ADP release in our model.
An interesting difference between the 4- and the 5-state model is also that the
5-state model allows backward steps at high loads (above the stall force),
while the 4-state model predicts rapid dissociation. In general, there are
three possibilities how backward steps can occur: (i) The motor hydrolyzes ATP,
but runs backwards. (ii) The motor slips backwards without hydrolyzing ATP ---
this is the case in our model. (iii) The motor synthesizes ATP from ADP and
phosphate while being pulled backwards, as assumed by tightly coupled
stochastic stepper models \citep[e.g.,][]{Kolomeisky.Fisher2003}. It is
possible to test these three possibilities experimentally: If (i) is the case,
the backward sliding velocity should show a Michealis-Menten type dependence on
ATP concentration. This mechanism would, however, require an even looser
mechanochemical coupling, so that not only the release of Pi, but also the
release of ADP and binding of ATP would be possible without completing the
power stroke. In case (iii) it should depend on ADP as well as on Pi
concentration, but not on ATP. In case (ii), which is favored by our study,
the backward stepping occurs when both heads have ADP bound on them and they
successively release actin at the lead position and rebind it at the new trail
position. Even though this stepping requires no net reaction between the
nucleotides, a certain (low) ADP concentration is still required to prevent the
heads from staying locked in the rigor (no nucleotide) state.
The application of the elastic lever-arm approach developed here should not be
limited to simple geometries and longitudinal loads. A natural extension of
the present work will be the influence of perpendicular forces on the activity
of the motor. One will also be able to study the stepping behavior in more
complex geometries, for example when passing a branching site induced by the
Arp2/3 complex \cite{Machesky.Gould1999}.
After completion of this manuscript, it has been brought to my attention that
\citet{Lan.Sun2005} have also published a model for myosin V, based on the
elasticity of the lever arm. In contrast to our model, they do not describe it
as an isotropic rod, but use a weaker in-plane stiffness, combined with a
strong (phenomenological) azimuthal term that prevents binding of both heads to
adjacent sites on actin. Another difference is that their study explicitly
excludes dissociation events, whereas we use the dissociation rate to determine
some of the model parameters.
\section*{Acknowledgment}
I would like to thank Erwin Frey and Jaime Santos for help with calculating the
lever-arm shape, Peter Knight for help with the geometry of the molecule, and
Matthias Rief and Mojca Vilfan for helpful discussions. This work was
supported by the Slovenian Office of Science (Grants No.~Z1-4509-0106-02 and
P0-0524-0106).
\section*{Appendix}
\subsection*{Numerical solution for the lever arm shape}
The aim of this calculation is to determine the shape of the dimeric
molecule for a given set of binding sites (trailing head bound on the
site with the index $i_1$, leading head with $i_2$), nucleotide states,
which determine the lever arm starting angles $\phi_1$ and $\phi_2$, and a
given external load $F$.
We start this task by deriving a function that numerically determines the
endpoint of a lever arm as a function of the force acting on it:
$\mathbf{x}_{j}(\mathbf{F}_{j}, \phi_{j})$ ($j=1,2$). The shape of the whole
molecule can then be determined numerically from the conditions that the
endpoints of the two lever arms coincide, $\mathbf{x}_1=\mathbf{x}_2$, and from
the force equilibrium in that point
\begin{equation}
\label{eq:15}
\mathbf{F}_1+\mathbf{F}_2 = -F \hat{e}_x\;.
\end{equation}
In many cases the function $\mathbf{x}_{j}$ will have more than one solution.
Then we solve the system with all possible combinations and then choose the
solution with the lowest energy $U=U_1+U_2+Fx$, where $U_1$ and $U_2$ denote
the energy stored in the distortion of each lever arm and $Fx$ the work
performed against the applied load.
For a head bound at site $i$, the position of the proximal end of its lever arm
in Cartesian coordinates reads
\begin{equation}
\label{eq:16}
\mathbf{x}^0=\left( \begin{array}{c}
i a+\delta\\
- R \sin(\theta) \\
R \cos(\theta)
\end{array}
\right)
\end{equation}
and its initial tangent
\begin{equation}
\label{eq:17}
\hat{t}^0=\left( \begin{array}{c}
\cos(\phi) \\
- \sin(\phi) \sin(\theta) \\
\sin(\phi) \cos(\theta)
\end{array}
\right)
\end{equation}
where $\phi$ is the lever arm tilt (a function of the nucleotide state),
$\delta$ is the relative position of the lever arm proximal end ($0$ or
$3.5\,{\rm nm}$) and $\theta$ is the azimuthal angle of the actin subunit to
which the head is bound, $\theta=\theta_0 i$ with $\theta_0\approx \frac{6}{13}
\times 360^{\circ} \approx 166^{\circ}$. The helix rise per subunit is $a=
2.75\,{\rm nm}$.
If the force $\mathbf{F}$ acts on a lever arm that leaves the head in the
direction $\hat{t}^0$, the whole lever arm will be bent in a plane spanned by
the vectors $\hat{t}^0$ and $\mathbf{F}$. We can introduce a new
two-dimensional orthogonal coordinate system in this plane, so that
\begin{align}
\tilde{\hat{t}}^0&=\left( \begin{array}{c}0\\1\end{array} \right) &
\tilde{\mathbf{F}}&=\left( \begin{array}{c}\tilde F _x \\ \tilde F _y
\end{array}\right) \\ \tilde F_y&=\mathbf{F} \hat{t}_0 & \tilde F_x&=\left|
\mathbf{F}-\hat{t}_0 ( \mathbf{F} \hat{t}_0 ) \right|
\end{align}
In this coordinate system the shape can be determined by solving the equations
\begin{align}
\label{eq:20}
M(s)&=\tilde{\mathbf{F}} \wedge ( \tilde{\mathbf{x}}(L)-\tilde{\mathbf{x}}(s)
)= EI \frac{d\phi(s)}{ds} \\ \frac{d\tilde{\mathbf{x}}}{ds} &= \hat{\tilde{t}}
\qquad \hat{\tilde{t}}=\left( \begin{array}{c} \sin(\phi) \\ \cos(\phi)
\end{array} \right)
\end{align}
with the boundary condition $\phi(0)=0$. The symbol ``$\wedge$'' denotes the
outer product, which is the out-of-plane component of the vector product. If we
differentiate Eq.~\ref{eq:20} by $\phi$ we get
\begin{equation}
\label{eq:22}
EI \frac{d^2 \phi}{ds^2} = - \tilde{F}_x \cos(\phi) + \tilde{F}_y \sin(\phi)
\end{equation}
Through partial integration and taking into account the boundary condition
$M(L)=0$, we finally obtain
\begin{equation}
\label{eq:23}
\begin{split}
&\frac{EI}{2}\left( \frac{d\phi}{ds} \right)^2 \\&= \tilde{F}_x
(\sin\phi_L-\sin\phi)+\tilde{F}_y (\cos \phi_L - \cos \phi )\\
&\equiv F \sin\left( \frac{\phi_L-\phi}{2}\right) \sin \left( \phi_F
-\frac{\phi_L+\phi}{2} \right)
\end{split}
\end{equation}
Here we introduced the force angle $\phi_F$, so that $\tilde{F}_x=F
\sin(\phi_F)$ and $\tilde{F}_y=F \cos(\phi_F)$.
\begin{figure}[tbp]
\figurecontents{ \includegraphics{Figure15} }
\mycaption{ The shapes of an elastic beam anchored at one end and pulled by a
given force $\mathbf F$ on its other end. The dashed line shows the
unloaded beam. According to the sign of the initial curvature and the final
angle $\phi_L$ the solutions can be divided into 4 classes. The beam
corresponds to the myosin V lever arm, which is anchored in the head at one
end and connected to a flexible joint at the other end. Note that the
bending shown is exaggerated in comparison with realistic dimer
configurations. }
\label{fig:15}
\end{figure}
Because of the ambiguity of a quadratic equation, Eq.~\ref{eq:23} generally has
two solutions for a given set of values for $\phi(s)$, $F$, $\phi_L$ and
$\phi_F$. As we have defined the coordinate system in a way that $\tilde{F}_x
\ge 0$, we have $0 \le \phi_F \le \pi$. We also restrict ourselves to
solutions with $\left| \phi(s) \right|<2\pi$, i.e., we do not consider any
spiraling solutions, because they always have a higher bending energy than the
straighter solution with the same endpoint. There are four classes of
functions $\phi(s)$ that satisfy the condition that the RHS of Eq.~\ref{eq:23}
be positive:
\begin{center}
\begin{tabular}{cccc}
Solution& $\phi_L$ & $\phi(s \to 0)$ & conditions \\
\hline
I & + & + & $0 \le \phi_L \le \phi_F$ \\
II & - & - & $\phi_F-2\pi \le \phi_L \le 2\phi_F-2\pi$ \\
III & + & - & $0\le \phi_L \le \phi_F$ \\
IV & - & + & $\phi_F-2\pi \le \phi_L \le 2\phi_F-2\pi$ \\
\end{tabular}
\end{center}
The solutions III and IV have a turning point at $\phi_0=-2
(\pi-\phi_F) -\phi_L$, where $d\phi/ds$ changes sign.
Eq.~\ref{eq:23} can finally be transformed to
\begin{align}
\label{eq:24}
L&=\frac{1}{2} \sqrt{\frac{EI}{F}} I(\phi_L) \qquad (\text{cases I and II})\\
L&=\frac{1}{2} \sqrt{\frac{EI}{F}} (2 I(\phi_0)+I(\phi_L)) \qquad (\text{cases III and IV})\\
I(\phi_x)&=\left| \int_0^{\phi_x} \left( \sin\left(
\frac{\phi_L-\phi}{2}\right) \sin \left( \phi_F -\frac{\phi_L+\phi}{2}
\right) \right) ^{-1/2} d\phi \right| \nonumber
\end{align}
Note that for classes II and III the RHS of Eq.~\ref{eq:24} is not monotonous
in $\phi_L$ and there can be two solutions for a given $L$. Taking this into
account, we obtain a total of up to 6 solutions. A situation in which all
cases are represented is shown in Fig.~\ref{fig:15}.
The configuration of the dimer is determined by solving Eq.~\ref{eq:15} for all
possible combinations of modes and taking the one with the lowest potential.
The numerical integration and solution were performed using NAG libraries
(Numerical Algorithms Group) and the 3-d graphical representation of the
calculated shapes was made with POV-Ray (www.povray.org).
|
{
"timestamp": "2005-03-14T13:44:10",
"yymm": "0503",
"arxiv_id": "physics/0503109",
"language": "en",
"url": "https://arxiv.org/abs/physics/0503109"
}
|
\section{Introduction}
The relative entropy of states of quantum systems
is a measure of how well one quantum state can be operationally
distinguished from another. Defined as
\begin{eqnarray}\nonumber
S(\rho||\sigma)=
\mathop{\rm Tr}\nolimits[\rho(\log \rho - \log \sigma) ]
\end{eqnarray}
for states $\rho$ and $\sigma$, it quantifies the extent to which one
hypothesis $\rho$ differs from an alternative hypothesis $\sigma$
in the sense of quantum hypothesis testing
\cite{ohya,Wehrl,disti,disti2,disti3}.
Dating back to work by Umegaki \cite{umegaki}, the relative entropy is a
quantum generalisation
of the Kullback-Leibler relative entropy for probability
distributions in mathematical statistics \cite{Kullback}.
The quantum relative entropy plays an
important role in quantum statistical mechanics \cite{Wehrl}
and in quantum information
theory, where it appears as a central notion in the
study of capacities of quantum channels
\cite{Schumacher,Holevo,Schumacher2,prop}
and in entanglement theory \cite{prop,Plenio,Plenio2}.
In {\it finite-dimensional Hilbert spaces},
the relative entropy functional
is manifestly continuous \cite{Wehrl},
see also footnotes \footnote{
For states on infinite-dimensional Hilbert spaces
the relative entropy functional is not trace norm
continuous any more,
but -- as the von-Neumann entropy -- lower semi-continuous. That is,
for sequences of states $\{\sigma_n\}_n$ and
$\{\rho_n\}_n$
converging in trace norm to states $\sigma$ and $\rho$, i.e.,
$\lim_{n\rightarrow\infty} \| \sigma_n-\sigma\|_1 =0$
and $\lim_{n\rightarrow\infty} \| \rho_n-\rho\|_1 =0$,
we merely have
$ S(\rho||\sigma)\leq \liminf_{n\rightarrow\infty}
S(\rho_n||\sigma_n)$.
However, for systems for which the Gibbs state exists, these
discontinuities can be tamed \cite{Wehrl} when considering compact
subsets of state space with finite mean energy.
In a similar manner, entropic
measures of entanglement can become
trace norm continuous on
subsets with bounded energy \cite{Jensito}.}, \footnote{
For considerations of the continuity of the relative entropy
in classical contexts, see Ref.\ \cite{naudts}.}.
In particular, if
$\{\sigma_n\}_{n}$ is a sequence of states
of fixed
finite dimension satisfying
\begin{equation}\nonumber
\lim_{n\rightarrow \infty} || \sigma_n - \sigma ||_1 =
\lim_{n\rightarrow \infty} \mathop{\rm Tr}\nolimits | \sigma_n - \sigma |
=0
\end{equation}
for a given state $\sigma$, then
\begin{eqnarray}\nonumber
\lim_{n\rightarrow\infty} S(\sigma_n|| \sigma)=0.
\end{eqnarray}
In practical contexts, however, more precise estimates can be
necessary, in particular in an asymptotic setting.
Consider a state $\rho$ on a Hilbert space ${\cal H}$, and a sequence
$\{\sigma_n\}_n$,
where $\sigma_n$ is a state on ${\cal H}^{\otimes n}$, the $n$-fold tensor product of ${\cal H}$.
The sequence is said to asymptotically approximate $\rho$ if $\sigma_n$ tends to $\rho^{\otimes n}$ for
$n\rightarrow \infty$.
More precisely, one typically requires that
\begin{equation}\nonumber
\lim_{n\rightarrow\infty}
\| \sigma_n - \rho^{\otimes n}\|_1 =0.
\end{equation}
Now, as an alternative to the trace norm distance,
one can consider the use of the Bures distance.
The Bures distance $D$ is defined as
$$
D(\rho_1,\rho_2) = 2\left(1-F(\rho_1,\rho_2)\right)^{1/2},
$$
in terms of the Uhlmann fidelity
$$
F(\rho_1,\rho_2) = \mathop{\rm Tr}\nolimits( \rho_1^{1/2} \rho_2 \rho_1^{1/2})^{1/2}.
$$
Because of the inequalities \cite{Hayden}
\begin{equation}\label{hayden}
1-F(\rho_1,\rho_2) \le \mathop{\rm Tr}\nolimits|\rho_1-\rho_2| \le
\left(1-F^2(\rho_1,\rho_2)\right)^{1/2},
\end{equation}
the trace norm distance tends to zero if and only if the Bures distance tends to zero, which shows that,
for the purpose of state discrimination, both distances are essentially equivalent and one can use whichever
is most convenient.
A natural question that now immediately arises
is whether the same statement is true for the relative entropy.
To find an answer to that one would need inequalities like (\ref{hayden}) connecting
the quantum relative entropy, used as a distance measure,
to the trace norm distance, or similar distance measures.
In this paper, we do just that: we find upper bounds on the relative entropy functional
in terms of various norm differences of the two states. As such, the presented
bounds are very much in the same spirit as Fannes' inequality, sharpening the
notion of continuity for the von Neumann entropy \cite{fannes}.
It has already to be noted here that one of the main stumbling blocks in this undertaking
is the well-known fact that the relative entropy is not a very good distance measure,
as it gives infinite distance between non-identical pure states. However, we will present
a satisfactory solution, based on using the minimal eigenvalue of the state
that is the second argument of the relative entropy.
Apart from the topic of upper bounds, we also study lower bounds on the relative entropy, giving a
complete picture of the relation between norm based distances and relative entropy.
We start in Section II
with presenting a short motivation of how this paper came about.
Section III contains the relevant notations, definitions and basic results that will be used
in the rest of the paper. In Section IV we discuss some properties of unitarily invariant (UI) norms
that will allow us to consider all UI norms in one go.
The first upper bounds on the relative entropy $S(\rho||\sigma)$ are presented in Section V, one bound being quadratic
in the trace norm distance of $\rho$ and $\sigma$ and the other logarithmic in the minimal eigenvalue of
$\sigma$. Both bounds separately capture an essential behaviour of the relative entropy, and it is
argued that finding a single bound that captures both behaviours at once is not a trivial undertaking.
Nevertheless, we will succeed in doing this in Section VII by constructing upper bounds that are as sharp
as possible for given trace norm distance \textit{and} minimal eigenvalue of $\sigma$.
In Section VI we use similar techniques to derive lower bounds that are as sharp as possible.
Finally, in Section IX, we come back to the issue of state discrimination mentioned at the beginning.
\section{Background}
In Ref.\ \cite{brat2}
(Example 6.2.31, p.\ 279) we find the following upper bound on the relative entropy, valid
for all $\rho$ and for non-singular $\sigma$:
\begin{equation}\label{bound_brat}
S(\rho||\sigma) \le \frac{||\rho-\sigma||_\infty}{\lambda_{\min}(\sigma)}.
\end{equation}
This bound is linear in the operator norm distance between $\rho$ and $\sigma$
and has a $1/x$ dependence on $\lambda_{\min}(\sigma)$.
For several purposes, such a bound is not necessarily sharp enough.
The impetus for the present paper was given by the observation that
sharper upper bounds on the relative entropy should be possible than (\ref{bound_brat}).
Specifically, there should exist bounds that are
\begin{enumerate}
\item {\em quadratic} in $\rho-\sigma$, and/or
\item depend on $\lambda_{\min}(\sigma)$ in a {\em logarithmic} way.
\end{enumerate}
A simple argument shows that a logarithmic dependence on $\lambda_{\min}(\sigma)$ can be
achieved instead of an $1/x$ dependence.
Note that $0\ge\log\sigma\ge\mathbbm{1}\cdot\log\lambda_{\min}(\sigma)$.
Thus,
\begin{eqnarray}
S(\rho||\sigma) &=& \mathop{\rm Tr}\nolimits[\rho(\log\rho-\log\sigma)] \nonumber \\
&\le& -S(\rho)-\log\lambda_{\min}(\sigma) \nonumber \\
&\le& -\log\lambda_{\min}(\sigma). \label{bound1}
\end{eqnarray}
Concerning the quadratic dependence on $\rho-\sigma$, we can put $\rho=\sigma+\varepsilon\Delta$,
with $\mathop{\rm Tr}\nolimits[\Delta]=0$, and calculate the derivative
$$\lim_{\varepsilon\rightarrow0} S(\sigma+\varepsilon\Delta||\sigma) / \varepsilon
$$
and find that this turns out to be zero for any non-singular $\sigma$.
Indeed, the gradient of the relative entropy $S(\rho||\sigma)$ with respect to $\rho$ is
$\mathbbm{1}+\log\rho-\log\sigma$ (see Lemma \ref{lemma1}).
Hence, for $\rho=\sigma$ and
$\mathop{\rm Tr}\nolimits[\Delta]=0$,
\begin{eqnarray}\nonumber
\lim_{\varepsilon\rightarrow0} S(\sigma+\varepsilon\Delta||\sigma)/\varepsilon
=\mathop{\rm Tr}\nolimits[\Delta(\mathbbm{1}+\log\sigma-\log\sigma)]=0.
\end{eqnarray}
This seems to imply that for small $\varepsilon$, $S(\sigma+\varepsilon\Delta||\sigma)$ must at least be
quadratic in $\varepsilon$, and, therefore, upper bounds might exist that indeed are quadratic in $\varepsilon$.
Furthermore, Ref.\ \cite{ohya} contains the following quadratic lower bound
(Th.\ 1.15)
\begin{equation}\nonumber
\label{bound_ohya}
S(\rho||\sigma) \ge \frac{1}{2}||\rho-\sigma||_1^2.
\end{equation}
The rest of the paper will be devoted to finding firm evidence for these intuitions, by exploring
the relation between relative entropy and norm based distances, culminating in a number of bounds
that are the sharpest possible.
\section{Notation}
In this paper, we will use the following notations.
We will use the standard vector and matrix bases:
$e^i$ is the vector with the $i$-th element equal to 1, and all other elements
being equal to $0$.
$e^{i,j}$ is the matrix with $i,j$ element equal to 1 and all other elements 0.
For any diagonal matrix $A$, we write $A_i$ as a shorthand for $A_{i,i}$,
and $\mathop{\rm Diag}\nolimits(a_1,a_2,\ldots)$ is the diagonal matrix with $a_i$ as diagonal elements.
We reserve two symbols for the following special matrices:
\begin{equation}\nonumber
E:=\mathop{\rm Diag}\nolimits(1,0,\ldots,0) = e^{1,1},
\end{equation}
and
\begin{equation}\nonumber
F:=\mathop{\rm Diag}\nolimits(1,-1,0,\ldots,0) = e^{1,1}-e^{2,2}.
\end{equation}
The positive semi-definite order is denoted using the $\ge$ sign: $A\ge B$ iff
$A-B\ge 0$ (positive semi-definite).
The (quantum) relative entropy is denoted as
$S(\rho||\sigma) = \mathop{\rm Tr}\nolimits[\rho(\log\rho-\log\sigma)]$.
All logarithms in this paper are natural logarithms.
When $\rho$ and $\sigma$ are both diagonal (i.e., when we encounter the
commutative, classical case) we use the shorthand
\begin{eqnarray*}
\lefteqn{S((r_1,r_2,\ldots)||(s_1,s_2,\ldots))} \\
&:=& S(\mathop{\rm Diag}\nolimits(r_1,r_2,\ldots)||\mathop{\rm Diag}\nolimits(s_1,s_2,\ldots)).
\end{eqnarray*}
\begin{lemma}\label{lemma1}
The gradient of the relative entropy $S(\rho||\sigma)$ with respect to its first argument $\rho$, being non-singular, is given by
$\mathbbm{1}+\log\rho-\log\sigma$.
\end{lemma}
\textit{Proof.}
The calculation of this derivative is straightforward.
Since the classical entropy function
$x\longmapsto h(x) := -x \log x $ is continuously differentiable
on $(0,1)$, and therefore,
\begin{eqnarray*}
\frac{\partial}{\partial\varepsilon}\Big|_{\varepsilon=0}
S(\rho+\varepsilon \Delta ) =
\mathop{\rm Tr}\nolimits[\Delta h'(\rho) ],
\end{eqnarray*}
we can write
\begin{eqnarray}
\label{log_deriv}
\lim_{\varepsilon\rightarrow0} S(\rho+\varepsilon\Delta||\sigma)/\varepsilon
&=& \mathop{\rm Tr}\nolimits[\Delta(\mathbbm{1}+\log\rho-\log\sigma)].\nonumber
\end{eqnarray}
\hfill$\square$\par\vskip24pt
Finally, we recall a number of series expansions related to the logarithm, which are valid for
$-1<y<1$,
\begin{eqnarray}
\log(1-y) &=& -\sum_{k=1}^\infty \frac{y^k}{k},\nonumber\\
\log(1+y)+\log(1-y) &=& -\sum_{k=1}^\infty\frac{y^{2k}}{k} ,\nonumber\\
\log(1+y)-\log(1-y) &=& 2\sum_{k=0}^\infty\frac{y^{2k+1}}{2k+1}.
\nonumber
\end{eqnarray}
These expansions will be made extensive use of.
\section{Unitarily Invariant Norms}
In this section we collect the main definitions and known results about unitarily invariant norms
along with a number of refinements that will prove to be very useful for the rest of the paper.
A \textit{unitarily invariant norm} (UI norm), denoted with
$|||.|||$, is a norm on square matrices
that satisfies the property
\begin{equation}\nonumber
|||UAV|||=|||A|||
\end{equation}
for all $A$ and for unitary $U$, $V$ (\cite{bhatia}, Section IV.2).
Perhaps the most important property of UI norms is that they only depend on the singular values of the matrix $A$.
If $A$ is positive semi-definite, then $|||A|||$ depends only on the eigenvalues of $A$.
A very important class of UI norms are the \textit{Ky Fan norms} $||.||_{(k)}$, which are defined as follows:
for any given square $n\times n$ matrix $A$, with singular values $s_j^\downarrow(A)$ (sorted in non-increasing order)
and $1\le k\le n$, the $k$-Ky Fan norm is the sum of the $k$ largest singular values of $A$:
$$
||A||_{(k)} = \sum_{j=1}^k s_j^\downarrow(A).
$$
Two special Ky Fan norms are the \textit{operator norm}
and the \textit{trace norm},
\begin{equation}\nonumber
||A||_\infty = ||A||_{(1)},\,\,
||A||_{\mbox{Tr}} = ||A||_1 = ||A||_{(n)}.
\end{equation}
The importance of the Ky Fan norms derives from their leading role in Ky Fan's \textit{Dominance Theorem}
(Ref.\ \cite{bhatia}, Theorem IV.2.2):
\begin{theorem}[Ky Fan Dominance]
Let $A$ and $B$ be any two $n\times n$ matrices. If $B$ majorises $A$ in all the Ky Fan norms,
$$
||A||_{(k)}\le ||B||_{(k)},
$$
for all $k=1,2,...$ ,
then it does so in
all other UI norms as well,
$$
|||A|||\le|||B|||.
$$
\end{theorem}
From Ky Fan's Dominance Theorem follows the following well-known
norm dominance statement.
\begin{lemma}
For any matrix $A$, and any unitarily invariant norm $|||.|||$,
$$
||A||_\infty \le \frac{|||A|||}{|||E|||} \le ||A||_1.
$$
\end{lemma}
\textit{Proof.}
We need to show that, for every $A$,
$$
|||(||A||_\infty)E||| \le |||A||| \le |||(||A||_1)E|||,
$$
holds for every unitarily invariant norm.
By Ky Fan's dominance theorem, we only need to show this for the Ky Fan norms.
All the Ky Fan norms of $E$ are 1, and
$$
||A||_\infty =||A||_{(1)} \le ||A||_{(k)} \le ||A||_{(d)}=||A||_1
$$
follows from the definition of the Ky Fan norms.
\hfill$\square$\par\vskip24pt
The main mathematical object featuring in this paper is not the state, but rather the difference $\Delta$ of
two states, $\Delta: =\rho-\sigma$, and for that object a stronger dominance result obtains.
We first show that the largest norm difference between two states occurs for orthogonal pure states.
Indeed, by convexity of norms, $|||\rho-\sigma|||$ is maximal in pure $\rho$ and $\sigma$.
A simple calculation then reveals that, for any unitarily invariant norm,
$$
|||\,\ket{\psi}\bra{\psi} - \ket{\phi}\bra{\phi}\,||| = \left(
1-|\langle\psi|\phi\rangle|^2
\right)^{1/2}
|||F|||.
$$
This achieves its maximal value $|||F|||$ for $\psi$ orthogonal to $\phi$,
showing that it makes sense to normalise a norm distance $|||\rho-\sigma|||$ by division by $|||F|||$.
We will call this a \textit{rescaled} norm.
We now have the following dominance result for rescaled norms of differences of states:
\begin{lemma}\label{lemma:dom}
For any Hermitian $A$, with $\mathop{\rm Tr}\nolimits[A]=0$,
$$
\frac{||A||_\infty}{||F||_\infty} \le \frac{|||A|||}{|||F|||} \le \frac{||A||_1}{||F||_1}.
$$
\end{lemma}
Note that equality can be obtained for any value of $|||A|||$, by setting $A=cF$.
\textit{Proof.}
We need to show, for all traceless Hermitian $A$, that
\begin{equation}\label{eq:dom}
|||(||A||_\infty)F||| \le |||A||| \le |||(||A||_1/2) F |||
\end{equation}
holds for every unitarily invariant norm.
Again by Ky Fan's dominance theorem,
we only need to do this for the Ky Fan norms $||.||_{(k)}$.
Since
$$
||F||_{(k)} = \left\{\begin{array}{cc}1,&k=1,\\2,&k>1,\end{array}\right.
$$
and
\begin{eqnarray*}
||X||_\infty = ||X||_{(1) }\le ||X||_{(k)} \le ||X||_{(d)}=||X||_1,
\end{eqnarray*}
the inequalities (\ref{eq:dom}) follow trivially for $k>1$.
The case $k=1$ is covered by Lemma \ref{lemma:k1} below.
\hfill$\square$\par\vskip24pt
\begin{lemma}\label{lemma:k1}
For any Hermitian $A$, with $\mathop{\rm Tr}\nolimits[A]=0$,
$$
||A||_1 \ge 2||A||_\infty.
$$
\end{lemma}
\textit{Proof.}
Let the Jordan decomposition of $A$ be
\begin{equation}\nonumber
A=A_+ - A_-,
\end{equation}
with $A_+$, $A_-\ge0$.
Since $\mathop{\rm Tr}\nolimits[A]=0$, clearly $\mathop{\rm Tr}\nolimits[A_+]=\mathop{\rm Tr}\nolimits[A_-]$ holds.
Thus, $||A||_1=\mathop{\rm Tr}\nolimits|A|=\mathop{\rm Tr}\nolimits[A_+]+\mathop{\rm Tr}\nolimits[A_-]=2\mathop{\rm Tr}\nolimits[A_+]$.
Also,
\begin{equation}\nonumber
||A||_\infty = \max(||A_+||_\infty,||A_-||_\infty).
\end{equation}
Hence,
$||A||_\infty \le \max(||A_+||_1,||A_-||_1) = \mathop{\rm Tr}\nolimits[A_+] = ||A||_1/2$.
\hfill$\square$\par\vskip24pt
In this paper, we will also be dealing with $\Delta=\rho-\sigma$ under the constraint $\sigma\ge\beta\mathbbm{1}$.
Obviously we have
$$
\beta\le 1/d.
$$
We now show that under this constraint, any rescaled norm of $\Delta$ is upper
bounded by $1-\beta$.
\begin{lemma}\label{lem:maxt}
For any state $\rho$, and states $\sigma$ such that $\sigma\ge\beta\mathbbm{1}$,
\begin{equation}\nonumber
T:=|||\rho-\sigma|||/|||F||| \le 1-\beta.
\end{equation}
\end{lemma}
\textit{Proof.}
We proceed by maximising $T$ under the constraint $\sigma\ge\beta\mathbbm{1}$.
Convexity of norms yields that $T$ is maximal when $\rho$ and $\sigma$ are extremal
\cite{Convex},
hence in $\rho$ being a pure state
$\ket{\phi}\bra{\phi}$ and $\sigma$ being
of the form
\begin{equation}\nonumber
\sigma = \beta\mathbbm{1} + (1-\beta d)\ket{\psi}\bra{\psi}.
\end{equation}
Fixing $\phi=e^1$, we need to maximise
$$
|||e^{1,1}-\beta\mathbbm{1}-(1-\beta d)\ket{\psi}\bra{\psi}\, |||
$$
over all $\psi$.
Put $\psi=(\cos\alpha,\sin\alpha,0,\ldots,0)^T$, then
the eigenvalues of the matrix are
$$
\lambda_\pm=\left((d-2)\beta\pm \left( {(\beta d)^2+4(1-\beta
d)\sin^2\alpha} \right)^{1/2} \right)/2
$$
and $-\beta$ (with multiplicity $d-2$).
One finds that, for $d>2$, $\lambda_+,|\lambda_-|\ge\beta$, for any value of $\alpha$,
and both $\lambda_+$ and $|\lambda_-|$ are maximal for $\alpha=\pi/2$, as would be expected.
The maximal Ky Fan norms of this matrix are therefore
\begin{eqnarray*}
||.||_{(1)} &=& \lambda_+ = 1-\beta ,\\
||.||_{(k)} &=& \lambda_+ + |\lambda_-|+(k-2)\beta = (2-\beta d) + (k-2)\beta,
\end{eqnarray*}
for $k>1$.
Hence, for every Ky Fan norm, the maximum norm value is obtained for orthogonal $\phi$ and $\psi$.
By the Ky Fan dominance theorem, this must then hold for any UI norm.
In case of the trace norm, as well as of the operator norm, the rescaled value of the maximum
is $1-\beta$. By Lemma \ref{lemma:dom}, this must then be the maximal value for any rescaled norm.
\hfill$\square$\par\vskip24pt
\textit{Remark.}
For the Schatten $q$-norm, $|||F||| = 2^{1/q}$.
The largest value of $|||F|||$ is 2, obtained for the trace norm,
and the smallest value is 1, for the operator norm.
\section{Some simple upper bounds}
In this Section we present our first attempts at finding upper bounds that capture the essential features
of relative entropy. In Subsection A we present a bound that is indeed quadratic in the trace norm distance,
the existence of which was already hinted at in Section II.
Likewise, in Subsection B, we find a bound that is logarithmic in the minimal eigenvalue of $\sigma$,
again in accordance with previous intuition.
Combining the two bounds into one that has both of these features turns out to be not so easy.
In fact, in Subsection C a number of arguments are given that initially hinted at the impossibility
of realising sich a combined bound.
Nevertheless, we will succeed in finding a combined bound later on in the paper, by using techniques
from optimisation theory \cite{Convex}.
\subsection{A quadratic upper bound}
\begin{lemma}
For any positive definite matrix $A$ and Hermitian $\Delta$
such that $A+\Delta $ is positive definite,
$$
\log(A+\Delta)-\log(A) \le \int_0^\infty dx(A+x)^{-1}\Delta(A+x)^{-1}.
$$
\end{lemma}
{\em Proof.}
Since the logarithm is strictly matrix concave \cite{HJII},
for all $t\in[0,1]$:
$$
\log((1-t)A+tB) \ge (1-t)\log(A)+t\log(B).
$$
Setting $B=A+\Delta$ and rearranging terms
then gives
$$
\frac{\log(A+t\Delta)-\log(A)}{t} \ge \log(A+\Delta)-\log(A),
$$
for all $t\in[0,1]$. A fortiori,
this holds in the limit for $t$ going to zero, and then the left-hand side is just the
Fr\'echet derivative of $\log$ at $A$ in the direction $\Delta$.
\hfill$\square$\par\vskip24pt
This Lemma allows us to give a simple upper bound on
$S(\sigma+\Delta||\sigma)$. Note that if $A\ge B$ then
$\mathop{\rm Tr}\nolimits [CA]\ge\mathop{\rm Tr}\nolimits [CB]$ for any $C\ge 0$. Therefore, we arrive at
\begin{eqnarray*}
S(\rho||\sigma) &=&
\mathop{\rm Tr}\nolimits\left[(\sigma+\Delta)(\log(\sigma+\Delta)-\log(\sigma))\right] \\
&\le& \int_0^\infty dx\mathop{\rm Tr}\nolimits[(\sigma+\Delta)(\sigma+x)^{-1}\Delta(\sigma+x)^{-1}] \\
&=& \int_0^\infty dx\mathop{\rm Tr}\nolimits[(\sigma+x)^{-1}\sigma(\sigma+x)^{-1}\Delta] \\
& + &\int_0^\infty dx\mathop{\rm Tr}\nolimits[\Delta(\sigma+x)^{-1}\Delta(\sigma+x)^{-1}].
\end{eqnarray*}
The first integral evaluates to $\mathop{\rm Tr}\nolimits[\Delta]$, because
$$
\int_0^\infty dx \frac{s}{(s+x)^2} = 1
$$ for any $s>0$, and therefore gives the value $0$.
The second integral can be evaluated most easily in a basis in which $\sigma$ is diagonal.
Denoting by $s_i$ the eigenvalues of $\sigma$, we get
\begin{eqnarray}
&&\int_0^\infty dx\mathop{\rm Tr}\nolimits[\Delta(\sigma+x)^{-1} \Delta(\sigma+x)^{-1}] \nonumber \\
&=& \sum_{i,j} \Delta_{i,j}\Delta_{j,i} \int_0^\infty dx (s_i+x)^{-1}
(s_j+x)^{-1} \nonumber\\
&=& \sum_{i\neq j} \Delta_{i,j}\Delta_{j,i} \frac{\log s_i -\log s_j }{s_i-s_j} +
\sum_i (\Delta_{i,i})^2 \frac{1}{s_i}.\label{dec}
\end{eqnarray}
The coefficients of $\Delta_{i,j}\Delta_{j,i}$ are easily seen to be always positive, and furthermore,
bounded from above by $1/\lambda_{\min}(\sigma)$. Hence we get the upper bound
$$
\int_0^\infty dx\mathop{\rm Tr}\nolimits[\Delta(\sigma+x)^{-1}\Delta(\sigma+x)^{-1}] \le \frac{\mathop{\rm Tr}\nolimits[\Delta^2]}{\lambda_{\min}(\sigma)},
$$
yielding an upper bound on the relative entropy which is, indeed, quadratic in $\Delta$:
\begin{theorem}
For states $\rho$ and $\sigma$ with $\Delta=\rho-\sigma$, $T=||\Delta||_2$ and
$\beta=\lambda_{\min}(\sigma)$,
\begin{equation}\nonumber
\label{bound_quad}
S(\rho||\sigma) \le \frac{T^2}{\beta}.
\end{equation}
\end{theorem}
\subsection{An upper bound that is logarithmic in the minimum eigenvalue of $\sigma$}
We have already found a sharper bound than (\ref{bound_brat}) concerning its dependence on $\lambda_{\min}(\sigma)$.
However, the bound (\ref{bound1}) is not sharp at all concerning its
dependence on $\rho-\sigma$.
A slight modification can greatly improve this.
First note
\begin{eqnarray*}
|\mathop{\rm Tr}\nolimits[\Delta\log\sigma]| &\le& ||\Delta||_1 \cdot
||\log\sigma||_{\infty} \\
&=& \mathop{\rm Tr}\nolimits|\Delta|\cdot|\log\lambda_{\min}(\sigma)|.
\end{eqnarray*}
This inequality can be sharpened, since $\mathop{\rm Tr}\nolimits[\Delta]=0$ and $\sigma$ is a state.
Let $\Delta=\Delta_+ - \Delta_-$ be the Jordan decomposition of $\Delta$, then
\begin{equation}\nonumber
|\mathop{\rm Tr}\nolimits[\Delta\log\sigma]| \le
||\Delta_+||_1\cdot|\log\lambda_{\min}(\sigma)|,
\end{equation}
and hence
\begin{eqnarray*}
|\mathop{\rm Tr}\nolimits[\Delta\log\sigma]| &\le&
\mathop{\rm Tr}\nolimits|\Delta|/2 \cdot|\log\lambda_{\min}(\sigma)|.
\end{eqnarray*}
Furthermore, we have Fannes' continuity of the von Neumann entropy \cite{fannes},
$$
|S(\sigma+\Delta)-S(\sigma)| \le T \log d +\min\left(-T\log
T,\frac{1}{e}\right),
$$
where $d$ is the dimension of the underlying Hilbert space
and $T:=\mathop{\rm Tr}\nolimits |\Delta| $.
Combining all this with
$$
S(\sigma+\Delta||\sigma) = -(S(\sigma+\Delta)-S(\sigma)) - \mathop{\rm Tr}\nolimits[\Delta\log\sigma]
$$
gives rise to the subsequent upper bound, logarithmic in the smallest eigenvalue of $\sigma$.
\begin{theorem}
For all states $\rho$ and $\sigma$ on a $d$-dimensional
Hilbert space, with $T=||\rho-\sigma||_1$ and $\beta=\lambda_{\min}(\sigma)$,
\begin{equation}
S(\rho||\sigma) \le T \log d +
\min\biggl (-T\log T,\frac{1}{e}\biggr) - \frac{T\log\beta}{2}.
\label{bound_log}
\end{equation}
\end{theorem}
\subsection{A combination of two bounds?}
The following question comes to mind almost automatically:
can we combine the two bounds (\ref{bound_quad}) and (\ref{bound_log})
into a single bound that is both quadratic in $\Delta$
and logarithmic in $\lambda_{\min}(\sigma)$? This would certainly be a very
desirable feature for a good upper bound.
For instance, could it be true that
$$
S(\rho||\sigma) \le C\cdot \mathop{\rm Tr}\nolimits[(\rho-\sigma)^2]\cdot |\log\lambda_{\min}(\sigma)|,
$$
for some constant $C>0$?
Unfortunately, the answer to this first attempt is negative.
In fact, the proposed inequality is violated no matter how large the value of $C$.
\begin{proposition}
For any $r>0$ there exist states $\sigma$ and $\rho$ such that
\begin{equation}
\label{bound_bad}
S(\rho||\sigma) > r\cdot \mathop{\rm Tr}\nolimits[(\rho-\sigma)^2]\cdot |
\log\lambda_{\min}(\sigma)|.
\end{equation}
\end{proposition}
\textit{Proof.}
It suffices to consider the case that $\sigma,\rho$ are states acting on
the Hilbert space ${\mathbb{C}}^2$, and that $\sigma$ and $\rho$ commute. Hence, the
statement must only be shown for two probability distributions
\begin{eqnarray*}
P=(p,1-p),\,\,\,Q=(q,1-q).
\end{eqnarray*}
Without loss of generality we can require $q$ to be in $[0,1/2]$.
Then, one has to show that for any $r>0$ there exist $p,q$
such that
the $C^\infty$-function $f$,
defined as
\begin{eqnarray*}
f(p,q,r) &=& r \left((p-q)^2 + (2-p-q)^2\right)\, | \log(q)| \\
&-& \left(
p \log(p/q) +(1-p) \log[(1-p)/(1-q)]
\right),
\end{eqnarray*}
assumes a negative value. Now, for any $r>1$, fix a $q\in(0,1/2)$ such that
$- 4 r (q \log q)<1$.
Clearly,
$$
f(q,q,r) =0,\,\,\,\,\,\,
\frac{\partial}{\partial p}\bigr|_{p=q} f(p,q,r)=0.
$$
Then
\begin{eqnarray*}
\frac{\partial^2}{\partial p^2} \bigr|_{p=q} f(p,q,r)
&=&
-\frac{1}{1-q} - \frac{1}{q} - 4 r \log(q) \\
&<& - \frac{1}{q} - 4 r \log(q)< 0 .
\end{eqnarray*}
This means that there exists an
$\varepsilon>0$ such that $f(p,q,r)<0$ for $p\in[q,q+\varepsilon]$,
which in turn proves the validity of
(\ref{bound_bad}).
\hfill$\square$\par\vskip24pt
The underlying reason for this failure is that the two bounds
(\ref{bound_log}) and (\ref{bound_quad}) are incompatible, in the sense that there are two different regimes
where either one or the other dominates.
To see when the logarithmic dependence dominates, let us again take the basis where
$\sigma$ is diagonal, with $s_i$ being the main diagonal elements.
When keeping $\Delta=\rho-\sigma$ fixed while $s_1=\lambda_{\text{min}}(\sigma)$
tends to zero, then
\begin{eqnarray}\nonumber
\lim_{s_1\rightarrow 0} S(\sigma+\Delta||\sigma)/ |\log s_1| = \Delta_{1,1}<\infty.
\end{eqnarray}
Hence, in the regime where $\lambda_{\min}(\sigma)$ tends to zero and $\rho-\sigma$ is fixed,
the bound (\ref{bound_log}) is the appropriate one.
The other regime is the one where $\sigma$ is fixed and $\rho-\sigma$ tends to zero.
This can be intuitively
seen by considering the case where the states $\rho$ and $\sigma$ commute
(the classical case).
Let $p_i$ and $q_i$ be the diagonal elements of $\rho$ and $\sigma$, respectively,
in a diagonalising basis, and $r_i=p_i-q_i$. Then
\begin{eqnarray}\nonumber
S(\rho||\sigma) = \sum_i (q_i+r_i)\log(1+r_i/q_i).
\end{eqnarray}
We can develop $S(\rho||\sigma)$ as a Taylor series in the $r_i$, giving
$$
S(\rho||\sigma) = \sum_i \frac{r_i^2}{2q_i} + O(r_i^3).
$$
Hence, in the regime where $\rho-\sigma$ tends to zero and $\sigma$ is otherwise fixed,
the relative entropy exhibits the behaviour of bound (\ref{bound_quad}).
In terms of the matrix derivatives,
this notion can be made more precise as follows. Denote the
bound (\ref{bound_log}) as
$$
g(\rho||\sigma) = \frac{\mathop{\rm Tr}\nolimits[(\rho-\sigma)^2]}{\lambda_{\min}(\sigma)}
$$
for states $\rho,\sigma$,
then clearly
$$
\lim_{\varepsilon \rightarrow 0} g(\sigma +\varepsilon \Delta
||\sigma)/\varepsilon=0.
$$
On using the integral representation of the second Fr\'echet derivative of
the matrix logarithm \cite{ohya},
\begin{eqnarray*}
\lefteqn{\frac{\partial^2}{\partial \varepsilon^2}\Big|_{\varepsilon=0}
\log(\sigma + \varepsilon \Delta)} \\
& =& -2 \int_0^\infty dx (\sigma +x)^{-1} \Delta (\sigma+x)^{-1} \Delta (\sigma+x)^{-1},
\end{eqnarray*}
one obtains
\begin{eqnarray*}
\lefteqn{\frac{\partial^2}{\partial \varepsilon^2}\Big|_{\varepsilon=0} S(\sigma+\varepsilon \Delta||\sigma)} \\
&=& -2 \mathop{\rm Tr}\nolimits\biggl[\sigma \int_0^\infty dx (\sigma +x)^{-1} \Delta (\sigma+x)^{-1} \Delta (\sigma+x)^{-1}\biggr]\\
& & +2 \mathop{\rm Tr}\nolimits\left[\Delta\int_0^\infty dx (\sigma+x)^{-1} \Delta (\sigma+x)^{-1}\right].
\end{eqnarray*}
The right hand side is bounded from above by
\begin{eqnarray*}
\lefteqn{\frac{\partial^2}{\partial \varepsilon^2}\Big|_{\varepsilon=0} S(\sigma+\varepsilon \Delta||\sigma)} \nonumber\\
&\leq& \int_0^\infty dx\mathop{\rm Tr}\nolimits[\Delta(\sigma+x)^{-1}\Delta(\sigma+x)^{-1}],
\end{eqnarray*}
see Ref.\ \cite{lieb,ohya}.
This bound can be written as in Eq.\ (\ref{dec}). Therefore,
one can conclude that
\begin{eqnarray*}
\frac{\partial^2}{\partial \varepsilon^2}\Big|_{\varepsilon=0}
S(\sigma+\varepsilon \Delta
||\sigma) = \frac{\partial^2}{\partial \varepsilon^2}\Big|_{\varepsilon=0}
g(\sigma+\varepsilon \Delta
||\sigma)
\end{eqnarray*}
holds for all $\Delta$ satisfying
$\mathop{\rm Tr}\nolimits[\Delta]=0$ if and only if
$\sigma=\mathbbm{1}/d$, where $d$ is the dimension of the underlying
Hilbert space.
These considerations seem to spell doom for any attempt at ``unifying'' the two kinds of upper bounds.
However, below we will see how a certain change of perspective will allow us to get out of the dilemma.
\section{A sharp lower bound in terms of norm distance}
We define $S_{\min}(T)$ with respect to a norm to be
the smallest relative entropy between two states that have
a distance of exactly $T$ in that norm,
that is
\begin{equation}
\label{eq:def_smin}
S_{\min}(T) = \min_{\rho,\sigma} \left\{S(\rho||\sigma): |||\rho-\sigma||| =
T \right\}.
\end{equation}
When one agrees to assign $S(\rho||\sigma)=+\infty$ for non-positive $\rho$,
the definition of $S_{\min}$ can be rephrased as
\begin{equation} \label{eq:def_smin2}
S_{\min}(T) = \min_{\Delta,\sigma} \left\{S(\sigma+\Delta||\sigma): |||\Delta||| =
T, \mathop{\rm Tr}\nolimits[\Delta]=0 \right\}.
\end{equation}
Intuitively one would guess that $S_{\min}$ is monotonously increasing with $T$. The following lemma shows
that this is true, but some care is required in proving it.
\begin{lemma}\label{lem:mono}
For $T_1\le T_2$, $S_{\min}(T_1)\le S_{\min}(T_2)$.
\end{lemma}
\textit{Proof.}
Keep $\sigma$ fixed and define
$$
f_\sigma(T) = \min_{\Delta} \left\{S(\sigma+\Delta||\sigma): |||\Delta||| =
T, \mathop{\rm Tr}\nolimits[\Delta]=0\right\},
$$
so that $S_{\min}(T) = \min_\sigma f_\sigma(T)$.
Considering $S(\sigma+\Delta||\sigma)$ as a function of $\Delta$, it is convex and minimal in
the origin $\Delta=0$.
Furthermore, for the norm balls
\begin{equation}\nonumber
{\cal B}(T):=\{\Delta: |||\Delta|||\le T, \mathop{\rm Tr}\nolimits[\Delta]=0\}
\end{equation}
we have
\begin{equation}\nonumber
\{0\}={\cal B}(0)\subseteq{\cal B}(T_1)\subseteq{\cal B}(T_2).
\end{equation}
This is sufficient to prove that
$0=f_\sigma(0)\le f_\sigma(T_1)\le f_\sigma(T_2)$.
Now, since this holds for any $\sigma$, it also holds when minimising over $\sigma$,
and that is just the statement of the Lemma.
\hfill$\square$\par\vskip24pt
As a direct consequence, a third equivalent definition of $S_{\min}(T)$ is
\begin{equation} \label{eq:def_smin3}
S_{\min}(T) = \min_{\Delta,\sigma} \left \{S(\sigma+\Delta||\sigma): |||\Delta|||
\ge T, \mathop{\rm Tr}\nolimits[\Delta]=0
\right\}.
\end{equation}
We now show that one can restrict oneself to the commutative case.
\begin{lemma}
The minimum in Eq.\ (\ref{eq:def_smin2}) is obtained for $\sigma$ and $\Delta$ commuting.
\end{lemma}
\textit{Proof.}
Fix $\Delta$ and consider a basis in which $\Delta$ is diagonal. Let
$\rho\longmapsto\mathop{\rm Diag}\nolimits(\rho)$ be the
completely positive trace-preserving map which, in that basis, sets all off-diagonal elements of $\rho$ equal to zero.
Thus $\mathop{\rm Diag}\nolimits(\Delta)=\Delta$.
By monotonicity of the relative entropy,
$$
S(\sigma+\Delta||\sigma) \ge S(\mathop{\rm Diag}\nolimits(\sigma)+\Delta||\mathop{\rm Diag}\nolimits(\sigma)).
$$
Minimising over all states $\sigma$ then gives
\begin{eqnarray*}
\min_\sigma S(\sigma+\Delta||\sigma) &\ge& \min_\sigma S(\mathop{\rm Diag}\nolimits(\sigma)+\Delta||\mathop{\rm Diag}\nolimits(\sigma)) \\
&=& \min_\sigma\left \{S(\sigma+\Delta||\sigma):[\sigma,\Delta]=0\right \}.
\end{eqnarray*}
On the other hand, the states $\sigma$ that commute with $\Delta$ are included in the domain of minimisation
of the left-hand side, hence equality holds.
\hfill$\square$\par\vskip24pt
For later reference we define the auxiliary function
\begin{equation}\label{eq:sofx}
s(x) := \min_{0<r<1-x} S((r+x,1-r-x)||(r,1-r)),
\end{equation}
for $0\le x< 1$.
An equivalent expression for this function is given by
\begin{equation}\nonumber
s(x) := \min_{x<r<1} S((r-x,1-r+x)||(r,1-r)).
\end{equation}
The first three non-zero terms in its series expansion around $x=0$ are given by:
\begin{equation}\nonumber
s(x) =
2 x^2 + \frac{4}{9}x^4 + \frac{32}{135}x^6 + O(x^{8}) \label{bound_ohya_better}
\end{equation}
(obtained using a computer algebra package).
Further calculations reveal that some of the higher-order coefficients are negative, the first one being
the coefficient of $x^{62}$.
One can easily prove \cite{csiszar} that
the lowest order expansion $2x^2$ is actually a lower bound. It is,
therefore, the sharpest quadratic lower bound.
For values of $x$ up to $1/2$,
the error incurred by considering only the lowest order term in (\ref{bound_ohya_better})
is at most 6.5\%. For larger values of $x$, the error increases rapidly.
In fact, when $x$ tends to its maximal value of 1,
$s(x)$ tends to infinity, as can easily be seen from the minimisation expression ($r$ tends to 0);
accordingly, the series expansion diverges.
For values of $x>4/5$,
$s(x)$ is well approximated by its upper bound
\begin{eqnarray*}
s(x) &\le & \lim_{r\rightarrow 1-x} S((r+x,1-r-x)||(r,1-r))\nonumber\\
& = & -\log(1-x).
\end{eqnarray*}
This is illustrated in Figure \ref{fig1}.
\begin{figure}
\includegraphics[width=3.4in]{s2.eps}
\caption{\label{fig1}The function $s$ defined in Eq.\ (\ref{eq:sofx})
(middle curve), the lower bound $2x^2$ (lower curve),
and the upper bound $-\log(1-x)$ (upper curve).}
\end{figure}
Let us now come back to Eq.\
(\ref{eq:def_smin2}), with $\sigma$ and $\Delta$ diagonal, and
$|||.|||$ any unitarily invariant norm.
Let $\sigma$ and $\Delta$ have diagonal elements $\sigma_k$ and $\Delta_k$, respectively.
Fixing $\Delta$, we minimise first over $\sigma$.
This is a convex problem and any local minimum is automatically a global minimum
\cite{Convex}. The corresponding Lagrangian
is
\begin{equation}\nonumber
{\cal L} = \sum_k \sigma_k (1+\Delta_k/\sigma_k)\log(1+\Delta_k/\sigma_k) - \nu
\left(\sum_k\sigma_k-1\right).
\end{equation}
The derivative of the Lagrangian with respect to $\sigma_k$ is
\begin{equation}\label{eq:deriv2}
\frac{\partial{\cal L}}{\partial \sigma_k} = \log(1+\Delta_k/\sigma_k)-\Delta_k/\sigma_k-\nu.
\end{equation}
This must vanish in a critical point, giving the expression
\begin{equation}\nonumber
\log(1+\Delta_k/\sigma_k) = \Delta_k/\sigma_k+\nu. \label{eq:de2}
\end{equation}
Now note that the equation $\log(1+x)-x=b$, for $b<0$ has only two real solutions, one positive and one negative,
and none for $b>0$.
Therefore, for any $k$ $\Delta_k/\sigma_k$ can assume only one of these two possible values.
Let $K$ be an integer between 1 and $d-1$.
Without loss of generality we can set
\begin{equation}\label{eq:ds}
\Delta_k/\sigma_k = \left\{\begin{array}{ll}c_p,& 1\le k\le K,
\\ -c_m,& K<k\le d,\end{array} \right.
\end{equation}
where $c_p$ and $c_m$ are positive numbers, to be determined along with $K$.
The requirement $\sum_k \Delta_k=0$ imposes
$$
c_p \sum_{k=1}^K \sigma_k - c_m \sum_{k=K+1}^d \sigma_k = 0,
$$
which upon defining
\begin{equation}\label{eq:r}
r: = \sum_{k=1}^K \sigma_k,
\end{equation}
turns into
\begin{equation}\label{eq:cc}
c_p r = c_m (1-r) =: c.
\end{equation}
Substituting Eqs.\
(\ref{eq:ds}) and (\ref{eq:r}), the function to be minimised becomes
$$
r (1+c_p)\log(1+c_p) +(1-r)(1-c_m)\log(1-c_m),
$$
which, given Eq.\
(\ref{eq:cc}), can be rewritten as
$$
S((r+c,1-r-c)||(r,1-r)).
$$
The one remaining constraint $|||\Delta|||=T$ likewise becomes
$$
|||(c_p\sigma_1,\ldots,c_p\sigma_K,-c_m\sigma_{K+1},\ldots,-c_m\sigma_d)|||=T.
$$
Defining
\begin{eqnarray*}
\tau' &:=& (\sigma_1,\ldots,\sigma_K)/r , \\
\tau'' &:=& (\sigma_{K+1},\ldots,\sigma_d)/(1-r),
\end{eqnarray*}
this turns into
$$
T=|||(c_p r\tau' ; -c_m(1-r)\tau'')|||=c |||(\tau';\tau'')|||,
$$
where we have exploited the homogeneity of a norm.
Note that by their definition, $\tau'$ and $\tau''$ are vectors consisting of positive numbers adding up to 1.
The minimisation itself thus turns into
$$
S_{\min}(T) = \min_{r,\tau',\tau''} S((r+c,1-r-c)||(r,1-r)),
$$
where $c :=T/|||(\tau';\tau'')|||$.
Quite obviously, the minimum over $c$ is obtained for the smallest possible $c$, hence
\begin{eqnarray*}
S_{\min}(T) & =& \min_{r}
S((r+T/\gamma,1-r-T/\gamma)||(r,1-r))\nonumber\\
& = & s(T/\gamma),
\end{eqnarray*}
with
$$
\gamma=\max_{\tau',\tau''} |||(\tau';\tau'')|||.
$$
By convexity of a norm, this maximum is obtained in an extreme point, so
$$
\gamma=|||F|||.
$$
Incidentally, by Lemma \ref{lem:maxt}, this value is also the maximum
$$
\max_{\rho,\sigma}|||\rho-\sigma|||,
$$
over all possible states $\rho$ and $\sigma$, i.e., $\gamma$ is the largest possible value of $T$ for the
given norm.
We have thus proven
\begin{theorem}
For any unitarily invariant norm $|||.|||$, we have the sharp lower bound
\begin{equation}\nonumber
S(\rho||\sigma) \ge s(|||\rho-\sigma|||/|||F|||).
\end{equation}
\end{theorem}
A few remarks are in order at this point:
\begin{enumerate}
\item Within the setting of finite-dimensional systems, this theorem generalises a result of Hiai, Ohya and Tsukada
\cite{hot,ohya} for the trace norm to all unitarily invariant norms.
This paper also uses the technique of getting lower bounds
by projecting on an abelian subalgebra and then
exploiting the case of a two-dimensional
support as the worst
case scenario.
\item If we take the Hiai-Ohya-Tsukada result for granted
and combine it with Lemma \ref{lemma:dom},
we immediately get
\begin{eqnarray*}
S(\rho||\sigma) &\ge & s(||\rho-\sigma||_1/||F||_1)\nonumber\\
& \ge & s(|||\rho-\sigma|||/|||F|||).
\end{eqnarray*}
\item The divergence of $s$ at $x=1$ is easily understood.
The largest norm difference between two states occurs for orthogonal pure states, in which case
their relative entropy is infinite.
\end{enumerate}
\section{Sharp upper bounds in terms of norm distance}
Let now $S_{\max}(T,\beta)$ be the largest relative entropy between $\rho$ and $\sigma$ that have
a normalised distance of exactly $T$ and $\lambda_{\min}(\sigma)=\beta$,
so let
\begin{equation} \label{eq:def_smax}
S_{\max}(T,\beta) := \max_{\rho,\sigma} \left \{S(\rho||\sigma):
\frac{|||\rho-\sigma|||}{|||F|||} = T, \lambda_{\min}(\sigma)=\beta\right \}.
\end{equation}
The need for the extra parameter $\beta$ arises because for $\beta=0$, $S_{\max}$ is infinite, as can be seen
by taking different pure states for $\rho$ and $\sigma$.
We can rephrase this definition as
\begin{eqnarray} \label{eq:def_smax2}
S_{\max}(T,\beta) &=& \max_{\Delta,\sigma} \Biggl \{S(\sigma+\Delta||\sigma):
\frac{|||\Delta|||}{|||F|||} = T, \mathop{\rm Tr}\nolimits[\Delta]=0, \nonumber \\
&& \sigma+\Delta\ge0,\lambda_{\min}(\sigma)=\beta\Biggr \}.
\end{eqnarray}
Because $\Delta$ commutes with the identity matrix, there is a unique common least upper bound on $\beta\mathbbm{1}$
and $-\Delta$, which we will denote by $\max(\beta\mathbbm{1},-\Delta)$. In the eigenbasis of $\Delta$, this is a diagonal
matrix with diagonal elements $\max(\beta,-\Delta_{i})$.
The constraints $\sigma\ge\beta$ and $\sigma+\Delta\ge 0$ can therefore be combined into the single constraint
\begin{equation}\label{eq:sigcon}
\sigma \ge \max(\beta\mathbbm{1},-\Delta).
\end{equation}
The extremal $\sigma$ obeying this constraint are
\begin{equation}\nonumber
\sigma = \max(\beta\mathbbm{1},-\Delta) + \eta \ket{\psi}\bra{\psi},
\end{equation}
where $\psi$ is any state vector,
and
\begin{equation}\nonumber
\eta := 1-\mathop{\rm Tr}\nolimits[\max(\beta\mathbbm{1},-\Delta)].
\end{equation}
Therefore, the constrained maximisation over $\sigma$ can be replaced by an unconstrained maximisation over all
pure states of the function
\begin{eqnarray}
S(\Delta+\max(\beta\mathbbm{1},-\Delta) + \eta \ket{\psi}\bra{\psi}\,\, || \nonumber \\
\qquad \max(\beta\mathbbm{1},-\Delta) + \eta \ket{\psi}\bra{\psi}). \label{eq:maximand}
\end{eqnarray}
Of course, all of this puts constraints on $\Delta$ as well. Indeed, in order that states $\sigma$
obeying (\ref{eq:sigcon}) exist,
$\max(\beta\mathbbm{1},-\Delta)$ must obey the condition
\begin{equation}\nonumber
\mathop{\rm Tr}\nolimits[\max(\beta\mathbbm{1},-\Delta)]\le 1.
\end{equation}
We now have to distinguish between two cases: the case $d=2$, and the case $d>2$.
\subsection{The case $d=2$}
For the $d=2$ case, the maximisation over $\Delta$ is trivial. In its eigenbasis,
$\Delta$ is a multiple of $\mathop{\rm Diag}\nolimits(1,-1)=F$. Hence, fixing the eigenbasis of $\Delta$ (which we can do because of unitary
invariance of the relative entropy), and fixing
\begin{equation}\nonumber
|||\Delta|||/|||F|||=T,
\end{equation}
actually leaves just one possibility
for $\Delta$, namely $\Delta=TF$.
The term $\max(\beta\mathbbm{1},-\Delta)$ leads to two cases: $T\le\beta$ and $T>\beta$.
The condition $T\le\beta$ implies, by Lemma \ref{lemma:dom}, that $||\Delta||_\infty\le\beta$ and, hence,
\begin{eqnarray*}
\max(\beta\mathbbm{1},-\Delta) &=& \mathop{\rm Diag}\nolimits(\beta,\beta) ,\\
\eta &=& 1-2\beta.
\end{eqnarray*}
The remaining maximisation of (\ref{eq:maximand}) is therefore given by
\begin{eqnarray}
\max_\psi S(\mathop{\rm Diag}\nolimits(\beta+T,\beta-T)+(1-2\beta)\ket{\psi}\bra{\psi} \, || \nonumber\\
\qquad \mathop{\rm Diag}\nolimits(\beta,\beta)+(1-2\beta)\ket{\psi}\bra{\psi}).\label{eq:maxi1a}
\end{eqnarray}
Positivity of $\eta$ requires $\beta\le 1/2$.
By unitary invariance of the relative entropy, and invariance of diagonal states under
diagonal unitaries (phase factors),
we can restrict ourselves to vectors
$\psi$ of the form $\psi=(\cos\alpha,\sin\alpha)^T$.
\begin{lemma}\label{lem:convex}
For a state vector $\psi=(\cos\alpha,\sin\alpha)^T$, the function to be maximised
in (\ref{eq:maxi1a}) is convex in $\cos(2\alpha)$.
\end{lemma}
\textit{Proof.}
Let $D_1$ be the determinant of the first argument.
It is linear in $t:=\cos(2\alpha)$:
$$
D_1 = \beta^2-T^2+(1-2\beta)(\beta-T t).
$$
After some
basic algebra involving eigensystem decompositions of the states,
the function to be maximised in (\ref{eq:maxi1a}) is found to be
given by
\begin{eqnarray*}
f(x) &:=& ((1-x)\log(1-x)+(1+x)\log(1+x))/2 \\
&+& (-1+2\beta-2T t)(\log(1-\beta)-\log\beta)/2 \\
&-& (\log(4-4\beta)+\log\beta)/2,
\end{eqnarray*}
where $x=(1-4D_1)^{1/2}$.
We will now show that this function is convex in $t$.
Since the second and third terms are linear in $t$, we only need to show convexity for the first term.
The series expansion of the first term is
$$
((1-x)\log(1-x)+(1+x)\log(1+x))/2 = \sum_{k=1}^\infty \frac{x^{2k}}{2k(2k-1)}.
$$
Every term in the expansion is a positive power of $x^2$ with positive coefficient
and is therefore convex in $x^2$, which itself is linear in
$t$. The sum is therefore also convex in $t$.
\hfill$\square$\par\vskip24pt
By the above Lemma, the maximum of the maximisation over $\psi$ is obtained
for extremal values of $t$, that is: either $\psi=(1,0)^T$ or $\psi=(0,1)^T$.
Evaluation of the maximum is now straightforward and it can be checked that the choice
$\psi=(1,0)^T$ always yields the largest value of the relative entropy.
We will now more specifically look at the case where $T>\beta$.
In this case, we get
\begin{eqnarray*}
\max(\beta\mathbbm{1},-\Delta) &=& \mathop{\rm Diag}\nolimits(\beta,T), \\
\eta &=& 1-\beta-T,
\end{eqnarray*}
and the remaining maximisation of (\ref{eq:maximand}) is given by
\begin{eqnarray}
\max_\psi S(\mathop{\rm Diag}\nolimits(\beta+T,0)+(1-\beta-T)\ket{\psi}\bra{\psi} \, || \nonumber\\
\qquad \mathop{\rm Diag}\nolimits(\beta,T)+(1-\beta-T)\ket{\psi}\bra{\psi}).\label{eq:maxi1}
\end{eqnarray}
Positivity of $\eta$ requires $\beta\le 1/2$ and $T\le 1-\beta$.
Again, we can restrict ourselves to states $\psi=(\cos\alpha,\sin\alpha)^T$.
We also have the equivalent of Lemma
\ref{lem:convex}, which needs more work in this case:
\begin{lemma}
For a state vector $\psi=(\cos\alpha,\sin\alpha)^T$, the function to be maximised
in (\ref{eq:maxi1}) is convex in $\cos(2\alpha)$.
\end{lemma}
\textit{Proof.}
Let $D_1$ and $D_2$ be the determinant of the first and second argument, respectively.
Both are linear in $t:=\cos(2\alpha)$:
\begin{eqnarray*}
D_1 &=& (1-\beta-T)(\beta+T)(1-t)/2 \\
D_2 &=& ((\beta+T-\beta^2-T^2)+(1-\beta-T)(T-\beta)t)/2.
\end{eqnarray*}
In the $(D_1,D_2)$-plane, this describes a line segment with gradient
$$
K:= -\frac{T-\beta}{T+\beta},
$$
which lies in the interval $[-1,0]$.
Again, after some
basic algebra,
the function to be maximised in (\ref{eq:maxi1}) is
identified to be
$f((1-4D_1)^{1/2}, (1-4D_2)^{1/2})$, where
\begin{eqnarray*}
f(x,y) &:=& ((1-x)\log(1-x)+(1+x)\log(1+x))/2 \\
&+& ((x^2+y^2-2y-4T^2)\log(1-y) \\
&-& (x^2+y^2+2y-4T^2)\log(1+y))/4 y.
\end{eqnarray*}
We will now show that $f((1-4D_1)^{1/2},
(1-4D_2)^{1/2})$ is convex in $t$.
First, note that
\begin{eqnarray*}
f(x,y)=f_0(x,y)+T^2 f_1(y).
\end{eqnarray*}
The term $f_1(y)$ is itself convex in $t$: its series expansion is
$$
f_1(y) = (\log(1+y)-\log(1-y))/y = 2\sum_{k=0}^\infty \frac{y^{2k}}{2k+1},
$$
which by the positivity of all its coefficients is convex in $y^2$, and $y^2$ is linear in $t$.
The other term, $f_0(x,y)$ is given by a sum of three terms
\begin{eqnarray*}
f_0(x,y) &=& \frac{1}{2}((1-x)\log(1-x)+(1+x)\log(1+x)) \\
&+& \frac{1}{4}((y-2)\log(1-y)-(y+2)\log(1+y)) \\
&-& \frac{x^2}{4}(\log(1+y)-\log(1-y))/y.
\end{eqnarray*}
Replacing each of the three terms by its series expansion yields
\begin{eqnarray*}
f_0(x,y) &=& \sum_{k=1}^\infty \frac{x^{2k}}{2k(2k-1)} \\
&+& \sum_{k=1}^\infty (k-1)\frac{y^{2k}}{2k(2k-1)} \\
&-& \frac{x^2}{2}\sum_{k=0}^\infty \frac{y^{2k}}{2k+1}.
\end{eqnarray*}
To show that this function is convex in $t$, we will evaluate it along the curve
\begin{eqnarray*}
x^2 &=& u+p \\
y^2 &=& v+Kp,
\end{eqnarray*}
with gradient $K$ between 0 and $-1$,
and $u$ and $v$ lying in the interval $[0,1]$,
and check positivity of its second derivative with respect to
$p$ at $p=0$:
\begin{eqnarray*}
&&{\left.
\frac{\partial^2}{\partial p^2}\right |_{p=0}
f_0(x,y)}
=\sum_{k=2}^\infty \frac{k-1}{2k-1}u^{k-2} \\
&&+(k-1)\left(K\frac{(k-1)K-2}{2k-1} -
K^2 \frac{k}{2k+1}u \right)v^{k-2}.
\end{eqnarray*}
The coefficient of $u^{k-2}$ is clearly positive, hence
the derivative is positive if the coefficient of $v^{k-2}$ is positive for all allowed values of $u$ and $K$.
The worst case occurs for $u=1$, yielding a coefficient
$$
K\frac{(k-1)K-2}{2k-1} - K^2 \frac{k}{2k+1} = \frac{-K (2 + 4 k + K)}{(2k-1)(2k+1)}.
$$
For values of $K$ between 0 and $-1$, this is indeed positive.
\hfill$\square$\par\vskip24pt
By the above Lemma, the maximum of the maximisation over $\psi$ is obtained
for extremal values of $t$, that is: either $\psi=(1,0)^T$ or $\psi=(0,1)^T$.
Evaluation of the maximum is again straightforward, and calculations show that
sometimes $\psi=(1,0)^T$ yields the larger value, and sometimes $\psi=(0,1)^T$.
In this way we have obtained the upper bounds:
\begin{theorem}\label{th:ub2b}
Let $\Delta=\rho-\sigma$, $T=|||\Delta|||/|||F|||$ and $\beta=\lambda_{\min}(\sigma)$.
For $d=2$, and $T\le\beta$,
\begin{eqnarray}
S(\rho||\sigma) &\le& (T+1-\beta)\log\frac{T+1-\beta}{1-\beta} \nonumber \\
&+& (\beta-T)\log(1-T/\beta) \label{bound_quad2}.
\end{eqnarray}
For $d=2$, and $T>\beta$,
\begin{eqnarray}
S(\rho||\sigma) &\le& \max(-\log(1-T) , \nonumber\\
&& (\beta+T)\log(1+T/\beta)+\nonumber\\
&& (1-\beta-T)\log(1-T/(1-\beta))).
\end{eqnarray}
\end{theorem}
It is interesting to study the behaviour of the bound in the
case of large
$\beta$. More specifically,
an approximation for bound (\ref{bound_quad2}), valid for $T \ll \beta$, is
\begin{eqnarray}
S(\rho||\sigma) &\le& \sum_{k=2}^\infty \frac{T^k}{k(k-1)}
\left(\frac{1}{\beta^k}-\frac{(-1)^k}{(1-\beta)^k}\right) \nonumber \\
&\approx& \frac{T^2}{2\beta(1-\beta)}, \label{bound_quad2_approx}
\end{eqnarray}
Figure \ref{fig2} illustrates the combined upper bounds of Theorem \ref{th:ub2b}
($d=2$) for various values of $\beta$.
\begin{figure*}
\begin{tabular}{cc}
\includegraphics[width=3.4in]{ud2b0_1.eps} & \includegraphics[width=3.4in]{ud2b0_2.eps} \\
(a) & (b) \\
\includegraphics[width=3.4in]{ud2b0_3.eps} & \includegraphics[width=3.4in]{ud2b0_5.eps} \\
(c) & (d)
\end{tabular}
\caption{\label{fig2}Upper bounds of Theorem \ref{th:ub2b}
on $S=S(\rho||\sigma)$ vs.\ the rescaled norm distance $T=|||\rho-\sigma|||/|||F|||$,
for $d=2$, and for values of smallest eigenvalue of $\sigma$ (a) $\beta=0.1$, (b) $0.2$, (c) $0.3$, and (d) $0.5$.
The two regimes $T\le\beta$ and $\beta\le T\le 1-\beta$ can be clearly identified.
For ease of
comparison, each curve is shown superimposed on the curves for
$\beta=0.1$, $0.2$, $0.3$, $0.4$ and $0.5$ (in grey).}
\end{figure*}
\subsection{The case $d>2$}
In case $d$ is larger than 2, it is not clear how to proceed in the most general setting, for general UI norms,
as the maximisation over $\Delta$ must explicitly be performed.
In the following, we will restrict ourselves to using the trace norm, which is in some sense the most important
one anyway. That is, the requirements on $\Delta$ are
\begin{eqnarray}
||\Delta||_1 &=& 2T, \label{eq:delta1} \\
\mathop{\rm Tr}\nolimits[\Delta] &=& 0 ,\label{eq:delta2} \\
\mathop{\rm Tr}\nolimits[\max(\beta\mathbbm{1},-\Delta)] &\le& 1. \label{eq:delta3}
\end{eqnarray}
The following very simple Lemma will prove to be a powerful tool.
\begin{lemma}\label{lem:sabc}
For all $A$, $B$, and $C$, positive semi-definite operators,
$$
S(A+C||B+C) \le S(A||B).
$$
\end{lemma}
\textit{Proof.}
First note that for any $a>0$,
\begin{eqnarray*}
S(aA||aB) &=& \mathop{\rm Tr}\nolimits[aA(\log(aA)-\log(aB))] \\
&=& a S(A||B).
\end{eqnarray*}
This, together with joint convexity of the relative entropy in its arguments (which
need not be normalised to trace 1), leads to
\begin{eqnarray*}
S(A+C||B+C) &=& 2 S(\frac{A+C}{2} || \frac{B+C}{2}) \\
&\le& S(A||B) + S(C||C) \\
&=& S(A||B).
\end{eqnarray*}
\hfill$\square$\par\vskip24pt
The Lemma immediately yields an upper bound on (\ref{eq:maximand}):
letting
$$
\sigma:=\max(\beta\mathbbm{1},-\Delta) + \eta \ket{\psi}\bra{\psi},
$$
such that we obtain
\begin{eqnarray}
S(\Delta+\sigma \,||\,\sigma)
&\le& S(\Delta+\max(\beta\mathbbm{1},-\Delta)\,||\, \max(\beta\mathbbm{1},-\Delta)) \nonumber \\
&=& S((\Delta+\beta\mathbbm{1})_+ \,||\, \beta\mathbbm{1} + (\Delta+\beta\mathbbm{1})_-).\label{eq:s11}
\end{eqnarray}
To continue, we consider two cases.
{\it Case 1 :}
When $T\le\beta$, the requirement (\ref{eq:delta3}) is automatically satisfied,
and $\max(\beta\mathbbm{1},-\Delta)=\beta\mathbbm{1}$.
Let $\Delta_+$ and $\Delta_-$ be the positive and negative part of $\Delta$, respectively.
That is, $\Delta = \Delta_+ - \Delta_-$, with $\Delta_+$ and $\Delta_-$ non-negative and orthogonal.
Because we are using the trace norm we can rewrite the conditions on $\Delta$ as
\begin{eqnarray*}
||\Delta||_1 &=& \mathop{\rm Tr}\nolimits[\Delta_+] + \mathop{\rm Tr}\nolimits[\Delta_-] = 2T ,\\
\mathop{\rm Tr}\nolimits[\Delta] &=& \mathop{\rm Tr}\nolimits[\Delta_+ ]- \mathop{\rm Tr}\nolimits[\Delta_- ]= 0,
\end{eqnarray*}
hence
$$
\mathop{\rm Tr}\nolimits[\Delta_+ ]= \mathop{\rm Tr}\nolimits[\Delta_-]= T.
$$
By Lemma \ref{lem:sabc}, (\ref{eq:maximand}) is upper bounded by $S(\Delta+\beta\mathbbm{1}||\beta\mathbbm{1})$.
By convexity, its maximum over $\Delta_+$, $\Delta_-\ge0$, with $\mathop{\rm Tr}\nolimits[\Delta_+ ]=
\mathop{\rm Tr}\nolimits[\Delta_-] = T$, is obtained
in $\Delta_+$ and $\Delta_-$ of rank 1, giving as upper bound
$$
S(\Delta + \sigma||\sigma) \leq
(\beta+T)\log\frac{\beta+T}{\beta}
+(\beta-T)\log\frac{\beta-T}{\beta}.
$$
The upper bound can be achieved in dimensions $d\ge3$
for all values of $T\le\beta$ by setting $\Delta=TF$ and $\psi=e^3$.
{\it Case 2:}
In the other case, when $T>\beta$, we have to deal with condition (\ref{eq:delta3}).
To do that we split $\Delta$ into three non-negative parts,
\begin{equation}\nonumber
\Delta = \Delta_+ -\Delta_0-\Delta_-,
\end{equation}
with $\Delta_+$, $\Delta_0$ and $\Delta_-$,
operating on orthogonal subspaces $V_+$, $V_0$ and $V_-$, respectively,
with
\begin{eqnarray*}
\phantom{-}\Delta_+ &\ge& 0, \\
0 &\ge& -\Delta_0\, \ge -\beta\mathbbm{1}_0, \\
-\beta\mathbbm{1}_- &\ge& -\Delta_-.
\end{eqnarray*}
We denote the projectors on these subspaces by $\mathbbm{1}_+$, $\mathbbm{1}_0$, and $\mathbbm{1}_-$.
Then
$$
(\Delta+\beta)_+ = \Delta_+ - \Delta_0 + \beta\mathbbm{1}_{+0},
$$
where $\mathbbm{1}_{+0}:=\mathbbm{1}_++\mathbbm{1}_0$.
The conditions on $\Delta$, $\mathop{\rm Tr}\nolimits[\Delta]=0$ and $\mathop{\rm Tr}\nolimits[ |\Delta| ]=2T$ translate to
$$
\mathop{\rm Tr}\nolimits[\Delta_+ ]= \mathop{\rm Tr}\nolimits[\Delta_0]+\mathop{\rm Tr}\nolimits[\Delta_- ]= T.
$$
Due to the orthogonality of positive and negative part, (\ref{eq:s11}) can be simplified to
$
S((\Delta+\beta\mathbbm{1})_+ \,||\, \beta\mathbbm{1}_{+0})$.
After subtracting $\beta\mathbbm{1}_0-\Delta_0$ from both arguments,
we get
$$
S(\Delta_++\beta\mathbbm{1}_+ || \beta\mathbbm{1}_++\Delta_0),
$$
which is an upper bound on (\ref{eq:s11}), by Lemma \ref{lem:sabc}.
Ignoring condition (\ref{eq:delta3}) on $\Delta$, we get
$$
S_{\max} \le \max_{\Delta_+\ge 0 \atop \mathop{\rm Tr}\nolimits[\Delta_+ ]= T}
S(\Delta_++\beta\mathbbm{1}_+ || \beta\mathbbm{1}_+).
$$
By convexity, the maximum is obtained for $\Delta_+$ rank 1, giving
the upper bound
$$
S_{\max}\le(T+\beta)\log((T+\beta)/\beta).
$$
To see that this bound is sharp for (almost) any value of $T$,
consider the two states
\begin{eqnarray*}
\rho &=& \mathop{\rm Diag}\nolimits(T+\beta,0,0^{\times J},\beta^{\times K},\beta+\eta) , \\
\sigma &=& \mathop{\rm Diag}\nolimits(\beta,T-J\beta,\beta^{\times J},\beta^{\times
K},\beta+\eta) ,\\
\eta &:=& 1-T-(d-1-J)\beta.
\end{eqnarray*}
Here, $J$ is an integer between 0 and $d-3$ and $k=d-3-J$.
Conditions on $J$ are $J\beta\le T$ (so that $\sigma\ge0$) and $T\le 1-(d-1-J)\beta$
(so that $\eta\ge0$).
This choice of states can thus be obtained for $\beta\le T\le 1-2\beta$.
It can be seen that $||\rho-\sigma||_1=2T$ and
\begin{equation}\nonumber
S(\rho||\sigma)=(T+\beta)\log((T+\beta)/\beta).
\end{equation}
The result of the foregoing can be subsumed into the following theorem.
\begin{theorem}
Let $\Delta=\rho-\sigma$, $T=||\Delta||_1/2$ and $\beta=\lambda_{\min}(\sigma)$.
If $T \le \beta$ then
\begin{equation}\nonumber
S(\rho||\sigma) \le (\beta+T)\log\frac{\beta+T}{\beta}
+(\beta-T)\log\frac{\beta-T}{\beta}, \label{bound_quad2_bis}
\end{equation}
and this upper bound is sharp when $d>2$.
If $\beta\le T \le 1-\beta$ then
\begin{equation}\nonumber
S(\rho||\sigma) \le (\beta+T)\log\frac{\beta+T}{\beta}. \label{bound_quad3}
\end{equation}
When $d>2$, this bound is sharp for (at least) $\beta\le T\le 1-2\beta$.
\end{theorem}
Figure \ref{fig3} illustrates these bounds and shows their superiority to the previously obtained bound
(\ref{bound_log}).
\begin{figure}
\includegraphics[width=3.4in]{ud3b2.eps}
\caption{\label{fig3}Comparison between upper bounds (\ref{bound_log}) and (\ref{bound_quad2_bis})-(\ref{bound_quad3})
on $S=S(\rho||\sigma)$ vs.\ the trace norm distance $T=||\rho-\sigma||_1/2$,
for various values of $\beta$, the smallest eigenvalue of $\sigma$.
The upper set of dashed curves depict bound (\ref{bound_log}) (with $d=3$)
for $\beta=0.1$ (lower curve), $0.2$, $0.3$, $0.4$ and $0.5$ (upper curve).
The lower set of full line curves depict bounds (\ref{bound_quad2_bis})-(\ref{bound_quad3})
for $\beta=0.1$ (upper curve), $0.2$, $0.3$, $0.4$ and $0.5$ (lower curve).
The two regimes $T\le\beta$ and $\beta\le T\le 1-\beta$ can be clearly seen.
}
\end{figure}
Again, it is interesting to look at the bound for large $\beta$.
An approximation for bound (\ref{bound_quad2_bis}), valid for $T \ll \beta$, is
given by
\begin{eqnarray}
S(\rho||\sigma) \le \sum_{k=1}^\infty \frac{T^{2k}}{k(2k-1) \beta^{2k-1}}
\approx
\frac{T^2}{\beta}. \label{bound_quad2_bis_approx}
\end{eqnarray}
\section{Application to state approximation}
In the following paragraph we will give an application of our bounds
to state approximation. Consider a state $\rho$ on a Hilbert space ${\cal H}$, and
a sequence $\{\sigma_n\}_n$
where $\sigma_n$ is a state on ${\cal H}^{\otimes n}$. As before,
the sequence is said to asymptotically approximate $\rho$ if for
$n$ tending to infinity, $
\| \sigma_n-\rho^{\otimes n}\|_1= \mathop{\rm Tr}\nolimits|\sigma_n-\rho^{\otimes n}|$
tends to zero.
Let us define $T_n$ as
$$
T_n := \mathop{\rm Tr}\nolimits|\rho^{\otimes n}-\sigma_n|/2.
$$
Because of the lower bound (\ref{bound_ohya}), we get
$$
S_n:=S(\rho^{\otimes n}||\sigma_n) \ge 2 T_n^2,
$$
and this bound is sharp.
Hence, $T_n$ goes to zero if $S_n$ does.
On the other hand, $T_n$ going to zero does not necessarily imply $S_n$ going to zero.
Indeed, $S_n$ can be infinite for any finite value of $n$ when $\rho^{\otimes n}$ is not restricted to the
range of $\sigma_n$. In particular, the relative entropy distance between two pure states is infinite unless
the states are identical. At first sight, this seems to render the relative entropy useless as a distance measure.
Nevertheless, sense can be made of it by imposing an additional requirement that the range of $\sigma_n$ must
contain the range of $\rho^{\otimes n}$.
Let us then restrict $\sigma_n$ to the range of $\rho^{\otimes n}$, as the relative entropy
only depends on that part of $\sigma_n$.
Letting $d$ be the rank of $\rho$, the dimension of the range of $\rho^{\otimes n}$ is $d^n$.
Let $\beta_n$ be the smallest non-zero eigenvalue of $\sigma_n$ on that range;
$\beta_n$ is at most $1/d^n$.
The behaviour of the relative entropy then very much depends on the relation between $\beta_n$ and $T_n$.
Since $\beta_n$ decreases at least exponentially, we only need to consider the case $T_n\ge\beta_n$, and
use the bound (\ref{bound_quad3})
$$
S_n \le (\beta_n+T_n)\log \biggl (1+\frac{T_n}{\beta_n}\biggr).
$$
In the worst-case behaviour of $T_n$ ($T_n/\beta_n$ tending to infinity) the bound can be approximated by
\begin{eqnarray*}
S_n & \le & T_n\log\frac{T_n}{\beta_n} = T_n(\log
T_n-\log\beta_n)\nonumber\\
& \approx & T_n|\log\beta_n|.
\end{eqnarray*}
To guarantee convergence of $S_n$ we therefore need $T_n$ to converge to 0 at least as fast as
$1/|\log\beta_n|$, which in the best case goes as $1/n$.
Note that bound (\ref{bound_log}) yields the same requirement, but as this bound is not a sharp one
it could have been too strong a requirement.
This gives us the subsequent theorem.
\begin{theorem}
Consider a state $\rho$ on a finite-dimensional Hilbert space ${\cal H}$
and a sequence $\{\sigma_n\}_n$ of states $\sigma_n$ on ${\cal H}^{\otimes n}$.
The sequence $\{\sigma_n\}_n$ asymptotically approximates $\rho$ in the trace norm,
if
\begin{equation}\nonumber
\lim_{n\rightarrow\infty}
S(\rho^{\otimes n}||\sigma_n)=0.
\end{equation}
Conversely, if the range of $\sigma_n$ includes the range of $\rho^{\otimes n}$
and $||\rho^{\otimes n}-\sigma_n||_1$ converges to zero faster than $1/|\log\beta_n|$,
where $\beta_n$ is the minimal eigenvalue of $\sigma_n$ restricted to the range of $\rho^{\otimes n}$,
then $\lim_{n\rightarrow\infty} S(\rho^{\otimes n}||\sigma_n)=0$.
\end{theorem}
\section{Summary}
In this paper, we have discussed several lower and upper bounds on the
relative entropy functional, thereby sharpening the notion of
continuity of the relative entropy for states which are
close to each other in the trace norm sense.
The main results are the sharp lower bound from Theorem 4, and the sharp upper bounds of
Theorems 5 ($d=2$) and 6 ($d>2$). Theorems 4 and 5 give the relation between relative entropy
and norm distances based on any unitarily invariant norm, while Theorem 6 holds only for the
trace norm distance. These results have been obtained employing methods from
optimisation theory.
\begin{acknowledgments}
This work was supported by the Alexander-von-Humboldt Foundation,
the European Commission (EQUIP, IST-1999-11053), the DFG
(Schwerpunktprogramm QIV), the EPSRC QIP-IRC,
and the European Research Councils
(EURYI).
\end{acknowledgments}
|
{
"timestamp": "2005-10-14T18:15:20",
"yymm": "0503",
"arxiv_id": "quant-ph/0503218",
"language": "en",
"url": "https://arxiv.org/abs/quant-ph/0503218"
}
|
\subsection*{Isospectral billiards.}
All known isospectral billiards can be obtained by unfolding
triangle-shaped tiles \cite{BusConDoySem94, OkaShu01}.
The way the tiles are unfolded can be specified by three
permutation matrices $M^{(\mu)}$, $1\leq \mu\leq 3$, associated to the
three sides of the triangle: $M^{(\mu)}_{ij}=1$ if tiles $i$ and $j$
are glued by their side $\mu$ (and $M^{(\mu)}_{ii}=1$ if the side $\mu$
of tile $i$ is the boundary of the billiard),
and 0 otherwise \cite{OkaShu01, Tha04, Gir04}.
Following \cite{OkaShu01}, one can sum up the action of the $M^{(\mu)}$
in a graph with coloured edges: each copy of the base
tile is associated to a vertex, and vertices $i$ and $j$, $i\neq j$,
are linked by an edge of colour $\mu$ if and only if $M^{(\mu)}_{ij}=1$
(see \ref{graphs}).
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.66\linewidth]{fig2.eps}
\end{center}
\caption{The graphs corresponding to a pair of isospectral billiards:
if we label the sides of the triangle by $\mu=1,2,3$, the unfolding rule
by symmetry with respect to side $\mu$ can be represented by edges made
of $\mu$ braids in the graph. From a given pair of graphs, one can construct
infinitely many pairs of isospectral billiards by applying the unfolding
rules to any triangle. Note that a different labeling of the tiles would just
induce a permutation of the labelings of points and blocks in the Fano plane.}
\label{graphs}
\end{figure}
In the same way, in the second member of the pair, the tiles are
unfolded according to
permutation matrices $N^{(\mu)}$, $1\leq \mu\leq 3$. Two billiards are
said to be transplantable if there exists an invertible matrix $T$
(the {\it transplantation matrix})
such that $\forall\mu\ \ T M^{(\mu)}= N^{(\mu)} T$. One can show that
transplantability implies isospectrality (if the matrix $T$ is not
merely a permutation matrix, in which case the two domains would just
have the same shape). The underlying
idea is that if $\psi^{(1)}$ is
an eigenfunction of the first billiard and $\psi^{(1)}_i$ its restriction
to triangle $i$, then one can build an eigenfunction $\psi^{(2)}$
of the second billiard by taking $\psi^{(2)}_i=\sum_j T_{ij}\psi^{(1)}_j$.
Obviously $\psi^{(2)}$
verifies Schr\"odinger equation; it can be checked from the
commutation relations that the function is smooth at all edges
of the triangles, and that boundary conditions at the boundary of
the billiard are fulfilled \cite{OkaShu01}.\\
Suppose we want to construct a pair of isospectral billiards, starting
from any polygonal base shape. Our idea is to start from the
transplantation matrix, and choose it in such a way that the existence
of commutation relations $T M^{(\mu)}= N^{(\mu)} T$ for some
permutation matrices $M^{(\mu)}, N^{(\mu)}$ will be known {\it a priori}.
As we will see, this is the case if $T$ is taken to be
the incidence matrix of a FPS; the matrices $M^{(\mu)}$ and
$N^{(\mu)}$ are then permutations on the points and the hyperplanes
of the FPS.
\subsection*{Finite projective spaces.}
For $n\geq 2$ and $q=p^h$ a power of a prime number, consider
the $(n+1)$-dimensional vector space $\mathbb{F}_{q}^{n+1}$, where
$\mathbb{F}_{q}$ is the finite field of order $q$.
The {\it finite projective space} $PG(n,q)$ of {\it dimension} $n$
and {\it order} $q$ is the set of subspaces of $\mathbb{F}_{q}^{n+1}$:
the points of $PG(n,q)$ are the 1-dimensional subspaces of $\mathbb{F}_{q}^{n+1}$,
the lines of $PG(n,q)$ are the 2-dimensional subspaces of $\mathbb{F}_{q}^{n+1}$,
and more generally $(d+1)$-dimensional subspaces of $\mathbb{F}_{q}^{n+1}$ are
called {\it $d$-spaces} of $PG(n,q)$; the $(n-1)$-spaces of $PG(n,q)$
are called hyperplanes or {\it blocks}.
A $d$-space of $PG(n,q)$ contains $(q^{d+1}-1)/(q-1)$ points. In particular,
$PG(n,q)$ has $(q^{n+1}-1)/(q-1)$ points. It also has
$(q^{n+1}-1)/(q-1)$ blocks \cite{Hir79}.
As an example, \ref{fano} shows the finite projective plane (FPP)
of order $q=2$, or Fano plane, $PG(2,2)$.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.58\linewidth]{fig3.eps}
\caption{The Fano plane $PG(2,2)$ and its corresponding incidence
matrix $T$. The Fano plane has $(q^3-1)/(q-1)=7$ points
and 7 lines; each line contains $q+1=3$ points and each point belongs
to 3 lines. Any pair of points belongs to one and only one line.}
\label{fano}
\end{center}
\end{figure}
A $(N,k, \lambda)-${\it symmetric balanced incomplete block design}
(SBIBD) is a set of
$N$ points, belonging to $N$ subsets (or {\it blocks});
each block contains $k$ points, in such a way that any two points belong
to exactly $\lambda$ blocks, and each point is contained in
$k$ different blocks \cite{DinSti92}. One can show that
$PG(n,q)$ is a $(N,k, \lambda)$-SBIBD with $N=(q^{n+1}-1)/(q-1)$,
$k=(q^{n}-1)/(q-1)$ and $\lambda=(q^{n-1}-1)/(q-1))$.
For example, the Fano plane is a $(7,3,1)-$SBIBD.
The points and the blocks can be labeled from $0$ to $N-1$.
For any $(N,k, \lambda)-$SBIBD one can define an $N\times N$
{\it incidence matrix}
$T$ describing to which block each point belongs. The entries $T_{ij}$
of the matrix are $T_{ij}=1$ if point $j$ belongs to line $i$, $0$
otherwise. The matrix $T$ verifies the relation
$T T^{t}=\lambda J+(N-k)\lambda/(k-1)I$,
where $J$ is the $N\times N$ matrix with all entries equal to 1
and $I$ the $N\times N$ identity matrix \cite{DinSti92}.
In particular, the incidence matrix of $PG(n,q)$ verifies
\begin{equation}
\label{tt}
T T^{t}=\lambda J+(k-\lambda)I
\end{equation}
with $k$ and $\lambda$ as given above. For example, the incidence matrix
of the Fano plane given in
\ref{fano} corresponds to a labeling of the lines such that line 0
contains points 0,1,3, and line 1 contains points 1,2,4, etc.
A {\it collineation} of a FPS is a
bijection that preserves incidence, that
is a permutation of the points that takes $d$-spaces to
$d$-spaces (in particular, it takes blocks to blocks).
Any permutation $\sigma$ on the points
can be written as a $N\times N$ {\it permutation matrix} $M$ defined by
$M_{i\sigma(i)}=1$ and the other entries equal to zero. If $M$ is a
permutation matrix associated to a collineation, then there exists a
permutation matrix $N$ such that
\begin{equation}
\label{commutation}
TM=NT.
\end{equation}
In words, (\ref{commutation}) means that permuting the
columns of $T$ (i.e. the blocks of the
space) under $M$ is equivalent to permuting the rows of $T$
(i.e. the points of the space) under $N$.
The commutation relation (\ref{commutation}) is a
related to an important feature of projective geometry, the so-called
''principle of duality'' \cite{Hir79}.
This principle states that for any theorem which is
true in a FPS, the dual theorem obtained by exchanging
the words ''point'' and ''block'' is also true. As we will see now,
this symmetry between points and blocks in FPSs is the central reason
which accounts for known pairs of isospectral billiards.\\
Let us consider a FPS ${\mathcal P}$ with incidence matrix $T$.
To each block in ${\mathcal P}$ we associate a tile in the first billiard,
and to each point in ${\mathcal P}$ we associate a tile in the second billiard.
If we choose permutations $M^{(\mu)}$ in the collineation
group of ${\mathcal P}$, then the commutation relation (\ref{commutation})
will ensure that there exist permutations $N^{(\mu)}$ verifying
$TM^{(\mu)}=N^{(\mu)}T$. These commutation relations imply
transplantability, and thus isospectrality, of the billiards
constructed from the graphs corresponding to $M^{(\mu)}$ and $N^{(\mu)}$.
If the base tile has $r$ sides, we need to choose $r$
elements $M^{(\mu)}$, $1\leq\mu\leq r$, in the
collineation group of ${\mathcal P}$. This choice is
constrained by several factors. Since
$M^{(\mu)}$ represents the reflexion of a tile with respect to one of
its sides, it has to be of order 2 (i.e. an involution).
In order that the billiards be connected, no point should be left out by
the matrices $M^{(\mu)}$ (in other words, the graph associated to the
matrices $M^{(\mu)}$ should be connected).
Finally, if we want the base tile to be of any shape, there should
be no loop in the graph. We now need to characterize
collineations of order 2.
\subsection*{Collineations of finite projective spaces.}
Let $q=p^h$ be a power of a prime number.
Each point $P$ of $PG(n,q)$ is a 1-dimensional subspace of $\mathbb{F}_{q}^{n+1}$,
spanned by some vector $v$. We write $P=P(v)$.
An {\it automorphism} is a bijection of the points $P(v_i)$
of $PG(n,q)$ obtained
by the action of an automorphism of $\mathbb{F}_{q}$ on the coordinates
of the $v_i$. If $q=p^h$, the automorphisms of $\mathbb{F}_{q}$ are
$t\mapsto t^{p^i}$, $0\leq i \leq h-1$.
A {\it projectivity} is a bijection of the points $P(v_i)$
of $PG(n,q)$ obtained by the action of a linear map $L$ on the $v_i$.
There are $q-1$ matrices $t L$, with $t\in\mathbb{F}_{q}\setminus\{0\}$ and
$L\in GL_{n+1}(\mathbb{F}_{q})$ (the group of $(n+1)\times (n+1)$ invertible
matrices with coefficients in $\mathbb{F}_{q}$), yielding the same projectivity
$P(v_i)\mapsto P(L v_i)$.
The {\it Fundamental theorem of projective geometry } states
that any collineation of $PG(n,q)$ can be written as
the composition of a projectivity by an automorphism \cite{Hir79}.
The converse is obviously true since projectivities and automorphisms
are collineations. Therefore the set of all collineations is obtained
by taking the composition of all the non-singular $(n+1)\times(n+1)$
matrices with coefficients in $\mathbb{F}_{q}$ by all the $h$ automorphisms of $\mathbb{F}_{q}$.
The collineation group of $PG(n, q)$ has
$[h\prod_{k=0}^{n}(q^{n+1}-q^k)]/(q-1)$ elements, among which we only
want to consider elements of order 2. In the case of FPPs ($n=2$),
there are various known properties characterizing
collineations of order 2.
A {\it central collineation}, or {\it perspectivity}, is a collineation fixing
each line through a point $C$ (called the centre). By ''fixing'' we
mean that the line is invariant but the points can be permuted within the
line. One can show that the fixed points of a non-identical perspectivity
are the centre itself
and all points on a line (called the axis), while the fixed lines are
the axis and all lines through the centre.
If the centre lies off the axis a perspectivity is called a
{\it homology} (and has $q+2$ fixed points), whereas if the centre
lies on the axis it is called an {\it elation} (and has $q+1$ fixed points).
Perspectivities in dimension $n=2$ have following properties \cite{Bon04}:
{\scshape Proposition 1.} A perspectivity of order two
of a FPP of order $q$ is an elation or a
homology according to whether $q$ is even or odd.
{\scshape Proposition 2.} A collineation of order two of a FPP
of order $q$ is a perspectivity if $q$
is not a square; it is a collineation fixing all points and lines
in a subplane if $q$ is a square (a subplane is a subset of points
having all the properties of a FPP).
When the order of the FPP is a square, there is
the following result \cite{Hir79}:
{\scshape Proposition 3.} $PG(2, q^2)$ can be partitioned into $q^2-q+1$
subplanes $PG(2, q)$.
\subsection*{Generating isospectral pairs.}
Let us assume we are looking for a pair of isospectral billiards with
$N=(q^3-1)/(q-1)$ copies of a base tile having the
shape of a $r$-gon, $r\geq 3$.
We need to find $r$ collineations of order 2 such that the associated graph is
connected and without loop. Such a graph connects $N$ vertices and thus
requires $N-1$ edges. From propositions 1-3, we can deduce the
number $s$ of fixed points of a collineation of order 2 for any FPP.
Since a collineation is a permutation, it has a cycle decomposition as
a product of transpositions. For permutations of order 2 with $s$ fixed
points, there are $(N-s)/2$ independent transpositions in this decomposition.
Each transposition is represented by an edge in the graph. As a consequence,
$q$, $r$ and $s$ have to fulfill the following condition:
$r(q^2+q+1-s)/2=q^2+q$. Let us examine the various cases.
{\it If $q$ is even and not a square},
propositions 1 and 2 imply that
any collineation of order 2 is an elation and therefore has $q+1$
fixed points. Therefore, $q$ and $r$ are constrained by the relation
$r q^2/2=q^2+q$.
The only integer solution with $r\geq 3$ and $q\geq 2$ is $(r=3, q=2)$.
These isospectral billiards correspond to $PG(2,2)$ and will be made
of $N=7$ copies of a base triangle.
{\it If $q$ is odd and not a square},
propositions 1 and 2 imply that
any collineation of order 2 is a homology and therefore has $q+2$
fixed points. Therefore, $q$ and $r$ are constrained by the relation
$r (q^2-1)/2=q^2+q$.
The only integer solution with $r\geq 3$ and $q\geq 2$ is $(r=3, q=3)$.
These isospectral billiards correspond to $PG(2,3)$ and will be made
of $N=13$ copies of a base triangle.
{\it If $q=p^2$ is a square},
propositions 2 and 3 imply that
any collineation of order 2 fixes all points in a subplane $PG(2,p)$
and therefore has $p^2+p+1$ fixed points. Therefore, $p$ and $r$ are
constrained by the relation $r (p^4-p)/2=p^4+p^2$.
There is no integer solution with $r\geq 3$ and $q\geq 2$. However,
one can look for isospectral billiards with loops: this will require
the base tile to have a shape such that the loop does not make the
copies of the tiles come on top of each other when unfolded.
If we tolerate one loop
in the graph describing the isospectral billiards, then there are $N$
edges in the graph instead of $N-1$ and the equation for $p$ and $r$
becomes $r (p^4-p)/2=p^4+p^2+1$, which has the only integer solution
$(r=3, p=2)$. These isospectral billiards correspond to $PG(2,4)$
and will be made of $N=21$ copies of a base triangle.
We can now generate all possible pairs of isospectral billiards
whose transplantation matrix is the incidence matrix of $PG(2,q)$,
with $r$ and $q$ restricted by the previous analysis. All pairs
must have a triangular base shape ($r=3$).
$PG(2, 2)$ provides 3 pairs (made of 7 tiles),
$PG(2, 3)$ provides 9 pairs (made of 13 tiles),
$PG(2, 4)$ provides 1 pair (made of 21 tiles).
It turns out that the pairs obtained here are exactly those obtained by
\cite{BusConDoySem94} and \cite{OkaShu01} by other methods.
Let us now consider spaces $PG(n,q)$ of higher dimensions. The smallest
one is $PG(3,2)$, which contains 15 points.
Since the base field for $PG(3,2)$ is $\mathbb{F}_{2}$, the Fundamental
Theorem of projective geometry states that the collineation group of
$PG(3,2)$ is essentially the group $GL_4(\mathbb{F}_{2})$ of
$4\times 4$ non-singular matrices with coefficients in $\mathbb{F}_{2}$.
Generating all possible graphs from the 316
elements of order 2 in $GL_4(\mathbb{F}_{2})$, we obtain four pairs of
isospectral billiards with 15 triangular tiles, which completes
the list of all pairs found in \cite{BusConDoySem94} and \cite{OkaShu01}.\\
Our method explicitly gives the transplantation matrix $T$ for all these
pairs: each one is the incidence matrix of a FPS.
The transplantation matrix explicitly provides the mapping between
eigenfunctions of both billiards. The inverse mapping is given by
$T^{-1}=(1/q^{n-1})(T^{t}-(\lambda/k)J)$.
For all pairs, isospectrality can therefore be explained
by the symmetry between points and blocks in FPSs.
We do not know if it is possible to find isospectral billiards for
which isospectrality would not rely on this symmetry.
Our construction furthermore allows to generalize the results we obtained
in \cite{Gir04}, where a relation between the Green functions of the
billiards in an isospectral pair was derived.
A similar relation can be found for all other pairs
obtained by point-block duality. Let $M^{(\mu)}$ and $N^{(\mu)}$
be the matrices describing the gluings of the tiles.
To any path $p$ going from a point to another on the first billiard,
one can associate the sequence $(\mu_1,...,\mu_n)$, $1\leq\mu_i\leq 3$,
of edges of the triangle hit by the path. The matrix $M=\prod M^{(\mu_i)}$
is such that $M_{ij}=1$ if path $p$ can be drawn from $i$ to $j$.
If $N=T^{-1}MT$, relations (\ref{tt}) and (\ref{commutation}) imply
$\sum_{k,l}T_{ik}T_{jl}M_{kl}=\lambda+(k-\lambda)N_{ij}$.
Since Green functions can be written as a sum
over all paths, the relation between the Green functions $G^{(2)}(a,i;b,j)$
and $G^{(1)}(a,i';b,j')$ is
$\sum_{i',j'}T_{i,i'}T_{j,j'}G^{(1)}(a,i';b,j')=
(k-\lambda)G^{(2)}(a,i;b,j)+\lambda G^{t}(a;b)$, where $G^{t}$ is the
Green function of the triangle, and a point $(a,i)$ is specified
by a tile number $i$ and its position $a$ inside the tile
(see \cite{Gir04} for further detail).
More precisely, the term $\sum_{k,l}T_{ik}T_{jl}M_{kl}$
can be interpreted as the number of pairs of
tiles $(i,j)$ in the first billiard such that a path identical to
$p$ can go from $i$ to $j$.
A given diffractive orbit going from $i$ to $j$ in the second
billiard corresponds to a matrix $N$ such that $N_{ij}=1$:
it is therefore constructed from a superposition
of $k=(q^n-1)/(q-1)$ identical diffractive orbits in the first billiard.
(Note that these results correspond to Neumann boundary conditions. It is
easy to obtain similar relations for Dirichlet
boundary conditions by
conjugating all matrices with a diagonal matrix $D$ with entries
$D_{ii}=\pm 1$ according to whether tile $i$ is like the initial tile
or like its mirror inverse.)
|
{
"timestamp": "2005-03-31T15:25:36",
"yymm": "0503",
"arxiv_id": "nlin/0503069",
"language": "en",
"url": "https://arxiv.org/abs/nlin/0503069"
}
|
\section{Introduction}
Nematic liquid crystals (nematics) represent the simplest
anisotropic fluid. The description of the dynamic behavior of the
nematics is based on well established equations. The description
is valid for low molecular weight materials as well as nematic
polymers.
The coupling between the preferred molecular orientation (director
$\Vec{\Hat{n}}$) and the velocity field leads to interesting flow
phenomena. The orientational dynamics of nematics in flow strongly
depends on the sign of the ratio of the Leslie viscosity
coefficients $\lambda = \alpha_3 / \alpha_2$.
In typical low molecular weight nematics $\lambda$ is positive
({\em flow-aligning materials}). The case of the initial director
orientation perpendicular to the flow plane has been clarified in
classical experiments by Pieranski and Guyon
\cite{Pieranski:SSC:1973, Pieranski:PRA:1974} and theoretical
works of Dubois-Violette and Manneville (for an overview see
\cite{PF:Ch4}). An additional external magnetic field could be
applied along the initial director orientation. In Couette flow
and low magnetic field there is a homogeneous instability
\cite{Pieranski:SSC:1973}. For high magnetic field the homogeneous
instability is replaced by a spatially periodic one leading to
rolls \cite{Pieranski:PRA:1974}. In Poiseuille flow, as in Couette
flow, the homogeneous instability is replaced by a spatially
periodic one with increasing magnetic field
\cite{Manneville:JPh:1979}. All these instabilities are
stationary.
Some nematics (in particular near a nematic-smectic transition)
have negative $\lambda$ ({\em non-flow-aligning materials}). For
these materials in steady flow and in the geometry where the
initial director orientation is perpendicular to the flow plane
only spatially periodic instabilities are expected
\cite{Pieranski:CPh}. These materials demonstrate also tumbling
motion \cite{Cladis:PRL:1975} in the geometry where the initial
director orientation is perpendicular to the confined plates that
make the orientational behavior quite complicated.
Most previous theoretical investigations of the orientational
dynamics of nematics in shear flow were carried out under the
assumption of strong anchoring of the nematic molecules at the
confining plates. However, it is known that there is substantial
influence of the boundary conditions on the dynamical properties
of nematics in hydrodynamic flow \cite{Kedney:LC:V24:P613:Y1998,
Nasibullayev:MCLC:V351:P395:Y2000,
Tarasov:LC:V28:N6:P833:Y2001,Nasibullayev:CR:2001}. Indeed, the
anchoring strength strongly influences the orientational behavior
and dynamic response of nematics under external electric and
magnetic fields. This changes, for example, the switching times in
bistable nematic cells \cite{Kedney:LC:V24:P613:Y1998}, which play
an important role in applications \cite{Chigrinov:1999}. Recently
the influence of the boundary anchoring on the homogeneous
instabilities in steady flow was investigated theoretically
\cite{Tarasov:LC:V28:N6:P833:Y2001}.
In this paper we study the combined action of steady flow (Couette
and Poiseuille) and external fields (electric and magnetic) on the
orientational instabilities of the nematics with initial
orientation perpendicular to the flow plane. We focus on {\em
flow-aligning} nematics. The external electric field is applied
across the nematic layer and the external magnetic field is
applied perpendicular to the flow plane. We analyse the influence
of weak azimuthal and polar anchoring and of external fields on
both homogeneous and spatially periodic instabilities.
In section II the formulation of the problem based on the standard
set of Ericksen-Leslie hydrodynamic equations
\cite{Leslie:MCLC:1976} is presented. Boundary conditions and the
critical Fre\'edericksz field in case of weak anchoring are
discussed. In section III equations for the homogeneous
instabilities are presented. Rigorous semi-analytical expressions
for the critical shear rate $a_c^2$ for Couette flow (section III
A), the numerical scheme for finding $a_c^2$ for Poiseuille flow
(section III B) and approximate analytical expressions for both
types of flows (section III C) are presented. In section IV the
analysis of the spatially periodic instabilities is given and in
section V we discuss the results. In particular we will be
interested in the boundaries in parameter space (anchoring
strengths, external fields) for the occurrence of the different
types of instabilities.
\section{Basic equations}
\begin{figure}
\epsfig{file=cell.eps,width=8cm}
\caption{Geometry of NLC cell ($a$). Couette ($b$) and Poiseuille ($c$) flows.}
\label{fig:cell}
\end{figure}
Consider a nematic layer of thickness $d$ sandwiched between two
infinite parallel plates that provide weak anchoring
(Fig. \ref{fig:cell} $a$). The origin of the Cartesian coordinates is
placed in the middle of the layer with the $z$ axis perpendicular
to the confining plates ($z = \pm d / 2$ for the upper/lower plate).
The flow is applied along $x$.
Steady Couette flow is induced by moving the upper plate with
a constant speed (Fig. \ref{fig:cell} $b$). Steady Poiseuille flow is induced
by applying a constant pressure difference along $x$ (Fig. \ref{fig:cell} $c$).
An external electric field $E_0$ is applied along $z$
and a magnetic field $H_0$ along $y$.
The nematodynamic equations have the following form \cite{deGennes}
\begin{eqnarray}
\label{eqn:EL:velocity}
&&\rho (\partial_t + \Vec{v} \cdot \Vec{\nabla}) v_i = - p_{,i} + [T^v_{ji} + T^e_{ji}]_{,j},\\
\label{eqn:EL:director}
&&\gamma_1 \Vec{N} = - (1 - \Vec{n} \Vec{n} \cdot) (\gamma_2 A \cdot \Vec{n} + \Vec{h}),
\end{eqnarray}
where $\rho$ is the density of the NLC and $p_{,i} = \Delta P /
\Delta x$ the pressure gradient; $\gamma_1 = \alpha_3 - \alpha_2$
and $\gamma_2 = \alpha_3 + \alpha_2$ are rotational viscosities;
$\Vec{N} = \Vec{n}_{,t} + \Vec{v} \cdot \Vec{\nabla} \Vec{n} -
(\nabla \times \Vec{v}) \times \Vec{n}/ 2$ and $A_{ij} = (v_{i,j}
+ v_{j,i}) / 2$, $h_i = \delta F / \delta n_i$. The notation
$f_{,i}\equiv \partial_i f$ is used throughout. The viscous tensor
$T^v_{ij}$ and elastic tensor $T^e_{ij}$ are
\begin{eqnarray}
&T^v_{ij} = &\alpha_1 n_i n_j A_{km} n_k n_m + \alpha_2 n_i N_j +
\alpha_3 n_j N_i \nonumber\\
&& + \alpha_4 A_{ij} + \alpha_5 n_i n_k A_{ki} + \alpha_6 A_{ik} n_k n_j,\\
&T^e_{ij} = & - \frac{\partial F}{\partial n_{k,i}} n_{k,j},
\end{eqnarray}
where $\alpha_i$ are the Leslie viscosity coefficients. The bulk free energy density $F$ is
\begin{eqnarray}
&F = & \frac12 \Bigl\{ K_{11} (\nabla \cdot \Vec{n})^2 + K_{22} [\Vec{n} \cdot
(\nabla \times \Vec{n})]^2 \nonumber\\
&& + K_{33} [\Vec{n} \times (\nabla \times \Vec{n})]^2 -
\varepsilon_0 \varepsilon_a (\Vec{n} \cdot \Vec{E}_0)^2 \\
&& - \mu_0 \chi_a (\Vec{n} \cdot \Vec{H}_0)^2
\Bigr\}.\nonumber
\end{eqnarray}
Here $K_{ii}$ are the elastic constants, $\varepsilon_a$ the anisotropy of
the dielectric permittivity and $\chi_a$ is the anisotropy of the magnetic susceptibility.
In addition one has the normalization equation
\begin{equation}
\label{eqn:normalization}
\Vec{n} = 1
\end{equation}
and incompressibility condition
\begin{equation}
\label{eqn:incompressibility}
\nabla \cdot \Vec{v} = 0.
\end{equation}
The basic state solution of equations \eqref{eqn:EL:velocity} and \eqref{eqn:EL:director} has the following form
\begin{equation}
\label{eqn:base}
\Vec{n}_0=(0,\:1,\:0),\:\Vec{v}_0=(v_{0x}(z),\:0,\:0),
\end{equation}
where $v_{0x}=V_0(1/2+z/d)$ for Couette and
$v_{0x} = (\Delta P/\Delta x)[d^2 / \alpha_4][1 / 4 - z^2 / d^2]$ for Poiseuille flow.
In order to investigate the stability of the basic state \eqref{eqn:base} with
respect to small perturbations we write:
\begin{equation}
\label{def:hom}
\Vec{n}=\Vec{n}_0+\Vec{n}_1(z) e^{\sigma t} e^{i q y},\:
\Vec{v}=\Vec{v}_0+\Vec{v}_1(z) e^{\sigma t} e^{i q y};
\end{equation}
We do not expect spatial variation along $x$ for steady flow. The
case $q = 0$ corresponds to a homogeneous instability. Here we
analyse stationary bifurcations, thus the threshold condition is
$\sigma = 0$.
Introducing the dimensionless quantities in terms of layer
thickness $d$ (typical length) and director relaxation time
$\tau_d = (-\alpha_2) d^2 / K_{22}$ (typical time) the linearised
equations \eqref{eqn:EL:velocity} and \eqref{eqn:EL:director} can
be rewritten in the form
\begin{subequations}
\label{eqn:rolls}
\begin{align}
\label{eqn:rolls:1}&(\eta_{13} - 1) q^2 S n_{1z} + i q (\eta_{13} q^2 - \partial_z^2) v_{1x} = 0,\\
\label{eqn:rolls:2}&\partial_z [\eta_{52} q^2 + (1 - \eta_{32}) \partial_z^2] (S n_{1x}) \nonumber\\
&\quad + (\eta_{12} q^4 - \eta_{42} q^2 \partial_z^2 + \partial_z^4) v_{1y} = 0,\\
\label{eqn:rolls:3}&(\partial_z^2 - k_{32} q^2 - h) n_{1x} + S n_{1z} + i q v_{1x} = 0,\\
\label{eqn:rolls:4}&\partial_z(k_{12} \partial_z^2 - k_{32} q^2 - h + k_{12} e) n_{1z} \nonumber\\
&\quad + \lambda \partial_z (S n_{1x}) - (q^2 + \lambda \partial_z^2) v_{1y}= 0,\\
\label{eqn:rolls:5}&v_{1z,z} = - i q v_{1y}.
\end{align}
\end{subequations}
where $\eta_{ij} = \eta_i / \eta_j$, $\eta_1 = (\alpha_4 +
\alpha_5 - \alpha_2) / 2$, $\eta_2 = (\alpha_3 + \alpha_4 +
\alpha_6) / 2$, $\eta_3 = \alpha_4 / 2$, $\eta_4 = \alpha_1 +
\eta_1 + \eta_2$, $\eta_5 = - (\alpha_2 + \alpha_5) / 2$, $k_{ij}
= K_{ii} / K_{jj}$, $\lambda = \alpha_3 / \alpha_2$, $h = \pi^2
H_0^2 / H_F^2$, $e = \sgn(\varepsilon_a) \pi^2 E_0^2/ E_F^2$ and
$H_F = (\pi / d) \sqrt{K_{22} / (\mu_0 \chi_a)}$, $E_F = (\pi / d)
\sqrt{K_{11} / (\varepsilon_0 |\varepsilon_a|)}$ are the critical
Fr\'eedericksz transition fields for strong anchoring.
For the shear rate $S$ one has, for Couette flow,
\begin{equation}
S = a^2,\: a^2 = \dfrac{V_0 \tau_d}{d}
\end{equation}
and for Poiseuille flow
\begin{equation}
S = -a^2 z,\: a^2 = -\dfrac{\Delta P}{\Delta x} \dfrac{\tau_d d}{\eta_3}.
\end{equation}
The anchoring properties are characterised by a surface energy per
unit area, $F_s$, which has a minimum when the director at the
surface is oriented along the {\em easy} axis (parallel to the $y$
axis in our case). A phenomenological expression for the surface
energy $F_s$ can be written in terms of an expansion with respect
to $(\Vec{n} - \Vec{n}_0)$. For small director deviations from the
easy axis one obtains
\begin{equation}
\label{F_s}
F_s = \frac12 W_a n_{1x}^2 + \frac12 W_p n_{1z}^2,\quad
W_a > 0,\: W_p > 0,
\end{equation}
where $W_a$ and $W_p$ are the ``azimuthal'' and ``polar''
anchoring strengths, respectively. $W_a$ characterizes the surface
energy increase due to distortions within the surface plate and
$W_p$ relates to distortions out of the substrate plane.
The boundary conditions for the director perturbations
can be obtained from the torques balance equation
\begin{equation}
\pm \frac{\partial F}{\partial (\partial n_{1i}/\partial z)} +
\frac{\partial F_s}{\partial n_{1i}} = 0,
\end{equation}
with ``$\pm$'' for $z = \pm d/2$.
The boundary conditions \eqref{F_s} can be rewritten in dimensionless form as:
\begin{equation}
\label{bc:director:dim}
\pm \beta_a n_{1x,z} + n_{1x} = 0,\:
\pm \beta_p n_{1z,z} + n_{1z} = 0,
\end{equation}
with ``$\pm$'' for $z = \pm 1/2$.
Here we introduced dimensionless anchoring strengths as
ratios of the characteristic anchoring length ($K_{ii} / W_i$) over the
layer thickness $d$:
\begin{equation}
\label{eqn:anchoring}
\beta_a = K_{22} / (W_a d),\: \beta_p = K_{11} / (W_p d).
\end{equation}
In the limit of strong anchoring, $(\beta_a,\:\beta_p) \to 0$, one has
$n_{1x} = n_{1z} = 0$ at $z = \pm 1/2$. For torque-free boundary
conditions, $(\beta_a,\:\beta_p)\to \infty$, one has $n_{1x,z} = n_{1z,z} = 0$ at the boundaries. From \eqref{eqn:anchoring} one can see that by changing the thickness $d$, the dimensionless anchoring strengths $\beta_a$ and $\beta_p$ can be varied with the ratio $\beta_a/\beta_p$ remaining constant.
The boundary conditions for the velocity field (no-slip) are
\begin{align}
\label{bc:vx}
&v_{1x}(z = \pm 1 / 2) = 0;\\
\label{bc:vy}
&v_{1y}(z = \pm 1 / 2) = 0;\\
\label{bc:vz}
&v_{1z}(z = \pm 1 / 2) = v_{1z,z}(z = \pm 1 / 2) = 0.
\end{align}
The existence of a nontrivial solution of the linear ordinary
differential equations \eqref{eqn:rolls} with the boundary
conditions \eqref{bc:director:dim}, (\ref{bc:vx} -- \ref{bc:vz})
gives values for the shear rate $S_0(q)$ (neutral curve). The
critical value $S_c(q_c)$, above which the basic state
\eqref{eqn:base} becomes unstable, are given by the minimum of
$S_0$ with respect to $q$.
\begin{table}
\caption{\label{table:sym:all}Symmetry properties of the solutions
of equations \eqref{eqn:rolls} under $\{z \to -z\}$.}
\begin{ruledtabular}
\begin{tabular}{ccccc}
\multicolumn{1}{c}{}&\multicolumn{2}{c}{Couette flow}&\multicolumn{2}{c}{Poiseuille flow}\\
\multicolumn{1}{c}{Perturbation}& ``odd'' & ``even'' & ``odd'' & ``even''\\
\hline
$n_{1x}$ & odd & even & odd & even\\
$n_{1z}$ & odd & even & even & odd\\
$v_{1x}$ & odd & even & odd & even\\
$v_{1y}$ & even & odd & odd & even\\
$v_{1z}$ & odd & even & even & odd\\
\end{tabular}
\end{ruledtabular}
\end{table}
The symmetry properties of the solutions of equations
\eqref{eqn:rolls} under the reflection $z \to - z$ is shown in the
Table \ref{table:sym:all}. We will always classify the solutions
by the $z$ symmetry of the $x$ component of the director
perturbation $n_{1x}$ (first row in Table I).
In case of positive $\varepsilon_a$, for some critical value of
the electric field the basic state loses its stability already in
the absence of flow ({\em Fre\'edericksz transition}). Clearly the
Fre\'edericksz transition field depends on the polar anchoring
strength. There is competition of the elastic torque $K_{11}
n_{1z,zz}$ and the field-induced torque $\varepsilon_a
\varepsilon_0 E_0^2 n_{1z}$. The solution of Eq.
\eqref{eqn:rolls:4} with $n_{1x} = 0$, $v_{1y} = 0$ for $h = 0$
has the form
\begin{equation}
\label{def:Eweak} n_{1z} = C \cos(\pi \delta z / d),
\end{equation}
where $\delta = E_F^{weak} / E_F$ and $E_F^{weak}$ is the actual
Fr\'eedericksz field.
After substituting $n_{1z}$ into the boundary conditions
\eqref{bc:director:dim} we obtain the expression for $\delta$:
\begin{equation}
\label{eqn:Eweak} \tan\dfrac{\pi \delta}2 =
\dfrac1{\pi \beta_p \delta}.
\end{equation}
One easily sees that $\delta \to 1$ for $\beta_p \to 0$ and $\delta \to \sqrt{2 / \beta_p} / \pi$ for $\beta_p \to \infty$. For $\beta_p =1$ one gets $E_F^{weak} = 0.42 E_F$.
\section{Homogeneous instability}
In order to obtain simpler equations we use the renormalized
variables as in Ref. \cite{Tarasov:LC:V28:N6:P833:Y2001}:
\begin{align}
\label{def:renorm}
&\Tilde{S} = \beta^{-1} S,\: N_{1x} = \beta^{-1} n_{1x},\: N_{1z} = n_{1z},\: V_{1x} = \beta^{-1} v_{1x},\nonumber\\
& V_{1y} = (\beta^2 \eta_{23})^{-1} v_{1y},\: V_{1z} = (\beta^2 \eta_{23})^{-1} v_{1z}
\end{align}
with
\begin{equation}
\beta^2 = \alpha_{32} k_{21} \eta_{32},\: \alpha_{i j} = \dfrac{\alpha_i}{\alpha_j}.
\end{equation}
In the case of homogeneous perturbations ($q = 0$) Eqs.
\eqref{eqn:rolls} reduce to $V_{1z} = 0$ and
\begin{subequations}
\label{eqn:set:hom}
\begin{align}
\label{eqn:set:hom:1}
&V_{1y,zz} - (1 - \eta_{23}) (\Tilde{S} N_{1x})_{,z} = 0,\\
\label{eqn:set:hom:2}&\Tilde{S} N_{1z} - N_{1x,zz} + h N_{1x} = 0,\\
\label{eqn:set:hom:3}&\eta_{23} \Tilde{S} N_{1x} + N_{1z,zz} - V_{1y,z} - (k_{21} h - e) N_{1z} = 0.
\end{align}
\end{subequations}
\subsection{Couette flow}
For Couette flow we can obtain the solution of \eqref{eqn:set:hom}
semi-analytically. For the ``odd'' solution one gets
\begin{align}
&N_{1x} = C_1 \sinh(\xi_1 z) + C_2 \sin(\xi_2 z),\\
&N_{1z} = C_3 \sinh(\xi_1 z) + C_4 \sin(\xi_2 z),\\
&V_{1y} = C_5 \cosh(\xi_1 z) + C_6 \cos(\xi_2 z) + C_7.
\end{align}
Taking into account the boundary conditions
(\ref{bc:director:dim}, \ref{bc:vy}) the solvability condition for
the $C_i$ (``boundary determinant'' equal to zero) gives an
expression for the critical shear rate $a_c$:
\begin{multline}
\label{eqn:couette:odd}
(h + \xi_2^2) [\xi_1 \beta_a \cosh(\xi_1 / 2) +
\sinh(\xi_1 / 2)] \\
\times [\xi_2 \beta_p \cos(\xi_2 / 2) + \sin(\xi_2 / 2)] \\
- (h - \xi_1^2)[\xi_2 \beta_a \cos(\xi_2 / 2) + \sin(\xi_2 / 2)] \\
\times [\xi_1 \beta_p \cosh(\xi_1 / 2) + \sinh(\xi_1 / 2)] = 0.
\end{multline}
where
\begin{align}
&\xi_1^2 = \dfrac{[ (1 + k_{12}) h - k_{12} e] + \xi}{2 k_{12}},\\
&\xi_2^2 = \dfrac{- [(1 + k_{12}) h - k_{12} e] + \xi}{2 k_{12}},\\
&\xi = \sqrt{ [(1 - k_{12}) h - k_{12} e]^2 + 4 k_{12}^2 a^4}.
\end{align}
For the ``even'' solution one obtains:
\begin{align}
&N_{1x} = C_1 \cosh(\xi_1 z) + C_2 \cos(\xi_2 z) + C_3,\\
&N_{1z} = C_4 \cosh(\xi_1 z) + C_5 \cos(\xi_2 z) + C_6,\\
&V_{1y} = C_7 \sinh(\xi_1 z) + C_8 z.
\end{align}
The boundary conditions (\ref{bc:vx}-\ref{bc:vz}) now lead to the
following condition (``boundary determinant''):
\begin{widetext}
\begin{equation}
\label{eqn:couette:even}
\begin{vmatrix}
1 & h & \dfrac{\eta_{23}}{2}\left(\dfrac{h(h-k_{12}e)}{a^4k_{12}\eta_{23}}-1\right)\\
- \xi_2 \beta_a \sin(\xi_2 / 2) + \cos(\xi_2 / 2) &
(h + \xi_2^2)[ - \xi_2 \beta_p \sin(\xi_2 / 2) + \cos(\xi_2 / 2)] &
\dfrac{1 - \eta_{23}}{\xi_2}\sin(\xi_2 / 2)\\
\xi_1 \beta_a \sinh(\xi_1 / 2) + \cosh(\xi_1 / 2) &
(h - \xi_1^2)[\xi_1 \beta_p \sinh(\xi_1 / 2) + \cosh(\xi_1 / 2)] &
\dfrac{1 - \eta_{23}}{\xi_1}\sinh(\xi_1 / 2)
\end{vmatrix}=0.
\end{equation}
\end{widetext}
\subsection{Poiseuille flow}
In the case of Poiseuille flow the system \eqref{eqn:set:hom} with
$\Tilde{S} = - z a^2 / \beta$ admits an analytical solution only
in the absence of external fields (in terms of Airy functions)
\cite{Tarasov:LC:V28:N6:P833:Y2001}. In the presence of fields we
solve the problem numerically. In the framework of the Galerkin
method we expand $N_{1x}$, $N_{1z}$ and $V_{1y}$ in a series
\begin{align}
\label{poise:full:galerkin}
&N_{1x} = \sum\limits_{n=1}^{\infty} C_{1,n} f_n(z),\nonumber\\
&N_{1z} = \sum\limits_{n=1}^{\infty} C_{2,n} g_n(z),\\
&V_{1y} = \sum\limits_{n=1}^{\infty} C_{3,n} u_n(z),\nonumber
\end{align}
where the trial functions $f_n$, $g_n$ and $u_n$ satisfy the
boundary conditions \eqref{bc:director:dim}, \eqref{bc:vy}. For
the ``odd'' solution we write
\begin{equation}
f_n(z) = \zeta_n^o(z;\beta_a),\;
g_n(z) = \zeta_n^e(z;\beta_p),\:
u_n(z) = \nu_n^o(z)
\end{equation}
and for the ``even'' solution
\begin{equation}
f_n(z) = \zeta_n^e(z;\beta_a),\;
g_n(z) = \zeta_n^o(z;\beta_p),\:
u_n(z) = \nu_n^e(z).
\end{equation}
The functions $\zeta_n^o(z;\beta)$, $\zeta_n^e(z;\beta)$,
$\nu_n^o(z)$, $\nu_n^e(z)$ are given in Appendix A. In our
calculations we have to truncate the expansions
\eqref{poise:full:galerkin} to a finite number of modes.
After substituting \eqref{poise:full:galerkin} into the system
\eqref{eqn:set:hom} and projecting the equations on the trial functions
$f_n(z)$, $g_n(z)$ and $u_n(z)$ one gets a system of linear homogeneous algebraic equations for $\Vec{X} = \{C_{i,n}\}$ in the form
$(A - a^2 B) \Vec{X} = 0$.
We have solved this eigenvalue problem for $a^2$. The
lowest (real) eigenvalue corresponds to
the critical shear rate $a_c^2$.
According to the two types of $z$-symmetry of the solutions (and of the set
of trial functions) one obtains the threshold values of $a_c^2$ for the
``odd'' and ``even'' instability modes.
The number of Galerkin modes was chosen such that the accuracy
of the calculated eigenvalues was better than 1\% (we took ten
modes in case of ``odd'' solution and five modes for ``even'' solution).
\subsection{Approximate analytical expression for the critical shear rate}
In order to obtain an {\em easy-to-use} analytical expression for
the critical shear rate as a function of the surface anchoring
strengths and the external fields we use the lowest-mode
approximation in the framework of the Galerkin method. By
integrating \eqref{eqn:set:hom:1} over $z$ one can eliminate
$V_{1y,z}$ from \eqref{eqn:set:hom:3} which gives
\begin{equation}
\label{eqn:set:director}
\Tilde{S} N_{1x} + N_{1z,zz} + (k_{21} h - e) N_{1z} = K,
\end{equation}
where $K$ is an integration constant. Taking into account the
boundary conditions for $V_{1y}$ one has
\begin{equation}
\label{eqn:K}
K - (1 - \eta_{32}) \int\limits_{-1/2}^{1/2} S N_{1x}(z) \:dz = 0.
\end{equation}
We choose for the director components $N_{1x}$, $N_{1z}$
the one-mode approximation
\begin{equation}
\label{eqn:leading}
N_{1x} = C_1 f(z),\:
N_{1z} = C_2 g(z),
\end{equation}
Substituting \eqref{eqn:leading} into \eqref{eqn:set:hom:2} and
\eqref{eqn:set:director} and projecting the first equation on $f(z)$
and the second one on $g(z)$ we get algebraic equations for $C_i$.
The solvability condition [together with \eqref{eqn:K}]
gives the expression for the critical shear rate $a_c^2$
\begin{equation}
\label{eqn:one:ac}
a_c^2 = \sqrt{\dfrac{c_1 c_2}
{c_3}},
\end{equation}
where $c_1 =\langle ff''\rangle - h \langle f^2\rangle $, $c_2 =
\langle gg''\rangle - (h/k_{12} - e) \langle g^2\rangle $, $c_3 =
\langle sfg\rangle [\langle sfg\rangle - (1 - \eta_{23}) \langle
sf\rangle \langle g\rangle ]$, where $\langle \dots\rangle $
denotes a spatial average
\begin{equation}
\label{def:int} \langle \dots\rangle
=\int\limits_{-1/2}^{1/2}(\dots)\:dz.
\end{equation}
The values for the integrals $\langle \dots\rangle $ are given in
Appendix B. In Table \ref{table:trial_func:appr} and Appendix A
the trial functions used are given. Equation \eqref{eqn:one:ac}
can be used for both Couette and Poiseuille flow by choosing the
function $s(z)$ [where $s(z) = 1$ for Couette flow and $s(z) = -
z$ for Poiseuille flow] and the trial functions $f(z)$ and $g(z)$
with appropriate symmetry.
\begin{table}
\caption{\label{table:trial_func:appr}Trial functions for the homogeneous solutions.}
\begin{ruledtabular}
\begin{tabular}{ccccc}
\multicolumn{1}{c}{}&\multicolumn{2}{c}{Couette flow}&\multicolumn{2}{c}{Poiseuille flow}\\
\multicolumn{1}{c}{Function}& ``odd'' & ``even'' & ``odd'' & ``even''\\
\hline
$f(z)$ & $\zeta_1^o(z;\:\beta_a)$ & $\zeta_1^e(z;\:\beta_a)$ &
$\zeta_1^o(z;\:\beta_a)$ & $\zeta_1^e(z;\:\beta_a)$\\
$g(z)$ & $\zeta_1^o(z;\:\beta_p)$ & $\zeta_1^e(z;\:\beta_p)$ &
$\zeta_1^e(z;\:\beta_p)$ & $\zeta_1^o(z;\:\beta_p)$\\
\end{tabular}
\end{ruledtabular}
\end{table}
For the material MBBA in the case of Couette flow the one-mode
approximation \eqref{eqn:one:ac} for the ``odd'' solution gives an
error that varies from 2.5\% to 16\% when $H_0 / H_F$ varies from
0 to 4. The ``even'' solution has an error of $0.6\% \div 8\%$ for
$0 \leqslant H_0 / H_F \leqslant 3$ and of $0.6\% \div 12\%$ for
$0 \leqslant E_0 / E_F \leqslant 0.6$.
For Poiseuille flow for ``odd'' solution the error
is $29\%$ in the absence of fields.
For the ``even'' solution the error is $12\% \div 15\%$ for
magnetic fields $0 \leqslant H_0 / H_F \leqslant 0.5$.
For both Couette and Poiseuille flow the accuracy of the formula
\eqref{eqn:one:ac} decreases with increasing field strengths.
\section{Spatially periodic instabilities}
We used for Eqs. \eqref{eqn:rolls} again the renormalized
variables \eqref{def:renorm}. The system \eqref{eqn:rolls} has no
analytical solution. Thus we solved the problem numerically in the
framework of the Galerkin method:
\begin{eqnarray}
\label{Ansatz:rolls}
N_{1x} = e^{iqy} \sum\limits_{n=1}^{\infty} C_{1,n} f_n(z),\:
N_{1z} = e^{iqy} \sum\limits_{n=1}^{\infty} C_{2,n} g_n(z),\\
V_{1x} = e^{iqy} \sum\limits_{n=1}^{\infty} C_{3,n} u_n(z),\:
V_{1z} = e^{iqy} \sum\limits_{n=1}^{\infty} C_{4,n} w_n(z).
\end{eqnarray}
After substituting \eqref{Ansatz:rolls} into the system
\eqref{eqn:rolls} and projecting on to the trial functions
$\{f_n(z),\:g_n(z),\:u_n(z),\:w_n(z)\}$ we get a system
of linear homogeneous algebraic equations for $\Vec{X} = \{C_{i,n}\}$.
This system has the form $[A(q) - a^2(q) B(q)] \Vec{X} = 0$.
We have solved the eigenvalue problem numerically to find the marginal stability curve $a(q)$. For the numerical calculations
we have chosen the trial functions shown in Table \ref{table:trial_func:rolls} and Appendix A.
\begin{table}
\caption{\label{table:trial_func:rolls}Trial functions for the spatially periodic solutions.}
\begin{ruledtabular}
\begin{tabular}{ccccc}
\multicolumn{1}{c}{}&\multicolumn{2}{c}{Couette flow}&\multicolumn{2}{c}{Poiseuille flow}\\
\multicolumn{1}{c}{Function}& ``odd'' & ``even'' & ``odd'' & ``even''\\
\hline
$f(z)$ & $\zeta_n^o(z;\:\beta_a)$ & $\zeta_n^e(z;\:\beta_a)$ &
$\zeta_n^o(z;\:\beta_a)$ & $\zeta_n^e(z;\:\beta_a)$\\
$g(z)$ & $\zeta_n^o(z;\:\beta_p)$ & $\zeta_n^e(z;\:\beta_p)$ &
$\zeta_n^e(z;\:\beta_p)$ & $\zeta_n^o(z;\:\beta_p)$\\
$u(z)$ & $\nu_n^o(z)$ & $\nu_n^e(z)$ &
$\nu_n^o(z)$ & $\nu_n^e(z)$\\
$w(z)$ & $\varsigma_n^o(z)$ & $\varsigma_n^e(z)$ &
$\varsigma_n^e(z)$ & $\varsigma_n^o(z)$\\
\end{tabular}
\end{ruledtabular}
\end{table}
In order to get an approximate expression for the threshold we
use the leading-mode approximation in the framework of the Galerkin
method. We used the same scheme described above
for the single mode and get the following formula for the
critical shear rate:
\begin{equation}
\label{rolls:one:common}
a_c^2 = \sqrt{ \eta_{23} f_x f_z / (\tilde{\alpha}_2\tilde{\alpha}_3) },
\end{equation}
with
\begin{align}
&f_x = \langle ff''\rangle - (q^2 k_{32} + h) \langle f^2\rangle ,\\
&f_z = \langle gg''\rangle - (q^2 k_{31} + k_{12} h - e) \langle g^2\rangle ,\\
&\tilde{\alpha}_2 =
[ \langle fsg\rangle - q^2 (1 - \eta_{31}) \langle fu\rangle \langle gsu\rangle / \gamma ],\\
&\tilde{\alpha}_3 =
\langle fsg\rangle + [ \alpha_{23} q^2 \langle gw\rangle + \alpha_3 \langle gw''\rangle ] \\
& \quad \times [ (1 - \eta_{32}) \langle w[sf]''\rangle - \eta_{52} q^2 \langle wsf\rangle ] / r,\\
&\gamma = q^2 \langle uu\rangle - \eta_{31} \langle uu''\rangle ,\\
&r = \langle ww^{(4)}\rangle - \eta_{42} q^2 \langle ww''\rangle
+ \eta_{12} q^4 \langle ww\rangle .
\end{align}
The values of the integrals $\langle \dots\rangle $ appearing in
the expression \eqref{rolls:one:common} are given in Appendix C.
In the case of strong anchoring an approximate analytical
expression for $a_c^2 = a_c^2(q_c)$ was obtained by Manneville
\cite{Manneville:JdPh:1976:285} using test functions that satisfy
free-slip boundary conditions. The formula
\eqref{rolls:one:common} is more accurate because we chose for
$v_{1z}$ Chandrasekhar functions that satisfy the boundary
conditions \eqref{bc:vz}.
For calculations we used material parameters for MBBA. The
accuracy of \eqref{rolls:one:common} is better than 1\% for
Couette flow and better than 3\% for Poiseuille flow. Note, that
Eq. \eqref{eqn:one:ac} for the homogeneous instability is more
accurate than \eqref{rolls:one:common} for $q = 0$ because
\eqref{rolls:one:common} was obtained by solving four equations
\eqref{eqn:rolls} by approximating all variables, whereas
\eqref{eqn:one:ac} was obtained by solving the reduced equations
\eqref{eqn:set:hom} by approximating only two variables.
\section{Discussion}
For the calculations we used parameters for MBBA at 25 $^\circ$C
\cite{mp:MBBA}. Calculations were made for the range of anchoring
strengths $\beta_a = 0 \div 1$ and $\beta_p = 0 \div 1$.
\subsection{Couette flow}
\begin{figure}
\epsfig{file=couette_e.eps,height=21cm}
\caption{Contour plot of the critical shear rate $a_c^2$ for Couette flow
vs. $\beta_a$ and $\beta_p$. $a$: $E_0 = 0$;
$b$: $E_0 = E_F^{weak}$, $\varepsilon_a < 0$;
$c$: $E_0 = E_F^{weak}$, $\varepsilon_a > 0$.
$E_F$ is defined after Eq. \eqref{def:Eweak} and calculated in
Eq. \eqref{eqn:Eweak}.}
\label{fig:couette0}
\end{figure}
We found that without and with an additional electric field the
critical shear rate $a_c^2$ for the ``even'' type homogeneous
instability (EH) is systematically lower than the threshold for
other types of instability (Fig. \ref{fig:couette0}a--c). Note,
that in the presence of the field the symmetry with respect to the
exchange $\beta_a \leftrightarrow \beta_p$ is broken.
In Fig. \ref{fig:couette0} contour plots for the critical value
$a_c^2$ vs. anchoring strengths $\beta_a$ and $\beta_p$ for
different values of the electric field are shown. The differences
between $a_c^2$ obtained from the exact, semi-analytical solution
\eqref{eqn:couette:even} and from the one-mode approximation
\eqref{eqn:one:ac} are indistinguishable in the figure.
\begin{figure}
\epsfig{file=couette_h.eps,height=21cm}
\caption{Critical shear rates and phase diagram for the instabilities under Couette flow
with additional magnetic field.
$a$: $H_0 / H_F = 3$; $b$: $H_0 / H_F = 3.5$; $c$: $H_0 / H_F = 4$.
Boundaries for occurrence of instabilities are given by thick solid lines
(full numerical) and thick dashed lines (one-mode approximation).}
\label{fig:couette1}
\end{figure}
In Fig. \ref{fig:couette1} contour plots of $a_c^2$ (thin dashed
lines) and the boundaries where the type of instability changes
[the solid lines are obtained numerically, the thick dashed lines
from \eqref{eqn:one:ac}] for different values of magnetic field
are shown. For not too strong magnetic field in the region of weak
anchoring the ``odd'' type homogeneous instability (OH) takes
place (Fig. \ref{fig:couette1}a). Increasing the magnetic field
the OH region expands toward stronger anchoring strengths. Above
$H_0 \approx 3.2$ a region with lowest threshold corresponding to
the ``even'' roll mode (ER) appears. This region has borders with
both types of the homogeneous instability (Fig.
\ref{fig:couette1}b). With increasing magnetic field the ER region
increases (Fig. \ref{fig:couette1}c) and above $H_0 / H_F = 4$ the
ER instability has invaded the whole investigated parameter range.
For strong anchoring and $H_0 / H_F = 3.5$ the critical wave
vector is $q_c = 5.5$. It increases with increasing magnetic field
and decreases with decreasing anchoring strengths. With increasing
magnetic field the threshold for the EH instability becomes less
sensitive to the surface anchoring. Leslie has pointed out (using
an approximate analytical approach) that for strong anchoring a
transition from a homogeneous state without transverse flow (EH)
to one with such flow (OH) as the magnetic field is increased is
not possible in MBBA because of the appearance of the ER type
instability \cite{Leslie:MCLC:1976}. This is consistent with our
results. We find that the EH--OH transition in MBBA is possible
only in the region of weak anchoring (Figs.
\ref{fig:couette1}a--c).
\begin{figure}
\epsfig{file=aq.eps,width=8cm}
\caption{$a_c$ vs. $q$. Couette flow, $\beta_a = 0.1$, $\beta_p = 0.1$.
$a$: $H_0 / H_F = 3$; $b$: $H_0 / H_F = 3.4$; $c$: $H_0 / H_F = 4$.}
\label{fig:couette2}
\end{figure}
In Fig. \ref{fig:couette2} marginal stability curves for different
values of the magnetic field and fixed anchoring strengths is
shown (solid line for ER and dashed lines for OR). There are
always two minima for the even mode; one of them at $q = 0$ that
corresponds to the homogeneous instability EH. For small magnetic
field the absolute minimum is at $q = 0$ (line a). The OR curve is
systematically higher than ER. In a small range of $q$ (dotted
lines) a stationary ER solution does not exist but we have OR
instead. With increasing magnetic field the critical amplitude for
the EH minimum ($q = 0$) increases more rapidly then the one for
the ER minimum ($q \neq 0$) so that for $H_0 / H_F > 3.4$ the ER
solution is realized (lines b and c). The range of $q$ where ER is
replaced by OR expands with increasing magnetic field.
For the ER instability in the absence of fields and strong
anchoring we find $a_c^2 = 12.15$ from the semi-analytical
expression \eqref{eqn:couette:even} as well as from the one-mode
approximation \eqref{eqn:one:ac} and also \eqref{rolls:one:common}
with $q = 0$. The only available experimental value for $a_c^2$ is
$6.3 \pm 0.3$ \cite{Pieranski:SSC:1973}. We suspect that the
discrepancy is due to deviations from the strong anchoring limit
and the difference in the material parameters of the substance
used in the experiment. Assuming $\beta_a \ll 1$ one would need
$\beta_p \approx 1$ to explain the experimental value.
\subsection{Poiseuille flow}
\begin{figure}
\epsfig{file=poise_e.eps,height=21cm}
\caption{Critical shear rates and phase diagram for the instabilities in
Poiseuille flow. $a$: $E_0 = 0$;
$b$: $E_0 = E_0^{weak}$, $\varepsilon_a < 0$;
$c$: $E_0 = E_0^{weak}$, $\varepsilon_a > 0$.
Thin dashed lines: full numerical threshold;
dotted lines: one-mode approximation for threshold.
Boundaries for occurrence of instabilities are given by thick solid lines
(full numerical) and thick dashed lines (one-mode approximation).}
\label{fig:poise0}
\end{figure}
In Fig. \ref{fig:poise0} the contour plot for $a_c^2$ [thin dashed
lines from the full numerical calculation, dotted lines from the
one-mode approximations \eqref{eqn:one:ac} and
\eqref{rolls:one:common}] and the boundary for the various types
of instabilities [thick solid line: numerical; thick dashed line:
\eqref{eqn:one:ac} and \eqref{rolls:one:common}] are shown. In
Poiseuille flow the phase diagram is already very rich in the
absence of external fields. In the region of large $\beta_a$ one
has the EH instability. For intermediate anchoring strengths rolls
of type OR occur [Fig. \ref{fig:poise0}$a$]. Note, that even in
the absence of the field there is no symmetry under exchange
$\beta_a \leftrightarrow \beta_p$, contrary to Couette flow. The
one-mode approximations \eqref{eqn:one:ac} and
\eqref{rolls:one:common} not give the transition to EH for strong
anchoring. Here we should note that in that region the difference
between the EH and the OR instability thresholds is only about
5\%. By varying material parameters [increase $\alpha_2$ by 10\%
or decrease $\alpha_3$ by 20\% or $\alpha_5$ by 25\% or $K_{33}$
by 35\%] it is possible to change the type of instability in that
region.
Application of an electric field leads for $\varepsilon_a < 0$
($\varepsilon_a > 0$) to expansion (contraction) of the EH region
[Figs. \ref{fig:poise0}$b$ and \ref{fig:poise0}$c$]. At $E_0 / E_F
= 1$ and $\varepsilon_a < 0$ rolls vanish completely and the EH
instability occurs in the whole area investigated. For
$\varepsilon_a > 0$ the instability of OH type appears in the
region of large $\beta_p$. In this case, increasing the electric
field from $E_F^{weak}$ to $E_F$ cause an expansion of the OH
region. Note that for $\beta_p > 1$, which is in the OH region,
the Fre\'edericksz transition occurs first .
\begin{figure}
\centering
\epsfig{file=poise_h.eps,width=6.3cm}
\caption{Phase diagram for the instabilities under Poiseuille flow
with an additional magnetic field ($H_0 / H_F = 0.4$).}
\label{fig:poise1}
\end{figure}
An additional magnetic field suppresses the homogeneous
instability (Fig. \ref{fig:poise1}). Above $H_0 / H_F \approx 0.5$
the OR instability (Fig. \ref{fig:poise1}) occurs for all
anchoring strengths investigated.
The wave vector $q_c$ in the absence of fields is $1.4$.
Application of an electric field decreases $q_c$ whereas the
magnetic field increases $q_c$. The wave vector decreases with
decreasing anchoring strengths.
In the absence of fields and strong anchoring we find for the EH
instability $a_c = 102$ [Eq. \eqref{eqn:one:ac} gives 110 and Eq.
\eqref{rolls:one:common} with $q = 0$ gives 130]. The experimental
value is 92 \cite{Guyon:JdPh:1975}. Thus, theoretical calculations
and experimental results are in good agreement. Note, that in the
experiments \cite{Guyon:JdPh:1975} actually not steady but
oscillatory flow with very low frequency was used ($f = 5 \cdot
10^{-3}$ Hz).
In summary, the orientational instabilities for both steady
Couette (semi-analytical for homogeneous instability and numerical
for rolls) and Poiseuille flow (numerical) were analysed
rigorously taking into account weak anchoring and the influence
of external fields. Easy-to-use expressions for the threshold of
all possible types of instabilities were obtained and compared
with the rigorous calculations. In particular the region in
parameter space where the different types of instabilities
occurred were determined.
\acknowledgments
Financial support from DFG (project Kr690/22-1 and EGK
``Non-equilibrium phenomena and phase transition in complex
systems'').
|
{
"timestamp": "2005-03-10T23:19:31",
"yymm": "0503",
"arxiv_id": "physics/0503091",
"language": "en",
"url": "https://arxiv.org/abs/physics/0503091"
}
|
\section{Introduction}
\label{S:Intro}
The best-known application of quantum cryptography is quantum key
distribution (QKD) \cite{BB84}. The goal of QKD is to allow two parties, Alice
and Bob, to share a common string of secret in the presence of an
eavesdropper, Eve. Such a key can subsequently be used for,
for example, perfectly
secure communications via the so-called one-time-pad.
Unlike conventional cryptography, the security of QKD is guaranteed by the
fundamental law of physics---the Heisenberg uncertainty principle.
The best-known protocol for QKD is the Bennett-Brassard protocol (BB84) \cite{BB84}.
In BB84, Alice sends Bob a sequence of single photons in one of
the four polarizations (vertical, horizontal, 45-degree and 135-degree)
and Bob randomly performs a measurement in one of the two
conjugate bases. In principle, the security of QKD has been
proven in a number of papers including \cite{proofs,Ben-Or,ShorPreskill}.
For practical implementations, an attenuated laser pulse
(a so-called weak coherent state) is often used as the
source. The security of QKD with a rather generic class of
imperfect devices has been proven in GLLP \cite{GLLP}, following the
earlier work \cite{ilm}
Recently, Hwang \cite{HwangDecoy} has proposed a decoy state
idea for improving
the performance (i.e., the key generation rate and distance) of
QKD systems. We \cite{Decoy} have demonstrated rigorously
how the decoy state idea
can be combined with GLLP to obtain a key generation rate
(per pulse emitted by Alice) which
is
lower bounded by:
\begin{equation}\label{refinedkeyrate}
S \geq Q_{signal} \{- H_2(E_{signal}) + \Omega_1 [ 1- H_2(e_1)] \},
\end{equation}
where $Q_{signal}$ and $E_{signal}$ are respectively the gain and
quantum bit error rate (QBER) of the signal state, $\Omega_1$ and
$e_1$ are respectively the fraction and QBER of detection events by
Bob that have originated from single-photon signals emitted by
Alice. Here, the gain means the ratio of
Bob's detection events to Alice's total number of emitted signals.
[Decoy state QKD has subsequently been investigated by
Wang \cite{WangDecoy} and by Harrington \cite{Harrington}.]
The key goal of this paper is to increase the above key generation
rate in Eq.~\ref{refinedkeyrate} by a term $Q_{signal} \Omega_0$
where $\Omega_0$ is the fraction of detection events of Bob that
have originated from vacua emitted by Alice. More concretely,
we have the following main Theorem.
{\bf Theorem~1} The key generation of an efficient BB84 scheme is
given by:
\begin{equation}\label{newkeyrate}
S \geq Q_{signal} \{- H_2(E_{signal}) + \Omega_0 + \Omega_1 [ 1- H_2(e_1)] \},
\end{equation}
where $\Omega_0$ is the fraction of detection events by Bob that
has originated from the vacuum signals emitted by Alice.
In other words, we find that each detection event by Bob that has
originated from a
vacuum (i.e., nothing) emitted by Alice automatically contributes to a bit of
secure key over and above the prior art result
(Eq.~\ref{refinedkeyrate})
presented in \cite{GLLP}
and also \cite{Decoy}.
Before we embark on a detailed discussion, let us check for
the consistency of our result. Naively, one might think that
our suggestion that the vacuum will contribute to a secure
key is an insane idea because if nothing is emitted by Alice,
what is the origin of security? We remark that the vacuum {\it alone}
does {\it not} contribute to a secure key. More concretely, suppose
all the signals sent by Alice are vacua and there are no background
events. Then, $\Omega_0 =1$, $\Omega_1 =0$, and $E_{signal} = 1/2$.
Therefore, from Eq.~\ref{newkeyrate}, we get the lower bound $0$ for
the key generation rate. The reason is that the term $\Omega_0$ is
exactly cancelled by the error correction term $- H_2(E_{signal})$.
What Eq.~\ref{newkeyrate} does say is that no privacy amplification
is needed for the vacua state. This is intuitively clear because
Eve cannot have any a priori information on Alice's bit, if nothing is
emitted from Alice's laboratory.
Now, let us prove our main result (Eq.~\ref{newkeyrate}).
We shall use the method of
communication complexity.
As noted by by Ben-Or \cite{Ben-Or} and
by Renner and Koenig \cite{RennerKoenig}, the number of
rounds of universal hashing needed for privacy amplification
in QKD is at most given by any upper bound to the size of
Eve's quantum memory which contains information
on the key. In other words, we have, informally:
{\bf Theorem~2} \cite{Ben-Or,RennerKoenig}: The key generation rate in QKD
\begin{equation}
S \geq N-{\cal S}_{Eve}
\end{equation}
where $N$ is the size of the sifted key shared between Alice
and Bob and ${\cal S}_{Eve}$ is the size of Eve's quantum memory.
[A more formal definition involving the relevant $\epsilon$ and
$\delta$ can be found as Eq.~(11) in \cite{RennerKoenig}.]
{\it Remark}: Note that Theorem~2 only gives a lower bound to the key
generation rate because it does not consider the possibility of
advantage distillation in QKD \cite{twoway}.
In summary, all we need to compute (a lower bound to) the
key generation rate is to work out the size of Eve's
quantum memory.
{\bf Proof of Theorem~1}:
Now, note that Eve has two pieces of information on the key.
The first piece, which is strictly quantum,
comes from Eve's eavesdropping attack
during the quantum transmission from Alice to Bob.
The second piece is classical and comes from the classical
error correction part.
We argue that the first piece, from eavesdropping the
quantum transmission, consists of
two parts: single-photon part and multi-photon part.
It should be emphasized that the vacua signals do {\it not} contribute
at all. This is because, since Alice is emitting nothing,
Eve cannot possibly learn anything about Alice's key.
Eve can influence and, in fact, decide on Bob's key by
sending her own photons into Bob's detector. However,
Bob's key does not really tell Eve anything about Alice's key.
Let us consider the multi-photon part first.
We take the most conservative assumption that Eve has all the
information on all multi-photon signals. Her quantum memory size on
the multi-photon part is then given by
$Q_{signal} \Omega_m$ .
Here, $\Omega_m$ is the fraction of detection events of
Bob that have originated from multi-photon signals.
Note that $\Omega_0 + \Omega_1 + \Omega_m = 1$.
The single-photon part is given by simply
$ Q_{signal} \Omega_1 H_2 (e_1^{phase})$, where
$e_1^{phase}$ is the phase error rate of the single-photon
signals. From Shor-Preskill's proof \cite{ShorPreskill},
$e_1^{phase} = e_1$, which is the bit-flip error rate for
the single-photon signals. So, the quantum memory for
single-photon part is actually given by
$ Q_{signal} \Omega_1 H_2 (e_1)$.
Adding the two parts, the first piece of Eve's information has a
memory size $Q_{signal} [ \Omega_1 H_2 (e_1) + \Omega_m ]$.
The second piece of Eve's information, which comes from classical
error correction, is asymptotically
given by $Q_{signal} H_2(E_{signal})$.
In summary,
adding the two pieces together, the total quantum memory size of Eve
is given by ${\cal S}_{Eve}=
Q_{signal} [ H_2(E_{signal}) + \Omega_1 H_2 (e_1) + \Omega_m ]$.
Now, the length of the sifted key (per pulse emitted by Alice)
shared by Alice and Bob is
$N= Q_{signal} [ \Omega_0 + \Omega_1 + \Omega_m ]$.
Therefore, the number of secure key bits (per pulse emitted
by Alice) is given by
\begin{eqnarray}
S &\geq& N-{\cal S}_{Eve} \nonumber \\
&=& Q_{signal} \{ - H_2(E_{signal}) + \Omega_0 + \Omega_1 [ 1- H_2(e_1)] \}.
\end{eqnarray}
which is precisely Eq.~(\ref{newkeyrate}).
This concludes the proof of our Theorem~1.
In summary, we have increased the key generation from
Eq.~(\ref{refinedkeyrate}) in
the prior art result
\cite{GLLP} to Eq.~(\ref{newkeyrate}) by showing that, rather
counter-intuitively, the detection
events due to vacua contribute directly to the secure key.
What is interesting about this result is that it is based on a
communication complexity approach and is not entirely clear whether it
can be derived from an entanglement distillation approach.
In future, it will perhaps be interesting to rephrase this result in the
general framework of $\Gamma$ states \cite{Gammastates}, which
generalizes the entanglement distillation approach.
\section*{Acknowledgements} \noindent We thank helpful
discussions with colleagues including J. Batuwantudawe,
Jean-Christian Boileau, Debbie Leung,
John Preskill and Kiyoshi Tamaki.
This part is financially supported in part
by funding agencies including CFI, CIPI, CRC
program, NSERC, OIT, and PREA.
Parts of this paper were written during visits to the Institute of
Quantum Information (IQI) at Caltech
and to the Isaac Newton Institute, Cambridge, UK, whose kind hospitality is
acknowledged.
|
{
"timestamp": "2005-03-01T03:16:28",
"yymm": "0503",
"arxiv_id": "quant-ph/0503004",
"language": "en",
"url": "https://arxiv.org/abs/quant-ph/0503004"
}
|
\section*{1. Introduction}
The Faraday Experiment is probably the first non equilibrium
pattern forming system that has been investigated scientifically,
namely by Michael Faraday in 1831 \cite{Faraday}. Nevertheless it
was only recently that it was possible to determine the complete
Fourier spectrum of the deformed surface state \cite{Kityk04}.
While an experimental analysis of the full mode spectrum in other
pattern forming model systems like Rayleigh-Benard or
Taylor-Couette has been standard technique for a long time, it is
the refraction of light at the free surface of a liquid that
renders the analysis of surface waves so difficult. Quantitative
information about the patterned state up to higher orders is
important not only to verify the validity of theoretical
calculations \cite{kumar94,zhang96} but also to gain insight on
the resonance mechanisms that \textit{form} the patterns. The
Faraday experiment is especially known for its richness of
different patterns that are observed
\cite{miles90,milner90,Christiansen92,edwards94,binks97,kudrolli97}.
By using complex liquids \cite{wagner99}, very low fill heights
\cite{wagner00} or by introducing additional driving frequencies,
highly complex ordered states like superlattices
\cite{kudrolli98,arbell98} have been observed recently. But we
will demonstrate that even simple patterns like lines, squares and
hexagons observed in a single driving frequency experiment can
still unveil unknown surprising characteristics.
A discussion of the different attempts to reveal quantitative
information on the the surface elevation profile $\zeta(r,t)$ of
Faraday waves is given in \cite{Wernet01}. The main difficulties
in determining the surface elevation profile of capillary waves at
the the free surface are the difference in refractive indexes of
the liquid and the air, and the fact that the interface diffuses
almost no light but rather reflects or transmits incoming light
completely. To our knowledge there is only one optical method that
overcomes this problem with the use Polystyrene colloids to
provide light scatterers within the fluid \cite{Wright96}, but the
method was only used in a turbulent regime. Another powerful
method for the investigation of capillary waves on ferrofluids
based on x-ray absorption was presented in \cite{Richter01}, but
the related costs and efforts might be justified for fully opaque
liquids only. To bypass the problems associated with light
refraction and reflection on a liquid-air interface we chose to
study the interface between two index matched liquids. The upper
fluid is transparent, the lower one is dyed. In the presence of
surface deformations the instantaneous thickness can be deduced
from the intensity of the light transmitted trough the colored
layer. From a hydrodynamic point of view the replacement of the
air by a second liquid is nothing but a change of viscosity and
density, though the low kinematic viscosity of air simplifies the
theoretical calculations. However, the first exact theoretical
analysis of the linear stability problem by Kumar and Tuckerman
\cite{kumar94} was carried out for the more general case of a
system of two layers of liquid.
\section*{2. Experimental setup}
The experimental setup is shown in Fig.~\ref{fig1}. The container
consists of an aluminium ring (diameter $D=$18 cm) seperating two
parallel glass windows by a gap of 10 mm. It is filled by two
unmixible liquids: a silicone oil (SOIL, Dow Corning, viscosity
$\eta = 20$ mPas, density $\rho=949$ kg/m$^3$) and an aqueous
solution of sugar and NiSO$_4$(WSS, $\eta = 7.2 $ mPas,
$\rho=1185$ kg/m$^3$). The liquid liquid interfacial tension has
been determined with the pending droplet method to $35 \pm 2$
dynes/cm. The ratio of the filling heights SOIL/WSS was 8.4/1.6.
The choice of heights was made in order to obtain a variety of
different patterns, including a transition from squares to
hexagons \cite{binks97b}. The sugar concentration has been adapted
to match the refractive index to that of the covering silicone oil
($n \simeq 1.405$) to a precision of $5 \times 10^{-4}$. The
Ni$^{2+}$-ions produce a broad absorbtion band in the spectral
region 600-800 nm and provide high contrast patterns projected
onto the diffusive screen. The container is illuminated from below
with parallel light, and a band pass filter in front of the camera
was used to detect only wavelengths $\lambda = (655 \pm 5) nm$. By
varying the intensity of the lamp the flat interface has been set
to a level of about $50\%$ of the maximum optical transmission. At
a NiSO$_4$ concentration of $17 \%$ by weight the contrast between
the light intensity passing through crests and valleys of the wave
pattern was optimum. The associated coefficient of optical
absorption was measured as $\alpha = 5.2 \pm 0.1 cm^{-1}$. In
order to avoid uncontrolled changes of the viscosity, density and
interfacial tension of both liquids all the measurements were
performed at a constant temperature (23$\pm$0.1$^o$C). The Faraday
waves were excited by an electromagnetic shaker vibrating
vertically with an acceleration in the form $a(t) =
a_0$cos$(\Omega t)$. The driving signal came from a computer via a
D/A-converter and the acceleration has been measured by
piezoelectric sensor. A self developed closed-loop algorithm was
used to suppress higher harmonics $n(\Omega t)$ in the driving
signal to guarantee a purely harmonic driving. Faraday patterns
were recorded in the following way: a high speed (250 Hz) 8-bit
CCD camera was mounted some distance above the diffusive screen.
\begin{figure}
\includegraphics[ width=0.5\linewidth]{fig1.eps}
\caption{Experimental setup: L - halogen lamp, Ln -
lens, C - container filled by two liquids: SOIL and WSS with the
same refractive indices, DS - diffusive screen, IF - interference
filter, CCD - high speed CCD camera. } \label{fig1}
\end{figure}
Pictures were taken synchronous to the external driving. For a
certain instant $t_o$ the surface elevation of the Faraday
patterns $h (x,y,t_o)$ is given by:
\begin{equation}
h(x,y,t_o)= \frac {1}{\alpha}ln \frac{I_r(x,y)}{I_p(x,y,t_o)}
\label{eqn1}
\end{equation}
where $I_r(x,y)$ and $I_p(x,y,t_o)$ are 2D intensity distributions
captured by the camera for the reference picture (flat interface,
$a_o=0$) and for the Faraday pattern ($a_o \ne 0$), respectively.
Finally, the surface elevation function $h (x,y,t)$ are Fourier
transformed and the time evolution of the Fourier amplitudes and
phases of spatial modes is extracted. The use of a high speed
camera compared to the earlier measurements by some of the
co-authors \cite{Kityk04} allows for a better temporal resolution
and the method is not sensitive to distortions (defects) on time
scales of several periods. The logarithm of the intensity profile
renders the dynamic range nonlinear, and with an 8-bit dynamical
range the resolution is approximately $1\%$ ($2\%$) at small
(high) surface elevations, relative to the maximal surface
heights. The validity of the method has been also checked with
flat layers of colored liquids of different thicknesses and the
same accuracy was found. However, one should note the the Fourier
transformation integrates over many pixels and a significant
better resolution is to expect.
\section*{3. The linear regime}
The experiments have been performed by quasistatically ramping the
driving amplitude for the frequencies $f=\Omega/2\pi=12,16,20,29$
and $57 Hz$ from slightly below the critical acceleration $a_c$
($\varepsilon = (a-a_c)/a_c= -0.02)$ up to just below the
acceleration where the interface disintegrates and droplets form.
In the same form a ramp was driven down to check for hysteretic
effects, of which none were found. For each amplitude step a
series of pictures where taken and Fourier transformed. Typically
the pattern occurs in the center region of the container first but
evolves in a range of $\Delta\varepsilon = 0.02$. From the Fourier
transformation of the pictures at $-0.02<\varepsilon < 0.1$ (Fig.
\ref{squ}) the critical acceleration $a_c$ and the critical
wavenumber $k_c$ has been determined (Fig. \ref{lin1}).
\begin{figure}
\includegraphics[ width=0.95\linewidth]{fig2.eps}
\caption{Critical acceleration $a_c$ and critical wave number
$k_c$ for different driving frequencies $\Omega$. The symbols mark
experimental data, the lines the theoretical linear stability
analysis. The size of the symbols coincide with the size of the
error bars.}\label{lin1}
\end{figure}
The experimental data can be compared with the results from the
theoretical linear stability analysis that has been performed
using the algorithm proposed by Kumar and Tuckerman\cite{kumar94}.
The agreement between theory and experiment is very good, similar
to former studies at the liquid-air interface
\cite{bechhoefer95,wagner97}. But with our new technique we are
now able to verify more details of the predictions of the linear
theory, e.g. the temporal spectra at onset of the instability. It
is a particular feature of the Faraday-Experiment, that at onset
only one wave number $k_c$ becomes unstable, but the temporal
spectrum already contains multiples of the fundamental
oscillation frequency $\omega$ at onset. We are in the regime of
subharmonic response and the fundamental oscillation frequency at
onset is always $\omega=\Omega/2$ but the spectrum also contains
$(n+1/2) \Omega$ frequency components. More precisely we can write
the surface deformation $h({\bf r,t})$ as
\begin{equation}
h({\bf r}, t)= \frac{1}{4} \, \sum_{i=1}^{N} {(A_i e^{{\rm i} {\bf
k}_i \cdot {\bf r}} } + c.c.) \, \sum_{n=-\infty}^{+\infty}
\zeta_n e^{{\rm i} (n+1/2) \Omega t} \label{eqn2}
\end{equation}
Here ${\bf r}=(x,y)$ are the horizontal coordinates. The set of
\textit{complex} Fourier coefficients $\zeta_n$ are the components
of the eigenvector related to the linear stability problem and
determine the subharmonic time dependence. The spatial modes are
characterized by the wave vectors ${\bf k}_i$ , each carrying an
individual \textit{complex} amplitude $A_i$. These quantities are
determined by the nonlinearities of the problem. In principle the
${\bf k}_i$ can have any length and orientation but at onset the
relation $|{\bf k}_i|= k_c$ holds. The number N of participating
modes determines the degree of rotational symmetry of the pattern:
$N=1$ corresponds to lines, $N=2$ to squares, $N=3$ to hexagons or
triangles, etc. It can be shown \cite{douady89} that the $\zeta_n$
and $\zeta_{-n}$ are coupled in a way that $\zeta_n=\zeta_{-n}$ so
that heterodyning of right and left travelling waves always result
in \textit{standing} waves. Equation \ref{eqn1} then reads
\begin{multline}
h({\bf r}, t)= \sum_{i=1}^{N} {(
|A_i| cos{ {\bf k}_i \cdot {\bf r}+\phi_i} }) \\
\times \sum_{n=0}^{+\infty} {(|\zeta_n|
cos{(n+1/2) \Omega t+\psi_n})}
\label{eqn3}
\end{multline}
The complex eigenvectors $\zeta_n$ can be calculated modulo a
constant factor and the ratio of the amplitudes
$|\zeta_n|/|\zeta_{n+l}|$ as well as the temporal phases $\psi_n$
can be compared with experimental data. They are obtained in the
following way: For each step in the driving amplitude a series of
snapshots of the surface state (Fig. \ref{squ}) is taken. The
primary pattern consists of squares and their formation is
governed by the nonlinearities of the problem and one of our goals
is to identify how far from onset the predictions from the linear
theory hold.
An analysis of the Fourier transformation of the pictures yield
amplitudes $A(t)({\bf k}(ij))$ that are shown in Fig.
\ref{12hzsq1}. For the wave vectors ${\bf k}(ij)$ the nomenclature
from crystallography is used, e.g. ${\bf k}(10)$ and ${\bf k}(01)$
are the vectors that generate the simple unit cell of the square
pattern (Fig. \ref{squ}). The temporal evolution of the amplitude
of one of the critical modes $A(t){\bf k}(10)$ with $|{\bf
k}(10)|=\textit{k}_c$ (Fig. \ref{12hzsq1} is then again Fourier
transformed and a typical spectrum is shown in Fig.
\ref{12hzspec}a. These data are taken for all driving strengths
$\varepsilon$ (Fig. \ref{12hzbif1}a) and we always find the same
values for $A(t){\bf k}(10)$ and $A(t){\bf k}(01)$ within the
experimental resolution. In agreement with former investigations
\cite{Wernet01} in a system with a larger aspect ratio (container
size to wave length) our study reveals also that the fundamental
spatial mode $|{\bf k}(10)|=\textit{k}_c$ for all $\varepsilon$.
We can now extract the ratio of $A(\Omega3/2,{\bf
k}(10))/A(\Omega/2,{\bf k}(10))$ that is shown in the inset of
Fig. \ref{lin2}.
\begin{figure}
\includegraphics[ width=0.95\linewidth]{fig3.eps}
\caption{The ratio of the amplitudes
$A(\Omega3/2)/A(\Omega/2)$ of the ${\bf k}(10)$ mode at
$\varepsilon=0$ for different driving frequencies. The values are
extrapolated from measurements at $\varepsilon>0$ shown in the
inset: the amplitude ratios at $\Omega/2\pi=12Hz$ and $29 Hz$ as
a function of the driving strength $\varepsilon$.} \label{lin2}
\end{figure}
\begin{figure}
\includegraphics[ width=0.95\linewidth]{fig4.eps}
\caption{The temporal phases $\psi$ ($\Omega/2)$ and
$\psi(3\Omega/2)$ of the ${\bf k}(10)$ mode for two driving
frequencies versus driving strength $\varepsilon$. The symbols
mark experimental, the lines theoretical data. Squares and broken
line: $\Omega/2\pi=57 Hz$. Circles and dotted line:
$\Omega/2\pi=12 Hz$; in the range $0.2<\varepsilon<0.28$ a
transition from squares to hexagons takes place and in this
disordered state an extraction of phases is not
possible.}\label{lin3}
\end{figure}
The contribution of higher harmonics is in the order of $5$ to
$10\%$ and increases slightly with the driving strength. The
agreement between the experimental data and the linear theory is
again very good up to driving strength $\varepsilon
>0.5$, especially for lower driving frequencies. It is very
surprising that the agreement even holds up to secondary patterned
surface states, where a transition from a square to a hexagonal
state has been taken place and, as we show later, strong nonlinear
contributions participate in the dynamics of the system. The
experimental data show also that at driving strength as low as
$\varepsilon = 0.02$ the surface state consists of no measurable
higher spatial Fourier modes (see Fig. \ref{12hzbif1}) but of
higher temporal harmonics in perfect agreement with the linear
theory. This allows an extrapolation of $A(\Omega3/2,{\bf
k}(10))/A(\Omega/2,{\bf k}(10))$ to the neutral situation
$\varepsilon = 0$ for all driving frequencies $\Omega$ (Fig.
\ref{lin2}). The frequency ratio decreases first with increasing
frequency and has a minimum at $\Omega/(2 \pi)\approx 40 Hz$. This
characteristic shape reflects the amount of damping present in the
system. At low driving frequencies the ration between fill hight
and wave number is small. In this regime damping from the bottom,
that increases with decreasing frequency, is most significant. For
larger driving frequencies the damping from the bulk of the liquid
(a function increasing with the frequency) is the strongest
contribution. This behavior is also reflected in the critical
accelerations (compare with Fig. \ref{lin1}). The ratio
$A(\Omega5/2,{\bf k}(10))/A(\Omega/2,{\bf k}(10))$ has been
evaluated too, but the experimental resolution is not sufficient
here for a conclusive comparison between theory and experiment.
In the same way the temporal phases $\psi_n$ can be extracted from
the Fourier spectrum and once more a good agreement between the
theoretical predictions and experimental data is obtained, at
least for the fundamental $\Omega/2$ component. For the $3
\Omega/2$ component the scatter of the experimental data is very
large and we find significant differences between experiment and
theory. But besides the large scatter we observe a pronounced
nonmonotonic behavior of the $\psi_{3/2\Omega}$ component at $57
Hz$ and this part of the spectrum seems to be governed by
nonlinear interactions.
\section*{4. The nonlinear surface state at $\Omega =12 Hz$}
\subsection {The square state}
\begin{figure}
\includegraphics[ width=0.95\linewidth]{fig5.eps}
\vspace{0.3cm} \caption{Snapshots of the surface state and the
power spectra at $\Omega/2\pi=12Hz$ and $\varepsilon=$0.17
($a_0=$30.0 m/s$^2$) for two different temporal phases a) at
maximum and b) minimum surface elevation as indicated in Fig.
\ref{12hzsq1}}. \label{squ}
\end{figure}
\begin{figure}
\includegraphics[ width=0.95\linewidth]{fig6.eps}
\vspace{-0.3cm} \caption{The absolute amplitudes $A$ of different
spatial modes and the driving signal $a(t)$ in the square state at
$\Omega/2\pi=12Hz$ and $\varepsilon=$0.17 ($a_0=$30.0 m/s$^2$).
Please note that unlike in Ref. \cite{Kityk04} not the square root
of the power spectra but the amplitude A of a deformation $h({\bf
r},t)=A cos({\bf k}(ij) \cdot {\bf r})$ is shown.} \label{12hzsq1}
\end{figure}
The primary pattern near onset ($0<\varepsilon<0.28$) consists of
squares, shown in Fig. \ref{squ}. Their formation is determined by
the minimum of the Lyaponov functional of the according amplitude
equation of the critical modes \cite{cross93} and a quantitative
theoretical prediction of the expected pattern can be given by
inspection of the cubic coupling coefficient \cite{zhang96}. To
our knowledge, for a two liquid system there has not yet been an
attempt to calculate this coefficient, but squares are a common
pattern in free surface experiments with low viscous liquids. The
amplitude equations follow from a solvability condition of a
weakly nonlinear analysis of the underlying constitutive equation,
and its principal form is determined by the symmetries of the
system. For the subharmonic response one can write
\begin{equation}
\label{ampleqn1} \tau \partial_t A({\bf k}_i)= \epsilon A({\bf
k}_i) - \sum_{j=1}^{N} \Gamma(\theta_{ij})|{A({\bf k}_j)}|^2
A({\bf k}_i),
\end{equation}
with $\tau$ the linear relaxation time and $\Gamma(\theta_{ij})$
the cubic coupling coefficient that depends on the angle
$\theta_{ij}$ between the modes ${\bf k}_j$ and ${\bf k}_j$ with
$|{\bf k}_{i,j}|=k_c$. The amplitudes $A({\bf k}_i)$ are modulated
with a subharmonic $(n+1/2)\Omega$ time spectrum given by the
$\zeta_n$ from the linear eigenvectors. Equation \ref{ampleqn1}
predicts a pitchfork bifurcation and in order to study this
scenario one has to extract the different temporal Fourier modes
of the measured $A({\bf k}_i,t)$ (Fig. \ref{12hzsq1}) first. The
result is shown in Figs. \ref{12hzspec} and \ref{12hzbif1}a. As
long as the pattern consists of squares there are no harmonic time
dependencies in the basic spatial modes to observe, but a
continued growth of $\Omega/2$ and $3\Omega/2$ contributions. The
$5\Omega/2$ contribution is very weak and only slightly larger
than the noise. The square of the sum of the amplitudes
$A_s=A(\Omega/2) + A(3\Omega/2)$ yields a straight line if plotted
versus the driving strength $\varepsilon$ (Fig. \ref{12hzbif1}c)
as one would expect for the case of a pitchfork bifurcation. From
the slope we can extract the cubic coupling coefficient
$A_s=\varepsilon/\Gamma(90^\circ)$, and we find
$\Gamma(90^\circ)=0.179 mm^{-2}$.
\begin{figure}
\begin{flushleft}
\includegraphics[ width=1.05\linewidth]{fig7.eps}
\end{flushleft}
\vspace{-0.3cm} \caption{The temporal spectra of the amplitudes
$A({\bf k}(ij,\omega))$ of different spatial modes at
$\Omega/2\pi=12Hz$ and a driving strength: a) $\varepsilon$=0.18
in the square state and b) $\varepsilon=$0.38 in the hexagonal
state. } \label{12hzspec}
\end{figure}
\begin{figure}
\includegraphics[ width=0.52\linewidth]{fig8a.eps}
\hspace{-0.6cm}
\includegraphics[ width=0.52\linewidth]{fig8b.eps}
\includegraphics[ width=0.78\linewidth]{fig8c.eps}
\caption{The amplitudes $A(n/2\Omega)$ of the a) ${\bf k}(10)$ and
b) ${\bf k}(11)$ mode at $\Omega/2\pi=12Hz$ as a function of the
driving strength $\varepsilon$. c): the square of the sum of the
subharmonic components of the $A(10)$ mode versus $\varepsilon$.
As expected for a forward bifurcation the data can be linearly
fitted, at least up to driving strength of $\varepsilon \approx
0.1$ } \label{12hzbif1}
\end{figure}
Now we can inspect the next higher harmonic spatial modes $A({\bf
k}(11))$ and $A({\bf k}(20))$. Their temporal evolution is shown
in Fig. \ref{12hzsq1}. Both modes are a result of an interaction
of two fundamental modes, ${\bf k}(11)={\bf k}(10)+{\bf k}(01)$
and ${\bf k}(20)={\bf k}(10)+{\bf k}(10)$. Quadratic coupling does
not appear in the amplitude equations, but they are a natural
consequence of nonlinear spatial wave interaction and it is no
surprise that we find that they obey harmonic oscillations, shown
in Fig. \ref{12hzspec}a. The striking result of our analysis is
rather the constant offset that we find in the $A({\bf k}(11))$
and $A({\bf k}(20))$ spectrum (Fig. \ref{12hzsq1}). This means
that in addition to the oscillatory part, the interfacial profile
is also composed of contributions of constant deformations of the
form $h({\bf r}, t)= |A_i| cos{ {\bf k}_i \cdot {\bf r}}$. This
might surprisingly first, but please note that this does not
violate the mass conservation. Actually, it is a simple
consequence of the quadratic coupling of a \textit{real} standing
surface wave oscillation, $\Re (e^{i {\bf k}_i \cdot {\bf r}} e^{i
{\bf k}_i \cdot {\bf r}} e^{i\Omega/2}e^{-i\Omega/2})=cos{2 {\bf
k}_i \cdot {\bf r}} (1+cos{\Omega})$ (compare also Fig.
\ref{lines-ampl}c).
\begin{figure}
\includegraphics[ width=0.75\linewidth]{fig9.eps}
\caption{The amplitudes $A({\bf k}(20))$ and $A({\bf k}(11))$
versus the square of the amplitude of the fundamental mode $A({\bf
k}(10))$ or the product of $A({\bf k}(10))$ and $A({\bf k}(01))$
respectively ($\Omega/2\pi=12Hz$). } \label{A1versA2}
\end{figure}
This quadratic coupling scheme can be verified by plotting $A({\bf
k}(20))$ and $A({\bf k}(11))$ versus the square of the amplitude
of the fundamental mode $A({\bf k}(10))$ or the product $A({\bf
k}(10))\times A({\bf k}(01))$ respectively. The data can be
perfectly reproduced by a linear fit. From the slope one gets the
strength of this nonlinear coupling and we do find the same values
for all frequencies $\Omega/2\pi = 16,20,29$ Hz where squares are
to be observed. Finally our Fourier analysis yields that the
imaginary part of the coupling scheme obeys the same resonance
conditions, and the spatial phase of the higher harmonic modes is
given by $\phi({\bf k}(20))=2\phi({\bf k}(10))$ and $\phi({\bf
k}(10))+\phi({\bf k}(01))$.
\subsection {The hexagonal state}
\begin{figure}
\includegraphics[ width=0.99\linewidth]{fig10.eps}
\vspace{0.3cm} \caption{Snapshots of the surface state and the
Power spectra at $\Omega/2\pi=12Hz$ and $\varepsilon=$0.37
($a_0=$39.3 m/s$^2$) for three different temporal phases. a) down
hexagons, b) minimal surface elevation, c) up hexagons. See Refs.
\cite{wagner00,Kityk04} for further explanations on the switch
from up to down hexagons in the Faraday-Experiment. } \label{hex}
\end{figure}
In the range ($0.20<\varepsilon<0.28$) the pattern becomes
disordered and transforms at higher driving strength to a
hexagonal state (see Fig. \ref{hex}) that consists of three
fundamental spatial Fourier modes ${\bf k}_{1,2,3}$. But please
note that for the construction of the crystallographic simple unit
cell two vectors ${\bf k}(10)={\bf k}_1$ and ${\bf
k}(\bar{1}1)={\bf k}_2$ are sufficient (${\bf k}(\bar{1}1)+{\bf
k}(10)={\bf k}(0\bar{1})={\bf k}_3$, as indicated in Fig.
\ref{kvectorshex}).
\begin{figure}
\includegraphics[ width=0.5\linewidth]{fig11.eps}
\caption{Vector diagram of the interacting modes
for the hexagonal surface state. } \label{kvectorshex}
\end{figure}\begin{figure}
\includegraphics[ width=0.99\linewidth]{fig12.eps}
\vspace{0.3cm} \caption{Snapshots of the surface state and the
power spectra at a,b) $\Omega/2\pi=$20Hz, $\varepsilon=$0.6
($a_0=$54.4 m/s$^2$) c,d) $\Omega/2\pi=$20Hz, $\varepsilon=$0.08
($a_0=$36.7 m/s$^2$) e,f) $\Omega/2\pi=$57Hz, $\varepsilon=$0.11
($a_0=$116.3 m/s$^2$). } \label{sqline}
\end{figure}
\begin{figure}
\includegraphics[ width=0.75\linewidth]{fig13a.eps}
\includegraphics[ width=0.75\linewidth]{fig13b.eps}
\vspace{0.3cm} \caption{The amplitudes $A(\Omega/2)$ of the a)
${\bf k}(10)$ and ${\bf k}(01)$ mode at $\Omega/2\pi=29Hz$ and b)
${\bf k}(1)$ mode at $\Omega/2\pi=57Hz$ as a function of the
driving strength $\varepsilon$. The shaded region in a) indicates
the crossed rolls state.} \label{sqlinebif}
\end{figure}
An analysis of the temporal behavior of the amplitudes $A({\bf
k}(ij))$ of the spatial modes reveals, besides the the striking
offset with the according constant spatial sinusoidal surface
deformation, both harmonic and subharmonic time dependencies (see
Fig. \ref{12hzspec}). While harmonic ($n\Omega$) temporal
contributions in the higher spatial harmonics ${\bf k}(20,11)$
appear in a similar manner to that of the square pattern, the
resonance between ${\bf k}(10)+{\bf k}(0\bar{1})={\bf
k}(1\bar{1})$ results in harmonic ($n\Omega$) contributions in the
critical mode $|{\bf k}(1\bar{1})|=\textit{k}_c$. Consequently the
$n \Omega$ contributions couple with $(n+1/2) \Omega$
contributions back into the spectrum of the ${\bf k}(20,11)$ modes
and result in subharmonic contributions. Harmonic contributions do
not appear in the temporal spectra of the linear unstable modes
$\textit{k}_c$ and quadratic interactions of ($n \Omega/2$)
components do not appear in the amplitude equations. Nevertheless
the hexagonal state allows for a \textit{spatial} resonance
between linear unstable modes. In other words, this means that -
within the framework of the weakly nonlinear approximation - we
have here the interesting case where the system has a broken
temporal symmetry that is driven by spatial resonances. It is not
to be observed at any point in the quadratic state, where spatial
resonances between linear unstable modes are forbidden too. This
particular violation of the weakly nonlinear resonance conditions
can best be seen in Fig. \ref{12hzbif1} where clearly the
amplitudes of the $\Omega$ components of the $|{\bf
k(10)}|=\textit{k}_c$ mode grow from zero at the transition point
from squares to hexagons, and similarly the $\Omega/2$ components
of the $|{\bf k(11)}|=2\textit{k}_c$ modes.
\section*{5. Pattern dynamics at $\Omega/2\pi> 12 Hz$}
\begin{figure}
\includegraphics[ width=0.95\linewidth]{fig14a.eps}
\includegraphics[ width=0.95\linewidth]{fig14b.eps}
\vspace{0.3cm} \caption{a,b): The absolute amplitudes $A$ of
different spatial modes in the line state at $\Omega/2\pi=$57Hz
and $\varepsilon=$0.11 ($a_0=$116.3 m/s$^2$). In b) the temporal
constant offset $A(2k,0)$ is indicated by the dotted line. c) The
amplitude $A(2k,0)$ versus $A^2(k,\Omega/2)$} \label{lines-ampl}
\end{figure}
The pattern dynamics at driving frequencies $\Omega/2\pi> 12 Hz$
are characterized by a transition to lines. At $\Omega/2\pi= 16
Hz$ the pattern still consists only of squares, while at
$\Omega/2\pi= 20$ and $29 Hz$ the primary pattern consists of
(slightly distorted) lines (Fig. \ref{sqline}c,d). At higher
driving strengths $\varepsilon$, a second Fourier mode
perpendicular to the first one starts to grow (Fig.
\ref{sqlinebif}a) and the pattern evolves to a square state (Fig.
\ref{sqline}a,b). For $\Omega/2\pi= 57 Hz$ a pure line state is
stable for all driving strengths (Fig. \ref{sqline}e,f and
\ref{sqlinebif}b). The pronounced constant offset in the $A(2{\bf
k},t)$ (Fig. \ref{lines-ampl}b) mode is now larger than the
temporal oscillation period and $A(2{\bf k},t)$ never crosses the
zero line. Similar like for the $A({\bf k} 20,\Omega)$ or $A({\bf
k} 11,\Omega)$ modes in the squares state this quadratic coupling
scheme holds also for the zero frequency modes as shown in (Fig.
\ref{lines-ampl}c).
\section*{6. Conclusion}
We have demonstrated a new technique to measure quantitatively the
spatio-temporal Fourier spectrum of Faraday waves on a two liquid
interface. With this technique it is now possible to test
theoretical predictions, especially those from numerical
simulations. To our knowledge there are still no full Navier
Stokes numerical simulation of the 3D problem and quantitative
tests for future work are most important. In this sense we would
like to encourage such attempts. But with our technique we are
also able to verify known predictions from the linear stability
analysis and we find good agreement up to high driving strength of
$\varepsilon \approx 0.5$. In the nonlinear state the most
pronounced result is the identification of strong temporal
constant sinusodial surface deformations in the spectrum. And with
our possibility to access any Fourier component separately we can
identify several resonance mechanisms, including an interesting
case of a temporal resonance violation by use of spatial
resonances.
\begin{acknowledgments}
This work was supported by the German Science Foundation project
Mu 912.
\end{acknowledgments}
|
{
"timestamp": "2005-03-10T14:37:32",
"yymm": "0503",
"arxiv_id": "nlin/0503023",
"language": "en",
"url": "https://arxiv.org/abs/nlin/0503023"
}
|
\subsection*{Introduction}
This paper presents a quantum protocol based on public
key cryptogrpahy for secure transmission of data over
a public channel.
The security of the protocol derives from the fact that Alice and Bob each use
secret keys in the multiple exchange of the qubit.
Unlike the BB84 protocol [1] and its many variants (e.g. [2]-[4]),
where the qubits are transmitted
in only one direction and classical information exchanged thereafter, the communication
in the proposed protocol remains quantum in each stage.
In the BB84 protocol, each transmitted qubit is in one of four different states;
in the proposed protocol, the transmitted qubit can be in any arbitrary state.
\subsection*{The Protocol}
Consider the arrangement of Figure 1 to transfer state $X$ from Alice to Bob.
The state $X$ is one of two orthogonal states, such as $0\rangle$ and $|1\rangle$,
or $\frac{1}{\sqrt2} (| 0 \rangle + | 1\rangle )$
and $\frac{1}{\sqrt2} (| 0 \rangle - | 1\rangle )$, or
$\alpha |0\rangle + \beta |1\rangle$ and
$\beta |0\rangle - \alpha |1\rangle$.
The orthogonal states of $X$ represent $0$
and $1$ by prior mutual agreement of the parties, and this is the data
or the cryptographic key being transmitted over the public channel.
Alice and Bob apply secret transformations $U_A$ and $U_B$ which are commutative,
i.e., $U_A U_B = U_B U_A$.
An example of this would be $U_A = R(\theta)$ and $U_B = R (\phi)$, each of which
is the rotation operator:
\vspace{0.2in}
$R(\theta) = \left[ \begin{array}{cc}
cos \theta & - sin \theta \\
sin \theta & cos \theta \\
\end{array} \right]$
\vspace{0.2in}
\begin{figure}
\hspace*{0.2in}\centering{
\psfig{file=cryp.eps,width=10cm}}
\caption{Three-stage protocol for quantum cryptography where $U_A U_B = U_B U_A$}
\end{figure}
\vspace{0.2in}
\noindent
The sequence of operations in the protocol is as follows:
\begin{enumerate}
\item Alice applies the transformation $U_A$ on $X$ and sends the qubit to Bob.
\item Bob applies $U_B$ on the received qubit $U_A (X)$ and sends it
back to Alice.
\item
Alice applies $U_A^\dagger$ on the received qubit, converting it to
$U_B (X)$, and forwards it to Bob.
\item
Bob applies $U_B^\dagger$ on the qubit, converting it to $X$.
\end{enumerate}
At the end of the sequence, the state $X$, which was
chosen by Alice and transmitted
over a public channel, has reached Bob.
Eve, the eavesdropper, cannot obtain any information by intercepting the
transmitted qubits, although she could disrupt the exchange by replacing the
transmitted qubits by her own. This can be detected by
\begin{itemize}
\item appending parity bits, and/or
\item appending previously chosen bit sequences, which could be the destination and
sending addresses or their hashed values, or some other mutually agreed sequence.
\end{itemize}
Since the $U$ transformations can be changed as frequently as one pleases, Eve
cannot obtain any statistical clues to their nature by intercepting the qubits.
\subsection*{Key distribution protocol}
A related key distribution
protocol is given in Figure 2.
Unlike the previous case, $X$ is a fixed public state (say $|0\rangle$ or
$\frac{1}{\sqrt2} (|0\rangle + |1\rangle)$).
The objective is to generate a key that is a function of the
transformations involved, which is not chosen in advance by either party.
The protocol consists of two stages:
\begin{figure}
\hspace*{0.2in}\centering{
\psfig{file=cryp2.eps,width=8cm}}
\caption{Key distribution protocol, where $U_A U_B = U_B U_A$.}
\end{figure}
\begin{enumerate}
\item Alice and Bob use secret transformations, $U_A$ and $U_B$, on the known
state $X$, and exchange these qubits.
\item
They again apply the same transformations on the received qubits, thereby each
getting $U_A U_B (X)$, since $U_A U_B = U_B U_A$. It is assumed that neither
Alice or Bob will measure the received qubits, and will use them as the input
to a quantum register.
\end{enumerate}
In a variant of this scheme, two copies of the unknown state $X$ may be supplied
to Alice and Bob by a key registration authority.
\subsection*{Conclusion}
The three-stage protocol provides perfect security in the exchange of data over a public
channel under the assumptions that a separate classical protocol ensures the
identity of the two parties, and errors (deliberate or random) are detected
by means of parity check and confirming that
a known bit sequence that was appended to the bits has arrived correctly.
Since the proposed protocol does not use
classical communication, it is immune to
the man-in-the-middle attack on the classical
communication channel which BB84 type quantum cryptography protocols
suffers from [5].
On the other hand, implementation of this protocol may be harder because
the qubits get exchanged multiple times.
\section*{References}
\begin{description}
\item
[1]
M.A. Nielsen and I.L. Chuang, {\it Quantum Computation and Quantum Information}.
Cambridge University Press, 2000.
\item
[2]
A.K. Ekert, ``Quantum cryptography based on Bell's theorem.''
Phys. Rev. Lett., {\bf 67,} 661-663 (1991).
\item
[3] S. Kak, ``Quantum key distribution using three basis states.''
Pramana, {\bf 54,} 709-713 (2000); also quant-ph/9902038.
\item
[4]
A. Poppe {\it et al}, ``Practical quantum key distribution with polarization entangled
photons.'' quant-ph/0404115.
\item
[5] K. Svozil, ``The interlock protocol cannot save quantum cryptography from
man-in-the-middle attacks.'' quant-ph/0501062.
\end{description}
\end{document}
\end{enumerate}
\end{document}
|
{
"timestamp": "2005-03-02T23:27:29",
"yymm": "0503",
"arxiv_id": "quant-ph/0503027",
"language": "en",
"url": "https://arxiv.org/abs/quant-ph/0503027"
}
|
\section*{References}
|
{
"timestamp": "2005-03-09T13:33:26",
"yymm": "0503",
"arxiv_id": "quant-ph/0503087",
"language": "es",
"url": "https://arxiv.org/abs/quant-ph/0503087"
}
|
\section{Introduction}
In modern condensed matter research most interesting subjects are only
subtle effects which can be investigated only by thorough and
systematic studies of large numbers of samples. Even though first
investigations have to be done by hand, a lot of time can be saved with
automated measurement setups. Such automated systems are already
widely used in large scale experiments, but most small laboratory
experiments, even though computer controlled, do not allow for
automated measurements. Automation is present, to some extent,
in order to facilitate measurements in that there frequently are
possibilities to have the system execute a certain measurement
automatically, but covering easily large parameter spaces is
often not possible. Commercially available complete measurement
systems, on the other hand, seldom come with sophisticated control
software without possibilities for programming long measurement
sequences. Of course such software systems are the result of expensive
software development which is beyond the possibilities of an
ordinary research laboratory. Even though there are commercial
programs available for some of the instruments that constitute an
experimental setup, the important part is the interplay between
them. Consequently most control software is written by the
scientists themselves, who face the lack of time, money and
manpower to develop extensive automation software.
In this paper we present an easy way
of creating control software which offers possibilities of programming
complex sequences and automatically executes them\cite{sourcecode}.
This is shown
to be achieved with moderate development effort using a common
laboratory programming language. We will first present the different
architecture approach needed to achieve this goal, after which the
addition of automation is a small step.
\section{software for laboratory equipment}
Programs created for control of experiments need to perform several
tasks. Firstly, they have to be able to send control commands to the
instruments and receive the measured data. Secondly, these data are
to be processed and displayed and eventually user input needs to be
translated to control commands.
Various development platforms offer vast libraries of procedures
to interface instruments, create user interfaces and perform
complicated data processing. These help to reduce the workload
associated with creating such software. LabVIEW\texttrademark \cite{labview},
a development environment from National Instruments\texttrademark\
for creating programs (called virtual instruments or shortly VIs) in
its own graphical programming language ``G'', is probably best
known and most widely used for such applications. ``G'' offers all the
flow control structures like loops and conditional branches found in
any other programming language. Moreover, any VI can easily be used in
any other VI as a subVI.
LabVIEW VIs consist of a user interface (UI) and a block
diagram (BD) containing the actual code. Programming is done by modelling
data flow, where graphical representations of functions and procedures
are interconnected by lines, usually called wires. The designation VI
stems from the similarity of such a program to an actual instrument,
the UI obviously corresponding to the instrument's front panel and the
BD to its internal wiring.
A usual way of creating LabVIEW software for measurement
control is by writing a main VI containing the UI and the logic for
acting appropriately on the user input as well as processing,
displaying and saving the data. Communication with the instruments is
performed by driver subVIs which are regularly executed by the main VI.
When such a driver VI is called to perform a query on an instrument
it sends the necessary command to the instrument, waits some time for
the instrument to prepare the answer and finally reads this response
from the instrument. Usually this process takes tens to hundreds of
milliseconds. Assuming the whole measurement setup consists of several
instruments, the main VI may be organised in two different ways.
Either all driver VIs are called sequentially, causing the time needed
to collect all data to grow with the number of instruments. Another
apprach would be to call the driver VIs in parallel, which is possible
thanks to the inherently multithreading architecture of LabVIEW. In
this case, however, all drivers would attempt to access the
instruments at the same time. This would result in a ``traffic jam''
in case the instruments are connected to a single interface bus. Some
drivers would be forced to wait until the others have finished their
writing to the bus. Moreover, as some instruments take measurements
less often than others, many operations on the bus would be
unnecessary because no new data would be obtained.
In this paper we present the use of independent driver VIs, which we
call handlers, running in parallel
and communicating with a main VI by means offered by LabVIEW. This
allows for a more efficient use of the interface bus employed to connect
the instruments and results in a higher data acquisition rate.
Moreover, by employing a ``state machine'' (SM) architecture such
programs become easier to extend in functionality, to maintain and
most importantly allow for the control by a separate program and
consequently automation.
\section{experimental setup}
\begin{figure}
\center
\includegraphics[width=80mm]{fig1}
\caption{\label{torquesetup}Torque measurement setup overview,
which was automated using the presented software. A cryostat is placed
between the poles of an iron yoke magnet, which is freely rotatable.
The torque sensor is inserted into the cryostat and connected to
readout electronics. All instruments needed to control the experiment's
state are connected to a personal computer.}
\end{figure}
The programs presented here were developed to control and automatise a
torque magnetometry apparatus which was built in our group\cite{Willemin1998a,Willemin1998b}.
Such a device is used to measure a sample's magnetic moment $\vect{m}$
by the torque
\begin{equation}
\mbox{\eulerbold{\char28}} = \mu_0 \vect{m} \times \vect{H}
\end{equation}
it experiences due to a magnetic field $\vect{H}$. It is well suited
for investigation of anisotropic magnetic phenomena as found in most
high temperature superconductors. Torque magnetometry is
complementary to most other magnetometry techniques in that it
is only sensitive to the part $m_\perp$ of \vect{m} perpendicular to
the applied field. A torque measurement
is fast --- one measurement taking a fraction of a second only ---
and due to the proportionality $\tau \propto H$ reaches high
sensitivities for $m_\perp$ in high fields. Our home made torque
magnetometer system, shown schematically in Fig.~\ref{torquesetup},
consists of a flow cryostat between the poles of an iron yoke magnet
which is sitting on a rotatable support.
The torque sensor with a sample mounted on it is inserted into the
cryostat and connected to a Lock-In Amplifier (LIA) for read out.
Details of the measurement principle are beyond the scope of this
article and are described elsewhere \cite{Willemin1998a,Willemin1998b}.
All devices needed to control and measure the system's state
are connected to a Windows PC via an IEEE-488 General Purpose
Interface Bus (GPIB), RS-232 serial connections and indirectly
via additional analog and digital input and output ports present
in the LIA instrument. The main parts are the EG\&G Model 7265 LIA,
a Lakeshore DRC 93A temperature controller, and the Bruker BH-15
magnetic field controller. Additional devices such as a
pressure transducer with read out electronics for monitoring
the exchange gas pressure in the cryostat or current sources
and volt meters for specialized applications may also be connected
via the GPIB. The GPIB is an interface bus which is widely used in
scientific instruments. It features 8-bit parallel data transfer,
handshaking and real-time response capabilities.
\section{software system architecture}
\begin{figure}
\center
\includegraphics{fig2}
\caption{\label{overview}Architecture of the torque control
software system. All VIs (torque.vi, dataserver.vi and the
handler*.vis) execute in parallel. Commands are sent along the solid
right pointing arrows and data propagates back along the dashed left
pointing arrows.}
\end{figure}
The architecture of the newly developed control software is
shown in Fig.~\ref{overview}. Each instrument connected
to the system is represented by a VI counterpart called \emph{handler.vi}.
All handlers are managed by the \emph{dataserver.vi} VI which
communicates with the \emph{torque.vi} VI, which is the main
application. All these VIs run independently in parallel.
This way each \emph{handler.vi} can be optimised to take best advantage
of the instrument it is built for. This includes the waiting times
needed for communication, an optimized data rate based on varying
needs as well as the use of each instruments ability to signal special
events via the GPIB. Since all \emph{handler.vis} run in parallel, their
individual write--wait--read cycles needed to talk to the instruments
are interlaced, thus reducing the bus' idle time.
Moreover each instrument is talked to only when necessary thus
reducing the bus occupation while retaining data quality. This can be
optimised particularly well by exploiting the service request (SRQ)
functionality of the GPIB. Each instrument can signal a number of
events to the GPIB controller by asserting the special SRQ line. Such
events might be error conditions but can also be indicators of data
availability. As an example the Lakeshore temperature controller is
programmed to assert the SRQ line whenever a new temperature reading
is ready. As this occurs only every two seconds, the instrument is
read only when really necessary instead of reading the same data
several times per second. Even instruments not offering such
functionality can be optimised by reducing the rate at which the
\emph{handler.vi} is instructed to read the instrument. This enables the
more crucial measurements to be read more often resulting in data
taken at a higher rate and resulting in better quality.
Because the \emph{handler.vis} are not called as subVIs by the main
VI a special means of communication needs to be established.
Here we present the use of \emph{queues} for sending commands to
the \emph{handler.vis} and \emph{DataSockets} for receiving the
measured data. A \emph{queue} is a first--in--first--out style memory
construct which is offered by LabVIEW. It may contain a fixed or unlimited
number of string entries, in our case commands. By use
of special subVIs any VI can append commands to a \emph{queue's}
end or retrieve the oldest commands. Any read entry is automatically
removed. \emph{Queues} are identified by a name, making access
to them fairly easy. In most applications a given \emph{queue} is read
by only one VI whereas several VIs may write to it.
\emph{DataSockets} are memory constructs as well, identified
by a unique name, but only contain the most recent datum. Their
data type can be freely chosen among the data types in LabVIEW.
The \emph{DataSockets} used in our case are arrays of floating
point numbers containing a \emph{handler.vi's} main data.
The \emph{dataserver.vi} mentioned above serves as an intermediate
VI which collects all the \emph{handler.vi's} data and puts
all together in a separate \emph{DataSocket} which is then read
by the \emph{torque.vi} main VI. Thus the main VI needs no knowledge
about which data to obtain from which instrument.
\begin{figure}
\center
\includegraphics{fig3}
\caption{\label{FIGparser}Schematic illustration of the VI's
basic structure. An all enclosing main loop executes infinitely.
The logic inside consists of a command
stack whose first element is divided into instruction and argument.
The instruction is used as the selector value into
a case structure containing the code for the individual
instructions. This results in the command parsing functionality needed
for the operation. Internal data needed for the VI's execution is
passed through each iteration and can be read and modified
by each command case.}
\end{figure}
In order for the VIs to be able to act accordingly
on the possible commands they must be given some command parsing
functionality. In fact such a command parser is every VI's
core part: Even the regular operations performed by the VIs are
put into commands which are executed repeatedly. Essentially, all
VIs are designed as command driven state machines (SM).
The use of the SM paradigm
in LabVIEW programs was already proposed at several occasions
and given LabVIEW's capabilities this is not surprising.
Nevertheless, to our knowledge only few applications make use of this
architecture. The basic idea is that by being executed, a program goes
through various named states. The order in which these states are
visited may be fixed and defined in advance or the state to follow
might be determined based on the current state's result. The
implementation in LabVIEW is fairly simple and schematically shown
in Fig.~\ref{FIGparser}. An infinitely running loop
contains a case structure consisting of all the states. These
states are identified by character strings and are therefore
easily human readable. In contrast to other methods, where the
identification is by numbers or special enumeration data types,
this makes the structure easy to extend and maintain.
Additionally to these structures the VIs contain a command stack
and some internal data needed for execution. Upon startup, when
the command stack is empty, a default case (state) is executed.
Usually this is the ``GetCommands'' case. This case contains
the code needed to empty this VI's \emph{queue} and a set of default
commands which are put onto the command stack. When the main loop
is iterated for the second time, the oldest command is removed
from the stack, split into an instruction and optional
arguments, whereupon this instruction is fed into the case
structure selector, defining the case to be executed. This case
may add more commands to the stack or simply perform a specific
task. When the case is finished, the main loop iterates again,
the next command is removed from the stack and so on. Whenever the
stack becomes empty, the default case ``GetCommands'' is
executed again and refills it.
Because the \emph{handler.vis} are independent programs not having
to rely on being called regularly by a master VI they can be
used to carry out more complex tasks than just talking to
the instruments. As an example \emph{handlerLakeshore.vi}, the
\emph{handler.vi} for the Lakeshore temperature
controller contains logic to control the temperature by software
through control of the coolant flow in the cryostat. The flow
controller is connected to a separate digital--to--analog converter
(DAC), thus enabling the \emph{handlerLakeshore.vi} to control
it by sending commands to the DAC's \emph{handler.vi}
(\emph{handlerDAC.vi}).
Keeping track of the last few seconds of measured data,
calculating their time trends and publishing it to the \emph{DataSocket}
is coded into a command and performed by the \emph{handler.vis} as well.
\section{Automation}
As mentioned earlier, all VIs are organized as state machines,
even the main VI \emph{torque.vi}. As shown in Fig.~\ref{automation}
every user action (button press, value change) on its user interface (UI)
is transformed into a command by the UI-handler which is then sent to
and processed in the SM. The SM then sends appropriate
commands to the \emph{dataserver.vi} and the \emph{handler.vis} (wide
arrow (1) in Fig.~\ref{automation}).
These two parts (UI-handler and SM) are independently running components
of \emph{torque.vi}. The communication between them is again ensured via
\emph{queues}. This enables other VIs, such as the \emph{sequencer.vi}
shown in Fig.~\ref{automation} to be used to control
the SM in \emph{torque.vi} programmatically by sending these commands
directly to the SM (wide arrow (2) in Fig.~\ref{automation}).
\begin{figure}
\center
\includegraphics{fig4}
\caption{\label{automation}All VIs consist of a User interface (UI) and
a block diagram (BD). In contrast to all other VIs the \emph{torque.vi's}
BD consists of the UI-handler part and the state machine (SM) itself,
both running in parallel. In normal, interactive operation of the torque
system, user actions on the \emph{torque.vi's} UI are translated by the
UI-handler into commands which are sent to the SM via a queue and then
propagate on to the dataserver and handler VIs (wide arrow (1)).
If an automated measurement is run, the \emph{sequencer.vi's} SM retrieves
commands from the text sequence on its UI, sends them via a queue to the
\emph{torque.vi's} SM from where they propagate on to the dataserver and
handler VIs (wide arrow (2)). The \emph{torque.vi's} SM sends confirmation
messages back to the \emph{sequencer.vi}. Solid black arrows
indicate direct access between the BD and the UI, whereas dotted arrows
represent data transmission via \emph{queues} and \emph{DataSockets}.}
\end{figure}
When automatic measurements are required, a sequence text file
is written containing the commands needed to accomplish
these measurements which is then read by the \emph{sequencer.vi}.
Additionally to the commands of \emph{torque.vi's} SM the
\emph{sequencer.vi} understands a set of flow control instructions
such as ``if'', ``while'' and ``for'' which are useful
for creating short sequences for repetitive tasks, as well as
the use of variables and their arithmetic manipulation and
comparison.
The \emph{sequencer.vi} parses through the sequence file by looking
for known keywords -- the commands. Any strings which are not
recognized as a keyword are treated as arguments to the preceding
keyword. The string \texttt{settemp 20 waittemp} present in a sequence
file would instruct the torque software to change the temperature
to 20\,K and wait for the cryostat to stabilize at this temperature.
In this example \texttt{settemp} and \texttt{waittemp} are keywords
and \texttt{20} is the argument to the keyword \texttt{settemp}.
Such sequencing possibilities are already well known in control
software of commercially available measurement equipment
(eg.\ SQUID magnetometers or the \emph{Quantum Design} Physical
Property Measurement System\cite{Quantum}).
Now such efficient and flexible data taking is also possible
with our home made torque magnetometer.
\section{Example of Application}
In order to demonstrate the possibilities of such an automatable
measurement system we present some results of a systematic
study\cite{Kohout2005}
of the so called lock-in transition in the high temperature
superconductor La$_{2-x}$Sr$_{x}$CuO$_{4}$. Details about this effect can be obtained
from various other sources and are not discussed
here\cite{Blatter1994,Steinmeyer1994}.
Most easily this effect is visible in angle dependent torque
measurements and manifests itself as a deviation from an otherwise
smooth behaviour. An example of such a measurement is shown in
Fig.~\ref{torque1}, where the measured data points close to
90$^\circ$ deviate from a theoretical curve\cite{kogan} which fits
well to the remaining angle range.
The same model can also be used to describe data taken as a
function of magnetic field magnitude $H$ at a fixed angle.
It is commonly accepted that in first approximation the
magnetic moment $m=\tau/H$ of a superconductor is proportional
to $\ln(H)$.
\begin{figure}
\center
\includegraphics{fig5}
\caption{\label{torque1}Angle dependent torque measurement (circles)
of an underdoped crystal of La$_{2-x}$Sr$_{x}$CuO$_{4}$\ with $x=0.07$ ($\ensuremath{T_\mathrm{c}}=17\U{K}$),
performed at $T=8\U{K}$ in a magnetic field $\mu_0H=1\U{T}$.
The solid line is a fit of a model derived by Kogan\cite{kogan}. The
deviation close to $\theta \approx 90\ensuremath{^\circ}$ stems from the lock-in
transition.}
\end{figure}
Within our study we measured six La$_{2-x}$Sr$_{x}$CuO$_{4}$\ single microcrystals with
varying Sr content $0.07 \le x \le 0.23$ and critical temperatures \ensuremath{T_\mathrm{c}}\
varying from 17\U{K} to 35\U{K}. They were mounted on
a highly sensitive torque sensor and cooled below \ensuremath{T_\mathrm{c}}.
Field dependent measurements
($\mu_0H=0\ldots1.5\U{T}$ at $5\U{mT}$ steps with increasing and
decreasing field) were
taken at 60 field orientations ($\theta=-90\ensuremath{^\circ}\ldots90\ensuremath{^\circ}$ with
varying steps) and at about ten temperatures below the critical
temperature \ensuremath{T_\mathrm{c}}.
We emphasize that such extensive measurements would hardly be
possible without our software's automation possibilities. As each
field scan takes about six minutes, without automation user interaction
would be necessary at this interval during \emph{one week} to collect
all these data for one crystal.
After writing the sequence and starting its execution, the
measurement system, on the other hand, finishes such a measurement
set within about \emph{three days} with no need of intervention.
The experiment is finished faster, because less time is lost
between consecutive field scans and because the measurement is
running day and night.
\begin{figure}
\center
\includegraphics{fig6}
\caption{\label{torque2}Field dependent measurement $\tau(H)$ of the
same La$_{2-x}$Sr$_{x}$CuO$_{4}$\ crystal as was used for the measurement in Fig.\ \ref{torque1}.
The angle of the magnetic field was fixed at $\theta=75\ensuremath{^\circ}$ and $\theta=80\ensuremath{^\circ}$. The measurements are plotted as $\tau/H$ vs.\ $\ln(H)$.
The lines are guides to the eye to show the two linear regions I
(low field) and II (high field).}
\end{figure}
We present here only one dataset of a single crystal taken at
one particular temperature. Such a dataset consists of 60
field scans taken at various orientations. The two field scans
shown in Fig.~\ref{torque2} illustrate the deviations of field
dependent data due to the lock-in transition. Clearly visible
are two regions (I and II) where $\tau/H$ is proportional to
$\ln(H)$. A comparison of these measurements to angle dependent
measurements at similar conditions indicate that region I corresponds
to the part where lock-in takes place, whereas data in
region II are well described by the theoretical curve
in Fig.~\ref{torque1}. By analysing the whole data set it is now
easy to investigate the evolution of these two regions as a
function of angle $\theta$. The result is shown in
Fig.~\ref{torque3}, where the extents of the two regions, obtained
from field dependent measurements, are plotted vs.\ the angle
$\theta$. The horizontal line A indicates the cut of the
measurement in Fig.~\ref{torque1} and the vertical lines B and
C the measurements shown in Fig.~\ref{torque2}.
The observed region, separating regions I and II manifests the
lock-in transition and can be understood in terms of a
model proposed by Feinberg and Villard\cite{Feinberg}.
\begin{figure}
\center
\includegraphics{fig7}
\caption{\label{torque3}Summary of field dependent measurements
performed on a La$_{2-x}$Sr$_{x}$CuO$_{4}$\ single crystal at $T=8\U{K}$. Only the extents
of the linear regions such as shown in Fig.\ \ref{torque2} as a function
of field orientation $\theta$ are shown. The enhancement of the low-field
region I close to the ab-plane ($\theta \approx 90\ensuremath{^\circ}$) is
clearly visible. The horizontal line A indicates the position
of the measurement shown in Fig.~\ref{torque1}. The
vertical lines B and C indicate the position of the measurements
shown in Fig.~\ref{torque2}.}
\end{figure}
\section{Acknowledgements}
This work was supported in part by the Swiss National Science
Foundation.
\newpage
|
{
"timestamp": "2005-03-01T11:16:07",
"yymm": "0503",
"arxiv_id": "physics/0503005",
"language": "en",
"url": "https://arxiv.org/abs/physics/0503005"
}
|
\section{Introduction.}
In quantum probability there exist several natural notions of independence,
see \cite{muraki03} and the references therein. These allow to define new
convolutions for probability measures, cf.\
\cite{voiculescu+dykema+nica92,voiculescu97,speicher+woroudi93,muraki00}.
Bercovici \cite{bercovici04} defined multiplicative monotone convolutions for
probability measures on the unit circle and on the half line. He showed that
with an appropriate function of the Cauchy transform these multiplicative
convolutions can be calculated by composition of those functions, similar to
Muraki's result \cite[Theorem 3.1]{muraki00} for the additive monotone
convolution. In this paper we give a new proof of Bercovici's result based on
the combinatorics of moments, see Theorem \ref{thm-operators}. Using Berkson
and Porta's \cite{berkson+porta78} characterization of composition semigroups, one can deduce a
characterization of continuous convolution semigroups for the monotone
convolution, see \cite[Theorem 4.6]{bercovici04} or Theorem
\ref{thm-levy-khintchine-circle} for the case of probability measures on the
unit circle.
This paper is organized as follows.
In Section \ref{sec-mon} we recall the definition of monotone independence and
the monotone product of algebraic and quantum probability spaces. In Section
\ref{sec-mon-cond} we show that the monotone product is actually a special
case of the conditionally free product introduced in
\cite{bozejko+speicher91b,bozejko+leinert+speicher96}.
Sections \ref{sec-op}, \ref{sec-conv}, and \ref{sec-levy} contain the main
results on the multiplicative monotone convolution. We formulate a slightly
modified version of a theorem by Bercovici that shows that these convolutions
can be calculated by taking the composition of appropriate functions of the
Cauchy transform of the measures, see Theorem \ref{thm-operators} and
Corollaries \ref{cor-unitary} and \ref{cor-pos}. We also state a
L\'evy-Khintchine type characterization of all continuous convolution
semigroups for the monotone convolution of probability measures on the unit
circle, see Theorem \ref{thm-levy-khintchine-circle}.
In Section \ref{sec-galton}, we show that the problem of embedding a
probability measure on the unit circle into a continuous monotone convolution
semigroup is very similar to the problem of embedding a discrete-time
Markovian branching process (or Galton-Watson process) into a continuous-time
Markovian branching process. In Section \ref{sec-embed} we adapt a
characterization of embeddable branching processes due to Gorya\u{\i}nov
\cite{goryainov93} to our situation.
Finally, in the Appendix we discuss the multiplicative monotone convolution of
probability measures on the half line and show that there exist two natural, but
inequivalent definitions. One of them is equivalent to the definition due to
Bercovici and can be treated by similar methods as the
multiplicative monotone convolution of measures on the unit circle., cf.\
\cite{bercovici04}.
\section{Monotone Independence.}\label{sec-mon}
In this section we present the definition of monotone independence and its main properties.
By an {\em algebraic probability space} we mean a pair
$(\mathcal{A},\varphi)$ consisting of a
unital algebra $\mathcal{A}$ and a unital functional
$\varphi:\mathcal{A}\to\mathbb{C}$. Assume that we have two algebraic
probability spaces $(\mathcal{A}_1,\varphi_1)$ and
$(\mathcal{A}_2,\varphi_2)$, such that the first algebra has a
decomposition $\mathcal{A}_1=\mathbb{C}\mathbf{1}\oplus\mathcal{A}_1^0$ (direct
sum as vector spaces), where $\mathcal{A}_1^0$ is a subalgebra of
$\mathcal{A}_1$. Then we define the algebraic monotone product
$(\mathcal{A},\varphi)$ of $(\mathcal{A}_1,\varphi_1)$ and
$(\mathcal{A}_2,\varphi_2)$ as follows, see also \cite{muraki01,muraki03}. The algebra $\mathcal{A}=\mathcal{A}_1\coprod
\mathcal{A}_2$ is the free product of $\mathcal{A}_1$ and
$\mathcal{A}_2$ with identification of the units of $\mathcal{A}_1$ and
$\mathcal{A}_2$. The unital functional
$\varphi=\varphi_1\triangleright\varphi_2:\mathcal{A}\to\mathbb{C}$ is determined by the condition
\begin{equation}\label{def-alg-mon}
\varphi(b_1a_1b_2\cdots a_{n-1}b_n)=\varphi_1(a_1\cdots
a_{n-1})\varphi_2(b_1)\cdots \varphi_2(b_n)
\end{equation}
for $n\in\mathbb{N}$ and all $a_1,\ldots,a_{n-1}\in\mathcal{A}_1^0$,
$b_1,\ldots,b_n\in\mathcal{A}_2$.
Let now $\mathcal{A}_1,\mathcal{A}_2\subseteq\mathcal{B}$ be two such
algebras, which are contained in an algebraic probability space
$(\mathcal{B},\Phi)$ and denote by
$j_1:\mathcal{A}_1\to\mathcal{B}$, $j_2:\mathcal{A}_2\to\mathcal{B}$
the inclusion maps. Then the universal property of the free
product of algebras implies that there exists a unique homomorphism
$j:\mathcal{A}_1\coprod\mathcal{A}_2\to B$ such that the following
diagram commutes
\[
\xymatrix{
& \mathcal{B} & \\
\mathcal{A}_1\ar[ur]^{j_1}\ar[r]_{i_1} &
\mathcal{A}_1\coprod\mathcal{A}_2 \ar[u]|-j & \mathcal{A}_2\ar[ul]_{j_2}\ar[l]^{i_2}
}
\]
where are $i_1:\mathcal{A}_1\to\mathcal{A}_1\coprod\mathcal{A}_2$ and
$i_2:\mathcal{A}_2\to\mathcal{A}_1\coprod\mathcal{A}_2$ are the
canonical inclusion maps.
The subalgebras $\mathcal{A}_1,\mathcal{A}_2$ are called {\em monotonically
independent} w.r.t.\ $\Phi$, if
\[
\Phi\circ j = (\Phi\circ j_1)\triangleright (\Phi\circ j_2)
\]
cf.\ \cite{franz02}
We will call a triple $(\mathcal{A},\mathcal{H},\Omega)$ consisting of
a Hilbert space $\mathcal{H}$, a unit vector $\Omega\in\mathcal{H}$,
and a subalgebra $\mathcal{A}\subseteq\mathcal{B}(\mathcal{H})$ a {\em
quantum probability space}.
If we have an algebraic probability space $(\mathcal{A},\varphi)$, whose
algebra has an involution such that $\Phi$ is even a state, and if for
all $a\in\mathcal{A}$ there exists a constant $C_a\ge 0$ such that the
inequality
\[
\Phi(x^*a^*ax)\le C_a\Phi(x^*x)
\]
holds for all $x\in\mathcal{A}$, then the GNS representation
$(H_\varphi,\pi_\varphi,\Omega_\varphi)$ of $(\mathcal{A},\Phi)$
yields a quantum probability space
$(\pi_\varphi(\mathcal{A}),H_\varphi,\Omega_\varphi)$. If two
subalgebras
$\mathcal{A}_1=\mathbb{C}\mathbf{1}\otimes\mathcal{A}_1^0,\mathcal{A}_2\subseteq\mathcal{A}$
are monotonically independent in $(\mathcal{A},\varphi)$, then
$\pi_\varphi(\mathcal{A}_1^0)$ and $\pi_\varphi(\mathcal{A}_2)$ are
monotonically independent in
$(\pi_\varphi(\mathcal{A}),H_\varphi,\Omega_\varphi)$ in the sense of
the following definition.
\begin{definition}\label{def-mon-indep}
Let $\mathcal{H}$ be a Hilbert space, $\Omega\in\mathcal{H}$ a unit vector, and define a state $\Phi:\mathcal{B}(\mathcal{H})\to\mathbb{C}$ on the algebra of bounded operators on $\mathcal{H}$ by
\[
\Phi(X)=\langle \Omega, X\Omega\rangle, \qquad\mbox{ for } X\in\mathcal{B}(\mathcal{H}).
\]
Two subalgebras $\mathcal{A}_1,\mathcal{A}_2\subseteq\mathcal{B}(\mathcal{H})$ are called {\em monotonically independent} w.r.t.\ $\Omega$, if the following two conditions are satisfied.
\begin{description}
\item[(a)]
For all $X,Z\in\mathcal{A}_1$, $Y\in\mathcal{A}_2$, we have
\[
XYZ = \Phi(Y)XZ.
\]
\item[(b)]
For all $Y\in\mathcal{A}_1$, $X,Z\in\mathcal{A}_{2}$,
\[
\Phi(XYZ)=\Phi(X)\Phi(Y)\Phi(Z).
\]
\end{description}
Two operators $X,Y\in\mathcal{B}(\mathcal{H})$ are called
monotonically independent w.r.t.\ $\Omega$, if the subalgebras
$\mathcal{A}_1={\rm alg}(X)={\rm span}\{X^k|k=1,2,\ldots\}$ and $\mathcal{A}_2={\rm alg}(Y)={\rm span}\{Y^k|k=1,2,\ldots\}$ are monotonically independent.
\end{definition}
\begin{proposition}\label{prop-mon-prod}
Let $(\mathcal{A}_i,\mathcal{H}_i,\Omega_i)$, $i=1,2$, be two quantum probability spaces, and denote the states associated to $\Omega_1$ and $\Omega_2$ by $\Phi_1$ and $\Phi_2$, respectively.
Then there exists a quantum probability space $(\mathcal{A},\mathcal{H},\Omega)$ and two injective state-preser\-ving homomorphisms $J_i:\mathcal{A}_i\to\mathcal{A}$, $i=1,2$, such that the images $J_1(\mathcal{A}_1)$ and $J_2(\mathcal{A}_2)$ are monotonically independent w.r.t.\ $\Omega$.
\end{proposition}
\Proof
We set $\mathcal{H}=\mathcal{H}_1\otimes\mathcal{H}_2$ and $\Omega=\Omega_1\otimes\Omega_2$. Denote by $P_2$ the orthogonal projection on $\mathbb{C}\Omega_2\subseteq\mathcal{H}_2$.
We define the embeddings $J_i:\mathcal{A}_i\to\mathcal{B}(\mathcal{H})$ by
\begin{eqnarray*}
J_1(X) &=& X\otimes P_2, \qquad \mbox{ for } X\in\mathcal{A}_1, \\
J_2(X) &=& \mathbf{1}\otimes X, \qquad \mbox{ for } X\in\mathcal{A}_2.
\end{eqnarray*}
For $\mathcal{A}$ we take the subalgebra generated by $J_1(\mathcal{A}_1)$ and $J_2(\mathcal{A}_2)$. It is clear that $J_1$ and $J_2$ are injective, state-preserving homomorphisms.
A simple calculation shows that $J_1(\mathcal{A}_1)$ and $J_2(\mathcal{A}_2)$ are monotonically independent w.r.t.\ $\Omega$. E.g., for products of the form $J_1(X_1)J_2(Y)J_1(X_2)$, $X_1,X_2\in\mathcal{A}_1$, $Y\in\mathcal{A}_2$, we get
\begin{eqnarray*}
J_1(X_1)J_2(Y)J_1(X_2) &=& (X_1\otimes P_2)(\mathbf{1}\otimes Y)(X_1\otimes P_2) = (X_1X_2)\otimes P_2YP_2 \\
&=& \Phi\big(J_2(Y)\big) J_1(X_1)J_1(X_2).
\end{eqnarray*}
On the other hand, for $J_2(Y_1)J_1(X)J_2(Y_2)$, $X\in\mathcal{A}_1$, $Y_1,Y_2\in\mathcal{A}_2$, we get
\begin{eqnarray*}
\Phi\big(J_2(Y_1)J_1(X)J_2(Y_2)\big) &=& \langle\Omega_1\otimes\Omega_2,(\mathbf{1}\otimes Y_1)(X\otimes P_2)(\mathbf{1}\otimes Y_2)\Omega_1\otimes\Omega_2\rangle \\
&=& \langle\Omega_1\otimes\Omega_2,X\otimes (Y_1PY_2)\Omega_1\otimes\Omega_2\rangle \\
&=& \Phi_1(X)\Phi_2(Y_1)\Phi_2(Y_2) =\Phi\big(J_2(Y_1)\big)\Phi\big(J_1(X)\big)\Phi\big(J_2(Y_2)\big).
\end{eqnarray*}
\endproof
We will call the quantum probability space $(\mathcal{A},\mathcal{H},\Omega)$ constructed in the previous proposition the {\em monotone product} of $(\mathcal{A}_1,\mathcal{H}_1,\Omega_1)$ and $(\mathcal{A}_2,\mathcal{H}_2,\Omega_2)$. When there is no danger of confusion, we shall identify the algebras $\mathcal{A}_1$ and $\mathcal{A}_2$ with their images $J_1(\mathcal{A}_1)$ and $J_2(\mathcal{A}_2)$, respectively.
The monotone product is associative and can be extended to more than two factors, see \cite{franz01}. But it is not commutative.
The embedding $J_1:\mathcal{A}_1\to\mathcal{A}$ is not unital and the product is not trace-preserving. If $\Phi_1|_{\mathcal{A}_1}$ is not identically equal to zero, then the calculation
\[
\Phi_1(X)\Phi_2(Y_1Y_2)=\Phi(XY_1Y_2)=\Phi(Y_2XY_1)=\Phi_1(X)\Phi_2(Y_1)\Phi_2(Y_2)
\]
for all $X\in\mathcal{A}_1$, $Y_1,Y_2\in\mathcal{A}_2$ shows that $\Phi$ can only be a trace on $\mathcal{A}$, if $\Phi_2|_{\mathcal{A}_2}$ is a homomorphism.
\section{Relation of monotone independence and conditional free
independence.}\label{sec-mon-cond}
We recall now the definition of the conditional free product of algebraic
probability spaces and show that the monotone product is contained as a
special case.
Let $(\mathcal{A}_1,\varphi_1,\psi_1)$ and
$(\mathcal{A}_2,\varphi_2,\psi_2)$ be two unital algebras, equipped
with two unital functionals. Recall that the conditionally free
product\cite{bozejko+speicher91b,bozejko+leinert+speicher96} of $(\mathcal{A}_1,\varphi_1,\psi_1)$ and
$(\mathcal{A}_2,\varphi_2,\psi_2)$ is defined as the triple
$(\mathcal{A},\varphi,\psi)$, where $\mathcal{A}=\mathcal{A}_1\coprod
\mathcal{A}_2$ is the free product of $\mathcal{A}_1$ and
$\mathcal{A}_2$ with identification the units of $\mathcal{A}_1$ and
$\mathcal{A}_2$. The unital functionals $\varphi$ and $\psi$ on $\mathcal{A}=\mathcal{A}_1\coprod
\mathcal{A}_2$ can be defined by the conditions
\begin{equation}\label{def-cond-free}
\varphi(a_1a_2\cdots a_n)=\varphi_{\epsilon(1)}(a_1)\cdots\varphi_{\epsilon(n)}(a_n) \quad \mbox{
and }\quad \psi(a_1a_2\cdots a_n)=0
\end{equation}
for all $n\in \mathbb{N}$ and all $a_i\in\mathcal{A}_{\epsilon(i)}$
with $\epsilon(i)\in\{1,2\}$,
$\epsilon(1)\not=\epsilon(2)\not=\cdots\not=\epsilon(n)$ and
$\psi_{\epsilon(1)}(a_1)=\cdots=\psi_{\epsilon(n)}(a_n)=0$. The
functional $\psi$ is simply the free product $\psi_1*\psi_2$ of $\psi_1$ and $\psi_2$,
cf.\ \cite{voiculescu+dykema+nica92,voiculescu97}. We will denote
$\varphi$ by
\[
\varphi=\varphi_1\,{}_{\psi_1}\kern-.5em*\kern-.3em{}_{\psi_2}\,\varphi_2.
\]
The product defined in this way for triples
$(\mathcal{A},\varphi,\psi)$ can be shown to be commutative and
associative, cf. \cite{bozejko+speicher91b,bozejko+leinert+speicher96}.
Taking pairs of the form $(\mathcal{A}_1,\varphi_1,\varphi_1)$ and
$(\mathcal{A}_2,\varphi_2,\varphi_2)$, one obtains the free
product also for the first functional, i.e.
\[
\varphi_1\,{}_{\varphi_1}\kern-.5em*\kern-.3em{}_{\varphi_2}\,\varphi_2
= \varphi_1*\varphi_2.
\]
Suppose now that the algebras $\mathcal{A}_1$ and $\mathcal{A}_2$
have decompositions
$\mathcal{A}_i=\mathbb{C}\mathbf{1}\oplus\mathcal{A}_i^0$, $i=1,2$, as
a direct sum of vector spaces, such that the $\mathcal{A}_i^0$ are
even subalgebras. If one defines functionals
$\delta_i:\mathcal{A}_i\to\mathbb{C}$ by
\begin{equation}\label{delta}
\delta_i(\lambda\mathbf{1}+a_0)=\lambda
\end{equation}
for $\lambda\in\mathbb{C}$, $a_0\in\mathcal{A}_i^0$, $i=1,2$, then one
obtains the boolean product
\[
\varphi_1\,{}_{\delta_1}\kern-.5em*\kern-.3em{}_{\delta_2}\,\varphi_2=\varphi_1\diamond\varphi_2,
\]
cf.\ \cite{speicher+woroudi93,bozejko+leinert+speicher96}.
Since the conditionally free product of triples of the form
$(\mathcal{A},\varphi,\delta)$ can be shown to be again of the same
form, the commutativity and associativity of the boolean product
follow immediately from this construction.
One can also obtain the monotone product from the
conditionally free product.
\begin{proposition}
Let $(\mathcal{A}_1,\varphi_1)$ and $(\mathcal{A}_2,\varphi_2)$ be two
algebraic quantum probability spaces and assume $\mathcal{A}_1$ has a
decomposition $\mathcal{A}_1=\mathbb{C}\mathbf{1}\oplus\mathcal{A}_1^0$, where
$\mathcal{A}_1^0$ is a subalgebra of $\mathcal{A}_1$. Define a unital
functional $\delta_1:\mathcal{A}_1\to\mathbb{C}$ as in Equation
(\ref{delta}).
Then we have
\[
\varphi_1\triangleright\varphi_2=\varphi_1\,{}_{\delta_1}\kern-.5em*\kern-.3em{}_{\varphi_2}\,\varphi_2
\]
\end{proposition}
\Proof
Let $n\in\mathbb{N}$, $\epsilon(1),\ldots,\epsilon(n)\in\{1,2\}$ such
that $\epsilon(1)\not=\epsilon(2)\not=\cdots\not=\epsilon(n)$, and
$a_1\in\mathcal{A}_{\epsilon(1)},\cdots,a_n\in\mathcal{A}_{\epsilon(n)}$
such that $\delta_1(a_k)=0$ if $\epsilon(k)=1$ and $\varphi_2(a_k)=0$
if $\epsilon(k)=2$. This implies $a_k\in\mathcal{A}_1^0$ for
$\epsilon(k)=1$ and therefore by Equation (\ref{def-alg-mon})
\[
\varphi_1\triangleright\varphi_2(a_1a_2\cdots
a_n)=\prod_{k:\epsilon(k)=2} \varphi_2(a_k)=0
\]
(If the product $a_1a_2\cdots a_n$ does not begin or end with an
element of $\mathcal{A}_2$, add $\mathbf{1}\in\mathcal{A}_2$ in order
to apply Equation (\ref{def-alg-mon})).
Therefore $\varphi_1\triangleright\varphi_2$ satisfies condition
(\ref{def-cond-free}) that defines the conditionally free product
$\varphi_1\,{}_{\delta_1}\kern-.5em*\kern-.3em{}_{\varphi_2}\,\varphi_2$.
\endproof
With this observation, Muraki's formula \cite[Theorem 3.1]{muraki00} for the
additive monotone convolution can be deduced from the analytic theory of the
additive conditionally free convolution developed in \cite{bozejko+leinert+speicher96}.
\section{Products of monotonically independent operators.}\label{sec-op}
For a bounded operator $X$ in a quantum probability space
$(\mathcal{B}(\mathcal{H}),\mathcal{H},\Omega)$ we define
\[
\psi_X(z)=\left\langle \Omega, \frac{zX}{1-zX} \Omega\right\rangle
\]
and
\[
K_X(z)= \frac{\psi_X(z)}{1+\psi_X(z)}
\]
for $|z|<1/||X||$.
The following theorem is similar to \cite[Theorem 2.2]{bercovici04}. Below we
provide a new proof.
\begin{theorem}\label{thm-operators}
Let $(B(\mathcal{H}),\mathcal{H},\Omega)$ be a quantum probability
space and $\mathcal{A}_1,\mathcal{A}_2 \subseteq B(\mathcal{H})$ two
monotonically independent subalgebras. Let $V_1,V_2\in
\mathbb{C}\mathbf{1}+\mathcal{A}_1$, such that $V_2V_1-\mathbf{1}\in
\mathcal{A}_1$ and $W\in\mathcal{A}_2$.
Then we have
\[
K_{V_1WV_2}(z)=K_{V_1V_2}\big(K_W(z)\big)
\]
for all $|z|<\min(1/||V_1WV_2||,1/||W||)$.
\end{theorem}
\Proof
Let $M=\max\big(||V_1WV_2||,||W||(||V_1V_2||+2)\big)$ and $|z|<1/M$. Then we have
\begin{eqnarray*}
\frac{zV_1WV_2}{1-zV_1WV_2} &=& \sum_{n=1}^\infty (zV_1WV_2)^n = \sum_{n=1}^\infty z^n V_1 \underbrace{W (X+\mathbf{1}) W \cdots
W(X+\mathbf{1})} W V_2 \\
&&\hspace{55mm}n-1 \mbox{ times} \\
&=& \sum_{n=1}^\infty z^n \sum_{k=1}^n \sum_{{\nu_1,\ldots,\nu_k\ge 1}\atop{\nu_1+\cdots+\nu_k=n}} V_1
W^{\nu_1} X W^{\nu_2} X \cdots X W^{\nu_k} V_2,
\end{eqnarray*}
where $X=V_2V_1-\mathbf{1}$.
Using properties (a) and (b) in Definition \ref{def-mon-indep}, we get
\begin{eqnarray*}
\psi_{V_1WV_2}(z) &=& \left\langle\Omega,
\frac{zV_1WV_2}{1-zV_1WV_2}\Omega\right\rangle \\
&=& \sum_{n=1}^\infty z^n \sum_{k=1}^n \sum_{{\nu_1,\ldots,\nu_k\ge
1}\atop{\nu_1+\cdots+\nu_k=n}}\left\langle\Omega,V_1X^{k-1}V_2\Omega\right\rangle\left\langle\Omega,W^{\nu_1}\Omega\right\rangle\cdots\left\langle\Omega,W^{\nu_k}\Omega\right\rangle \\
&=& \sum_{k=1}^\infty
\left\langle\Omega,V_1(V_2V_1-\mathbf{1})^{k-1}V_2\Omega\right\rangle
\big(\psi_W(z)\big)^k \\
&=& \sum_{k=1}^\infty
\left\langle\Omega,V_1V_2(V_1V_2-\mathbf{1})^{k-1}\Omega\right\rangle
\big(\psi_W(z)\big)^k \\
&=&
\sum_{k=1}^\infty\left\langle\Omega,\psi_W(z)V_1V_2\frac{1}{\mathbf{1}-\psi_W(z)(V_1V_2-\mathbf{1})}\Omega\right\rangle
\\
&=&\sum_{k=1}^\infty\left\langle\Omega,\frac{\frac{\psi_W(z)}{1+\psi_W(z)}V_1V_2}{\mathbf{1}-\frac{\psi_W(z)}{1+\psi_W(z)}V_1V_2}\Omega\right\rangle
= \psi_{V_1V_2}\big(K_W(z)\big).
\end{eqnarray*}
By uniqueness of analytic continuation, we get
\[
K_{V_1WV_2}(z)=K_{V_1V_2}\big(K_W(z)\big)
\]
for all $|z|<\min(1/||V_1WV_2||,1/||W||)$.
\endproof
\begin{corollary}\label{cor-unitary}
Let $U,V$ be two unitary operators such that $U-\mathbf{1}$ and $V$
are monotonically independent with respect to $\Omega$. Then we have
\[
K_{UV}(z)=K_{VU}(z)=K_U\big(K_V(z)\big)
\]
for all $|z|\in\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$.
\end{corollary}
\begin{corollary}\label{cor-pos}
Let $X,Y$ be two positive operators such that $X-\mathbf{1}$ and $Y$
are monotonically independent with respect to $\Omega$. Then we have
\[
K_{\sqrt{X}Y\sqrt{X}}(z)=K_X\big(K_Y(z)\big)
\]
for all $|z|<\min(1/||\sqrt{X}Y\sqrt{X}||,1/||Y||)$.
\end{corollary}
\section{Multiplicative monotone convolution for probability measures on the
unit circle.}\label{sec-conv}
For a probability measure $\mu$ on $S^1$ we define
\[
\psi_\mu(z)=\int_{S^1} \frac{zx}{1-zx}{\rm d}\mu(x) \quad \mbox{ and }\quad K_\mu(z)=\frac{\psi_\mu(z)}{1+\psi_\mu(z)}
\]
for $z\in\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$.
We will call $K_\mu$ the {\em K-transform} of $\mu$, it characterizes the
measure $\mu$ completely. Furthermore, for a holomorphic function
$K:\mathbb{D}\to\mathbb{D}$ there exists a probability measure $\mu$ on the
unit circle $S^1$ such that $K=K_\mu$ if and only if $K(0)=0$. This follows
from the Herglotz representation theorem, the proof is similar to
\cite[Proposition 3.3]{franz04}.
It is clear that the composition of two K-transforms is again a K-transform of
some probability measure on the unit circle. In view of Corollary
\ref{cor-unitary} this suggests the following definition.
\begin{definition}
Let $\mu,\nu$ be two probability measures on $S^1$, with transforms $K_\mu$ and
$K_\nu$. Then the unique probability measure $\mu\kern0.3em\rule{0.04em}{0.52em}\kern-.35em\gtrdot\nu$ on $S^1$ with
\[
K_{\mu\kern0.17em\lower0.1ex\hbox{\rule{0.025em}{0.43em}}\kern-.105em\gtrdot\nu}=K_\mu\circ K_\nu
\]
is called the {\em monotone convolution} of $\mu$ and $\nu$.
\end{definition}
\begin{remark}
\begin{enumerate}
\item
The monotone convolution is weakly continuous.
\item
The monotone convolution is associative,
i.e.
\[
(\lambda\kern0.3em\rule{0.04em}{0.52em}\kern-.35em\gtrdot\mu)\kern0.3em\rule{0.04em}{0.52em}\kern-.35em\gtrdot\nu=\lambda\kern0.3em\rule{0.04em}{0.52em}\kern-.35em\gtrdot(\mu\kern0.3em\rule{0.04em}{0.52em}\kern-.35em\gtrdot\nu)
\]
for all
$\lambda,\mu,\nu$, but not commutative, i.e., in general
$\mu\kern0.3em\rule{0.04em}{0.52em}\kern-.35em\gtrdot\nu\not=\nu\kern0.3em\rule{0.04em}{0.52em}\kern-.35em\gtrdot\mu$.
\item
The Dirac measure $\delta_1$ at $1$ is a two-sided unit,
$\delta_1\kern0.3em\rule{0.04em}{0.52em}\kern-.35em\gtrdot\mu=\mu\kern0.3em\rule{0.04em}{0.52em}\kern-.35em\gtrdot\delta_1=\mu$ for all $\mu$. Right convolution by a
Dirac measure $\delta_x$ acts
as translation, i.e.\ $\mu\kern0.3em\rule{0.04em}{0.52em}\kern-.35em\gtrdot\delta_x=T_x\mu$, where $T_x:S^1\to S^1$ is
defined by $T_x(y)=xy$ for $x\in S^1$. But $\delta_x\kern0.3em\rule{0.04em}{0.52em}\kern-.35em\gtrdot\mu\not=T_x\mu$ in
general.
\item
The monotone convolution is affine in the first argument. Together with weak
continuity this implies the
following formula
\[
\mu\kern0.3em\rule{0.04em}{0.52em}\kern-.35em\gtrdot\nu = \int_{S^1} {\rm d}\mu(x) \delta_x\kern0.3em\rule{0.04em}{0.52em}\kern-.35em\gtrdot\nu.
\]
\end{enumerate}
\end{remark}
\section{L\'evy-Khintchine formula for monotone convolution semigroups.}\label{sec-levy}
We call a weakly continuous one-parameter family $(\mu_t)_{t\ge0}$ of
probability measures on the unit circle a
continuous monotone convolution semigroup, if
\[
\mu_0=\delta_1 \qquad \mbox{ and } \qquad \mu_s\kern0.3em\rule{0.04em}{0.52em}\kern-.35em\gtrdot\mu_t=\mu_{s+t}
\]
for all $s,t\ge 0$. By definition a one-parameter family
$(\mu_t)_{t\ge}$ is a continuous monotone convolution semigroup if and only if
the $K$-transforms $K_t=K_{\mu_t}$, $t\ge 0$ form a continuous semigroup
w.r.t.\ to composition. The continuity of the $K$-transforms is uniform in $z$
on compact sets. Our main tool for characterizing continuous monotone
convolution semigroups will be Berkson and Porta's \cite{berkson+porta78}
characterisation of composition semigroups of holomorphic maps.
\begin{theorem}\label{thm-levy-khintchine-circle} \cite[Theorem 4.6]{bercovici04}
Let $(\mu_t)_{t\ge 0}$ be a weakly continuous family of probability measures
on the unit circle, with K-transforms $(K_t)_{t\ge 0}$. Then the following are
equivalent.
\begin{description}
\item[(a)]
$(\mu_t)_{t\ge 0}$ is a continuous monotone convolution semigroup.
\item[(b)]
$(K_t)_{t\ge 0}$ is a continuous semigroups w.r.t.\ to composition.
\item[(c)]
There exists a holomorphic function $u:\mathbb{D}\to\mathbb{C}$ with $\Re\,
u(z)\ge 0$ for $z\in\mathbb{D}$ such that
$(K_t)_{t\ge 0}$ is the (unique) solution of
\[
\frac{{\rm d}K_t(z)}{{\rm d}t} = -K_t(z)u\big(K_t(z)\big)
\]
for $z\in\mathbb{D}$ and $t\ge 0$, with initial condition $K_0(z)=z$.
\end{description}
\end{theorem}
\Proof
The equivalence between (a) and (b) follows from the definition
and the continuity properties of the monotone convolution.
The equivalence between (b) and (c) is an immediate consequence of
\cite[Theorem (3.3)]{berkson+porta78}, it suffices to identify the fixed point
at zero as the Denjoy-Wolff point of the $K_t$.
\endproof
\begin{remark}
\begin{enumerate}
\item
The function $u$ in (c) can be computed from the derivative of $(K_t)_{t\ge
0}$ in $t=0$ by
\[
u(z) = -\frac{1}{z}\left.\frac{{\rm d}}{{\rm d}t}\right|_{t=0} K_t(z),
\]
we will call it the {\em generator} of $(K_t)_{t\ge 0}$.
\item
Such a function $u$ has a unique Herglotz representation
\[
u(z)=ib + \int_{S^1} \frac{w+z}{w-z}{\rm d}\rho(w),
\]
where $b$ is a real number and $\rho$ a finite measure on $S^1$.
\end{enumerate}
\end{remark}
\section{Relation to Galton-Watson processes.}\label{sec-galton}
A probability measure $\mu$ on the unit circle is called infinitely divisible
w.r.t.\ to the monotone convolution, if for all $n\in\mathbb{N}$ there exists
a probability measure $\mu_n$ on the unit circle such that
\begin{eqnarray*}
\mu&=&\underbrace{\mu_n\kern0.3em\rule{0.04em}{0.52em}\kern-.35em\gtrdot\cdots\kern0.3em\rule{0.04em}{0.52em}\kern-.35em\gtrdot\mu_n}. \\
&& \qquad n\mbox{ times}
\end{eqnarray*}
Bercovici has shown in \cite[Theorem 4.7]{bercovici04} that all infinitely
divisible probability measures can be embedded into a continuous monotone
convolution semigroup, i.e.\ if $\mu$ is infinitely divisible w.r.t.\ to the
monotone convolution, then there exists a continuous monotone convolution
semigroup $(\mu_t)_{t\ge 0}$ such that $\mu=\mu_t$ for some $t\ge0$. And from
the previous
section it is clear this implies that the
K-transform $K_\mu$ can be embedded into a continuous composition semigroup of K-transforms.
A similar problem has been studied in the theory of Galton-Watson processes.
Let $X_{n,k}$, $n,k=1,2,\ldots$ be independent, identically distributed random
variables with values in $\mathbb{N}$ with generating function
\[
\varphi(z)=\mathbb{E}(z^{X_{n,k}})=\sum_{m=0}^\infty p_m z^m,\quad \text{ for }z\in\mathbb{D}
\]
where $p_m=\mathbb{P}(X_{n,k}=m)$. Then the associated Galton-Watson process $(Y_n)_{n\ge 0}$ is defined by
$Y_0=1$, and
\[
Y_{n+1}=\sum_{k=1}^{Y_n} X_{n,k}, \qquad \mbox{ for } n\ge 1.
\]
This process describes the evolution of a population where after each step
each individual produces a random number of offspring according to the
probabilities $(p_m)_{m\ge 0}$.
Its generating functions form a discrete composition semigroup,
\[
\mathbb{E}(z^{Y_n})=\varphi^n(z),\qquad \text{ for }z\in\mathbb{D},\quad n\in\mathbb{N}.
\]
If $\mathbb{P}(X_{n,k}=0)=0$ (i.e.\ no individual dies without offspring), then
$\varphi(0)=0$ and $\varphi$ is the $K$-transform of a probability measure
$\mu$ on $S^1$. If $(Y_n)_{n\ge 0}$ can be embedded into a continuous-time
Markovian branching process (or equivalently, if $(\varphi^n)_{n\ge 0}$ can be embedded into a
continuous composition semigroup $(\varphi_t)_{t\ge0}$ of generating
functions), then $\mu$ is infinitely divisible for the monotone convolution
and can be embedded into a continuous monotone convolution semigroup. The problem of embedding Galton-Watson processes has been studied by Gorya\u{\i}nov\cite{goryainov93,goryainov00}.
\begin{example}
Continuous-time Markovian branching processes with extinction probability $0$ can be obtained by choosing infinitesimal offspring probabilities $\lambda_j\ge 0$ for $j\ge 2$ such
that $\alpha=\sum_{j=2}^\infty \lambda_j<\infty$, setting
\[
v(z)=\sum_{j=2}^\infty \lambda_jz^j-\alpha z, \qquad \mbox{ for } |z|\le 1,
\]
and solving the differential equation
\[
\frac{{\rm d}}{{\rm d}t}\varphi_t(z) = v\big(\varphi_t(z)\big)
\]
with initial condition $\varphi_0(z)=z$, cf.\ \cite[Theorem 4]{goryainov93}.
A simple example is the Yule process, where $v(z)=\alpha(z^k-z)$ and
\[
\varphi_t(z) =\frac{ze^{-\alpha
t}}{\sqrt[k-1]{1-\left(1-e^{-\alpha(k-1)t}\right)z^{k-1}\,}}, \qquad t\ge 0,
\]
for some $k\in\mathbb{N}$, $k\ge 2$. This process describes a population were
the individuals are replaced by $k$ new individuals after an exponentially
distributed random time.
\end{example}
\section{On the embedding of probability measures into continuous monotone
convolution semigroups.}\label{sec-embed}
\cite[Theorem 6]{goryainov93} and \cite[Theorem 7]{goryainov93} characterize
probability generating functions that can be embedded into composition
semigroups of probability generating functions. In this section we give a similar characterization for K-transforms
of probability measures on the unit circle that can be embedded into continuous
monotone convolution semigroups.
Let $(K_t)_{t\ge 0}$ be a continuous composition semigroups of
K-transforms. By \cite{berkson+porta78}, $K_t$ is differentiable w.r.t.\ $t$
and satisfies the differential equation
\begin{equation}\label{eq-4}
\frac{{\rm d}}{{\rm d}t} K_t(z) = v\big(K_t(z)\big) = v(z)K'_t(z)
\end{equation}
for $t\ge 0$, $z\in\mathbb{D}$, with $v$ given by
\[
v(z)=\left.\frac{{\rm d}}{{\rm d}t}\right|_{t=0} K_t(z).
\]
This equation follows from the semigroup property $K_{s+t}=K_s\circ
K_t=K_t\circ K_s$ by differentiation w.r.t.\ $s$ at $s=0$.
By Theorem \ref{thm-levy-khintchine-circle}, the function $v$ is of the form
$v(z)=-zu(z)$, with a holomorphic function $u:\mathbb{D}\to\mathbb{C}$ such
that $\Re\,u(z)\ge 0$ for $z\in\mathbb{D}$.
We will need the following lemma.
\begin{lemma}\label{lemma-unique}
Let $u:\mathbb{D}\to\mathbb{C}$, $u\not\equiv0$, be a holomorphic function such
that $\Re\,u(z)\ge 0$ for $z\in\mathbb{D}$ and set $\beta=u(0)$, $v(z)=-zu(z)$
for $z\in\mathbb{D}$.
Then, for all $t\ge 0$, the equation
\begin{equation}\label{eq-5}
v\big(f(z)\big) = v(z)f'(z), \qquad z\in\mathbb{D},
\end{equation}
has a unique solution $f$ with $f'(0)=e^{-t\beta}$.
\end{lemma}
\Proof
The proof of this lemma is borrowed from \cite[Lemma 2]{goryainov93}.
Let $(K_t)_{t\ge 0}$ be a composition semigroup of K-transforms with generator
$u$. Then all $K_t$, $t\ge 0$ satisfy Equation (\ref{eq-5}). Furthermore, the
differential equation that the $K_t$ satisfy, implies
\[
\frac{{\rm d}}{{\rm d}t}K'_t(0)=-u(0)K'_z(0)
\]
and therefore $K'_t(0)=e^{-t\beta}$, since $K_0(z)=z$ and $K'_0(0)=1$. This proves existence.
Let now $f$ be an arbitrary solution of Equation (\ref{eq-5}) with
$f'(0)=e^{-t\beta}$. Since $v$ has no zeros inside $\mathbb{D}$ other than
$z=0$, we get $f(0)=0$ by substituting $z=0$ into Equation (\ref{eq-5}).
Differentiation Equation (\ref{eq-5}) $k$ times, we can calculate the higher
derivatives of $f$ at zero from $f'(0)=e^{-t\beta}$ and the derivatives of $v$
at zero. This proves uniqueness.
\endproof
\begin{remark}
Let $(K_t)_{t\ge0}$ be the K-transforms of a continuous monotone convolution semigroup
$(\mu_t)_{t\ge 0}$ with generator $u$. Then $K'_t(0)=e^{-tu(0)}$ is the first
moment of $\mu_t$, i.e.\
\[
e^{-tu(0)}=\int_{S^1} x {\rm d}\mu_t, \qquad \mbox{ for }t\ge 0.
\]
\end{remark}
We come to the main result of this section.
\begin{theorem}
Let $\mu$ be a probability measure on the unit circle $S^1$ that is not
concentrated in one point.
Then $\mu$ can be embedded into a continuous monotone convolution semigroup if and only
if $K'_\mu(z)\not=0$ for all $z\in\mathbb{D}$ and there exists a locally
uniform limit
\[
\lim_{n\to\infty} -\frac{K^n_\mu(z)}{(K^n_\mu)'(z)}=v(z),
\]
in $\mathbb{D}$ that is of the form $v(z)=\alpha z u(z)$ with a non-zero constant
$\alpha\in\mathbb{C}$ and a holomorphic function
$u:\mathbb{D}\to\mathbb{C}$ such that $\Re\,u(z)\ge 0$ for $z\in\mathbb{D}$
and $K'_\mu(0)=e^{-t_0u(0)}$ for some some $t_0\ge 0$.
\end{theorem}
\Proof
The proof of this theorem is similar to that of \cite[Theorem 6]{goryainov93}.
Suppose that $\mu$ can be embedded into a continuous monotone convolution semigroup. Then $K_{\mu}$ can be embedded into a
composition semigroup of K-transforms $(K_t)_{t\ge 0}$. Therefore all $K_t$ are
injective and $K'_t(z)\not=0$ for all $z\in\mathbb{D}$, $t\ge0$, cf.\
\cite{berkson+porta78}. Denote by $u$ the generator of $(K_t)_{t\ge 0}$ and
define $v$ by $v(z)=-zu(z)$ for $z\in\mathbb{D}$. By the Denjoy-Wolff theorem we get $\lim_{t\to\infty}K_t(z)=0$ and $\lim_{t\to\infty}K'_t(z)=0$
locally uniformly for all $z\in\mathbb{D}$. Therefore
\[
\lim_{t\to \infty} \frac{v\big(K_t(u)\big)}{K_t(z)}=v'(0)=-u(0).
\]
With the right-hand-side of Equation (\ref{eq-4}) this implies
\[
\lim_{n\to\infty} -\frac{K^n_\mu(z)}{(K^n_\mu)'(z)}=\lim_{t\to\infty}
-\frac{K_t(z)}{K'_t(z)}
=\lim_{t\to\infty}-\frac{K_t(z)v(z)}{v\big(K_t(z)\big)}= -\frac{v(z)}{v'(0)}=-z\frac{u(z)}{u(0)}.
\]
The limit is of the form required in the theorem with the constant $\alpha=-1/u(0)$.
To show the converse, let now $K_\mu$ be a K-transform satisfying the
conditions of the theorem with $v(z)=\alpha z u(z)$, $\alpha$ and $u$ as described in the theorem.
Let $(K_t)_{t\ge 0}$ be the composition semigroup of K-transforms with
generator $u$. Then the $K_t$ satisfy
\[
v\big(K_t(z)\big) = v(z)K'_t(z), \qquad \mbox{ for } t\ge 0, \quad z\in\mathbb{D}.
\]
The conditions of the theorem imply that $K_\mu$ is also a solution of the same
equation,
\[
v(z)=\lim_{t\to\infty} -\frac{K_\mu^{n+1}(z)}{(K_\mu^{n+1})'(z)}
=\lim_{t\to\infty} -\frac{K_\mu^n\big(K_\mu(z)\big)}{K'_\mu(z)(K_\mu^n)'(z)} =
\frac{v\big(K_\mu(z)\big)}{K'_\mu(z)}.
\]
The uniqueness in Lemma \ref{lemma-unique} now implies $K_\mu=K_{t_0}$.
\endproof
\begin{remark}
Let $\mu=\delta_x$ be concentrated in one point $x=e^{i\varphi}\in S^1$. Then
we get $\psi_{\delta_x}(z)=\frac{xz}{1-xz}$ and $K_\mu(z)=e^{i\varphi}z$ and
$\mu$ can be embedded into the continuous convolution semigroups
$(\mu_t^{(k)})_{t\ge 0}$ given by $\mu^{(k)}_t=\delta_{e^{it(\varphi+2\pi
k)}}$, $k\in\mathbb{Z}$.
\end{remark}
\section{Appendix: Multiplicative monotone convolution for
probability measures on $\mathbb{R}_+$.}
Just as there are many different ways to define multiplicatively a positive
operator from two given positive operators, there are different definitions of multiplicative monotone
convolutions of two probability measures $\mu$ and $\nu$ on
$\mathbb{R}_+$. Two possible choices are to take positive self-adjoint operators $X$ and $Y$, whose distributions
are given by $\mu$ and $\nu$, resp., such that $X-\mathbf{1}$ and
$Y-\mathbf{1}$ are monotonically independent, and to define the
convolution of $\mu$ and $\nu$
as the distributions of $\sqrt{X}Y\sqrt{X}$ or
$\sqrt{Y}X\sqrt{Y}$.
By Corollary \ref{cor-pos} the K-transform of $\sqrt{X}Y\sqrt{X}$ is equal to
the composition of the K-transforms of $X$ and $Y$. Therefore this gives a
definition which is equivalent to the one chosen by Bercovici, cf.\
\cite{bercovici04}.
We will show below that choosing the distribution of $\sqrt{Y}X\sqrt{Y}$
as the convolution of the distributions of $X$ and $Y$ leads to an
inequivalent definition.
The operators $\sqrt{X}Y\sqrt{X}$ and
$\sqrt{Y}X\sqrt{Y}$ have the same spectrum, except for $0$. More
precisely,
$\sigma(\sqrt{X}Y\sqrt{X})\backslash\{0\}=\sigma(\sqrt{Y}X\sqrt{Y})\backslash\{0\}$,
since $\sqrt{X}Y\sqrt{X}=AB$ and $\sqrt{Y}X\sqrt{Y}=BA$ with
$A=\sqrt{X}\sqrt{Y}$ and $B=\sqrt{Y}\sqrt{X}$.
But the following example shows that, unlike in the free case where one
works with tracial states, here the distributions of
$\sqrt{X}Y\sqrt{X}$ and $\sqrt{Y}X\sqrt{Y}$ are in general different
and therefore we have two different multiplicative monotone
convolutions for probability measures on $\mathbb{R}_+$.
\begin{example}
Consider the positive definite $2\times 2$-matrix
\[
M(a)=\left(\begin{array}{cc} 1 & a \\ a &
1\end{array}\right)=\frac{1}{\sqrt{2}}\left(\begin{array}{cc} 1 & 1
\\ 1 & -1 \end{array}\right)\left(\begin{array}{cc} 1+a & 0
\\ 0 & 1-a \end{array}\right)\frac{1}{\sqrt{2}}\left(\begin{array}{cc} 1 & 1
\\ 1 & -1 \end{array}\right),
\]
with $a\in(0,1)$. Then we have
\[
\left\langle\left(\begin{array}{c} 0 \\ 1 \end{array}\right), A^k
\left(\begin{array}{c} 0 \\ 1 \end{array}\right)\right\rangle =
\frac{1}{2}\left((1-a)^k+(1+a)^k\right)
\]
for $k\in\mathbb{N}$, i.e.\ the distribution of $A$ in the vector
state given by $\omega=\left(\begin{array}{c} 0 \\ 1
\end{array}\right)$ is equal to
$\frac{1}{2}(\delta_{1-a}+\delta_{1+a})$.
A simple calculation yields
\begin{equation}\label{sqrt}
\sqrt{M(a)}=\frac{1}{2}\left(\begin{array}{ccc} \sqrt{1+a}+\sqrt{1-a} & & \sqrt{1+a}-\sqrt{1-a}
\\ \sqrt{1+a}-\sqrt{1-a} & & \sqrt{1+a}+\sqrt{1-a} \end{array}\right).
\end{equation}
Let $a,b\in(0,1)$ and consider the pair of positive definite matrices
\begin{eqnarray*}
X&=&\mathbf{1}\otimes\mathbf{1}+\big(M(a)-\mathbf{1}\big)\otimes
P_\omega = \left(\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 1 & a \\
0 & 0 & a & 1
\end{array}\right), \\
Y&=&\mathbf{1}\otimes M(b)
= \left(\begin{array}{cccc}
1 & 0 & b & 0 \\
0 & 1 & 0 & b \\
b & 0 & 1 & 0 \\
0 & b & 0 & 1
\end{array}\right),
\end{eqnarray*}
in $\mathcal{M}_2(\mathbb{C})\otimes
\mathcal{M}_2(\mathbb{C})\cong\mathcal{M}_4(\mathbb{C})$ where
$P_\omega$ denotes the orthogonal projection onto
$\omega=\left(\begin{array}{c} 0 \\ 1 \end{array}\right)$. With
respect to the vector state given by $\omega\otimes \omega$, $X-\mathbf{1}\otimes\mathbf{1}$ and $Y-\mathbf{1}\otimes\mathbf{1}$ are monotonically independent, with
distributions given by $\frac{1}{2}(\delta_{1-a}+\delta_{1+a})$ and
$\frac{1}{2}(\delta_{1-b}+\delta_{1+b})$, respectively.
As in Equation (\ref{sqrt}), we compute
\begin{eqnarray*}
\sqrt{X} &=& \left(\begin{array}{ccccccc}
1 & & 0 & & 0 & & 0 \\
0 & & 1 & & 0 & & 0 \\
0 & & 0 & & \frac{\sqrt{1+a}+\sqrt{1-a}}{2} & & \frac{\sqrt{1+a}-\sqrt{1-a}}{2} \\
0 & & 0 & & \frac{\sqrt{1+a}-\sqrt{1-a}}{2} & &
\frac{\sqrt{1+a}+\sqrt{1-a}}{2}\end{array}\right), \\
\sqrt{Y} &=& \frac{1}{2}\left(\begin{array}{ccccccc}
\sqrt{1+b}+\sqrt{1-b} & & 0 & & \sqrt{1+b}-\sqrt{1-b} & & 0 \\
0 & & \sqrt{1+b}+\sqrt{1-b} & & 0 & & \sqrt{1+b}-\sqrt{1-b} \\
\sqrt{1+b}-\sqrt{1-b} & & 0 & & \sqrt{1+b}+\sqrt{1-b} & & 0 \\
0 & & \sqrt{1+b}-\sqrt{1-b} & & 0 & &
\sqrt{1+b}+\sqrt{1-b}\end{array}\right).
\end{eqnarray*}
The eigenvalues of both $\sqrt{X}Y\sqrt{X}$ and $\sqrt{X}Y\sqrt{X}$ are
\begin{eqnarray*}
\lambda_1 &=& 1+\frac{a}{2}+\frac{1}{2}\sqrt{a^2+4(1+a)b^2}, \\
\lambda_2 &=& 1+\frac{a}{2}-\frac{1}{2}\sqrt{a^2+4(1+a)b^2}, \\
\lambda_3 &=& 1-\frac{a}{2}+\frac{1}{2}\sqrt{a^2+4(1-a)b^2}, \\
\lambda_4 &=& 1-\frac{a}{2}-\frac{1}{2}\sqrt{a^2+4(1-a)b^2},
\end{eqnarray*}
and therefore their distributions have the same support. But
their distributions in the
vector state $\omega=\left(\begin{array}{c} 0 \\ 0 \\ 0 \\ 1
\end{array}\right)$ are different.
For example their second moments differ,
\begin{eqnarray*}
\langle\omega,\left(\sqrt{X}Y\sqrt{X}\right)^2\omega\rangle &=&
1+b^2+a^2, \\
\langle\omega,\left(\sqrt{Y}X\sqrt{Y}\right)^2\omega\rangle &=&
1+b^2+\frac{a^2}{2}\left(1+\sqrt{1-b^2}\right),
\end{eqnarray*}
(recall that we assumed $a\not=0$, $b\not=0$).
\end{example}
\section*{Acknowledgements.}
I presented the results of this paper, in particular Theorems \ref{thm-operators} and \ref{thm-levy-khintchine-circle} at
the conference ``Quantum Probability and Infinite Dimensional Analysis'' in
B\c{e}dlewo, Poland, in June 2004. I wish to thank Marek Bo{\.z}ejko and Janusz
Wysoczanski who indicated Reference \cite{bercovici04} to me.
I am also indebted to W.\ Hazod for bringing Gorya\u{\i}nov's work
\cite{goryainov93,goryainov00} to my attention.
|
{
"timestamp": "2005-03-25T16:28:22",
"yymm": "0503",
"arxiv_id": "math/0503602",
"language": "en",
"url": "https://arxiv.org/abs/math/0503602"
}
|
\section{INTRODUCTION}
In recent years entanglement has been recognized as a physical resource central to quantum information processing. As a result, a remarkable research effort has been devoted to classifying and quantifying it. The first achievement in this direction was the identification of the entropy of entanglement \cite{bennett1}, $E_{E}$, as the unique measure of entanglement for pure bipartite states in the asymptotic limit. It was shown that that $m$ copies of a pure state $\ket{\psi}$ can be reversibly converted into $n$ copies of $\ket{\phi}$ by local operations and classical communication (LOCC) if, and only if, $mE_{E}(\ket{\psi}) = nE_{E}(\ket{\phi})$. This reversibility is lost however when one considers the more general picture of mixed states. In this case two different entanglement measures, associated with the formation and distillation processes, respectively, have to be taken into account. On one hand the entanglement cost, $E_{C}(\rho)$ \cite{bennett1}, is the minimal number of singlets necessary to create the state $\rho$ by LOCC in the asymptotic regime. On the other, the distillable entanglement, $E_{D}(\rho)$ \cite{bennett1}, is the maximum number of singlets that can be extracted by LOCC from $\rho$. Another important measure connected to asymptotic properties is the relative entropy of entanglement, $E_{R}$ \cite{vedral1}. It is related to how distinguishable an entangled state is from a separable one and gives bounds to $E_{C}$ and $E_{F}$.
The finite copy case is more complex and the entropic quantities considered above are not applicable anymore. For bipartite pure states, where the reversibility is already lost, the minimum set of entanglement measures characterizing deterministic and probabilistic transformations were derived \cite{nielsen1,vidal1,jonathan1}. The mixed case, however, is known only for very restricted situations and remains mainly unsolved.
Another approach for the quantification of entanglement is to measure the usefulness of a state to perform a given quantum information task. For example, the maximal fidelity of teleportation achieved by single copy LOCC \cite{horodecki1}, the maximal secret-key rate attainable by local measurements in a cryptographic protocol \cite{curty1} and the capacity of dense coding \cite{bruss1}, despite not being equal to any of the measures discussed so far, are clearly the best quantifiers when one of these protocols is analyzed.
Entanglement in multi-partite systems exhibits a much richer structure than the bipartite case and its study is even more challenging. Already in the pure three qubits case, there are two different manners for a state to be entangled, in the sense that there are states that cannot be converted, even with a certain probability, in each other \cite{dur1}. From the measures considered above only $E_{R}$ is unambiguously defined to multi-partite systems, although it is not the only one.
It is thus clear that entanglement is a highly complex phenomenon, which cannot be quantified by only one measure. Then, a natural way to measure it is to use any quantie which satisfies some particular properties, being the monotonicity under LOCC the most important \cite{vidal2,vedral1}. In this axiomatic approach any measure which does not increase, on average, under LOCC, called an entanglement monotone \cite{vidal2}, is a good measure of entanglement and, conversely, any meaningful quantifier has to be an entanglement monotone, or at least has some sort of weaker monotonicity under LOCC.
A closely related problem to the quantification problem is the characterization of entanglement. The very fundamental question whether a given mixed state is entangled or not is extremely difficult, being actually NP-HARD \cite{gurvits}. A possible approach is then to consider sufficient criteria for entanglement, such as the Peres-Horodecki \cite{peres} and the alligment \cite{chen} tests. Nonetheless, the strongest manner to characterize entanglement is using entanglement witnesses (EW) \cite{horodecki2,terhal1}. They are Hermitian operators whose expectation value is positive in every separable state. Therefore, a negative expectation value in a measurement of a witness operator in an arbitrary state is a direct indication of entanglement in this state. Furthermore, it was shown by the Horodeckis that a state is entangled if, and only if, it is detected by an EW \cite{horodecki2}. A great deal of research has been devoted to the study of EWs, varying from the their classification and optimization \cite{lewenstein1,terhal2,terhal1} to their use in the characterization of entanglement in important, even macroscopic, physical systems \cite{toth1,brukner1,wu1}. Also optimal set-ups for local measurements of witnesses \cite{toth2,guhne1} and experimental realizations of witnessing entanglement were realized \cite{bou}. In spite of the determination of EWs for all states being also computationally intractable \cite{brandao1}, different methods from convex optimization theory can be applied to the problem, leading to efficient approximative procedures to determine and even optimize EWs for arbitrary states \cite{brandao1,doherty1,eisert1}.
The main objective of this paper is to show that EWs can be very helpful also in the quantification of entanglement. The first measure related to EWs was due to Bertlmann {\it et al} \cite{bert} and was shown to be equal to the Hilbert-Schmidt distance from the set of separable states. Brandao and Vianna \cite{brandao2} took another significant step in this direction, showing that a measure derived from optimal EWs of the most studied group of witnesses so far, the group of EWs with unit trace, was in fact equal to the random robustness, which led to the establishment of properties still unknown for the later, such as its monotonicity under separable trace-preserving superoperators.
Besides the obvious benefit of increasing the number of entanglement measures known, EWs based quantifiers are particular interesting due to the possibility of performing experimental measurements of them, which could be important to the extension of entanglement to other areas of physics, such as thermodynamics and statistical mechanics. Moreover, despite being necessary in general a complete tomography of a state to the determination of its degree of entanglement based on an EW measure, any EW provides a lower bound to it, even when no information at all about the state is available.
The paper is structured as follows. In Sec. II we briefly review the basic properties of multi-partite optimal entanglement witnesses. In Sec. III we define a class of entanglement measures based on EWs, which includes several important already known quantities such as the negativity and the concurrence, and introduce a new infinite family of entanglement monotones having the generalized robustness and the best separable approximation measure as its limits. In Sec IV we present further properties of the considered measures and relate them to the localizable entanglement. In Sec. V it is shown how the methods developed in the last years to the characterization of entanglement based on convex optimization can be used to calculate approximately a large number of measures based on EWs. In Sec. VI possible extensions of our approach to Gaussian states are discussed. In Sec. VII we consider how the measures and their calculation are modified in states with symmetries. In Sec. VIII the questions of the amount of entanglement and of nonlocality in the presence of a super-selection rule are answered from the perspective of the studied measures. In Sec. IX it is shown that the three most successful approaches to the quantification of entanglement in systems of indistinguishable particles can be easily accessed from the EWs based quanties. In Sec. X and XI the questions of bounds on the teleportation distance and on the distillable entanglement of a given quantum state is review using our measures. It is shown that they provide sharper bounds than the negativity for the majority of states. In Sec. XII we derive lower bounds to the entanglement of formation with any EW. Possible applications of the measures are exemplified in Sec. XIII, where the derivation of two thermodynamic \textit{equations of state} which take into account entanglement is presented. Finally, in Sec. XIV we summarize our results and discuss future perspectives.
\section{MULTIPARTITE SYSTEMS AND OPTIMAL ENTANGLEMENT WITNESSES}
We consider a system shared by $N$ parties ${\cal f} A_{i} {\cal g}_{i=1}^{N}$. Following \cite{dur2}, we call a $k-$partite split a partition of the system into $k \leq N$ sets ${\cal f} S_{i} {\cal g}_{i=1}^{k}$, where each may be composed of several original parties. Given a density operator $\rho_{1...k} \in {\cal B}(H_{1} \otimes ... \otimes H_{k})$ (the Hilbert space of bounded operators acting on $H_{1}\otimes ... \otimes H_{k}$) associated with some $k-$partite split, we say that $\rho_{1...k}$ is a $m$-separable state if it is possible to find a convex decomposition for it such that in each pure state term at most $m$ parties are entangled among each other, but not with any member of the other group of $n - m$ parties. For example, every 1-separable state, also called fully-separable, can be written as:
\begin{equation}
\rho_{1...k} = \sum_{i}p_{i} \ket{\psi_{i}}_{1}\bra{\psi_{i}}\otimes ... \otimes \ket{\psi_{i}}_{k}\bra{\psi_{i}}.
\end{equation}
Another example is the 2-separable states of a 3-partite split given by:
\begin{equation}
\rho_{1:2:3} = \sum_{i}p_{i}\rho_{i},
\end{equation}
where each $\rho_{i}$ is separable with respect to at least one of the three possible partitions (A:BC, AB:C and AC:B). For each kind of separable state there is a different kind of entanglement associated to it. We will say that a state is $(m+1)$-partite entangled if it is not $m$-separable. It is clear that if a state is $m$-separable it cannot be $n$-entangled for all $n > m$.
It is possible to detect ($m$+1)-partite entanglement using entanglement witnesses. In order to do that, consider the index set $P = {\cal f}1, 2, ..., k{\cal g}$. Let $P^{m}$
be a subset of $P$ which has at most $m$ elements. Then $W$ is a $(m+1)$-partite entanglement witness if
\begin{center}
\begin{equation}
\begin{array}{c}
_{P^{m}_{v}}\bra{\psi}\otimes ... \otimes \hspace{0.07
cm}_{P^{m}_{1}}\bra{\psi}W\ket{\psi}_{P^{m}_{1}} \otimes ... \otimes
\ket{\psi}_{P^{m}_{v}} \geq 0 \\ \\
\forall \hspace{0.2 cm} P^{m}_{1}, ..., P^{m}_{v} \hspace{0.2 cm} $such
that$ \\ \\
\bigcup_{k=1}^{v}P^{m}_{k} = P \hspace{0.2 cm} $and$ \hspace{0.2 cm}
P^{m}_{k} \bigcap P^{m}_{l} = {\cal f}{\cal g}.
\end{array}
\end{equation}
\end{center}
Equation $(3)$ assures that the operator $W$ is positive for all
$m$-separable states. Thus, as the subspace of $m$-separable density operators is convex and closed, a state $\rho$ is $(m+1)$-entangled if and only if there is a Hermitian operator satisfying equation (3) such that $Tr(W\rho) < 0$ \cite{horodecki3}.
Usually one is interest in a selected group of witnesses operators called {\it optimal}. Two different definitions of optimal entanglement witness (OEW) exist. The first, introduced by Lewenstein {\it et al} \cite{lewenstein1}, is based on how much entangled states a given entanglement witness (EW) $W$ is able to detect: $W$ is optimal iff there is no other EW which detects all the states detected by $W$ and some other states not detected by $W$. The second definition, due to Terhal \cite{terhal2}, establish the concept of OEW relative to a chosen entangled state $\rho$. The $\rho-$optimal entanglement witness $W_{\rho}$ is given by
\begin{equation}
Tr(W_{\rho}\rho) = \min_{W \in {\cal M}} Tr(W\rho),
\end{equation}
where ${\cal M}$ is the intersection of the set of entanglement witnesses, denoted by ${\cal W}$, with some other set ${\cal C}$ such that ${\cal M}$ is compact \cite{foot1}. Note that every $\rho$-OEW is also an OEW accordingly to the first definition, whereas the converse may not be true.
A general expression for entanglement witnesses was presented in \cite{lewenstein2}. Every EW acting on $k$-partite Hilbert space can be written as:
\begin{equation}
W = P + \sum_{i=1}^{k}Q_{i}^{T_{i}} - \epsilon I,
\end{equation}
where $P$ and the $Q_{i}$'s are positive semi-definite, $\epsilon \geq 0$, $I$ is the identity operator and $T_{i}$ is the partial transposition with respect to partie $i$. Note that even ($m$+1)-partite EWs can be written in the form of equation (5) \cite{foot2}. An important class of EW is the decomposable entanglement witnesses (d-EW), which can always be written as:
\begin{equation}
W = P + \sum_{i=1}^{k}Q_{i}^{T_{i}}.
\end{equation}
This class will be particularly important in our discussion, since the set of entangled states detected by d-EW is invariant under LOCC \cite{doherty2}.
\section{Definitions and basic properties}
In this section we show how $\rho$-optimal EWs can be used to quantify all the different kinds of multipartite entanglement. First, an unifying approach, which includes several important entanglement measures (EM), will be presented. Then we will consider a new infinite family of entanglement monotones \cite{vidal2}.
A general expression for the quantification of entanglement via EWs is defined as:
\begin{equation}
E(\rho) = \max {\cal f}0, -\min_{W \in {\cal M}} Tr(W\rho) {\cal g},
\end{equation}
where ${\cal M} = {\cal W} \cap {\cal C}$, and the set ${\cal C}$ is what distinguish the quantities. We call {\it witnessed entanglement} any measure expressed by equation (7).
Some well known EM can be expressed as (7). The first, introduced by Bertlmann {\it et al} \cite{bert}, is:
\begin{equation}
B(\rho) = \max_{||W - I||_{2} \leq 1}[\min_{\sigma \in {\cal S}}Tr(W\sigma) - Tr(W\rho)],
\end{equation}
where $W \in {\cal W}$. $B(\rho)$ was shown to be monotonic decreasing under mixing enhancing maps \cite{hayden1} and to be equal to the ${\cal H}_{s}$-distance of $\rho$ to the set ${\cal S}$ of fully-separable states:
\begin{equation}
B(\rho) = D(\rho) = \min_{\sigma \in {\cal S}}||\rho - \sigma||_{2}.
\end{equation}
The second is the negativity, i.e., the sum of the negative eigenvalues of $\rho^{T_{A}}$ (the partial transpose of $\rho$ with respect to subsystem A) \cite{vidal3,eisert2,zyc}. It is easily seen that ${\cal N}$ can be written as:
\begin{equation}
{\cal N}(\rho) = \max {\cal f} 0, - \min_{0 \leq W \leq I}Tr(W^{T_{A}}\rho) {\cal g}.
\end{equation}
Another quantie is the maximal fidelity of distillation under PPT-protocols, introduced by Rains \cite{rains1},
\begin{equation}
F_{d}(\rho) = \frac{I}{d} + \max {\cal f} 0, - \min_{W \in {\cal M}}Tr(W^{T_{A}}\rho) {\cal g},
\end{equation}
where ${\cal M} = {\cal f}W \hspace{0.1 cm}| \hspace{0.1 cm} (1 - d)I/d \leq W \leq I/d, \hspace{0.3 cm} 0 \leq W^{T_{A}} \leq 2I/d {\cal g}$ \cite{foot3}.
The last is the celebrated Wootter's concurrence of two qubits, which can be written, accordingly to Verstraete \cite{verstraete1}, as
\begin{equation}
C(\rho) = \max {\cal f} 0, -\min_{A \in SL(2,C)}Tr((\ket{A}\bra{A})^{T_{B}}\rho) {\cal g},
\end{equation}
where $\ket{A}$ denotes the unnormalized state $(A \otimes I)\ket{I}$ with $\ket{I} = \sum_{i} \ket{ii}$, det$(A) = 1$.
Assuming that the set ${\cal C}$ is also convex, which is the case of all the quantities considered in this paper, except the concurrence, it is possible to apply the concept of Lagrange duality from the theory of convex optimization to the problems represented by equation (7) \cite{boyd}. Remarkably, the {\it dual} measures obtained are those related to mixing properties, such as the robustness of entanglement \cite{vidal4}, introduced by Vidal and Tarach, and the best separable approximation measure \cite{karnas}, introduced by Karnas and Lewenstein. Moreover, since in all the cases considered here there always exist a strictly feasible point, i.e, a $W \in \textbf{relint} {\cal M}$ (denoted in the convex optimization literature by Slater condition), the optimal solution of the primal and dual problems are the same, i.e., the primal and dual measures are equal \cite{boyd, ree}.
We now show that the dual representation of the generalized robustness of entanglement $R_{G}(\rho)$ \cite{steiner}, i.e., the minimal $s$ such that
\begin{equation}
\frac{\rho + s\pi}{1 + s}
\end{equation}
is separable, where $\pi$ is any, not necessarily separable, density matrix, is given by (7) with ${\cal M} = {\cal f} W \in {\cal W} \hspace{0.1 cm} | \hspace{0.1 cm} W \leq I {\cal g}$. Following \cite{boyd}, the Lagrangian of the problem is given by
\begin{equation}
\begin{split}
L(W, g(\sigma), Z) = Tr(W\rho) + Tr(WZ) \\
- Tr(Z) - \int\limits_{\sigma \in {\cal S}} g(\sigma)Tr(W\sigma)d\sigma,
\end{split}
\end{equation}
where $Z$, $g(\sigma)$ are the Lagrange multipliers associated with the constraints $W \leq I$ and $Tr(W\sigma) \geq 0 \hspace{0.1 cm} \forall \sigma \in {\cal S}$, respectively. Note that since the definition of EW is a composition of infinite constraints, its Lagrange multiplier is a generalized function \cite{ree}. The dual problem is then
\begin{eqnarray}
\quad \mbox{minimize} \quad & Tr(Z) \\
\quad \mbox{subject to} \quad & Z \geq 0 \nonumber \\
& g(\sigma) \geq 0, \hspace{0.2 cm} \forall \sigma \in {\cal S} \nonumber \\
& \rho + Z = {\displaystyle \int\limits_{\sigma \in {\cal S}}} g(\sigma)\sigma d\sigma. \nonumber
\end{eqnarray}
Since $g(\sigma) \geq 0$, the integral in the constraints above is a separable state. Conversely, any separable state $\sigma_{o}$ is obtained with the choice of $g(\sigma) = \delta(\sigma - \sigma_{o})$. It is then easily seen that the result of (15) is the generalized robustness. The dual representation of the robustness of entanglement, $R(\rho)$, has, instead of $W \leq I$, the constraint $Tr(W\sigma) \leq 1, \hspace{0.1 cm} \forall \hspace{0.1 cm} \sigma \in S$.
The best separable approximation measure $BSA(\rho)$ \cite{karnas} is the minimum $\lambda$ such that there exist a separable state $\sigma$ and a density operator $\delta \rho$ satisfying
\begin{equation}
\rho = (1 - \lambda)\sigma + \lambda \delta \rho.
\end{equation}
It can been seen that the dual representation of $BSA(\rho)$ is given by (7) with ${\cal M} = {\cal f} W \in {\cal W} \hspace{0.1 cm} | \hspace{0.1 cm} W \geq -I {\cal g}$.
In \cite{brandao2} it was shown that the random robustness $R_{r}(\rho)$ \cite{vidal4}, i.e., to the minimal $s$ such that
\begin{equation}
\frac{\rho + s(I/D)}{1 + s}
\end{equation}
is separable, is equal to equation (7), with ${\cal M} = {\cal f} W \in {\cal W} \hspace{0.1 cm} | \hspace{0.1 cm} Tr(W) = D {\cal g}$. This result can also be easily derived using the concept of Lagrange duality.
In the next subsection we will introduce our new family of entanglement monotones
\subsection{A New Family of Entanglement Monotones}
If we let ${\cal C}$ be the set of Hermitian matrices $W$ such that $-nI \leq W \leq mI$, where $n, m \geq 0$, then the quantity derived from (7) will be denoted by $E_{n:m}$.
\begin{proposition}
$E_{n:m}$ is an entanglement monotone for every $n, m \geq 0$, i.e.
\begin{equation}
\sum_{i}p_{i}E_{n:m}(\rho_{i}') \leq E_{n:m}(\rho)
\end{equation}
where $\rho_{i}'$ is the final state conditional on the occurrence of the classical variable $i$, which occurs with probability $p_{i}$ at the end of a LOCC protocol.
\end{proposition}
\begin{proof}
It suffices to consider final states of the form
\begin{equation}
\rho_{i}' = A_{i}\rho A_{i}^{\cal y} / p_{i}
\end{equation}
with $p_{i} = Tr[A_{i}\rho A_{i}^{\cal y}]$, where the Kraus operators $A_{1}, ..., A_{M}$ are given by $A_{i} = A_{i}^{1} \otimes ... \otimes A_{i}^{k}$ and satisfy $\sum_{i=1}^{M}A_{i}^{\cal y}A_{i} \leq I$:
\begin{equation}
\begin{array}{c}
\sum_{i}p_{i}E_{W}(\rho_{i}') = \sum_{i}p_{i} \max{\cal f}0, -Tr(W_{\rho_{i}'}\rho_{i}') {\cal g} \\
= \sum_{k} -Tr(A_{k}^{\cal y}W_{\rho_{k}'}A_{k}\rho) \leq -Tr(W_{\rho}\rho) = E_{W}(\rho)
\end{array}
\end{equation}
where $k$ sums only the terms such that $\max{\cal f}0, -Tr(W_{\rho_{i}'}\rho_{i}'){\cal g}$ is different from zero. In the last inequality we used that $W = \sum_{k}A_{k}^{\cal y}W_{\rho_{k}'}A_{k} \leq m\sum_{k}A_{k}^{\cal y}A_{k} \leq mI$, $W = \sum_{k}A_{k}^{\cal y}W_{\rho_{k}'}A_{k} \geq -n\sum_{k}A_{k}^{\cal y}A_{k} \geq -nI$, and that $W_{\rho}$ is optimal.
\end{proof}
Note that the proof of proposition (2), with minors modifications, applies also to ${\cal N}$ and $F_{d}$.
The dual representation of $E_{m:n}(\rho)$ is
\begin{eqnarray}
\quad \mbox{minimize} \quad & ms + nt \\
\quad \mbox{subject to} \quad & \rho + s\pi_{1} = (1 + s - t)\sigma + t\pi_{2} \nonumber
\end{eqnarray}
where $\pi_{i}$ are density matrices, $\sigma$ is a separable state and $s, t \geq 0$. From (24) we find that
\begin{equation*}
\lim_{m \rightarrow \infty}E_{n:m}(\rho) = nBSA(\rho), \hspace{0.4 cm}
\lim_{n \rightarrow \infty}E_{n:m}(\rho) = mR_{G}(\rho)
\end{equation*}
Actually, the equalities above are already valid when one of the numbers is sufficiently larger than the other. The elements of this new family of EMs can be interpreted as intermediate measures between the generalized robustness and the best separable approximation. Note that for every distinct rational number $n/m$ within a certain finite interval, the $E_{m:n}$ are genuine different EMs, meaning that there is no positive number $c$ such that $E_{m:n} = cE_{m':n'}$ if $n/m \neq n'/m'$.
If we consider that ${\cal C}$ is the intersection of set of Hermitian matrices $W$ such that $-nI \leq W \leq mI$ with the set of decomposable entanglement witnesses, a new family of entanglement monotones, denoted by $E^{PPT}_{n:m}$ is defined. To see that they are indeed EMs, all we have to note is that for every $A_{i} = A^{1}_{i} \otimes ... \otimes A^{k}_{i}$, $A^{\cal y}_{i}WA_{i}$ is a decomposable EW whenever $W$ is. Therefore, proposition (2) also applies to them.
It is possible to derive several other families of EMs considering intersections of the sets ${\cal C}$ of different entanglement measures which can be written as equation (7), such as those given by equations (10-12).
\section{Multipartite Entanglement Hierarchy}
We now discuss more about the different kinds of multipartite entanglement introduced in the second section. Usually the set of separable states is regarded to be composed of all sates which can be created by LOCC protocols. In this sense, given a specific split and considering that each part of the split can perform global quantum operations on its subsystems, only 1-separable states can be properly identified as separable. However, one might also be interest in the situation where some of the parties are allowed to perform join operations. In this case, the different types of entanglement play an important role. Consider, for example, the situation where $k$ parties want to create a common quantum state and each one is connected to the others via a quantum channel. If they all agree in using their channels, every state can be prepared and the situation becomes trivial. However, suppose that they agree that only $m \leq k$ parties will use their quantum channels, where the probabilities of which parties will be involved are given by $p_{i}$. At the end of the protocol they will share an ensemble of states, ${\cal f}\rho_{i}, p_{i}{\cal g}$, which clearly does not have $m + 1$-partite entanglemnt. Now, since erasing classical information cannot create entanglement, we are lead to consider the different kinds of entanglement discussed before. This property is reflected in the condition that every goog entanglement measure should be convex, which we show for every quantity defined according to equation (7)
\begin{proposition}
$E$ is a convex function for any choice of ${\cal C}$, i.e.,
\begin{equation}
E\left(\sum_{i}p_{i}\rho_{i} \right) \leq \sum_{i}p_{i}E(\rho_{i}),
\end{equation}
whenever the $\rho_{i}$ are Hermitian, and $p_{i} \geq 0$ with $\sum_{i}p_{i} = 1$.
\end{proposition}
Proposition (3) follows from the convexity and the concavity of the \textit{max} and \textit{min} functions, respectively.
Consider a given $k$-partite split of a multi-partite system $\rho$. It is possible to attribute $(k - 1)$ numbers, $E^{m}$, $1 \leq m \leq k - 1$, where each one quantifies one type of multi-partite entanglement of the system. It is easy to see from equation (3) that all constraints imposed to an EW which detects $m$-partite entanglement ($m$-EW), are also imposed to every $n$-EW, with $n \geq m$. Hence, the following order between the $E^{m}$ holds:
\begin{equation}
E^{m}(\rho) \geq E^{n}(\rho), \hspace{0.4 cm} \forall \hspace{0.2 cm} n \geq m.
\end{equation}
$E^{1}(\rho)$, formed by the OEW with respect to the fully separable states is an upper-bound to all other $E(\rho)$, including those with respect to other splits formed by grouping several original parties into one. This means, for example, that in a 3-split, $E^{1}(\rho)$ is greater or equal to the bipartite entanglement of any of the three 2-splits, namely A-BC, AB-C and AC-B. Actually, it is possible to establish a complete hierarchy in the proposed measures \cite{brandao3}.
An interesting measure of entanglement for multi-partite systems is the {\it localizable entanglement}, introduced by Verstraete {\it et al} \cite{verstraete2}. Given a quantum system of $n$ parties $\rho$, the {\it localizable entanglement} $E_{ij}(\rho)$ is the maximal amount of entanglement that can be created, on average, between the parties $i$ and $j$ by performing a single-copy LOCC protocol in the system \cite{foot4}. More specifically, if at the end of a LOCC protocol we have an ensemble of states $\mu = {\cal f}p_{k}, \rho_{k}^{ij}{\cal g}$, where $p_{k}$ is the probability that the reduced state of the parties $i$ and $j$ is $\rho_{k}^{ij}$, the LE is then given by
\begin{equation}
E^{ij} = \max_{\mu}\sum_{k}p_{k}E(\rho_{k}^{ij}),
\end{equation}
where $E(\rho)$ represents, in this paper, one measure based on OEWs. The LE has the operational meaning which applies to situations in which out of some multipartite entangled state one would like to concentrate as much entanglement as possible in two particular parties \cite{verstraete2}, which could be used later, for instance, in some quantum information task.
\begin{proposition}
Consider a multi-partite state $\rho$. Then
\begin{equation}
E^{ij}_{n:m} \leq E^{1}_{n:m}(\rho), \hspace{0.4 cm} \forall \hspace{0.2 cm} i, j, n, m.
\end{equation}
\end{proposition}
\begin{proof}
As in the proof of proposition 2, it suffices to consider final states of the form
\begin{equation}
\rho_{l}^{ij} = Tr_{/ij}(A_{l}\rho A_{l}^{\cal y} / p_{l})
\end{equation}
with $p_{l} = Tr[A_{l}\rho A_{l}^{\cal y}]$, where $Tr_{/ij}$ stands for the partial trace of all parties, except i and j. The Kraus operators $A_{1}, ..., A_{M}$ are given by $A_{l} = A_{l}^{1} \otimes ... \otimes A_{l}^{k}$ and satisfy $\sum_{i=1}^{M}A_{i}^{\cal y}A_{i} \leq I$.
\begin{equation}
\begin{array}{c}
E^{ij}_{n:m} = \sum_{l}p_{l}E_{n:m}(\rho_{l}^{ij}) = \sum_{l} \max{\cal f}0, -Tr(I\otimes W_{\rho_{l}^{ij}}\rho_{l}) {\cal g} \\
= \sum_{k} -Tr(A_{k}^{\cal y}I\otimes W_{\rho_{k}^{ij}}A_{k}\rho) \leq -Tr(W_{\rho}\rho) = E_{n:m}(\rho)
\end{array}
\end{equation}
where $k$ sums only the terms such that $\max{\cal f}0, -Tr(W_{\rho_{l}'}\rho_{l}'){\cal g}$ is different from zero. In the last inequality we used that the EW $W = \sum_{k}A_{k}^{\cal y}I\otimes W_{\rho_{k}}^{ij}A_{k} \leq m\sum_{k}A_{k}^{\cal y}A_{k} \leq mI$, $W = \sum_{k}A_{k}^{\cal y}I\otimes W_{\rho_{k}}^{ij}A_{k} \geq -n\sum_{k}A_{k}^{\cal y}A_{k} \geq -nI$ and that $W_{\rho}$ is optimal. Note that proposition (5) also applies to $E_{n:m}^{PPT}$.
\end{proof}
The following relation between the negativity, ${\cal N}(\rho)$, and $E_{\infty:1}^{PPT}(\rho) = R_{G}^{PPT}(\rho)$ holds:
\begin{proposition}
\begin{equation}
{\cal N}(\rho) \leq E_{\infty:1}^{PPT}(\rho) \leq d{\cal N}(\rho)
\end{equation}
\end{proposition}
\begin{proof}
For every positive operator $M$, we have
\begin{equation}
\lambda_{max}(M^{T_{A}}) \leq \lambda_{max}(M) \leq d\lambda_{max}(M^{T_{A}})
\end{equation}
where the first (second) inequality is saturated for separable (singlet) states. Hence, as $0 \leq W \leq 1$ implies $W^{T_{A}} \leq 1$, we find
\begin{eqnarray}
{\cal N}(\rho) = -\min_{0 \leq W \leq I}Tr(W^{T_{A}}\rho) \nonumber
\\ \leq -\min_{\substack{W^{T_{A}} \leq I \\ W \geq 0}}Tr(W^{T_{A}}\rho) = E_{\infty,1}^{PPT}(\rho)
\end{eqnarray}
where we have used that the optimal decomposable EW for a bipartite system has always the form $W^{T_{A}}$, $W \geq 0$. From equation (32) we also find that $W \geq 0$, $W^{T_{A}} \leq I$ implies $0 \leq W \leq dI$. Thus,
\begin{eqnarray}
E_{\infty:1}^{PPT}(\rho) = -\min_{\substack{W^{T_{A}} \leq I \\ W \geq 0}}Tr(W^{T_{A}}\rho) \nonumber
\\ \leq -\min_{0 \leq W \leq dI}Tr(W^{T_{A}}\rho) = d{\cal N}(\rho)
\end{eqnarray}
\end{proof}
The second inequality is strict for example on the state
\begin{equation}
\rho = \frac{I - d(P^{+})^{T_{A}}}{d^{2} - d}
\end{equation}
where $P^{+}$ is the maximal $d$ x $d$ entangled state.
\section{Numerical calculation}
The lack of an operational procedure to calculate entanglement measures in general is ultimately related to the complexity of distinguish entangled from separable mixed states, which was shown to be NP-HARD \cite{gurvits}. Since an operational measure, which has positive value in every entangled state, would also be a necessary and sufficient test for separability, we should not expect to find one. Nonetheless, some approximative numerical methods based on convex optimization have been proposed to the separability problem \cite{brandao1,doherty1,eisert1}. What we will show in this section is that these methods can also be used to calculate, approximately, the {\it witnessed entanglement}.
The first one, proposed in \cite{brandao1} by Brandao and Vianna, linked the optimization of EWs with a class of convex optimization problems known as robust semidefinite programs (RSDP). Although RSDPs belong to NP-HARD, some well known probabilistic relaxations, which transforms the problem in a semidefinite program (SDP), were applied, leading to a method of optimizing {\it pseudo}-EWs (operators which are positive in almost all separable states) to every multipartite state and with respect to all types of entanglement.
The second approach, due to Doherty {\it et al} \cite{doherty1,doherty2} , was actually the first method to the separability problem based on SDP. Using the existence of symmetric extensions for separable states and the concept of duality in convex optimization, a hierarchy of SDPs, where the $(k + 1)^{th}$ test is at least as powerful as $(k)^{th}$ (but demands more computational effort), was constructed to detect entanglement. In each step k, an OEW with respect to a restricted set of EWs, which converges to the whole set of EWs in the limit of $k \rightarrow \infty $, is obtained. This method can be used, however, only for the entanglement with respect to the fully-separable states, $E^{1}$. Note that the further constraints that we demand to the EWs can be incorporated in the SDP, since they are linear matrix (in)equalities.
The last method, introduced by Eisert {\it et al} \cite{eisert1}, is based on recently developed relaxations of non-convex polynomial problems of degree three in a hierarchy of SDPs, which converges to the solution of the original problem as the dimension of the SDP reaches the infinity \cite{lassere}. One of the applications of this method is the minimization of the expectation values of EWs with respect to pure product states. Therefore, it can be used together with the second method discussed to lower the value of $Tr(W\rho)$, where $W$ is a non-optimal EW determined by some step of the hierarchy.
We would like to stress the complementary character of these methods. Whereas the first method usually provides upper bounds to $E(\rho)$, since it only determines {\it pseudo}-EWS, the second and third provides lower bounds to $E(\rho)$, as the EWs resulting from them are non-optimal. Although only the first one can calculate $E^{m}(\rho)$, for $m > 1$, the number of constrains imposed grows exponentially with $m$. Thus, in most of cases, we will restrict ourselves to the determination of $E^{1}$, which is an upper bound to all other types of multi-partite entanglement (see section IV).
Note that measures restricted to decomposable EWs can always be exactly calculated, in the worst case, by a semidefinite program.
\subsection{Example I}
\begin{figure}
\begin{center}
\includegraphics[scale=0.45]{WGHZ1.eps}
\caption{(Color online) $E^{1}_{n:1}(\rho_{q})$ for $0 \leq n \leq 4$ and $0 \leq q \leq 1$.}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[scale=0.45]{WGHZ2.eps}
\caption{(Color online) $E^{2}_{n:1}(\rho_{q})$ for $0 \leq n \leq 4$ and $0 \leq q \leq 1$.}
\end{center}
\end{figure}
As a first example we calculated $E_{n:1}^{1}$ and $E_{n:1}^{2}$ , $0 \leq n \leq 1$, for the following family of states
\begin{equation}
\rho_{q} = q\ket{W}\bra{W} + (1- q)\ket{GHZ}\bra{GHZ}, \hspace{0.2 cm} 0 \leq q \leq 1
\end{equation}
where $\ket{W} = (\ket{001} + \ket{010} + \ket{100})/\sqrt{3}$ and $\ket{GHZ} = (\ket{000} + \ket{111})/\sqrt{2}$. The results are plotted in figures (1) and (2). When $n << 1$, $E_{n:1}^{1} = nBSA$ and we see that for all q there is not product vectors and even biseparable vectors in the range of $\rho_{q}$. In the other limit, where $E_{n:1}^{1} = R_{G}$, we find that the generalized robustness of entanglement with respect to biseparable states is the same for all $\rho_{q}$ with $q \leq 0.7$.
\subsection{Example II}
The classes of entangled states equivalent by SLOCC for 2 X 2 X n systems were determined in \cite{miyake} and can be represented by the states (1-5) of figure (3). The arrows indicate which transformations are probabilistic possibles.
\begin{figure}
\begin{center}
\includegraphics[scale=0.4]{verstraete1.eps}
\caption{Representative states of the five distinct classes of 3-entangled states.}
\end{center}
\end{figure}
$E_{n:1}^{1}$ , $0 \leq n \leq 1$, was calculated for each of these and plotted in figure (4).
\begin{figure}
\begin{center}
\includegraphics[scale=0.4]{2x2x4.eps}
\caption{$E^{1}_{n:1}(\rho_{q})$, $0 \leq n \leq 4.5$, for the states (1-5) of figure (3).}
\end{center}
\end{figure}
Note that for all n considered, the incomparable states (2-3) and (4-5) with respect to state transformations have approximately the same $E^{1}_{n:1}$.
\subsection{Example III}
As a final example, we present a numerical comparison between ${\cal N}$ and $E_{\infty,1} = R_{G}^{PPT}$. The bipartite PPT-generalized robustness can be determined as easily as the negativity. Actually, it can be written as
\begin{equation}
R_{G}^{PPT}(\rho) = \frac{1}{\lambda_{max}(P^{T_{A}})}{\cal N}(\rho)
\end{equation}
where $\lambda_{max}(P^{T_{A}})$ is the maximum eigenvalue of the partial transposed projector onto the negative eigenspace of $\rho^{T_{A}}$. We have generated $10^{5}$ random states using the algorithm presented in \cite{zyc} and plotted in figures (5) and (6).
\begin{figure}
\begin{center}
\includegraphics[scale=0.45]{NEw1.eps}
\caption{(Color online) $R_{G}^{PPT} x \hspace{0.1 cm} {\cal N}$ for $10^{5}$ 4 x 4 random sates.}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[scale=0.45]{NEw2.eps}
\caption{(Color online) $R_{G}^{PPT} x \hspace{0.1 cm} {\cal N}$ for $10^{5}$ 6 x 6 random sates.}
\end{center}
\end{figure}
Although $d{\cal N} \geq R_{G}^{PPT} \geq {\cal N}$ (see section IV), we see from figures 5 and 6 that $R_{G}^{PPT} \leq 2{\cal N}$ for the majority of states.
\section{Gaussian states}
We considered $n$ distinguishable infinite dimensional subsystems, each with local Hilbert space ${\cal H} = {\cal L}^{2}({\cal R}^{n})$. A Gaussian state is characterized by a density operator whose characteristic function $\chi_{\rho}(x) = Tr[\rho W(x)]$ is a Gaussian function \cite{giedke1}. We can write, for every Gaussian state $\rho$,
\begin{equation}
\rho = \pi^{-n}\int_{{\cal R}^{2n}}dx e^{-\frac{1}{4}x^{T}\gamma x + id^{T}x}W(x),
\end{equation}
where $W(x) = \exp[-ix^{T}R]$ are the displacement operators and $R = (X_{1}, P_{1}, X_{2}, ..., P_{n})$, with $[X_{k}, P_{l}] = i\delta_{kl}$. The matrix $\gamma \geq iJ_{n}$ is a 2$n$ x 2$n$ real matrix called correlation matrix (CM) and $d$ is an 2$n$ real vector called displacement \cite{giedke1}. The symplectic matrix is given by
\begin{equation}
J_{n} = \bigoplus^{n}_{k = 1}J_{1}, \hspace{0.3 cm} J_{1} = \left(
\begin{array}{cc}
0 & -1 \\ 1 & 0
\end{array}
\right)
\end{equation}
Note that the displacement of a state can always be adjusted to $d = 0$ by a sequence of unitaries applied to individual modes. This implies that d is irrelevant for the study of entanglement. Thus, we set $d = 0$ for now on without loss of generality.
The optimization of EWs for states of infinite dimension is completely infeasible. Nonetheless, we can still obtain meaningful quanties if we restrict it to a simpler, but sufficiently large, set of operators. An obvious choice would be the restriction to Gaussian entanglement witnesses (GEW), i.e., Gaussian operators which are positive in separable Gaussian states. Unfortunately, none Gaussian entangled state is detected by a GEW. Assuming that G is a GEW with covariance matrix $\Gamma$, we find
\begin{equation}
Tr(\rho G) = \int_{{\cal R}^{2n}}dx e^{-\frac{1}{4}x^{T}(\Gamma + \gamma)x + c} \geq 0.
\end{equation}
Another possible class of operators then is given by
\begin{equation}
{\cal W}_{\cal G} = {\cal f} Q \in {\cal B}({\cal H} \otimes {\cal H}) \hspace{0.1 cm} | \hspace{0.1 cm} Q = 2^{n}I - G {\cal g},
\end{equation}
where $G$ is a Gaussian operator and $I$ the identity operator \cite{foot5}. In the next proposition we show that $E_{\infty:m}^{G}(\rho)$ given by
\begin{equation}
E_{\infty:1}^{G}(\rho) = \max {\cal f} 0, -\min_{Q \in {\cal W}_{\cal G}, \hspace{0.1 cm} Q \leq I} Tr(Q \rho) {\cal g}
\end{equation}
can be very efficiently numerically calculated by a simple semidefinite program.
\begin{proposition}
\begin{equation}
E_{\infty:m}^{G}(\rho) = \max {\cal f}0, \hspace{0.1 cm} \det(\Gamma + \gamma)^{-1/2} - 2^{n} {\cal g}
\end{equation}
where the matrix $\Gamma \in M_{2n}({\cal R})$ is obtained by the following SDP determinant maximization problem
\begin{equation*}
\max_{\Gamma, \tau, \Delta, s_{ij}} \hspace{0.1 cm} \det(\Gamma + \gamma)^{-1/2}
\end{equation*}
\begin{center}
subject to \hspace{0.3 cm} $-1 \leq \tau \leq 1, \hspace{0.6 cm} \Gamma \geq 0$
\end{center}
\begin{eqnarray}
\left(
\begin{array}{cc}
\tilde{\Gamma}_{2} + i\tau J & \tilde{\Gamma}_{12}^{T} \\
\tilde{\Gamma}_{12} & \tilde{\Gamma}_{1} - iJ
\end{array} \right) \geq 0
\end{eqnarray}
\begin{equation*}
\left(
\begin{array}{cc}
J_{n} + \Gamma & \Delta \\
\Delta & D(\Delta)
\end{array} \right) \geq 0, \hspace{0.4 cm} \left(
\begin{array}{cc}
s_{k-1,2l-1} & s_{k,l} \\
s_{k,l} & s_{k-1,2l}
\end{array} \right) \geq 0
\end{equation*}
\begin{center}
$s_{k-1,2l-1} \geq 0, \hspace{0.11cm} s_{k-1,2l} \geq 0, \hspace{0.11cm} s_{kl} \geq 2$, \hspace{0.11 cm} {\small $k = 1..l, i = 1...2^{l-k}$}
\end{center}
where $\Delta$ is a n x n lower triangular matrix comprised of additional variables, D($\Delta$) is a diagonal matrix with same diagonal entries as those of $\Delta$, l is the smallest number such that $2^{l} \geq n$, and $s_{0,i} = \Delta_{ii}$ if $1 \leq i \leq n$ and $s_{0,i} = 2$ if $n \leq i \leq 2^{l}$.
\end{proposition}
\begin{proof}
Consider the following structure for the bipartite Gaussian operator $G$
\begin{equation}
G = \int_{{\cal R}^{2n}}dx e^{-\frac{1}{4}x^{T}\Gamma x}W(x),
\end{equation}
where $\Gamma^{T} = \Gamma \geq 0 \in M_{2n}({\cal R})$, with modes $1$ to $m$ and $m + 1$ to $n$ belonging to Alice and Bob, respectively. The optimization objective $Tr(Q \rho)$, where $Q = 2^{n}I - G$, can be written as
\begin{equation}
2^{n} - \pi^{-n}\int_{{\cal R}^{2n}}dx e^{-\frac{1}{4}x^{T}(\Gamma + \gamma)x}
= 2^{n} - \det(\Gamma + \gamma)^{-1/2}
\end{equation}
From the Jamiolkowski isomorphism, $Q$ is an EW iff the map ${\cal Q}$ defined as $Q = I \otimes {\cal Q}(P^{+}) = 2^{n}I - I \otimes {\cal G}(P^{+})$ \cite{foot6} is positive, which is equivalent to $\rho' = I \otimes {\cal G}(\rho) \leq 2^{n}I$, for every density operator $\rho$. The covariance matrix of $\rho'$, $\gamma'$, can be written as \cite{simon}
\begin{equation}
\gamma' = S^{T}\sigma S ,
\end{equation}
where $S \in Sp(2n,{\cal R})$ and $\sigma$ is the covariance matrix
\begin{equation*}
\sigma = \mbox{diag}(\mu_{1},\mu_{1},...,\mu_{n},\mu_{n})
\end{equation*}
corresponding to a tensor product of states diagonal in the number basis given by
\begin{equation}
M' = \bigotimes_{i}\frac{2}{\mu_{i} + 1}\sum_{k=0}^{\infty}\left(\frac{\mu_{i} - 1}{\mu_{i} + 1}\right)\ket{k}_{i}{}_{i}\bra{k},
\end{equation}
$\ket{k}_{i}$ being the {\it k}-th number state of the Fock space ${\cal H}_{i}$ \cite{adesso}. The symplectic transformation (47) is reflected in the Hilbert space level by an unitary transformation: $G = U(S)^{\cal y}G'U(S)$. Since we are considering bounded operators, the $\mu_{i}$ must be non-negative. We also see that the positiveness of ${\cal Q}$ is equivalent to
\begin{equation}
\lambda_{max}(\rho') = \prod_{j = 1}^{n} \left( \frac{1}{1 + \mu_{j}} \right) \leq 1, \hspace{0.2 cm} \forall \rho'
\end{equation}
Thus, since we are only considering Gaussian operators, equation (49) is satisfied iff $\gamma' \geq 0$ for every $\gamma \geq iJ$, where $\gamma'$ and $\gamma$ are the covariance matrices of $\rho'$ and $\rho$, respectively. Following Giedke and Cirac \cite{giedke1}, one finds that ${\cal Q}$ is positive iff
\begin{equation}
\min_{z \in{\cal C}^{2n}}\max{\cal f}z^{\cal y}(M + iJ)z, z^{\cal y}(M - iJ)z{\cal g} \geq 0
\end{equation}
where $M = \tilde{\Gamma}_{2} - \tilde{\Gamma}_{12}^{T}(\tilde{\Gamma}_{1})^{-1}\tilde{\Gamma}_{12}$ and $\tilde{\Gamma} = (I \oplus \Lambda)\Gamma(I \oplus \Lambda)$, with $\Lambda = $ diag$(1,-1,1,-1,...,-1)$. The matrices $\Gamma_{i}$ are such that
\begin{equation*}
\Gamma = \left(
\begin{array}{cc}
\Gamma_{1} & \Gamma_{12} \\
\Gamma_{12}^{T} & \Gamma_{2}
\end{array} \right)
\end{equation*}
We now express condition (50) as a linear matrix inequality. Equation (50) is equivalent to
\begin{eqnarray}
z^{\cal y}(M + iJ)z \geq 0 \quad \mbox{$\forall \hspace{0.1 cm} z \in {\cal C}^{2n}$ \hspace{0.1 cm}s.t.} \quad z^{\cal y}(M - iJ)z \leq 0 \nonumber \\
z^{\cal y}(M - iJ)z \geq 0 \quad \mbox{$ \forall \hspace{0.1 cm} z \in {\cal C}^{2n}$ \hspace{0.1 cm} s.t.} \quad z^{\cal y}(M + iJ)z \leq 0 \nonumber
\end{eqnarray}
An important theorem of matrix analysis, known as {\cal S}-procedure, can be stated as follows: a quadratic function in the variable $x$, $G(x)$, is positive for all $x$ such that $H(x) \geq 0$, where $H(x)$ is another quadratic function, iff there exists a positive real number $\tau$ such that $G - \tau H \geq 0$, for all $x$ \cite{boyd2}.
Applying it to the two conditions above we find that equation (50) holds iff there exists a positive number $\tau$ such that
\begin{equation}
M + \tau iJ \geq 0, \hspace{0.3 cm} -1 \leq \tau \leq 1
\end{equation}
We now use another fact of matrix analysis which says that the constraints on the Schur complement $R > 0, \hspace{0.3 cm} Q - SR^{-1}S^{T} \geq 0$ and $\ker(R) \subseteq \ker(Q)$ are equivalent to
\begin{equation*}
\left( \begin{array}{cc}
Q & S \\ S^{T} & R
\end{array} \right) \geq 0
\end{equation*}
Hence, applying it to equation (51), we find that a Gaussian operator ${\cal G}$ is an EW iff there exists a real number $-1 \leq \tau \leq 1$ such that equation (44) holds.
From the Williason decomposition, we see that $Q \leq I$ is equivalent to
\begin{equation}
\prod_{i=1}^{n}\left(\frac{2}{\eta_{i} + 1}\right) = 2^{n}\det(I + (S^{T})^{-1}\Gamma S^{-1})^{-1} \leq 1,
\end{equation}
where $\eta_{i}$ ate the symplectic eigenvalues of $Q$. Since $S$ is symplectic, one has $S^{T}J_{n}S = J_{n}$, so that $\det(I + (S^{T})^{-1}\Gamma S^{-1})^{-1} = \det[{J_{n}^{T}(S^{T})^{-1}(J_{n} + \Gamma)S^{-1}}]^{-1} = \det(J_{n} + \Gamma)^{-1}$ . The proposition then follows from reference \cite{ben}, which presents a LMI representation for the inequality $\det(A)^{1/m} \geq t$, where $A$ is a positive $m$ x $m$ real matrix.
\end{proof}
\section{States with symmetry}
Entanglement measures have usually their calculation greatly simplified when the state in question has certain symmetries. Following \cite{vollbrecht1}, let $G$ be a closed group of product unitary operators of the form $U = U_{1} \otimes U_{2}$. Defining the projection
\begin{equation}
\textbf{P}(A) = \int dU \hspace{0.2 cm} UAU^{*},
\end{equation}
for any operator $A$ on $H_{1}\otimes H_{2}$ \cite{foot7}, where $dU$ is the Haar measure of $G$, we say that an operator $M$ is invariant under $G$ if $\textbf{P}(M) = M$, which is equivalent to $[U, M] = 0$ for all $U \in G$. Consider now the determination of any measure expressed by equation (7). If the state in question has the property $\textbf{P}(\rho) = \rho$, then one can restrict the optimization in (7) to operators with the this same symmetry. More specifically,
\begin{eqnarray}
E(\rho) = -\min_{W \in {\cal M}} Tr(W\rho) = -\min_{W \in {\cal M}} Tr(W\textbf{P}(\rho)) \nonumber
\\ = -\min_{W \in {\cal M}} Tr(\textbf{P}(W)\rho) = -\min_{W \in \textbf{P}({\cal M})} Tr(W\rho),
\end{eqnarray}
where $\textbf{P}({\cal M}) = {\cal f}W \in {\cal M} \hspace{0.1 cm} | \hspace{0.1 cm} W = \textbf{P}(W){\cal g}$ \cite{foot10}.
As an example, consider the isotropic states $\rho_{p}$ on ${\cal C}^{d}\otimes{\cal C}^{d}$
\begin{equation}
\rho_{p} = pP^{+} + (1-p)\frac{I}{d^{2}},
\end{equation}
where $P^{+} = \ket{\Phi^{+}}\bra{\Phi^{+}}$ is the maximally entangled state. It can be shown that $P^{+}$ and the identity are the only operators which commute with all unitaries of the form $U\otimes U^{*}$. Hence, the OEWs for $\rho_{p}$ can be written as
\begin{equation}
W(\rho_{p}) = a(p)P^{+} + b(p)I.
\end{equation}
Since $\bra{\psi}P^{+}\ket{\psi} \leq 1/d$, for every separable state $\ket{\psi}$, we find
\begin{eqnarray}
E_{n:1}(\rho_{p}) = \begin{cases}
(n + 1)p + \frac{(1 - p)(n + 1)}{d^{2}} - 1 & n \leq d-1, \\
dp + \frac{1 - p}{d} - 1 & n \geq d.
\end{cases}
\end{eqnarray}
As the OEWs for this family of states are decomposable, equation (57) is also valid to $E^{PPT}_{n:1}$.
\section{Superselection Rules}
The effect of superselection rules (SSR) in theory of entanglement has been studied recently under a number of different perspectives \cite{verstraete3,bartlett1,terhal3,schuch}. Two striking features emerge from the existence of a SSR. The entanglement of a given state under SSR is usually reduced \cite{bartlett1} and the notion of nonlocality has to be redefined, as there exists separable states that cannot be created by LOCC \cite{verstraete3}. In this section we show how the {\it witnessed entanglemet} fit in each of these scenarios.
Following Bartlett and Wiseman \cite{bartlett1}, we define a SSR as a restriction on the allowed local operations on a system, associated with a group of physical transformations $G$. An operation ${\cal O}$ is $G$-covariant if
\begin{equation}
{\cal O}[T(g)\rho T^{\cal y}(g)] = T(g){\cal O}[\rho]T^{\cal y}(g),
\end{equation}
for all group elements $g \in G$ and all density operators $\rho$. Then the SSR associated to $G$ is the restriction on the allowed operations on the system to those $G$-invariant. As these restrictions make a state $\rho$ indistinguishable from the states $T(g)\rho T^{\cal y}(\rho)$ for all $g \in G$, it is convenient to describe $\rho$ by the $G$-invariant state
\begin{equation}
{\cal G}(\rho) = \int\limits_{G}dgT(g)\rho T^{\cal y}(g),
\end{equation}
where $dg$ is the group-invariant Haar measure. For multipartite systems, where the SSRs are local, we have ${\cal G}[\rho] = {\cal G}\otimes ... \otimes{\cal G}[\rho]$. As it was shown in \cite{bartlett1}, the maximal amount of entanglement which can be produced by LOCC in a register shared by all the parties, initially in a product state and not subjected to SSRs, from a state $\rho$, constrained by a $G$-SSR, is given by the entanglement they can produce from ${\cal G}(\rho)$ by unconstrained LOCC. If $E$ is an entanglement monotone, any LOCC applied to ${\cal G}(\rho)$, can, on average, at most maintain $E({\cal G}[\rho])$. Since it is always possible to reach this bound applying local swap operators, we have that the maximal amount of entanglement produced by SSR is exactly $E({\cal G}[\rho])$. Hence, from section (VII), it follows that, under a $G$-SSR
\begin{equation}
E(\rho) = \max {\cal f}0, -\min_{W \in {\cal G}[{\cal M}]} Tr(W\rho) {\cal g},
\end{equation}
where ${\cal G}[{\cal M}] = {\cal f}W \in {\cal M} \hspace{0.1 cm} | \hspace{0.1 cm} W = {\cal G}[W]{\cal g}$ \cite{foot8}.
We now consider the effect of SSRs in the notion of locality. The states that can be prepared locally in the presence of a $G$-SSR are those which can be written as (1), with each $\ket{\psi_{i}}_{k}$ being $G$-invariant. It is possible to detect nonlocal states with witness operators, defining a $G$-nonlocality witness (GW) as a Hermitian operator which satisfies equation (3), with $\ket{\psi}_{P_{k}^{m}}{}_{P_{k}^{m}}\bra{\psi} = {\cal G}[\ket{\psi}_{P_{k}^{m}}{}_{P_{k}^{m}}\bra{\psi}]$, for all $i$ and $k$. This nonlocal character of some states in the presence of a SSR can be quantified \cite{schuch}. We can then, as it was done with entanglement, use GWs to perform this quantification. A {\it witnessed nonlocality measure}, $N_{G}$, will be any quantie given by equation (7), with the set of EWs substituted by the set of nonlocality witnesses. It is easy to see that all properties discussed for $E$ are valid for $N_{G}$.
As an example, considered the following state
\begin{eqnarray}
\rho = \frac{1}{4}(\ket{0}_{A}\bra{0}\otimes \ket{0}_{B}\bra{0} + \ket{1}_{A}\bra{1}\otimes \ket{1}_{B}\bra{1}) \nonumber \\ + \frac{1}{2}\ket{\Psi_{+}}_{AB}\bra{\Psi_{+}},
\end{eqnarray}
where $\ket{\Psi_{+}}_{AB} = (\ket{0}_{A}\ket{1}_{B} + \ket{1}_{A}\ket{1}_{B})/\sqrt(2)$. Verstraete and Cirac have shown \cite{verstraete3} that, although this state has a separable decomposition, it is not local when a particle number SSR exists, since all possible separable decompositions have local states involving superpositions of a different number of particles. Any Hermitian matrix with positive diagonal entries is a G-nonlocality witness in this case. This should be contrasted with the case of a general EW, where an infinite number of inequalities are necessary for its characterization. Calculating, for example, the nonlocal measure equivalent to $E_{\infty:1} = R_{G}$,
\begin{equation*}
N_{G}(\rho) = \max{\cal f}0, -\min_{G \in {\cal G}}Tr(G\rho){\cal g},
\end{equation*}
where ${\cal G} = {\cal f}G \hspace{0.1 cm} | \hspace{0.1 cm} G_{ii} \geq 0, G \leq I {\cal g}$, we find $N_{G})(\rho) = 1/2$, with $G = -\ket{01}\bra{10} - \ket{10}\bra{01}$.
\section{Indistinguishable particles}
The study of entanglement in systems of indistinguishable particles has been the subject of recent controversy \cite{zanardi1,gittings,pask,schliemann,wiseman1}. At least three different approaches to the problems have been proposed, namely, the {\it entanglement of modes} \cite{zanardi1}, the {\it quantum correlations} \cite{schliemann} and the {\it entanglement of particles} \cite{wiseman1}. Each of these has its own advantages and drawbacks, and no consensus has been reached on which one is the most suitable. In this section we show how the proposed measures based on EW can be used to quantify entanglement in each of the three methods.
We start with the {\it entanglement of modes}, proposed by Zanardi \cite{zanardi1}, which suggests that the entanglement of indistinguishable particles should be calculated by any regular entanglement measure, using the density matrix in the mode-occupation, or Fock, representation. In this case it is clear that the determination of the witnessed entanglement follows straightforwardly.
The {\it quantum correlations}, introduced by Schliemann {\it et al} \cite{schliemann}, is motivated by the believe that no quantum correlations due to symmetrization (for bosons) or anti-symmetrization (for fermions) should be considered as true entanglement. Then, the characterization of entanglement, for pure states, is determined by the Slater rank of the state, as opposed to the Schimdt rank usually considered in distinguishable particles. Furthermore, Schliemann {\it et al} have shown that the concept of entanglement witness is also applicable to multipartite systems of indistinguishable particles \cite{foot11} and, thus, the witnessed entanglement are well defined in this case too.
The last approach, due to Wiseman and Vaccaro \cite{wiseman1}, is probably the best motivated one. The {\it entanglement of particles} is defined as the maximal amount of entanglement, computed by a standard measure, which Alice and Bob can produce between a quantum register, shared by them, composed of distinguishable particles by local operations. The amount of entanglement will clearly depends on the physical constraints imposed, which are in most of cases expressed as a SSR. Therefore the approach presented in the previous section to SSR can also be applied in this case.
\section{Teleportation distance}
In this section we derive lower bounds to the teleportation distance, using $E_{n:m}$. We consider a quantum state $\rho$ shared by $k$ parties and ask what is the best possible teleportation distance attained by a LOCC protocol when the parties form two groups and teleport a quantum state from one group to the other.
Consider a teleportation protocol where a bipartite state $\rho_{AB}$ is used as a quantum channel between Alice and Bob. Following the approach of Vidal and Werner to the negativity \cite{vidal3}, we will first consider the {\it single distance} of a bipartite state defined as:
\begin{equation}
\Delta(P_{+}, \rho) \equiv \inf_{P}||P_{+} - P(\rho)||_{1},
\end{equation}
where $P_{+}$ is the maximally entangled state and the infimum is taken over LOCC protocols $P$. Using the convexity and the invariance under unitary transformations in the two terms of the absolute distance, and the invariance of $P_{+}$ under unitary transformations of the form $U\otimes U^{*}$, we may assume that the optimal state which minimizes equation (62), $P_{opt}(\rho)$, has undergone a twirling operation \cite{foot9} and, therefore, is a {\it noise singlet},
\begin{equation}
\rho_{p} = pP_{+} + (1-p)\frac{I\otimes I}{d^{2}}.
\end{equation}
The absolute distance of $\rho_{p}$ is given by $||P_{+} - \rho_{p}||_{1} = 2(1 - p)(d^{2} - 1)/d^{2}$. From equation (57),
\begin{equation}
||P_{+} - \rho_{p}||_{1} = 2(1 - \frac{1 + E_{n:1}^{PPT}(\rho_{p})}{d}).
\end{equation}
From the monotonicity of $E_{n:1}^{PPT}$ under LOCC, we find that, for $n \geq d$,
\begin{proposition}
\begin{equation}
\Delta(P_{+}, \rho) \geq 2(1 - \frac{1 + E_{n:1}^{PPT}(\rho)}{d}).
\end{equation}
\end{proposition}
Since $E_{n:1}^{PPT}(\rho_p) = 2{\cal N}(\rho_{p})$, for $n \geq d$, we see that $E_{n:1}^{PPT}$ provides, when $E_{n:1}^{PPT} \leq 2{\cal N}$, a sharper bound than the one derived from the negativity. In the limit case $n \rightarrow \infty$ already, where $E_{n:1}^{PPT}$ is equal to the PPT-generalized robustness, we see from section (V) that the new bound is indeed sharper for the majority of states.
A measure of the degree of performance of a quantum channel is the teleportation distance
\begin{equation}
d(\Lambda) = \int d\phi ||\phi - \Lambda(\phi)||_{1}.
\end{equation}
As it was shown by the Horedecki family \cite{horodecki1}, the minimal teleportation distance that can be achieved when using the bipartite state $\rho$ to construct an arbitrary teleportation channel is given by
\begin{equation}
d_{min}(\rho) = \frac{d}{d + 1}\Delta(P_{+}, \rho).
\end{equation}
Therefore,
\begin{equation}
d_{min}(\rho) \leq \frac{2d}{d + 1}(1 - \frac{1 + E_{n:1}^{PPT}(\rho)}{d}), \hspace{0.1 cm} n \geq d.
\end{equation}
Until now we have just adapted Vidal and Werner's reasoning for the negativity to the $E_{n:1}^{PPT}$. Nevertheless, as opposed to ${\cal N}$, $E_{n:1}^{PPT}$ are also defined to multi-partite systems.
\begin{proposition}
Consider a quantum state $\rho$ shared by $k$ parties. Let $\rho^{1..m:(m+1)..k}$ denote a bipartite split of the system, where the parties 1 to $m$ and $m$ + 1 to $k$ form two groups. Then,
\begin{equation}
d_{min}(\rho^{1..m:(m+1)..k}) \leq \frac{2D}{D + 1}(1 - \frac{1 + (E_{n:1}^{PPT})^{1}(\rho)}{D}),
\end{equation}
$\forall \hspace{0.1 cm} 1 < m < k$, where $D$ stands for the minimum of the dimensions of the two groups.
\end{proposition}
Proposition (8) follows from the upper bound to all types of entanglement provided by $E^{1}$. Equation (57) is saturated, for example, on the $k$-partite GHZ state $\ket{\Psi_{GHZ}} = 1/\sqrt{2}(\ket{00...0} + \ket{11...1})$.
\section{Upper Bounds for the Distillable Entanglement}
We now move on to show another application of the family $E_{1:m}$, namely bounds to the distillable entanglement of bipartite mixed states. We first derive the following additivity property
\begin{proposition}
\begin{equation}
E_{n:1}(\rho \otimes \rho) \leq E_{n:1}(\rho)^{2} + 2E_{n:1}(\rho) , \hspace{0.2 cm} \forall \hspace{0.1 cm} n \geq 1.
\end{equation}
\end{proposition}
\begin{proof}
Consider the dual representation (24) of $E_{n:1}$. Let s, t, $\sigma$, $\pi_{1}$ and $\pi_{2}$ be variables which minimize (24). Then we find $E_{n:1}(\rho) = s + nt$, with
\begin{equation}
\rho = (1 + s - t)\sigma + t\pi_{2} - s\pi_{2}.
\end{equation}
Thus,
\begin{eqnarray}
\rho \otimes \rho = (1 + s - t)^{2}\sigma \otimes \sigma + t(1 + s - t)\sigma \otimes \pi_{2} \nonumber \\
- s(1 + s - t) \sigma \otimes \pi_{1} + t(1 + s - t)\pi_{2} \otimes \sigma + t^{2}\pi_{2} \otimes \pi_{2} \nonumber \\
- st\pi_{2}\otimes \pi_{1} - s(1 + s - t)\pi_{1}\otimes \sigma - st\pi_{1}\otimes \pi_{2} + s^{2}\pi_{1}\otimes \pi_{1} \nonumber \\
= [1 + (2s + s^{2} + t^{2}) - (2t + 2st)]\sigma \otimes \sigma \nonumber \\
+ [t(\rho \otimes \pi_{2} + \pi_{2} \otimes \rho) + st(\pi_{1}\otimes \pi_{2} + \pi_{2}\otimes \pi_{1})] \nonumber \\
- [ s(\rho \otimes \pi_{1} + \pi_{1} \otimes \rho) + s^{2} \pi_{1}\otimes \pi_{1} + t^{2}\pi_{2} \otimes \pi_{2}], \nonumber
\end{eqnarray}
where in the last two lines we used that
\begin{equation*}
\sigma = \frac{1}{1 + s - t}\left(\rho + s\pi_{1} - t\pi_{2}\right).
\end{equation*}
It is therefore easily seen that if $E_{n:1}(\rho \otimes \rho) = s' + nt'$, then $s' + nt'\leq s^{2} + t^{2} + 2s + n(2t + 2st)$. Hence, as $n \geq 1$,
\begin{eqnarray}
E_{n:1}(\rho)^{2} + 2E_{n:1}(\rho) - E_{n:1}(\rho \otimes \rho) \nonumber
\\ = s^{2} + n^{2}t^{2} + 2nst + 2s + 2nt - s' - nt' \nonumber \\
\geq s^{2} + n^{2}t^{2} + 2nst + 2s + 2nt - s^{2} - t^{2} - 2s - 2nt - 2nst \nonumber \\
= t^{2}(n^{2} - 1) \geq 0. \nonumber
\end{eqnarray}
\end{proof}
We can define a family of quantities close related to $E_{n:1}$ by
\begin{equation}
LE_{n:1}(\rho) = \log_{2}(1 + E_{n:1}(\rho)).
\end{equation}
The $LE_{n:1}(\rho)$ are non-increasing under trace preserving separable operations. From proposition (9) we find that they are also weakly-subadditive. Indeed, for $n \geq 1$,
\begin{equation}
LE_{n:1}(\rho \otimes \rho) \leq \log_{2}((1 + E_{n:1}(\rho))^{2}) = 2LE_{n:1}(\rho).
\end{equation}
Note that the same results are also valid to $E^{PPT}_{n:1}$. We now can state the main result of this section
\begin{proposition}
\begin{equation}
E_{D}(\rho) \leq LE_{n:1}(\rho) , \hspace{0.1 cm} \forall \hspace{0.1 cm} n \geq 1.
\end{equation}
where $E_{D}(\rho)$ is the distillable entanglement of the bipartite state $\rho$.
\end{proposition}
\begin{proof}
The proof of proposition (10) is basically an application of a theorem due to the Horodeckis \cite{horodecki5} which can be stated as follows: any function B satisfying the conditions 1)-3) below is an upper bound for the entanglement of distillation.
\begin{enumerate}
\item Weak monotonicity: $B(\rho) \geq B(\Lambda(\rho))$ where $\Lambda$ is any trace-preserving superoperator realizable by means of LOCC operations.
\item Partial subadditivity: $B(\rho^{n}) \leq nB(\rho)$.
\item Continuity for isotropic states $\rho_{p}$ given by equation (55). Suppose that we have a sequence of isotropic states $\rho_{p}$\ such that $Tr(\rho_{p}P^{+}) \rightarrow 1$, if $d \rightarrow \infty$. Then we require
\begin{equation}
\lim_{d \rightarrow \infty}\frac{1}{\log_{2}d}B(\rho_{p}) \rightarrow 1
\end{equation}
\end{enumerate}
We have already shown that $LE_{n:1}(\rho)$, for $n \geq 1$, satisfies conditions (1) and (2). From equation (54)
\begin{equation}
LE_{n:1}(\rho_{p}) = \log_{2}(dp + \frac{1 - p}{d}), \hspace{0.1 cm} \forall \hspace{0.1 cm} n \geq 1
\end{equation}
By evaluating this expression now for large $d$, we easily obtain that condition (3) is satisfied.
\end{proof}
It is also possible to state a proposition like (8) to the bounds on the distillable entanglement. $E^{1}$ will in this case provide an upper bound to $E_{D}$ of all bipartite partitions.
\section{Lower bounds for the entanglement of formation}
One of the most celebrated entanglement measures is the entanglement of formation \cite{bennett2}
\begin{equation}
E_{F}(\rho) = \min_{p_{1}, \psi_{i}}\sum_{i}p_{i}E_{E}(\ket{\psi_{i}}),
\end{equation}
where $E_{E}$ is the entropy of entanglement. Although this measure has a very meaningful physical interpretation and good properties, its calculation has been done only for a very class of states \cite{terhal5}. We show in this section that any entanglement witness can be used to provide lower bounds to the entanglement of formation.
Let $\rho = \ket{\Psi}\bra{\Psi}$ be a pure bipartite state with the following Schimdt decomposition:
\begin{equation}
\ket{\Psi} = \sum_{j=1}^{d} c_{j} \ket{jj} , \hspace{0.4 cm} c_{1} \geq c_{2} \geq ... \geq c_{d}.
\end{equation}
An analytic expression for the random robustness $R_{r}$ and the generalized robustness $R_{G}$ of a pure state given by equation (78) is \cite{vidal4}
\begin{equation}
R_{G}(\rho) = \left(\sum_{j=1}^{d}c_{j} \right)^{2} - 1,
\end{equation}
\begin{equation}
R_{r}(\rho) = c_{1}c_{2}.
\end{equation}
We start with two bounds for the entropy of entanglement, i.e., the von Neumann entropy of the reduced density matrix of a pure state $\ket{\psi}$. In the case of two qubits, Wootters has shown that \cite{wootters1}
\begin{equation*}
H\left( \frac{1 + \sqrt{1 - 4c_{1}^{2}c_{2}^{2}}}{2}\right) = E_{E}(\ket{\psi}),
\end{equation*}
where $H(x) = -x\log(x) - (1 - x)\log(1 - x)$. That is a particular case of the more general inequality
\begin{equation}
H\left(\frac{1 + \sqrt{1 - 4c_{1}^{2}c_{2}^{2}}}{2}\right) \leq -\sum_{i}^{d}c_{i}^{2}\log(c_{i}^{2}), \hspace{0.2 cm} \sum_{i}^{d}c_{i}^{2} = 1.
\end{equation}
Another similar inequality is
\begin{equation}
\frac{\log(d) - 1}{d}\left[\left(\sum_{i}^{d}c_{i}\right)- 1\right] \leq -\sum_{i}^{d}c_{i}^{2}\log(c_{i}^{2}).
\end{equation}
Equations (72-73) can be proved maximizing the L.H.S. minus the R.H.S. and noting that the maximum is null in both cases.
Choosing ${\cal f}p_{i}, \ket{\psi_{i}}{\cal g}$ to be an optimal ensemble in equation (71), we have
\begin{eqnarray}
E_{F}(\rho) = \sum_{i}p_{i}E_{E}(\ket{\psi_{i}}) \geq \sum_{i}p_{i}H\left(\frac{1 + \sqrt{1 - 4(c_{1}^{2})_{i}(c_{2}^{2})_{i}}}{2}\right) \nonumber \\
\geq H\left(\frac{1 + \sqrt{1 - 4R_{r}^{2}(\rho)}}{2}\right), \nonumber
\end{eqnarray}
where we have used the convexity of $R_{r}$ and $f(x) = H\left(\frac{1 + \sqrt{1 - 4x^{2}}}{2}\right)$. Similarly, we find
\begin{equation}
E_{F}(\rho) \geq \frac{\log(d) - 1}{d}R_{G}(\rho).
\end{equation}
The bound derived from $R_{r}$ is suitable for slightly entangled states, where the Schimdt coefficients of the optimal $\ket{\psi_{i}}$ decay fast enough, making the truncation in the second Schimdt coefficient a good approximation.
As an first example, we consider the Horodecki 3x3 states \cite{horodecki6}. These states exhibit bound entanglement, since they have positive partial transposition. They are given by
\begin{equation}
\rho(a) = \left[
\begin{array}{ccccccccc}
a & 0 & 0 & 0 & a & 0 & 0 & 0 & a \\
0 & 0 & a & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & a & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & a & 0 & 0 & 0 & 0 \\
a & 0 & 0 & 0 & a & 0 & 0 & 0 & a \\
0 & 0 & 0 & 0 & 0 & a & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & \frac{1 + a}{2} & 0 & \frac{\sqrt{1 - a^{2}}}{2}\\
0 & 0 & 0 & 0 & 0 & 0 & 0 & a & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & \frac{\sqrt{1 - a^{2}}}{2} & 0 & \frac{1 + a}{2}
\end{array}
\right]
\end{equation}
This family of states is interesting to test the first bound since their entanglement of formation was numerically calculated by Audenaert {\it et al} \cite{audenaert} and was found to be very low. Figure (7) shows the bound provided by $R_{r}$ for the states $\rho' = e\rho(a) + (1 - e)(I/D)$.
\begin{figure}
\begin{center}
\includegraphics[scale=0.45]{horodecki.eps}
\caption{(Coloronline) Lower bound for the Horodecki states using $R_{r}$.}
\end{center}
\end{figure}
For our second example we consider one of the unique states for which a analytic formula for $E_{F}$ is know. It was shown by Terhal and Vollbrecht \cite{terhal5} that for the isotropic states \cite{foot13}
\begin{equation*}
\rho_{F} = \frac{1 - F}{d^{2} - 1}\left( I - P^{+} \right) + FP^{+}
\end{equation*}
\begin{equation}
E_{F}(\rho_{F}) = \frac{d\log(d - 1)}{d - 2}(F - 1) + \log(d), \hspace{0.2 cm} F \in \left[\frac{4(d - 1)}{d^{2}}, 1\right]
\end{equation}
From section (VII) we find
\begin{equation}
E_{F}(\rho_{F}) \geq (\log(d) - 1)\left(F - \frac{1}{d} \right)
\end{equation}
We see that, in this case, for sufficiently large d, the difference of the bound and the actual value of $E_{F}$ is always less than F.
Note that every entanglement witness $W$, after being normalized such that either $Tr(W) = 1$ or $W \leq I$ holds, can be used to deliver lower bounds to the entanglement of formation.
\section{Entanglement, Thermodynamics and Lattice Systems}
The study of entanglement properties of many-body systems, manly condensed matter, has received much attention recently \cite{nielsen2,arnesen,wang1,wang5,bose1,connor1,osborne1,osterloh,toth1,vidal6,ghose,brukner1}. Several important models have been analyzed and connections with thermodynamic variables, such as internal energy and magnetization, have been raised \cite{wang5,bose1,connor1,osborne1,toth1}. The negativity and concurrence have been the most used measures, partly due their easy computation, but also because they made possible the derivation of some interesting simple {\it thermodynamics like} equations. This can be understood from the view of the {\it witnessed entanglement}. Every quantie derived from (7) not only defines a measure of the degree of entanglement, but also gives a Hermitian operator, which vary for each state, whose expectation value quantifies the entanglement of the state in question. It is exactly the possibility of measure experimentally the amount of entanglement, which is a feature shared by all common thermodynamics variables, that makes quantities expressed by (7) useful to the study of entanglement thermodynamical properties.
In this section we present, as an example, the study of entanglement in the XXX Heisenberg model with and without a magnetic field using $E_{\infty:1} = R_{G}$. The corresponding Hamiltonian is given by
\begin{equation}
H_{XXX} = J\sum_{i=1}^{N}\vec{\sigma}_{i}.\vec{\sigma}_{i+1} + B\sum_{i=1}^{N}\sigma_{i}^{z}.
\end{equation}
We first consider $B = 0$, in which case both Hamiltonians have $SU(2)$ symmetry. According to section (VII), we can restrict the EWs in (7) to the ones that also have $SU(2)$ symmetry. Then, from a standard result from representation theory \cite{eggeling1,weyl}, we find that all EWs with this symmetry can be written as
\begin{equation}
W = \sum_{i}\mu_{i}V_{i},
\end{equation}
where $V_{i}$ are unitary permutation operators. From analytic and numerical studies for the $XXX$ Heisenberg model of odd $N$ in the fundamental state and in the thermodynamical limit \cite{sakai}, we find that all other correlators are very small compared to the first neighbor correlators. We, thus, use the following ansatz for the optimal entanglement witness for the thermal states, at very low temperatures, of Hamiltonians (79) and (80)
\begin{equation}
W = \left(NI + \sum_{i=1}^{N}\left( \sigma_{i}^{x}\sigma_{i+1}^{x} + \sigma_{i}^{y}\sigma_{i+1}^{y} + \sigma_{i}^{z}\sigma_{i+1}^{z}\right)\right)/2N,
\end{equation}
where the factor 2$N$ in the denominator comes from $W \leq I$. Note that this is the EW introduced by Toth {\it et al} \cite{toth5}. Assuming that $|B| << |J|$, and using the continuity of OEWs, we find that for the XXX Heisenberg model, at temperatures sufficiently close to zero,
\begin{equation}
R_{G} \approx \frac{U - BM}{2NJ} - \frac{1}{2},
\end{equation}
where the magnetization and the internal energy are given, respectively, by $M = \sum_{i}\left<\sigma_{i}^{z}\right>$ and $U = \left<H\right>$.
We now proceed analysing the relation between entanglement and the magnetic susceptibility ($\chi$) in thermal states of $H_{2}$. According to Brukner, Vedral and Zeilinger \cite{brukner1}, under temperatures close to zero and at zero external magnetic field, $\chi = (g^{2}\mu_{B}^{2}/kT)[NI + (1/3)\sum_{i}\vec{\sigma_{i}}.\vec{\sigma_{i+1}}]$. Thus,
\begin{equation}
\chi \approx \frac{2Ng^{2}\mu_{B}^{2}}{3kT} + \frac{2NR_{G}}{3kT}.
\end{equation}
Remarkably, we see that the susceptibility is given by a term which resembles the classical Curie law more a term which takes into account the entanglement presented in the state. The equation above can be seen as a quantitative version of the experimental result of Ghose {\it et al} \cite{ghose}, who shown that at very low temperatures the magnetic susceptibility of certain materials is affected by the existence of entanglement.
\section{Conclusion}
Summarizing, we have presented a new perspective to the quantification of entanglement based on witness operators. Several important EMs were shown to fit into this scenario and a new infinite family of EMs was introduced. The usefulness of the {\it witnessed entanglement} was illustrated by the study of diverse features of entanglemnt, including super-selection rules constraints and efficiency of quantum information protocols. Finally, we have shown some interesting preliminary results in the study of thermodynamical properties of entanglement in macroscopic systems.
We believe the results presented in this paper are only preliminary. The quantification on entanglement with EWs might be a very fruitful approach to development of the theory of entanglement, specially in the new applications of entanglement, such as in identifying quantum phase transitions and improving the approximation of mean field theories \cite{vedral5}.
\section{acknowledgment}
The author would like to thank J. Eisert and D. Santos for helpful comments and especially Michal Horodecki for pointing out a mistake in a earlier version of this draft. Financial support from CNPQ is also acknowledged.
|
{
"timestamp": "2006-07-29T10:18:03",
"yymm": "0503",
"arxiv_id": "quant-ph/0503152",
"language": "en",
"url": "https://arxiv.org/abs/quant-ph/0503152"
}
|
\section{Introduction}
Recent Au+Au collision experiments at the Relativistic Heavy Ion
Collider (RHIC) saw a dramatic suppression of hadrons with high
transverse momenta (``high-$p_T$ suppression'') \cite{Whitepapers},
and the quenching of jets in the direction opposite
to a high-$p_T$ trigger particle \cite{STARjetqu,PHENIXjetqu}, when
compared with p+p and d+Au collisions. This is taken
as evidence for the creation of a very dense, color opaque medium
of deconfined quarks and gluons \cite{QGP3jetqu}. Independent
evidence for the creation of dense, thermalized quark-gluon
matter, yielding comparable estimates for its initial energy density
($\langle e\rangle\agt 10$\,GeV/fm$^3$ at time
$\tau_{\rm therm}\alt0.6$\,fm/$c$ \cite{QGP3v2}), comes
from the observation of strong elliptic flow in non-central Au+Au
collisions \cite{Whitepapers}, consistent with ideal fluid dynamical
behaviour of the bulk of the matter produced in these collisions.
These two observations raise the question what happens, in the small
fraction of collision events where a hard scattering produces a pair
of high-$p_T$ partons, to the energy lost by the parton travelling
through the medium. The STAR Collaboration has shown that, while
in central Au+Au collisions there are no {\em hard} particles left in the
direction opposite to a high-$p_T$ trigger particle, one sees enhanced
production (compared to p+p) of {\em soft} (low-$p_T$) particles, broadly
distributed over the hemisphere diametrically opposite to the trigger
particle \cite{STARjet_therm}. This shows that the energy of the fast
parton originally emitted in the direction opposite to the trigger
particle is not lost, but severely degraded by interactions with
the medium. As the impact parameter of the collisions decreases,
the average momentum of the particles emitted opposite to
the trigger particle approaches the mean value associated with
{\em all} soft hadrons, i.e. the $\langle p_T\rangle$ of the
thermalized medium \cite{STARjet_therm}. This suggests that the
energy lost by the fast parton has been largely thermalized.
Nevertheless, this energy is deposited locally along the fast
parton's trajectory, leading to local energy density inhomogeneities
which, if thermalized, should in turn evolve hydrodynamically.
This would modify the usual hydrodynamic expansion of the
collision fireball as observed in the overwhelming number of soft
collision events where no high-$p_T$ partons are created.
Since the fast parton moves at supersonic speed,
it was suggested in Ref.~\cite{shuryak} that a Mach shock (``sonic
boom'') should develop, resulting in {\em conical flow} and preferred
particle emission at a specific angle away from the direction of the
fast parton which lost its energy. This Mach angle is sensitive to
the medium's speed of sound $c_s$ and thus offers the possibility to
measure one of its key properties. A recent analysis by the PHENIX
Collaboration \cite{Adler:2005ee} of azimuthal di-hadron correlations
in $200\,A$\,GeV Au+Au collisions revealed structures in the angular
distribution which might be suggestive of conical flow.
The idea of Mach shock waves travelling through compressed nuclear
matter was first advocated 30 years ago \cite{shock,SGM74},
but RHIC collisions for the first time exhibit \cite{QGP3v2}
the kind of ideal fluid behaviour which might make an extraction
of the speed of sound conceivable. An alternate scenario, in which
the color wake field generated by
the fast colored parton travelling through a quark-gluon plasma
accelerates soft colored plasma particles in the direction
perpendicular to the wake front \cite{Stocker,RM05}, leads to an
emission pattern which is sensitive to the propagation of
{\em plasma} rather than {\em sound} waves \cite{RM05}. In a
strongly coupled plasma with overdamped plasma oscillations,
which seems to be the preferred interpretation of RHIC data
\cite{SQCD,sQGP}, the wake field scenario should reduce to the
hydrodynamic Mach cone picture. We here study the dynamical
consequences of the latter, going beyond the discussion of
linearized hydrodynamic equations in a static background
offered in \cite{shuryak}.
We assume that just before hydrodynamics become applicable, a pair
of high-$p_T$ partons is produced near the surface of the fireball.
One of them moves outward and escapes, forming the trigger jet, while
the other enters into the fireball along, say, the $-x$ direction.
The fireball is expanding and cooling. The ingoing parton travels
at the speed of light and loses energy in the fireball which
thermalizes and acts as a source of energy and momentum for
the fireball medium. We model this medium as an ideal fluid with
vanishing net baryon density. Its dynamics is controlled by
the energy-momentum conservation equations
\begin{equation}
\label{1}
\partial_\mu T^{\mu\nu} =J^\nu,
\end{equation}
where the energy-momentum tensor has the ideal fluid form
$T^{\mu\nu}\eq(\varepsilon{+}p)u^\mu u^\nu -p g^{\mu\nu}$,
with energy density $\varepsilon$ and pressure $p$ being
related by the equation of state (EOS) $p{\eq}p(\varepsilon)$,
$u^\mu\eq\gamma(1,v_x,v_y,0)$ is the fluid 4-velocity, and
the source current $J^\nu$ is given by
\begin{eqnarray}
\label{2}
&&J^\nu(x)=J(x)\,\bigl(1,-1,0,0\bigr),\\
\label{3}
&&J(x) = \frac{dE}{dx}(x)\, \left|\frac{dx_{\rm jet}}{dt}\right|
\delta^3(\bm{r}-\bm{r}_{\rm jet}(t)).
\end{eqnarray}
Massless partons have light-like 4-momentum, so the current $J^\nu$
describing the 4-momentum lost and deposited in the medium by the
fast parton is taken to be light-like, too. $\bm{r}_{\rm jet}(t)$ is
the trajectory of the jet moving with speed $|dx_{\rm jet}/dt|\eq{c}$.
$\frac{dE}{dx}(x)$ is the energy loss rate of the parton as it moves
through the liquid. It depends on the fluid's local rest
frame particle density. Taking guidance from the
phenomenological analysis of parton energy loss observed in Au+Au
collisions at RHIC \cite{Eloss} we take
\begin{equation}
\label{4}
\frac{dE}{dx} = \frac{s(x)}{s_0} \left.\frac{dE}{dx}\right|_0
\end{equation}
where $s(x)$ is the local entropy density without the jet.
The measured suppression of high-$p_T$ particle production in Au+Au
collisions at RHIC was shown to be consistent with a parton energy
loss of $\left.\frac{dE}{dx}\right|_0\eq14$\,GeV/fm at a reference
entropy density of $s_0\eq140$\,fm$^{-3}$ \cite{Eloss}. For comparison,
we also perform simulations with ten times larger energy loss,
$\left.\frac{dE}{dx}\right|_0\eq140$\,GeV/fm.
For the hydrodynamic evolution we use AZHYDRO \cite{AZHYDRO,QGP3v2},
the only publicly available relativistic hydrodynamic code for
anisotropic transverse expansion. This algorithm is formulated in
$(\tau,x,y,\eta)$ coordinates, where $\tau{=}\sqrt{t^2{-}z^2}$ is the
longitudinal proper time,
$\eta{=}\frac{1}{2}\ln\left[\frac{t{+}z}{t{-}z}\right]$
is space-time
rapidity, and $\bm{r}_\perp{\,=\,}(x,y)$ defines the plane transverse to the
beam direction $z$. AZHYDRO employs longitudinal boost invariance
along $z$ but this is violated by the source term (\ref{3}). We
therefore modify the latter by replacing the $\delta$-function
in (\ref{3}) by
\begin{eqnarray}
\label{5}
\delta^3(\bm{r}-\bm{r}_{\rm jet}(t)) &\longrightarrow&
\frac{1}{\tau}\,\delta(x-x_{\rm jet}(\tau))\,\delta(y-y_{\rm jet}(\tau))
\nonumber\\
&\longrightarrow&\frac{1}{\tau} \,
\frac{e^{-(\bm{r}_\perp-\bm{r}_{\perp,{\rm jet}}(\tau))^2/(2\sigma^2)}}
{2\pi\sigma^2}
\end{eqnarray}
with $\sigma{\,=\,}0.35$\,fm. Intuitively, this replaces the ``needle''
(jet) pushing through the medium at one point by a ``knife'' cutting the
medium along its entire length along the beam direction. The resulting
``wedge flow'' is expected to leave a stronger signal in the azimuthal
particle distribution $dN/d\phi$ than ``conical flow'' induced
by a single parton, since in the latter case one performs an implicit
$\phi$-average when summing over all directions of the cone normal
vector. While a complete study of this would require a full
(3+1)-dimensional hydrodynamic calculation, the present boost-invariant
simulation should give a robust upper limit for the expected angular
signatures. We show that the angular structures predicted from
wedge flow are too weak to explain the experimentally observed
$\phi$-modulation \cite{Adler:2005ee}.
The modified hydrodynamic equations in $(\tau,x,y,\eta)$ coordinates
read \cite{AZHYDRO}
\begin{eqnarray}
\label{6}
\partial_\tau \tilde{T}^{\tau \tau} +
\partial_x(\tilde{v}_x \tilde{T}^{\tau \tau}) +
\partial_y(\tilde{v}_y \tilde{T}^{\tau \tau})
&=& - p + \tilde{J},
\\
\label{7}
\partial_\tau \tilde{T}^{\tau x} +
\partial_x(v_x \tilde{T}^{\tau x}) +
\partial_y(v_y \tilde{T}^{\tau x})
&=& - \partial_x \tilde{p} - \tilde{J}, \quad
\\
\label{8}
\partial_\tau \tilde{T}^{\tau y} +
\partial_x(v_x \tilde{T}^{\tau y}) +
\partial_y(v_y \tilde{T}^{\tau y})
&=& -\partial_y \tilde{p}, \quad
\end{eqnarray}
where $\tilde{T}^{\mu\nu}\eq\tau T^{\mu\nu}$,
$\tilde{v}_i{\eq}T^{\tau i}/T^{\tau\tau}$,
$\tilde p\eq\tau p$, and $\tilde{J}\eq\tau J$.
To simulate central Au+Au collisions at RHIC, we use the standard
initialization described in \cite{QGP3v2} and provided in the
downloaded AZHYDRO input file \cite{AZHYDRO}, corresponding to a
peak initial energy density of $\varepsilon_0\eq30$\,GeV/fm$^3$ at
$\tau_0\eq0.6$\,fm/$c$. We use the equation of state EOS-Q described
in \cite{QGP3v2,AZHYDRO} incorporating a first order phase transition
and hadronic chemical freeze-out at a critical temperature
$T_c{\,=\,}164$\,MeV. The hadronic sector of EOS-Q is soft with a
squared speed of sound $c_s^2 \approx 0.15$.
\begin{figure}[t]
\includegraphics[bb=14 50 581 820,width=0.99\linewidth,clip]{Fig1.eps}
\caption{Contours of constant local energy density in the $x$-$y$ plane
at three different times, $\tau{\,=\,}4.6,$ 8.6, and 12.6 fm/$c$.
In each case the position of the fast parton, along with the integrated
energy loss $\Delta E=\int J(x) dxdy d\tau$ up to this point, is indicated
at the top of the figure. Diagrams (a)-(c) in the left column were
calculated with a reference energy loss $dE/dx|_0{\,=\,}14$\,GeV/fm,
those in the right column (panels (d)-(f)) with a 10 times larger value.
}
\label{F1}
\end{figure}
In our study the quenching jet starts from $x_{\rm jet}\eq6.4$\,fm
at $\tau_0\eq0.6$\,fm, moving left towards the center with constant
speed $v_{\rm jet}\eq{c}$. For an upper limit on conical flow effects,
the fast parton is assumed to have sufficient initial energy to emerge
on the other side of the fireball. To simulate cases where the fast
parton has insufficient energy to fully traverse the medium we have
also done simulations where the parton loses all its energy within an
(arbitrarily chosen) distance of 6.4 fm. We further compared with a
run where the parton moved at (constant) subsonic speed
($v_\mathrm{jet}{\,=\,}0.2\,c{\,<\,}c_s$).
The resulting evolution of the energy density of the QGP fluid is
shown in Figs.~\ref{F1} and \ref{F2}. The left column of Fig.~\ref{F1}
shows results for a phenomenologically acceptable value
$\left.\frac{dE}{dx}\right|_0\eq14$\,GeV/fm \cite{Eloss} for the
reference parton energy loss whereas in all other columns we use a
ten times larger energy loss. The width of the Gaussian source
(see Eq.~(\ref{5})) is $\sigma$=0.7 fm. In the left column of
Fig.~\ref{F1} the effects of the energy deposition from the fast
parton are hardly visible. Only for a much larger energy loss (right
column) we recognize a clear conical flow pattern. The accumulating
wave fronts from the expanding energy density waves build up a
``sonic boom'' shock front which creates a Mach cone. The right columns
in Figs.~\ref{F1} and \ref{F2} show that the cone normal vector forms
an angle $\theta_M$ with the direction of the quenching jet that is
qualitatively consistent with expectations from the theoretical relation
$\cos\theta_M{\eq}c_s/v_\mathrm{jet}$. However, this angle is not
sharply defined since the cone surface curves due to
inhomogeneity and radial expansion of the underlying medium.
This differs from the static homogeneous case \cite{shuryak}.
\begin{figure}[t]
\includegraphics[bb=14 50 581 820,width=0.99\linewidth,clip]{Fig2.eps}
\caption{
As in Fig.\ref{F1} but for a parton moving at subsonic speed
$v_\mathrm{jet}{\,=\,}0.2\,c$ (left column) and for a fast parton
($v_\mathrm{jet}{\,=\,}c$) which loses all its energy within
the first 6.4\,fm (right column).
}
\label{F2}
\end{figure}
When the parton travels at a subsonic speed
$v_\mathrm{jet}{\,=\,}0.2\,c{\,<\,}c_s$ (left column in Fig.~\ref{F2}),
it doesn't get very far before the fireball freezes out due to
longitudinal expansion. In this case one only observes an accumulation
of energy around the parton but no evidence of Mach cone formation.
When the parton travels with $v_\mathrm{jet}{\,=\,}c$ but looses all
its energy after 6.4\,fm before fully traversing the fireball (right
column in Fig.~\ref{F2}), the fireball evolution beyond $\tau$=7\,fm
is not affected by the fast parton directly but only indirectly through
the propagation of earlier deposited energy. Still, Figs.~\ref{F1}f
and \ref{F2}f show that the late time evolution of the fireball is
quite similar in both cases, demonstrating that energy deposition by
the fast parton during the late fireball stages is small, due to
dilution of the matter, and can almost be neglected.
Tests with different values for the width $\sigma$ of the Gaussian
profile in Eq.~(\ref{5}) for the deposited energy show
that the cone angle gets better defined for smaller source size
$\sigma$. Note that the quenching jet destroys the azimuthal symmetry
of the initial energy density distribution but leaves the azimuthally
symmetric energy contours to the left of the jet unaffected.
We close with a discussion of observable consequences of conical
flow. One expects \cite{shuryak} azimuthally anisotropic particle
emission, peaking at angles $\phi\eq\pi{\,\pm\,}\theta_M$ relative to
the trigger jet where $\theta_M$ is the Mach angle. Using the standard
Cooper-Frye prescription, we have computed the angular distribution of
directly emitted pions at a freeze-out temperature
$T_\mathrm{fo}{\,=\,}100$\,MeV \cite{QGP3v2}.
\begin{figure}[t]
\includegraphics[bb=14 13 550 800,width=0.99\linewidth,clip=]%
{Fig3.eps}
\vspace{-4.5cm}
\caption{Azimuthal distribution $dN/dy d\phi$ of negative pions
per unit rapidity. In the upper panel we integrate over all $p_T$
while the lower panel shows only pions with $p_T{\,>\,}1$\,GeV/$c$.
Different symbols refer to different parameters as indicated. For
better visibility the $\phi$-independent rate in the absence
of the quenching jet has been subtracted. Filled symbols show the
realistic case $dE/dx|_0{\,=\,}14$\,GeV/fm, enhanced by a factor 10
for visibility.
}
\label{F3}
\end{figure}
Figure~\ref{F3} shows the azimuthal distribution of $\pi^-$ from a
variety of different simulations. [The $\phi$-independent constant
$(dN_{\pi^-}/dy d\phi)_\mathrm{no\ jet}=27$ from central collisions
without jets has been subtracted.] For $dE/dx|_0=14$\,GeV/fm the
azimuthal modulation is very small. In none of the cases studied we
find peaks at the predicted Mach angle with an associated dip in the
direction of the quenched jet at $\phi\eq\pi$. One rather sees a
{\em peak} at $\phi\eq\pi$, broadened by shoulders on both sides
which turn into small peaks at (relative to the quenching jet)
backward angles if thermal smearing is reduced by considering only
high-$p_T$ pions. The peak at $\phi\eq\pi$
is absent when the parton loses all its energy halfway through the
fireball or is too slow to get to the other side before freeze-out,
suggesting that it reflects the directed momentum imparted on the
medium by the fast parton. It is slightly more accentuated for smaller
$\sigma$ and higher $p_T$. We also found that the width of the shoulders
is almost independent of the speed of sound of the medium and can not be
used to diagnose the stiffness of its equation of state. The shoulders
exist even for subsonic parton propagation ($v_\mathrm{jet}\eq0.2\,c$,
upright triangles), showing that other mechanisms (such as backsplash
from the hard parton hitting the fluid and a general bias of the energy
deposition towards the right side of the fireball due to the higher
density of the medium at early times) have a strong influence on the
angular distribution of the emitted particles which interferes
with the position of the Mach peaks. The shoulder resulting from
this combination of effects is much broader than the angular
structures seen in the data \cite{Adler:2005ee}. The absence of
a clear dip at $\phi\eq\pi$ in our simulations is all the more troubling
since it should have been stronger for the ``wedge flow'' studied
here than for real ``conical'' flow.
Our calculation does not average over the initial production points
of the trigger particle, i.e. it ignores that in most cases its
quenching partner does not travel right through the middle of the
fireball cylinder, but traverses it semi-tangentially. This should
further decrease the prominence of the shoulders in $dN/d\phi$. We
conclude that conical flow may be able to explain the broadening
of the away-side peak in the hadron angular correlation function
around $\phi\eq\pi$ pointed out by the STAR Collaboration
\cite{STARjet_therm}, but is unlikely to be responsible for the
relatively sharp structures near $\phi\eq\pi{\pm}1$
seen by PHENIX \cite{Adler:2005ee}. This conclusion extends to
other conical flow phenomena, such as those generated by color wake
fields \cite{Stocker,RM05}.
This work was supported by the U.S. Department of Energy under contract
DE-FG02-01ER41190.
|
{
"timestamp": "2006-04-10T14:47:39",
"yymm": "0503",
"arxiv_id": "nucl-th/0503028",
"language": "en",
"url": "https://arxiv.org/abs/nucl-th/0503028"
}
|
\section{Introduction}
Let $(X,o)$ be a normal complex surface singularity germ.
Let $\Gamma$ and $\Sigma$ denote the resolution graph and the link
of $(X,o)$, respectively.
We assume that $X$ is homeomorphic to the cone over $\Sigma$.
It is known that the resolution graph and
the link of the singularity determine each other
(\cite{neumann.plumbing}).
Assume that the link $\Sigma$ is a $\mathbb Q$-homology sphere,
or equivalently, that
the exceptional set of a good resolution is a tree of
rational curves.
Then $H_1(\Sigma, \mathbb Z)$ is finite.
A morphism $(Y,o) \to (X,o)$ of germs of normal surface
singularities is called a {\itshape universal abelian
covering} if it induces an unramified Galois covering
$Y \setminus \{o\} \to X \setminus \{o\}$ with
covering transformation group $H_1(\Sigma,\mathbb Z)$.
By our assumption, the universal abelian
covering $(Y,o) \to (X,o)$ must exist; in fact, the link of $Y$ is
the universal abelian cover of $\Sigma$ in the topological sense.
We are interested in the analytic properties of $Y$ and
a way to construct $Y$ explicitly.
In the case that $(X,o)$ is quasihomogeneous,
Neumann \cite{neumann.abel} proved that the universal abelian cover
$(Y,o)$ is a Brieskorn-Pham complete intersection, by writing down the
explicit equations from the data of $\Sigma$
(it is known that $\Gamma$ is star-shaped in this case).
Neumann and Wahl generalized Brieskorn-Pham complete intersections
and the way to construct them, and obtained numerous
interesting results;
see \cite{nw-uac},
\cite{nw-HSL}, \cite{nw-qcusp}, \cite{nw-CIuac}.
They introduced the {\itshape splice diagram equations}
(or {\itshape forms}) associated with a
weighted tree called a {\itshape splice diagram}
satisfying the ``semigroup condition''.
From an arbitrary resolution graph corresponding to a $\mathbb Q$-homology
sphere link, we can construct a splice diagram.
Let $\widetilde Y$ denote the singularity defined by the splice diagram
equations obtained from $\Gamma$.
They proved that $\widetilde Y$ is an isolated complete intersection surface
singularity, and that (under ``congruence condition'')
if the equations are chosen
so that the {\itshape discriminant
group} $G$ ($\cong H_1(\Sigma, \mathbb Z)$) for $\Gamma$ naturally acts on
$\widetilde Y$, then the quotient $\widetilde Y/G$ is a normal surface
singularity (it is called a {\itshape splice-quotient
singularity}) with resolution graph $\Gamma$, and the
quotient morphism is the universal abelian covering.
Neumann and Wahl conjectured that rational singularities and
minimally elliptic singularities with $\mathbb Q$-homology sphere links are
splice-quotient singularities.
However, it is not known whether the splice diagrams obtained
from the resolution graphs of those singularities satisfy
the semigroup condition.
In this paper we prove that
the universal abelian cover of a rational or minimally elliptic
singularity is a complete intersection singularity defined
by certain special functions.
Let $\pi\: M \to X$ be a good resolution with the exceptional set $A$.
Under a topological condition (\condref{c:A}),
we can associate a collection of certain
special polynomials and a system of weights with each node of $A$.
These polynomials are quasihomogeneous with respect to the weights.
We call the union of those collections over all nodes a
Neumann-Wahl system,
which is an analogue of the system of splice diagram equations.
Though the definition of a Neumann-Wahl system is very similar to
that of splice diagram equations,
they are not the same (see \remref{r:diff}).
We suspect that a Neumann-Wahl system is a special type of
system of splice diagram forms.
Our first result is the following (see \thmref{t:V}).
\begin{thm}\label{t:Vintro}
Let $(V,o)$ be a singularity defined by a Neumann-Wahl system.
Then $(V,o)$ is an isolated complete intersection surface
singularity.
A singularity defined by functions obtained by adding
``higher terms'' to the Neumann-Wahl system is an
equisingular deformation of $(V,o)$.
\end{thm}
We call a singularity defined by a Neumann-Wahl system a
Neumann-Wahl complete intersection.
If in addition a certain analytic condition (\condref{c:B}) and a
topological condition (\condref{c:C}),
which is stronger than \condref{c:A}, are
satisfied, then the universal abelian
cover $Y$ is an equisingular deformation of a Neumann-Wahl complete
intersection (\thmref{t:main-general}). The equations of
$Y$ are constructed on the resolution space $M$.
Our equations for the deformation are automatically
equivalent with respect to a natural
action of the discriminant group $G$, and
the action is free on nonsingular locus.
An important point is
that \condref{c:C} and \ref{c:B} are satisfied in case $(X,o)$ is
a rational or minimally elliptic singularity (it can be
easily verified!).
Thus we have the following (see \thmref{t:main} and
\proref{p:esquotient}).
\begin{thm}\label{t:mainintro}
If $(X,o)$ is a rational or minimally elliptic singularity, then its
universal abelian cover $(Y,o)$ is an equisingular deformation of a
Neumann-Wahl complete intersection singularity $(Y_0,o)$.
Moreover the $(X,o)$ is an equisingular deformation of $(Y_0/G,o)$.
\end{thm}
This paper is organized as follows. In \sref{s:pre}, we
briefly review fundamental results on the universal abelian
covers of normal surface singularities in \cite{o.uac-rat}.
We recall there that $\mathcal O_{Y,o}$ is
isomorphic to an $\mathcal O_{X,o}$-algebra $\mathcal A:=\bigoplus _{b \in
\mathcal B}H^0(-L^{(b)})$, where $\mathcal B$ is a group isomorphic to $G$
and $L^{(b)}$ are suitable divisors on $M$.
In \sref{s:NWS}, we first define monomial cycles.
The semigroup of monomial cycles is naturally
isomorphic to that of monomials; the variables are
associated with the ends of $A$.
We show that \condref{c:C} implies \condref{c:A}, and is
satisfied for rational or minimally elliptic singularities.
Then we introduce the Neumann-Wahl systems and the weights.
In \sref{s:VdefbyF}, we prove a slight generalization of
\thmref{t:Vintro}; there
we consider a system of polynomials obtained by substituting
some power of the variables into the variables of a
Neumann-Wahl system.
We will apply some ideas in \cite[\S 2]{nw-CIuac} for the proof.
In the last section, we prove \thmref{t:mainintro}.
We construct the equations by looking at certain relations
of sections of $H^0(-L^{(b)})$'s.
We can take a good basis of the algebra $\mathcal A$ by
\condref{c:B}, and can
explicitly obtain the relations by \condref{c:C}.
\section{Preliminaries}\label{s:pre}
In this section, we recall some fundamental results on the universal
abelian covers of surface singularities; see
\cite{o.uac-rat} for details.
Let $(X,o)$ be a normal complex surface singularity germ.
We assume that the link $\Sigma$ of $(X,o)$ is a $\mathbb Q$-homology sphere
and that $X$ is homeomorphic to a cone over $\Sigma$.
Then $X$ has a unique universal abelian cover.
Let $\pi\: M \to X$ be a resolution of the singularity, and let
$A=\bigcup _iA_i$ be the decomposition of the exceptional set
$A=\pi^{-1}(o)$ into irreducible components.
Assume that $\pi$ is a good resolution, i.e.,
$A$ is a divisor having only simple normal crossings.
Then the condition that $\Sigma$ is a $\mathbb Q$-homology sphere is
equivalent to that $H^1(\mathcal O_A)=0$, i.e., $A$ is a tree of
nonsingular rational curves.
We call a divisor supported in $A$ a cycle.
Let $A_{\mathbb Z}$ denote the group of cycles.
An element of $A_{\mathbb Q}:=A_{\mathbb Z}\otimes \mathbb Q$ is called a $\mathbb Q$-cycle.
Let $\du i \in A_{\mathbb Q}$ denote the dual cycle of $A_i$, i.e.,
the $\mathbb Q$-cycle
satisfying $\du i\cdot A_j=-\delta _{ij}$,
where $\delta _{ij}$ denotes the Kronecker delta.
We denote by $\du{\mathbb Z}$ the subgroup of $A_{\mathbb Q}$ generated by
$\du i$'s.
Recall that the first homology group $H_1(\Sigma, \mathbb Z)$, the Galois
group of the universal abelian covering of $X$, is isomorphic
to the group $\du {\mathbb Z}/A_{\mathbb Z}$ called the discriminant group.
The order of the group is $|\det(A_i\cdot A_j)|$.
Let $D$ be a $\mathbb Q$-divisor on $M$.
We denote by $\nu(D)$ the $\mathbb Q$-cycle
satisfying $(\nu(D)-D)\cdot A_i=0$ for all $A_i$.
We say that $D$ is $\pi$-anti-nef if $-D$ is $\pi$-nef,
i.e., $D\cdot A_i \le 0$ for all $A_i$.
Let $ \mathcal F (D)$ denote the set of $\pi$-anti-nef $\mathbb Q$-divisors $F$
satisfying $F-D\in A_{\mathbb Z}$.
Note that $\mathcal F(D)$ has the minimum with respect to ``$\ge$''.
We take effective $\mathbb Q$-cycles $E_1, \ldots , E_s$ such that
if $\mathcal E_i$
denotes the cyclic subgroup of $\du {\mathbb Z}/A_{\mathbb Z}$ generated by $E_i$,
then $\du {\mathbb Z}/A_{\mathbb Z}=\mathcal E_1 \oplus \cdots \oplus \mathcal E_s$.
Let $r_i$ be the order of $\mathcal E_i$.
Then for $1 \le i \le s$ there exists a divisor
$L_i$ and a function $f_i$ on a suitable neighborhood
of $A$ such that $r_iL_i- r_iE_i=\di (f_i)$.
For any $b=(b_1,\dots,b_s)\in \mathbb Z^s$, we define a divisor
$L^{(b)}$ by
$$
L^{(b)}=\sum_{j=1}^s b_jL_j-\left[\sum_{j=1}^sb_jE_j\right].
$$
Let $\bar b_i$ denote the smallest
nonnegative integer such that $r_i \mid b_i -\bar b_i$.
Let $\bar b= (\bar b_1, \dots , \bar b_s)$ and
$\mathcal B=\{\bar b| b \in \mathbb Z^s\}$; the set $\mathcal B$ is identified with the
discriminant group $\du
{\mathbb Z}/A_{\mathbb Z}$.
We define a sheaf $\bar \mathcal A$ of $\mathcal O_M$-modules by
$$
\bar \mathcal A=\bigoplus _{b \in \mathcal B}\mathcal O_M(-L^{(b)}).
$$
The $\mathcal O_M$-algebra structure of $\bar \mathcal A $ is given by the composite
$$
\mathcal O_M(-L^{(b)}) \otimes \mathcal O_M(-L^{(b')}) \to \mathcal O_M(-L^{(b)}-L^{(b')})
\subset \mathcal O_M(-L^{(b+b')})
$$
and the isomorphism
$$
\mathcal O_M(-L^{(b+b')}) \to \mathcal O_M(-L^{(\overline{b+b'})})
$$
given by multiplying by $\prod
_{b_i+b_i'\neq\overline{b_i+b'_i}}f_i^{-1}$.
Then the natural projection
$$
Y:=\specan_X \pi_*\bar\mathcal A \to X
$$
is the universal abelian covering
(see \cite[Theorem 3.4]{o.uac-rat}).
The local ring $\mathcal O_{Y,o}$ of the singularity $(Y,o)$ is
isomorphic to
$$
\mathcal A:=(\pi_*\bar\mathcal A )_o=\bigoplus _{b \in \mathcal B}H^0(-L^{(b)}),
$$
where
$H^0(-L^{(b)})=\dlim _U H^0(U,\mathcal O_M(-L^{(b)}))$,
$U$ varies over all open neighborhoods of $A$.
We write $\mathcal A_b=H^0(-L^{(b)})$.
Let $h \in \mathcal A_b$.
If $\di (h)-L^{(b)}-D$ has no component of $A$ for some
cycle $D\in A_{\mathbb Z}$,
then we write $(h)_A=\nu(L^{(b)})+D$.
\begin{lem}\label{l:product}
Let $\sigma_i \in \mathcal A_{b^i}$, $i=1,2$, and
let $\sigma_1\cdot \sigma_2\in \mathcal A_{\overline{b^1+b^2}}$ be
the product of $\sigma_1$ and $\sigma_2$ in the algebra $\mathcal A$.
Suppose that a divisor $L \in \mathcal F(L^{( \overline{b^1+b^2})})$
satisfies
$\nu(L)=(\sigma_1)_A+(\sigma_2)_A$.
Then $\sigma_1\cdot \sigma_2$
is a section of $H^0(-L) $
and $(\sigma_1\cdot \sigma_2)_A=\nu(L)$.
\end{lem}
\begin{proof}
It follows from the definition of the algebra structure of
$\mathcal A$.
\end{proof}
Let $D$ be a reduced and connected cycle.
A component $A_i$ of $D$ is called an {\itshape end} of $D$
if $(D-A_i)\cdot A_i\le 1$.
We denote by $\mathcal E(D)$ the set of ends of $D$.
Assume that $\mathcal E(A)=\{A_1, \dots ,A_m\}$.
For any $1 \le i \le m$, there uniquely exists a divisor
$L^i$ such that $\nu
(L^i)=\du i$ and $L^i \in \mathcal F(L^{(b)})$ for some $b \in \mathcal B$.
If $(X,o)$ is rational, then there exists $y_i \in H^0(-L^i)$
such that $(y_i)_A=\nu(L^i)$ (cf. \lemref{l:satisfyB}).
The next theorem follows from \cite[Theorem 7.5]{o.uac-rat} and its
proof.
\begin{thm}\label{t:repOY}
Assume that $(X,o)$ is rational.
Let $y_i \in H^0(-L^i)$, $1 \le i \le m$, be as above.
Let $S=\mathbb C\{x_1, \dots ,x_m\}$ be the convergent power series
ring.
We define a homomorphism $\psi \: S \to \mathcal A=\mathcal O_{Y,o}$ of
$\mathbb C$-algebras by $\psi(x_i)=y_i$.
Then $\psi$ is surjective.
\end{thm}
In the last section, we give a proof of \thmref{t:repOY},
which is different from that in \cite{o.uac-rat}.
In fact it is shown that the
assertion also holds true for minimally elliptic singularities.
\section{Neumann-Wahl systems}\label{s:NWS}
In this section we will introduce a Neumann-Wahl system
associated with the exceptional set $A$.
It is a set of certain polynomials,
and an analogue of the system of splice diagram
equations in Neumann and Wahl's work (\cite{nw-uac}, \cite{nw-HSL},
\cite{nw-CIuac}).
We use the notation of the preceding section, and keep
the assumption that $H^1(\mathcal O_A)=0$.
First assume that the set $\mathcal E(A)$ consists of the
components $A_1, \dots ,A_m$.
Let $\mathbb C[x_1, \ldots ,x_m]$ be the polynomial ring.
\thmref{t:repOY} suggests us the following
\begin{defn}
Let $D=\sum a_i\du i \in A_{\mathbb Q}$, $a_i \ge 0$.
If $a_i=0$ for all $A_i \notin \mathcal E(A)$,
then we call $D$ a {\itshape $\mathbb Q$-monomial cycle};
if in addition $a_i \in \mathbb Z$ for all $i$, we call $D$ a
{\itshape monomial cycle}.
For any monomial cycle $D=\sum _{i=1}^ma_i\du i$, we associate a
monomial
$$
x(D):=\prod _{i=1}^mx_i^{a_i} \in \mathbb C[x_1, \ldots ,x_m].
$$
The $x$ induces an isomorphism between the semigroup of
monomial cycles and that of monomials of $x_1,
\ldots, x_m$.
Formally, we may also consider the $\mathbb Q$-monomial $x(D)$ for a
$\mathbb Q$-monomial cycle $D$.
\end{defn}
\begin{defn}
For any $F=\sum a_kA_k \in A_{\mathbb Q}$, we write $m_{A_k}(F)=a_k$.
For any component $A_j$, we define the {\itshape
$A_{j}$-weight} of $x_i$ to be $m_{A_{j}}(\du i)$.
Then the {\itshape $A_{j}$-degree} of a monomial $x(D)$ is
defined to be $m_{A_{j}}(D)$.
\end{defn}
A connected component of
$A-A_i$ is called a {\itshape branch} of $A_i$.
A component $A_i$ is called a {\itshape node} if $(A-A_i)\cdot
A_i\ge 3$.
We consider the following conditions concerning the weighted
dual graph of $A$.
\begin{cond}\label{c:A}
For any branch $C$ of any node $A_i$,
there exists a monomial cycle $D$
such that $D-\du i$ is an effective integral cycle
supported on $C$;
in this case, we say that $D$ (or monomial
$x(D)$) belongs to the branch $C$.
\end{cond}
Note that in general there may exist more than one monomials
belonging to a branch.
\begin{cond}\label{c:C}
$A$ is star-shaped, or for any branch $C$ of any component
$A_i\notin \mathcal E(A)$, the fundamental cycle $Z_C$ supported
on $C$ satisfies $Z_C\cdot A_i=1$.
\end{cond}
\begin{lem}\label{l:existmonomials}
\condref{c:C} implies \condref{c:A},
and is satisfied in the following cases:
\begin{enumerate}
\item $(X,o)$ is a rational singularity;
\item $(X,o)$ is a minimally elliptic singularity, and
the minimally elliptic cycle is supported on $A$
(this condition is satisfied on the minimal good resolution).
\end{enumerate}
\end{lem}
\begin{proof}
Assume that the condition (1) or (2) is satisfied.
From basic results on the computation sequences for the
fundamental cycle
(\cite{la.rat}, \cite{la.me}), we obtain
\condref{c:C}.
Suppose that $C_{1}$ is a branch of a node $A_{i_1}$.
Then $D_1:=\du {i_1}+Z_{C_1}$ is $\pi$-anti-nef and $D_1 \cdot A_j=0$
for every $A_j \le A-C_1$.
If $D_1 \cdot A_{i_2}<0$ for $A_{i_2} \notin \mathcal E(A)$, then take a
branch $C_2$ of $A_{i_2}$, not
containing $A_{i_1}$, and put $D_2:=D_1+Z_{C_2}$.
In this manner we obtain a finite sequence $\{D_1, \ldots ,D_n\}$
of $\pi$-anti-nef cycles, which ends with a monomial cycle
belonging to $C_1$. Thus \condref{c:A} is satisfied.
These arguments also show that \condref{c:A} holds in case
$A$ is star-shaped.
\end{proof}
\begin{defn}\label{d:CSAM}
Assume that \condref{c:A} is satisfied, and that
$A_1, \ldots ,A_s$ are all of the nodes of $A$.
Let $C_1, \dots ,C_p$ be the branches of $A_{1}$.
(1) A monomial $x(D)$ is
called an {\itshape admissible monomial} at the node $A_{1}$
if it belongs to one of the branches of $A_1$.
A set of monomials $\{x(D_1), \dots ,x(D_p)\}$ is called a
{\itshape complete system} of admissible monomials
at $A_{1}$ if $D_i$ belongs to $C_i$ for
$i=1, \ldots ,p$.
(2) Let $\{m_1, \ldots ,m_p\}$ be any complete system of
admissible monomials at $A_{1}$.
Let $F=(c_{ij})$, $c_{ij} \in \mathbb C$,
be a $((p-2) \times p)$-matrix such
that every maximal minor of it has rank $p-2$.
We define polynomials $f_1, \ldots ,f_{p-2}$ by
$$
\begin{pmatrix}
f_1\\ \vdots \\ f_{p-2}
\end{pmatrix}
= F
\begin{pmatrix}
m_1\\ \vdots \\ m_p
\end{pmatrix}.
$$
We call each $f_i$ an {\itshape admissible form} at $A_1$ and
the set $\{f_1 , \dots ,f_{p-2}\}$ a {\itshape Neumann-Wahl
system} at $A_{1}$.
(3) Let $\mathcal F_i$ denote a Neumann-Wahl system at a node $A_{i}$.
Then we call the set
$ \bigcup _{i=1}^s\mathcal F_i$ a {\itshape Neumann-Wahl system} associated
with $A$; it is an empty set in case $A$ has no nodes.
\end{defn}
\begin{rem}\label{r:normal}
The matrix $F$ above can be reduced to the following matrix by
row operations:
$$
\begin{pmatrix}
1 & 0 & \dots & 0 & a_1 & b_1 \\
0 & 1 & \dots & 0 & a_2 & b_2 \\
\vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\
0 & 0 & \dots & 1 & a_{p-2} & b_{p-2}
\end{pmatrix},
$$
where $a_ib_j-a_jb_i\neq 0$ for $i\neq j$, and all $a_i$ and $b_i$
are nonzero.
\end{rem}
The admissible forms at a node $A_i$ are
quasihomogeneous polynomials with respect to the
$A_i$-weight.
The following lemma is needed in the next section.
\begin{lem}\label{l:higher}
Let $D$ be a $\pi$-anti-nef $\mathbb Q$-cycle such that
$m_{A_i}(D)>m_{A_i}(\du i)$ for some $i$.
Then for any component $A_j$, we have $m_{A_j}(D)>m_{A_j}(\du i)$.
\end{lem}
\begin{proof}
First we will show that $D \ge \du i$.
There exist effective $\mathbb Q$-cycles $D_1$ and $D_2$ such that
$D-\du i=D_1-D_2$ and that $D_1$ and $D_2$ have no common
components.
If $D_2>0$, then
$$
m_{A_i}(D_2)=-\du i\cdot D_2=-D\cdot D_2+D_1\cdot
D_2-D_2\cdot D_2>0.
$$
It contradicts the assumption of the lemma. Hence $D-\du
i\ge 0$.
Let $a=m_{A_i}(D-\du i)$ and $C$ a branch of $A_i$.
If a component $A_k$ of $C$ intersects $A_i$, then $(\du
i+aA_i)\cdot A_k=a>0$.
Since $D$ is $\pi$-anti-nef, there exists a $\mathbb Q$-cycle $C'$
supported on $C$ such that $C'\cdot A_l \le 0$ for all
$A_l \le C$ and that
$D \ge \du i+aA_i+C'$.
\end{proof}
\begin{rem}\label{r:diff}
The definition of a Neumann-Wahl system is very similar to that of
a system of splicing diagram equations in the Neumann and Wahl's work.
However those are not the same.
Let us consider a weighted graph $\Gamma$ represented as in
\figref{f:Gamma};
the vertex \rule{2mm}{2mm} has weight $-4$ and other vertices
$\bullet$ have weight $-2$.
\begin{figure}[htbp]
\begin{center}
\setlength{\unitlength}{0.5cm}
\begin{picture}(11,2)(0,0)
\multiput(0,0)(0,2){2}{\circle*{0.25}}
\multiput(5,1)(2,0){2}{\circle*{0.25}}
\put(2.8,0.8){\rule{2mm}{2mm}}
\multiput(9,0.5)(0,1){2}{\circle*{0.25}}
\multiput(11,0)(0,2){2}{\circle*{0.25}}
\put(3,1){\line(1,0){4}}
\put(0,0){\line(3,1){3}}
\put(0,2){\line(3,-1){3}}
\put(7,1){\line(4,1){4}}
\put(7,1){\line(4,-1){4}}
\end{picture}
\caption{\label{f:Gamma}}
\end{center}
\end{figure}
The graph $\Gamma$ is realized as the
resolution graph of an elliptic
singularity with $\mathbb Q$-homology sphere link.
The splice diagram $\Delta$ associated with $\Gamma$ is
represented as in \figref{f:Delta}.
\begin{figure}[htbp]
\begin{center}
\setlength{\unitlength}{0.5cm}
\begin{picture}(10,2)(0,0)
\multiput(0,0)(0,2){2}{\circle*{0.25}}
\multiput(3,1)(4,0){2}{\circle*{0.25}}
\multiput(10,0)(0,2){2}{\circle*{0.25}}
\put(3,1){\line(1,0){4}}
\put(0,0){\line(3,1){3}}
\put(0,2){\line(3,-1){3}}
\put(7,1){\line(3,1){3}}
\put(7,1){\line(3,-1){3}}
\put(2.2,1.5){$2$}
\put(2.2,0){$2$}
\put(7.5,1.5){$3$}
\put(7.5,0){$3$}
\put(3.5,1.2){$3$}
\put(6,1.2){$20$}
\end{picture}
\caption{\label{f:Delta}}
\end{center}
\end{figure}
We see that $\Delta$ satisfies semigroup condition.
However, $\Gamma$ does not satisfy \condref{c:A}.
Therefore we cannot define Neumann-Wahl systems in this
case, though the splice diagram
equations are defined.
That might indicate a Neumann-Wahl system is a special type
of
system of splice diagram equations.
\end{rem}
\section{Varieties defined by Neumann-Wahl systems}\label{s:VdefbyF}
In this section we prove that any Neumann-Wahl system
defines a complete intersection
surface with an isolated singularity at the origin;
so we will call such a singularity a {\itshape Neumann-Wahl complete
intersection singularity}.
In fact, we will prove the assertion for a slight generalization of
a Neumann-Wahl system.
We note that some of methods in this section are discussed in
\cite{nw-CIuac}.
We use the notation of the preceding section.
Assume that \condref{c:A} is satisfied and that $A$ has at
least one node.
As in the preceding section, we associate ends of $A$ with
the variables $x_1, \ldots
,x_m$, where $m=\# \mathcal E(A)$.
Suppose that $A_1, \ldots ,A_s$ are all of the nodes of $A$.
Let $d_i$ denote the number of branches of a node $A_i$.
Let $\mathcal M_i=\{m_{i1}, \ldots ,m_{id_i}\}$ denote a complete system of
admissible monomials and $\mathcal F_i=\{f_{i1}, \ldots ,f_{id_i-2}\}$
a Neumann-Wahl system at a node $A_i$, where each $f_{ij}$
is a linear form of monomials of $\mathcal M_i$.
By counting the numbers of the ends, the nodes and the edges
around the nodes of the dual graph of $A$, we see that
$$
\sum _{i=1}^s(d_i-2)=m-2.
$$
Let $C_1, \ldots ,C_{d_1}$ denote the branches of $A_1$.
Without loss of generality, we may assume the following.
\begin{enumerate}
\item For $1\le j \le d_1-1$, $C_j$ is a chain of curves;
in this case, $A_1$ is
an end of the minimal reduced connected cycle containing
all nodes of $A$.
\item For $1\le j \le d_1-1$, the variable $x_j$
corresponds to the end of $C_j$.
\item For $2\le i \le s$, the admissible monomial $m_{id_i}$
belongs to the branch of $A_i$ containing $A_1$.
\item For $1\le i \le s$, the admissible forms of $\mathcal F_i$ are
given by the
matrix as in \remref{r:normal}; we write
\begin{align*}
f_{ij}
&= m_{ij}+a_{ij}m_{id_i-1}+b_{ij}m_{id_i}, & &1\le i \le s, \;
1\le j \le d_i-2, \\
m_{1j} & = x_j^{\alpha_j}, & &1\le j \le d_1-1.
\end{align*}
\end{enumerate}
Now we slightly modify the admissible forms.
This modification is needed for the induction step of the
proof of the main theorem.
Let $\mathbb N$ denote the set of positive integers.
A vector $v \in \mathbb N^m$ is said to be {\itshape primitive} if
$v$ cannot be written as $v=c v'$
with $v' \in \mathbb N^m$ and $c \in \mathbb N$, $c>1$.
Fix an arbitrary vector $\delta=(\delta_1, \ldots
,\delta_m) \in \mathbb N^m$.
For each node $A_i$, let $e_i$ denote the positive integer
such that
$$
\mathbf w_i:=(\adeg i(x_1)\cdot e_i/\delta_1, \ldots ,
\adeg i(x_m)\cdot e_i/\delta_m) \in \mathbb N^m
$$
is primitive.
Let $S=\mathbb C\{x_1, \dots ,x_m\}$ be the convergent power series
ring.
\begin{defn}\label{d:wdeg}
Let $\mathbf w =(w_1, \ldots ,w_m)\in \mathbb N^m$.
We define the {\itshape $\mathbf w$-degree} of a $\mathbb Q$-monomial $m=\prod
_{k=1}^mx_k^{a_k}$ to be $\wdeg{}(m)=\sum a_kw_k$.
Let $f=\sum_{k\ge 1} f_k \in S$, where $f_1\neq 0$
and each $f_k$ is a quasihomogeneous polynomial with respect to
$\mathbf w$ such that $\wdeg {}(f_k)<\wdeg{}(f_{k+1})$.
We call $f_1$ the {\itshape leading form} of $f$, and
denote it by $\LF_{\mathbf w}(f)$.
Then $f-\LF_{\mathbf w}(f)$ is called the
{\itshape higher term} of $f$.
We define {\itshape $\mathbf w$-order} of $f$ to be
$\mathbf w\text{-}\!\ord (f)=\mathbf w\text{-}\!\deg
(\LF_{\mathbf w}(f))$. We set $\mathbf w\text{-}\!\ord (0)=\infty$.
\end{defn}
We write $\mathbf x_k=x_k^{\delta_k}$ and $\mathbf f=f(\mathbf x_1, \ldots ,\mathbf x_m)$ for
$f=f(x_1, \ldots ,x_m) \in S$.
We call $\mathbf f$ the {\itshape $\delta$-lifting} of $f$.
Any monomial can be thought as the $\delta$-lifting of a
$\mathbb Q$-monomial.
A polynomial $f$ is quasihomogeneous with respect to $A_i$-weight
if and only if so is $\mathbf f$ with respect to the weight $\mathbf w_i$; in
fact, for a $\mathbb Q$-monomial $m$, we have
\begin{equation}\label{eq:deg}
\wdeg i(\mathbf m)=\adeg
i(m)\cdot e_i.
\end{equation}
For each $\mathbf f_{ij}$, we take a convergent power series
$f_{ij}^+ \in S$
satisfying
$$
\wdeg i(\mathbf f_{ij})<\word i(f_{ij}^+).
$$
For each $t \in \mathbb C$, we set
$$
\mathcal F_t=\{\mathbf f _{ij}+tf_{ij}^+ | 1 \le i \le s, \; 1\le j \le d_i-2\}.
$$
We note that if $m_{ij}=x(D_{ij})$, then for $i \ge 2$ and
$1 \le j \le d_i-1$,
$$
m_{A_1}(D_{id_i})>m_{A_1}(D_{ij})=m_{A_1}(\du i).
$$
If the $\delta$-lifting of a $\mathbb Q$-monomial $x(D)$ is
contained in $f_{ij}^+$, then $m_{A_i}(D)>m_{A_i}(\du i)$.
By \lemref{l:higher}, $m_{A_1}(D)>m_{A_1}(\du i)$.
Thus we obtain the following:
\begin{align*}
\LF_{\mathbf w_1}(\mathbf f_{1j}+tf_{1j}^+)&=\mathbf f_{1j}, \\
\LF_{\mathbf w_1}(\mathbf f_{ij}+tf_{ij}^+)& =\mathbf m_{ij}+a_{ij}\mathbf m_{id_i-1},
\quad \text{for $i \ge 2$}.
\end{align*}
Therefore the set
$$
\LF_{\mathbf w_1}\mathcal F:=\{\LF_{\mathbf w_1}(f)|f \in \mathcal F_t\}
$$
is independent of $t \in \mathbb C$; it can be shown that
$\LF_{\mathbf w_i}\mathcal F$ is also
independent of $t$ for $2 \le i \le s$.
\begin{defn}[{Wahl \cite{wahl.es}, cf. \cite[V]{la.simul}}]
Let $\omega \: \widetilde X \to T$ be a deformation of a normal surface
singularity $\widetilde X_o=\omega^{-1}(o)$, $o \in T$.
Suppose that each fiber $\widetilde X_t$ has only one singular point
and that
there exists a simultaneous resolution $\bar \omega\: \widetilde M \to \widetilde X$
with the exceptional set $\widetilde A$.
If the restriction $(\omega \circ \bar \omega)|_{\widetilde A}$ is
a locally trivial deformation of the exceptional divisor
of $\widetilde M_o$, then we call
$\omega \circ \bar \omega$ (resp. $\omega$) an {\itshape
equisingular deformation}.
$\bar \omega$ is called a {\itshape weak simultaneous resolution} of
$\omega$.
\end{defn}
For a subset $B$ of any commutative ring,
let $I(B)$ denote the ideal generated by the elements of $B$.
Let $(V_t, o) \subset (\mathbb C^m,o)$ denote the singularity
defined by the ideal $I(\mathcal F_t) \subset S$.
The main result of this section is the following.
\begin{thm}[cf. {\cite[Theorem 2.6]{nw-CIuac}}]\label{t:V}
The singularity $(V_t ,o)$ is an isolated complete
intersection surface singularity for each $t \in \mathbb C$.
Furthermore, the family $\{V_t | t \in \mathbb C\}$ is an equisingular
deformation.
\end{thm}
First we show that every $V_t$ is a complete intersection.
\begin{lem}[cf. {\cite[Theorem 3.1]{nw-CIuac}}]\label{l:curveC}
For any variable $x_k$, let $A_{i_k}$ denote the node
nearest to the end corresponding to $x_k$.
Let $C \subset \mathbb C^{m}$ be the affine variety defined by
the ideal $I(\LF_{\mathbf w_{i_k}}\mathcal F\cup \{x_k\})$.
\begin{enumerate}
\item If $x_j=0$ ($j\neq k$) at $p \in C$, then $p$ is the origin.
\item $C$ is a complete intersection curve and smooth
except for the origin.
\end{enumerate}
\end{lem}
\begin{proof}
Without loss of generality, we may assume that $k=1$.
Let
$$
c_{j}=\begin{cases}
b_{11}/a_{11} & \text{if $j=d_1-1$}, \\
b_{1j}-a_{1j}b_{11}/a_{11} & \text{if $2 \le j\le d_1-2$}.
\end{cases}
$$
Then $c_j\neq 0$ and $C$ is a subvariety of the affine
space $\mathbb C^{m-1}$ with coordinates $x_2, \dots, x_m$
defined by the equations
\begin{equation}\label{eq:C}
\begin{split}
\mathbf x_j^{\alpha_j}+c_j\mathbf m_{1d_1}=0, \quad & 2 \le j \le d_1-1, \\
\mathbf m_{ij}+a_{ij}\mathbf m_{id_i-1}=0, \quad & 2\le i \le s,
1\le j \le
d_i-2.
\end{split}
\end{equation}
If a monomial appearing in \eqref{eq:C}
vanishes at $p \in C$,
then so does every monomial in the equations at the same node.
On the other hand, for each $2 \le i \le m$, some power of
$x_i$ appears in \eqref{eq:C}
because each end of $A$ is a unique end of a branch of the
nearest node.
Thus if a variable $x_i$ ($i \ge 2$) vanishes on $C$, then
so do all monomials appearing in \eqref{eq:C}, since $A$ is
a connected tree of curves.
Hence we obtain (1).
Since $\#\LF_{\mathbf w_1}\mathcal F=m-2$, it follows from (1) that
$\{x_1, x_i\} \cup \LF_{\mathbf w_1}\mathcal F$, where $i\neq 1$, is a
regular sequence.
Hence $C$ is one-dimensional and complete intersection.
The argument above also shows that an ideal $I(\{x_1, x_i-1\} \cup
\LF_{\mathbf w_1}\mathcal F)$
defines a nonsingular zero-dimensional variety.
Since $C $ is defined by
quasihomogeneous polynomials, $C$ is smooth except for the
origin.
\end{proof}
\begin{cor}[cf. {\cite[Corollary 3.4]{nw-CIuac}}]\label{c:CI}
$V_t$ is a complete intersection surface singularity, and
the support of $V_t \cap \{x_j=x_k=0\}$, $j\neq k$,
is the origin.
\end{cor}
\begin{proof}
By \lemref{l:curveC}, $\LF_{\mathbf w_{i_k}}\mathcal F\cup \{x_j,x_k\}$ is
a regular sequence.
Hence so are $\mathcal F_t$ and $\mathcal F_t\cup\{x_j,x_k\}$.
\end{proof}
\begin{defn}
We define the {\itshape weighted dual graph} of a normal
surface singularity to be
that of the
exceptional set of the minimal good resolution of the singularity.
\end{defn}
Let $(W,o)$ be a germ of a normal surface singularity and
$W' \to W$ the minimal resolution.
The {\itshape canonical cycle} on $W'$ is a
$\mathbb Q$-cycle which is
numerically equivalent to the canonical divisor $K_{W'}$.
The self-intersection number of the canonical cycle is an
invariant of the singularity and
determined by the weighted dual graph; we denote it by $K^2(W)$.
Let $\omega \: \widetilde X \to T \subset \mathbb C$ be
a deformation of surface singularities.
The invariance of $K^2(\widetilde X_t)$ implies the existence of the
simultaneous canonical model (or simultaneous RDP
resolution) of $\omega$; first the Gorenstein case was proved by
Laufer (\cite[Theorem 4.3]{la.simul}),
and the general case by Ishii (\cite[Corollary 1.10]{is.simul}).
By \cite{bri.simul}, the singularities of the simultaneous
canonical
model are simultaneously resolved after a suitable finite
base change.
Thus we obtain the following theorem by the arguments of
\cite[VI]{la.simul}.
\begin{thm}[Laufer, Ishii]\label{t:la}
Let $\omega \: \widetilde X \to T\subset \mathbb C$ be a deformation of a normal
surface singularity. If the weighted dual graphs of
$\widetilde X_t$, $t\in T$, are the same, then $\omega$ is an
equisingular deformation, and it
admits a simultaneous resolution such that each fiber is
the minimal good resolution.
\end{thm}
For a divisor $D$, we denote by $D_{red}$ the reduced
divisor with $\supp (D_{red})=\supp (D)$.
For the induction step of the proof of the main theorem,
we need the following.
\begin{lem}\label{l:es-div}
Let $\omega \: \widetilde X \to \mathbb C$ be an equisingular
deformation of a germ of a normal surface
singularity $\widetilde X_0=\omega^{-1}(0)$.
Let $\widetilde D$ be a reduced divisor on $\widetilde X$, which contains the
singular locus of $\widetilde X$.
Suppose that $\omega |_{\widetilde D}$ is a locally
trivial deformation of a reduced divisor $\widetilde D_0:=\t D|_{\widetilde X_0}$ on
$\widetilde X_0$.
Then there exists a simultaneous resolution $\bar \omega\:
\widetilde M \to \widetilde X$ with the exceptional set $\widetilde A$ such that
$(\omega \circ \bar \omega)|_{(\bar \omega^*\widetilde D)_{red}}$ is
a locally trivial deformation
(hence so is $(\omega \circ \bar \omega)|_{\widetilde A}$).
\end{lem}
\begin{proof}
There exists a weak simultaneous resolution $\bar \omega\:\widetilde M
\to \widetilde X$ with the exceptional set $\widetilde A$ such that each
$\widetilde A_t:=\widetilde A |_{\widetilde M_t}$ has only simple normal
crossing and $(\omega \circ \bar \omega)|_{\widetilde A}$ is a
locally trivial deformation.
Let $\widetilde F$ denote the strict transform of $\widetilde D$ on $\widetilde M$.
Let $\widetilde A^1$ (resp. $\widetilde F^1$) be a divisor on $\widetilde M$, which
is the total space of the deformation of an irreducible
component of $\widetilde A_0$ (resp. $\widetilde F_0:=\widetilde F |_{ \widetilde M_0}$).
Then the intersection number $\widetilde A^1_t\cdot \widetilde F_t^1$ is constant.
We may assume $\# (\widetilde A^1_t\cap \widetilde F_t^1) \le 1$.
If $\widetilde A^1_t\cdot \widetilde F_t^1\ge 2$, then take the blowing up of
$\widetilde M$ along the curve $\widetilde A^1\cap \widetilde F^1$.
By taking blowing ups successively in a similar way, we obtain a
simultaneous resolution $\bar \omega'\: \widetilde M' \to \widetilde X$ such that
each divisor $((\bar \omega'|_{\widetilde M'_t})^*\widetilde D_t)_{red}$ has
only normal crossings
and that the weighted dual graph of the divisor is
independent of $t \in \mathbb C$.
\end{proof}
\begin{lem}\label{l:G}
Let $\omega\: \widetilde X \to \mathbb C$ be as in \lemref{l:es-div}.
Suppose that $\widetilde X$ is embedded in an open subset of
$\mathbb C^n\times \mathbb C$ such that the singular locus of $\widetilde X$ is
$\{o\}\times \mathbb C$, and that $\omega$ is the composite of
this embedding and the projection $\mathbb C^n\times \mathbb C \to \mathbb C$.
Let $G$ be a finite subgroup of the unitary group $\mathrm
U(n)\subset \mathrm {GL}(\mathbb C^n)$.
Then $G$ acts on $\mathbb C^n\times \mathbb C$ by $g\cdot(z,t)=(g\cdot z,
t)$, $g \in G$.
Assume the action induces an action on
$\widetilde X$ which is free on $\widetilde X \setminus \{o\}\times \mathbb C$.
Then the morphism $\mathcal Omega\: \widetilde X /G \to \mathbb C$ obtained from $\omega$
is an equisingular deformation of $(\widetilde X_t/G,o)$, $t \in \mathbb C$.
\end{lem}
\begin{proof}
Let $\mathbf S_c \subset \mathbb C^n$, $c>0$, denote the $(2n-1)$-sphere
of radius $c$.
Let $t_0 \in \mathbb C$ be an arbitrary point.
Then there exist an open neighborhood $U$ of $t_0$ and a
positive number $\epsilon \in \mathbb R$ such that
$\Sigma_t:=\mathbf S_{\epsilon}\cap \widetilde X_t \subset \mathbb C^n$ is the
link of $\widetilde X_t$ for every $t \in U$ and the family
$\{\Sigma_t | t \in U\}$ is topologically trivial.
By the assumption, $G$ acts on $\{\Sigma_t | t \in U\}$ freely.
Thus we obtain a family $\{\Sigma_t/{G} | t \in U\}$ which is
topologically trivial.
Recall that the weighted dual graph of a surface singularity is
determined by its link (Neumann \cite{neumann.plumbing}).
By \thmref{t:la}, we obtain the assertion.
\end{proof}
We mention the weighted blowing up which is needed in the
proof of the theorem.
Let $\mathbf w=(w_1, \ldots ,w_m) \in \mathbb N^m$ be a primitive vector, and
let $\beta\: Z \to \mathbb C^m$ be the weighted blowing up with
respect to the weight $\mathbf w$.
It is a projective morphism inducing an isomorphism
$Z \setminus \beta^{-1}(o) \to \mathbb C^m \setminus \{o\}$, and
$\beta^{-1}(o)=\P(\mathbf w)$, the weighted projective space of
type $\mathbf w$.
The variety $Z$ is covered by affine varieties $Z_1, \dots ,Z_m$;
each $Z_i$ is a quotient of $W_i=\mathbb C^m$ by a cyclic group $\mathcal C_i$
of
order $w_i$ determined by the weight $\mathbf w$.
Let $\{x_1, \ldots ,x_m\}$ and $\{z_1, \ldots ,z_m\}$
be the coordinates of $\mathbb C^m$ and $W_1$, respectively.
The action of the group $\mathcal C_1$ on $W_1$ is given by the diagonal
matrix
$$
\diag [e(-1/w_1), e(w_2/w_1), \, \ldots \, ,e(w_m/w_1)],
$$
where $e(q)=\exp(2\pi \sqrt{-1}q)$. Since $\mathbf w$ is
primitive, $\mathcal C_1$ is trivial or the fixed
locus is a proper subvariety of the hyperplane $\{x_1=0\}$
which is the exceptional locus.
The morphism $W_1 \to \mathbb C^m$,
which is the composite of the quotient morphism $W_1 \to Z_1$
and $\beta\:Z_1 \to \mathbb C^m$,
is given by
$$
x_1=z_1^{w_1}, \quad x_i=z_1^{w_i}z_i \quad (i=2, \ldots ,m).
$$
\begin{proof}[Proof of \thmref{t:V}]
We prove the theorem by induction on the number of nodes $s$ of $A$.
We have to show the isolated singularity of each $V_t$ and the
equisingularity of the family $\{V_t | t \in \mathbb C\}$.
First assume that $s=1$.
Then $V_0$ is the so-called Brieskorn-Pham complete intersection
singularity; it is known that $V_0$ has an isolated
singularity
(it is also easily checked by using the Jacobian criterion).
We fix $t \in \mathbb C$.
Since $\LF_{\mathbf w_1}\mathcal F$ is a regular sequence,
it follows from the theory of filtered rings that there exists an
equisingular deformation of $V_0$
with general fiber $V_t$ (cf. \cite[\S 6]{tki-w},
\cite{wahl.defqh}).
Hence $V_t$ is an isolated complete intersection
singularity, and the
weighted dual graphs of $V_0$ and $V_t$ are the same.
By \thmref{t:la}, the family $\{V_t | t \in \mathbb C\}$ is
an equisingular deformation.
Next assume that $s\ge 2$.
Let $\beta\: Z\times \mathbb C \to \mathbb C^m\times \mathbb C$ be the trivial family of
the weighted blowing up $Z \to \mathbb C^m$ with
respect to the weight $\mathbf w_1=(w_1, \ldots ,w_m)$.
The family $\{V_t|t \in \mathbb C\}$ is naturally embedded in
$\mathbb C^m\times\mathbb C$.
Let $W_i=\mathbb C^m$ be as above; however we write $x_i$ instead of $z_i$.
Then the cyclic group $\mathcal C_i$ acts on $W_i\times \mathbb C$ as
in \lemref{l:G}.
Let $V_t^i\subset W_i\times {\{t\}}$ be the strict transform
of $V_t$.
Recall that $V_t\cap \{x_1=x_2=0\}=\{o\}$ by
\corref{c:CI} and that the action of the cyclic group $\mathcal C_i$
on $V_t^i$ is free outside the exceptional locus.
Thus to prove that $V_t$ has an isolated singularity at the
origin, it suffices to show that any component of singular
loci of $V_t^1$ and $V_t^2$,
intersecting the exceptional set, is an isolated point.
Note that the exceptional divisor is singular at singular
points of $V_t^i$.
By \thmref{t:la}, \lemref{l:es-div} and \ref{l:G}, it is
enough to prove the following three claims (the
claims on $V_t^2$ are proved in the same way).
\begin{clm}\label{clm:1}
The family of exceptional divisors $E_t \subset V_t^1$ is trivial.
\end{clm}
\begin{clm}\label{clm:2}
Let $p \in E_0$ be a singular point and let $p_t \in E_t$
denote the point $p$ under the identification $E_0=E_t$.
Then each $(V_t^1, p_t)$ is
an isolated singularity and the weighted dual graph of
$(V_t^1,x_t)$ is independent of $t$.
\end{clm}
\begin{clm}\label{clm:3}
If $q_1, \ldots ,q_k \in V_0^1$ be the fixed points of
$\mathcal C_1$-action,
then the fixed locus of the family $\{V_t^1|t \in \mathbb C\}$ is
$\bigcup _j\{q_j\}\times \mathbb C$.
If $\mathcal C_{1,j} \subset \mathcal C_1$ denotes
the isotropy group of $\{q_j\}\times \mathbb C$, then
\lemref{l:G} applies to
the family $\{(V_t^1,q_j)|t \in \mathbb C\}$ with $\mathcal C_{1,j}$-action
for every $q_j$.
\end{clm}
The exceptional divisor $E_t$ is defined by $x_1=0$ in $V_t^1$.
The ideal of $V_t^1 \subset \mathbb C^m= W_1\times {\{t\}}$ is
generated by the following functions:
\begin{align*}
F_{11} &
=1+a_{11} \mathbf x_{d_1-1}^{\alpha_{d_1-1}}+b_{11}
\mathbf m_{1d_1}+t\bar f_{11}^+, & & \\
F_{1j} &
=\mathbf x_j^{\alpha_j}+a_{1j} \mathbf x_{d_1-1}^{\alpha_{d_1-1}}+b_{1j}
\mathbf m_{1d_1}
+t\bar f_{1j}^+, & & 2 \le j \le d_1-2, \\
F_{ij} &
=\mathbf m_{ij}+a_{ij}\mathbf m_{id_i-1}+b_{ij}\bar \mathbf m_{id_i}+t\bar f_{ij}^+,
& & i \ge 2, \; 1 \le j \le d_i-2,
\end{align*}
where
$$
\bar f_{ij}^+=f_{ij}^+(x_1^{w_1}, x_1^{w_2}x_2,
\ldots ,x_1^{w_m}x_m)/x_1^{\word 1(\mathbf f_{ij})},
$$
and $\bar
\mathbf m_{id_i}$ is obtained in
the same way from $\mathbf m _{id_i}$.
Recall the condition on the order of $f_{ij}^+$, and that
$$
\adeg 1(m_{ij})<\adeg 1(m_{id_i}), \quad i \ge 2, \; 1 \le
j \le d_i-2.
$$
Then we see that the ideal of $E_t \subset \mathbb C^m$ is
generated by $x_1$ and the following polynomials:
\begin{align}
&1+a_{11} \mathbf x_{d_1-1}^{\alpha_{d_1-1}}+b_{11}
\mathbf m_{1d_1},\label{eq:tF1} & & \\
&\mathbf x_j^{\alpha_j}+a_{1j} \mathbf x_{d_1-1}^{\alpha_{d_1-1}}+b_{1j}
\mathbf m_{1d_1}, & & 2 \le j \le d_1-2, \label{eq:tFj}\\
&\mathbf m_{ij}+a_{ij}\mathbf m_{id_i-1},
& & i\ge 2, \; 1 \le j \le d_i-2. \label{eq:gij}
\end{align}
These polynomials do not contain $t$; thus \clmref{clm:1}
is verified.
Since the fixed points of the $\mathcal C_j$-action lie on $E_t$,
the first assertion of \clmref{clm:3} follows.
By looking at the action explicitly,
we see that the $\mathcal C_{1,j}$-action on $W_1\times \mathbb C$ is
unitary around $q_j$.
Thus \clmref{clm:3} follows from \clmref{clm:2}.
It is easy to see that there are $m-d_1$ polynomials in
\eqref{eq:gij} and any of them
contains no variables $x_1, \ldots ,x_{d_1-1}$.
As in the proof of \lemref{l:curveC},
we can show that the functions of \eqref{eq:gij} define a
complete intersection curve $C' \subset \mathbb C^{m-d_1+1}$
which is smooth except for the origin.
Furthermore \eqref{eq:tF1} and \eqref{eq:tFj} define a
tower of cyclic coverings over $C'$.
Hence the singularities of $E_t$ are lying above the point
$(0,\ldots ,0)$ of $C'$.
Therefore at each singular point of $E_t$, the only $d_1-2$
variables $x_2,\ldots ,x_{d_1-1}$
are nonzero, and others are zero.
We fix $t \in \mathbb C$. Let $p$ be a singular point of $E_t$.
At $p$, the Jacobian matrix
$$
\partial (F_{11}, \ldots ,F_{1d_1-2})/\partial(x_2, \ldots
,x_{d_1-1})
$$
is regular, and thus $x_2, \ldots ,x_{d_1-1}$ can be
expressed by a convergent power series with nonzero
constant terms
in $x_1, x_{d_1}, \ldots ,x_m$.
By substituting them into $F_{ij}$, $i\ge 2$,
we obtain new defining functions for the germ $(V_t^1, p)$ in
$(\mathbb C^{m-d_1+2}, o)$ as follows:
\begin{equation}\label{eq:neweq}
F'_{ij}:=\mathbf m_{ij}+a_{ij}\mathbf m_{id_i-1}+b_{ij}'\bar
\mathbf m_{id_i}'+r_{ij},
\quad i\ge 2, \quad 1 \le j \le d_i-2,
\end{equation}
where $\bar \mathbf m_{id_i}'$ is a monomial obtained by substituting $1$
into $x_2, \ldots, x_{d_1-1}$ of $\bar \mathbf m_{id_i}$.
We see that
\begin{equation}\label{eq:neq}
a_{ij_1}b'_{ij_2}\not=a_{ij_2}b'_{ij_1} \; (j_1\neq j_2)
\quad \text{and} \quad b'_{ij} \neq 0.
\end{equation}
By \eqref{eq:deg}, for every $\mathbb Q$-monomial $m=x(D)$, we have
$$
\wdeg 1(\mathbf m)=e_1\cdot \adeg 1(m)=e_1\cdot m_{A_1}(D).
$$
Suppose that $m_{id_i}=x(D_i)$ for $i\ge 2$.
Since $\adeg 1(m_{id_i-1})=m_{A_1}(\du i)$,
the exponent of $x_1$ in $\bar \mathbf m_{id_i}'$ is
\begin{equation}\label{eq:exponent}
\wdeg 1(\mathbf m_{id_i})- \wdeg 1(\mathbf m_{id_i-1})=
e_1m_{A_1}(D_i-\du i)>0.
\end{equation}
Since $D_i-\du i \in A_{\mathbb Z}$, it follows
that $\bar \mathbf m_{id_i}'$ is a monomial of
$x_1^{e_1}, \mathbf x_{d_1}, \ldots ,\mathbf x_m$.
We denote by $h_{i}$ the monomial
obtained by replacing $x_1^{e_1},\mathbf x_{d_1}, \ldots ,\mathbf x_m$ of
$\bar \mathbf m_{id_i}'$ with $x_1, x_{d_1}, \cdots ,x_m$, respectively.
Let $\delta'=(e_1,\delta_{d_1},\ldots ,\delta_m) \in
\mathbb N^{m-d_1+2}$ and let
$$
f'_{ij}=m_{ij}+a_{ij}m_{id_i-1}+b_{ij}'h_{i},
\quad i\ge 2, \quad 1 \le j \le d_i-2.
$$
Then the polynomial
$\mathbf m_{ij}+a_{ij}\mathbf m_{id_i-1}+b_{ij}'\bar \mathbf m_{id_i}'$ is the
$\delta'$-lifting of $f'_{ij}$.
Let $A_{e}$ be the component of the branch $C_{d_1}$ of the
node $A_1$,
which intersects $A_1$; see \figref{f:diag}.
\begin{figure}[htbp]
\begin{center}
\setlength{\unitlength}{0.5cm}
\begin{picture}(14,3)(-7,-2)
\put(0,-1.2){\framebox(6,2){}}
\put(-2,0){\line(1,0){4}}
\put(-2,0){\circle*{0.25}}
\put(-2.2,-1){$A_1$}
\put(0.8,-1){$A_e$}
\put(2.3,0){ . . .}
\put(1,0){\circle*{0.25}}
\put(6.5,-0.5){$C_{d_1}$}
\put(-4,-1){\line(2,1){2}}
\put(-6,-2){\framebox(2,1){}}
\put(-8,-2){$C_{d_1-1}$}
\put(-4,1){\line(2,-1){2}}
\put(-6,1){\framebox(2,1){}}
\put(-7.5,1){$C_1$}
\put(-5,-0.5){$\vdots$}
\end{picture}
\caption{\label{f:diag}}
\end{center}
\end{figure}
We may assume that the $A_e$ is an end of $C_{d_1}$;
take the blowing up at $A_1 \cap A_e$ if
necessary.
Let $A'=C_{d_1}$.
Then $A'\subset M$ can be blown down to a normal surface
singularity.
We consider admissible forms concerning $A'$.
We associate the end $A_e$ with the variable $x_1$, while
keeping the correspondence between the
other ends and the variables
$x_{d_1}, \ldots, x_m$.
\begin{clm}\label{clm:smallgraph}
We have the following.
\begin{enumerate}
\item $A'$ has $s-1$ nodes $A_2, \cdots, A_s$.
\item $A'$ satisfies \condref{c:A},
and for each $2\le i \le s$, the set of polynomials
$$
\{f'_{i1}, \ldots ,f'_{i d_i-2}\}
$$
is a Neumann-Wahl system concerning $A'$ at a node $A_i$.
\item In this new situation, we define a weight $\mathbf w_i' \in
\mathbb N^{m-d_1+2}$
($2\le i \le s$) with respect to $\delta'$ as in the
sentence before \defref{d:wdeg}.
Then
$$
\wpdeg i(\mathbf m_{ij}+a_{ij}\mathbf m_{id_i-1}+b_{ij}'\bar \mathbf m_{id_i}')
<\wpord i(r_{ij}).
$$
\end{enumerate}
\end{clm}
The inductive hypothesis and \clmref{clm:smallgraph} imply
that $(V_t^1,p)$ is an isolated singularity
and is an equisingular
deformation of the singularity defined by the $\delta'$-lifting of
the Neumann-Wahl system
$\{f'_{ij}\}$; thus \clmref{clm:2} follows.
Now we prove \clmref{clm:smallgraph}.
First, the assertion (1) is obvious.
It is clear that the number of branches of a node $A_i$ ($i\ge 2$)
in $A'$ is the same as that of $A_i$ in $A$.
For $D=\sum a_jA_j \in A_{\mathbb Q}$, we write $\res(D)=
\sum _{A_j\le A'}a_jA_j$.
Let $\t A_k$ denote the dual cycle of $A_k$ on $A'$.
Let $D_{ij}$ ($i \ge 2$) be the monomial
cycle such that $x(D_{ij})=m_{ij}$,
and let
$$
E_{ij}=\res(D_{ij}-\du i).
$$
Since $(D_{ij}-\du i)\cdot A_e=0$, we have $-E_{ij}\cdot
A_e=m_{A_1}(D_{ij}-\du i)$; this is the exponent of $x_1$
in $h_{i}$
(see \eqref{eq:exponent}).
By computing the intersection numbers $(E_{ij}+\t A_i)\cdot
A_l$ for every $A_l \le A'$,
we see that $E_{ij}+\t A_i$ is a monomial cycle belonging
to a branch of $A_i$ and that
$$
x(E_{id_i}+\t A_i)=h_{i} \quad \text{and} \quad x(E_{ij}+\t
A_i)=m_{ij} \quad \text{for $1\le j <d_i$}.
$$
By \eqref{eq:neq}, we have (2) of \clmref{clm:smallgraph}.
We can generalize the argument above as follows.
Let $m$ be any monomial in $\mathbf f_{ij}+tf_{ij}^+$.
Suppose that $m$ is
the $\delta$-lifting of a $\mathbb Q$-monomial $x(D)$ associated with a
$\mathbb Q$-monomial cycle $D \in A_{\mathbb Q}$.
Let $m'$ be the $\delta'$-lifting of the monomial
$x(\res(D-\du i)+\t A_{i})$.
Then, by the operation which changes
$\mathbf f_{ij}+tf_{ij}^+$ into the function $F'_{ij}$, the monomial
$m$ is changed into a function of the form $u m'$,
where $u \in S$ is a unit.
We have that $m_{A_i}(\res(D-\du i)+\t A_{i})=m_{A_i}(D)$
for a node $A_i$ $(i \ge 2)$.
Now it is easy to see that (3) holds.
Thus we have proved \clmref{clm:smallgraph}.
\end{proof}
Let $\hat S=\mathbb C[[x_1, \ldots ,x_m]]$ denote the formal power series
ring.
By a similar argument as in the proof of \thmref{t:V}, we
obtain the following.
\begin{thm}\label{t:formal}
Let $\mathbf f_{ij}$ be as above.
Take a formal power series $g_{ij} \in \hat S$ such that
$$
\wdeg i(\mathbf f_{ij})<\word i(g_{ij})\quad \text{ for} \quad
1 \le i\le s\;, 1 \le j \le d_i-2.
$$
Let $J \subset \hat S$ be the ideal generated by all
$\mathbf f_{ij}+g_{ij}$'s.
Then $\hat S/J$ is a two-dimensional complete intersection ring,
and has only a singularity at the maximal ideal.
\end{thm}
\section{The main results}\label{s:mainresults}
In this section we will prove the following.
\begin{thm}\label{t:main}
If $(X,o)$ is a rational or minimally elliptic singularity, then its
universal abelian cover $(Y,o)$ is an equisingular deformation of a
Neumann-Wahl complete intersection singularity.
The deformation is defined by the functions of the form
$f+tf^+$, where $\{f\}$ is a Neumann-Wahl
system associated with the exceptional set $A$,
$f^+$ is a function with order greater than the degree of
$f$ and $t$ is the parameter.
\end{thm}
We start without the assumption of the theorem.
We use the notation of \sref{s:pre}.
Write $\mathcal A_b=H^0(-L^{(b)})$
and $\mathcal O_{Y,o}=\bigoplus _{b \in \mathcal B}\mathcal A_b$.
Recall that for any component $A_i$ of $A$,
there exist a divisor $L^i$ and an element $b^i \in \mathcal B$ such that
$\nu (L^i)=\du i$ and $L^i - L^{(b^i)} \in A_{\mathbb Z}$.
We consider the
following condition, which depends on the resolution $\pi\: M \to X$.
\begin{cond}\label{c:B}
For each end $A_i \in \mathcal E(A)$,
there exists a section $y_i \in H^0(-L^i)$ such that
$(y_i)_A=\nu(L^i)$.
\end{cond}
\begin{lem}\label{l:satisfyB}
If $(X,o)$ is rational, or if $(X,o)$ is minimally elliptic and the
minimally elliptic cycle $E$ is supported on $A$,
then \condref{c:B} is satisfied.
\end{lem}
\begin{proof}
Let $D$ be any $\pi$-nef divisor on $M$.
If $(X,o)$ is rational, then $D$ is $\pi$-free; see
\cite[4.17]{chap}.
Assume that $(X,o)$ is minimally elliptic and
the minimally elliptic cycle $E$ is supported on $A$.
Since $E$ is 2-connected,
$H^0(D)$ has no fixed component on $A$ if $D\cdot A\ge 1$ by
\cite[4.23, Remark]{chap}.
Thus the assertion follows.
\end{proof}
Let $\frak m_Y$ (resp. $\frak m_X$) denote the maximal ideal of $\mathcal O_{Y,o}$
(resp. $\mathcal O_{X,o}$).
We may identify $\frak m_Y$ as $\frak m_X \bigoplus (\bigoplus _{b
\neq 0}\mathcal A_b)$ (cf. \cite[\S 6]{o.uac-rat}).
\begin{lem}\label{l:powerofmax}
Let $\{Z_k \in A_{\mathbb Z}|k \in \mathbb N\}$ be a sequence of cycles such that
$Z_{k+1}>Z_k>0$ for every $k \in \mathbb N$.
Then there exists a function $\alpha\:\mathbb N \to \mathbb N$ such that for
each $b \in \mathcal B$,
$$
H^0(-L^{(b)}-Z_k)\subset \frak m_Y^{\alpha(k)} \quad \text{and}
\quad \lim_{k \to \infty} \alpha (k)= \infty.
$$
\end{lem}
\begin{proof}
We only give an outline.
We can take a positive integer $a$ so that for any $\pi$-nef
divisor $D$ on $M$ and a cycle
$Z:=a \sum_{A_i \le A} \du i$, the natural map
$$
H^0(D-Z)\otimes H^0(-Z) \to H^0(D-2Z)
$$
is surjective (cf. \cite[III]{la.simul}).
Let $\beta$ be a nonnegative integer such that $-L^{(b)}-\beta Z$ is
$\pi$-nef for every $b \in \mathcal B$.
Then we obtain $H^0(-L^{(b)}-(\beta+k)Z)\subset \frak m_Y^k$.
We may assume $Z_1>(\beta+1)Z$.
Now define $\alpha(l)= \max\{k \in \mathbb N |(\beta+k)Z \le Z_l\}$.
\end{proof}
\begin{ass}
From now on, we assume that \condref{c:C} and
\ref{c:B} are satisfied.
\end{ass}
If $A$ is a chain of curves, then $(X,o)$ is a cyclic quotient
singularity and $\mathcal O_{Y,o}=\mathbb C\{y_1,y_2\}$,
where $y_i$'s are as in \condref{c:B}
(if $A$ is irreducible, then $y_1,y_2 \in H^0(-L^1)$).
Let $m=\#\mathcal E(A)$.
Assume that $m\ge 3$.
Then we can define admissible monomials at each node
by associating each end with a variable as in
\sref{s:NWS}.
We define the homomorphism
$$
\psi\: S=\mathbb C\{x_1, \ldots ,x_m\} \to \mathcal A=\mathcal O_{Y,o}
$$
of $\mathbb C$-algebras by $\psi (x_i)=y_i$.
We denote by $\hat{}$
the maximal-ideal-adic completions of local rings.
Let $\hat \psi\:\hat S=\mathbb C[[x_1, \ldots ,x_m]] \to \hat {\mathcal O}_{Y,o}$
be the
induced homomorphism.
By the definition of the set $\mathcal B$,
we may regard $\mathcal B$ as the discriminant
group $G:=\du {\mathbb Z}/A_{\mathbb Z}$ in the natural way.
Let $S_b \subset S$ (resp. $\hat S_b \subset \hat S$),
$b \in \mathcal B$, denote the set of power series
represented as the sum of monomials $x(D)$ satisfying
$D\pmod{A_{\mathbb Z}}=b$.
Then we have $S=\bigoplus _{b \in \mathcal B}S_b$ and the $\psi$ becomes a
homomorphism of $\mathcal B$-graded (or $G$-graded) algebras.
The same holds for $\hat S$ and $\hat{\psi}$.
Let $A_1, \ldots ,A_s$ be all of the nodes of $A$,
and $d_i$ the number of branches of a node $A_i$.
Let $\mathcal M_i=\{m_{i1}, \ldots ,m_{id_i}\}$ denote a complete system of
admissible monomials at a node $A_i$.
Let $\mathbb C\mathcal M_i \subset S$ denote the $\mathbb C$-linear subspace spanned by
the monomials of $\mathcal M_i$.
\begin{lem}\label{l:constructCSAF}
For any node $A_i$, let $\mu_i$ denote the composite of
homomorphisms
$$
\mathbb C\mathcal M_{i} \overset{\psi}\longrightarrow
H^0( \mathcal O_M(-L^{i})) \to H^0(\mathcal O_{A_{i}}(-L^{i})).
$$
Then $\mu_i$ is surjective.
We have $h^0(\mathcal O_{A_{i}}(-L^{i}))=2$ and $\dim \Ker \mu_i=d_i-2$.
Let $\mathcal F_i=\{f_{i1}, \ldots ,f_{id_i-2}\}$ be a basis of
$\Ker \mu_i$.
Then $\mathcal F_i$ is a Neumann-Wahl system at the node $A_i$.
\end{lem}
\begin{proof}
Since $L^i\cdot A_i=-1$, we have $h^0(\mathcal O_{A_{i}}(-L^{i}))=2$.
Suppose $m_{ij}=x(D_{ij})$ for a monomial cycle $D_{ij}$ belonging
to a branch $C_{ij}$ of $A_i$.
Since $(D_{ij}-\du {i})\cdot A_{i}=1$,
it follows from \lemref{l:product} that
$\mu_i(x(D_{ij}))$ has a zero of order one at $A_{i} \cap C_{ij}$.
Thus $\mu_i(x(D_{id_i-1}))$ and $\mu_i(x(D_{id_i}))$ generate
$H^0(\mathcal O_{A_{i}}(-L^{i}))$.
Therefore we obtain a complete system of admissible forms
expressed by
a $((p-2) \times p)$-matrix as in \remref{r:normal}, which
is a basis of
$\Ker \mu_i$.
\end{proof}
Let us recall that polynomials of $\mathcal F_i$ are quasihomogeneous with
respect to the $A_i$-weight.
\begin{lem}\label{l:higherterms}
Let $A_i$ be a node and $h\in H^0(-L^i)$.
Then there exists $\bar h \in \hat S_{b^i}$
such that $\hat \psi(\bar h)=h$ in $\hat {\mathcal O}_{Y,o}$.
Suppose that $h=\psi (\bar h_0)$ for an
admissible form $\bar h_0 \in \Ker \mu _i$.
Then we can take the $\bar h$ so that
$\adeg i (\bar h_0)<\aord i(\bar h)$.
If in addition $\psi $ is surjective,
such $\bar h$ can be taken from $S_{b^i}$.
\end{lem}
\begin{proof}
We use the notation of \lemref{l:constructCSAF}.
Suppose that $h \neq 0$.
Let $F_0$ be the divisor such that $\nu (F_0)=(h)_A$ and
$F_0-L^i \in A_{\mathbb Z}$.
Let $c_0=-F_0\cdot A_i$. Then $c_0\ge 0$.
By \condref{c:C} and the proof of \lemref{l:existmonomials},
there exists a cycle $F'_0\ge F_0$ such that
$m_{A_i}(F'_0-F_0)=0$, $(F'_0-F_0)\cdot A_i=0$ and
$F'_0\cdot A_j=0$ if $j\neq i$ and $A_j$ is not an end.
Let $D'_{ij}=D_{ij}-\du i$.
Then an arbitrary cycle of the form $E_{\mathbf
a}:=F'_0+\sum a_{j}D'_{ij}$ is a monomial cycle,
where $\mathbf a=(a_1, \ldots ,a_m) \in \mathbb Z^m$ with
$\sum a_j=c_0$ and $a_j \ge 0$.
It is easy to see that the $\mu_i(x(E_{\mathbf a}))$'s span
$H^0(\mathcal O_{A_i}(-F_0))$.
Thus we have a quasihomogeneous polynomial $\bar h_1$,
which is a linear form of $x(E_{\mathbf a})$'s, with respect to
$A_i$-weight such that $h-\psi(\bar h_1) \in H^0(-F_0-A_i)$.
Let $F_1$ be the divisor such that $\nu (F_1)=(h-\psi(\bar h_1))_A$
and $F_1-L^i \in A_{\mathbb Z}$.
Then it follows from the argument above that there exists
a quasihomogeneous polynomial $\bar h_2$
such that
$$
h-\psi(\bar h_1)-\psi(\bar h_2) \in H^0(-F_1-A_i).
$$
Thus we obtain a sequence $\{\bar h_k|k \in \mathbb N\}$
of quasihomogeneous polynomials and a sequence
$\{F_k | k \in \mathbb N\}$
of divisors satisfying the following: for all $k \in \mathbb N$,
\begin{enumerate}
\item $h-\psi(\bar h_1+\cdots +\bar h_k) \in H^0(-F_k)$,
\item $F_{k+1}>F_k$,
\item $\adeg i(\bar h_k)<\adeg i(\bar h_{k+1})$.
\end{enumerate}
By \lemref{l:powerofmax}
there exists a function $\alpha\: \mathbb N \to \mathbb N$ such
that $H^0(-F_k) \subset \frak m_Y^{\alpha(k)}$ and
that $\lim_{k\to \infty}\alpha(k) = \infty$.
Now put $\bar h=\sum \bar h_i \in \hat S_{b^i}$.
Then $\hat \psi(\bar h)=h$.
Suppose that $h=\psi(\bar h_0)$ with an admissible form
$\bar h_0 \in \Ker \mu_i$.
Since $\psi(\bar{h_0}) \in H^0(-L^i-A_i)$,
we have
$$
\adeg i(\bar h_0)<m_{A_i}(F_0)=\adeg i(\bar h_{1}).
$$
If $\psi$ is surjective, then the maps $(x_1, \cdots ,x_m)^k
\to \frak m_Y^k$ and $S_{b} \to \mathcal A_b$ are surjective
for every $k \in \mathbb N$ and $b \in \mathcal B$.
Therefore, for sufficiently large $k$, there exists $\bar h'
\in S_{b^i}$ such that $\aord i(\bar h')>\adeg i(\bar h_0)$
and $h=\psi(\bar h_1+\cdots +\bar h_k+\bar h')$.
\end{proof}
\begin{prop}\label{p:surj}
The homomorphism $\psi\: S \to \mathcal O_{Y,o}$ is surjective.
\end{prop}
\begin{proof}
We fix a node $A_i$.
By \cite[Proposition 5.1]{nw-CIuac}, the group
$G=\du {\mathbb Z}/A_{\mathbb Z}$ is generated by
$\{\du j|A_j \in \mathcal E(A)\}$.
Hence for each $b \in \mathcal B$,
there exists a monomial $m_b \in S$ such that
$\mathcal A_b\cdot m_b \subset H^0(-L^i)$.
By \lemref{l:higherterms},
we have $\mathcal A_b \subset \hat\psi(\hat S_b)\cdot
m_b^{-1}$. Therefore
$$
\hat \psi (\hat S) \subset \hat {\mathcal O}_{Y,o}
\subset \sum_{b \in \mathcal B} \hat \psi (\hat S) \cdot m_b^{-1}.
$$
Then it follows that $\hat S /\Ker \hat \psi$ is a
two-dimensional domain.
By \lemref{l:higherterms}, for any $f_{kl} \in \bigcup \mathcal F_j$,
there exists $\tilde f_{kl} \in \hat S$ such that
$\LF_{A_k}(\tilde f_{kl})=f_{kl}$
and $\tilde f_{kl} \in \Ker \hat \psi$.
Let $\tilde I \subset \hat S$ denote the ideal generated by
all $\tilde f_{kl}$'s.
Then it follows from \thmref{t:formal}
that $\hat S/\tilde I$ is a two-dimensional normal domain.
Since $\tilde I \subset \Ker \hat{\psi}$,
we obtain that $\hat S/\tilde I\cong \hat \psi (\hat S)$.
Since $\hat S$ is Noetherian, $ \hat {\mathcal O}_{Y,o} $ is finitely
generated $\hat \psi(\hat S)$-module.
By the normality of $\hat \psi (\hat S)$,
we obtain $\hat \psi (\hat S)=\hat {\mathcal O}_{Y,o}$.
This implies that $\psi$ is surjective.
\end{proof}
It follows from \lemref{l:higherterms} and \proref{p:surj}
that for each
$f_{ij} \in \mathcal F_i$, there exists $f_{ij}^+ \in S_{b^i}$
such that $f_{ij}+f_{ij}^+
\in \Ker \psi$ and $\adeg i( f_{ij})<\aord i(f_{ij}^+)$.
For each $t \in \mathbb C$, we denote by $I_t \subset S$ the ideal
generated by the functions
$$
f_{ij}+tf_{ij}^+, \quad 1\le i \le s, 1 \le j \le d_{i}-2.
$$
As in the proof of \proref{p:surj}, by \thmref{t:V},
we see that $\Ker \psi=I_1$.
\begin{cor}
$\mathcal O_{Y,o} \cong S/I_1$.
\end{cor}
Again by \thmref{t:V}, we obtain the following.
\begin{thm}\label{t:main-general}
If \condref{c:C} and \ref{c:B} are satisfied, then the universal
abelian cover $(Y,o)$ of $(X,o)$
is an equisingular deformation of a
Neumann-Wahl complete intersection singularity.
The deformation is defined by the functions
$$
f_{ij}+T f_{ij}^+ \in S[T], \quad 1\le i \le s, 1 \le j \le
d_{i}-2,
$$
where $S[T]$ is the polynomial ring over $S$.
\end{thm}
Now \thmref{t:main} follows from \thmref{t:main-general},
\lemref{l:existmonomials} and
\ref{l:satisfyB}.
\begin{cor}\label{c:QGor}
If \condref{c:C} and \ref{c:B} are satisfied, then $(X,o)$ is
$\mathbb Q$-Gorenstein.
If in addition the link of $(X,o)$ is a homology sphere,
then $(X,o)$ is an equisingular deformation of a
Neumann-Wahl complete intersection singularity.
\end{cor}
\begin{rem}
There is ambiguity in the choice of the complete systems of
admissible monomials $\mathcal M_i$.
The proof of \lemref{l:higherterms} shows that
if $m$ and $m'$ belong to the same branch of a node $A_i$,
then there exists $h \in S_{b^i}$ such that $x(m)-x(m')-h \in \Ker
\psi$ and $\aord i(h)>\adeg i(x(m))$.
Therefore, whether $(Y,o)$ is a Neumann-Wahl complete
intersection depends not only on the choice of the sections
$\{y_i\}$, but also on the choice of the complete
systems of admissible monomials.
\end{rem}
In the rest of this section, we describe the action of the
Galois group of the universal abelian covering.
Recall that the Galois group $H_1(\Sigma, \mathbb Z)$ of the universal
abelian covering $(Y,o) \to (X,o)$ is isomorphic
to the discriminant group $G=\du {\mathbb Z}/A_{\mathbb Z}$.
For any $\mathbb Q$-cycle $D=\sum a_i\du i=\sum b_iA_i$,
we have $D\cdot A_j=-a_j$ and $D\cdot \du j=-b_j$.
Thus we obtain the following
\begin{lem}\label{l:du}
Let $D \in A_{\mathbb Q}$. Then
\begin{enumerate}
\item $D \in \du {\mathbb Z}$ if and only if $\{D\cdot A_i|A_i \le
A\} \subset \mathbb Z$,
\item $D \in A_{\mathbb Z}$ if and only if $\{D\cdot \du i|A_i \le
A\} \subset \mathbb Z$.
\end{enumerate}
\end{lem}
For any $D \in \du {\mathbb Z}$, we denote by $(D)$ the class of
$D$ modulo $A_{\mathbb Z}$.
Then the action of $G$ on $S=\mathbb C\{x_1,\ldots ,x_m\}$ is
expressed as follows (cf. \cite[\S 5]{nw-CIuac}).
For $(D) \in G$ and a monomial cycle $F$,
\begin{equation}\label{eq:action}
(D)\cdot x(F):=\exp (2\pi\sqrt{-1}D\cdot F)x (F).
\end{equation}
By \lemref{l:du} (1), this is well-defined.
Let $(Y_t,o)$ denote the singularity defined by
the ideal $I_t \subset S$.
We have seen that $\{Y_t|t \in \mathbb C\}$ is an equisingular family.
Since $I_t$ is generated by homogeneous elements
of $G$-graded algebra $S$, the group $G$ naturally acts on $S/ I_t
=\mathcal O_{Y_t,o}$.
It follows from \lemref{l:du} (2) that $\mathcal
O_{X,o}=(\mathcal O _{Y,o})^{G}$.
By \corref{c:CI} and \cite[Proposition 5.2]{nw-CIuac},
the action is free on $Y_t\setminus \{o\}$ (see also \cite[Theorem
7.2 (2)]{nw-CIuac}).
Therefore $X_t:=Y_t/G$ is a normal singularity.
The linear action of $G$ on $\mathbb C^m$ determined by
\eqref{eq:action} is unitary.
By \lemref{l:G} and the uniqueness of the universal abelian
covering, we obtain the following
\begin{thm}\label{p:esquotient}
The family $\{X_t|t \in \mathbb C\}$ is an equisingular
deformation, and each $Y_t \to X_t$ is the universal abelian
covering.
\end{thm}
|
{
"timestamp": "2006-03-16T10:51:12",
"yymm": "0503",
"arxiv_id": "math/0503733",
"language": "en",
"url": "https://arxiv.org/abs/math/0503733"
}
|
\section{Introduction}
The theoretical framework of self-consistent
mean-field models enables a description of the nuclear
many-body problem in terms of universal energy density functionals.
By employing global effective interactions,
adjusted to empirical properties of symmetric and asymmetric
nuclear matter, and to bulk properties of
spherical nuclei, the current generation of
self-consistent mean-field models has achieved a high level of
accuracy in the description of ground states and
properties of excited states in arbitrarily heavy nuclei,
exotic nuclei far from $\beta$-stability,
and in nuclear systems at the nucleon drip-lines \cite{BHR.03}.
The relativistic mean-field (RMF) models,
in particular, are based on concepts of non-renormalizable effective
relativistic field theories and density functional theory.
They have been very successfully applied in studies of nuclear
structure phenomena at and far from the valley of $\beta$-stability.
For a quantitative analysis of open-shell
nuclei it is necessary to consider also pairing correlations.
Pairing has often been taken into account in a very
phenomenological way in the BCS model with the monopole pairing
force, adjusted to the experimental odd-even mass differences. This
approach, however, presents only a poor approximation for
nuclei far from stability. The physics of
weakly-bound nuclei necessitates a unified and self-consistent
treatment of mean-field and pairing correlations. This has led to
the formulation and development of the relativistic
Hartree-Bogoliubov (RHB) model, which represents a relativistic
extension of the conventional Hartree-Fock-Bogoliubov framework,
and provides a basis for a consistent microscopic
description of ground-state properties of
medium-heavy and heavy nuclei, low-energy excited states,
small-amplitude vibrations, and
reliable extrapolations toward the drip lines \cite{VALR.05}.
In most applications of the RHB model \cite{VALR.05}
the pairing part of the well
known and very successful Gogny force~\cite{BGG.84} has be employed in the
particle-particle ($pp$) channel:
\begin{equation}
V^{pp}(1,2)~=~\sum_{i=1,2}e^{-((\mathbf{r}_{1}-\mathbf{r}_{2})/{\mu_{i}})^{2}%
}\,(W _{i}~+~B_{i}P^{\sigma}-H_{i}P^{\tau}-M_{i}P^{\sigma}P^{\tau})\;,
\end{equation}
with the set D1S \cite{BGG.91} for the parameters $\mu_{i}$,
$W_{i}$, $B_{i}$, $H_{i}$, and $M_{i}$ $(i=1,2)$. This force has
been very carefully adjusted to the pairing properties of finite
nuclei all over the periodic table. In particular, the basic
advantage of the Gogny force is the finite range, which
automatically guarantees a proper cut-off in momentum space.
However, the resulting pairing field is non-local and the solution of the
corresponding Dirac-Hartree-Bogoliubov integro-differential equations
can be time-consuming, especially in the case of deformed nuclei.
Another possibility is the use of a zero-range, possibly density-dependent,
$\delta$-force in the $pp$-channel of the RHB model \cite{Meng.98}. This
choice, however, introduces an additional cut-off parameter in
energy and neither this parameter, nor the strength of the
interaction, can be determined in a unique way. The effective range of
the interaction is determined by the energy cut-off, and the
strength parameter must be chosen accordingly in order to reproduce
empirical pairing gaps.
In a series of recent papers \cite{Bul1,Bul2,Bul3} A. Bulgac and Y. Yu
have introduced a simple scheme for the renormalization of the
Hartree-Fock-Bogoliubov equations in the case of zero-range
pairing interaction. The scheme is equivalent to a simple energy cut-off
with a position dependent coupling constant. In this work we use
the prescription of Refs.~\cite{Bul1,Bul2} to implement a
regularization scheme for the relativistic Hartree-Bogoliubov
equations with zero-range pairing. We analyze the resulting
$^1S_0$ pairing gap in isospin-symmetric nuclear matter and apply
the RHB model to the calculation of ground-state pairing properties of
finite spherical nuclei.
In Sec.~\ref{SecRHB} we present an outline of the RHB model and
introduce the renormalization scheme for the case of zero-range pairing.
The model is applied in Sec.~\ref{SecNM} to pairing in
isospin-symmetric nuclear matter. Ground-state
pairing properties of Sn nuclei are analyzed in Sec.~\ref{SecSn}.
Sec.~\ref{SecSum} contains the summary and conclusions.
\section{\label{SecRHB} Relativistic Hartree-Bogoliubov model with
zero-range pairing}
A detailed review of the relativistic Hartree-Bogoliubov model
can be found, for instance, in Ref.~\cite{VALR.05}.
In this section we include those features which are essential
for the discussion of the renormalization of the RHB equations.
The model can be derived within the framework of
covariant density functional theory. When pairing correlations are
included, the energy functional depends not only on the density
matrix $\hat{\rho}$ and the meson fields $\phi_{m}$, but in addition
also on the anomalous density $\hat{\kappa}$
\begin{equation}
E_{RHB}[\hat{\rho},\hat{\kappa},\phi_{m}]=E_{RMF}[\hat{\rho},\phi
_{m}]+E_{pair}[\hat{\kappa}]\;,\label{ERHB}%
\end{equation}
where $E_{RMF}[\hat{\rho},\phi]$ is the RMF energy density functional
and the pairing energy $E_{pair}[\hat{\kappa}]$
is given by
\begin{equation}
E_{pair}[\hat{\kappa}]=\frac{1}{4}\mathrm{Tr}\left[ \hat{\kappa}^{\ast}%
V^{pp}\hat{\kappa}\right] .
\end{equation}
$V^{pp}$ denotes a general two-body pairing interaction. The
equation of motion for the generalized density matrix
\begin{equation}
\mathcal{R}=\left(
\begin{array}
[c]{cc}%
\rho & \kappa\\
-\kappa^{\ast} & 1-\rho^{\ast}
\end{array}
\right) \;,
\end{equation}
reads
\begin{equation}
i\partial_{t}\mathcal{R}=\left[
\mathcal{H}(\mathcal{R}),\mathcal{R}\right]
\;.\label{TDRHB}
\end{equation}
The generalized Hamiltonian $\mathcal{H}$ is a functional derivative of the
energy with respect to the generalized density
\begin{equation}
\mathcal{H}_{RHB}~=~\frac{\delta
E_{RHB}}{\delta\mathcal{R}}~=~\left(
\begin{array}
[c]{cc}%
\hat{h}_{D}-m-\mu & \hat{\Delta}\\
-\hat{\Delta}^{\ast} & -\hat{h}^{\ast}_{D}+m+\mu
\end{array}
\right) \;.\label{HFB-hamiltonian}%
\end{equation}
The self-consistent mean field $\hat{h}_{D}$ is the Dirac
Hamiltonian, and the pairing field reads
\begin{equation}
\Delta_{ab}(\mathbf{r},\mathbf{r}^{\prime})={\frac{1}{2}}\sum\limits_{c,d}%
V_{abcd}^{pp}(\mathbf{r},\mathbf{r}^{\prime})\kappa_{cd}(\mathbf{r}%
,\mathbf{r}^{\prime}),\label{equ.2.5}%
\end{equation}
where $a,b,c,d$ denote quantum numbers that specify the Dirac
indices of the spinors, and
$V_{abcd}^{pp}(\mathbf{r},\mathbf{r}^{\prime})$ are the matrix
elements of a general two-body pairing interaction.
Pairing effects in nuclei are restricted to an energy window of a few MeV
around the Fermi level, and their scale
is well separated from the scale of binding energies, which
are in the range of several hundred to thousand MeV.
There is no experimental evidence for any
relativistic effect in the nuclear pairing field $\hat{\Delta}$.
Therefore, pairing can be treated as a non-relativistic phenomenon, and a
hybrid RHB model with a non-relativistic pairing interaction can be employed.
For a general two-body interaction, the matrix elements of
the relativistic pairing field read
\begin{equation}
\hat{\Delta}_{a_1 p_1, a_2 p_2} =
{\frac{1}{2}}\sum\limits_{a_3 p_3, a_4 p_4}
\langle a_1 p_1, a_2 p_2 |V^{pp}|a_3 p_3, a_4 p_4\rangle_a~
\kappa_{a_3 p_3, a_4 p_4}\; ,
\end{equation}
where the indices ($p_1,p_2,p_3,p_4 = +,-$) refer to the large and
small components of the quasiparticle Dirac spinors. In most
applications of the RHB model, only the large components of the
spinors $U_{k}({\bf r})$ and $V_{k}({\bf r})$ have been included in
the non-relativistic pairing tensor $\hat{\kappa}$ in Eq.
(\ref{kappa0}). The resulting pairing
field reads
\begin{equation}
\hat{\Delta}_{a_1 +, a_2 +} =
{\frac{1}{2}}\sum\limits_{a_3 +, a_4 +}
\langle a_1 +, a_2 + |V^{pp}|a_3 +, a_4 +\rangle_a~
\kappa_{a_3 +, a_4 +}\; .
\end{equation}
The other components: $\hat{\Delta}_{+-}$,
$\hat{\Delta}_{-+}$, and $\hat{\Delta}_{--}$ are neglected,
in accordance with the results that are obtained with
a relativistic zero-range force \cite{SR.02}.
The ground state of an open-shell nucleus is described by the solution of
the relativistic Hartree-Bogoliubov equations
\begin{equation}
\left(
\begin{array}
[c]{cc}%
\hat{h}_{D}-m-\mu & \hat{\Delta}\\
-\hat{\Delta}^{\ast} & -\hat{h}^{\ast}_{D}+m+\mu
\end{array}
\right) \left(
\begin{array}
[c]{c}%
U_{k}(\mathbf{r})\\
V_{k}(\mathbf{r})
\end{array}
\right) =E_{k}\left(
\begin{array}
[c]{c}%
U_{k}(\mathbf{r})\\
V_{k}(\mathbf{r})
\end{array}
\right) \;, \label{eqhb}%
\bigskip
\end{equation}
which correspond to the stationary limit of Eq. (\ref{TDRHB}).
The chemical potential $\mu$ is determined by the particle
number subsidiary condition in order that the expectation value of
the particle number operator in the ground state equals the number
of nucleons. The column vectors denote the quasiparticle wave
functions, and $E_{k}$ are the quasiparticle energies.
The RHB wave functions determine the hermitian single-particle density matrix
\begin{equation}
\hat{\rho}_{ll^{\prime}}=(V^{\ast}V^{T})_{ll^{\prime}},%
\label{rho0}%
\end{equation}
and the antisymmetric anomalous density
\begin{equation}
\hat{\kappa}_{ll^{\prime}}=(V^{\ast}U^{T})_{ll^{\prime}}. \label{kappa0}%
\end{equation}
The calculated nuclear ground-state properties sensitively depend
on the choice
of the effective Lagrangian and pairing interaction.
Over the years many parameter sets of the
mean-field Lagrangian have been derived that provide a satisfactory
description of nuclear properties along the $\beta $-stability line.
The most successful RMF effective interactions are purely
phenomenological, with parameters adjusted to reproduce the nuclear
matter equation of state and a set of global properties of spherical
closed-shell nuclei. This framework has recently been
extended to include effective
Lagrangians with explicit density-dependent meson-nucleon
couplings. In a number of studies
it has been shown that this class of global
effective interactions provides an improved description of asymmetric
nuclear matter, neutron matter and finite nuclei far from stability.
In the present analysis of ground-state properties of Sn isotopes
the density-dependent effective
interaction DD-ME1~\cite{Nik1.02} will be employed in the
particle-hole ($ph$) channel of the RHB model.
In the following we extend the regularization scheme of Bulgac and Yu
\cite{Bul1,Bul2,Bul3} to the solution of the relativistic
Hartree-Bogoliubov equations for a zero-range
pairing interaction
\begin{equation}
V^{pp}(\mathbf{r},\mathbf{r}^{\prime}) = g \delta(\mathbf{r} -
\mathbf{r}^{\prime})\; .
\label{pair_int}
\end{equation}
In Refs.~\cite{Bul1,Bul2} it has been shown that in this
case the renormalized pairing field can be expressed as
\begin{equation}
\Delta (\mathbf{r}) =
-g_{\mathit{eff}}(\mathbf{r}) \kappa_c (\mathbf{r}) \;,
\label{eq:Delta}
\end{equation}
where $\kappa_c(\mathbf{r})$ denotes the cut-off anomalous density
\begin{equation}
\kappa_c(\mathbf{r}) =\sum\limits_{E_k>0}^{E_c}
V_{k}^{\dagger}(\mathbf{r})U_{k}(\mathbf{r}) \;.
\label{pair_c}
\end{equation}
The cut-off energy $E_c$ defines the two corresponding momenta $k_c$ and $l_c$
\begin{eqnarray}
\sqrt{k_c^2(\mathbf{r}) + {m^*}^2(\mathbf{r})}+ V(\mathbf{r}) - m &=&E_c +\mu
\label{cut1} ,\\
\sqrt{l_c^2(\mathbf{r}) + {m^*}^2(\mathbf{r})}+ V(\mathbf{r}) - m &=&-E_c+\mu
\label{cut2}\;.
\end{eqnarray}
$m^*(\mathbf{r}) = m +S(\mathbf{r})$ is the Dirac mass, and $S(\mathbf{r})$
and $V(\mathbf{r})$ are, respectively,
the scalar and vector single-nucleon potentials contained
in the Dirac Hamiltonian $\hat{h}_{D}$. The chemical potential $\mu$ determines
the local Fermi momentum
\begin{equation}
\sqrt{k_f^2(\mathbf{r}) + {m^*}^2(\mathbf{r})} + V(\mathbf{r}) - m =\mu\;.
\label{kfermi}
\end{equation}
The effective, position-dependent coupling in Eq.~(\ref{eq:Delta}) reads
\begin{equation}
\frac{1}{ g_{\mathit{eff}}(\mathbf{r})} =
\frac{1}{g} + F_1(\mathbf{r}) + F_2(\mathbf{r}) \;,
\label{g_eff}
\end{equation}
with
\begin{eqnarray}
F_1(\mathbf{r}) &=& -{{k_c(\mathbf{r})\sqrt{k_f^2(\mathbf{r})+
{m^*}^2(\mathbf{r})}}\over{4\pi^2}} \left [ 1 -
{{k_f(\mathbf{r})}\over{k_c(\mathbf{r})}}~{\rm Ar~cth}
{{k_c(\mathbf{r})}\over{k_f(\mathbf{r})}} \right ] \nonumber \\
&&-{{k_c(\mathbf{r})\sqrt{k_c^2(\mathbf{r})+
{m^*}^2(\mathbf{r})}}\over{8\pi^2}} +
{{\sqrt{k_f^2(\mathbf{r})+
{m^*}^2(\mathbf{r})}}\over{4\pi^2}}~k_f(\mathbf{r}) ~{\rm Ar~cth}
{{k_c(\mathbf{r})\sqrt{k_f^2(\mathbf{r})+
{m^*}^2(\mathbf{r})}}\over {k_f(\mathbf{r})\sqrt{k_c^2(\mathbf{r})+
{m^*}^2(\mathbf{r})}}} \nonumber \\
&&-{{2k_f^2(\mathbf{r})+
{m^*}^2(\mathbf{r})}\over {8\pi^2}} ~{\rm ln}
{{k_c(\mathbf{r})+ \sqrt{k_c^2(\mathbf{r})+
{m^*}^2(\mathbf{r})}}\over {{m^*}(\mathbf{r})}}
\label{F1Re}
\end{eqnarray}
\begin{eqnarray}
F_2(\mathbf{r}) &=& -{{l_c(\mathbf{r})\sqrt{k_f^2(\mathbf{r})+
{m^*}^2(\mathbf{r})}}\over{4\pi^2}} \left [ 1 -
{{k_f(\mathbf{r})}\over{l_c(\mathbf{r})}}~{\rm Ar~th}
{{l_c(\mathbf{r})}\over{k_f(\mathbf{r})}} \right ] \nonumber \\
&&-{{l_c(\mathbf{r})\sqrt{l_c^2(\mathbf{r})+
{m^*}^2(\mathbf{r})}}\over{8\pi^2}} +
{{\sqrt{k_f^2(\mathbf{r})+
{m^*}^2(\mathbf{r})}}\over{4\pi^2}}~k_f(\mathbf{r}) ~{\rm Ar~th}
{{l_c(\mathbf{r})\sqrt{k_f^2(\mathbf{r})+
{m^*}^2(\mathbf{r})}}\over {k_f(\mathbf{r})\sqrt{l_c^2(\mathbf{r})+
{m^*}^2(\mathbf{r})}}} \nonumber \\
&&-{{2k_f^2(\mathbf{r})+
{m^*}^2(\mathbf{r})}\over {8\pi^2}} ~{\rm ln}
{{l_c(\mathbf{r})+ \sqrt{l_c^2(\mathbf{r})+
{m^*}^2(\mathbf{r})}}\over {{m^*}(\mathbf{r})}}
\label{F2Re}
\end{eqnarray}
$F_1 + F_2$ is the relativistic generalization of the corresponding
correction to the coupling constant $g$, as defined in
Eq. (16) of Ref.~\cite{Bul1}.
\section{\label{SecNM}Pairing properties of symmetric nuclear matter}
A zero-range pairing interaction leads to a particularly simple
expression for the gap equation in symmetric nuclear matter
\begin{equation}
\frac{1}{ g_{\mathit{eff}}} =
- \frac{1}{4\pi^2}\int_{l_c}^{k_c} dk~
{k^2\over \sqrt{\left [ \sqrt{k^2+{m^*}^2} -
\sqrt{k_f^2+{m^*}^2}~ \right ]^2 + \Delta^2}} \; ,
\label{gap}
\end{equation}
The momenta $k_c$ and $l_c$ are determined by the cut-off energy $E_c$
Eqs. (\ref{cut1},~\ref{cut2}), and the effective coupling $g_{\mathit{eff}}$
is defined in Eq.~(\ref{g_eff}).
In the left panel of Fig.~\ref{FigA} we display the density dependence
of the resulting pairing gap in nuclear matter (dashed curve).
The single-particle spectrum has been calculated with the
relativistic effective interaction DD-ME1~\cite{Nik1.02}, and
the coupling constant of the zero-range pairing
interaction Eq.~(\ref{pair_int}) $g = -330$ MeV fm$^{3}$ is
typical for the values used by Bulgac and Yu in their
analyses. The pairing gap is shown in comparison
to the gap calculated with the effective Gogny
interaction D1S \cite{BGG.91} (dots). The corresponding
single-particle spectrum has been computed in the
Hartree-Fock approximation for the Gogny interaction. The
density dependence of the two gaps is completely different.
The pairing gap of the renormalized zero-range interaction
increases uniformly with density, whereas the gap of the
Gogny interaction display the characteristic maximum of
$\approx 2.5$ MeV at low density $\rho = 0.03 - 0.04$ fm$^{-3}$
(corresponding to a Fermi momentum of approximately 0.8 fm$^{-1}$)
and decreases at higher densities. The bell-shaped form of the pairing gap
as a function of the density was, in fact,
obtained already more than forty years ago \cite{ES.60}.
This density dependence
is not characteristic only of the phenomenological
finite-range interactions,
but is also obtained when the gap is calculated with
bare nucleon-nucleon potentials adjusted to
the empirical nucleon-nucleon phase shifts and
deuteron properties
(for a recent review see Ref.~\cite{DH-J.03}).
The decrease of the gap at Fermi momenta $k_f > 0.8$ fm$^{-1}$
simply reflects the repulsive character of the nucleon-nucleon
interaction at short distances \cite{SRR.02}.
Of course there is no repulsive component in the zero-range
force with constant coupling Eq.~(\ref{pair_int}), and
the corresponding pairing gap displays the unphysical
uniform increase with density. We notice, however, that
in the range of densities shown in Fig.~\ref{FigA}, i.e. up
to nuclear matter saturation density, the values of the pairing
gap of the renormalized zero-range interaction are comparable
with those of the Gogny pairing gap. As will be shown in the
next section, this means that the renormalization scheme
for the zero-range interaction with constant coupling can be
safely applied to the calculation of pairing correlations in
finite nuclei, provided an appropriate choice is made
for the strength parameter $g$.
On the other hand, there is no particular reason why the
strength parameter $g$ of the zero-range pairing interaction should
be a constant. In fact, in many applications to finite nuclei
an explicit density dependence is introduced, and in this
way pairing correlations partially include finite-range effects. For instance,
in one of the first applications \cite{BE.91} Bertsch and Esbensen
used a density-dependent contact interaction, together with
a simple energy cut-off, in a description of pairing correlations in
weakly bound neutron-rich nuclei. They also compared the corresponding
pairing gap in symmetric nuclear matter with the result of a Hartree-Fock
calculation using the Gogny interaction. In the present anaysis we have
adjusted a density-dependent strength parameter $g(\rho)$
of the zero-range pairing interaction Eq.~(\ref{pair_int}),
in such a way that the pairing gap of the renormalized
zero-range interaction Eq.~(\ref{gap}), reproduces the density
dependence of the Gogny pairing gap. The resulting density dependence
can be approximated by the following analytic expression
\begin{equation}
g(\rho) = \frac{1}{a_0 + a_1 \rho^{1/3} + a_2 \rho^{2/3}} \; ,
\label{g_rho}
\end{equation}
with $a_0 = -0.064$ fm$^{-2}$, $a_1 = 0.447$ fm$^{-1}$, and $a_2 =-3.693$.
The resulting pairing gap, displayed in the left panel of
Fig.~\ref{FigA} (solid line),
is in very good agreement with the one calculated using the Gogny interaction.
A very similar procedure was employed in Ref.\cite{YuB.03}, where the density
dependence of the ``bare coupling constant" $g(\rho)$ was adjusted to a
specific formula for the pairing gap in low-density homogeneous neutron matter.
In the right panel of Fig.~\ref{FigA} we display the effective couplings
$g_{\mathit{eff}}$ calculated using the constant
$g = -330$ MeV fm$^{3}$, and the density dependent coupling
of Eq. (\ref{g_rho}). The density dependence of the two
effective couplings is completely different.
In order to prevent an unphysical growth of
the pairing gap with density, the density dependence of the pairing
strength Eq. (\ref{g_rho}) ensures that the effective coupling
becomes weaker with increasing nucleon density. A very strong
effective coupling in the low-density region produces a peak
in the corresponding pairing gap shown in the left panel.
On the other hand, $g_{\mathit{eff}}$ calculated using the constant
coupling increases in absolute value with density, i.e. the
resulting pairing gap increases uniformly with density. However,
rather similar values for the two effective couplings
$g_{\mathit{eff}}$ are calculated
in the region of densities characteristic for the bulk of finite nuclei.
One should not, therefore, expect very different results for
the pairing properties of finite nuclei calculated with the
zero-range interaction with constant coupling, or with the
density-dependent coupling of Eq. (\ref{g_rho}). In the next
section we will show that this is really not true in
weakly-bound nuclei far from stability.
The renormalization prescription must, of course, lead to a pairing field
which is independent of the cut-off energy $E_c$, if the latter is
chosen large enough. This is illustrated in Fig.~\ref{FigB},
where we plot the pairing gap,
calculated using the density-dependent coupling of Eq.~(\ref{g_rho}),
for a number of
characteristic values $E_c$ in the interval between 5 MeV and 60 MeV.
The pairing gap shows a weak dependence on the cut-off energy only for the
two lowest values of $E_c$. When the cut-off is
increased beyond 10 MeV, the corresponding pairing gaps cannot
be distinguished. Thus already for $E_c \geq 10$ MeV the pairing
gap of the renormalized zero-range interaction in symmetric
nuclear matter converges. This is in agreement with the results
obtained in the analysis of the pairing gap in homogeneous neutron
matter~\cite{Bul1}.
\section{\label{SecSn}Ground-state pairing properties of spherical nuclei}
In this section the renormalization scheme is tested in the calculation
of ground-state pairing properties of Sn isotopes.
The DD-ME1 mean-field Lagrangian is employed for the $ph$ channel, and the
zero-range interaction Eq.~(\ref{pair_int}) is used in the $pp$ channel.
The renormalization procedure described in the previous section is
carried out for the zero-range interaction with constant pairing strength
$g = -330$ MeV fm$^{3}$, and and for the density-dependent
coupling of Eq.~(\ref{g_rho}). In the latter case
the density dependence of the pairing strength has been
adjusted to reproduce the Gogny D1S pairing gap
in symmetric nuclear matter. In the following we denote by
RCC the case of the renormalized constant coupling, and by RDDC the
results obtained with the renormalized density-dependent coupling.
While in the symmetric nuclear matter the Fermi momentum is always real
(see Eq. (\ref{kfermi})), in the surface region of finite nuclei it becomes
imaginary. In Ref.~\cite{Bul1} it has been shown that
also in this case the renormalized anomalous density is
real. The effective coupling $g_{\mathit{eff}}$ is still given by
Eq.~(\ref{g_eff}), but
\begin{eqnarray}
F_1(\mathbf{r}) &=& -{{k_c(\mathbf{r})\sqrt{-|k_f(\mathbf{r})|^2+
{m^*}^2(\mathbf{r})}}\over{4\pi^2}} \left [ 1 -
{{k_f(\mathbf{r})}\over{k_c(\mathbf{r})}}~{\rm Ar~ctg}
{{k_c(\mathbf{r})}\over{k_f(\mathbf{r})}} \right ] \nonumber \\
&&-{{k_c(\mathbf{r})\sqrt{k_c^2(\mathbf{r})+
{m^*}^2(\mathbf{r})}}\over{8\pi^2}} +
{{\sqrt{-|k_f(\mathbf{r})|^2+
{m^*}^2(\mathbf{r})}}\over{4\pi^2}}~k_f(\mathbf{r}) ~{\rm Ar~ctg}
{{k_c(\mathbf{r})\sqrt{-|k_f(\mathbf{r})|^2+
{m^*}^2(\mathbf{r})}}\over {k_f(\mathbf{r})\sqrt{k_c^2(\mathbf{r})+
{m^*}^2(\mathbf{r})}}} \nonumber \\
&&-{{(-2|k_f(\mathbf{r})|^2+
{m^*}^2(\mathbf{r}))}\over {8\pi^2}} ~{\rm ln}
{{k_c(\mathbf{r})+ \sqrt{k_c^2(\mathbf{r})+
{m^*}^2(\mathbf{r})}}\over {{m^*}(\mathbf{r})}} \;,
\label{F1Im}
\end{eqnarray}
and
\begin{equation}
F_2(\mathbf{r}) = 0\;.
\label{F2Im}
\end{equation}
However, if either $k_c$ or $l_c$ becomes imaginary,
the corresponding terms in the effective coupling should be omitted.
The rate of convergence of the renormalization scheme is illustrated
in Fig.~\ref{FigC} where, for the nucleus $^{114}$Sn, we display the
average pairing gaps and the pairing energies as functions
of the cut-off energy $E_c$. The average gaps shown in the
left panel, are defined as
\begin{equation}
< \Delta_N > = {{\sum_{nlj} \Delta_{nlj} v_{nlj}^2}\over
{\sum_{nlj} v_{nlj}^2}} \; ,
\label{ang}
\end{equation}
where $v_{nlj}^2$ are the occupation probabilities
of the neutron states in the canonical basis. Both the
pairing gaps and the pairing energies converge already for $E_c \geq 10$ MeV.
We also notice that, even though the renormalized constant coupling
and the renormalized density-dependent coupling lead to very
different pairing gaps in symmetric nuclear matter, in $^{114}$Sn
they produce similar average pairing gaps and virtually identical
pairing energies.
The corresponding pairing fields as functions of the radial
coordinate, and $g_{\mathit{eff}}$ Eq.~(\ref{g_eff}) as functions
of the density, are plotted in Fig.~\ref{FigD} for a series of values
of the energy cut-off. In both cases the calculation of the pairing
field and $g_{\mathit{eff}}$ shows convergence for $E_c > 10$ MeV.
While the renormalized constant coupling and the renormalized
density-dependent coupling produce very similar average pairing gaps
and pairing energies, the dependence of the corresponding
pairing fields on the radial coordinate is rather different.
The RCC pairing field (upper left panel) is concentrated in the
bulk of the nucleus, whereas the RDDC pairing field (lower left panel)
exhibits a pronounced peak on the surface. This behavior
reflects the difference between the effective couplings $g_{\mathit{eff}}$,
already shown in the right panel of Fig.~\ref{FigA} for the case of
symmetric nuclear matter. In the panels on the right of Fig.~\ref{FigD} we
plot the effective couplings $g_{\mathit{eff}}(r(\rho))$
as functions of the nucleon
density in $^{114}$Sn. The $g_{\mathit{eff}}$ which corresponds to
the density-dependent coupling of Eq.~(\ref{g_rho})
decreases steeply in the region of very low density, i.e.,
on the surface of the nucleus. Consequently, also the pairing field
displays a peak in the surface region.
In both the RCC and RDDC cases the pronounced discontinuity of
the effective coupling $g_{\mathit{eff}}$ at very low density corresponds
to the transition from real to imaginary Fermi momentum $k_f$.
This is illustrated in Fig.~\ref{FigE}, where we plot the
effective single-nucleon
potential (left panel) and the correction to the coupling originating from
the renormalization of the anomalous density (right panel). The effective
single-nucleon potential is determined by the sum of the vector and
scalar potentials $V_{cen}(\mathbf{r})=S(\mathbf{r})+V(\mathbf{r})$.
For real values of the Fermi momentum (the effective potential is below the
chemical potential $\mu$) the correction to the coupling
$F_1(\mathbf{r})+F_2(\mathbf{r})$ is calculated from Eqs. (\ref{F1Re})
and (\ref{F2Re}), and for imaginary values of the Fermi momentum
(the effective potential is above the chemical potential $\mu$) from
Eqs. (\ref{F1Im}) and (\ref{F2Im}).
In the region where the Fermi momentum changes from real to imaginary
the correction $F_1(\mathbf{r})+F_2(\mathbf{r})$ displays a very
sharp peak, which is reflected in the discontinuities of the
effective couplings.
The importance of possible surface effects is illustrated in Fig.~\ref{FigF},
where we plot the calculated average pairing gaps and pairing energies
for the chain of even-even Sn isotopes with $110 \le A \le 160$.
Although both the RCC and RDCC schemes lead to comparable values
of the average pairing gaps for the entire isotopic chain, the pairing energies
differ significantly for isotopes beyond the doubly closed-shell $^{132}$Sn.
For example, the pairing energy of $^{150}$Sn calculated with renormalized
density-dependent coupling (RDDC) is almost 25 MeV larger than the
one calculated with the renormalized constant coupling (RCC).
The large increase in the pairing energy for the RDDC case
is caused by the dominant role of the surface region for the
very neutron-rich Sn isotopes, and because the effective coupling
is especially strong at very low densities.
In the panels on the left of Figs.~\ref{FigG} and~\ref{FigH}
we plot the self-consistent solutions for the cut-off anomalous
densities Eq.~(\ref{pair_c}) for the isotopes
$^{114}$Sn, $^{124}$Sn and $^{150}$Sn, calculated
using the RDDC and RCC effective couplings, respectively.
The corresponding effective couplings $g_{\mathit{eff}}$
are shown in the panels on the right of Figs.~\ref{FigG} and~\ref{FigH}.
The anomalous densities for $^{114}$Sn and $^{124}$Sn are concentrated in the
nuclear volume ($r \le 6$ fm), where the effective couplings
$g_{\mathit{eff}}$ have comparable values. Therefore, the corresponding pairing
energies are similar for the RDDC and RCC cases. In $^{150}$Sn,
on the other hand, the anomalous densities extend to the region
$r\ge 8$ fm, where the RDDC effective
coupling becomes much stronger than the one calculated with the
constant coupling (RCC). Hence, the pairing energy for $^{150}$Sn,
calculated using the renormalized
density-dependent coupling is much larger than the one obtained with
the renormalized constant coupling.
\section{\label{SecSum}Conclusions}
A simple renormalization scheme for the Hartree-Fock-Bogoliubov
equations with zero-range pairing has recently been introduced
\cite{Bul1,Bul2,Bul3}. In the present work we have implemented this
renormalization scheme for the
relativistic Hartree-Bogoliubov equations with a zero-range
pairing interaction. The procedure is equivalent to a simple
energy cut-off with a position dependent coupling constant.
We have verified that the resulting average pairing gaps and
pairing energies do not depend on the cut-off energy $E_c$, if
the latter is chosen large enough. Convergence is achieved already for
values $E_c \ge 10$ MeV, both in nuclear matter and for finite nuclei.
If the strength parameter of the zero-range pairing is a constant,
the resulting pairing gap in symmetric nuclear matter displays an
unphysical increase with density. We have therefore
adjusted a density-dependent strength parameter of the zero-range pairing in
such a way that the renormalization procedure reproduces
in symmetric nuclear matter the pairing gap of the phenomenological
Gogny interaction. In this sense the present study goes beyond the
simple extension of the renormalization scheme of Ref.~\cite{Bul1}
to the relativistic framework. However, the resulting effective coupling is
too strong in the region of low density, and this leads to
large pairing energies in open-shell nuclei with very diffuse surfaces,
e.g. in neutron-rich Sn isotopes. One must therefore be careful
when applying the renormalized HFB or RHB models with zero-range pairing
to nuclei far from stability. Adjusting the strength parameter
to the pairing gap in symmetric nuclear matter obviously
does not provide enough information about the density dependence
of the zero-range pairing to be used in very neutron-rich nuclei.
\leftline{\bf ACKNOWLEDGMENTS}
This work has been supported in part by the Bundesministerium
f\"ur Bildung und Forschung - project 06 MT 193, by
the Alexander von Humboldt Stiftung,
and by the Croatian Ministry of Science - project 0119250.
\bigskip
|
{
"timestamp": "2005-03-30T10:04:40",
"yymm": "0503",
"arxiv_id": "nucl-th/0503078",
"language": "en",
"url": "https://arxiv.org/abs/nucl-th/0503078"
}
|
\section{Introduction}
Search for spatiotemporal solitons in diverse optical media, alias
``light bullets" (LBs) \cite{Yaron}, is a challenge to fundamental
and applied research in nonlinear optics, see original works
\cite{KR,Koshiba,chi2,Miriam,tandem,Frank,Isaac,saturable,Wagner}
and a very recent review \cite{review}. Stationary solutions for
LBs can easily be found in the cubic ($\chi ^{(3)}$)
multi-dimensional nonlinear Schr\"{o}dinger (NLS) equation
\cite{Yaron}, but their stability is a problem, as they are
unstable against spatiotemporal collapse \cite{Berge}. The problem
may be avoided by resorting to milder nonlinearities, such as
saturable \cite{saturable}, cubic-quintic \cite{CQ}, or quadratic
($\chi ^{(2)}$) \cite{KR,Koshiba,chi2,Miriam,tandem,Frank,Isaac}.
Despite considerable progress in theoretical studies, three-dimensional (3D)
LBs in a bulk medium have not yet been observed in an experiment. The only
successful experimental finding reported thus far was a stable quasi-2D
spatiotemporal soliton in $\chi ^{(2)}$ crystals \cite{Frank} (the
tilted-wavefront technique \cite{Paolo}, used in that work, precluded
achieving self-confinement in one transverse direction). On the other hand,
it was predicted \cite{Isaac} that a spatial cylindrical soliton may be
stabilized in a bulk medium composed of layers with alternating signs of the
Kerr coefficient. Similar stabilization was then predicted for what may be
regarded as 2D\ solitons in Bose-Einstein condensates (BECs), with the
coefficient in front of the cubic nonlinear term subjected to periodic
modulation in time via the \textit{Feshbach resonance} in external ac
magnetic field \cite{FR,Castilla}. However, no stable 3D soliton could be
found in either realization (optical or BEC) of this setting.
Serious difficulties encountered in the experimental search for
LBs in 3D media is an incentive to look for alternative settings
admitting stable\emph{\/} 3D optical solitons. With the Kerr
nonlinearity, a possibility is to use a layered structure that
periodically reverses the sign of the local group-velocity
dispersion (GVD), without affecting the $\chi ^{(3)}$ coefficient.
This resembles a well-known scheme in fiber optics, known as
\textit{dispersion management}\ (DM), see, e.g., Refs. \cite{DM}
and review \cite{Progress}. A 2D generalization of the DM scheme
was recently proposed, assuming a layered planar waveguide of this
type, uniform in the transverse direction \cite{we,Spain}. As a
result, large stability regions for the 2D spatiotemporal solitons
were identified, including double-peaked breathers; however, a 3D
version of the same model could not give rise to any stable
soliton \cite{we}. It was also shown in Ref. \cite{we} that no
stable 3D soliton could be found in a more sophisticated model,
which combines the DM and periodic modulation of the Kerr
coefficient in the longitudinal direction.
Another approach to the stabilization of multidimensional solitons
was developed in the context of the self-attracting BEC. It is
based on the corresponding Gross-Pitaevskii equation which
includes a periodic potential created as an optical lattice (OL,
i.e., an interference pattern produced by illuminating the
condensate by counter-propagating coherent laser beams). It has
been demonstrated that 2D \cite{BBB,Yang,Estoril} and 3D
\cite{BBB} solitons can be easily stabilized by the OL of the same
dimension. Moreover, stable solitons can also be readily supported
by \emph{low-dimensional\/} OLs, i.e., 1D and 2D ones in the 2D
\cite{Estoril,BBB2} and 3D \cite{Estoril,BBB2,Barcelona} cases,
respectively; additionally, a 3D soliton can be stabilized by a
cylindrical (\textit{Bessel}) lattice \cite{Dumitru}, similar to
one introduced, in the context of 2D models, in Ref.
\cite{Bessel}. On the other hand, 3D solitons cannot be stabilized
by a 1D periodic potential \cite{BBB2}; however, the 1D lattice
potential in combination with the above-mentioned time-periodic
modulation of the nonlinearity, provided by the Feshbach resonance
in the ac magnetic field, supports single- and multi-peaked stable
3D solitons in vast areas of the respective parameter space
\cite{we-new}.
The above results suggest a possibility of existence of stable 3D
``bullets" in a $\chi ^{(3)}$ medium with the DM in the
longitudinal direction ($z$), additionally equipped with an
effective lattice potential (i.e., periodic modulation of the
refractive index) in one transverse direction ($y$), while in the
remaining transverse direction ($x$) the medium remains uniform.
If this is possible, stable LBs will be definitely possible too in
a medium with the periodic modulation of the refractive index in
both transverse directions; however, the setting with one uniform
direction is more interesting in terms of steering solitons and
studying collisions between them \cite{Estoril,BBB2}. The
objective of the present work is to predict such 3D spatiotemporal
solitons and investigate their stability. Our first consideration
of this possibility is based on the variational approximation
(VA); systematic simulations of the 3D model are quite
complicated, and will be presented in a follow-up work. It is
relevant to mention that the existence and stability of 3D
solitons in the Gross-Pitaevskii equation with the quasi-2D
periodic potential, which were originally predicted by the
VA\cite{Estoril,BBB2}, was definitely confirmed by direct
simulations \cite{Estoril,BBB2,Barcelona}, which suggests that in
the present model the 3D solitons may easily be stable too.
The model is based on the normalized NLS equation describing the evolution
of the local amplitude $u$ of the electromagnetic wave, which is a
straightforward extension of the 2D model put forward in Ref. \cite{we}:
\begin{equation}
i\frac{\partial u}{\partial z}+\left[ \frac{1}{2}\left(
\frac{\partial ^{2}}{\partial x^{2}}+\frac{\partial ^{2}}{\partial
y^{2}}+D(z)\frac{\partial ^{2}}{\partial \tau ^{2}}\right)
+\varepsilon \cos (2y)+|u|^{2}\right] u=0. \label{general}
\end{equation}Here, $\varepsilon $ is the strength of the transverse modulation (the
modulation period is normalized to be $\pi $), while $\tau $ and $D(z)$ are
the same reduced temporal variable and local GVD coefficient as in the
fiber-optic DM models \cite{DM,Progress}.
Equation (\ref{general}) implies the propagation of a linearly polarized
wave, with the single component $u$; a more general situation will be
described by a two-component (vectorial) version of Eq. (\ref{general}),
with the two polarization coupled, as usual, by the cubic
cross-phase-modulation terms. We do not expect that the vectorial model will
produce results qualitatively different form those presented below. As
usual, the NLS equation assumes the applicability of the paraxial
approximation, i.e., the spatial size of solitons (see below) must be much
larger than the underlying wavelength of light, which is definitely a
physically relevant assumption \cite{review}, and the temporal part of the
equation implies that the higher-order GVD is negligible (previous
considerations have demonstrated that the higher-order dispersion does not
drastically alter DM solitons \cite{TOD}).
As is commonly adopted, we assume a symmetric \textit{DM map}, with equal
lengths $L$ of the normal- and anomalous-GVD segments (usually, the results
are not sensitive to the map's asymmetry),
\begin{equation}
D(z)=\left\{
\begin{array}{l}
\overline{D}+D_{\mathrm{m}}>0,\,0<z<L, \\
\overline{D}-D_{\mathrm{m}}<0,\,L<z<2L,\end{array}\right.
\label{D(z)}
\end{equation}the average dispersion being much smaller than the
modulation amplitude,
$\left\vert \overline{D}\right\vert \ll D_{\mathrm{m}}$. Using the scaling
invariances of Eq. (\ref{general}), we fix $L\equiv 1$ and
$D_{\mathrm{m}}\equiv 1$.
Recently, a somewhat similar 2D model was introduced in Ref.
\cite{Salerno}. The most important difference is that it has the
variable coefficient $D(z)$ multiplying \emph{both} the GVD and
diffraction terms, $u_{\tau \tau }$ and $u_{xx}$. Actually, that
model was motivated by a continuum limit of some discrete systems;
in the present context, it would be quite difficult to implement
the periodic reversal of the sign of the transverse diffraction.
\section{The variational approximation}
Aiming to apply the VA for the search of LB solutions (a review of the
variational method can be found in Ref. \cite{Progress}), we adopt the
Gaussian \textit{ansatz},
\begin{eqnarray}
u &=&A(z)\exp \left\{ \mathrm{i}\phi (z)-\frac{1}{2}\left[
\frac{x^{2}}{W^{2}(z)}+\frac{y^{2}}{V^{2}(z)}+\frac{\tau
^{2}}{T^{2}(z)}\right] \right. +
\nonumber \\
&&+\left. \frac{\mathrm{i}}{2}\left[ b(z)\,x^{2}+c(z)\,y^{2}+\beta (z)\,\tau
^{2}\right] \right\} , \label{ansatz}
\end{eqnarray}where $A$ and $\phi $ are the amplitude and phase of the soliton,
$T$ and $W,V$ are its temporal and two transverse spatial widths, and $\beta $
and $b,c$ are the temporal and two spatial chirps. The Lagrangian from which
Eq. (\ref{general}) can be derived is
\begin{eqnarray}
L &=&\frac{1}{2}\int_{-\infty }^{+\infty }\,\mathrm{d}x\int_{-\infty
}^{+\infty }\,\mathrm{d}y\int_{-\infty }^{+\infty }\,\mathrm{d}\tau \left[
\mathrm{i}\left( u_{z}u^{\ast }-u_{z}^{\ast }u\right) -\left\vert
u_{x}\right\vert ^{2}-\left\vert u_{y}\right\vert ^{2}-D\left\vert u_{\tau
}\right\vert ^{2}\right. \\
&&\left. +2\varepsilon \cos (2y)|u|^{2}+|u|^{4}\right] .
\end{eqnarray}The substitution of the ansatz (\ref{ansatz}) in this expression and
integrations lead to an \textit{effective Lagrangian}, with the prime
standing for $d/dz$:
\begin{eqnarray}
(4/\pi ^{3/2})L_{\mathrm{eff}} &=&A^{2}WVT\left[ 4\phi ^{\prime }-b^{\prime
}W^{2}-c^{\prime }V^{2}-\beta ^{\prime }T^{2}-W^{-2}-V^{-2}-DT^{-2}\right.
\nonumber \\
&&\left. -b^{2}W^{2}-c^{2}V^{2}+\varepsilon \exp \left( -V^{2}\right)
-D(z)\beta ^{2}T^{2}+A^{2}/\sqrt{2}\right] , \label{effL}
\end{eqnarray}
The first variational equation, $\delta L_{\mathrm{eff}}/\delta \phi =0$,
applied to Eq. (\ref{effL}) yields the energy conservation, $dE/dz=0$, with
\begin{equation}
E\equiv A^{2}WVT. \label{E}
\end{equation}The conservation of $E$ is used to eliminate $A^{2}$ from the set of
subsequent equations, $\delta L_{\mathrm{eff}}/\delta \left( W,V,T,b,c,\beta
\right) =0$. They can be arranged so as, first, to eliminate the chirps,
\begin{equation}
b=W^{\prime }/W,\,c=V^{\prime }/V,\beta =D^{-1}T^{\prime }/T. \label{betab}
\end{equation}the remaining equations for the spatial and temporal widths being
\begin{eqnarray}
W^{\prime \prime } &=&\frac{1}{W^{3}}-\frac{E}{2\sqrt{2}W^{2}VT},
\label{variat1} \\
V^{\prime \prime } &=&\frac{1}{V^{3}}-4\varepsilon V\exp \left(
-V^{2}\right) -\frac{E}{2\sqrt{2}WV^{2}T}, \label{variat2} \\
\left( \frac{T^{\prime }}{D}\right) ^{\prime }
&=&\frac{D}{T^{3}}-\frac{E}{2\sqrt{2}WVT^{2}}. \label{variat3}
\end{eqnarray}{The Hamiltonian of these equations, which is a dynamical invariant in the
case of constant $D$, is
\[
{\mathcal{H}}=\left( W^{\prime }\right) ^{2}+\left( V^{\prime
}\right) ^{2}+\frac{\left( T^{\prime }\right)
^{2}}{D}+\frac{1}{W^{2}}+\frac{1}{V^{2}}+\frac{D}{T^{2}}-4\varepsilon
\exp (-V^{2})-\frac{E}{\sqrt{2}WVT}
\]} In the case of the piece-wise constant modulation, such as in (\ref{D(z)}),
the variables $W$, $W^{\prime }$, $V$, $V^{\prime }$, $T$ and $\beta $
must be continuous at junctions between the segments with $D_{\pm }\equiv
\overline{D}\pm D_{\mathrm{m}}$. As it follows from Eq. (\ref{betab}), the
continuity of the temporal chirp $\beta (z)$ implies a jump of the
derivative $T^{\prime }$ when passing from $D_{-}$ to $D_{+}$, or vice
versa:
\begin{equation}
\left( T^{\prime }\right) _{D=D_{+}}=\left( D_{+}/D_{-}\right) \left(
T^{\prime }\right) _{D=D_{-}}. \label{jump}
\end{equation}
In the case of a continuous DM map, rather than the one
(\ref{D(z)}), Eq. (\ref{variat3}) has a formal singularity at the
points where $D(z)$ vanishes, changing its sign. However, it is
known that there is no real singularity in this case, as
$T^{\prime }$ vanishes at the same points, which cancels the
singularity out \cite{Progress}.
In the absence of the DM and transverse modulation, i.e., $D\equiv
+1$ and $\varepsilon =0$, three equations (\ref{variat1}) -
(\ref{variat3})\ reduce to one, which is tantamount to the
variational equation derived in Ref. \cite{Sweden} from the
spatiotemporally isotropic ansatz [cf. Eq. (\ref{ansatz})],
$u=A\exp \left[ \mathrm{i}\phi -(1/2)\left(
W^{-2}+\mathrm{i}b\right) \left( x^{2}+y^{2}+\tau ^{2}\right)
\right] $. In particular, this single equation correctly predicts
the asymptotic law of the \textit{strong collapse} in the 3D case,
which is stable against anisotropic perturbations \cite{Russia},
$V=W=T\approx \left( 5E/3\sqrt{2}\right) ^{1/5}\left(
z_{0}-z\right) ^{2/5}$, $z=z_{0}$ being the collapse point. The
location of this point is determined by initial conditions, but,
in any case, it belongs to an interval $D>0$, where the GVD is
anomalous.
Another possible collapse scenario is an effectively
two-dimensional (weak) one, with two widths shrinking to zero as
$z_{0}-z\rightarrow 0$, while the third one remains finite. For
instance, the corresponding asymptotic law may be\begin{equation}
V=T=A\left( z_{0}-z\right)
^{1/2},~W=\frac{\sqrt{2}E}{4+A^{4}}-\frac{\left( 4+A^{4}\right)
^{2}}{4\sqrt{2}EA^{2}}\left( z_{0}-z\right) \ln \left(
z_{0}-z\right) , \label{2Dcollapse}
\end{equation}where $A$ is a positive constant or else $V=2/T\sim (z_{0}-z)^{1/2}$
and $W\rightarrow W_{0}$ (in this case too, the collapse point
$z_0$ must belong to a segment with $D>0$). In direct simulations
of Eqs. (\ref{variat1}) - (\ref{variat3}), we actually observed
only the latter scenario. However, we did not specially try to
find initial conditions that could initiate a solution
corresponding to the strong 3D collapse, as our objective is not
the study of the collapse, but rather search for solitons stable
against collapse. In fact, known results for the solitons in the
3D Gross-Pitaevskii equation with the OL potential suggest that,
while the VA may be incorrect in the description of the collapse,
as a singular solution, it provides for quite accurate predictions
for the stability of solitons as \emph{regular solutions}
\cite{Estoril,BBB2}.
If the DM is absent, and the constant GVD is normal, i.e.,
$D\equiv -1$, only the 2D collapse in the transverse plane would
be possible, so that (cf. Eq. (\ref{2Dcollapse}))\[ V=W=A\left(
z_{0}-z\right) ^{1/2},~T=\frac{\sqrt{2}E}{4+A^{4}}+\frac{\left(
4+A^{4}\right) ^{2}}{4\sqrt{2}EA^{2}}\left( z_{0}-z\right) \ln
\left( z_{0}-z\right) .
\]However, we did not observed this collapse scenario in our simulations. The
same comment as one given above pertains to this case as well.
A possibility of the stabilization of the 3D soliton by a
sufficiently strong lattice can be understood noticing that, for
large $\varepsilon $, one may keep only the first two terms on the
right-hand side of Eq. (\ref{variat2}). This approximation yields
a nearly constant value $V_{0}$ of $V$, which is a smaller root of
the corresponding equation,
\begin{equation}
4\varepsilon V_{0}^{4}\exp \left( -V_{0}^{2}\right) =1 \label{static}
\end{equation}(a larger root corresponds to an unstable equilibrium).
The two roots exist provided that
\begin{equation}
\varepsilon >\varepsilon _{\mathrm{\min }}=\mathrm{e}^{2}/16\approx
\allowbreak 0.46, \label{min}
\end{equation}the relevant one being limited by $V_{0}<2$.
Then, the substitution of $V=V_{0}$ in the remaining
equations (\ref{variat1}) and (\ref{variat3})
leads to essentially the same VA-generated dynamical system as
derived for the 2D DM model in Ref. \cite{we}, which was shown to
give rise to stable spatiotemporal solitons. On the other hand, it
was demonstrated in Ref. \cite{we} too that, in the case of
$\varepsilon =0$, the 3D VA equations, as well as the full
underlying 3D model, have no stable soliton solutions.
The stabilization of the LB in the present model for large
$\varepsilon $ can also be understood in a different way, without
resorting to VA: in a very strong lattice, the soliton is trapped
entirely in a single ``valley" of the periodic potential, and the
problem thus reduces to a nearly 2D one, where spatiotemporal
solitons may be stable, cf. a similar stabilization mechanism for
the solitons in the Gross-Pitaevskii equations developed in
\cite{Castilla}. From this point of view, a really interesting
issue is to find an \emph{actual} minimum $\varepsilon _{\min }$
of the lattice's strength which is necessary for the stabilization
of the 3D solitons, as at $\varepsilon $ close enough to
$\varepsilon _{\mathrm{\min }} $ the stabilized solitons are truly
3D objects, rather than their nearly-2D counterparts.
\section{Results}
We explored the parameter space of the variational system
(\ref{variat1}) - (\ref{variat3}), $\left( E,\varepsilon
,\overline{D}\right) $, by means of direct simulations of the
equations (with regard to the jump condition (\ref{jump})). As a
result, it was possible to identify regions where the model admits
\emph{stable} solitons featuring regular oscillations in $z$ with
the DM-map period. An example of such a regime is shown in Fig.
\ref{fig1} (oscillations in the evolution of $W$ are not visible
in the figure because, as an estimate demonstrates, their
amplitude is $\simeq 0.001$).
\begin{figure}[tbp]
\includegraphics[width=13.5cm]{fig1.eps}
\caption{An example of the stable evolution of solutions to the
variational equations (\protect\ref{variat1}) -
(\protect\ref{variat3}). The soliton's widths in the direction
$x,y$ and $\protect\tau $, i.e., $W,V$ and $T$, are shown as
functions of $z$, for $E=0.5$, $\protect\varepsilon =1$, and
$\overline{D}=0$.} \label{fig1}
\end{figure}
Systematic results obtained from the simulations are summarized in stability
diagrams displayed in Figs. \ref{fig2} and \ref{fig3}. A remarkable fact,
apparent in Fig. \ref{fig2}, is that the minimum value of the lattice's
strength, $\varepsilon _{\min }=0.46$, at which the solitons may be stable,
coincides with the analytical prediction (\ref{min}), up to the available
numerical accuracy.
\begin{figure}[tbp]
\includegraphics[width=13.5cm]{fig2.eps}
\caption{The stability area for the 3D spatiotemporal solitons in
the $\left( E,\protect\varepsilon \right) $ plane, with
$\overline{D}=0$, is shown by light-gray shading. In gray and
dark-gray regions, the 3D soliton is predicted, respectively, to
spread out and collapse. The vertical line corresponds to the
analytically predicted threshold (\protect\ref{min}).}
\label{fig2}
\end{figure}
\begin{figure}[tbp]
\includegraphics[width=13.5cm]{fig3.eps}
\caption{The stability area in the $\left( E,\overline{D}\right) $ plane,
with $\protect\varepsilon =1$, is shown by light--gray shading. In the gray
region, the 3D soliton is predicted to spread out.}
\label{fig3}
\end{figure}
The existence of a maximum value $E_{\max }$ of the energy
admitting the stable LBs is, essentially, a quasi-2D feature,
which can be understood assuming that the potential lattice is
strong. Indeed, as explained above, in such a case the value of
$V$ is approximately fixed as the smaller root of Eq.
(\ref{static}). Within a segment where the GVD coefficient keeps
the constant value, $D_{+}=\overline{D}+D_{\mathrm{m}}>0$, which
corresponds to anomalous dispersion (see Eq. (\ref{D(z)}), the
remaining equations (\ref{variat1}) and (\ref{variat3}) are
tantamount to those for a uniform 2D Kerr-self-focusing medium,
hence the energy is limited by the value $E_{\max }$ corresponding
to the \textit{Townes soliton}; the soliton will collapse if
$E>E_{\max }$ \cite{Berge}.
The fact that the region of stable solitons is also limited by a
minimum energy, $E_{\min }$, except for the case of
$\overline{D}=0$, when $E_{\min }=0$ (see Fig. \ref{fig3})), is
actually a quasi-1D feature, which is characteristic to the DM
solitons in optical fibers. In that case, the term $\sim E$ in the
evolution equation for $T(z)$, cf. Eq. (\ref{variat3}), is
necessary to balance the average GVD coefficient $\overline{D}$,
so that $E_{\min }$ and $\overline{D}$ vanish simultaneously
\cite{DM}. It is noteworthy too that, as well as in the case of
the 1D DM solitons in fibers, the stability area in Fig.
\ref{fig3} includes a part with \emph{normal} average GVD,
$\overline{D}<0$, which seems counterintuitive, but can be
explained \cite{DM}. This part extends in Fig. \ref{fig3} up to
$\left( -\overline{D}\right) _{\max }\approx 0.005$.
\section{Conclusions}
In this work, we have proposed a possibility to stabilize
spatiotemporal solitons (``light bullets") in three-dimensional
self-focusing Kerr media by means of the dispersion management
(DM), which means that the local group-velocity dispersion
coefficient alternates between positive and negative values along
the propagation direction, $z$. Recently, it was shown that the DM
alone can stabilize solitons in 2D (planar) waveguides, but in the
bulk (3D) DM medium the ``bullets" are unstable. In this work, we
have demonstrated that the complete stabilization can \ be
provided if the longitudinal DM is combined with periodic
modulation of the refractive index in one transverse direction
($y$), out of the two. The analysis was based on the variational
approximation (systematic results of direct simulations will be
reported in a follow-up paper). A stability area for the light
bullets was identified in the model's parameter space. Its salient
features are a necessary minimum strength of the transverse
modulation of the refractive index, and minimum and maximum values
$E_{\min ,\max }$ of the soliton's energy. The former feature can
be accurately predicted (see Eq. (\ref{min})) in an analytical
form from the evolution equation for the width of the soliton in
the $y$-direction. The existence of $E_{\min }$, which vanishes
when we assume zero average dispersion, can be explained in the
same way as for the temporal solitons in DM optical fibers. Also,
similar to the case of DM solitons in fibers, we find that the
stability area extends to a region of \emph{normal} average
dispersion \cite{DM}. On the other hand, the existence of $E_{\max
}$ can be understood similarly to as it was recently done in the
2D counterpart of the present model (the strong transverse lattice
can squeeze the system to a nearly 2D shape).
The results presented in this work suggest a new approach to the challenging
problem of the creation of 3D spatiotemporal optical solitons. The model
also opens a way to address advanced issues, such as collisions between the
LBs, and the existence and stability of solitons with different symmetries
(for instance, LBs which are odd in the longitudinal and/or transverse
directions). These issues will be considered elsewhere.
\section{Acknowlegdements}
M.M., M.T. and E.I. acknowledge support from KBN Grant No. 2P03 B4325.
B.A.M. acknowledges the hospitality of the Physics Department and Soltan
Institute for Nuclear Studies at the Warsaw University, and partial support
from the Israel Science Foundation grant No. 8006/03. This author also
appreciates the help of A. Desyatnikov in making Ref. \cite{Estoril}
available on the internet.
|
{
"timestamp": "2005-03-08T12:03:11",
"yymm": "0503",
"arxiv_id": "physics/0503058",
"language": "en",
"url": "https://arxiv.org/abs/physics/0503058"
}
|
\section{Notations}
Let ${p}=e^{-\frac{\pi K'}{K}}$, $q=-e^{-\frac{\pi \lambda}{2K}}$ and $\zeta=e^{-\frac{\pi \lambda u}{2K}}$.
We introduce $x$, $\tau$ and $r$ by $x=-q$,
$\tau=\frac{2iK}{K'}$ and $r=\frac{K'}{\lambda}$. Then $p=e^{-\frac{2\pi i}{ \tau}}=x^{2r}$.
Through this paper, we assume ${\rm Im}\tau >0$.
Let ${\tilde{p}}=e^{2\pi i \tau}=e^{-\pi \frac{I'}{I}}$, where $I=\frac{K'}{2},\
I'=2K$. We use the theta functions
\begin{eqnarray*}
&&\vartheta_1(u|\tau)=2\tilde{p}^{1/8}(\tilde{p};\tilde{p})_\infty\sin\pi u
\prod_{n=1}^\infty(1-2\tilde{p}^n\cos2\pi u+\tilde{p}^{2n}),\\
&&\vartheta_0(u|\tau)=-ie^{\pi i(u+\tau/4)}\vartheta_1\left(u+\frac{\tau}{2}\Big|\tau\right),\\
&&\vartheta_2(u|\tau)=\vartheta_1\left(u+\frac{1}{2}\Big|\tau\right),\\
&&\vartheta_3(u|\tau)=e^{\pi i(u+\tau/4)}\vartheta_1\left(u+\frac{\tau+1}{2}\Big|\tau\right)
\end{eqnarray*}
and Jacobi's elliptic functions
\begin{eqnarray*}
&&{\rm sn}\lambda u=\frac{\vtf{3}{0}{\tau}\vtf{1}{\frac{\lambda u}{2I}}{\tau}}{\vtf{2}{0}{\tau}\vtf{0}{\frac{\lambda u}{2I}}{\tau}}
=\frac{\vtf{3}{0}{\tau}\vtf{1}{\frac{u}{r}}{\tau}}{\vtf{2}{0}{\tau}\vtf{0}{\frac{u}{r}}{\tau}},\\
&&{\rm cn}\lambda u=\frac{\vtf{0}{0}{\tau}\vtf{2}{\frac{\lambda u}{2I}}{\tau}}{\vtf{2}{0}{\tau}\vtf{0}{\frac{\lambda u}{2I}}{\tau}}
=\frac{\vtf{0}{0}{\tau}\vtf{2}{\frac{u}{r}}{\tau}}{\vtf{2}{0}{\tau}\vtf{0}{\frac{u}{r}}{\tau}},\\
&&{\rm dn}\lambda u=\frac{\vtf{0}{0}{\tau}\vtf{3}{\frac{\lambda u}{2I}}{\tau}}{\vtf{3}{0}{\tau}\vtf{0}{\frac{\lambda u}{2I}}{\tau}}
=\frac{\vtf{0}{0}{\tau}\vtf{3}{\frac{u}{r}}{\tau}}{\vtf{3}{0}{\tau}\vtf{0}{\frac{u}{r}}{\tau}}.
\end{eqnarray*}
We also use the symbol $[u]$ defined by
\begin{eqnarray*}
&&[u]=x^{\frac{u^2}{r}-u}\Theta_{x^{2r}}(x^{2u})=C\vt{1}{u}{\tau}, \quad C=x^{-\frac{r}{4}}e^{-\frac{\pi i}{4}}\tau^{\frac{1}{2}}
\end{eqnarray*}
and abbreviation
\begin{eqnarray*}
&&\vth{1,2}{u}=\vth{1}{u}\vth{2}{u}=\vtf{0}{0}{\tau}\vt{1}{u}{\tau},
\end{eqnarray*}
etc..
\section{Fusion of Baxter's $R$-matrix}
Baxter's elliptic $R$-matrix is given by\cite{Baxter}
\begin{eqnarray}
&&{R}(u)=
{R}_0(u)\left(\matrix{a(u)&&&d(u)\cr
&b(u)&c(u)&\cr
&c(u)&b(u)&\cr
d(u)&&&a(u)\cr}\right),
\end{eqnarray}
where
\begin{eqnarray}
R_0(u)&=&z^{-\frac{r-1}{2r}}\frac{(px^2z;x^4,p)_\infty(x^2z;x^4,p)_\infty
(p/z;x^4,p)_\infty(x^4/z;x^4,p)_\infty}{(px^2/z;x^4,p)_\infty
(x^2/z;x^4,p)_\infty(pz;x^4,p)_\infty(x^4z;x^4,p)_\infty},\lb{evR}\\
a(u)&=&\frac{\vtf{2}{\frac{1}{2r}}{\frac{\tau}{2}}\vtf{2}{\frac{ u}{2r}}{\frac{\tau}{2}}}{\vtf{2}{0}{\frac{\tau}{2}}\vtf{2}{\frac{1+u}{2r}}{\frac{\tau}{2}}},\qquad
b(u)=\frac{\vtf{2}{\frac{1 }{2r}}{\frac{\tau}{2}}\vtf{1}{\frac{ u}{2r}}{\frac{\tau}{2}}}
{\vtf{2}{0}{\frac{\tau}{2}}\vtf{1}{\frac{1+u}{2r}}{\frac{\tau}{2}}},\\
c(u)&=&\frac{\vtf{1}{\frac{ 1}{2r}}{\frac{\tau}{2}}\vtf{2}{\frac{ u}{2r}}{\frac{\tau}{2}}}
{\vtf{2}{0}{\frac{\tau}{2}}\vtf{1}{\frac{1+u}{2r}}{\frac{\tau}{2}}},\qquad
d(u)=-\frac{\vtf{1}{\frac{ 1}{2r}}{\frac{\tau}{2}}\vtf{1}{\frac{ u}{2r}}{\frac{\tau}{2}}}
{\vtf{2}{0}{\frac{\tau}{2}}\vtf{2}{\frac{1+u}{2r}}{\frac{\tau}{2}}}
\end{eqnarray}
with $z=\zeta^2=x^{2u}$.
Let $V=\C v_{\varepsilon_1}\oplus \C v_{\varepsilon_2},\ \varepsilon_1,\varepsilon_2=+,-$. We regard $R(u)\in {\rm End}(V\otimes V)$.
The $R$-matrix \eqref{evR} satisfies
\begin{eqnarray}
&&R(u)PR(u)P=\hbox{id},\\
&&R(-u-1)=(\sigma^{y }\otimes 1)^{-1}\ (PR(u)P)^{t_1} \ \sigma^y\otimes 1,\\
&&R(0)=P, \qquad \lim_{u\to -1}R(u)=P-\hbox{id}.
\end{eqnarray}
Here ${^{t_1}}$ denotes the transposition with respect to the first vector space in the tensor product and $P(\varepsilon_1\otimes \varepsilon_2)=\varepsilon_2\otimes \varepsilon_1$.
Fusion of $R(u)$ was considered systematically in \cite{DJKMO}.
Let $V^{(2)}$ be the space of the symmetric tensors in $V\otimes V$
spanned by $v^{(2)}_{2}\equiv v_+\otimes v_+,\ v^{(2)}_{0}\equiv
\frac{1}{2}(v_+\otimes v_-+v_-\otimes v_+),\ v^{(2)}_{-2}\equiv v_-\otimes v_-$.The projection operator of the space $V\otimes V$ on $V^{(2)}$
is given by $\Pi=\frac{1}{2}(P+\hbox{id})$.
Let $V_1, V_2, V_{{\bar{1}}}, V_{{\bar{2}}}$ be the copies of V.
Define
\begin{eqnarray}
&&R^{(2,1)}_{12, \bar{j}}(u)=\Pi_{12}R_{1{\bar{j}}}(u+1)R_{2{\bar{j}}}(u) \ \in {\rm End}(V^{(2)} \otimes V_{{\bar{j}}}).
\lb{hfusion}
\end{eqnarray}
It follows that
\begin{eqnarray}
&&R^{(2,1)}_{12,{\bar{j}}}(u)\Pi_{12}=R^{(2,1)}_{12,{\bar{j}}}(u).
\end{eqnarray}
The 2$\times$2 fusion of the $R$-matrix is then given by
\begin{eqnarray}
&&R^{(2,2)}(u)=\Pi_{{\bar{1}}{\bar{2}}}R^{(2,1)}_{12,{\bar{2}}}(u)R^{(2,1)}_{12,{\bar{1}}}(u-1) \ \in {\rm End}(V^{(2)} \otimes V^{(2)}).\lb{vhfusion}
\end{eqnarray}
This satisfies the Yang-Baxter equation on $V^{(2)} \otimes V^{(2)} \otimes V^{(2)}$.
We calculate the matrix elements of $R^{(2,1)}(u)$ and $R^{(2,2)}(u)$ defined by
\begin{eqnarray}
&&R^{(2,1)}(u) v^{(2)}_{\mu}\otimes v^{}_{\varepsilon}=\sum_{\mu'=2,0,-2 \atop \varepsilon'=+,-} R^{(2,1)}(u)^{\mu \varepsilon}_{\mu' \varepsilon'}\ v^{(2)}_{\mu'}\otimes v^{}_{\varepsilon'},\\
&&R^{(2,2)}(u) v^{(2)}_{\mu_1}\otimes v^{(2)}_{\mu_2} =\sum_{\mu'_1,\mu'_2=2,0,-2} R^{(2,2)}(u)^{\mu_1 \mu_2}_{\mu'_1 \mu'_2}\ v^{(2)}_{\mu'_1}\otimes v^{(2)}_{\mu'_2}.
\end{eqnarray}
From \eqref{hfusion}, we have
\begin{eqnarray}
&&R^{(2,1)}(u)^{\mu\ \bar{\varepsilon}}_{\mu'\ \bar{\varepsilon}'}=\sum_{\varepsilon_2',\bar{\varepsilon}''=\pm 1}R(u+1)^{\mu-\varepsilon_2\ \bar{\varepsilon}''}_{\mu'-\varepsilon_2'\ \bar{\varepsilon}'}R(u)^{\varepsilon_2\ \bar{\varepsilon}}_{\varepsilon_2'\ \bar{\varepsilon}''},
\end{eqnarray}
where we set $\mu=\varepsilon_1+\varepsilon_2,\ \mu'=\varepsilon_1'+\varepsilon_2'$. Evaluating the summation explicitly, we obtain
\begin{prop}
\begin{eqnarray*}
&&R^{(2,1)}(u)=R^{(2,1)}_0(u)\left(\matrix{R^{+2+}_{+2+}&0&0&R^{\ 0-}_{+2+}&R^{-2+}_{+2+}&0\cr
0&R^{+2-}_{+2-}&R^{\ 0+}_{+2-}&0&0&R^{-2-}_{+2-}\cr
0&R^{+2-}_{\ 0+}&R^{0+}_{0+}&0&0&R^{-2-}_{\ 0+}\cr
R^{+2+}_{\ 0-}&0&0&R^{\ 0-}_{\ 0-}&R^{-2+}_{\ 0-}&0\cr
R^{+2+}_{-2+}&0&0&R^{\ 0-}_{-2+}&R^{-2+}_{-2+}&0\cr
0&R^{+2-}_{-2-}&R^{\ 0+}_{-2-}&0&0&R^{-2-}_{-2-}\cr}\right),
\end{eqnarray*}
where
\begin{eqnarray*}
&&R^{(2,1)}_0(u)=R_0(u+1)R_0(u)=-\frac{[u+1]}{[u]},\\
&&R(u)^{+2+}_{+2+}=R(u)^{-2-}_{-2-}=\frac{\vtf{2}{\frac{1}{2r}}{\frac{\tau}{2}}^2\vtf{2}{\frac{u}{2r}}{\frac{\tau}{2}}}{\vtf{2}{0}{\frac{\tau}{2}}^2\vtf{2}{\frac{u+2}{2r}}{\frac{\tau}{2}}},\\
&&R(u)^{\ 0-}_{+2+}=R(u)^{\ 0+}_{-2-}
=-\frac{\vtf{1}{\frac{1}{2r}}{\frac{\tau}{2}}\vtf{2}{\frac{1}{2r}}{\frac{\tau}{2}}
\vtf{1}{\frac{u}{2r}}{\frac{\tau}{2}}}{\vtf{2}{0}{\frac{\tau}{2}}^2\vtf{2}{\frac{u+2}{2r}}{\frac{\tau}{2}}},\\
&&R(u)^{-2+}_{+2+}=R(u)^{+2-}_{-2-}=-\frac{\vtf{1}{\frac{1}{2r}}{\frac{\tau}{2}}^2\vtf{2}{\frac{u}{2r}}{\frac{\tau}{2}}}{\vtf{2}{0}{\frac{\tau}{2}}^2\vtf{2}{\frac{u+2}{2r}}{\frac{\tau}{2}}},\\
&&R(u)^{+2-}_{+2-}=R(u)^{-2+}_{-2+}=\frac{\vtf{2}{\frac{1}{2r}}{\frac{\tau}{2}}^2\vtf{1}{\frac{u}{2r}}{\frac{\tau}{2}}}{\vtf{2}{0}{\frac{\tau}{2}}^2\vtf{1}{\frac{u+2}{2r}}{\frac{\tau}{2}}},\\
&&R(u)^{\ 0+}_{+2-}=R(u)^{\ 0-}_{-2+}=\frac{\vtf{1}{\frac{1}{2r}}{\frac{\tau}{2}}\vtf{2}{\frac{1}{2r}}{\frac{\tau}{2}}
\vtf{2}{\frac{u}{2r}}{\frac{\tau}{2}}}{\vtf{2}{0}{\frac{\tau}{2}}^2\vtf{1}{\frac{u+2}{2r}}{\frac{\tau}{2}}},\\
&&R(u)^{-2-}_{+2-}=R(u)^{+2+}_{-2+}=-
\frac{\vtf{1}{\frac{1}{2r}}{\frac{\tau}{2}}^2\vtf{1}{\frac{u}{2r}}{\frac{\tau}{2}}}{\vtf{2}{0}{\frac{\tau}{2}}^2\vtf{1}{\frac{u+2}{2r}}{\frac{\tau}{2}}},\\
&&R(u)^{+2-}_{0\ +}=R(u)^{-2+}_{0\ -}=
\frac{\vtf{1}{\frac{1}{r}}{\frac{\tau}{2}}\vtf{2}{\frac{u+1}{2r}}{\frac{\tau}{2}}^2}
{\vtf{2}{0}{\frac{\tau}{2}}
\vtf{1}{\frac{u+2}{2r}}{\frac{\tau}{2}}\vtf{2}{\frac{u+2}{2r}}{\frac{\tau}{2}}},\\
&&R(u)^{+2+}_{0\ -}=R(u)^{-2-}_{0\ +}=-
\frac{\vtf{1}{\frac{1}{r}}{\frac{\tau}{2}}\vtf{1}{\frac{u+1}{2r}}{\frac{\tau}{2}}^2}
{\vtf{2}{0}{\frac{\tau}{2}}
\vtf{1}{\frac{u+2}{2r}}{\frac{\tau}{2}}\vtf{2}{\frac{u+2}{2r}}{\frac{\tau}{2}}},\\
&&R(u)^{0\ +}_{0\ +}=R(u)^{0\ -}_{0\ -}=
\frac{\vtf{2}{\frac{1}{r}}{\frac{\tau}{2}}\vth{1,2}{u+1}}
{\vtf{2}{0}{\frac{\tau}{2}}\vth{1,2}{u+2}}.
\end{eqnarray*}
\end{prop}
Similarly, from \eqref{vhfusion}, we obtain
\begin{eqnarray*}
&&R^{(2,2)}(u)^{\mu_1\ \mu_2}_{\mu_1'\ \mu_2'}=\sum_{\mu''=0, \pm2 \atop \bar{\varepsilon}_1'=\pm1}R^{(2,1)}(u)^{ \mu''\ {\mu}_2-\bar{\varepsilon}_1}_{\mu_1'\ {\mu}_2'-\bar{\varepsilon}_1'}R^{(2,1)}(u-1)^{ \mu_1\ \bar{\varepsilon}_1}_{\mu''\ \bar{\varepsilon}_1'},
\end{eqnarray*}
where $\mu_1=\varepsilon_1+\varepsilon_2,\ \mu_1'=\varepsilon_1'+\varepsilon_2'$ and
$\mu_2=\bar{\varepsilon}_{{1}}+\bar{\varepsilon}_{{2}},\ \mu_2'=\bar{\varepsilon}_{{1}}'
+\bar{\varepsilon}_{{2}}'$.
\begin{prop}
\begin{eqnarray*}
&&R^{(2,2)}(u)=R^{(2,2)}_0(u)\left(\matrix{
G&0&A&0&B&0&A&0&H\cr
0&F&0&C&0&C^*&0&D&0\cr
A^*&0&G^*&0&B^*&0&H^*&0&A^*\cr
0&C&0&F&0&D&0&C^*&0\cr
I&0&I^*&0&E&0&I^*&0&I\cr
0&C^*&0&D&0&F&0&C&0\cr
A^*&0&H^*&0&B^*&0&G^*&0&A^*\cr
0&D&0&C^*&0&C&0&F&0\cr
H&0&A&0&B&0&A&0&G\cr
}\right),
\end{eqnarray*}
where
\begin{eqnarray*}
R^{(2,2)}_0(u)&=&R^{(2,1)}_0(u-1)R^{(2,1)}_0(u)=\frac{[u+1]}{[u-1]},\\
A&=&R(u)^{+2-2}_{+2+2}=R(u)^{-2+2}_{-2-2}=R(u)^{-2+2}_{+2+2}=R(u)^{+2-2}_{-2-2}{\nonumber}\\
&=&-
\frac{
\vtf{1}{\frac{1}{r}}{\frac{\tau}{2}}
\vtf{2}{\frac{u}{2r}}{\frac{\tau}{2}}
\vtf{1,2}{\frac{1}{2r}}{\frac{\tau}{2}}
\vtf{1,2}{\frac{u}{2r}}{\frac{\tau}{2}}
}{
\vtf{2}{0}{\frac{\tau}{2}}^3
\vtf{2}{\frac{u+2}{2r}}{\frac{\tau}{2}}
\vtf{1,2}{\frac{u+1}{2r}}{\frac{\tau}{2}}
},\\
B&=&R(u)^{\ 0\ 0}_{+2+2}=R(u)^{\ 0\ 0}_{-2-2}=-
\frac{
\vtf{2}{\frac{1}{r}}{\frac{\tau}{2}}
\vtf{1}{\frac{u}{2r}}{\frac{\tau}{2}}
\vtf{1,2}{\frac{1}{2r}}{\frac{\tau}{2}}
\vtf{1,2}{\frac{u}{2r}}{\frac{\tau}{2}}
}{
\vtf{2}{0}{\frac{\tau}{2}}^3
\vtf{2}{\frac{u+2}{2r}}{\frac{\tau}{2}}
\vtf{1,2}{\frac{u+1}{2r}}{\frac{\tau}{2}}
},\\
C&=&R(u)^{+2\ 0}_{\ 0+2}=R(u)^{-2\ 0}_{\ 0-2}=R(u)^{\ 0+2}_{+2\ 0}=R(u)^{\ 0-2}_{-2\ 0}=
\frac{
\vtf{2}{\frac{u}{2r}}{\frac{\tau}{2}}^2 \vtf{1,2}{\frac{1}{r}}{\frac{\tau}{2}}
}
{
\vtf{2}{0}{\frac{\tau}{2}}^2 \vth{1,2}{u+2}
},\\
D&=&R(u)^{+2\ 0}_{-2\ 0 }=R(u)^{-2\ 0}_{+2\ 0 }=R(u)^{\ 0+2}_{\ 0-2}=R(u)^{\ 0 -2}_{\ 0+2}
=-\frac{
\vtf{1}{\frac{1}{r}}{\frac{\tau}{2}}^2 \vth{1,2}{u}
}
{
\vtf{2}{0}{\frac{\tau}{2}}^2 \vth{1,2}{u+2}
},\\
E&=&R(u)^{\ 0\ 0 }_{\ 0\ 0 }{\nonumber}\\
&=&\frac{
\vtf{1}{\frac{1}{r}}{\frac{\tau}{2}}
\vtf{1,2}{\frac{1}{2r}}{\frac{\tau}{2}}
\left(
\vtf{2}{\frac{u+1}{2r}}{\frac{\tau}{2}}^3
\vtf{2}{\frac{u-1}{2r}}{\frac{\tau}{2}}+
\vtf{1}{\frac{u+1}{2r}}{\frac{\tau}{2}}^3
\vtf{1}{\frac{u-1}{2r}}{\frac{\tau}{2}}
\right)
}{
\vtf{2}{0}{\frac{\tau}{2}}^3
\vtf{1,2}{\frac{u+2}{2r}}{\frac{\tau}{2}}
\vtf{1,2}{\frac{u+1}{2r}}{\frac{\tau}{2}}}{\nonumber}\\
&&+\frac{
\vtf{2}{\frac{1}{r}}{\frac{\tau}{2}}^2\vth{1,2}{u}
}{
\vtf{2}{0}{\frac{\tau}{2}}^2\vth{1,2}{u+2}
}
,\\
F&=&R(u)^{\ 0+2}_{\ 0 +2}=R(u)^{\ 0-2}_{\ 0 -2}=R(u)^{+2\ 0}_{+2\ 0}=R(u)^{-2\ 0}_{-2\ 0}
=\frac{
\vtf{2}{\frac{1}{r}}{\frac{\tau}{2}}^2 \vth{1,2}{u}
}
{
\vtf{2}{0}{\frac{\tau}{2}}^2 \vth{1,2}{u+2}
},
\end{eqnarray*}
\begin{eqnarray*}
G&=&R(u)^{+2+2}_{+2+2}=R(u)^{-2-2}_{-2-2}{\nonumber}\\
&=&
\frac{
\vtf{2}{\frac{u}{2r}}{\frac{\tau}{2}}
\left(\vtf{1}{\frac{1}{2r}}{\frac{\tau}{2}}^4\vtf{1}{\frac{u-1}{2r}}{\frac{\tau}{2}}
\vtf{2}{\frac{u+1}{2r}}{\frac{\tau}{2}}+
\vtf{2}{\frac{1}{2r}}{\frac{\tau}{2}}^4\vtf{2}{\frac{u-1}{2r}}{\frac{\tau}{2}}
\vtf{1}{\frac{u+1}{2r}}{\frac{\tau}{2}}
\right)
}
{
\vtf{2}{0}{\frac{\tau}{2}}^4 \vtf{2}{\frac{u+2}{2r}}{\frac{\tau}{2}}
\vtf{1,2}{\frac{u+1}{2r}}{\frac{\tau}{2}}
}
,
\end{eqnarray*}
\begin{eqnarray*}
H&=&R(u)^{+2+2}_{-2-2}=R(u)^{-2-2}_{+2+2}=
\frac{
\vtf{1}{\frac{1}{r}}{\frac{\tau}{2}}
\vtf{1}{\frac{u}{2r}}{\frac{\tau}{2}}^3
\vth{1,2}{1}
}{
\vtf{2}{0}{\frac{\tau}{2}}^3
\vtf{2}{\frac{u+2}{2r}}{\frac{\tau}{2}}
\vth{1,2}{u+1}
},
\end{eqnarray*}
\begin{eqnarray*}
I&=&R(u)^{+2+2}_{\ 0\ 0 }=R(u)^{-2-2}_{\ 0\ 0}{\nonumber}\\
&=&-\frac{
\vtf{1}{\frac{1}{r}}{\frac{\tau}{2}}
\left(
\vtf{2}{\frac{1}{2r}}{\frac{\tau}{2}}^2
\vtf{1}{\frac{u+1}{2r}}{\frac{\tau}{2}}^3
\vtf{2}{\frac{u-1}{2r}}{\frac{\tau}{2}}+
\vtf{1}{\frac{1}{2r}}{\frac{\tau}{2}}^2
\vtf{2}{\frac{u+1}{2r}}{\frac{\tau}{2}}^3
\vtf{1}{\frac{u-1}{2r}}{\frac{\tau}{2}}
\right)
}{
\vtf{2}{0}{\frac{\tau}{2}}^3
\vtf{1,2}{\frac{u+2}{2r}}{\frac{\tau}{2}}
\vtf{1,2}{\frac{u+1}{2r}}{\frac{\tau}{2}}}{\nonumber}\\
&&-\frac{
\vtf{1}{\frac{u}{2r}}{\frac{\tau}{2}}^2\vtf{1,2}{\frac{1}{r}}{\frac{\tau}{2}}
}{
\vtf{2}{0}{\frac{\tau}{2}}^2\vth{1,2}{u+2}
}.
\end{eqnarray*}
The $*$-ed matrix element is obtained from a corresponding non-$*$-ed element by replacing
the theta functions only depending on $u$ in the following rule.
\begin{eqnarray*}
&&\vartheta_1 \to -\vartheta_2,\quad \vartheta_2 \to \vartheta_1.
\end{eqnarray*}
\end{prop}
From this expression, one can easily see the following symmetries.
\begin{eqnarray*}
&&R^{(2,2)}(u)^{\varepsilon_1 \varepsilon_2}_{\varepsilon_1' \varepsilon_2'}=R^{(2,2)}(u)^{\varepsilon_2 \varepsilon_1}_{\varepsilon_2' \varepsilon_1'}, \qquad\qquad ( P{\rm-invariance})\\
&&R^{(2,2)}(u)^{\varepsilon_1 \varepsilon_2}_{\varepsilon_1' \varepsilon_2'}=R^{(2,2)}(u)^{-\varepsilon_1 -\varepsilon_2}_{-\varepsilon_1' -\varepsilon_2'} \qquad\qquad ({\rm \Z_2-symmetry}).
\end{eqnarray*}
In 1980, Fateev proposed the 21-vertex model as the spin one extension of Baxter's eight vertex model\cite{Fateev}.
Solving the Yang-Baxter equation, he obtained the following $R$-matrix.
\begin{eqnarray*}
&&R_{F}(u)=\tilde{F}(u)\left(\matrix{
s_1&0&0&0&\mu&0&0&0&\nu\cr
0&t&0&r&0&0&0&0&0\cr
0&0&T&0&0&0&R&0&0\cr
0&r&0&t&0&0&0&0&0\cr
\mu&0&0&0&s_2&0&0&0&\rho\cr
0&0&0&0&0&a&0&q&0\cr
0&0&R&0&0&0&T&0&0\cr
0&0&0&0&0&q&0&a&0\cr
\nu&0&0&0&\rho&0&0&0&s_3\cr
}
\right)
\end{eqnarray*}
where
\begin{eqnarray*}
&&s_1=\cn2\lambda+\frac{{\rm sn}\lambda\ \sn2\lambda}{{\rm sn}\lambda u\ {\rm sn}\lambda (u+1)},\\
&&s_2=\cn2\lambda+\dn2\lambda-1+\frac{{\rm sn}\lambda\ \sn2\lambda}{{\rm sn}\lambda u\ {\rm sn}\lambda( u+1)},\\
&&s_3=\dn2\lambda+\frac{{\rm sn}\lambda\ \sn2\lambda}{{\rm sn}\lambda u\ {\rm sn}\lambda( u+1)},\\
&&T=1,\qquad\qquad t=\cn2\lambda,\qquad\qquad a=\dn2\lambda,\\
&&r=\frac{{\rm cn}\lambda u\ \sn2\lambda}{{\rm sn}\lambda u},\qquad\qquad \mu=-\frac{{\rm cn}\lambda( u+1)\ \sn2\lambda}{{\rm sn}\lambda( u+1)},\\
&&R=\frac{ \sn2\lambda}{{\rm sn}\lambda u},\qquad\qquad \nu=-\frac{ \sn2\lambda}{{\rm sn}\lambda( u+1)},\\
&&q=\frac{{\rm dn}\lambda u\ \sn2\lambda}{{\rm sn}\lambda u},\qquad\qquad \rho=-\frac{{\rm dn}\lambda( u+1)\ \sn2\lambda}{{\rm sn}\lambda( u+1)}
\end{eqnarray*}
and $\tilde{F}(u)$ satisfies
\begin{eqnarray*}
&&\tilde{F}(u)=\tilde{F}(-u-1),\\
&&\tilde{F}(u)\tilde{F}(-u)=\frac{{\rm sn}^2\lambda u}{{\rm sn}^2\lambda u-{\rm sn}^22\lambda}.
\end{eqnarray*}
$R_{F}(u)$ has the following symmetries.
\begin{eqnarray*}
&&R_{F}(u)^{ij}_{kl}=R_{F}(u)^{ji}_{lk} \qquad\qquad ( P{\rm-invariance})\\
&&R_{F}(u)^{ij}_{kl}=R_{F}(u)_{ij}^{kl},\qquad\qquad (T{\rm-invariance})\\
&&R_{F}(u)^{ij}_{kl}=R_{F}(-u-1)_{il}^{kj}\qquad\qquad ({\rm Crossing\ symmetry}).
\end{eqnarray*}
We find
\begin{prop}
\begin{eqnarray*}
&&\tilde{F}(u)=R^{(2,2)}_0(u)\frac{\vt{0}{2}{\tau}\vt{1}{u}{\tau}}{\vtf{0}{0}{\tau}\vt{1}{u+2}{\tau}}.
\end{eqnarray*}
\end{prop}
Then the following theorem is essentially due to Jimbo\cite{Jimbo}.
\begin{thm}
$R^{(2,2)}(u)$ is gauge equivalent to $R_{F}(u)$. Namely,
\begin{eqnarray*}
&&R_{F}(u)=U\otimes U \ R^{(2,2)}(u)\ (U\otimes U)^{-1},
\end{eqnarray*}
where
\begin{eqnarray*}
&&U=\left(\matrix{1&0&0\cr 0&x&0\cr 0&0&y\cr}\right)\left(\matrix{\frac{1}{\sqrt{2}}&0&\frac{1}{\sqrt{2}}\cr 0&1&0\cr
\frac{1}{\sqrt{2}}&0&-\frac{1}{\sqrt{2}}\cr}\right),
\end{eqnarray*}
and
\begin{eqnarray*}
&&x^2={-\frac{1}{2}\frac{\vtf{0}{0}{\tau}\vt{3}{1}{\tau}}{\vt{0}{1}{\tau}\vtf{3}{0}{\tau}}},\quad
y^2={-\frac{\vtf{2}{0}{\tau}\vt{3}{1}{\tau}}{\vt{2}{1}{\tau}\vtf{3}{0}{\tau}}}.
\end{eqnarray*}
\end{thm}
Combining the crossing symmetry of $R_{F}(u)$ and the $P$-invariance of $R^{(2,2)}(u)$, we find
the following crossing symmetry formula for $R^{(2,2)}(u)$.
\begin{cor}
\begin{eqnarray}
&&R^{(2,2)}(-u-1)=Q^{-1}\otimes 1\ (P^{(2)}R^{(2,2)}(u)P^{(2)})^{t_1}\ Q\otimes 1,\lb{crossing}
\end{eqnarray}
where
\begin{eqnarray*}
&&Q=U^t U=\frac{1}{2}\left(\matrix{1+y^2&0&1-y^2\cr 0&x^2&0\cr 1-y^2&0&1+y^2\cr}\right)
\end{eqnarray*}
and $P^{(2)}$ is the permutation operator $P^{(2)}( v^{(2)}_{\varepsilon_1}\otimes v^{(2)}_{\varepsilon_2})= v^{(2)}_{\varepsilon_2}\otimes v^{(2)}_{\varepsilon_1}$.
\end{cor}
\noindent
{\it Remark :}\
\noindent
The crossing symmetry of the elliptic $R$-matrix is related to the dual module of the finite
dimensional module of the elliptic algebra ${\cal A}_{q,p}(\widehat{\goth{sl}}_2)$, or the module of the Sklyanin algebra. See \cite{JM} for the case $U_q(\widehat{\goth{sl}}_2)$.
Let $V_\zeta$ be the 3-dimensional module of ${\cal A}_{q,p}(\widehat{\goth{sl}}_2)$, and $V_\zeta^*$ its dual module.
The above $Q$-matrix gives an isomorphism between $V_\zeta$ and $V_\zeta^*$.
\section{The Vertex-Face Correspondence }
The vertex-face correspondence is a relationship between Baxter's $R$-matrix and the SOS face weight $W\BW{a_1}{a_2}{a_4}{a_3}{u}$ given by
\begin{eqnarray}
&&{W}\left(\left.
\begin{array}{cc}
n&n\pm 1\\
n\pm1&n\pm2
\end{array}\right|u\right)={R}_0(u),{\nonumber}\\
&&{W}\left(\left.
\begin{array}{cc}
n&n\pm 1\\
n\pm1&n
\end{array}\right|u\right)={R}_0(u)\frac{[n\mp u][1]}{[ n][ 1+u]},\lb{face}\\
&&{W}\left(\left.
\begin{array}{cc}
n&n\pm 1\\
n\mp 1&n
\end{array}\right|u\right)=
{R}_0(u)
\frac{[ n\pm 1][u] }{[ n][1+u]}.{\nonumber}
\end{eqnarray}
Let us consider the following vectors in $V$
\begin{eqnarray}
&&\psi(u)^a_b=\psi_+(u)^a_b\ v_+ + \psi_-(u)^a_b\ v_-,\qquad \\
&&{\psi}_+(u)^a_b=\vtf{0}{\frac{(a-b)u+a}{2r}}{\frac{\tau}{2}},\qquad \psi_-(u)=\vtf{3}{\frac{(a-b)u+a}{2r}}{\frac{\tau}{2}} \lb{intertwinvec}{\nonumber}
\end{eqnarray}
with $|a-b|=1$. Baxter showed the following identity\cite{Baxter}.
\begin{eqnarray}
&&\sum_{\varepsilon_1',\varepsilon_2'}
R(u-v)_{\varepsilon_1 \varepsilon_2}^{\varepsilon_1' \varepsilon_2'}\
\psi_{\varepsilon_1'}(u)_{b}^{a}
\psi_{\varepsilon_2'}(v)_{c}^{b}
=\sum_{b' \in {\mathbb{Z}}}
\psi_{\varepsilon_2}(v)_{b'}^{a}
\psi_{\varepsilon_1}(u)_{c}^{b'}
W\left(\left.
\begin{array}{cc}
a&b\\
b'&c
\end{array}\right|u-v\right).\lb{vertexface}
\end{eqnarray}
\subsection{Fusion}
Following Date et al.\cite{DJKMO}, we consider the fusion of the Vertex-Face correspondence relation \eqref{vertexface}.
The fusion of the SOS weights is briefly summarized as follows. The SOS weight \eqref{face} satisfies
\begin{eqnarray}
&&W\BW{a}{b}{d}{c}{0}=\delta_{b,d},\\
&&W\BW{a}{b}{d}{c}{-1}=0\qquad {\rm if}\ |a-c|=2,\\
&&W\BW{a}{a\pm 1}{a\pm 1}{a}{-1}=-W\BW{a}{a\pm 1}{a\mp 1}{a}{-1}.
\end{eqnarray}
Then if one defines
\begin{eqnarray}
&&W_{21}\BW{a}{b}{d}{c}{u}=\sum_{d'}W\BW{a}{a'}{d}{d'}{u+1}W\BW{a'}{b}{d'}{c}{u}, \lb{fhfusion}
\end{eqnarray}
one can verify the following statements.
(i) The RHS of \eqref{fhfusion} is independent of the choice of $a'$ provided $|a-a'|=|a'-b|=1$.
(ii) For all $a, b, c, d$,
\begin{eqnarray*}
&&W_{21}\BW{a}{b}{d}{c}{-1}=0.
\end{eqnarray*}
The $2\times 2$ fusion of the SOS weight is then given by the formula
\begin{eqnarray}
&&W_{22}\BW{a}{b}{d}{c}{u}=\sum_{a'}W_{21}\BW{a}{b}{a'}{b'}{u-1}W_{21}\BW{a'}{b'}{d}{c}{u}
\lb{fvhfusion}.
\end{eqnarray}
Here the RHS is independent of the choice of $b'$ provided $|b-b'|=|b'-c|=1$. Now the dynamical variables $a, b, c, d$ satisfies the extended admissible condition; $a_j-a_k\in \{2, 0, -2\}$ for any two adjacent local heights $a_j, a_k$. Furthermore the resultant SOS weight $W_{22}$ satisfies the face type YBE and defines the $2\times 2$ fusion SOS model. Explicit expressions of $W_{22}\BW{a}{b}{d}{c}{u}$ are given, for example, in \cite{KKW}. It satisfies the unitarity and crossing symmetry relations
\begin{eqnarray}
&&\sum_{s}W_{22}\BW{a}{s}{d}{c}{-u} W_{22}\BW{a}{b}{s}{c}{u}=\delta_{b,d},\label{SOSunitarity}\\
&&W_{22}\BW{d}{c}{a}{b}{u}=\frac{(b,c)_2 \ g_a g_c}{(a,d)_2\ g_b g_d} \,
W_{22}\BW{a}{d}{b}{c}{-1-u}.
\label{SOScrossing}
\end{eqnarray}
Here $g_a=\varepsilon_a\sqrt{[a]}$\, $\varepsilon_a=\pm1,\ \varepsilon_a\varepsilon_{a+1}=(-)^a$ and
\begin{eqnarray*}
&&(a,b)_M=(b,a)_M=\left[\matrix{M\cr
\frac{a-b+M}{2}\cr}\right]^{-1}
\frac{\left[\frac{a+b-M}{2}, \frac{a+b+M}{2}
\right]}{\sqrt{[a][b]}},\\
&&\left[\matrix{A\cr B\cr}\right]=\frac{[A][A-1]\cdots [A-B+1]}{[B][B-1]\cdots
[1]},\\
&&[A,B]=[A][A+1]\cdots [B] \quad (A<B),\qquad [A,A-1]=1.
\end{eqnarray*}
The fusion of the intertwining vectors is given by
\begin{eqnarray}
&&\psi^{(2)}(u)^a_b=\Pi\ \psi(u+1)^a_c\otimes \psi(u)^c_b \ \in V\otimes V.\lb{fusionpsi}
\end{eqnarray}
The RHS is independent of the choice of $c$ provided $|a-c|=|c-b|=1$.
Then using \eqref{vertexface}, \eqref{hfusion}, \eqref{vhfusion}, \eqref{fhfusion}, \eqref{fvhfusion}
and \eqref{fusionpsi}, pne can show
\begin{eqnarray}
&&\sum_{\mu_1',\mu_2'}
R^{(2,2)}(u-v)_{\mu_1 \mu_2}^{\mu_1' \mu_2'}\
\psi^{(2)}_{\mu_1'}(u)_{b}^{a}
\psi^{(2)}_{\mu_2'}(v)_{c}^{b}
=\sum_{b' \in {\mathbb{Z}}}
\psi^{(2)}_{\mu_2}(v)_{b'}^{a}
\psi^{(2)}_{\mu_1}(u)_{c}^{b'}
W_{22}\left(\left.
\begin{array}{cc}
a&b\\
b'&c
\end{array}\right|u-v\right).{\nonumber}\\
&&\lb{fusionvertexface}
\end{eqnarray}
Explicitly, the vector $\psi^{(2)}(u)^a_b$ is calculated as follows\cite{KKW}.
\begin{prop}
\begin{eqnarray*}
\left(\begin{array}{c}
\psi^{(2)}_2(u)_{n+2}^n\\
\psi^{(2)}_0(u)_{n+2}^n\\
\psi^{(2)}_{-2}(u)_{n+2}^n
\end{array}\right)=
\left(\begin{array}{c}
\vartheta_0\left(\left.
\frac{u-n+1
}{2r}
\right|\frac{\tau}{2}
\right)
\vartheta_0
\left(
\left.\frac{u-n-1}{2r}\right|\frac{\tau}{2}
\right)\\
{2}
\vartheta_0\left(\left.
\frac{u-n}{r}\right|
\tau\right)
\vartheta_0\left(
\left.\frac{1}{r}\right|\tau
\right)
\\
\vartheta_3\left(\left.
\frac{u-n+1
}{2r}
\right|\frac{\tau}{2}
\right)
\vartheta_3
\left(
\left.\frac{u-n-1}{2r}\right|\frac{\tau}{2}
\right)
\end{array}\right),
\end{eqnarray*}
\begin{eqnarray*}
\left(\begin{array}{c}
\psi^{(2)}_2(u)_{n}^n\\
\psi^{(2)}_0(u)_{n}^n\\
\psi^{(2)}_{-2}(u)_{n}^n
\end{array}\right)=
\left(\begin{array}{c}
\vartheta_0\left(\left.
\frac{u-n+1}{2r}\right|\frac{\tau}{2}
\right)
\vartheta_0\left(
\left.\frac{u+n+1}{2r}\right|\frac{\tau}{2}
\right)\\
{2}
\vartheta_0\left(\left.
\frac{n}{r}\right|
\tau
\right)
\vartheta_0
\left(
\left.\frac{u+1}{r}
\right|
\tau
\right)\\
\vartheta_3\left(\left.
\frac{u-n+1}{2r}\right|\frac{\tau}{2}
\right)
\vartheta_3\left(
\left.\frac{u+n+1}{2r}\right|\frac{\tau}{2}
\right)
\end{array}
\right),
\end{eqnarray*}
\begin{eqnarray*}
\left(\begin{array}{c}
\psi^{(2)}_2(u)_{n-2}^n\\
\psi^{(2)}_0(u)_{n-2}^n\\
\psi^{(2)}_{-2}(u)_{n-2}^n
\end{array}\right)=
\left(\begin{array}{c}
\vartheta_0\left(\left.
\frac{u+n+1}{2r}\right|\frac{\tau}{2}
\right)
\vartheta_0\left(
\left.\frac{u+n-1}{2r}\right|\frac{\tau}{2}
\right)
\\
{2}\vartheta_0\left(\left.
\frac{u+n}{r}\right|
\tau
\right)
\vartheta_0\left(
\left.\frac{1}{r}\right|
\tau
\right)
\\
\vartheta_3\left(\left.
\frac{u+n+1}{2r}\right|\frac{\tau}{2}
\right)
\vartheta_3\left(
\left.\frac{u+n-1}{2r}\right|\frac{\tau}{2}
\right)\end{array}\right).
\end{eqnarray*}
\end{prop}
\subsection{The dual intertwining vectors and their fusion}
Let us consider the dual vector $\psi^*(u)^a_b$ defined by
\begin{eqnarray}
&&\psi^*(u)^a_b\ v_{\varepsilon}= \psi^*_\varepsilon(u)^a_b,\qquad
\psi^*_\varepsilon(u)^a_b=-\varepsilon\frac{a-b}{2[b][u]}C^2\ \psi_{-\varepsilon}(u-1)^a_b \lb{dualintvec}
\end{eqnarray}
with $|a-b|=1$.
By a direct calculation, we verify the inversion relations
\begin{eqnarray}
&&\sum_{\varepsilon=\pm}\psi_\varepsilon^*(u)^a_b\psi_\varepsilon(u)^b_c=\delta_{a,c},\lb{inversiona}\\
&&\sum_{a=b\pm1}\psi_{\varepsilon'}^*(u)^a_b\psi_\varepsilon(u)^b_a=\delta_{\varepsilon',\varepsilon}.\lb{inversionb}
\end{eqnarray}
Hence we call $\psi^*(u)^a_b$ the dual intertwining vector.
From the crossing symmetry properties of $R$ and $W$ the following vertex-face correspondence is held.
\begin{eqnarray}
&&\sum_{\varepsilon_1',\varepsilon_2'}
R(u-v)^{\varepsilon_1 \varepsilon_2}_{\varepsilon_1' \varepsilon_2'}\
\psi^*_{\varepsilon_1'}(u)_{b}^{a}
\psi^*_{\varepsilon_2'}(v)_{c}^{b}
=\sum_{s \in {\mathbb{Z}}}
\psi^*_{\varepsilon_2}(v)_{b'}^{a}
\psi^*_{\varepsilon_1}(u)_{c}^{b'}
W\left(\left.
\begin{array}{cc}
c&b'\\
b&a
\end{array}\right|u-v\right).\lb{vertexfacedual}
\end{eqnarray}
The fusion of the dual intertwining vectors is given by\cite{KKW}
\begin{eqnarray}
\psi^{*(2)}(u)_a^b=
\sum_{c=a\pm 1} \psi^*(u+1)_a^c \otimes \psi^*(u)_c^b. \lb{fusiondualint}
\end{eqnarray}
Then $\psi^{*(2)}(u)_a^b$ satisfies
\begin{eqnarray}
&&\Pi\ \psi^{*(2)}(u)_a^b=\psi^{*(2)}(u)_a^b\ \Pi. \lb{prdualint}
\end{eqnarray}
In the components, \eqref{fusiondualint} yields
\begin{eqnarray}
\psi^{*(2)}_{\mu}(u)_a^b=
\sum_{c=a\pm 1} \psi^*_{\varepsilon_1}(u+1)_a^c \psi^*_{\varepsilon_2}(u)_c^b. \lb{compfusiondualint}
\end{eqnarray}
The relation \eqref{prdualint} indicates that the RHS of \eqref{compfusiondualint}is independent
of the choice of $\varepsilon_1, \varepsilon_2$ proivided $\mu=\varepsilon_1+\varepsilon_2$.
Then using \eqref{inversiona} and \eqref{inversionb}, it is easy to verify the following inversion relations.
\begin{prop}
\begin{eqnarray}
&&\sum_{\varepsilon=0,\pm2}\psi^{*(2)}_\varepsilon(u)^a_b\psi^{(2)}_\varepsilon(u)^b_c=\delta_{a,c},
\\
&&\sum_{a=b,b\pm 2}\psi^{*(2)}_{\varepsilon'}(u)^a_b\psi^{(2)}_\varepsilon(u)^b_a=\delta_{\varepsilon', \varepsilon}.
\end{eqnarray}
\end{prop}
Furthermore, in the similar way to the derivation of \eqref{fusionvertexface}, we obtain the fused form of \eqref{vertexfacedual}
\begin{eqnarray}
&&\sum_{\mu_1',\mu_2'}
R^{(2,2)}(u-v)_{\mu_1 \mu_2}^{\mu_1' \mu_2'}\
\psi^{*(2)}_{\mu_1'}(u)_{b}^{a}
\psi^{*(2)}_{\mu_2'}(v)_{c}^{b}
=\sum_{b' \in {\mathbb{Z}}}
\psi^{*(2)}_{\mu_2}(v)_{b'}^{a}
\psi^{*(2)}_{\mu_1}(u)_{c}^{b'}
W_{22}\left(\left.
\begin{array}{cc}
c&b'\\
b&a
\end{array}\right|u-v\right).{\nonumber}\\
&&\lb{fusionvertexfacedual}
\end{eqnarray}
The expression of $\psi^{*(2)}_\mu(u)_a^b\ (\mu=2, 0, -2)$ is evaluated as follows.
\begin{prop}
\begin{eqnarray}
\left(\begin{array}{c}
\psi^{*(2)}_2(u)_{n+2}^n\\
\psi^{*(2)}_0(u)_{n+2}^n\\
\psi^{*(2)}_{-2}(u)_{n+2}^n
\end{array}\right)=\frac{C^4}{4[n+1][n+2][u][u+1]}
\left(\begin{array}{c}
\vartheta_3\left(\left.
\frac{u-n-1
}{2r}
\right|\frac{\tau}{2}
\right)^2\\
-\vartheta_3\left(\left.
\frac{u-n-1}{2r}\right|
\frac{\tau}{2}\right)
\vartheta_0\left(
\left.\frac{u-n-1}{2r}\right|\frac{\tau}{2}
\right)
\\
\vartheta_0\left(\left.
\frac{u-n-1
}{2r}
\right|\frac{\tau}{2}
\right)^2
\end{array}\right),{\nonumber}
\end{eqnarray}
\begin{eqnarray}
&&\left(\begin{array}{c}
\psi^{*(2)}_2(u)_{n}^n\\
\psi^{*(2)}_0(u)_{n}^n\\
\psi^{*(2)}_{-2}(u)_{n}^n
\end{array}\right)=-\frac{C^5}{4[n][n+1][n-1][u][u+1]}{\nonumber}\\
&&\qquad\times \left(\begin{array}{c}
\vartheta_3\left(\left.
\frac{u+n+1}{2r}\right|\frac{\tau}{2}
\right)
\vartheta_3\left(
\left.\frac{u-n-1}{2r}\right|\frac{\tau}{2}
\right)\vt{1}{n-1}{\tau}+\vartheta_3\left(\left.
\frac{u-n+1}{2r}\right|\frac{\tau}{2}
\right)
\vartheta_3\left(
\left.\frac{u+n-1}{2r}\right|\frac{\tau}{2}
\right)\vt{1}{n+1}{\tau}\\
-\vartheta_1\left(\left.
\frac{n}{r}\right|
\frac{\tau}{2}
\right)
\vartheta_2
\left(
\left.\frac{1}{r}
\right|
\frac{\tau}{2}
\right)
\vartheta_0
\left(
\left.\frac{u}{r}
\right|{\tau}
\right)\\
\vartheta_0\left(\left.
\frac{u+n+1}{2r}\right|\frac{\tau}{2}
\right)
\vartheta_0\left(
\left.\frac{u-n-1}{2r}\right|\frac{\tau}{2}
\right)\vt{1}{n-1}{\tau}+\vartheta_0\left(\left.
\frac{u-n+1}{2r}\right|\frac{\tau}{2}
\right)
\vartheta_0\left(
\left.\frac{u+n-1}{2r}\right|\frac{\tau}{2}
\right)\vt{1}{n+1}{\tau}
\end{array}
\right),{\nonumber}
\end{eqnarray}
\begin{eqnarray}
\left(\begin{array}{c}
\psi^{*(2)}_2(u)_{n-2}^n\\
\psi^{*(2)}_0(u)_{n-2}^n\\
\psi^{*(2)}_{-2}(u)_{n-2}^n
\end{array}\right)=\frac{C^4}{4[n-1][n-2][u][u+1]}
\left(\begin{array}{c}
\vartheta_3\left(\left.
\frac{u+n-1}{2r}\right|\frac{\tau}{2}
\right)^2
\\
-\vartheta_3\left(\left.
\frac{u+n-1}{2r}\right|
\frac{\tau}{2}
\right)
\vartheta_0\left(
\left.\frac{u+n-1}{2r}\right|
\frac{\tau}{2}
\right)
\\
\vartheta_0\left(\left.
\frac{u+n-1}{2r}\right|\frac{\tau}{2}
\right)^2
\end{array}\right).{\nonumber}
\end{eqnarray}
\end{prop}
Applying the crossing symmetry relations \eqref{crossing} and \eqref{SOScrossing} twice to \eqref{fusionvertexface}, we obtain the relation which should be compared with \eqref{fusionvertexfacedual}. Then fixing the suitable normalization function, we obtain\begin{thm}
\begin{eqnarray*}
&&\psi^{*(2)}_{\varepsilon}(u)^a_b=-\frac{C^4}{4[u][u+1]}\frac{\vtf{3}{0}{\tau}}{\vt{3}{1}{\tau}}\frac{g_a}{g_b (a,b)_2} \sum_{\varepsilon'=0,\pm2}Q^{\varepsilon'}_{\varepsilon}\psi^{(2)}_{\varepsilon'}(u-1)^a_b.
\end{eqnarray*}
\end{thm}
\vspace{3mm}
~\\
{\Large\bf Acknowledgements}~~
\noindent
The author would like to thank Michio Jimbo for sending his note and for stimulating discussions.
He is also grateful to Takeo Kojima and Robert Weston for their collaboration in the work \cite{KKW}.
|
{
"timestamp": "2005-03-31T06:47:18",
"yymm": "0503",
"arxiv_id": "math/0503726",
"language": "en",
"url": "https://arxiv.org/abs/math/0503726"
}
|
\section{Introduction}
The one dimensional Discrete Nonlinear Schr\"odinger Equation (DNLS),
\begin{eqnarray}
\label{DNLS}
i\dot{\psi}_n+\epsilon(\psi_{n-1}-2\psi_n+\psi_{n+1})+\gamma |\psi_n|^2\psi_n=0,
\end{eqnarray}
represents an infinite ($n\in\mathbb{Z}$), or a finite ($|n|\leq K$), one-dimensional array of coupled anharmonic oscillators, coupled to their nearest neighbors with a coupling strength $\epsilon$. Here $\psi_n(t)$ stands for the complex mode amplitude
of the oscillator at lattice site $n$, and $\gamma$ denotes an anharmonic parameter. Setting $\epsilon=1/(\Delta x)^2$,
reminds that the model includes a finite spacing between molecules, and the formal continuum limit, the NLS partial differential equation, is obtained by taking $\Delta x\rightarrow 0$. The DNLS equation is one of the most inportant inherently discrete models, having a crucial role in modelling of a great variety of phenomena, ranging from solid state and condensed matter physics to biology, \cite{Aubry,Eil,FlachWillis,HennigTsironis,Yuri}. Depending on the size of the lattice, we have to deal with an infinite or finite system of ordinary differential equations, respectively.
The gauge invariance of the nonlinearity, allows for the support of special solutions of (\ref{DNLS}) of the form
$\psi_n(t)=\phi_n\exp (i\omega t)$, $\omega>0$. These solutions are called {\em breather solutions}, due to their periodic time behavior. Inserting the ansatz of a breather solution into (\ref{DNLS}), it follows that $\phi_n$ satisfies the nonlinear system of algebraic equations
\begin{eqnarray}
\label{breather}
-\epsilon(\phi_{n-1}-2\phi_n+\phi_{n+1})+\omega\phi_n=\gamma |\psi_n|^2\psi_n.
\end{eqnarray}
The problem of existence and stability properties of breather solutions of coupled oscillators, has been developed as a fascinating sublect of research, from the derivation of the stationary DNLS equation \cite{Holstein}, the derivation of stationary solutions for the (coupled) DNLS, by numerical continuation from the so-called anticontinuum limit (the case $\epsilon\rightarrow 0$) \cite{Eil2}, to the ingenious construction of localized time-periodic or quasiperiodic solutions of general discrete systems, starting from periodic solutions of the corresponding anticontinuum limit equations \cite{Aubry, RSMackayAubry}. We refer to \cite{Eil,Kevrekidis} for a review of the existing results and the history of the problem as for a long list of references.
Motivated by \cite[Section 3]{Eil} and \cite{bang}, in this work we consider higher dimensional generalizations of DNLS, involving an arbitrary power law nonlinearity,
and site dependence of the anharmonic parameter $\gamma$. For this particular case of nonlinearity, we also refer to \cite{Mol1,Mol2,Mol3}. For instance, we seek for breather solutions of the DNLS equation in infinite higher dimensional lattices ($n=(n_1,n_2,\ldots,n_N)\in\mathbb{Z}^N$),
\begin{eqnarray}
\label{DNLSh}
i\dot{\psi}_n+(\mathbf{A}\psi)_n+ \gamma_n|\psi_n|^{2\sigma}\psi_n=0,
\end{eqnarray}
where
\begin{eqnarray}
\label{DiscLap}
(\mathbf{A}\psi)_{n\in\mathbb{Z}^N}&=&\psi_{(n_{1}-1,n_2,\ldots ,n_N)}+\psi_{(n_1,n_{2}-1,\ldots ,n_N)}+\cdots+
\psi_{(n_1,n_{2},\cdots ,n_N-1)}\nonumber\\
&&-2N\psi_{(n_{1},n_2,\ldots ,n_N)}
+\psi_{(n_{1}+1,n_2,\ldots ,n_N)}\nonumber\\
&&+\psi_{(n_1,n_{2}+1,\ldots ,n_N)}+\cdots+
\psi_{(n_1,n_{2},\cdots ,n_N+1)},
\end{eqnarray}
In this case, equation (\ref{DNLSh}), could be viewed as the discretization of the NLS partial differential equation
\begin{eqnarray}
\label{NLSh}
i\psi_t+\Delta\psi+\gamma(x)|\psi|^{2\sigma}\psi=0,\;\;x\in\mathbb{R}^N.
\end{eqnarray}
As in the one dimensional case, it can be easily seen that any breather solution $\psi_n(t)=\phi_n\exp (i\omega t)$,
of the DNLS equation (\ref{DNLSh}), satisfies the infinite nonlinear system of agebraic equations
\begin{eqnarray}
\label{swe}
-(\mathbf{A}\phi)_{n}+\omega\phi_n=\gamma_n|\phi_n|^{2\sigma}\phi_n,\;\;n\in\mathbb{Z}^N.
\end{eqnarray}
Based on a variational approach, which makes use of the famous Mountain Pass Theorem (MPT), we give a simple proof on the existence of (nontrivial) breather solutions for (\ref{DNLSh}), by showing that the energy functional associated to (\ref{swe}), has a critical point of ``mountain pass type''.
Our main assumption is that $\gamma_n$ decays in an appropriate rate, in the sense that $\gamma_n$ is in an appropriate sequence space. This restriction enables us to use a compact inclusion between ordinary sequence spaces and {\em weighted} sequence spaces, in order to justify one of the crucial steps needed for for the application of MPT, namely the Palais-Smale condition. This is an important difference with the case of constant anharmonic parameter as the analysis of our recent work \cite{AN} shows: The latter is associated with lack of compactness, and restricted our study for a finite dimensional problem (in 1-D lattice, assuming Dirichlet boundary conditions). The application of MPT to (\ref{swe}) gives rise to some restrictions, which possibly have some physical interpretation, if viewed as local estimates for some ``energy quantities'' associated with the breather solution.
On the other hand, it is shown that nontrivial solutions of (\ref{swe}) do not exist, in a sufficiently small ball of the energy space. The proof is based on a fixed point argument used also in \cite{AN}. This result could have the implementation, that we should not expect the existence of breather solutions, if the energy of the excitations of the lattice is sufficiently small.
If the estimates derived by the application of the MPT, do not appear just as a technical step for the proof, they could be combined with that of the non-existence result, to derive a ``dispersion relation'' of nonlinearity exponent $\sigma$ vs the frequency $\omega$ of the breather solution, providing indication on the behavior of the associated energy quantities.
For a detailed discussion on the breather problem in higher dimensional lattices and the dependence of the frequency $\omega$ on the conserved quantities of DNLS, we refer to \cite[Section 6]{Eil}.
\section{Preliminaries}
In this introductory section, we describe the functional setting needed for the treatment of the infinite nonlinear system (\ref{swe}). We also introduce some weighted sequence spaces, and we prove a compact inclusion between the ordinary sequence spaces and their weighted counterparts.
For some positive integer $N$, we consider the complex sequence spaces
\begin{equation}
\label{ususeqs}
\ell^p=\left\{
\begin{array}{ll}
&\phi=\{\phi_n\}_{n\in\mathbb{Z}^{N}},\;n=(n_1,n_2,\ldots,n_N)\in\mathbb{Z}^N,\;\;\phi_n\in\mathbb{C},\\
&||\phi||_{\ell^p}=\left(\sum_{n\in\mathbb{Z}^N}|\phi_n|^p\right)^{\frac{1}{p}}<\infty
\end{array}
\right\}.
\end{equation}
Between $\ell^p$ spaces the following elementary embedding relation \cite{ree79} holds,
\begin{eqnarray}
\label{lp1}
\ell^q\subset\ell^p,\;\;\;\; ||\phi||_{\ell^p}\leq ||\phi||_{\ell^q}\,\;\; 1\leq q\leq p\leq\infty.
\end{eqnarray}
Note that the contrary holds for the $L^p(\Omega)$-spaces if $\Omega\subset\mathbb{R}^N$ has finite measure. For $p=2$, we get the usual Hilbert space of square-summable sequences, which becomes a real Hilbert space if endowed with the real scalar product
\begin{eqnarray}
\label{lp2}
(\phi,\psi)_{\ell^2}=\mathrm{Re}\sum_{{n\in\mathbb{Z}^N}}\phi_n\overline{\psi_n},\;\;\phi,\,\psi\in\ell^2.
\end{eqnarray}
For a sequence {\em of positive real numbers}\ \ $\delta=\{\delta_n\}_{n\in\mathbb{Z}^{N}}$, we define the weighted sequence spaces $\ell^2_{\delta}$
\begin{equation}
\label{ususeqsw}
\ell^p_{\delta}=\left\{
\begin{array}{ll}
&\phi=\{\phi_n\}_{n\in\mathbb{Z}^{N}},\;n=(n_1,n_2,\ldots,n_N)\in\mathbb{Z}^N,\;\;\phi_n\in\mathbb{C},\\
&||\phi||_{\ell^p_{\delta}}=\left(\sum_{n\in\mathbb{Z}^N}\delta_n|\phi_n|^p\right)^{\frac{1}{p}}<\infty
\end{array}
\right\}.
\end{equation}
For the case $p=2$, it is not hard to see that $\ell^2_{\delta}$ is a Hilbert space, with scalar product
\begin{eqnarray}
\label{weightscal}
(\phi,\psi)_{\ell^2_{\delta}}=\mathrm{Re}\sum_{n\in\mathbb{Z}^N}\delta_n\phi_n\overline{\psi_n},\;\;\phi,\,\psi\in\ell^2_\delta.
\end{eqnarray}
For a certain class of weight $\delta$, we have the following lemma which shall play a crucial role in our analysis.
\begin{lemma}
\label{compactness}
We assume that the positive sequence of real numbers $\delta\in\ell^{\rho}$, $\rho=\frac{q-1}{q-2}$ for some $q>2$. Then
$\ell^2\hookrightarrow\ell^2_{\delta}$ with compact inclusion.
\end{lemma}
{\bf Proof:} We use the ideas of \cite[Lemma 2.3, pg. 79]{KJB} and (\ref{lp1}). We consider a bounded sequence $\phi_k\in\ell^2$ and we denote by $(\phi_k)_n$ the $n$-th coordinate of this sequence. It suffices to show that the sequence $\phi_k$ is a Cauchy sequence in $\ell^2_{\delta}$. For some $q>2$ we consider its H\"older conjugate through the relation $p^{-1}+q^{-1}=1$. Then for all positive integers $k,l$, we have
\begin{eqnarray}
\label{lem1}
||\phi_k-\phi_l||^2_{\ell^2_{\delta}}&=&\sum_{n\in\mathbb{Z}^{N}}\delta_n|(\phi_k)_n-(\phi_l)_n|^2\\
&\leq&
\left(\sum_{n\in\mathbb{Z}^{N}}\delta_n|(\phi_k)_n-(\phi_l)_n|^p\right)^{\frac{1}{p}}
\left(\sum_{n\in\mathbb{Z}^{N}}\delta_n|(\phi_k)_n-(\phi_l)_n|^q\right)^{\frac{1}{q}}.\nonumber
\end{eqnarray}
Since $\phi_k$ is a bounded sequence in $\ell^2$, it follows from (\ref{lp1}) that $\phi_k$ is bounded in $\ell^q$.
Then from (\ref{lem1}) we have that there exists a positive constant $c$, such that
\begin{eqnarray}
\label{lem2}
||\phi_k-\phi_l||^2_{\ell^2_{\delta}}\leq c \left(\sum_{n\in\mathbb{Z}^{N}}\delta_n|(\phi_k)_n-(\phi_l)_n|^p\right)^{\frac{1}{p}}.
\end{eqnarray}
Since $\delta\in\ell^{\rho}$, it holds that for any $\epsilon_1>0$, there exists $K_0(\epsilon_1)$ such that for all $K>K_0(\epsilon_1)$
$$\sum_{|n|> K}|\delta_n|^{\rho}<\epsilon_1.$$
Thus, using the boundedness of $\phi_k$ in $\ell^q$ once again, we have
\begin{eqnarray}
\label{lem5}
\sum_{|n|> K}\delta_n|(\phi_k)_n-(\phi_l)_n|^p&\leq& \left(\sum_{|n|> K}|\delta_n|^{\rho}\right)^{\frac{1}{\rho}}
\left(\sum_{|n|> K}|(\phi_k)_n-(\phi_l)_n|^q\right)^{\frac{p}{q}}\nonumber\\
&<& c\epsilon_1^{\frac{1}{\rho}}.
\end{eqnarray}
On the other hand, since
the sequence $\phi_k$ is a Cauchy sequence in the finite dimensional space $\mathbb{C}^{(2K+1)^N}$, we get that for $k$ and $l$ sufficiently large and for any $\epsilon_2>0$, holds that
\begin{eqnarray}
\label{lem4}
\sum_{|n|\leq K}\delta_n|(\phi_k)_n-(\phi_l)_n|^p < \epsilon_2.
\end{eqnarray}
Inequality (\ref{lem2}) can be rewritten as
\begin{eqnarray}
\label{lem3}
||\phi_k-\phi_l||^{2p}_{\ell^2_{\delta}}\leq c\left\{\sum_{|n|\leq K}\delta_n|(\phi_k)_n-(\phi_l)_n|^p
+\sum_{|n|> K}\delta_n|(\phi_k)_n-(\phi_l)_n|^p\right\}.
\end{eqnarray}
Now from (\ref{lem5})-(\ref{lem3}), and appropriate choices of $\epsilon_1$ and $ \epsilon_2$, we may derive that for sufficiently large $k$ and $l$, $$||\phi_k-\phi_l||_{\ell^2_{\delta}}<\epsilon,\;\;\mbox{for any}\;\;\epsilon>0.$$
That is $\phi_k$ is a Cauchy sequence in $\ell^2_{\delta}$.\ \ $\diamond$
\vspace{0.2cm}
Let $\mathbf{A}:D(\mathbf{A})\subseteq X\rightarrow X$ a $\mathbb{C}$-linear, self-adjoint\ $\leq 0$ operator with dense domain $D(\mathbf{A})$ on the Hilbert space $X$, equipped with the scalar product $(\cdot ,\cdot)_{X}$. The space
$X_{\mathbf{A}}$ is the completion of $D(\mathbf{A})$ in the norm $||u||_{\mathbf{A}}^2=||u||^2_X-(\mathbf{A}u,u)_X$, for $u\in X_{\mathbf{A}}$, and we denote
by $X_{\mathbf{A}}^*$ its dual and by $\mathbf{A}^*$ the extension of $\mathbf{A}$ to the dual of $D(\mathbf{A})$, denoted by $D(\mathbf{A})^*$ (Friedrichs extension theory \cite{Davies1}, \cite[Vol. II/A]{zei85}).
Considering the operator $\mathbf{A}$ defined by (\ref{DiscLap}), we observe that for any $\phi\in\ell^2$
\begin{eqnarray}
\label{preA}
||\mathbf{A}\phi||_{\ell^2}^2\leq 4N||\phi||_{\ell^2}^2,
\end{eqnarray}
that is, $\mathbf{A}:\ell^2\rightarrow\ell^2$ is a continuous operator. Now we consider the discrete operator $\mathbf{L}^+:\ell^2\rightarrow\ell^2$ defined by
\begin{eqnarray}
\label{discder1}
(\mathbf{L}^+\psi)_{n\in\mathbb{Z}^N}&=&\left\{\psi_{(n_{1}+1,n_2,\ldots ,n_N)}-\psi_{(n_{1},n_2,\ldots ,n_N)}\right\}\nonumber\\
&+&\left\{\psi_{(n_{1},n_2+1,\ldots ,n_N)}-\psi_{(n_{1},n_2,\ldots ,n_N)}\right\}\nonumber\\
&\vdots&\nonumber\\
&+&\left\{\psi_{(n_{1},n_2,\ldots ,n_N+1)}-\psi_{(n_{1},n_2,\ldots ,n_N)}\right\},
\end{eqnarray}
and $\mathbf{L}^{-}:\ell^2\rightarrow\ell^2$ defined by
\begin{eqnarray}
\label{discder2}
(\mathbf{L}^-\psi)_{n\in\mathbb{Z}^N}&=&\left\{\psi_{(n_{1}-1,n_2,\ldots ,n_N)}-\psi_{(n_{1},n_2,\ldots ,n_N)}\right\}\nonumber\\
&+&\left\{\psi_{(n_{1},n_2-1,\ldots ,n_N)}-\psi_{(n_{1},n_2,\ldots ,n_N)}\right\}\nonumber\\
&\vdots&\nonumber\\
&+&\left\{\psi_{(n_{1},n_2,\ldots ,n_N-1)}-\psi_{(n_{1},n_2,\ldots ,n_N)}\right\}.
\end{eqnarray}
Setting
\begin{eqnarray}
\label{discder3}
(\mathbf{L}^+_{\nu}\psi)_{n\in\mathbb{Z}^N}=\psi_{(n_{1},n_2,\ldots , n_{\nu-1},n_{\nu}+1,n_{\nu+1},\ldots ,n_N)}-\psi_{(n_{1},n_2,\ldots ,n_N)},\\
\label{discder4}
(\mathbf{L}^-_{\nu}\psi)_{n\in\mathbb{Z}^N}=\psi_{(n_{1},n_2,\ldots , n_{\nu-1},n_{\nu}-1,n_{\nu+1},\ldots ,n_N)}-\psi_{(n_{1},n_2,\ldots ,n_N)},
\end{eqnarray}
we observe that the operator $\mathbf{A}$ satisfies the relations
\begin{eqnarray}
\label{diffop2}
(-\mathbf{A}\psi_1,\psi_2)_{\ell^2}&=&\sum_{\nu=1}^N(\mathbf{L}_\nu^+\psi_1,\mathbf{L}_\nu^+\psi_2)_{\ell^2},\;\;\mbox{for all}\;\;\psi_1,\psi_2\in\ell^2,\\
\label{diffop3}
(\mathbf{L}_\nu^+\psi_1,\psi_2)_{\ell^2}&=&(\psi_1,\mathbf{L}_\nu^-\psi_2)_{\ell^2},\;\;\mbox{for all}\;\;\psi_1,\psi_2\in\ell^2.
\end{eqnarray}
From (\ref{diffop2}), it is clear that $\mathbf{A}:\ell^2\rightarrow\ell^2$ defines a self adjoint operator on $\ell^2$, and $\mathbf{A}\leq 0$. The graph-norm
\begin{eqnarray*}
||\phi||_{D(\mathbf{A})}^2=||\mathbf{A}\phi||_{\ell^2}^2+||\phi||_{\ell^2}^2,
\end{eqnarray*}
is an equivalent with that of $\ell^2$, since
\begin{eqnarray*}
||\phi||_{\ell^2}^2\leq ||\phi||^2_{D(\mathbf{A})}\leq (4N+1)||\phi||_{\ell^2}^2.
\end{eqnarray*}
In our case, it appears that $X_{\mathbf{A}}=\ell^2$
equipped with the norm
$$||\phi||_{\mathbf{A}}^2=||\phi||_X^2-(\mathbf{A}\phi,\phi)_X=\sum_{\nu=1}^{N}||\mathbf{L}^+_{\nu}\phi||_{\ell_2}^2+ ||\phi||^2_{\ell^2},$$
for $\phi\in\ell^2$, and is an equivalent norm with the usual one of $\ell^2$. Moreover,
$D(\mathbf{A})=X=\ell^2=D(\mathbf{A})^*$. Obviously $\mathbf{A}^*=\mathbf{A}$ and the operator
$i\mathbf{A}:\ell^2\rightarrow \ell^2$ defined by $(i\mathbf{A})\phi=i\mathbf{A}\phi$ for $\phi\in
\ell^2$, is $\mathbb{C}$-linear and skew-adjoint and $i\mathbf{A}$
generates a group $(\mathcal{T}(t))_{t\in\mathbb{R}}$, of
isometries on $\ell^2$ (see \cite{cazS}).
The analysis of the operator $\mathbf{A}$ is useful if one would like to consider the DNLS equation (\ref{DNLSh}) as an abstract evolution equation \cite{AN}, and holds for other discrete operators
which are not necessary discretizations of the Laplacian (for example as those of \cite{SZ2}).
\subsection{Existence of non trivial breather solutions in the case of decaying anharmonic parameter}
We shall seek for nontrivial breather solutions as critical points of the functional
\begin{eqnarray}
\label{Enegfun}
\mathbf{E}(\phi )=\frac{1}{2}\sum_{\nu=1}^{N}||\mathbf{L}^+_{\nu}\phi||_{\ell_2}^2+\frac{\omega^2}{2}\sum_{n\in\mathbb{Z}^N}|\phi_n|^2-\frac{1}{2\sigma +2}\sum_{n\in\mathbb{Z}^N}\gamma_n|\phi_n|^{2\sigma +2}.
\end{eqnarray}
To establish differentiability of the functional
$\mathbf{E}:\ell^2\rightarrow\mathbb{R}$, we shall use the
following discrete version of the dominated convergence Theorem,
provided by \cite{Bates2}.
\begin{theorem}
\label{dc}
Let $\{\psi_{i,k}\}$ be a double sequence of summable functions, $$\sum_{i\in\mathbb{Z}^N}|\psi_{i,k}|<\infty,$$
and $\lim_{k\rightarrow\infty}\psi_{i,k}=\psi_{i}$, for all
$i\in\mathbb{Z}^N$. If there exists a summable sequence $\{g_{i}\}$ such that $|\psi_{i,k}|\leq g_{i}$ for all $i,k$'s, we have that
$\lim_{k\rightarrow\infty}\sum_{i\in\mathbb{Z}^N}\psi_{i,k}=\sum_{i\in\mathbb{Z}^N}\psi_{i}$.
\end{theorem}
We then have the following Lemma.
\begin{lemma}
\label{derivative}
Let $(\phi_n)_{n\in\mathbb{Z}^N}=\phi\in\ell^{2\sigma+2}$ for some $0<\sigma <\infty$. Moreover we assume that $\gamma_n\in\ell^{\rho}$, $\rho=\frac{q-1}{q-2}$ for some $q>2$. Then the functional
$$\mathbf{F}(\phi)=\sum_{n\in\mathbb{Z}^N}\gamma_n|\phi_n|^{2\sigma +2},$$
is a $\mathrm{C}^{1}(\ell^{2\sigma +2},\mathbb{R})$ functional and
\begin{eqnarray}
\label{gatdev}
<\mathbf{F}'(\phi),\psi>=(2\sigma +2)\mathrm{Re}\sum_{n\in\mathbb{Z}^N}\gamma_n|\phi_n|^{2\sigma}\phi_n\overline{\psi_n},\;\;\psi=(\psi_n)_{n\in\mathbb{Z}^N}\in\ell^{2\sigma +2}.
\end{eqnarray}
\end{lemma}
{\bf Proof:}\ \ We assume that $\phi,\,\psi\in\ell^{2\sigma +2}$.
Then for any $n\in\mathbb{Z}^N$, $0<s<1$, we get
\begin{eqnarray}
\label{mv}
&&\frac{\mathbf{F}(\phi +s\psi)-\mathbf{F}(\psi)}{s}=\frac{1}{s}\mathrm{Re}\sum_{n\in\mathbb{Z}^N}\gamma_n\int_{0}^{1}\frac{d}{d\theta}|\phi_n +
\theta s\psi_n|^{2\sigma +2}d\theta\\
&&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=(2\sigma +2)\mathrm{Re}\sum_{n\in\mathbb{Z}^N}\gamma_n\int_{0}^{1}|\phi_n+s\theta\psi_n|^{2\sigma}
(\phi_n+s\theta\psi_n)\overline{\psi_n} d\theta.\nonumber
\end{eqnarray}
Since $\gamma_n$ is in $\ell^{\rho}$ it follows from (\ref{lp1}) that
\begin{eqnarray}
\label{boundV}
\mathrm{sup}_{n\in\mathbb{Z}^N}|\gamma_n|=M<\infty.
\end{eqnarray}
On the other hand we have the inequality
\begin{eqnarray}
\label{mv1}
&&\sum_{n\in\mathbb{Z}^N}|\phi_n+\theta s\psi_n|^{2\sigma +1}|\psi_n|
\leq
\sum_{n\in\mathbb{Z}^N}\left(|\phi_n|+|\psi_n|\right)^{2\sigma +1}|\psi_n|\;\;\;\;\;\;\;\nonumber\\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;&&\leq\left(\sum_{n\in\mathbb{Z}^N}(|\phi_n|+|\psi_n|)^{2\sigma +2}\right)^{\frac{2\sigma +1}{2\sigma +2}}
\left(\sum_{n\in\mathbb{Z}^N}|\psi_n|^{2\sigma +2}\right)^{\frac{1}{2\sigma +2}}.
\end{eqnarray}
Now by using (\ref{boundV}) and inserting (\ref{mv1}) into (\ref{mv}), we see that Lemma \ref{dc} is applicable: Letting $s\rightarrow 0$, we get the existence of the Gateaux derivative (\ref{gatdev}) of the functional $\mathbf{F}:\ell^{2\sigma +2}\rightarrow\mathbb{R}$.
We show next that the functional $\mathbf{F}':\ell^{2\sigma +2}\rightarrow\ell^{\frac{2\sigma +2}{2\sigma +1}}$ is
continuous. For $\phi\in\ell^{2\sigma +2}$, we set $(F_1(\phi))_{n\in\mathbb{Z}^N}=|\phi_n|^{2\sigma}\phi_n$.
Let us note that for any $F\in \mathrm{C}(\mathbb{C},\mathbb{C})$ which takes the form
$F(z)=g(|z|^2)z$, with $g$ real and sufficiently smooth, holds
\begin{eqnarray}
\label{GL}
F(\phi_1)-F(\phi_2)=\int_{0}^{1}\left\{(\phi_1-\phi_2)(g(r)+rg'(r))+(\overline{\phi}_1-\overline{\phi}_2)\Phi^2 g'(r)\right\}d\theta,
\end{eqnarray}
for any $\phi_1,\;\phi_2\in \mathbb{C}$,where $\Phi=\theta \phi_1+(1-\theta)\phi_2$, $\theta\in (0,1)$ and $r=|\Phi|^2$ (see \cite[pg. 202]{GiVel96}).
Applying (\ref{GL}) for the case of $F_1$ $(g(r)= r^{\sigma})$ one obtains that
\begin{eqnarray*}
F_1(\phi_1)-F_1(\phi_2)=\int_0^1[(\sigma
+1)(\phi_1-\phi_2)|\Phi|^{2\sigma}
+\sigma(\overline{\phi}_1-\overline{\phi}_2)\Phi^2|\Phi|^{2\sigma
-2}]d\theta,
\end{eqnarray*}
which implies the inequality
\begin{eqnarray}
\label{GL2}
|F_1(\phi_1)-F_1(\phi_2)|\leq (2\sigma +1)(|\phi_1|+|\phi_2|)^{2\sigma}|\phi_1-\phi_2|.
\end{eqnarray}
We consider a sequence $\phi_m\in\ell^{2\sigma +2}$ such that $\phi_m\rightarrow \phi$ in $\ell^{2\sigma +2}$. Using (\ref{boundV}), we get the inequality
\begin{eqnarray}
\label{hoin}
\left|\left<\mathbf{F}'(\phi_m)-\mathbf{F}'(\phi),\,\psi\right>\right|&\leq& c(M)||F_1(\phi_m)-F_1(\phi)||_{\ell^{q}}||\psi||_{\ell^p},\\
&&q=\frac{2\sigma +2}{2\sigma +1},\;\;p=2\sigma +2.\nonumber
\end{eqnarray}
We denote by $(\phi_m)_n$ the $n$-th coordinate of the sequence $\phi_m\in\ell^2$. By setting $\Phi_n=(|(\phi_m)_n|+|\phi_n|)^{2\sigma}$, we get from (\ref{GL2}), that for some constant $c>0$
\begin{eqnarray*}
&&\sum_{n\in\mathbb{Z}^N}|F_1((\phi_m)_n)-F_1(\phi_n)|^{q}\leq c\sum_{n\in\mathbb{Z}^N}(\Phi_n)^q|(\phi_m)_n-\phi_n|^{q}\nonumber\\
&&\leq c\left(\sum_{n\in\mathbb{Z}^N}|(\phi_m)_n-\phi_n|^{2\sigma +2}\right)^{\frac{1}{2\sigma +1}}
\left(\sum_{n\in\mathbb{Z}^N}(\Phi_n)^{\frac{\sigma +1}{\sigma}}\right)^{\frac{2\sigma}{2\sigma +1}}\rightarrow 0,
\end{eqnarray*}
as $m\rightarrow\infty$.\ \ \ $\diamond$
By using (\ref{diffop2}), we may easily get that the rest of the terms of the functional $\mathbf{E}$ given by (\ref{Enegfun}), define $\mathrm{C}^1(\ell^2,\mathbb{R})$ functionals. Since Lemma \ref{derivative} holds for any $\phi\in\ell^2$ (by (\ref{lp1})), we finally obtain that the functional $\mathbf{E}$ is $\mathrm{C}^1(\ell^2,\mathbb{R})$. Moreover, by using the analysis of Section 1 for the self-adjoint operator $\mathbf{A}:\ell^2\rightarrow\ell^2$, it appears that any solution of
(\ref{swe}), satisfies the formula
\begin{eqnarray*}
(-\mathbf{A}\phi,\psi)_{\ell^2}+\omega(\phi,\psi)_{\ell^2}=(\gamma_nF_1(\phi),\psi)_{\ell^2},\;\;\mbox{for all}\;\;\psi\in\ell^2,
\end{eqnarray*}
and vice versa. Equivalently, due to the differentiability of the functional $\mathbf{E}$, any solution of (\ref{swe}) is a critical point of $\mathbf{E}$. For convenience, we recall \cite[Definition 4.1, pg. 130]{CJ}
(PS-condition) and \cite[Theorem 6.1, pg. 140]{CJ} or
\cite[Theorem 6.1, pg. 109]{struwe} (Mountain Pass Theorem of Ambrosseti-Rabinowitz \cite{Amb}).
\begin{definition}
\label{condc} Let $X$ be a Banach space and $\mathbf{E}:X\rightarrow\mathbb{R}$ be $\mathrm{C}^1$. We say that
$\mathbf{E}$ satisfies condition $(PS)$ if, for any sequence $\{\phi_n\}\in X$ such that $|\mathbf{E}(\phi_n)|$ is bounded and $\mathbf{E}'(\phi_n)\rightarrow 0$ as $n\rightarrow\infty$,
there exists a convergent subsequence. If this condition is only satisfied in the region where $\mathbf{E}\geq\alpha >0$ (resp $\mathbf{E}\leq -\alpha <0$) for all $\alpha >0$, we say $\mathbf{E}$ satisfies condition $(PS^+)$ (resp. $(PS^-)$).
\end{definition}
\begin{theorem}
\label{mpass}
Let $\mathbf{E}:X\rightarrow\mathbb{R}$ be $C^1$ and satisfy (a) $\mathbf{E}(0)=0$, (b) $\exists\rho >0$, $\alpha >0:\;||\phi||_X=\rho$ implies $\mathbf{E}(\phi)\geq\alpha$, (c) $\exists \phi_1\in X :\;||\phi_1||_X\geq\rho$ and
$\mathbf{E}(\phi_1)<\alpha$. Define $$\Gamma=\left\{\gamma\in \mathrm{C}^0([0,1],X):\;\gamma (0)=0,\;\;\gamma (1)=\phi_1\right\}.$$
Let $F_{\gamma}=\{\gamma(t)\in X:\;0\leq t\leq 1\}$ and $\mathcal{L}=\{F_\gamma :\;\gamma\in \Gamma\}$. If $\mathbf{E}$ satisfies condition $(PS)$, then $$\beta:=\inf_{F_{\gamma}\in \mathcal{L}}\sup\{\mathbf{E}(v):v\in F_{\gamma}\}\geq\alpha$$ is a critical point of the functional $\mathbf{E}$.
\end{theorem}
For fixed $\omega>0$, we shall consider a norm in $\ell^2$ defined by
\begin{eqnarray}
\label{moup1}
||\phi||_{\ell^2_{\omega}}^2=\sum_{\nu=1}^{\nu=N}||\mathbf{L}^+_{\nu}\phi||_{\ell_2}^2+\omega ||\phi||^2_{\ell^2},\;\;\phi\in\ell^2.
\end{eqnarray}
The norm (\ref{moup1}) is an equivalent norm with the usual one of $\ell^2$, since
\begin{eqnarray}
\label{moup2}
\omega ||\phi||^2_{\ell^2}\leq ||\phi||_{\ell^2_\omega}^2\leq (2N+\omega)||\phi||^2_{\ell^2}.
\end{eqnarray}
We first check the behavior of the functional $\mathbf{E}$. Using (\ref{moup2}), we observe that
\begin{eqnarray}
\label{moup3}
|\mathbf{F}(\phi)|&\leq& M\sum_{n\in\mathbb{Z}^N}|\phi_n|^{2\sigma+2}
\leq M||\phi||_{\ell^2}^{2\sigma+2}\nonumber\\
&\leq&\frac{M}{\omega^{\sigma+1}}||\phi||^{2\sigma+2}_{\ell^2_{\omega}}.
\end{eqnarray}
Now setting $M_0=M/\omega^{\sigma+1}$
we observe that
\begin{eqnarray}
\label{moup4}
\mathbf{E}(\phi)&=&\frac{1}{2}||\phi||^2_{\ell^2_{\omega}}-\frac{1}{2\sigma+2}\mathbf{F}(\phi)\nonumber\\
&&\geq \frac{1}{2}||\phi||^2_{\ell^2_{\omega}}-\frac{M_0}{2\sigma+2}||\phi||^{2\sigma+2}_{\ell^2_\omega}.
\end{eqnarray}
Now we select some $\phi\in\ell^2$ such that $||\phi||_{\ell^2_{\omega}}=R>0$. Then, if
\begin{eqnarray}
\label{disp1}
0<R < \left(\frac{\sigma +1}{M_0}\right)^{\frac{1}{2\sigma}}=\left(\frac{(\sigma +1)\omega^{\sigma+1}}{M}\right)^{\frac{1}{2\sigma}}:=E_{\ell^2_{\omega}}^*(\sigma,\omega,M),
\end{eqnarray}
it follows from (\ref{moup4}) that
\begin{eqnarray}
\label{moup5}
\mathbf{E}(\phi) \geq \alpha>0,\;\;\alpha=R^2\left(\frac{1}{2}-\frac{M_0}{2\sigma +2}R^{2\sigma}\right).
\nonumber
\end{eqnarray}
We assume that $\gamma_n>0$ for all $n\in\mathbf{S}_+\subseteq\mathbb{Z}^N$. We shall consider next, some $\psi\in\ell^2$ such that $||\psi||_{\ell^2_{\omega}}=1$ and
\begin{eqnarray*}
\{\psi_n\}_{n\in\mathbb{Z}}=\{\psi_n\}_{n\in\mathbf{S}_+}+\{\psi_n\}_{n\in(\mathbb{Z}^N\setminus\mathbf{S}_+)},\;\;\mbox{where}\;\;
\left\{
\begin{array}{rlr}
&\{\psi_n\}_{n\in\mathbf{S}_+}&>0,\nonumber \\
&\{\psi_n\}_{n\in(\mathbb{Z}^N\setminus\mathbf{S}_+)}&=0.
\end{array}
\right.
\end{eqnarray*}
For some $t>0$ we considet the element $\chi=t\psi\in\ell^2$. We have that
\begin{eqnarray}
\label{moup6}
\mathbf{E}(\chi)=\frac{t^2}{2}-\frac{1}{2\sigma +2}t^{2\sigma +2}\sum_{n\in\mathbf{S}_+}\gamma_n|\psi_n|^{2\sigma+2}.
\end{eqnarray}
Now letting $t\rightarrow +\infty$ we get that $\mathbf{E}(t\psi)\rightarrow -\infty$.
For fixed $\phi\neq 0$ and choosing $t$ sufficiently large, we may set $\phi_1=t\phi$ to satisfy the second condition of Theorem \ref{mpass}. To conclude with the existence of a non-trivial breather solution, it remains to show that the functional $\mathbf{E}$ satisfies Lemma \ref{condc}.
To this end, we consider a sequence $\phi_m$ of
$\ell^2$ be such that $|\mathbf{E}(\phi_m)|<M'$ for some $M'>0$ and
$\mathbf{E}'(\phi_m)\rightarrow 0$ as $m\rightarrow\infty$. By using
(\ref{Enegfun}) and Lemma \ref{derivative}, we observe that for $m$ sufficiently large
\begin{eqnarray}
\label{boundP.S}
M'\geq \mathbf{E}(\phi_m)-\frac{1}{2\sigma +2}\left<\mathbf{E}'(\phi_m),\phi_m\right>=
\left(\frac{1}{2}-\frac{1}{2\sigma +2}\right)||\phi_m||^2_{\ell^2_{\omega}}.
\end{eqnarray}
Therefore the sequence $\phi_m$ is bounded. Thus, we may extract a subsequence, still denoted by $\phi_m$, such that
\begin{eqnarray}
\label{weakcon}
\phi_m\rightharpoonup \phi\;\;\mbox{in}\;\;\ell^2,\;\;\mbox{as}\;\;m\rightarrow\infty.
\end{eqnarray}
For this subsequence it follows once again from (\ref{Enegfun}) and Lemma \ref{derivative} that
\begin{eqnarray}
\label{moup8}
||\phi_m-\phi||_{\ell^2_{\omega}}^2&=&\left<\mathbf{E}'(\phi_m)-\mathbf{E}'(\phi),\phi_m-\phi\right>\nonumber\\
&&+\sum_{n\in\mathbb{Z}^N}\gamma_n[|(\phi_m)_n|^{2\sigma}(\phi_m)_n-|\phi_n|^{2\sigma}\phi_n]((\phi_m)_n-\phi_n)).\;\;\;
\end{eqnarray}
Another assumption on the sequence $\gamma_n$ is that the sequence $|\gamma_n|=(\delta_n)_{n\in\mathbb{Z}^N}$ satisfies the assumptions of Lemma \ref{compactness}. We consider the associated Hilbert space $\ell^2_{\delta}$. Now by using the inequality (\ref{GL2}), we get for the second term of right hand side of (\ref{moup8}), the estimate
\begin{eqnarray}
\label{moup9}
&&\sum_{n\in\mathbb{Z}^N}\gamma_n[|(\phi_m)_n|^{2\sigma}(\phi_m)_n-|\phi_n|^{2\sigma}\phi_n]((\phi_m)_n-\phi_n))\;\;\;\;\;\nonumber\\
&&\;\;\;\;\leq c\sum_{n\in\mathbb{Z}^N}\Phi_n|\gamma_n|\,\,|(\phi_m)_n-\phi_n|^2\nonumber\\
&&\;\;\;\;\leq c\sup_{n\in\mathbb{Z}^N}\Phi_n\sum_{n\in\mathbb{Z}^N}|\gamma_n|\,|(\phi_m)_n-\phi_n|^2
=c_2||\phi_m-\phi||_{\ell^2_\delta}^2,
\end{eqnarray}
where $c_2=c\sup_{n\in\mathbb{Z}^N}\Phi_n$.
Obviously, $\phi_m$ is bounded in $\ell^2_{\delta}$ and by Lemma \ref{compactness} it follows that
\begin{eqnarray}
\label{weakcon2}
\phi_m\rightarrow \phi\;\;in\;\;\ell^2_{\delta},\;\;\mbox{as}\;\;m\rightarrow\infty.
\end{eqnarray}
Combining (\ref{moup8}), (\ref{moup9}) and (\ref{weakcon2}), we obtain that $$||\phi_m-\phi||_{\ell^2_{\omega}}\rightarrow 0,\;\;\mbox{as}\;\;m\rightarrow\infty.$$ Hence $\phi_m$ has a (strongly) convergent subsequence. The assumptions of Theorem \ref{mpass} are satisfied, and we may summarize in the following
\begin{theorem}
\label{COMP}
Assume that the site-dependent anharmonic parameter $\gamma_n>0$ in some $\mathbf{S}_+\subseteq\mathbb{Z}^N$. Moreover, we assume that $|\gamma_n|=\delta_n\in\ell^{\rho}$, $\rho=\frac{q-1}{q-2}$ for some positive integer $q>2$. Then for any $\omega>0$, there exists a nontrivial breather solution $\psi_n(t)=\phi_n\exp (i\omega t)$ of the DNLS equation (\ref{DNLSh}).
\end{theorem}
\vspace{0.2cm}
We remark here that the assumptions on the sequence of anharmonic parameters $\gamma_n$, are crucial for the derivation of the strong convergence of the subsequence $\phi_m$. If $\gamma_n$ is constant for all $n\in\mathbb{Z}^N$, then due to the lack of the Schur property for the space $\ell^{2}$ (in contrast with the space $\ell^1$ which posses this property-weak convergence coincides with strong convergence), we may not conclude the strong convergence of the subsequence, from its weak convergence. Of course the strong convergence, is valid in the case of a finite lattice: In this case, the problem is formulated in finite dimensional spaces where weak is equivalent to strong convergence \cite{AN}.
Inequality (\ref{disp1}) could have some physical interpretation with respect to the nontrivial breather solutions of frequency $\omega>0$, if one considers (\ref{disp1}) as a possible local upper bound for the ``energy'' quantity defined by (\ref{moup1}). It contains information on the type of nonlinearity and the sequence of anharmonic parameters, through its dependence on the nonlinearity exponent $\sigma$ and $M$. Such type of relations seem to be reasonable, as the next result concerning nonexistence of nontrivial breather solutions shows. The restriction on the energy of the excitations for nonexistence, combined with the upper bound (\ref{disp1}) above, could provide us with some indicative information, on the behavior of energy quantities, associated with the nontrivial breather solution.
For the shake of completeness, we recall
\cite[Theorem 18.E, pg. 68]{zei85}
(Theorem of Lax and Milgram). This theorem will be used to establish existence of solutions for an auxiliary infinite linear system of algebraic equations related to (\ref{swe}).
\begin{theorem}
\label{LMth}
Let $X$ be a Hilbert space and $\mathbf{A}:X\rightarrow X$ be a linear continuous operator. Suppose that there exists $c^*>0$ such that
\begin{eqnarray}
\label{strongmonot}
\mathrm{Re}(\mathbf{A}u,u)_X\geq c^*||u||^2_X,\;\;\mbox{for all}\;\;u\in X.
\end{eqnarray}
Then for given $f\in X$, the operator equation $\mathbf{A}u=f,\;\;u\in X$, has a unique solution
\end{theorem}
The non existence result can be stated as follows
\begin{theorem}
\label{notri}
There exist no nontrivial breather solution of energy less than
\begin{eqnarray}
\label{disp2}
E_{\mathrm{min}}(\omega,\sigma,M):=\frac{1}{2}\left(\frac{\omega}{M(2\sigma +1)}\right)^{1/2\sigma}.
\end{eqnarray}
\end{theorem}
{\bf Proof:}\ \ For some $\omega> 0$, we consider the operator $\mathbf{A}_{\omega}:\ell^2\rightarrow\ell^2$, defined by
\begin{eqnarray}
\label{strongop1}
(\mathbf{A}_{\omega}\phi)_{n\in\mathbb{Z}^N}&=&(\mathbf{A}\phi)_{n\in\mathbb{Z}^N}+\omega\phi_n.
\end{eqnarray}
It is linear and continuous and satisfies assumption (\ref{strongmonot}) of Theorem \ref{LMth}: Using (\ref{diffop2}), we get that
\begin{eqnarray}
\label{check}
(\mathbf{A}_{\omega}\phi,\phi)_{\ell^2}=\sum_{\nu=1}^N|\mathbf{L}^+_{\nu}\phi||^2_{\ell^2}+\omega ||\phi||^2\geq \omega ||\phi||^2_{\ell^2}\;\;\mbox{for all}\;\;\phi\in\ell^2.
\end{eqnarray}
For given $z\in\ell^2$, we consider the linear operator equation
\begin{eqnarray}
\label{linear}
(\mathbf{A}_{\omega}\phi)_{n\in\mathbb{Z}^N}=\gamma_n|z_n|^{2\sigma}z_n.
\end{eqnarray}
For the map
\begin{eqnarray}
\label{nolimap}
(\mathbf{T}(z))_{n\in\mathbb{Z}^N}=\gamma_n|z_n|^{2\sigma}z_n,
\end{eqnarray}
we observe that
\begin{eqnarray*}
||\mathbf{T}(z)||^2_{\ell^2}\leq M^2\sum_{n\in\mathbb{Z}^N}|z_n|^{4\sigma +2}
\leq M^2||z||_{\ell^2}^{4\sigma +2}.
\end{eqnarray*}
Hence, the assumptions of Theorem \ref{LMth} are satisfied, and (\ref{linear}) has a unique solution $\phi\in\ell^2$. For some $R>0$, we consider the closed ball of $\ell^2$, $B_R:=\{z\in\ell^2\;:||z||_{\ell^2}\leq R\}$, and we define the map
$\mathcal{P}:\ell^2\rightarrow\ell^2$, by $\mathcal{P}(z):=\phi$ where $\phi$ is the unique solution of the operator equation (\ref{linear}). Clearly the map $\mathcal{P}$ is well defined.
Let $\zeta$, $\xi\in B_R$ such that $\phi=\mathcal{P}(\zeta)$, $\psi=\mathcal{P}(\xi)$. The difference $\chi:=\phi-\psi$ satisfies the equation
\begin{eqnarray}
\label{claim2}
(\mathbf{A}_{\omega}\chi)_{n\in\mathbb{Z}^N}=(\mathbf{T}(z))_{n\in\mathbb{Z}^N}-(\mathbf{T}(\xi))_{n\in\mathbb{Z}^N}.
\end{eqnarray}
The map $\mathbf{T}:\ell^2\rightarrow\ell^2$ is locally Lipschitz, since we may use (\ref{GL2}) once again, to get
\begin{eqnarray}
\label{claim3}
||\mathbf{T}(\zeta)-\mathbf{T}(\xi)||_{\ell^2}^2&\leq& (2\sigma+1)^2M^2\sum_{n\in\mathbb{Z}^N}(|\zeta_n|+|\xi_n|)^{2\sigma})^2|\zeta_n-\xi_n|^2\nonumber\\
&\leq&(2\sigma+1)^2M^2[\sup_{n\in\mathbb{Z}^N}(|\zeta_n|+|\xi_n|)^{2\sigma})]^2\sum_{n\in\mathbb{Z}^N}|\zeta_n-\xi_n|^2\nonumber\\
&\leq& M_1^2R^{4\sigma}||\zeta-\xi||^2_{\ell^2},
\end{eqnarray}
whith $M_1=2^{2\sigma}M(2\sigma+1)$. Taking now the scalar product of (\ref{claim2}) with $\chi$ in $\ell^2$ and using (\ref{claim3}), we have
\begin{eqnarray}
\label{cmap1a}
\sum_{\nu=1}^N||\mathbf{L}_{\nu}^+\chi||^2_{\ell^2}+\omega ||\chi||^2_{\ell^2}&\leq& ||\mathbf{T}(\zeta)-\mathbf{T}(\xi)||_{\ell^2}||\chi||_{\ell^2}\nonumber\\
&\leq&M_1R^{2\sigma}||\zeta-\xi||_{\ell^2}||\chi||_{\ell^2}\nonumber\\
&\leq&\frac{\omega}{2}||\chi||_{\ell^2}^2+\frac{1}{2\omega}M^2_1R^{4\sigma}||z-\xi||_{\ell^2}^2.
\end{eqnarray}
From (\ref{cmap1a}), we obtain the inequality
\begin{eqnarray}
\label{claim4}
||\chi||_{\ell^2}^2=||\mathcal{P}(z)-\mathcal{P}(\xi)||_{\ell^2}^2
\leq \frac{1}{\omega^2}M^2_1R^{4\sigma}||z-\xi||^2_{\ell^2}.
\end{eqnarray}
Since $\mathcal{P}(0)=0$, from inequality (\ref{claim4}), we derive that for $R< E_{\mathrm{min}}$, the map $\mathcal{P}:B_R\rightarrow B_R$ and is a contraction. Therefore $\mathcal{P}$, satisfies the assumptions of Banach Fixed Point Theorem and has a unique fixed point, the trivial one. Hence, for $R<E_{\mathrm{min}}$ the only breather solution is the trivial. \ \ $\diamond$
\vspace{0.2cm}
\newline
If the energy of the excitation
is less that $E_{\mathrm{min}}$ the lattice may not support a standing wave of frequency $\omega$. This time, relation (\ref{disp2}) could be seen as some kind of dispersion relation of frequency vs energy for the nonexsistence of breather solutions of the DNLS equation (\ref{DNLSh}). The dependence $E_{\ell^2_{\omega}}^*$ and $E_{min}$ on $\omega,\sigma, M$ as it appears from inequalities (\ref{disp1}), (\ref{disp2}), could be a point of departure for investigations on the relation of the energy quantity defined by (\ref{moup1}) and the $\ell^2$-norm of the nontrivial breather solution (the power), as well as on their behavior. For example, the inequality
$E_{\mathrm{min}}<E_{\ell^2_{\omega}}^*$,
is satisfied if
\begin{eqnarray}
\label{disp3}
\left(\frac{1}{2^{2\sigma}(\sigma+1)(2\sigma+1)}\right)^{\frac{1}{\sigma}}<\omega.
\end{eqnarray}
In the case $\sigma=1$ (cubic nonlinearity) we get a lower bound $\omega>24^{-1}\sim 0.04166$, for the frequency of the nontrivial breather solution $\psi_n(t)=\phi_n\exp (i\omega t)$, satisfying
\begin{eqnarray}
\label{disp4}
||\phi||_{\ell^2_{\omega}}>E_{\mathrm{min}}.
\end{eqnarray}
Let us also remark that a similar nonexistence result as Theorem \ref{notri}, can be proved in the case (a) of an infinite lattice with $\gamma=\mathrm{const}, \cite{AN}$ and (b) the case of finite lattice (assuming Dirichlet boundary conditions).
Numerical simulations, for testing restriction (\ref{disp2}) or (\ref{disp3})-(\ref{disp4}), could be of interest. Further developments could consider DNLS equations with site dependence on the coupling strength, or operators which are not necessarily discretizations of the Laplacian (for examples of such operators see \cite{SZ2}). \vspace{0.2cm}
{\bf Acknowledgements}. I would like to thank Professors J. C. Eilbeck, and J. Cuevas, for their valuable discussions (especially for resolving the significance of relation (\ref{disp3})) and their interest, improving considerably the presentation of the final version of the manuscript, and my colleagues A. N. Yannacopoulos and H. Nistazakis for their suggestions. I would like also to thank the referee for his useful comments. This work was partially supported by the research project proposal ``Pythagoras I-Dynamics of Discrete and Continuous Systems and Applications''- National Technical University of Athens and University of the Aegean.
|
{
"timestamp": "2005-06-27T18:01:20",
"yymm": "0503",
"arxiv_id": "nlin/0503031",
"language": "en",
"url": "https://arxiv.org/abs/nlin/0503031"
}
|
\section{Introduction}
\label{intro} In this paper, we study the spectrum of
one-dimensional perturbed periodic Schr{\"o}dinger operators.
Precisely, we consider the Schr{\"o}dinger operator defined on
$L^{2}(\mathbb{R})$ by:
\begin{equation}
\label{eqpa}
H_{\varphi,\varepsilon}=-\frac{d^{2}}{dx^{2}}+[V(x)+W(\varepsilon
x+\varphi)],
\end{equation}
where $\varepsilon>0$ is a small positive parameter, $\varphi$ is
a real parameter, and $V$ is a real valued 1-periodic function. We
also assume that $V$ is $L^{2}_{\textrm{loc}}$ and that $W$ is a
fast-decaying function.\\
The operator $H_{\varphi,\varepsilon}$ can be regarded as an
adiabatic perturbation of the periodic operator $H_{0}$:
\begin{equation}
\label{opper} H_{0}=-\triangle+V.
\end{equation}
The spectrum of the periodic operator $H_{0}$ is absolutely
continuous and consists of intervals of the real axis called the
spectral bands, separated by the gaps.\\
If the perturbation $W$ is relatively compact with respect to
$H_{0}$, there are in the gaps of $H_{0}$ some eigenvalues
\cite{Zhe, RB}. We intend to locate these eigenvalues, called
impurity levels.\\
The equation
\begin{equation}
\label{eqp} H_{\varphi,\varepsilon}\psi=E\psi
\end{equation}
depends on two parameters $\varepsilon$ et $\varphi$. We study the
operator $H_{\varphi,\varepsilon}$ in the adiabatic limit, i.e as
$\varepsilon\rightarrow 0$. The periodicity of $V$ implies that
the eigenvalues of $H_{\varphi,\varepsilon}$ are
$\varepsilon$-periodic in $\varphi$. We shall shift $\varphi$ in
the complex plane and we shall assume that $W$ is analytic in a
strip of the complex plane.\\
If $V=0$, there are many results. The case when $W$ is a well has
been studied; in the interval $]\inf\limits_{\mathbb{R}} W,0[$, there is a
quantified sequence of eigenvalues \cite{Fe}. We shall give an
analogous description of the eigenvalues of
$H_{\varphi,\varepsilon}$ in an interval $J$ out of the spectrum
of $H_{0}$. Precisely, when $W$ and $J$ satisfy some additional
conditions described in sections \ref{assW1}, \ref{assW2} et
\ref{assJ}, we show that the eigenvalues of
$H_{\varphi,\varepsilon}$ oscillate around some quantized
energies. The quantization is given by a Bohr-Sommerfeld
quantization rule; the amplitude of oscillation is exponentially
small and is determined by a tunneling coefficient.\\
\subsection{Physical motivation}
The operator $H_{\varphi,\varepsilon}$ is an important model of
solid state physics. The function $\psi$ is the wave function of
an electron in a crystal with impurities. $V$ represents the
potential of the perfect crystal; as such it is periodic. The
potential $W$ is the perturbation created by impurities. In the
semiconductors, this perturbation is slow-varying \cite{Zi}. It is
natural to consider the semi-classical limit.
\subsection{Perturbation of periodic operators}
In $\mathbb{R}^{d}$, the spectral theory of the perturbations of a
periodic operator
\begin{equation}
\label{hp} H_{P}=H_{0}+P
\end{equation}
has motivated numerous studies with different view points.\\
The characterization of the existence of eigenvalues is not easy:
particularly, in any dimension, \cite{KuVa2} deals with the
existence of embedded eigenvalues in the bands. On the real axis,
the situation is simpler. When the perturbation is integrable, the
eigenvalues are necessarily in the adherence of the gaps (\cite{RoBe, HiSh1}).\\
To count the eigenvalues in the gaps, many results have been
obtained thanks to trace formulas. In the large coupling constant
limit, i.e when $P=\lambda U$, with $\lambda\rightarrow+\infty$,
\cite{ADH, Bi1, So2} have studied
$\lim\limits_{\lambda\mapsto+\infty}\textrm{tr}(P_{[E,E']}^{(\lambda)})$,
where $P_{[E,E']}^{(\lambda)}=1_{[E,E']}H_{\lambda}$ (spectral
projector of $H_{\lambda}$ on an interval $[E,E']$ of a gap of
$H_{0}$). In the semi-classical case, \cite{Di1} has given, under
assumptions close to mine, an asymptotic expansion of $\textrm{tr}
[f(H_{\varphi,\varepsilon})]$, for $f\in C_{0}^{\infty}(\mathbb{R})$ and
$\textrm{Supp }f$ in a gap of $H_{0}$. These formulas are valid in
any dimension but are less accurate. For example, in the expansion
obtained in \cite{Di1}, the accuracy depends on the successive
derivatives of the function $f$; the formula does not
give an exponentially precise localization of the eigenvalues.\\
In the one-dimensional case, the scattering theory, well-known in
the case $V=0$, has been developed in \cite{Fi1, New} for the
periodic case. Precisely, we construct some particular solutions
of equation \eqref{hp}, which tend to zero as $x$ tends to
infinity. We call these functions recessive functions. The
eigenvalues of equation \eqref{eqp} are given by a relation of
linear dependence between these solutions.
\subsection{Main steps of the study}
\label{ppalet} We give here the main ideas of the paper. An
important difficulty is the dependence of the equation on the
parameters $\varepsilon$ and $\varphi$; particularly, one has to
decouple the ``fast'' variable $x$ and the ``slow'' variable
$\varepsilon x$. The new idea developed in \cite{FK1, FK2} is the
following : we construct some particular solutions of \eqref{eqp},
satisfying an additional relation called the consistency
condition:
\begin{equation}
\label{coh}
f(x+1,\varphi,E,\varepsilon)=f(x,\varphi+\varepsilon,E,\varepsilon).
\end{equation}
This condition relates their
behavior in $x$ and their behavior in $\varphi$.\\
To find a recessive solution of \eqref{eqp}, it suffices to
construct a solution of \eqref{eqp} which satisfies \eqref{coh}
and which tends to $0$ as $|\mbox{Re }\varphi|$ tends to $+\infty$.
First, we build on the horizontal half-strip $\{\varphi\in \mathbb{C}\ ;\
\varphi\in]-\infty,-A]+i[-Y,Y]\}$ a solution $h_{-}^{g}$ of
equation \eqref{eqp} which is consistent and which tends to $0$ as
$\mbox{Re }\varphi$ tends to $-\infty$. Similarly, we construct
$h_{+}^{d}$ for $\{\varphi\in \mathbb{C}\ ;\
\varphi\in[A,+\infty[+i[-Y,Y]\}$ (Theorem \ref{jostthm}). These
functions are recessive for the variable $x$. The characterization
of the eigenvalues is given by the relation of linear dependence
between $h_{-}^{g}$ and $h_{+}^{d}$:
$$w(h_{-}^{g},h_{+}^{d})=0.$$
In the above-mentioned equation, $w$ represents the Wronskian
whose definition is recalled in \eqref{wronsk}.\\
It remains to compute $w(h_{-}^{g},h_{+}^{d})$. To do that, we use
the complex WKB method developed by A. Fedotov and F. Klopp. This
method consists in describing some complex domains, called
canonical domains, on which we construct some functions satisfying
\eqref{coh} and having a particular asymptotic behavior:
\begin{equation}
\label{stdas}
f_{\pm}(x,\varphi,E,\varepsilon)=e^{\pm\frac{i}{\varepsilon}\int^{\varphi}\kappa}(\psi_{\pm}(x,\varphi,E)+o(1)),\quad
\varepsilon\rightarrow 0.
\end{equation}
In equation \eqref{stdas}, the function $\kappa$ is a analytic
multi-valued function, defined in \eqref{momcompa}; the functions
$\psi_{\pm}$ are some particular solutions of equation
$$H_{0}\psi=(E-W(\varphi))\psi,$$ analytic in $\varphi$ on these canonical domains and called Bloch solutions. We will prove the existence of such functions
in section \ref{cansolbloch}.\\
A. Fedotov and F. Klopp prove the existence of functions with
standard asymptotic only on compact domains of the complex plane.
We shall extend some results on infinite strips of the complex
plane. The consistency condition implies that the function
$h_{-}^{g}$ satisfies the standard asymptotic \eqref{stdas} to the
left of $-A$ and that $h_{+}^{d}$ satisfies an analogous property
to the right of $A$. Thus, the computation of
$w(h_{-}^{g},h_{+}^{d})$ is similar to the calculations of A.
Fedotov and F. Klopp. We must find a sufficiently large domain of
the complex plane, in which we know the Wronskian of $h_{-}^{g}$ and $h_{+}^{d}$.\\
The methods used in their works underline some topological
obstacles, which change the standard asymptotic \eqref{stdas};
these obstacles depend on $W$ and $E$. We give precise assumptions
in sections \ref{assW} and \ref{assJ}.
\section{The main results}
\label{resppaux} In this section, we describe the general context
and the main results of the paper.\\
First, we present the assumptions on the potentials $V$ and $W$,
and on the interval $J$. There are mainly three kinds of
assumptions. Firstly, the study requires some assumptions on the
decay of $W$ to develop the scattering theory. Then, in view of
the hypotheses of the complex WKB method of \cite{FK1}, we assume
that $W$ is analytic in some domain of the complex plane. Finally,
we shall depict the geometric framework and particularly the
subset
$(E-W)^{-1}(\mathbb{R})$.\\
We obtain an equation for the eigenvalues in terms of geometric
objects depending on $H_{0}$, $W$ and $E$: the phases and action
integrals, defined in sections \ref{splecross}.
\subsection{The
potential $V$}
\label{assV} We assume that $V$ has the following properties:\\
\\
{\bf ($\mathbf{H_{V,p}}$) $\mathbf{V}$ is
$\mathbf{L^{2}_{\textrm{loc}}}$, $\mathbf{1}$-periodic.}\\
\\
We consider \eqref{eqp} as a perturbation of the periodic
equation:
\begin{equation}
-\frac{d^{2}}{dx^{2}}\psi(x)+V(x)\psi(x)=(E-W(\varphi))\psi(x).\label{esp}
\end{equation}
We shall use some well known facts about periodic Schr{\"o}dinger
operators. They are described in detail in section \ref{opepera}.\\
We just recall elementary results on $H_{0}$. The operator $H_{0}$
defined in \eqref{opper} is a self-adjoint operator on
$H^{2}(\mathbb{R})$. The spectrum of $H_{0}$ consists of intervals of the
real axis:
\begin{equation}
\label{band}
\sigma(H_{0})=\bigcup\limits_{n\in\mathbb{N}}[E_{2n+1},E_{2n+2}],
\end{equation}
such that:
$$ E_{1}<E_{2}\leq E_{3}< E_{4}...E_{2n}\leq E_{2n+1}<
E_{2n+2}...,\quad E_{n}\rightarrow + \infty,n\rightarrow
+\infty.$$
These intervals $[E_{2n+1},E_{2n+2}]$ are called the
{\it spectral bands}. We set $E_{0}=-\infty$. The intervals
$(E_{2n},E_{2n+1})$ are called the {\it spectral gaps}. If
$E_{2n}\neq E_{2n+1}$, we say that the gap is open.\\
Furthermore, we assume that $V$ satisfies:
\\
{\bf ($\mathbf{H_{V,g}}$) Every gap of $\mathbf{H_{0}}$ is not empty.}\\
\\
This assumption is ``generic'', we refer to \cite{ReSi4} section
XIII.16. An important object of the theory of one-dimensional
periodic operators is the Bloch quasi-momentum $k$ (see section
\ref{qm}). This function is a multi-valued analytic function; its
branch points are the ends of the spectrum, they are of square
root type. We shall give a few details about this function in
section
\ref{opepera}. Finally, we suppose:\\
\\
{\bf ($\mathbf{H_{V}}$) $\mathbf{V}$ satisfy ($\mathbf{H_{V,p}}$)
and
($\mathbf{H_{V,g}}$).}\\
\subsection{The perturbation $W$}
\label{assW}
\subsubsection{Smoothness assumptions}
\label{assW1} We assume that $W$ is such that:\\
\\
{\bf ($\mathbf{H_{W,r}}$) There exists $\mathbf{Y>0}$ such that
$\mathbf{W}$ is analytic in the strip
$\mathbf{S_{Y}=\{|\mbox{Im }(\xi)|\leq Y\}}$ and there exists
$\mathbf{s>1}$ et $C>0$ such that for $\mathbf{z\in S_{Y}}$, we
have:}
\begin{equation}
\mathbf{|W(z)|\leq\frac{C}{1+|z|^{s}}}.\end{equation} These
assumptions are essential to develop the complex WKB method. The
analyticity of the perturbation is crucial in the theory of
\cite{FK1}. The decay of $W$ replaces the
compactness resulting from periodicity in \cite{FK1}.\\
We begin with presenting the complex momentum. This main object of
the complex WKB method shows the importance of $W^{-1}(\mathbb{R})$.
\subsubsection{The complex momentum and its branch points}
\label{momcompb} We put:
$$\mathbb{C}_{+}=\{\varphi\in\mathbb{C}\ ;\ \mbox{Im }\varphi\geq
0\}\textrm{ and } \mathbb{C}_{-}=\{\varphi\in\mathbb{C}\ ;\ \mbox{Im }\varphi\leq 0\}.$$
For equation \eqref{eqp}, we consider the analytic function
$\kappa$ defined by
\begin{equation}
\label{momcompa} \kappa(\varphi)=k(E-W(\varphi)).
\end{equation}
We recall that the function $k$ is presented in section
\ref{assV}. The function $\kappa$ is called the complex momentum.
It plays a
crucial role in adiabatically perturbed problems, see \cite{Bu, FK1}.\\
$\mathbb{N}$ is the set of non-negative integers. We define:
\begin{equation}
\label{nupsilon} \Upsilon(E)=\{\varphi\in S_{Y}\ ;\ \exists\
n\in\mathbb{N}^{*}\ /\ E-W(\varphi)=E_{n}\}
\end{equation}
The set of branch points of $\kappa$ is clearly a subset of
$\Upsilon(E)$. The following result gives a characterization of
the branch points of $\kappa$ among the points of $\Upsilon(E)$:
\begin{lem}
Let $\varphi$ be a point of $\Upsilon(E)$. If $\inf\{q\ ;\
W^{(q)}(\varphi)\neq 0\}\in 2\mathbb{N}+1$, then $\varphi$ is a branch
point of $\kappa$.
\end{lem}
This result follows from the fact that the ends of the spectrum
are of square root type.
\subsubsection{Geometric assumptions}
\label{assW2} \label{descw} The spectrum $\sigma(H_{0})$ consists
of real intervals. Fix $E\in\mathbb{R}$. If $E-W(\varphi)$ is in the
spectrum $\sigma(H_{0})$, then $W(\varphi)$ is real. The spectral
study of (\ref{eqp}) is then tightly connected with the geometry
of $W^{-1}(\mathbb{R})$.\\
We state now the geometric assumptions for $W$. These assumptions
are mainly a description of $W^{-1}(\mathbb{R})$ in a strip containing the
real axis. We call strictly vertical a line whose slope does not
vanish; for precise definitions, we refer to section
\ref{vertdef}.
\\
{\bf ($\mathbf{H_{W,g}}$)\begin{enumerate}
\item $\mathbf{W_{|\mathbb{R}}}$ is real and has a finite number of extrema, which are non-degenerate. \item There exists $\mathbf{Y>0}$
and a finite sequence of strictly vertical lines containing an
extremum of $\mathbf{W}$, such that:
\begin{equation}
\mathbf{W^{-1}(\mathbb{R})\cap S_{Y}=\bigcup\limits_{i\in \{1\ldots
p\}}(\Sigma_{i})\cup\mathbb{R}}.
\end{equation}\end{enumerate}}
\subsection{Some remarks}
\begin{itemize}
\item Since $W$ is real analytic, we know that $W(\overline{\varphi})=\overline{W(\varphi)}$; this implies that $W^{-1}(\mathbb{R})$ is symmetric with respect to the real axis.
\item
We define $\Sigma_{i}^{+}=\Sigma_{i}\cap\mathbb{C}_{+}$ and
$\Sigma_{i}^{-}=\Sigma_{i}\cap\mathbb{C}_{-}$. \item Figure \ref{exW1}
shows an example of the pre-image of the real axis by such a
potential.
\end{itemize}
As we have explained in section \ref{ppalet}, we cover the strip
$S_{Y}$ with local canonical domains. On these domains, we
construct consistent functions with standard behavior (ie
satisfying
\eqref{coh} and \eqref{stdas}).\\
To compute the connection between the bases associated with
different domains, we get round the branch points (for analog
studies, we refer the reader to \cite{FR, FK2}). We will now state
some more accurate assumptions about the configuration of the
branch points; in particular, these assumptions specify
$(E-W)^{-1}(\sigma(H_{0}))$ when $E$ is real. The spectral results
of A. Fedotov and F. Klopp on perturbed periodic equation have
shown the
importance of the relative positions of $J$ and $\sigma(H_{0})$.\\
\input{fig1}
\subsection{Assumptions on the interval $J$} \label{assJ}
Now, we describe the interval $J$ on which we study equation
\eqref{eqp}.
\subsubsection{Hypotheses}
We assume that the interval $J$ is a compact interval satisfying:\\
\\
{\bf $\mathbf{(H_{J})}$ \begin{enumerate}
\item For any $\mathbf{E\in J}$, there exists only one band $\mathbf{B}$ of
$\mathbf{\sigma(H_{0})}$ such that the pre-image $\mathbf{C:=(E-W)^{-1}(B)}$ is not empty.
\item For any $\mathbf{E\in J}$, $\mathbf{C:=(E-W)^{-1}(B)}$ is connected and compact and $\mathbf{(E-W)^{-1}(\stackrel{\circ}{B})}$ contains exactly one real extremum of $\mathbf{W}$.
\end{enumerate}}
\subsubsection{Consequences}
\begin{itemize}
\item $(H_{J})$ implies that $J$ is included in a gap.\item The band $B$ in $(H_{J})$ (1) depends a priori on $E$.
But, since $J$ is connected, the band $B$ is fixed for any $E\in
J$.\item Similarly, the extremum of $W$ in assumption $(H_{J})$
(2) depends on $E$, but by connectedness, it is the same for any
$E\in J$.\end{itemize}
\subsubsection{Notations}
Put $B=[E_{2n-1},E_{2n}]$, for $n\in\mathbb{N}^{*}$. Moreover, we can
always change $W$ or
$\varphi$ so that the extremum of $W$ in $(2)$ is $0$.\\
Then $(H_{J})$ has the following consequences:
\begin{enumerate}\item For any $E\in J$, $(E-W)^{-1}(\sigma(H_{0}))\cap
S_{Y}=(E-W)^{-1}(B)\cap S_{Y}$\item Let
$E_{r}\in\{E_{2n-1},E_{2n}\}$ be the end of $B$ satisfying
$E_{r}\in(E-W)(\mathbb{R})$ for any $E\in J$. We define $E_{i}$ such that
$\{E_{i},E_{r}\}=\{E_{2n-1},E_{2n}\}$.\item There are exactly four
branch points $(\varphi_{r}^{-},\varphi_{r}^{+})\in\mathbb{R}^{2}$ and
$(\varphi_{i}, \overline{\varphi_{i}})$ in $S_{Y}$ related to
$E_{r}$ and $E_{i}$. They satisfy:
$$E-W(\varphi_{r}^{+})=E_{r},E-W(\varphi_{r}^{-})=E_{r},\
\varphi_{r}^{-}<0<\varphi_{r}^{+},$$
$$E-W(\varphi_{i})=E-W(\overline{\varphi_{i}})=E_{i},\
\mbox{Im }\varphi_{i}>0.$$\item There exists a strictly vertical line
$\sigma$ containing $0$ and connecting $\overline{\varphi_{i}}$
to $\varphi_{i}$, such that $(E-W)^{-1}(B)\cap
S_{Y}=[\varphi_{r}^{-},\varphi_{r}^{+}]\cup\sigma$. We define
$\sigma_{+}=\sigma\cap\mathbb{C}_{+}$ and $\sigma_{-}=\sigma\cap\mathbb{C}_{-}$.
We let $\Sigma=(E-W)^{-1}(\mathbb{R})\backslash\mathbb{R}$, $\sigma\subset\Sigma$.
\end{enumerate} These objects are described in figure \ref{pbf}.
\input{fig2}
\subsubsection{Remarks and examples}
We first give a few comments on assumption $(H_{J})$.
\begin{itemize}
\item
We call $C$ the cross. \item This assumption means intuitively
that, in $S_{Y}$, we see the band $B$ only near the extremum $0$.
\end{itemize}
To illustrate these technical assumptions, we give a few examples
of potentials $W$ and intervals $J$. We have depicted some
examples in figure \ref{exW2}.\begin{itemize}\item The simplest
case is when $W$ has only a non-degenerate minimum
$W_{-}$ (see figure \ref{exW2} A).\\
in concrete terms, we can think of the example:
$$ W(x)=\frac{-\alpha}{1+x^{2}},\quad \alpha>0,$$
Then, if we fix $B=[E_{2n-1},E_{2n}]$ and $Y<1$, we can choose
$J=[a,b]$ such that:
$$\max\{E_{2n-2},E_{2n-1}-\alpha,E_{2n}-\frac{\alpha}{1-Y^{2}}\}< a<b<\min\{E_{2n-1},E_{2n}-\alpha,E_{2n+1}-\frac{\alpha}{1-Y^{2}}\}$$
\item We can assume that $W$ has a maximum $W_{+}$ and
a minimum $W_{-}$, if $J$ is chosen to see the band only near the maximum (see figure \ref{exW2} B).\\
$$W(x)=\frac{2}{1+x^{2}}-\frac{1}{1+(x-5)^{2}} $$
$$J\subset]E_{2n-1}+W_{+},E_{2n-2}+W_{+}[\cup]E_{2n},E_{2n+1}+W_{-}[,\quad |J|\leq|E_{2n-2}-E_{2n-1}|$$
Consider this example a little further. The choice of $Y$ is more
complicated in this case. The study of equation $W(u)=w$ for
$w>W_{+}$ shows that there exists only one solution in the strip
$\{\mbox{Im } u\in]0,1[\}$ that we call $Z(w)$ ; we choose
$Y\in]\sup\limits_{E\in J}Z(E-E_{2l-1}),\inf\limits_{E\in
J}Z(E-E_{2l-2})[$.
\item In fact, we could adapt our method to weaker assumptions. For example, we can assume that we do not see the branch points $\varphi_{i}$
and $\overline{\varphi_{i}}$ (incomplete cross), which means that
the vertical line $\sigma$ does not contain any branch points of
$\kappa$. We refer to section \ref{unccross} for some details.
\item For the sake of simplicity, we have assumed that all the
extrema of $W$ are non degenerate. Actually, it suffices to assume
that only the extremum of $W$ in $0$ is non degenerate. \item
Similarly, we could weaken assumption $(H_{V,g})$. We only have to
assume that the gaps adjoining the band $B$ of $(H_{J})$ are not
empty.
\end{itemize}
\input{fig3}
\subsection{Phases and action}
\label{splecross} In this section, we define the tunneling
coefficient $t$ and the phases $\Phi$ et $\Phi_{d}$; these
analytic objects play an essential role in the location of the
eigenvalues. These coefficients are represented as integrals of
the complex momentum $\kappa$ in the $\varphi$ plane.\\
In the strip $S_{Y}$, we consider $\kappa$ a branch of the complex
momentum, continuous on $C$.
\subsubsection{Definition and properties}
\label{chem} We introduce the action $S$ and the phases $\Phi$ and
$\Phi_{d}$ related to the branch $\kappa$.
\begin{defn}
We define the phase:
\begin{equation}
\label{phi}
\Phi(E)=\int_{\varphi_{r}^{-}}^{\varphi_{r}^{+}}(\kappa(u)-\kappa(\varphi_{r}^{-}))du,
\end{equation}
the action:
\begin{equation}
\label{action}
S(E)=i\int_{\sigma}(\kappa(u)-\kappa(\varphi_{i}))du,
\end{equation}
the second phase: \begin{equation} \label{phid}
\Phi_{d}(E)=\int_{\varphi_{r}^{-}}^{0}(\kappa(u)-\kappa(\varphi_{r}^{-}))du+\int_{\varphi_{r}^{+}}^{0}(\kappa(u)-\kappa(\varphi_{r}^{+}))du+\int_{\sigma_{+}}(\kappa(u)-\kappa(\varphi_{i}))du-\int_{\sigma_{-}}(\kappa(u)-\kappa(\overline{\varphi_{i}}))du.
\end{equation}
\end{defn}
In section \ref{anares}, we prove the following result on the
behavior of the coefficients $\Phi$, $S$ and $\Phi_{d}$.
\begin{lem}
\label{phaseactint} There exists a branch $\tilde{\kappa}_{i}$
such that the phases and action integrals have the following
properties:
\begin{enumerate}
\item $\Phi$, $S$, $\Phi_{d}$ are analytic in $E$ in a complex neighborhood of the interval $J$.
\item $\Phi$, $S$, $\Phi_{d}$ take real values on $J$. $\Phi$ and $S$ are positive on $J$.
\item $\forall E\in J,\quad \Phi'(E)(E_{i}-E_{r})>0,\quad S(E)\leq
2\pi\ \mbox{Im }(\varphi_{i}(E)).$
\end{enumerate}
\end{lem}
We define {\it the tunneling coefficient} :
\begin{equation}
t(E,\varepsilon)=\exp(-S(E)/\varepsilon).
\end{equation}
$t$ is exponentially small.
\subsubsection{Remark} The phase and action are simply a generalization
of the coefficients of the form $\int\sqrt{E-W(\varphi)}d\varphi$,
well-known in the case $V=0$ (we refer to \cite{Fe, FR, Ra}).\\
We point out that the coefficient $\Phi$ depend only on the value
of $W$ on the real axis, whereas $S$ and $\Phi_{d}$ depend on the
values of $W$ in the complex plane. The phase $\Phi$ is
independent of the analyticity of $W$ unlike $S$ and $\Phi_{d}$.\\
Now, we state the equation for eigenvalues for \eqref{eqp}.
\subsubsection{The main result}
\begin{thm}
\label{eigenloc}
Equation for eigenvalues.\\
Let $V$, $W$ and $J$ satisfy assumptions $(H_{V})$, $(H_{W,r})$, $(H_{W,g})$ and $(H_{J})$. Fix $Y_{0}\in]0,Y[$.\\
There exists a complex neighborhood $\mathcal{V}$ of $J$, a real
number $\varepsilon_{0}>0$ and two functions $\widetilde{\Phi}$
and $\widetilde{\Phi_{d}}$ with complex values, defined on
$\mathcal{V}\times]0,\varepsilon_{0}[$ such that:
\itemize{\item The functions $\widetilde{\Phi}(\cdot,\varepsilon)$
and $\widetilde{\Phi_{d}}(\cdot,\varepsilon)$ are analytic on
$\mathcal{V}$. Moreover, $\widetilde{\Phi}$ and
$\widetilde{\Phi_{d}}$ satisfy:
$$\widetilde{\Phi}(E,\varepsilon)=\Phi(E)+h_{0}(E,\varepsilon)\quad\textrm{
and
}\quad\widetilde{\Phi_{d}}(E,\varepsilon)=\Phi_{d}(E)+h_{1}(E,\varepsilon),$$
where $\rho$ is a real coefficient,
$h_{0}(E,\varepsilon)=o(\varepsilon)$ and
$h_{1}(E,\varepsilon)=o(\varepsilon)$ uniformly in
$E\in\mathcal{V}$. \item If we define the energy levels
$\{E^{(l)}(\varepsilon)\}$ in $J$ by:
\begin{equation}
\frac{\widetilde{\Phi}(E^{(l)}(\varepsilon),\varepsilon)}{\varepsilon}=l\pi+\frac{\pi}{2},\quad\quad\forall
l\in\{L_{-}(\varepsilon),\ldots,L_{+}(\varepsilon)\},
\end{equation}
then, for any $\varepsilon\in]0,\varepsilon_{0}[$,\itemize{\item
the spectrum of $H_{\varphi, \varepsilon}$ in $J$ consists in a
finite number of eigenvalues, that is to say
\begin{equation} \sigma(H_{\varphi, \varepsilon})\cap J
=\bigcup\limits_{l\in\{L_{-}(\varepsilon),\ldots,L_{+}(\varepsilon)\}}\{E_{l}(\varphi,\varepsilon)\},
\end{equation}\item these eigenvalues satisfy
{\footnotesize\begin{equation}\label{compcross}
E_{l}(\varphi,\varepsilon)=E^{(l)}(\varepsilon)+\varepsilon(-1)^{l+1}\frac{t(E^{(l)}(\varepsilon),\varepsilon)}{\Phi'(E^{(l)}(\varepsilon))}\left[\cos\left(\frac{\widetilde{\Phi_{d}}(E^{(l)}(\varepsilon),\varepsilon)+2\pi\varphi+\rho\varepsilon}{\varepsilon}\right)+t(E^{(l)}(\varepsilon),\varepsilon)r(E^{(l)}(\varepsilon),\varphi,\varepsilon)\right],
\end{equation}}
where there exists $c>0$ such that
$$\sup\limits_{E\in\mathcal{V},\varphi\in\mathbb{R}}r(E,\varphi,\varepsilon)<\frac{1}{c}e^{-\frac{c}{\varepsilon}}.$$}}
\end{thm}
We prove this result in section \ref{anares}.
\subsubsection{Remark}
\label{unccross} If we only assume that $\sigma$ does not contain
any branch points, asymptotic \eqref{compcross} is replaced by the
estimate:
$$|E_{l}(\varphi,\varepsilon)-E^{(l)}(\varepsilon)|< C e^{-\frac{2\pi Y}{\varepsilon}}$$
where $2 Y$ is the width of the strip $S_{Y}$.
\subsubsection{Application : asymptotic expansion of the trace} By using the previous result,
we can compute the first terms in the asymptotic expansion of the
trace formula , and partially recover a result of \cite{Di1}.
\begin{cor}
\label{cordim} Let $f\in C_{0}^{\infty}(\mathbb{R})$ be a real function
such that $\textrm{Supp }f\in J$. Then the function
$f(H_{\varphi,\varepsilon})$ is $\varepsilon$-periodic in
$\varphi$ and its Fourier expansion satisfies:
\begin{equation}
\textrm{tr
}[f(H_{\varphi,\varepsilon})]=\frac{1}{\varepsilon}\int_{0}^{\varepsilon}[f(H_{u,\varepsilon})]du+O(e^{-S/\varepsilon})
\end{equation}
\begin{equation}
\int_{0}^{\varepsilon}[f(H_{u,\varepsilon})]du=\frac{1}{2\pi}\int_{\mathbb{R}_{u}}\int_{[-\pi,\pi]}f(W(u)+E(\kappa))d\kappa
du+o(\varepsilon)
\end{equation}
where $S=\inf\limits_{e\in\textrm{Supp }f}S(e)>0$
\end{cor}
We give more details and the proof of this corollary in section
\ref{trform2}.
\section{Main steps of the study}
\label{schem} Here, we explain the main ideas of the paper.
\subsection{One-dimensional perturbed periodic operators}
\subsubsection{} \label{rapp}
We consider equation \eqref{eqp} as a perturbation of the periodic
equation
\begin{equation}
\label{espa}
H_{0}\psi=E\psi
\end{equation}
where the operator $H_{0}$ is defined in \eqref{opper}. To do
that, we shall describe the spectral theory of periodic operators
in section \ref{opepera}.
\begin{itemize}
\item
\label{rappa} For the moment, we simply introduce the Bloch
solutions of equation \eqref{esp}. We call a {\it Bloch solution}
of \eqref{esp} a function $\Psi$ satisfying \eqref{esp} and:
\begin{equation}
\label{blochsola} \forall
x\in\mathbb{R},\quad\Psi(x+1,E)=\lambda(E)\Psi(x,E),
\end{equation}
with $\lambda\neq 0$ independent of $x$. The coefficient
$\lambda(E)$ is called {\it Floquet multiplier}. We represent
$\lambda(E)$ in the form $\lambda(E)=e^{i k(E)}$; $k$ is the
quasi-momentum presented in section \ref{assV} and described in
section \ref{qm}. If $E\notin\sigma(H_{0})$, there exist two
linearly independent Bloch solutions of \eqref{espa}(see section
\ref{bloch}). We call them $\widetilde{\Psi}_{+}$ et
$\widetilde{\Psi}_{-}$; the associated Floquet multipliers are
inverse of each other and the functions $\widetilde{\Psi}_{\pm}$
are represented in the form:
$$ \widetilde{\Psi}_{\pm}(x,E)=e^{\pm ik(E)x}p_{\pm}(x,E)\quad\textrm{avec}\quad p_{\pm}(x+1,E)=p_{\pm}(x,E).$$
For $\mbox{Im } k(E)>0$, the function $\widetilde{\Psi}_{+}(x,E)$ tends to
$0$ as $x$ tends to $+\infty$ and the function
$\widetilde{\Psi}_{-}(x,E)$ tends to $0$ as $x$ tends to
$-\infty$. Actually, equation \eqref{blochsola} defines the
functions $\widetilde{\Psi}_{+}$ and $\widetilde{\Psi}_{-}$ except
for a multiplicative coefficient. Precisely, equation
\eqref{blochsola} defines two one-dimensional vector spaces that
we call {\it Bloch sub-spaces}.\\
To study the eigenvalues of perturbations of periodic operators,
\cite{Fi1} and \cite{New} introduce, for $\mbox{Im } k(E)>0$, two
functions $(x,\varphi,E,\varepsilon)\mapsto
F_{+}(x,\varphi,E,\varepsilon)$ and
$(x,\varphi,E,\varepsilon)\mapsto F_{-}(x,\varphi,E,\varepsilon)$
solutions of \eqref{eqp} satisfying:
\begin{equation}
\label{jostcond} \lim\limits_{x\rightarrow
+\infty}[F_{+}(x,\varphi,E,\varepsilon)-\widetilde{\Psi}_{+}(x,E)]=0,\quad
\lim\limits_{x\rightarrow
-\infty}[F_{-}(x,\varphi,E,\varepsilon)-\widetilde{\Psi}_{-}(x,E)]=0
\end{equation}
Condition \eqref{jostcond} guarantees the uniqueness of $F_{+}$
(resp. of $F_{-}$) since the function $\widetilde{\Psi}_{+}$
(resp. $\widetilde{\Psi}_{-}$) tends to $0$ as $x$ tends to
$+\infty$ (resp. $-\infty$). These functions are called Jost
functions; they are generally constructed as solutions of a
Lippman-Schwinger integral equation. This construction is an
adaptation of the usual theory of scattering (chapter XI of
\cite{ReSi3}) for a perturbation of laplacian; it consists in
looking for particular solutions of
\eqref{eqp} from the solutions of the periodic equation.\\
We call {\it Jost sub-spaces} the sub-spaces $\mathcal{J}_{+}$ and
$\mathcal{J}_{-}$ generated by $F_{+}$ and $F_{-}$.\\
$\mathcal{J}_{+}$ (resp $\mathcal{J}_{-}$) is the set of solutions
of \eqref{eqp} being a member of $L^{2}([0,\infty))$ (resp.
$L^{2}((-\infty,0])$).
\item Let $f$ and $g$ be two derivable functions, the {\it
Wronskian} of $f$ and $g$ called $w(f,g)$ is defined by:
\begin{equation}
\label{wronsk} w(f,g)=f'g-fg'
\end{equation}
We recall that if $f$ and $g$ are the solutions of a second-order
differential equation, their Wronskian is independent of $x$. The
spectral interest of the Jost sub-spaces is the following:
\begin{prop}\label{carvp} We assume that $\mbox{Im } k(E)>0$. Let $h^{-}_{g}\in\mathcal{J}_{-}$
and $h^{+}_{d}\in\mathcal{J}_{+}$ be two nontrivial Jost solutions
of \eqref{eqp}. $E$ is an eigenvalue of $H_{\varphi,\varepsilon}$
if and only if: \begin{equation}
w(h^{+}_{d},h^{-}_{g})=0
\end{equation}
\end{prop}
To compute the eigenvalues, it suffices to construct the Jost
sub-spaces.
\end{itemize}
\subsection{Construction of consistent Jost solutions} We denote by $(H_{J}^{0})$ the following assumption:
\\ \\
{\bf $\mathbf{(H_{J}^{0})}$ There exists $n\in\mathbb{N}$ such that
$\mathbf{J}$ is a compact interval of $\mathbf{]E_{2n},E_{2n+1}[}$.}\\
\\
Clearly, $(H_{J}^{0})$ is weaker than $(H_{J})$.\\
We introduce
a new notation.\\
For a function $f:\ \mathcal{U}\subset\mathbb{C}^{n}\rightarrow\mathbb{C}^{p}$, we
define the function $f^{*}:\
\overline{\mathcal{U}}\rightarrow\mathbb{C}^{p}$:
\begin{equation}
\label{eqconj1} f^{*}(Z)=\overline{f(\overline{Z})}.
\end{equation}
As we have explained in section \ref{intro}, an useful idea to
study \eqref{eqp} is the construction of consistent solutions,
i.e. satisfying \eqref{coh}. First, we choose in $\mathcal{J}_{-}$
and $\mathcal{J}_{+}$ some consistent bases. We shall prove the
following result:
\begin{thm}\label{jostthm}
We assume that $(H_{V})$, $(H_{W,r})$ and
$(H_{J}^{0})$ are satisfied. Fix $X>1$. Then, there exist a
complex neighborhood $\mathcal{V}=\overline{\mathcal{V}}$ of $J$,
a real $\varepsilon_{0}>0$, two points $m_{g}$ and $m_{d}$ in
$\mathbb{C}$, two real numbers $A_{g}$ and $A_{d}$ and two functions
$(x,\varphi,E,\varepsilon)\mapsto
h_{-}^{g}(x,\varphi,E,\varepsilon)$,
$(x,\varphi,E,\varepsilon)\mapsto
h_{+}^{d}(x,\varphi,E,\varepsilon)$ such that:
\begin{itemize}
\item The functions $(x,\varphi,E,\varepsilon)\mapsto
h_{-}^{g}(x,\varphi,E,\varepsilon)$ and
$(x,\varphi,E,\varepsilon)\mapsto
h_{+}^{d}(x,\varphi,E,\varepsilon)$ are defined and consistent on
$\mathbb{R}\times S_{Y}\times \mathcal{V}\times]0,\varepsilon_{0}[$.
\item For any $x\in[-X,X]$ and $\varepsilon\in]0,\varepsilon_{0}[$, $(\varphi,E)\mapsto
h_{-}^{g}(x,\varphi,E,\varepsilon)$ and $(\varphi,E)\mapsto
h_{+}^{d}(x,\varphi,E,\varepsilon)$ are analytic on $S_{Y}\times
\mathcal{V}$.
\item The function $x\mapsto h_{-}^{g}(x,\varphi,E,\varepsilon)$ (resp. $x\mapsto h_{+}^{d}(x,\varphi,E,\varepsilon)$)
is a basis of $\mathcal{J}_{-}$ (resp. $\mathcal{J}_{+}$).
\item The functions $h_{-}^{g}$ and $h_{+}^{d}$ have the following asymptotic behavior:
\begin{equation}
\label{asj1}
h_{-}^{g}(x,\varphi,E,\varepsilon)=e^{\frac{-i}{\varepsilon}\int_{m_{g}}^{\varphi}\kappa(u)du}\psi_{-}(x,\varphi,E)(1+R_{g}(x,\varphi,E,\varepsilon)),
\end{equation}
and
\begin{equation}
\label{asjda1}
h_{+}^{d}(x,\varphi,E,\varepsilon)=e^{\frac{i}{\varepsilon}\int_{m_{d}}^{\varphi}\kappa(u)du}\psi_{+}(x,\varphi,E)(1+R_{d}(x,\varphi,E,\varepsilon)),
\end{equation}
where
\begin{itemize}
\item $R_{g}$ and $R_{d}$ satisfy:
$$\sup\limits_{x\in]-X,X[,\ \mbox{Re }\varphi<A_{g},\\ E\in\mathcal{V}}\max\{|R_{g}(x,\varphi,E,\varepsilon)|,|\partial_{x}R_{g}(x,\varphi,E,\varepsilon)|\}\leq
r(\varepsilon).$$
$$\sup\limits_{x\in]-X,X[,\ \mbox{Re }\varphi>A_{d},\\ E\in\mathcal{V}}\max\{|R_{d}(x,\varphi,E,\varepsilon)|,|\partial_{x}R_{d
}(x,\varphi,E,\varepsilon)|\}\leq r(\varepsilon),$$ with
$$\lim\limits_{\varepsilon\rightarrow 0}r(\varepsilon)=0.$$
\item The functions $\psi_{+}$ and $\psi_{-}$ are the Bloch canonical solutions
of the periodic equation \eqref{espa} defined in section
\ref{cansolbloch}.
\end{itemize}
\item There exist two real numbers $\sigma_{g}\in\{-1,1\}$, $\sigma_{d}\in\{-1,1\}$, an integer $p$ and two functions $E\mapsto\alpha_{g}(E)$ and $E\mapsto\alpha_{d}(E)$ such
that:
\begin{enumerate}
\item For any $\varepsilon\in]0,\varepsilon_{0}[$, $x\in\mathbb{R}$,
$E\in\mathcal{V}$,et $\varphi\in S_{Y}$ ,we have:
\begin{equation}
\label{stargj}
\alpha_{g}^{*}(E)(h_{-}^{g})^{*}(x,\varphi,E,\varepsilon)=i\sigma_{g}e^{-\frac{i}{\varepsilon}2p\pi
x} \alpha_{g}(E)h_{-}^{g}(x,\varphi,E,\varepsilon)
\end{equation}
\begin{equation}
\label{stardj}
\alpha_{d}^{*}(E)(h_{+}^{d})^{*}(x,\varphi,E,\varepsilon)=i\sigma_{d}e^{\frac{i}{\varepsilon}2p\pi
x} \alpha_{d}(E)h_{+}^{d}(x,\varphi,E,\varepsilon)
\end{equation}
\item The functions $\alpha_{g}$ and $\alpha_{d}$ are analytic
and given by \eqref{renormconstg} and \eqref{renormconstd}. They
do not vanish on $\mathcal{V}$.
\end{enumerate}
\end{itemize}
\end{thm}
We immediately deduce from Theorem \ref{jostthm} and Proposition
\ref{carvp} that the eigenvalues of $H_{\varphi,\varepsilon}$ are
characterized by:
\begin{equation}
w(h_{-}^{g}(\cdot,\varphi,E,\varepsilon),h_{+}^{d}(\cdot,\varphi,E,\varepsilon))=0
\end{equation}
Theorem \ref{jostthm} is the consequence of two main ideas:
\begin{itemize}
\item First, we adapt the construction of Jost functions developed
by \cite{Fi1,Ne}. Indeed, this construction proves that asymptotic
\eqref{as1} is only satisfied on domains which depend on
$\varepsilon$ (see section \ref{scattheory}).
\item We must understand how this asymptotic evolves on a domain
which does not depend on $\varepsilon$. To do that, we extend the
continuation results of Fedotov and Klopp in a non compact frame
(see section \ref{infwkb}).
\end{itemize}
\subsection{Conclusion}
To finish the computations, it suffices to apply the methods of
\cite{FK1}. We have to go through the cross (see figure
\ref{pbf}). We will show that there exists in the neighborhood of
the cross a consistent basis $f_{\pm}^{i}$ with standard
asymptotic. We shall express the functions $h_{-}^{g}$ and
$h_{+}^{d}$ on this basis (section \ref{calcmattransf}).
\section{Periodic Schr{\"o}dinger operators on the real line}
\label{opepera} We now discuss the periodic operator \eqref{opper}
where $V$ is a 1-periodic, real-valued,
$\mathbf{L^{2}_{\textrm{loc}}}$-function. We collect known results
needed in the present paper (see \cite{Ma, McK, Ti}).
\subsection{Bloch solutions}
\label{bloch} Let $\widetilde{\Psi}$ be a solution of the equation
\begin{equation}
\label{espc} H_{0}\widetilde{\Psi}=\mathcal{E}\widetilde{\Psi}
\end{equation}
satisfying the relation
\begin{equation}
\widetilde{\Psi}(x+1)=\lambda\widetilde{\Psi}(x),\quad\forall
\in\mathbb{R}
\end{equation}
for some complex number $\lambda\neq 0$ independent of $x$. Such a
solution is called a {\it Bloch solution}, and the number
$\lambda$ is called the {\it Floquet multiplier}. Let us discuss
the analytic properties of Bloch solutions.\\
In \eqref{band}, we have denoted by $[E_{1}, E_{2}],\ldots
,[E_{2n+1}, E_{2n+2}],\ldots$ the spectral bands of the periodic
Schr{\"o}dinger equation. Consider $\Gamma_{\pm}$ two copies of
the complex plane $\mathcal{E}\in \mathbb{C}$ cut along the spectral bands. Paste
them together to get a Riemann surface with square root
branch points. We denote this Riemann surface by $\Gamma$.\\
One can construct a Bloch solution $\widetilde{\Psi}(x,\mathcal{E})$ of
equation \eqref{espc} meromorphic on $\Gamma$. The poles of this
solution are located in the spectral gaps. Precisely, each
spectral gap contains precisely one simple pole. This pole is
situated either on $\Gamma_{+}$ or on $\Gamma_{-}$. The position
of the pole is
independent of $x$. For the details, we refer to \cite{Fi1}.\\
Except at the edges of the spectrum (i.e. the branch points of
$\Gamma$), the restrictions $\widetilde{\Psi}_{\pm}$ of
$\widetilde{\Psi}$ on $\Gamma_{\pm}$ are linearly independent
solutions of \ref{espc}. Along the gaps, these functions are real
and satisfy :
\begin{equation}
\label{gaprel}
\overline{\widetilde{\Psi}_{\pm}(x,\mathcal{E}-i0)}=\widetilde{\Psi}_{\pm}(x,\mathcal{E}+i0),\quad\forall
\mathcal{E}\in]E_{2n},E_{2n+1}[,\ n\in\mathbb{N}.
\end{equation}
Along the bands, we have :
\begin{equation}
\label{bandrel}
\overline{\widetilde{\Psi}_{\pm}(x,\mathcal{E}-i0)}=\widetilde{\Psi}_{\mp}(x,\mathcal{E}+i0),\quad\forall
\mathcal{E}\in]E_{2n+1},E_{2n+2}[,\ n\in\mathbb{N}.
\end{equation}
\subsection{Bloch quasi-momentum}
\subsubsection{}
Consider the Bloch solution $\widetilde{\Psi}(x,\mathcal{E})$ introduced in
the previous subsection. The corresponding Floquet multiplier
$\lambda(\mathcal{E})$ is analytic on $\Gamma$. Represent it in the form:
\begin{equation}
\lambda(\mathcal{E})=\exp(ik(\mathcal{E})).
\end{equation}
The function $k(\mathcal{E})$ is called Bloch quasi-momentum. It has the
same branch points as $\widetilde{\Psi}(x,\mathcal{E})$, but the
corresponding Riemann surface is more complicated. \\ To describe
the main properties of $k$, consider the complex plane cut along
the real line from $E_{1}$ to $+\infty$. Denote the cut plane by
$\mathbb{C}_{0}$. One can fix there a single valued branch of the
quasi-momentum by the condition
\begin{equation}
ik_{0}(\mathcal{E})<0,\quad \mathcal{E}<E_{1}.
\end{equation}
All the other branches of the quasi-momentum have the form $\pm
k_{0}(\mathcal{E})+2\pi m, m\in\mathbb{Z}$. The $\pm$ and the number $m$ are
indexing these branches. The image of $\mathbb{C}_{0}$ by $k_{0}$ is
located in the upper half of the complex plane.
\begin{equation}
\mbox{Im } k_{0}(\mathcal{E})>0,\quad \mathcal{E}\in\mathbb{C}_{0}.
\end{equation}
In figure \ref{qm}, we drew several curves in $\mathbb{C}_{0}$ and their
images under transformation $E\mapsto k_{0}(E)$. The
quasi-momentum $k_{0}(E)$ is real along the spectral zones, and,
along the spectral gaps, its real part is constant; in particular,
we have
\begin{equation}\label{qm1}
k_{0}(E_{1})=0\quad k_{0}(E_{2l}\pm i0)=k_{0}(E_{2l+1}\pm
i0)=\pm\pi l,\ \ \ l\in\mathbb{N}.
\end{equation}
All the branch points of $k$ are of square root type. Let $E_{m}$
be one of the branch points of $k$. Then, each function:
\begin{equation} \label{qm2}
f_{m}^{\pm}(\mathcal{E})=(k_{0}(\mathcal{E}\pm i0)-k_{0}(E_{m}\pm
i0))/\sqrt{\mathcal{E}-E_{m}},\ \ \ E\in\mathbb{R}
\end{equation}
can be analytically continued in a small vicinity of the branch
point $E_{m}$.\\ Finally, we note that
\begin{equation}
k_{0}(\mathcal{E})=\sqrt{\mathcal{E}}+O(1/\sqrt{\mathcal{E}}),\ \ \ |\mathcal{E}|\rightarrow\infty
\end{equation}
where $E\in\mathbb{C}_{0}$ and $0<\arg E<2\pi$.
The values of the quasi-momentum $k_{0}$ on the
two sides of the cut $[E_{1},+\infty)$ are related to each other
by the formula:
\begin{equation}
\label{ko}
\forall \mathcal{E}\in]E_{1},+\infty[,\quad k_{0}(\mathcal{E}+i0)=-\overline{k_{0}(\mathcal{E}-i0)},\ \ \ E_{1}\leq
\mathcal{E}.
\end{equation}
Consider the spectral gap $(E_{2l},E_{2l+1}),l \in\mathbb{N}$. Let
$\mathbb{C}_{l}$ be the complex plane cut from $-\infty$ to $E_{2l}$ and
from $E_{2l+1}$ to $+\infty$. Denote by $k_{l}$ the branch of the
quasi-momentum analytic on $\mathbb{C}_{l}$ and coinciding with $k_{0}$
for $\mbox{Im } E >0$. Then, one has:
\begin{equation}
\label{kl}
\forall\mathcal{E}\in]-\infty,E_{2l}[\cup]E_{2l+1},+\infty[,\quad\quad k_{l}(\mathcal{E}+i0)+\overline{k_{l}(\mathcal{E}-i0)}=2\pi
l.
\end{equation}
\input{fig4}
\subsection{Periodic components of the Bloch solution}
Let $D$ a simply connected domain that does not contain any branch
point of $k$. On $D$, we fix an analytic branch of $k$. Consider
two copies of $D$, denoted by $D_{\pm}$, corresponding to two
sheets of $\mathcal{G}$. Now we redefine $\widetilde{\Psi}_{\pm}$ to be the
restrictions of $\widetilde{\Psi}$ to $D_{\pm}$. They can be
represented in the form:
\begin{equation}
\widetilde{\Psi}_{\pm}(x,\mathcal{E})=e^{\pm i k(\mathcal{E})x}p_{\pm}(x,\mathcal{E}),\ \ \
\mathcal{E}\in D
\end{equation}
where $p_{l}^{\pm}(x,\mathcal{E})$ are 1-periodic in $x$,
\begin{equation}
p_{\pm}(x+1,\mathcal{E})=p_{\pm}(x,\mathcal{E}),\ \ \ \forall x\in\mathbb{R}
\end{equation}
\subsection{Analytic solutions of \eqref{espc}}
To describe the asymptotic formulas of the complex WKB method for
equation \eqref{eqp}, one needs specially normalized Bloch
solutions of the equation \eqref{espc}.\\
Let $\mathcal{D}$ be a simply connected domain in the complex
plane containing no branch point of the quasi-momentum $k$. We fix
on $\mathcal{D}$ a continuous determination of $k$. We fix
$\mathcal{E}_{0}\in\mathcal{D}$. We recall the following result
(\cite{FK1, FK4}).
\begin{lem}\label{anasol}
We define the functions $g_{\pm}$ :
\begin{equation}
\label{fu} g_{\pm}\ :\ \mathcal{D}\rightarrow\mathbb{C} \ ;\
\mathcal{E}\mapsto
-\frac{\int_{0}^{1}p_{\mp}(x,\mathcal{E})\partial_{\mathcal{E}}p_{\pm}(x,\mathcal{E})dx}{\int_{0}^{1}p_{+}(x,\mathcal{E})p_{-}(x,\mathcal{E})dx},
\end{equation}
and the functions $\psi_{\pm}^{0}$ :
\begin{equation}
\label{blochsol} \psi_{\pm}^{0}\ :\
\mathbb{R}\times\mathcal{D}\rightarrow\mathbb{C}\ ;\
(x,\mathcal{E})\mapsto\sqrt{k'(\mathcal{E})}e^{\int_{\mathcal{E}_{0}}^{\mathcal{E}}g_{\pm}(e)de}\widetilde{\Psi}_{\pm}(x,\mathcal{E}).
\end{equation}
The functions $\mathcal{E}\mapsto\psi_{\pm}^{0}(x,\mathcal{E})$ are analytic on
$\mathcal{D}$, for any $x\in\mathbb{R}$. The functions $\psi_{\pm}^{0}$
are called {\it analytic Bloch solutions normalized at the point
$\mathcal{E}_{0}$} of \eqref{espc}.
\end{lem}
Sometimes, we shall denote $\psi_{\pm}^{0}(x,\mathcal{E},\mathcal{E}_{0})$
to specify the normalization. We refer to section 1.4.4 of
\cite{FK4} for the details of the proof. The proof follows from
the study of the poles of $\widetilde{\Psi}_{\pm}$ and the zeros
of $k'$. The poles of $g_{\pm}$ are simple and exactly situated at
the singularities of $\sqrt{k'}\widetilde{\Psi}_{\pm}$. The
computation of the residues of $g_{\pm}$ at these points completes
the proof.
\subsection{Useful formulas}
We end this section with some useful formulas. We recall that the
functions $g_{\pm}$ are given in \eqref{fu}. Fix $n\in\mathbb{N}$.
Equations \eqref{gaprel} and \eqref{bandrel} lead to the following
relations:
\begin{equation}\label{symgap}
g_{\pm}^{*}(\mathcal{E})=g_{\pm}(x,\mathcal{E}),\quad\forall \mathcal{E}\in]E_{2n},E_{2n+1}[.
\end{equation}
\begin{equation}\label{symband}
g_{\pm}^{*}(\mathcal{E})=g_{\mp}(x,\mathcal{E}),\quad\forall
\mathcal{E}\in]E_{2n+1},E_{2n+2}[.
\end{equation}
\section{Main tools of the complex WKB method}
\label{wkbconst} In this section, we recall the main tools of the
complex WKB method on compact domains. The idea of the method is
to construct some consistent functions of \eqref{eqp} with
asymptotic behavior \eqref{stdas}. This construction is not
possible on any domain of the complex plane but on some domains called canonical.\\
We apply the results of \cite{FK1, FK2, FK3} to the assumptions
$(H_{W,g})$ and $(H_{J})$. We build a neighborhood of the cross,
in which we construct a consistent basis with standard behavior
\eqref{stdas}. In this section, we fix $Y$ such that the
assumptions $(H_{W,g})$ and $(H_{J})$ are satisfied in the strip
$S_{Y}$.
\subsection{Canonical domains}
The canonical domain is the main geometric notion of the complex
WKB method.
\subsubsection{The complex momentum}
\label{vertdef}
The canonical domains can be described in terms of
the complex momentum $\kappa(\varphi)$. Remind that this function
is defined by formula \eqref{momcompa}. We have described $\kappa$
in section \ref{momcompb}. The properties of $\kappa$ depend on
the spectral
parameter $E$ and of the analytic properties of $W$.\\
We first formulate some definitions (\cite{FK1}).
\subsubsection{Vertical, strictly vertical curves}
\begin{defn}
We say that a curve $\gamma$ is $\textit{vertical}$ if it
intersects the lines $\mbox{Im } z=\textrm{Const}$ at non-zero angles
$\theta$.\\
We say that a curve $\gamma$ is $\textit{strictly vertical}$ if
there is a positive number $\delta$ such that, at any point of
$\gamma$, the intersection angle $\theta$ satisfies the
inequality:
\begin{equation}
\delta<\theta<\pi-\delta.
\end{equation}
\end{defn}
\subsubsection{Canonical, strictly canonical curves}
\label{lc} Let $\gamma$ be a vertical curve which does not contain
any branch point. On $\gamma$, fix a continuous branch of the
momentum of $\kappa$.
\begin{defn}
We call $\gamma$ $\textit{canonical}$ if, along $\gamma$,
\begin{itemize}
\item $\mbox{Im }\varphi\mapsto\mbox{Im }\int^{\varphi}\kappa(u)du$ is strictly increasing. \item
$\mbox{Im }\varphi\mapsto\mbox{Im }\int^{\varphi}(\kappa(u)-\pi)du$ is strictly
decreasing.
\end{itemize}
\end{defn}
Assume that $\gamma$ is strictly vertical. If there is a positive
number $\delta$ such that, along $\gamma$:
\begin{equation}
\label{integra}
\mbox{Im }\int_{\varphi}^{\varphi'}\kappa(u)du\geq\delta\mbox{Im }(\varphi'-\varphi)\quad\forall(\varphi,\varphi')\in\gamma^{2},
\end{equation}
and
\begin{equation}\label{integrb}
\mbox{Im }
\int_{\varphi}^{\varphi'}(\pi-\kappa(u))du\geq\delta\mbox{Im }(\varphi'-\varphi)\quad\forall(\varphi,\varphi')\in\gamma^{2},
\end{equation}
we call $\gamma$ $\delta-\textit{strictly canonical}$.\\ We
identify the complex numbers with vectors in $\mathbb{R}^{2}$. To
construct canonical lines, we have to study the vector fields
$\kappa$ and $\kappa-\pi$, or rather their integral curves. For
$\varphi\in D$, $S(\varphi)$ denotes the sector of apex $\varphi$
such that, for any vector $z\in S(\varphi)$, we have:
\begin{equation}
\mbox{Im }(i\overline{\kappa(\varphi)}(z-\varphi))>0\textrm{ et }
\mbox{Im }(i(\overline{\kappa(\varphi)}-\pi)(z-\varphi))<0.
\end{equation}
Let $\gamma\in D$ a curve which does not contain any branch point.
For all $\varphi\in\gamma$, we denote $t(\varphi)$ the vector
tangent to $\gamma$ in $\varphi$ and oriented upward. The curve
$\gamma\in D$ is canonical for the determination $\kappa$ if and
only if for any $\varphi\in\gamma$, the vector $t(\varphi)$
belongs to $S(\varphi)$ (see figure \ref{can}). The cone
$S(\varphi)$ depends on the determination of $\kappa$. For
example, if $\kappa$ satisfies $\mbox{Re }\kappa\in]0,\pi[$, this cone is
not empty.
\subsubsection{}
In what follows, $\xi_{1}$ and $\xi_{2}$ are two points in $\mathbb{C}$
such that
$$ \mbox{Im } \xi_{1}<\mbox{Im }\xi_{2}.$$
We shall denote by $\gamma$ a smooth curve going from $\xi_{1}$ to
$\xi_{2}$; this curve will always be oriented from $\xi_{1}$ to
$\xi_{2}$.
\input{fig5}
\subsubsection{Definition of the canonical domain}
Let $K$ be a simply connected domain in
$\left\{\mbox{Im }\varphi\in[\mbox{Im }\xi_{1},\mbox{Im }\xi_{2}]\right\}$ containing no
branch points of the complex momentum. On $K$, fix a continuous
branch $\kappa$.
\begin{defn}
We call $K$ a $\textit{canonical domain for }\kappa,\
\xi_{1}\textit{ and }\xi_{2}$ if it is the union of curves that
are connecting $\xi_{1}$ and $\xi_{2}$ and that are canonical with respect to $\kappa$.\\
If there is $\delta>0$ such that $K$ is a union of
$\delta$-strictly canonical curves, we call $K$
$\delta-\textit{strictly canonical}$.
\end{defn}
\subsubsection{}
Assume that $K$ is a canonical domain. Denote by $\partial K$ its
boundary. Fix a positive number $\delta$. We call the domain
$$\mathcal{C}=\{z\in K\ ;\ \textrm{dist}(z,\partial K)>\delta\}$$
an admissible sub-domain of $K$.\\
Note that the branch points of the complex momentum are outside of
$\mathcal{C}$, at a distance greater than $\delta$.
\subsection{Canonical Bloch solutions}
\label{cansolbloch}To describe the asymptotic formulas of the
complex WKB method for equation \eqref{eqp}, we shall use the
analytic Bloch solutions of \eqref{espc}, defined in Lemma
\ref{anasol} for the parameter $\mathcal{E}=E-W(\varphi)$. Precisely, we
consider the unperturbed periodic equation:
\begin{equation}
\label{espb} H_{0}\psi=(E-W(\varphi))\psi.
\end{equation}
\subsection{}
Let $D$ be a simply connected domain in $S_{Y}$, containing no
branch points of $\kappa$. The mapping $\varphi\mapsto
E-W(\varphi)$ maps $D$ onto a domain $\mathcal{D}\subset\mathbb{C}$. The
domain $\mathcal{D}$ does not contain any branch point of $k$.\\
Fix $\varphi_{0}\in D$, such that $k'(E-W(\varphi_{0}))\neq 0$. In
Lemma \ref{anasol}, we have built the analytic Bloch solutions
$\{\psi_{\pm}^{0}\}$ of equation (\ref{espc}), normalized in
$E-W(\varphi_{0})$. For $\varphi\in D$, we define:
\begin{equation}
\psi_{\pm}(x,\varphi,E)=\psi_{\pm}^{0}(x,E-W(\varphi)),\quad\forall
u\in\mathbb{R},\quad\forall \varphi\in D.
\end{equation}
In \cite{FK1}, it is proved that the functions
$\varphi\mapsto\psi_{\pm}(x,\varphi,E)$ can be analytically
continued to $D$. $\psi_{\pm}$ are called the {\it canonical Bloch
solutions} of equation \eqref{espb}. Sometimes, we shall precise $\psi_{\pm}(x,\varphi,E,\varphi_{0})$ to specify the normalization.\\
We define
\begin{equation}
\label{omega}
\omega_{\pm}(\varphi,E)=-W'(\varphi)g_{\pm}(E-W(\varphi)).
\end{equation}
We also define:
\begin{equation}
\label{racq}
q(\varphi)=\sqrt{k'(E-W(\varphi)}
\end{equation}
\subsection{The consistency relation}
\subsubsection{Consistent functions and consistent bases}
We recall that we say that $f$ is a {\it consistent} function if
it satisfies \eqref{coh}.\\
We say also that a basis $\{f_{\pm}\}$ of solutions of \eqref{eqp}
is {\it a consistent basis} if:
\begin{itemize}
\item The functions $f_{+}$ and $f_{-}$ are consistent.
\item Their Wronskian is independent of $\varphi$.
\end{itemize}
\subsubsection{Analyticity and consistency}
First, we define the width of a set.
\begin{defn}
Fix $Y_{0}>0$ and $M\subset S_{Y_{0}}$ a set of points. We define
$l(M,Y_{0})$:
\begin{equation}
\label{larga}
l(M,Y_{0})=\inf\limits_{y\in[-Y_{0},Y_{0}]}\sup\left\{|\mbox{Re }\varphi-\mbox{Re }\varphi'|
; (\varphi,\varphi')\in M^{2}\textrm{ such that }
\mbox{Im }\varphi=\mbox{Im }\varphi'=y \right\}
\end{equation}
$l(M,Y_{0})$ is called the {\it width} of $M$ in $S_{Y_{0}}$
\end{defn}
One has:
\begin{lem}
\label{anacoh} Fix $E$. We consider $X>0$, $\tilde{Y}\in]0,Y[$,
$\varepsilon_{0}>0$ and $K$ a complex domain such that
$l(K,\tilde{Y})>\varepsilon_{0}$. We assume that for any
$\varepsilon\in]0,\varepsilon_{0}[$,
$f(\cdot,\varphi,E,\varepsilon)$ is a consistent solution of
\eqref{eqp} for $\varphi\in K$ and that for any $x\in[-X,X]$, the
function $\varphi\mapsto f(x,\varphi,E,\varepsilon)$ is analytic
on $K$. Then, for any $\varepsilon\in]0,\varepsilon_{0}[$ and any
$x\in[-X,X]$, the function $\varphi\mapsto
f(x,\varphi,E,\varepsilon)$ is analytic on $S_{\tilde{Y}}$.
\end{lem}
This result is proved in \cite{FK3, FK4}.
\subsection{The theorem of the complex WKB method on a compact domain}
\label{wkbth} In this section, we recall the main result of the
complex WKB method.
\subsubsection{Standard asymptotic behavior}
\label{cptmtasstd} We briefly introduce the notion of standard
asymptotic behavior (see \cite{FK4}). Speaking about a solution
having standard asymptotic behavior, we mean first of all that
this solution has the asymptotics \eqref{stdas} and other
properties that we present now.\\
Fix $E_{0}\in\mathbb{C}$. Let $D\subset\mathbb{C}$ a simply connected domain
containing no branch points. Let $\kappa$ be a branch of the
complex momentum continuous in $D$ and $\psi_{\pm}$ the canonical
Bloch solutions normalized in $\varphi_{0}\in D$.\\
We say that a consistent solution $f$ has standard behavior $f\sim
e^{\frac{i}{\varepsilon}\int^{\varphi}\kappa(u)du}\psi_{+}(x,\varphi,E)$,
respectively $f\sim
e^{-\frac{i}{\varepsilon}\int^{\varphi}\kappa(u)du}\psi_{-}(x,\varphi,E)$
in $D$ if
\begin{itemize}\item there exists a complex neighborhood $\mathcal{V}_{0}$ of $E_{0}$ and
$X>0$ such that $f$ is a consistent solution of equation
\eqref{eqp} for any $(x,\varphi,E)\in[-X,X]\times D\times
\mathcal{V}_{0}$;
\item for any $x\in[-X,X]$, the function $((\varphi,E)\mapsto
f(x,\varphi,E,\varepsilon))$ is analytic on $D\times
\mathcal{V}_{0}$;
\item for any $A$, a sub-admissible domain of $D$, there is a
neighborhood $\mathcal{V}_{A}$ of $E_{0}$ such that
\begin{equation}
\label{cptmtasstda} \forall(x,\varphi,E)\in[-X,X]\times D\times
\mathcal{V}_{A},\quad
f(x,\varphi,E,\varepsilon)=e^{\frac{i}{\varepsilon}\int^{\varphi}\kappa(u)du}(\psi_{+}(x,\varphi,E)+o(1)),\quad\varepsilon\rightarrow
0
\end{equation}
respectively
\begin{equation}
\label{cptmtasstdb} \forall(x,\varphi,E)\in[-X,X]\times D\times
\mathcal{V}_{A},\quad
f(x,\varphi,E,\varepsilon)=e^{-\frac{i}{\varepsilon}\int^{\varphi}\kappa(u)du}(\psi_{-}(x,\varphi,E)+o(1)),\quad\varepsilon\rightarrow
0
\end{equation}
\item the asymptotics \eqref{cptmtasstda} and \eqref{cptmtasstdb}
are uniform on $[-X,X]\times D\times \mathcal{V}_{A}$;
\item the asymptotics \eqref{cptmtasstda} and \eqref{cptmtasstdb}
can be differentiated once in $x$.
\end{itemize}
\subsubsection{}
Let us formulate the Theorem WKB on a compact domain.
\begin{thm}\cite{FK1,FK4}\\
\label{finwkbthm} We assume that $V$ satisfies $(H_{V})$ and that
$W$ satisfies $(H_{W,r})$. Fix $X>1$ and $E_{0}\in\mathbb{C}$. Let
$K\subset S_{Y}$ be a bounded canonical domain with respect to
$\kappa$. There exists $\varepsilon_{0}>0$ and a consistent basis
$\{f_{+}(x,\varphi,E,\varepsilon),f_{-}(x,\varphi,E,\varepsilon)\}$
of solutions of \eqref{eqp}, having the standard behavior
\eqref{cptmtasstda} et \eqref{cptmtasstdb} in $K$.\\
For any fixed $x\in\mathbb{R}$, the functions $\varphi\mapsto
f_{\pm}(x,\varphi,E,\varepsilon)$ are analytic in $K$.
\end{thm}
\subsection{The main geometric tools of the complex WKB method}
\label{constgeo} In this section, we introduce the main geometric
tools of the complex WKB method. To do that, we recall some ideas
of \cite{Fe, FK1, FK3, Wa}.
\subsubsection{Stokes lines}
The definition of the Stokes lines is fairly standard,
\cite{Fe,FK1}. The integral
$\varphi\mapsto\int^{\varphi}\kappa(u)du$ has the same branch
points as the complex momentum. Let $\varphi_{0}$ be one of them.
Consider the curves beginning at $\varphi_{0}$, and described by
the equation
\begin{equation}
\mbox{Im }\int_{\varphi_{0}}^{\varphi}(\kappa(\xi)-\kappa(\varphi_{0}))d\xi=0
\end{equation}
These curves are the {\it Stokes lines} beginning at
$\varphi_{0}$. According to equation \eqref{ko} and equation
\eqref{kl}, the Stokes line definition is independent of the
choice of the branch of $\kappa$.\\
Assume that $W'(\varphi_{0})\neq 0$. Equation \eqref{qm2} implies
that there are exactly three Stokes lines beginning at
$\varphi_{0}$. The angle between any two of them at this point is
equal to $\frac{2\pi}{3}$.
\subsection{Lines of Stokes type}
\label{ligtypsto} We recall that $D\subset S_{Y}$ is a simply
connected domain containing no branch points. Let $\gamma\subset
D$ be a smooth curve. We say that $\gamma$ is a line of Stokes
type with respect to $\kappa$ if, along $\gamma$, we have
$$\textrm{ either }\mbox{Im }\left(\int^{\varphi}\kappa(u)
du\right)=\textrm{Const}\quad\textrm{ or
}\quad\mbox{Im }\left(\int^{\varphi}(\kappa(u)-\pi)
du\right)=\textrm{Const}$$
\subsection{Pre-canonical lines}
Let $\gamma\subset D$ be a vertical curve. We call $\gamma$
$\textit{pre-canonical}$ if it consists of union of bounded
segments of canonical curves and/or lines of Stokes type.
\subsection{Some branches of the complex momentum}
In this section, we describe different branches of $\kappa$ near
the branch points described in \ref{assJ}. The geometrical
configuration is similar to the one studied in \cite{FK2}.
\subsubsection{Different cases}
\label{poss} We assume that $(H_{W,r})$, $(H_{W,g})$, and
$(H_{J})$ are satisfied. To study the geometrical tools of the WKB
complex method, one needs to specify the properties of $\mbox{Im }\kappa$
and $\mbox{Re }\kappa$. We know that $\kappa(\varphi_{r}^{\pm})\equiv
0[\pi]$ (see section \ref{opepera}). We consider two cases: either
$\kappa(\varphi_{r}^{\pm})\equiv 0[2\pi]$ or
$\kappa(\varphi_{r}^{\pm})\equiv \pi[2\pi]$.\\
We define $S_{-}$ the open domain delimited by the real line at
the bottom and by $\Sigma_{+}$ to the right:
\begin{equation}
\label{smoins} S_{-}=\{\varphi-r\ ;\
\varphi\in\Sigma_{+}^{*},r\in\mathbb{R}_{+}^{*}\}\cap S_{Y}
\end{equation}
Similarly, we define $S_{+}$ the open domain delimited by the real
line at the bottom and by $\Sigma_{+}$ to the left:
\begin{equation}
\label{splus} S_{+}=\{\varphi+r\ ;\
\varphi\in\Sigma_{+}^{*},r\in\mathbb{R}_{+}^{*}\}\cap S_{Y}
\end{equation}
The domains $S_{+}$ and $S_{-}$ are shown in figure
\ref{smoinsa}.\\ We prove the following result.
\begin{lem}
\label{detpos}\label{dtepos}There exists a branch $\kappa_{i}$ of
the complex momentum such that
\begin{enumerate}
\item $\mbox{Im }\kappa_{i}(\varphi)>0$ for $\varphi\in S_{-}$, $\kappa_{i}(\varphi_{r}^{-}+i0)=0$ and $\kappa_{i}(\varphi_{i}-0)=\pi$,\\
ou
\item $\mbox{Im }\kappa_{i}(\varphi)<0$ for $\varphi\in S_{-}$, $\kappa_{i}(\varphi_{r}^{-}+i0)=\pi$ and $\kappa_{i}(\varphi_{i}-0)=0$.
\end{enumerate}
\end{lem}
\begin{dem}
\begin{itemize}\item First, we specify the sign of $\mbox{Im }\kappa_{i}$. The set
$(E-W)(\mathbb{R}-[\varphi_{r}^{-},\varphi_{r}^{+}]))$ belongs to a gap
$G$. We define $\Lambda_{-}=(E-W)(S_{-})$. We prove that
$\Lambda_{-}$ is a connected domain which intersects with $\mathbb{R}$
only in the gap $G$. According to assumption $(H_{W,g})$
(\ref{assW2}), there exists a sequence of vertical curves
$\widetilde{\Sigma}_{k}$ such that:
$$\Lambda_{-}\cap\mathbb{R}=(E-W)((-\infty,\varphi_{r}^{-}]\cup[\varphi_{r}^{+},+\infty))\cup(E-W)(\widetilde{\Sigma}_{k}^{+}).$$
$(E-W)(\widetilde{\Sigma}_{k}^{+})$ is a connected domain of $\mathbb{R}$;
it contains at least a point of $G$ and does not intersect with
$\partial\sigma(H_{0})$. Consequently,
$(E-W)(\widetilde{\Sigma}_{k}^{+})$ belongs to $G$ and:
$$\Lambda_{-}\cap\mathbb{R}=G.$$
We fix on $\Lambda_{-}$ a continuous branch of the quasi momentum
$k$. The sign of $\mbox{Im } k$ does not change since $(E-W)(S_{-})$ does
not intersect with $\sigma(H_{0})$. If we define
$\kappa_{i}(\varphi)=k(E-W(\varphi))$, $\mbox{Im }\kappa_{i}$ we can
assume that $\mbox{Im }\kappa_{i}>0$ on $S_{-}$.
\item Now, we consider $\mbox{Re }\kappa_{i}$. According to section
\ref{qm}, we can choose the branch $\kappa_{i}$ such that
$\kappa_{i}(\varphi_{r}^{-}+i0)\in\{0,\pi\}$. First, we study the
case $\kappa_{i}(\varphi_{r}^{-})=0$; this assumption implies two
possibilities.
\begin{enumerate}\item The point $0$ is a minimum for $W$ and the
band $B$ in $(H_{J})$ is in the form $[E_{4l+1},E_{4l+2}]$. The
points $E_{r}$ and $E_{i}$ satisfy $E_{r}=E_{4l+1}$ and
$E_{i}=E_{4l+2}$. There exists a neighborhood $V$ of
$[\varphi_{r}^{-},0]\cup\sigma$ such that $(E-W)(S_{-}\cap
V)\subset\mathbb{C}_{+}\backslash\mathbb{R}$. Actually, in the neighborhood of
$0$, we have $\mbox{Im }(E-W(\varphi))\geq 0$; according to $(H_{W,g})$,
there exists a neighborhood $V$ of $[\varphi_{r}^{-},0]\cup\sigma$
such that $(E-W)(S_{-}\cap V)$ does not intersect $\mathbb{R}$. By
continuity of the mapping $\varphi\mapsto\mbox{Im }(E-W(\varphi))$, the
sign of $\mbox{Im }(E-W(\varphi))$ remains positive on $S_{-}\cap V$.
There exists a branch $k$ of the quasi-momentum such that $$\mbox{Im }
k(\mathcal{E})>0 \textrm{ for }\mbox{Im }\mathcal{E}>0 \textrm{ and }
k(E_{n}+i0)=0,\ k(E_{p}+i0)=\pi.$$ We define
$\kappa_{i}(\varphi)=k(E-W(\varphi))$.
\item The point $0$ is a maximum for $W$ and the band $B$ is in
the form $[E_{4l+3},E_{4l+4}]$; the points $E_{r}$ and $E_{i}$
satisfy $E_{r}=E_{4l+4}$ and $E_{i}=E_{4l+3}$. Let $k$ be the
branch of the quasi-momentum such that $\mbox{Im } k(\mathcal{E})>0$ for
$\mbox{Im }\mathcal{E}<0$, and $k(E_{n})=0$; then $k(E_{p})=\pi$. We
define $\kappa_{i}(\varphi)=k(E-W(\varphi))$.
\end{enumerate}
The case $\kappa_{i}(\varphi_{r}^{-})=\pi$ is similar.
\end{itemize}
This completes the proof of the lemma.
\end{dem}\\
For the sake of clarity, for all the proofs, we shall consider the
case:
\begin{equation}
\label{premcassc} \kappa_{i}(\varphi_{r}^{-}+i0)=0\textrm{ et
}\kappa_{i}(\varphi_{i}-0)=\pi
\end{equation}
The arguments in the second case are similar and we will not give
the details.
\subsubsection{Other branches of the complex momentum}
\label{compmom} \label{detsc} The properties of the complex
momentum near the branch points $\varphi_{i},\
\overline{\varphi_{i}},\ \varphi_{r}^{\pm}$ are
determined by the behavior of $k$ near $E_{r}$ and $E_{i}$.\\
Now, we describe other branches of $\kappa$, which are obtained
from the branch $\kappa_{i}$ (Lemma \ref{dtepos}) by analytic
continuation. We consider the case \eqref{premcassc}.
\begin{itemize}
\item We denote by $\kappa_{g}$ the continuation of $\kappa_{i}$
to the domain $\{\mbox{Re }(\varphi)<\mbox{Re }(\varphi_{r}^{-})\}$.
$\kappa_{g}$ satisfies
$$\begin{array}{c}\mbox{Im }(\kappa_{g}(\varphi))>0\textrm{ for }\{\mbox{Re }(\varphi)<\mbox{Re }(\varphi_{r}^{-})\}\\\mbox{Re }(\kappa_{g})(\varphi)\rightarrow 0\textrm{ as }\mbox{Re }(\varphi)\rightarrow -\infty. \end{array}$$
$\kappa_{g}$ is the continuation of $\kappa_{i}$ through
$(-\infty,\varphi_{r}^{-}]$.
\item We consider the strip $S_{Y}$ cut along
$(\Sigma\backslash\sigma)\cup(\overline{\Sigma}\backslash\overline{\sigma})\cup(-\infty,\varphi_{r}^{-})\cup(\varphi_{r}^{+},+\infty)$.
We always denote by $\kappa_{i}$ the continuation of $\kappa_{i}$
through $C$.
\item On $\{\mbox{Re }(\varphi)>\mbox{Re }(\varphi_{r}^{+})\}$, we fix a
continuous branch $\kappa_{d}$ with the conditions:
$$\begin{array}{c}\mbox{Im }(\kappa_{d}(\varphi))>0\textrm{ for }\{\mbox{Re }(\varphi)>\mbox{Re }(\varphi_{r}^{+})\}\\\mbox{Re }(\kappa_{d})(\varphi)\rightarrow 0\textrm{ as }\mbox{Re }(\varphi)\rightarrow +\infty. \end{array}$$
$\kappa_{d}$ is the continuation of $\kappa_{i}$ through
$\overline{S_{+}}$.
\end{itemize}
Here, we describe the behavior of the different branches of
$\kappa$.
\begin{equation}
\label{kgki} \forall\varphi\in
S_{-},\quad\kappa_{g}(\varphi)=\kappa_{i}(\varphi)\quad;\quad\forall\varphi\in\overline{S_{-}},\quad\kappa_{g}(\varphi)=-\kappa_{i}(\varphi).
\end{equation}
\begin{equation}
\label{kdki} \forall\varphi\in
S_{+},\quad\kappa_{d}(\varphi)=-\kappa_{i}(\varphi)\quad;\quad\forall\varphi\in\overline{S_{+}},\quad\kappa_{d}(\varphi)=\kappa_{i}(\varphi).
\end{equation}
\input{fig6}
\subsection{Stokes lines}
\label{stline} This section is devoted to the description of the
Stokes lines under assumptions $(H_{W,g})$ and $(H_{J})$. We
describe the Stokes lines beginning at $\varphi_{r}^{-}$,
$\varphi_{r}^{+}$, $\varphi_{i}$ and $\overline{\varphi_{i}}$.
Since $W$ is real on the real line, the set of the Stokes lines is
symmetric with respect to the real line.\\
First, $\kappa_{i}$ is real on the interval
$[\varphi_{r}^{-},\varphi_{r}^{+}]\subset\mathbb{R}$; therefore,
$[\varphi_{r}^{-},\varphi_{r}^{+}]$ is a part of a Stokes line
beginning at $\varphi_{r}^{-}$. The two other Stokes lines
beginning at $\varphi_{r}^{-}$ are symmetric with respect to the
real line. We denote by $b$ the Stokes line going upward and by
$\bar{b}$ its symmetric. Similarly, we denote by $a$ and $\bar{a}$
the two other Stokes lines beginning at $\varphi_{r}^{+}$; $a$
goes upwards.\\
Consider the Stokes lines beginning at $\varphi_{i}$. The angles
between the Stokes lines at this point are equal to $2\pi/3$. So,
one of the Stokes lines is situated between $\Sigma$ and
$e^{\frac{2i\pi}{3}}\Sigma$. It is locally going to the right of
$\Sigma$; we denote by $d$ this line. Similarly, we denote by $e$
the Stokes line between $\Sigma$ and $e^{-\frac{2i\pi}{3}}\Sigma$.
Finally, we denote by $c$ the third Stokes line beginning at
$\varphi_{i}$; $c$ is going upwards.\\
By symmetry, we denote by $\bar{c}$, $\bar{d}$ and $\bar{e}$ the
Stokes lines beginning at $\overline{\varphi_{i}}$.\\
We describe the behavior of $a$, $b$, $c$, $d$ and $e$ in the
strip $S_{Y}$. We have represented these lines in figure \ref{ls}.
In this figure, we have precised the values of $\kappa$ in the
branch points.
\begin{lem}
\label{stlinea} We assume that $V$, $W$ and $J$ satisfy $(H_{V})$,
$(H_{W})$ and $(H_{J})$. Then, the Stokes lines described in
figure \ref{ls} have the following properties:
\begin{enumerate}
\item $a$ stays vertical; it intersects $\{\mbox{Im }(\varphi)=Y\}$.
\item $b$ stays vertical; it intersects $\{\mbox{Im }(\varphi)=Y\}$.
\item $d$ intersects $a$ above $\varphi_{r}^{+}$; the segment
between $\varphi_{i}$ and this intersection with $a$ is vertical.
\item $e$ intersects $b$ above $\varphi_{r}^{-}$; the segment
between $\varphi_{i}$ and this intersection with $b$ is vertical.
\item $c$ stays vertical; it intersects $\{\mbox{Im }(\varphi)=Y\}$ and does
not intersect $\sigma$.
\item $a$ and $c$ do not intersect one another in the strip $S_{Y}$.
\item $b$ and $c$ do not intersect one another in the strip $S_{Y}$.
\end{enumerate}
\end{lem}
\begin{dem}
First, we note that a Stokes line can become horizontal only at a
point where $\mbox{Im } \kappa=0$, i.e. at a point of the pre-image of a
spectral band. Besides, a Stokes line beginning at
$\varphi_{1}^{\pm}$ (respectively at $\varphi_{2}$ or
$\overline{\varphi_{2}}$) is locally orthogonal to $i\
\overline{\kappa(\varphi)}$ (respectively $i\
\overline{(\pi-\kappa(\varphi))}$).\\
We first prove 1). According $(H_{_{J}})$, the pre-image of the
spectrum is $[\varphi_{r}^{-},\varphi_{r}^{+}]\cup\sigma$. So, $a$
becomes horizontal only if it intersects $\sigma$. Let us prove by
contradiction that it is impossible. Let us assume that $a$
intersects $\sigma$ in $\varphi_{a}$, then:
$$\mbox{Im }\int_{\varphi_{r}^{+}}^{\varphi_{a}}\kappa(u)du=0=\mbox{Im }\int_{0,\textrm{ along }\sigma}^{\varphi_{a}}\kappa(u)du$$
$$=\int_{0}^{\varphi_{a}}(\mbox{Re }\kappa(u))d(\mbox{Im }(u))\leq-k_{1}(E-W_{-})\mbox{Im }\varphi_{a}<0$$
which is impossible. Therefore, $a$ stays vertical. Moreover, as
$\varphi\rightarrow\infty,\ \varphi\in S_{Y},\
\mbox{Im }(i\bar{\kappa})\rightarrow 0$. Thus, $a$ admits a vertical
asymptote and intersects $\{\mbox{Im }(\varphi)=Y\}$.\\
Similarly, we prove 2).\\
To prove 3), we consider the Stokes line $d$. If $a$ and $d$ do
not intersect one another, then $d$ intersect either $\sigma$ or
$[0,\varphi_{r}^{+}]$. In this case, we denote by $\varphi_{d}$
the intersection between $d$ and $\sigma$ and we have:
$$\mbox{Im }\int_{\varphi_{d}}^{\varphi_{i}}(\kappa(u)-\pi)du=0=\int_{\varphi_{d}}^{\varphi_{i}}\mbox{Re }(\kappa(u)-\pi)d(\mbox{Im }(u))<0$$
Consequently, $d$ and $a$ do not intersect one another. Before its
intersection with $a$, $d$ does not intersect the pre-image of a
spectral band and it stays vertical. We prove similarly the
properties of $e$.\\
We prove now 5). $c$ is going upwards. $c$ does not intersect the
pre-image of a spectral band in
$\{\mbox{Im }\varphi\in]\mbox{Im }\varphi_{i},Y[\}$ and $c$ stays vertical.\\
We prove 6) by contradiction. Let us assume that there is
$\varphi_{a}\in a\cap c$. Then, we compute:
$$\mbox{Im }\int_{0}^{\varphi_{a}}\kappa(u)du=0=\mbox{Im }\int_{\sigma}\kappa(u)du+\mbox{Im }\int_{\varphi_{i}}^{\varphi_{a}}\kappa(u)du$$
First,
$\mbox{Im }\int_{\varphi_{i}}^{\varphi_{a}}\kappa(u)du=\pi\mbox{Im }(\varphi_{a}-\varphi_{i})>0$
and $\mbox{Im }\int_{\sigma}\kappa(u)du=\int_{\sigma}\mbox{Re }\kappa(u)d(\mbox{Im } u)>0$.\\
which is impossible. So, $a$ and $c$ do not intersect one another
in $S_{Y}$.
\end{dem}\\
In the following, we choose $\widetilde{Y}\in]\sup\limits_{E\in
J}\mbox{Im }\varphi_{i}(E),Y[$.
\input{fig7}
\subsection{Construction of a consistent basis with standard
behavior in the neighborhood of the cross} In this section, we
begin with constructing a canonical line near the cross. To do
that, we follow the methods developed in \cite{FK4}.
\subsubsection{General constructions}
We first recall some general geometric tools presented in
\cite{FK4}, section 4.1.
\begin{itemize}
\item We first introduce the idea of enclosing canonical domain.
\begin{defn}
Let $\gamma\subset D$ be a line canonical with respect to
$\kappa$. Denote its ends by $\xi_{1}$ and $\xi_{2}$. Let a domain
$K\subset D$ be a canonical domain corresponding to the triple
$\kappa,\ \xi_{1}$ and $\xi_{2}$. If $\gamma\subset K$, then $K$
is called a canonical domain {\it enclosing $\gamma$}.
\end{defn}
We have the following property:
\begin{lem}\cite{FK2}\\
\label{enccan} One can always construct a canonical domain
enclosing any given compact canonical curve located in an
arbitrarily small neighborhood of that curve.
\end{lem}
Such canonical domains, whose existence is established using this
lemma are called {\it local}.
\item To construct a canonical domain, we need a canonical line to
start with. To construct such a line, we first build pre-canonical
lines made of some ``elementary'' curves. Let $\gamma\subset D$ be
a vertical curve. We call $\gamma$ {\it pre-canonical} if it is a
finite union of bounded segments of canonical lines and/or lines
of Stokes type. The interest of pre-canonical curves is the
following:
\begin{lem}\cite{FK2}\\
\label{precan} Let $\gamma$ be a pre-canonical curve. Denote the
ends of $\gamma$ by $\xi_{a}$ and $xi_{b}$. Fix $V\subset D$, a
neighborhood of $\gamma$ and $V_{a}\subset D$ a neighborhood of
$\xi_{a}$. Then, there exists a canonical line $\tilde{\gamma}$
connecting the point $\xi_{b}$ to some point in $V_{a}$.
\end{lem}
\end{itemize}
\subsubsection{Constructing a canonical line near the cross}
Here, we mimic the construction of \cite{FK4}, section 4.2. We
assume that assumptions $(H_{V})$, $(H_{W,r})$, $(H_{W,g})$ and
$(H'_{J})$ are satisfied. We now explain the construction of a
canonical line going from $\{\mbox{Im }\xi=-Y\}$ to $\{\mbox{Im }\xi=Y\}$. First,
we consider the curve $\beta$ which is the union of the Stokes
line $\bar{b}$, the segment $[\varphi_{r}^{-},0]$ of the real
line, the closed curve
$\sigma_{+}$ and the Stokes line $c$.\\
We now construct $\alpha$ a pre-canonical line close to the line
$\beta$. We prove:
\begin{prop}
\label{cancurva}
Fix $\delta>0$. In the
$\delta$-neighborhood of $\beta$, there exists $\alpha$ a
pre-canonical line with respect to the branch $\kappa$ connecting
$\xi_{1}$ to $\xi_{2}$ and having the following properties:
\begin{itemize}
\item at its upper end, $\mbox{Im }\xi_{2}=Y$,
\item at its lower end, $\mbox{Im }\xi_{1}=-Y$,
\item it goes around the branch points of the complex momentum as
the curve shown in figure \ref{cancurve};
\item it contains a canonical line which stays in $S_{-}$, goes
downward from a point in $S_{-}$ to the curve $\sigma$ and then
continues along this curve until it intersects the real line.
\end{itemize}
\end{prop}
\begin{dem}
The proof of this Proposition is completely similar to the proof
of Proposition 4.2 in \cite{FK4}. It consists in breaking down
$\alpha$ in ``elementary'' segments. We do not give the details.
\end{dem}\\
An immediate consequence of Proposition \ref{cancurva} is the
following result:
\begin{prop}
\label{cancurvb} In arbitrarily small neighborhood of the
pre-canonical line $\alpha$, there exists a canonical line
$\gamma$ which has all the properties of the line $\alpha$ listed
in Proposition \ref{cancurva}.
\end{prop}
\input{fig8}
\subsubsection{Some continuation tools}
In this section, we recall some continuation tools; these tools
are developed in \cite{FK4}.
\begin{enumerate}
\item
Now, we present the continuation lemma on compact domains. We
recall that $q$ is defined in \eqref{racq}.
\begin{lem}
\label{lemcontfin} \cite{FK1}
Let $\varphi_{-}, \varphi_{+},
\varphi_{0}$ be fixed points such that
\begin{itemize}
\item $\mbox{Im }\varphi_{-}=\mbox{Im }\varphi_{+}$;
\item there is no branch point of $\varphi\mapsto\kappa(\varphi)$ on the interval $[\varphi_{-},
\varphi_{+}]$;
\item $\varphi_{0}\in(\varphi_{-} \varphi_{+}),
q(\varphi_{0})\neq 0.$
\end{itemize}
Fix a continuous branch of $\kappa$ on $[\varphi_{-},
\varphi_{+}]$. Let $f(x,\varphi,E,\varepsilon)$, $ f
_{\pm}(x,\varphi,E,\varepsilon)$ be solutions of \eqref{eqp} for
$\varphi\in[\varphi_{-}, \varphi_{+}]$ and $x\in[-X,X]$ satisfying
\eqref{coh} and such that:
\begin{enumerate}
\item $f(x,\varphi,E,\varepsilon)=e^{\frac{i}{\varepsilon}\int_{\varphi_{0}}^{\varphi}\kappa(u)du}(\psi_{+}(x,\varphi,E)+o(1))$
pour $\varphi\in[\varphi_{-}, \varphi_{0}]$ for
$\varphi\in[\varphi_{-}, \varphi_{0}]$ when
$\varepsilon\rightarrow 0$ and the asymptotic is differentiable in
$x$;
\item $f_{\pm}(x,\varphi,E,\varepsilon)=e^{\pm\frac{i}{\varepsilon}\int_{\varphi_{0}}^{\varphi}\kappa(u)du}(\psi_{\pm}(x,\varphi,E)+o(1))$
for $\varphi\in[\varphi_{-}, \varphi_{+}]$ when
$\varepsilon\rightarrow 0$, and the asymptotic is differentiable
in $x$.
\end{enumerate}
Here, $\psi_{\pm}$ are canonical Bloch solutions associated to the
complex momentum $\kappa$.\\
Then,\\
\begin{itemize}
\item if $\mbox{Im }(\kappa(\varphi))>0$ for all $\varphi\in
[\varphi_{-}, \varphi_{+}]$, there exists $C>0$ such that, for
$\varepsilon>0$ small enough,
\begin{equation}
\left|\frac{df}{dx}(x,\varphi,E,\varepsilon)\right|+|f(x,\varphi,E,\varepsilon)|\leq
C
e^{\frac{1}{\varepsilon}\int_{\varphi}^{\varphi_{0}}|\mbox{Im }\kappa(u)|du},\quad\varphi\in
[\varphi_{0}, \varphi_{+}];
\end{equation}
\item if $\mbox{Im }(\kappa(\varphi))<0$ for all
$\varphi\in[\varphi_{-}, \varphi_{+}]$, then
\begin{equation}
f(x,\varphi,E,\varepsilon)=e^{\frac{i}{\varepsilon}\int_{\varphi_{0}}^{\varphi}\kappa(u)du}(\psi_{+}(x,\varphi,E)+o(1)),\quad\varphi\in[\varphi_{0},
\varphi_{+}],
\end{equation}
and the asymptotic is differentiable in $x$.
\end{itemize}
\end{lem}
Intuitively, this lemma means that a function $f$ has the standard
behavior along a horizontal line as long as the leading term of
its asymptotics is growing along that line. For analogous results
with real WKB method, we refer to \cite{Vo}.
\item
The estimate we obtained in Lemma \ref{lemcontfin} can be far from
optimal. The Adjacent Canonical Domain Principle gives a more
precise result:
\begin{prop}\cite{FK3}
\label{adjdom} Assume that a solution $f$ has standard behavior in
either the left hand side or the right hand side of a constant
neighborhood of a vertical curve $\gamma$. Assume that $\gamma$ is
canonical with respect to some branch of the complex momentum.
Then $f$ has standard behavior in any bounded canonical domain
enclosing $\gamma$.
\end{prop}
\item The last tool we shall need in the sequel is the Stokes
Lemma.\\
Notations and assumptions:\\
Assume that $\xi_{0}$ is a branch point of the complex momentum
such that $W'(\xi_{0})\neq 0$. There are three Stokes lines
beginning at $\xi_{0}$. The angles between them at $\xi_{0}$ are
equal to $2\pi/3$. We denote these lines by $\sigma_{1}$,
$\sigma_{2}$ and $\sigma_{3}$, so that $\sigma_{1}$ is vertical at
$\xi_{0}$. Let $V$ be a neighborhood of $\xi_{0}$; assume that $V$
is so small that $\sigma_{1}$, $\sigma_{2}$ and $\sigma_{3}$
divide it into three sectors. We denote them by $S_{1}$, $S_{2}$
and $S_{3}$ so that $S_{1}$ be situated between $\sigma_{1}$ and
$\sigma_{2}$, and the sector $S_{2}$ be between $\sigma_{2}$ and
$\sigma_{3}$ (see figure \ref{lsbp}).\\
We recall now the result:
\begin{lem}\cite{FK4}.\label{stoklemma} Let $V$ be sufficiently
small. Let $f$ be a solution that has standard behavior
$f=e^{\frac{i}{\varepsilon}\int^{\varphi}\kappa(u)du}(\psi_{+}(x,\varphi)+o(1))$
inside the sector $S_{1}\cup\sigma_{2}\cup S_{2}$ of $V$.
Moreover, assume that, in $S_{1}$ near $\sigma_{1}$, one has
$\mbox{Im }\kappa>0$ if $S_{1}$ is to the left of $\sigma_{1}$ and
$\mbox{Im }\kappa<0$ otherwise. Then, $f$ has standard behavior
$f=e^{\frac{i}{\varepsilon}\int^{\varphi}\kappa(u)du}(\psi_{+}(x,\varphi)+o(1))$
inside $V\backslash\sigma_{1}$, the asymptotics being obtained by
analytic continuation from $S_{1}\cup\sigma_{2}\cup S_{2}$ to
$V\backslash\sigma_{1}$.
\end{lem}
\end{enumerate}
\input{fig9}
\subsubsection{Construction of a basis with standard asymptotic
behavior near the cross} We prove the existence of a consistent
basis with standard asymptotic behavior near the canonical line
$\alpha$. Let $\alpha$ be the curve described in Proposition
\ref{cancurvb}. According to Lemma \ref{enccan}, we can construct
a local canonical domain $K_{i}$ enclosing $\alpha$.
\begin{prop}
\label{bcki} Assume that $(H_{V})$, $(H_{W,r})$, $(H_{W,g})$ and
$(H_{J})$ are satisfied. Fix $E_{0}\in J$, $X>1$ and
$\tilde{Y}\in]0,\tilde{Y}[$. Then, there exist a complex
neighborhood $\mathcal{U}_{0}$ of $E_{0}$, a real number
$\varepsilon_{0}>0$ and a function $f_{i}$ satisfying the
following properties:
\begin{itemize}
\item The function $(x,\varphi,E,\varepsilon)\mapsto
f_{i}(x,\varphi,E,\varepsilon)$ is defined on $\mathbb{R}\times
S_{\tilde{Y}}\times\mathcal{U}_{0}\times]0,\varepsilon_{0}[$.
\item For any $x\in\mathbb{R}$, for any
$\varepsilon\in]0,\varepsilon_{0}[$, the function
$((\varphi,E)\mapsto f_{i}(x,\varphi,E,\varepsilon))$ is analytic
on $S_{\tilde{Y}}\times\mathcal{U}_{0}$.
\item For $(x,\varphi,E)\in [-X,X]\times
K_{i}\times\mathcal{U}_{0}$, the function $f_{i}$ has the
asymptotic behavior:
\begin{equation}
\label{asca}
f_{i}(x,\varphi,E,\varepsilon)=e^{\frac{i}{\varepsilon}\int_{0}^{\varphi}\kappa(u)
du}\left(\psi_{\pm}(x,\varphi,E)+o(1)\right),\quad\varepsilon\rightarrow
0.
\end{equation}
\item The asymptotics \eqref{asca} are uniform in $(x,\varphi,E)\in [-X,X]\times K_{i}\times\mathcal{U}_{0}$.
\item The asymptotics \eqref{asca} can be differentiated once in $x$.
\item There exists a real number $\sigma_{i}\in\{-1,1\}$ such that the function $f_{i}$ satisfies the relation:
$$w(f_{i},f_{i}^{*})=w(f_{i}(\cdot,\varphi,E,\varepsilon),f_{i}(\cdot,\overline{\varphi},\overline{E},\varepsilon))=\sigma_{i}(k'_{i}w_{i})(E-W(0))$$
\end{itemize}
\end{prop}
The end of this section is devoted to the proof of Proposition
\ref{bcki}. This Proposition mainly follows from Theorem
\ref{finwkbthm}.
\subsubsection{Existence of $f_{i}$}
The domain $K_{i}$ is a local canonical domain. According to
Theorem \ref{finwkbthm}, we can build a function $f_{i}$ such
that, on $K_{i}$, $f_{i}$ has the following asymptotic behavior:
$$f_{i}\sim
e^{\frac{i}{\varepsilon}\int^{\varphi}\kappa_{i}}\psi_{+}.$$ Let
us normalize $f_{i}$ in $0$.
\subsubsection{Computation of the Wronskian $w(f_{i},f_{i}^{*})$}
\label{precsigma} To finish the proof of Proposition \ref{bcki},
it remains to
compute $w(f_{i},f_{i}^{*})$.\\
Let $R_{+}$ be a small enough rectangle to the left of
$\alpha_{+}$, so that $R_{+}\subset K_{i}\cap S_{\tilde{Y}}$. We
define $R=R_{+}\cup R_{-}$; we study the behavior of $f_{i}$ and
$(f_{i})^{*}$ in $R$.
\begin{itemize}
\item First, by construction, in $R_{+}$, the function $f_{i}$
satisfies:
$$ f_{i}\sim
e^{\frac{i}{\varepsilon}\int_{0}^{\varphi}\kappa_{i}}\psi_{+}^{i}.$$
\item To the right of $\alpha_{-}$, the function $f_{i}$ satisfies
$$ f_{i}\sim
e^{\frac{i}{\varepsilon}\int_{0}^{\varphi}\kappa_{i}}\psi_{+}^{i},$$
with $\mbox{Im }\kappa_{i}<0$. According to Lemma \ref{lemcontfin}, we
know that, in $\bar{S_{-}}$, the function $f_{i}$ admits the
asymptotic behavior:
$$ f_{i}\sim
e^{\frac{i}{\varepsilon}\int_{0}^{\varphi}\kappa_{i}}\psi_{+}^{i}.$$
\item Thus, the function $f_{i}$ has the standard asymptotic behavior in
$R$.
\item Now, we study the behavior of $f_{i}^{*}$. To do that, we
start with describing the main objects related to $\kappa_{i}$ in
$R$. Let $k_{i}$ be the branch of the quasi-moment of
\eqref{espc}, analytically continued through $[E_{r},E_{i}]$ and
satisfying:
\begin{equation*}
k_{i}(E_{r})=0\quad\textrm{ and }\quad k_{i}(E_{i})=\pi
\end{equation*}
$k_{i}$ is real on $[E_{r},E_{i}]$. Therefore, $k_{i}$ satisfies:
$$k_{i}(\overline{\mathcal{E}})=\overline{k_{i}(\mathcal{E})}.$$
The branch $\kappa_{i}$ satisfies
$\kappa_{i}(\varphi)=k_{i}(E-W(\varphi))$. The associated
canonical Bloch solutions $\Psi_{\pm}^{i}$ are such that:
$$\overline{\Psi_{+}^{i}(x,\overline{\varphi})}=\Psi_{-}^{i}(x,\varphi).$$
Therefore, we have in $R$:
\begin{equation}
\label{kappai} \kappa_{i}^{*}(\varphi)=\kappa_{i}(\varphi)\quad
(\Psi_{+}^{i})^{*}(\varphi)=\Psi_{-}^{i}(\varphi)\quad(\omega_{+}^{i})^{*}(\varphi)=\omega_{-}^{i}(\varphi)\quad
\forall\varphi\in R.
\end{equation}
Besides, since $k'_{i}$ is real on the band, there exists a real
number $\sigma_{i}\in\{-1,1\}$ such that:
\begin{equation}
q_{i}^{*}(\varphi)=\sigma_{i}q_{i}(\varphi).
\end{equation}
We shall precise this coefficient in section \ref{deco}.\\
We compute:
$$w(f_{+}^{i}(\cdot,\varphi,E,\varepsilon),(f_{+}^{i})^{*}(\cdot,\varphi,E,\varepsilon))=q_{i}(0)q_{i}^{*}(0)w(\Psi_{+}^{i}(\cdot,0),\Psi_{-}^{i}(\cdot,0))g(\varphi,E,\varepsilon).$$
Since
$\overline{w(\Psi_{+}^{i}(\cdot,0),\Psi_{-}^{i}(\cdot,0))}=-w(\Psi_{+}^{i}(\cdot,0),\Psi_{-}^{i}(\cdot,0))$,
the term $g(\varphi,E,\varepsilon)$ satisfies:
$$g^{*}(\varphi,E,\varepsilon)=\overline{g(\bar{\varphi},\bar{E},\varepsilon)}=g(\varphi,E,\varepsilon),$$
$$g(\varphi,E,\varepsilon)=[1+o(1)].$$
Since the Wronskian is analytic and $\varepsilon$-periodic, this
asymptotic is valid in $S_{\widetilde{Y}}$. \\
Since $g^{*}=g$ and $g=[1+o(1)]$, there exists an analytic
function $(\varphi,E)\mapsto h(\varphi,E,\varepsilon)$ on
$S_{\widetilde{Y}}\times\mathcal{U}$ such that:\\
- $g(\varphi,E,\varepsilon)=h(\varphi,E,\varepsilon)h^{*}(\varphi,E,\varepsilon)$,\\
- $h(\varphi,E,\varepsilon)=[1+o(1)]$.\\
We slightly deform $f_{i}$, i.e., we replace $f_{i}$ by
$\frac{f_{i}}{h(\varphi,E,\varepsilon)}$; the basis
$\{f_{i},f_{i}^{*}\}$ is consistent.
\end{itemize}
This ends the proof of Proposition \ref{bcki}.\\
\section{Consistent Jost solutions of \eqref{eqp}}
\label{scattheory} This section is devoted to the proof of the
following result.
\begin{prop}
\label{propconstinf} We assume that $(H_{V})$, $(H_{W,r})$ and
$(H_{J}^{0})$ are satisfied. Fix $X>1$ and $\lambda>1$. Then,
there exist a complex neighborhood
$\mathcal{V}=\overline{\mathcal{V}}$ of $J$, a real
$\varepsilon_{0}>0$, a constant $C>0$, two complex numbers
$m_{g},\ m_{d}$ and two functions
$(x,\varphi,E,\varepsilon)\mapsto
h_{-}^{g}(x,\varphi,E,\varepsilon)$,
$(x,\varphi,E,\varepsilon)\mapsto
h_{+}^{d}(x,\varphi,E,\varepsilon)$ such that, if we define
$$B_{\varepsilon}^{g}=\left\{\varphi\in S_{Y}\ ;\
\mbox{Re }\varphi<-C\varepsilon^{-\frac{\lambda}{s-1}}\right\}\textrm{ et
}B_{\varepsilon}^{d}=\left\{\varphi\in S_{Y}\ ; \
\mbox{Re }\varphi>C\varepsilon^{-\frac{\lambda}{s-1}}\right\},$$ then
\begin{itemize}
\item The functions $(x,\varphi,E,\varepsilon)\mapsto
h_{-}^{g}(x,\varphi,E,\varepsilon)$ and
$(x,\varphi,E,\varepsilon)\mapsto
h_{+}^{d}(x,\varphi,E,\varepsilon)$ are clearly defined and
consistent on $\mathbb{R}\times S_{Y}\times
\mathcal{V}\times]0,\varepsilon_{0}[$.
\item For any $x\in[-X,X]$ and $\varepsilon\in]0,\varepsilon_{0}[$, $(\varphi,E)\mapsto
h_{-}^{g}(x,\varphi,E,\varepsilon)$ and $(\varphi,E)\mapsto
h_{+}^{d}(x,\varphi,E,\varepsilon)$ are analytic on $S_{Y}\times
\mathcal{V}$.
\item The function $x\mapsto h_{-}^{g}(x,\varphi,E,\varepsilon)$ (resp. $x\mapsto h_{+}^{d}(x,\varphi,E,\varepsilon)$)
is a basis of $\mathcal{J}_{-}$ (resp. $\mathcal{J}_{+}$).
\item The functions $h_{-}^{g}$ and $h_{+}^{d}$ have the following asymptotic behavior:
\begin{equation}
\label{as1}
h_{-}^{g}(x,\varphi,E,\varepsilon)=e^{\frac{-i}{\varepsilon}\int_{m_{g}}^{\varphi}\kappa(u)du}\psi_{-}(x,\varphi,E)(1+R_{g}(x,\varphi,E,\varepsilon)),
\end{equation}
and
\begin{equation}
\label{asda1}
h_{+}^{d}(x,\varphi,E,\varepsilon)=e^{\frac{i}{\varepsilon}\int_{m_{d}}^{\varphi}\kappa(u)du}\psi_{+}(x,\varphi,E)(1+R_{d}(x,\varphi,E,\varepsilon)),
\end{equation}
where
\begin{itemize}
\item $R_{g}$ and $R_{d}$ satisfy:
$$\exists M>0,\quad\forall\varepsilon\in]0,\varepsilon_{0}[,\quad,\forall
x\in[-X,X],\quad\forall E\in\mathcal{V},\quad\forall\varphi\in
B_{\varepsilon}^{g},\quad|R_{g}(x,\varphi,E,\varepsilon)|\leq\frac{M}{\varepsilon|\mbox{Re }\varphi|^{s-1}},$$
$$\exists M>0,\quad\forall\varepsilon\in]0,\varepsilon_{0}[,\quad,\forall
x\in[-X,X],\quad\forall E\in\mathcal{V},\quad\forall\varphi\in
B_{\varepsilon}^{d},\quad|R_{d}(x,\varphi,E,\varepsilon)|\leq\frac{M}{\varepsilon|\mbox{Re }\varphi|^{s-1}}.$$
\item The functions $\psi_{+}$ and $\psi_{-}$ are the Bloch canonical solutions
of the periodic equation \eqref{espa} defined in section
\ref{cansolbloch}.
\end{itemize}
\item The asymptotics (\ref{as1}) and (\ref{asda1}) may be differentiated
once in $x$.
\item There exist two real numbers $\sigma_{g}\in\{-1,1\}$, $\sigma_{d}\in\{-1,1\}$, an integer $p$ and two functions $E\mapsto\alpha_{g}(E)$ and $E\mapsto\alpha_{d}(E)$ such that:
\begin{enumerate}
\item
For any $\varepsilon\in]0,\varepsilon_{0}[$, $x\in\mathbb{R}$,
$E\in\mathcal{V}$,et $\varphi\in S_{Y}$ ,we have:
\begin{equation}
\label{starg}
\overline{\alpha_{g}(E)h_{-}^{g}(x,\overline{\varphi},\overline{E},\varepsilon)}=i\sigma_{g}e^{-\frac{i}{\varepsilon}2p\pi
x} \alpha_{g}(E)h_{-}^{g}(x,\varphi,E,\varepsilon)
\end{equation}
\begin{equation}
\label{stard}
\overline{\alpha_{d}(E)h_{+}^{d}(x,\overline{\varphi},\overline{E},\varepsilon)}=i\sigma_{d}e^{\frac{i}{\varepsilon}2p\pi
x} \alpha_{d}(E)h_{+}^{d}(x,\varphi,E,\varepsilon)
\end{equation}
\item The functions $\alpha_{g}$ and $\alpha_{d}$ are analytic
and given by \eqref{renormconstg} and \eqref{renormconstd}. They
do not vanish on $\mathcal{V}$.
\end{enumerate}
\end{itemize}
\end{prop}
We shall construct some consistent Jost solutions of \eqref{eqp}.
To do that, we regard equation \eqref{eqp} as a perturbation of
equation \eqref{esp} with $\mathcal{E}=E$. We adapt the construction of
Jost functions developed in \cite{Fi1, New}. Precisely, we look
for solutions of \eqref{esp} in the form :
$$F_{-}^{g}=e^{-
ik(E)\varphi/\varepsilon}\psi_{-}^{0}(x,E)(1+o(1)),\quad
x\rightarrow -\infty,$$
$$ F_{+}^{d}=e^{ik(E)\varphi/\varepsilon}\psi_{+}^{0}(x,E)(1+o(1)),\quad x\rightarrow +\infty.$$
Since the functions $(x,\varphi,E,\varepsilon)\mapsto e^{\pm
ik(E)\varphi/\varepsilon}\psi_{\pm}(x,E)$ are consistent, they
allow us to construct a consistent resolvent for the periodic
equation. Using this property and the fact that equation
\eqref{eqp} is invariant by the consistency transformation
$(x,\varphi)\mapsto(x-1,\varphi+\varepsilon)$, we obtain the
consistency of the Jost functions.
\subsection{Construction of the Jost functions}
\label{scathyp} We start with constructing $F_{-}^{g}$. The
construction of $F_{+}^{d}$ is similar. Since the parameter $E$
lies in the neighborhood of a gap, $\mbox{Im } k(E)$ is non zero; the
function $F_{-}^{g}$ is therefore exponentially decreasing and
goes to zero as $x$ goes to $-\infty$. Such a solution is called
recessive.
\subsubsection{}
On a small enough complex neighborhood of $J$,
$\mathcal{V}=\overline{\mathcal{V}}$, one can fix a determination
$k$ of the quasi-momentum such that:
$$\mbox{Im } k(E)\geq\beta>0,\quad\forall E\in\mathcal{V}.$$
Fix $m_{g}$ in $S_{Y}$ such that:
\begin{itemize}
\item The point $m_{g}$ is not a branch point of $\kappa$.
\item It satisfies $\mbox{Im } m_{g}>0$, $k'_{E}(m_{g})\neq 0$.
\item The domain $\{\varphi\in S_{Y}\ ;\ \mbox{Re }(\varphi-m_{g})<0\textrm{
and }\mbox{Im }(\varphi-m_{g})>0\}$ does not contain any branch point of
$\kappa$.
\end{itemize}
We define $E_{g}=E-W(m_{g})$.
We denote by $\psi_{\pm}^{0}$
the analytic Bloch solutions of equation \eqref{esp} normalized at
the point $E_{g}$ ( $k'(E_{g}))\neq 0$). These solutions are
constructed in Lemma \ref{anasol}.
$$ \psi_{\pm}^{0}(x,E)=e^{\pm
ik(E)x}p_{\pm}^{0}(x,E)\quad\textrm{with}\quad
p_{\pm}^{0}(x+1,E)=p_{\pm}^{0}(x,E).$$
We define :
$$\widetilde{\psi_{\pm}}(x,\varphi,E,\varepsilon)=e^{\pm
ik(E)(x+\frac{\varphi}{\varepsilon})}p_{\pm}^{0}(x,E)=e^{\pm
ik(E)\frac{\varphi}{\varepsilon}}\psi_{\pm}^{0}(x,E).$$ We
consider the resolvent $R$ of $H_{0}$ :
$$(Rg)(x)=-\int_{-\infty}^{x}\frac{\psi_{+}^{0}(x,E)\psi_{-}^{0}(x',E)-\psi_{+}^{0}(x',E)\psi_{-}^{0}(x,E)}{(k'w_{0})(E_{g})}g(x')dx'$$
\subsection{}
Since $\widetilde{\psi_{-}}$ goes to zero as $x$ goes to
$-\infty$, we look for a recessive consistent solution
$\widetilde{f}$ of \eqref{eqp} in the form :
\begin{equation}
\label{eqres}
\tilde{f}(x,\varphi,E,\varepsilon)=\widetilde{\psi_{-}}(x,\varphi,E,\varepsilon)+R[W(\varepsilon
x+\varphi)\tilde{f}(x,\varphi,E,\varepsilon)].
\end{equation}
We define
$\tilde{f}(x,\varphi,E,\varepsilon)=e^{-ik(E)(x+\frac{\varphi}{\varepsilon})}f(x,\varphi,E,\varepsilon)$;
equation \eqref{eqres} is transformed into:
\begin{equation}
\label{eqres_a}
f(x,\varphi,E,\varepsilon)=p_{-}^{0}(x,E)+\int_{-\infty}^{x}A(x,x',E)W(\varepsilon
x'+\varphi)f(x',\varphi,E,\varepsilon)dx'
\end{equation}
where the function $A$ satisfies:
\begin{equation}
\label{noy}
A(x,x',E)=\frac{e^{2ik(E)(x-x')}p_{+}^{0}(x,E)p_{-}^{0}(x',E)-p_{+}^{0}(x',E)p_{-}^{0}(x,E)}{(k'w_{0})(E_{g})}.
\end{equation}
Since $\mbox{Im } k(E)\geq \beta>0$ for $E\in\mathcal{V}$, there exists a
constant $C>0$ such that:
\begin{equation}
\forall x>x',\quad\forall E\in\mathcal{V},\quad |A(x,x',E)|\leq C
\end{equation}
\subsubsection{}
Fix $X_{0}\in\mathbb{R}$ and $a>0$. If $I$ is a real interval, we define:
$$R_{I}=\{\varphi\in S_{Y}\ ;\ \mbox{Re }\varphi\in I\}.$$
Let $B((-\infty,X_{0}]\times R_{[-a,a]})$ the set of bounded
functions $\{f\ :\ (x,\varphi)\mapsto f(x,\varphi)\}$ on
$(-\infty,X_{0}]\times R_{[-a,a]}$. The set
$B((-\infty,X_{0}]\times R_{[-a,a]})$ equipped with the norm
$$
\|f\|_{\infty}=\sup\limits_{x\in(-\infty,X_{0}],\mbox{Re }\varphi\in[-a,a]}|f(x,\varphi)|$$
is a Banach space.\\
We define the integral operator $T_{E}$ by:
$$\begin{array}{ccccc}T_{E}:&B((-\infty,X_{0}]\times R_{[-a,a]})&\rightarrow &B((-\infty,X_{0}]\times R_{[-a,a]})&\\
&f&\mapsto &F
\end{array}$$
\begin{equation}
\label{opea} \textrm{ where }
F(x,\varphi)=\int_{-\infty}^{x}A(x,x',E)W(\varepsilon
x'+\varphi)f(x',\varphi)dx'.
\end{equation}
The operator $T_{E}$ is a bounded operator on
$B((-\infty,X_{0}]\times R_{[-a,a]})$ and satisfies the estimate:
$$\forall x\in(-\infty,X_{0}],\ \forall\varphi\in R_{[-a,a]},\quad |T_{E}(f)(x,\varphi)|\leq
C\|f\|_{\infty}\int_{-\infty}^{x}|W(\varepsilon x'+\mbox{Re }
(\varphi)+i\mbox{Im }(\varphi))|dx'.$$
$$\|T_{E}(f)\|_{\infty}\leq\frac{M}{\varepsilon}\sup\limits_{x\in(-\infty,X_{0}],\mbox{Re }\varphi\in[-a,a]}\frac{1}{|\varepsilon x+\mbox{Re }(\varphi)|^{s-1}}.$$
\subsubsection{} Fix $\lambda>1$. There exists a constant $C>0$ such that:
$$ |X_{0}|>
C\varepsilon^{-\frac{\lambda
s}{s-1}}\Rightarrow\||T_{E}\||<\varepsilon^{s(\lambda-1)}.$$ We
rewrite \eqref{eqres_a} in the form:
\begin{equation}
\label{eqres_b} (1-T_{E})f=p_{-}^{0}(x,E)
\end{equation}
The operator is then invertible on $B((-\infty,X_{0}]\times
R_{[-a,a]})$. We define:
\begin{equation}
\label{eqresc}
F_{-}^{g}(x,\varphi,E,\varepsilon)=(1-T_{E})^{-1}p_{-}^{0}(x,\varphi,E,\varepsilon)
\end{equation}
We now give some properties of $F_{-}^{g}$.
\subsection{Properties of $F_{-}^{g}$}
\subsubsection{Asymptotic behavior in $x$}
Substituting $$(1-T_{E})^{-1}=1-(1-T_{E})^{-1}T_{E}$$ in equation
(\ref{eqresc}), we obtain:
\begin{equation}
\label{eqresd}
F_{-}^{g}(x,\varphi,E,\varepsilon)=e^{-ik(E)\varphi/\varepsilon}\psi_{-}(x,E)(1+R_{g}(x,\varphi,E,\varepsilon)),
\end{equation}
with
$$
|R_{g}(x,\varphi,E,\varepsilon)|\leq\frac{M}{\varepsilon|\varepsilon
x|^{s-1}},$$ for $x\in(-\infty,X_{0}]$ and $\varphi\in
R_{[-a,a]}$.\\
The function $F_{-}^{g}$ is therefore in the Jost subspace
$\mathcal{J}_{-}$ of equation \eqref{eqp}.
\subsubsection{Study of the consistency}
We assume that $a>1$ and $\varepsilon<1$. We now prove that the
function $F_{-}^{g}$
is consistent.\\
We denote by $G$ the function:
$$G\ :\ (x,\varphi,E,\varepsilon)\mapsto
G(x,\varphi,E,\varepsilon)=F_{-}^{g}(x+1,\varphi-\varepsilon,E,\varepsilon)$$
$G$ is defined for $x\in(-\infty,X_{0}-1]$ and $\varphi\in
R_{[-a+1,a-1]}$. Moreover, the function $G$ belongs to $B((-\infty,X_{0}-1]\times R_{[-a+1,a-1]})$.\\
We define the operator:
$$\begin{array}{ccccc}\widetilde{T_{E}}:&B((-\infty,X_{0}-1]\times R_{[-a+1,a-1]})&\rightarrow &B((-\infty,X_{0}-1]\times R_{[-a+1,a-1]})&\\
&f&\mapsto &F
\end{array}$$
\begin{equation}
\label{opeb} \textrm{ where }
F(x,\varphi)=\int_{-\infty}^{x}A(x,x',E)W(\varepsilon
x'+\varphi)f(x',\varphi)dx'.
\end{equation}
Since $B((-\infty,X_{0}]\times R_{[-a,a]})\subset
B((-\infty,X_{0}-1]\times R_{[-a+1,a-1]})$ and according to
equations \eqref{opea} and \eqref{opeb}, the operator
$\widetilde{T_{E}}$ is an extension of the operator $T_{E}$. Let
us denote by $\widetilde{F_{-}^{g}}$ the restriction of
$F_{-}^{g}$ to $(-\infty,X_{0}-1]\times R_{[-a+1,a-1]}$.\\
We compute in $ B((-\infty,X_{0}-1]\times R_{[-a+1,a-1]})$:
$$(\widetilde{T_{E}}(G))(x,\varphi,E,\varepsilon)=(T_{E}(F_{-}^{g}))(x+1,\varphi-\varepsilon,E,\varepsilon)$$
This leads to:
$$((1-\widetilde{T_{E}})(G))(x+1,\varphi-\varepsilon,E,\varepsilon)=((1-T_{E})(F_{-}^{g}))(x+1,\varphi-\varepsilon,E,\varepsilon)=p^{0}_{-}(x+1,E)=p^{0}_{-}(x,E),$$
The functions $\widetilde{F_{-}^{g}}$ and $G$ satisfy the
relation:
$$((1-\widetilde{T_{E}})(G))=((1-\widetilde{T_{E}})(\widetilde{F_{-}^{g}})).$$
For a sufficiently small $\varepsilon_{0}$, the operator
$\widetilde{T_{E}}$ satisfies, for any
$\varepsilon\in]0,\varepsilon_{0}[$:
$$\||\widetilde{T_{E}}\||<\frac{1}{2}.$$
The operator $(1-\widetilde{T_{E}})$ is invertible in
$B((-\infty,X_{0}-1]\times R_{[-a+1,a-1]})$ and:
$$\widetilde{F_{-}^{g}}=G.$$
For $\varphi\in R_{[-a+1,a-1]}$, the functions $F_{-}^{g}$ and $G$
coincide on $(-\infty,X_{0}-1]$; according to the Cauchy-Lipschitz
Theorem, they coincide for $x\in\mathbb{R}$. Fix $x\in\mathbb{R}$; $F_{-}^{g}$ and
$G$ coincide for $\varphi\in R_{[-a+1,a-1]}$. By analyticity, they
are equal for $\varphi\in S_{Y}$.
\subsubsection{Asymptotic behavior in $\varphi$}
We use now the consistency of $F_{-}^{g}$ to compute its
asymptotics as $\mbox{Re }\varphi$ goes to $-\infty$. Fix $X>0$. We study
$F_{-}^{g}$ for $x\in[-X,X]$. The function $F_{-}^{g}$ is
consistent, and:
$$F_{-}^{g}(x,\varphi,E,\varepsilon)=F_{-}^{g}(x+\frac{[\mbox{Re }(\varphi)]}{\varepsilon},\varphi-[\mbox{Re }(\varphi)],E,\varepsilon)$$
$$=e^{-ik(E)(x+\varphi/\varepsilon)}p_{-}^{0}(x,E)\left(1+O(\frac{1}{\varepsilon|\varepsilon x+[\mbox{Re }\varphi]|^{s-1}})\right).$$
As a result, there exists a constant $C$ such that:
$$\mbox{Re }\varphi<
-C\varepsilon^{-\frac{\lambda}{s-1}}\Rightarrow
F_{-}^{g}(x,\varphi,E,\varepsilon)=e^{-ik(E)(x+\varphi/\varepsilon)}p_{-}(x,E)(1+\widetilde{R_{g}}(x,\varphi,E,\varepsilon)),$$
where
$$|\widetilde{R_{g}}(x,\varphi,E,\varepsilon)|\leq\frac{M}{\varepsilon|\mbox{Re }\varphi|^{s-1}},$$
for $x\in[-X,X]$ and $\mbox{Re }\varphi<
-C\varepsilon^{-\frac{\lambda}{s-1}}$.\\
We define $B_{\varepsilon}^{g}=\{\varphi\in S_{Y}\ ;\ \mbox{Re }\varphi<
-C\varepsilon^{-\frac{\lambda}{s-1}}\}$.
\subsection{Renormalization of $F_{-}^{g}$}
We now renormalize $F_{-}^{g}$. We define:
\begin{equation}
\label{renorm}
f_{-}^{g}(x,\varphi,E,\varepsilon)=e^{-\frac{i}{\varepsilon}\int_{m_{g}}^{-\infty}[\kappa(u)-k(E)]du}F_{-}^{g}(x,\varphi,E,\varepsilon),
\end{equation}
where the integral $\int_{m_{g}}^{-\infty}[\kappa(u)-k(E)]du$ is
taken in the upper half plane. The function
$E\mapsto\int_{m_{g}}^{-\infty}[\kappa-k(E)]$ is analytic on
$\mathcal{V}$. For $\varphi\in B_{\varepsilon}^{g}$, we have:
$$f_{-}^{g}(x,\varphi,E,\varepsilon)=e^{-\frac{i}{\varepsilon}\int_{m_{g}}^{\varphi}[\kappa(u)-k(E)]du}e^{-\frac{i}{\varepsilon}\int_{\varphi}^{-\infty}[\kappa-k(E)]}e^{-\frac{ik(E)\varphi}{\varepsilon}}\psi_{-}^{0}(x,E)(1+o(1))$$
Since the function $\psi_{-}$ is analytic and since
$W(\varphi)=O(\varepsilon^{\frac{\lambda s}{s-1}})$ for
$\varphi\in B_{\varepsilon}^{g}$, we get:
\begin{equation}
\forall\varphi\in B_{\varepsilon}^{g},\quad
\psi_{-}(x,\varphi,E)=\psi_{-}^{0}(x,E-W(\varphi))=\psi_{-}^{0}(x,E)(1+o(1))
\end{equation}
We finally obtain that, for $x\in[-X,X]$ and $\varphi\in
B_{\varepsilon}^{g}$:
\begin{equation*}
f_{-}^{g}(x,\varphi,E,\varepsilon)=e^{-\frac{i}{\varepsilon}\int_{m_{g}}^{\varphi}\kappa(u)du}\psi_{-}(x,\varphi,E)(1+o(1))
\end{equation*}
\subsubsection{Symmetries}
Let $\gamma$ be a complex path and $f$ be an analytic function on
$\gamma$. We have:
\begin{equation}
\label{symint}
\int_{\gamma}f(z)dz=\overline{\int_{\overline{\gamma}}f^{*}(z)dz}.
\end{equation}
Since $J$ satisfies $(H_{J}^{0})$, according to equation
\eqref{kl}, there exists an integer $p$ such that:
\begin{equation}
\label{entierp} k(E)+k^{*}(E)=2 p \pi.
\end{equation}
We recall that the functions $\omega_{\pm}$ associated to $\kappa$
are defined by equation \eqref{omega}. We consider a path
$\widetilde{\gamma}_{g}$ such that:
\begin{itemize}
\item The path $\widetilde{\gamma}_{g}$ connects $\overline{m_{g}}$ to
$m_{g}$ and is symmetric with respect to the real axis.
\item The path $\widetilde{\gamma}_{g}$ does not contain any
branch point of $\kappa$ and any pole of $\omega_{\pm}$.
\end{itemize}
We fix a continuous determination $q_{g}$ of $\sqrt{k'_{E}}$ on
$\gamma_{g}$. According to relation \eqref{kl}, we have
$(k^{*})'=-k'$, which implies that there exists
$\sigma_{g}\in\{-1,1\}$ such that:
\begin{equation}
\label{symraccarrg} q_{g}^{*}=i\sigma_{g}q_{g}
\end{equation}
The functions $\psi_{\pm}(x,\varphi,E,m_{g})$ satisfy the
relation:
\begin{equation}
\label{symblochnorm}
\psi_{\pm}^{*}(x,\varphi,E,m_{g})=i\sigma_{g}e^{\pm \frac{2i p\pi
x}{\varepsilon}}e^{\int_{\widetilde{\gamma_{g}}}\omega_{\pm}^{g}}\psi_{\pm}^{*}(x,\varphi,E,m_{g}).
\end{equation}
Besides, equations \eqref{symint} and \eqref{symgap} lead to the
following relations:
\begin{equation}
\overline{\int_{\widetilde{\gamma_{g}}}\omega_{+}}=-\int_{\widetilde{\gamma_{g}}}\omega_{+}\
;\
\overline{\int_{\widetilde{\gamma_{g}}}\omega_{-}}=-\int_{\widetilde{\gamma_{g}}}\omega_{-}.
\end{equation}
According to $(H_{W,r})$, $W^{*}=W$. By using \eqref{noy}, we
compute:
$$ A(x,x',E)=\overline{A(x,x',\bar{E})}.$$
The operator $T_{E}$ satisfies:
$$T_{E}(f^{*})=[T_{E}(f)]^{*}.$$
Consequently, according to \eqref{eqres_b} and
\eqref{symblochnorm}, we obtain that, for $E$ in $\mathcal{V}$,
$x$ in $\mathbb{R}$ and $\varphi$ in $B_{\varepsilon}^{g}$,
\begin{equation}
(F_{-}^{g})^{*}(x,\varphi,E,\varepsilon)=\overline{F_{-}^{g}(x,\bar{\varphi},\bar{E},\varepsilon)}=i\sigma_{g}e^{-\frac{i}{\varepsilon}2p\pi
x}e^{\int_{\widetilde{\gamma_{g}}}\omega_{-}^{g}}F_{-}^{g}(x,\varphi,E,\varepsilon).
\end{equation}
This leads to:
$$(h_{-}^{g})^{*}=i\sigma_{g}e^{-\frac{i}{\varepsilon}2p\pi x}\frac{\alpha_{g}(E)}{\alpha_{g}^{*}(E)}h_{-}^{g},$$
where
\begin{equation}
\label{renormconstg}
\alpha_{g}(E)=e^{-\frac{i}{2\varepsilon}\left(\int_{\widetilde{\gamma}_{g}}(\kappa(u)-p\pi)du+p\pi(m_{g}+\overline{m_{g}})\right)}e^{\frac{1}{2}\int_{\widetilde{\gamma}_{g}}\omega_{-}^{g}}
\end{equation}
Similarly, we fix $m_{d}$ in $S_{Y}$ such that:
\begin{itemize}
\item The point $m_{d}$ is not a branch point of $\kappa$.
\item It satisfies $\mbox{Im } m_{d}>0$, $k'_{E}(m_{d})\neq 0$.
\item The domain $\{\varphi\in S_{Y}\ ;\ \mbox{Re }(\varphi-m_{d})>0\textrm{
and }\mbox{Im }(\varphi-m_{d})>0\}$ does not contain any branch point of
$\kappa$.
\end{itemize}
We define $E_{d}=E-W(m_{d})$ and we define the function
$h_{+}^{d}$ by:
\begin{equation}
\label{renormd}
h_{+}^{d}(x,\varphi,E,\varepsilon)=e^{\frac{i}{\varepsilon}\int_{m_{d}}^{+\infty}[\kappa(u)-k(E)]du+\frac{ik(E)m_{d}}{\varepsilon}}F_{+}^{d}(x,\varphi,E,\varepsilon)
\end{equation}
where the integral $\int_{m_{d}}^{+\infty}[\kappa(u)-k(E)]du$ is
taken
in the upper half plane.\\
We consider the path $\widetilde{\gamma}_{d}$ such that:
\begin{itemize}
\item The path $\widetilde{\gamma}_{d}$ connects $\overline{m_{d}}$ to
$m_{d}$ and is symmetric with respect to the real axis.
\item The path $\widetilde{\gamma}_{d}$ does not contain any
branch point of $\kappa$ and any pole of $\omega_{\pm}$.
\end{itemize}
We fix a continuous branch $q_{d}$ of $\sqrt{k'_{E}}$ on
$\gamma_{d}$. There exists a real number $\sigma_{d}$ such that:
\begin{equation}
\label{symraccarrd} q_{d}^{*}=i\sigma_{d}q_{d}
\end{equation}
The function $h_{+}^{d}$ satisfies:
$$(h_{+}^{d})^{*}=i\sigma_{d}e^{\frac{i}{\varepsilon}2p\pi x}\frac{\alpha_{d}(E)}{\alpha_{d}^{*}(E)}h_{+}^{d},$$
where
\begin{equation}
\label{renormconstd}
\alpha_{d}(E)=e^{\frac{i}{2\varepsilon}\left(\int_{\widetilde{\gamma}_{d}}(\kappa(u)-p\pi)du+p\pi(m_{d}+\overline{m_{d}})\right)}e^{\frac{1}{2}\int_{\widetilde{\gamma}_{d}}\omega_{+}^{d}}
\end{equation}
We define {\it the transmission coefficient}:
\begin{equation}
\label{transmcoeff}
d(\varphi,E,\varepsilon)=w(\alpha_{g}h_{-}^{g}(\cdot,\varphi,E,\varepsilon),\alpha_{d}h_{+}^{d}(\cdot,\varphi,E,\varepsilon))
\end{equation}
We immediately deduce from Proposition \ref{carvp} and Proposition
\ref{propconstinf} that the eigenvalues of
$H_{\varphi,\varepsilon}$ are characterized by:
\begin{equation}
d(\varphi,E,\varepsilon)=0
\end{equation}
\subsection{Some remarks}
\subsubsection{}
The assumption $(H_{W,r})$ is not optimal. Actually, it suffices
to assume that $W$ is analytic real in $S_{Y}$ and that there
exists a function $f\in L^{1}(\mathbb{R})$ such that :
$$\forall x\in\mathbb{R}\quad\sup\limits_{y\in[-Y,Y]}|W(x+iy)|\leq f(x).$$
\subsubsection{} In equations \eqref{starg} and \eqref{stard}, we
could have included the numbers $i\sigma_{g}$ and $i\sigma_{d}$
into the functions $\alpha_{g}$ and $\alpha_{d}$, but we prefer
showing the relations between $q_{g}$ and $q_{g}^{*}$, $q_{d}$ and
$q_{d}^{*}$.
\subsubsection{} Note that this construction differs from the
constructions of canonical domains in \cite{FK1}. Indeed, the
domains on which we construct these functions depend on
$\varepsilon$. We shall extend these asymptotics on a fixed strip
in the neighborhood of the real line (section \ref{infwkb}).
\section{WKB Theorem on non compact domains}
\label{infwkb} In this section, we prove a continuation result on
non compact domains of $S_{Y}$. This result is a generalization on
non compact domains of the method developed in \cite{FK1} and
particularly of Lemma \ref{lemcontfin}.\\
We prove that the continuation of asymptotics stay valid on some
half-strips $\{\varphi\in S_{Y}\ ;\ |\mbox{Re }\varphi|>A\}$. To do that,
we cover these domains by a countable union of small local
overlapping canonical
domains, called $\delta$-chain (see section \ref{deltcha}).\\
This principle follows the recent developments and improvements of
the WKB method (see \cite{FK3}). The idea is to get over the local
notion of canonical domain in favor of maximal domains. These
domains, constructed as union of local canonical domains are some
domains on which a function keeps the standard behavior (see
\cite{FK3}).
\subsection{Continuation Theorem on non compact domains}
\subsubsection{The main result}
We shall prove the following result:
\begin{thm} Continuation Theorem on non compact domains.\\
\label{infcontle} Fix $\tilde{Y}\in]0,Y[$. Assume that $V$
satisfies $(H_{V})$, that $W$ satisfies $(H_{W,r})$ and that $J$
satisfies $(H_{J}^{0})$. Then, there exist a real
$\varepsilon_{0}>0$, a complex neighborhood $\mathcal{V}$ of $J$
and two real numbers $A_{g}$ and $A_{d}$ such that, if $f$ has the
following properties:
\begin{itemize}
\item The function $f(\cdot,\varphi,E,\varepsilon)$ is a
consistent solution of \eqref{eqp}.
\item The function $(\varphi,E)\mapsto
f(x,\varphi,E,\varepsilon)$ is analytic on $S_{\tilde{Y}}\times
\mathcal{V}$ for any $x\in[-X,X]$ and any
$\varepsilon\in]0,\varepsilon_{0}[$.
\end{itemize}
Then,
\begin{enumerate}
\item There exists $\kappa$ a continuous branch on
$\{\mbox{Re }\varphi<A_{g}\}$ such that $\mbox{Im }\kappa>0$. Moreover, for any
$C<B<A_{g}$, if the function $f$ satisfies the asymptotic behavior
\begin{equation}
\label{asprolac}
f(x,\varphi,E,\varepsilon)=e^{-\frac{i}{\varepsilon}\int^{\varphi}\kappa(u)du}(\psi_{-}(x,\varphi,E)+r_{C}(x,\varphi,E,\varepsilon))
\end{equation}
with $\lim\limits_{\varepsilon\rightarrow
0}\sup\limits_{[-X,X]\times
R_{(-\infty,C]}\times\mathcal{V}}\max\{|r_{C}(x,\varphi,E,\varepsilon)|,|\partial_{x}r_{C}(x,\varphi,E,\varepsilon)|\}=0$,\\
then, this behavior stays valid until $B$. Precisely:
\begin{equation}
\label{asprola}
f(x,\varphi,E,\varepsilon)=e^{-\frac{i}{\varepsilon}\int^{\varphi}\kappa(u)du}(\psi_{-}(x,\varphi,E)+r_{B}(x,\varphi,E,\varepsilon))
\end{equation}
with $\lim\limits_{\varepsilon\rightarrow
0}\sup\limits_{[-X,X]\times
R_{(-\infty,B]}\times\mathcal{V}}\max\{|r_{B}(x,\varphi,E,\varepsilon)|,|\partial_{x}r_{B}(x,\varphi,E,\varepsilon)|\}=0$.\\
\item There exists $\kappa$ a continuous branch on
$\{\mbox{Re }\varphi>A_{d}\}$ such that $\mbox{Im }\kappa>0$. Moreover, for any
$C>B>A_{d}$, if $f$ satisfies the asymptotic behavior
\begin{equation}
\label{asprolbc}
f(x,\varphi,E,\varepsilon)=e^{\frac{i}{\varepsilon}\int^{\varphi}\kappa(u)du}(\psi_{+}(x,\varphi,E)+r_{C}(x,\varphi,E,\varepsilon))
\end{equation}
with $\lim\limits_{\varepsilon\rightarrow
0}\sup\limits_{[-X,X]\times
R_{[C,+\infty)}\times\mathcal{V}}\max\{|r_{C}(x,\varphi,E,\varepsilon)|,|\partial_{x}r_{C}(x,\varphi,E,\varepsilon)|\}=0$,\\
then this behavior stays valid until $B$. Precisely:
\begin{equation}
\label{asprolb}
f(x,\varphi,E,\varepsilon)=e^{\frac{i}{\varepsilon}\int^{\varphi}\kappa(u)du}(\psi_{+}(x,\varphi,E)+r_{B}(x,\varphi,E,\varepsilon))
\end{equation}
with $\lim\limits_{\varepsilon\rightarrow
0}\sup\limits_{[-X,X]\times
R_{[B,+\infty)}\times\mathcal{V}}\max\{|r_{B}(x,\varphi,E,\varepsilon)|,|\partial_{x}r_{B}(x,\varphi,E,\varepsilon)|\}=0$.
\end{enumerate}
\end{thm}
Theorem \ref{infcontle} and Proposition \ref{propconstinf} clearly
imply Theorem \ref{jostthm}.
\subsubsection{Some remarks}
\label{hypW} We shall prove Theorem \ref{infcontle} as $W$
satisfies
the weaker assumptions:\\
{\bf (H1) $\mathbf{W}$ is an analytic real function in $\mathbf{S_{Y}}$}.\\
{\bf (H2) $\mathbf{\exists\ C>0,\quad \exists\ s>1\textrm{ such that }\forall\ z\in S_{Y},\quad |W'(z)|\leq\frac{C}{1+|z|^{s}}}$}\\
{\bf (H3) $\mathbf{\exists\ f \in L^{1}(\mathbb{R})\textrm{ such that }\forall x\in\mathbb{R}\quad \sup\limits_{y\in[-Y,Y]}|W(x+iu)|\leq f(x)}$}\\
The following lemma relates $(H_{W,r})$ and $(H1)$, $(H2)$ and
$(H3)$:
\begin{lem}
\label{hypfaib} Let $W$ satisfy $(H_{W,r})$ on $S_{Y}$. Fix
$\tilde{Y}\in]0,Y[$. Then $W$ satisfies $(H1)$, $(H2)$ and $(H3)$
on $S_{\tilde{Y}}$.
\end{lem}
\begin{dem}
Assume that $W$ satisfy $(H_{W,r})$ on $S_{Y}$. We prove that $W$
satisfies $(H2)$ on $S_{\tilde{Y}}$ by using the following lemma:
\begin{lem}
\label{der} Let $f$ be an analytic function on $S_{Y}$ such that $|f(z)|\leq\frac{C}{1+|z|^{s}}$, $C>0$.\\
Fix $\eta>0$. Then,
$$\forall p\in\mathbb{N}^{*}\quad \exists C_{p}>0/\quad \forall z\in S_{Y-\eta}\quad|f^{(p)}(z)|\leq\frac{C_{p}}{1+|z|^{s}}.$$
\end{lem}
\begin{dem}\\
This result is a consequence of the Cauchy formula. We do not give
the details.
\end{dem}\\
- Clearly, $W$ satisfies $(H1)$ on $S_{Y}$.\\
- $W$ satisfies $(H3)$ with $f(x)=\frac{C}{1+|x|^{s}}$.\\
This completes the proof of Lemma \ref{hypfaib}.
\end{dem}
\subsubsection{}
Let us briefly outline the ideas of the proof. We shall
concentrate on $B_{g}=\{\varphi\in S_{Y};\ \mbox{Re }(\varphi)<A_{g}\}$.
There are three steps.\\
First we cover $B_{g}$ with an union of overlapping local compact
canonical domains $K_{m}$.\\
In each canonical domain $K_{m}$, we can construct a consistent
local basis thanks to Theorem \ref{finwkbthm}. To compute the
connection between the consistent bases of $K_{m}$ and $K_{m+n}$,
it suffices to do the product of the $n$ transfer matrices between
the canonical bases of two successive domains. The accuracy of the
rest cannot be better than the sum of the accuracies obtained on
each domain. Theorem \ref{finwkbthm} gives an estimate in $o(1)$;
this accuracy is insufficient when $n$ goes to
infinity.\\
A refinement of the calculation of asymptotics in Theorem
\ref{finwkbthm} is therefore necessary. We prove it by using the
integrability of $W$.
\subsubsection{Branch points}
The following result specifies the location of the branch points
of $\kappa$. We recall that $\Upsilon(E)$ is defined in
\eqref{nupsilon}.
\begin{lem}
Let $\mathcal{V}$ be a complex neighborhood of the interval $J$.
Assume that $W$ satisfies
$$\lim\limits_{x\rightarrow
+\infty}\sup\limits_{y\in[-Y,Y]}|W(x+iy)|=0,$$ then:
\begin{equation*}
\label{bplem} \exists A>0\textrm{ such that }\forall\
E\in\mathcal{V},\quad\varphi\in\Upsilon(E)\cap
S_{Y}\Rightarrow|\mbox{Re }(\varphi)|<A.
\end{equation*}
\end{lem}
\begin{dem}
Since $\overline{\mathcal{V}}\cap\partial\sigma(H_{0})=\emptyset$,
there exists $\alpha>0$ such that: $$\forall\
E\in\mathcal{V},\quad\forall\
p\in\mathbb{N}^{*},\quad|E-E_{p}|\geq\alpha.$$ If $\varphi_{p}(E)$
satisfies $E-W(\varphi_{p}(E))=E_{p}$, we get:
$$\forall\
E\in\mathcal{V},\quad\forall\
p\in\mathbb{N}^{*},\quad|W(\varphi_{p}(E))|\geq\alpha.$$ Finally, $\{u\in
S_{Y}\ ;\ |W(u)|\geq\alpha\}$ is a subset of a compact of $S_{Y}$.
This completes the proof of Lemma \ref{bplem}.
\end{dem}
\subsubsection{Uniform asymptotics on a $\delta$-chain}
\label{deltcha} First, we introduce a new definition. We remind
that the width of a complex subset is defined in \eqref{larga}.
\begin{defn}$\delta$-$\textit{chain of strictly canonical domains}$\\
Fix $\widetilde{Y}\in]0,Y[$. Fix $E$. Let $D$ be a simply
connected domain of $S_{\widetilde{Y}}$ containing no branch
points of the complex momentum. We fix on $D$ a continuous branch
$\kappa$ of the complex momentum. Let $\{\tau_{n}\}_{n\in\mathbb{N}}$ be a
sequence of real numbers and $K$ be a compact of $S_{\widetilde{Y}}$.\\
$\{K+\tau_{n}\}_{n\in\mathbb{N}}$ is called a $\delta$-chain for $E$,
$\kappa$ and $D$ if it satisfies the following properties:
\begin{enumerate}
\item $\bigcup\limits_{n=0}^{\infty}(K+\tau_{n})=D.$
\item $\exists\tau>0\textrm{ such that }\forall n\in\mathbb{N}\quad
l((K+\tau_{n})\cap(K+\tau_{n+1}),\widetilde{Y})>\tau.$
\item The domain $K$ is an union of curves $\gamma$ such that, for
any $n$, $\gamma+\tau_{n}$ is a $\delta$-strictly canonical curve
for $\kappa$.
\end{enumerate}
\end{defn}
$K$ is called the fundamental domain of the $\delta$-chain. Now,
we have the intermediate result:
\begin{prop}
\label{infwkbprop} Assume that $V$ satisfies $(H_{V})$ and that
$W$ satisfies $(H_{1})$, $(H_{2})$ and $(H_{3})$. Fix
$\tilde{Y}\in]0,Y[$. Let $\mathcal{V}$ a complex neighborhood of
$J$ and $D\subset S_{\tilde{Y}}$ a domain with the following
properties:
\begin{itemize}
\item $\inf\limits_{p\in\mathbb{N}^{*},E\in\mathcal{V}}
\textrm{dist}\{D,\varphi_{p}(E)\}\geq C,$
\item there exists $\{\tau_{n}\}_{n\in\mathbb{N}}$ such that, for any $E\in\mathcal{V}$,
$\{K+\tau_{n}\}_{n\in\mathbb{N}}$ is a $\delta$-chain for $E$ and $D$.
\end{itemize}
Fix $\varphi_{0}\in D$.\\
Then, there exists $\varepsilon_{0}>0$ such that, for any
$n\in\mathbb{N}$, there exist two functions
$(x,\varphi,E,\varepsilon)\mapsto\psi_{\pm}^{n}(x,\varphi,E,\varepsilon)$
with the following properties:
\begin{itemize}
\item The functions $(x,\varphi,E,\varepsilon)\mapsto\psi_{\pm}^{n}(x,\varphi,E,\varepsilon)$
are defined on
$\mathbb{R}\times(K+\tau_{n})\times\mathcal{V}\times]0,\varepsilon_{0}[$
and form a consistent basis.
\item for any fixed $x\in\mathbb{R},\
\varepsilon\in]0,\varepsilon_{0}[$, the functions
$(\varphi,E)\mapsto\psi_{\pm}^{n}(x,\varphi,E,\varepsilon)$ are
analytic on $(K+\tau_{n})\times\mathcal{V}$.
\item for $x\in[-X,X]$, $\varphi\in (K+\tau_{n})$ and
$E\in\mathcal{V}$, the functions $\psi_{\pm}^{n}$ have the
asymptotic behavior:
\begin{equation}
\label{asc}
\psi_{\pm}^{n}(x,\varphi,E,\varepsilon)=e^{\pm\frac{i}{\varepsilon}\int_{\varphi_{0}}^{\varphi}\kappa
du}\left(\psi_{\pm}(x,\varphi,E)+\frac{1}{1+|\tau_{n}|^{s}}o(1)\right).
\end{equation}
\item The asymptotics (\ref{asc}) are uniform in $x,\
\tau$, $\varphi\in K+\tau_{n}$ et $E\in\mathcal{V}$.
\item The asymptotics can be differentiated once in $x$.
\end{itemize}
\end{prop}
The proof of Proposition \ref{infwkbprop} mimics this of Theorem
1.1 in \cite{FK1}. We omit the details and we refer to \cite{FK1},
section 4 for an analogous statement.
\subsection{Construction of a $\delta$-chain of strictly canonical
domains} In this section, we shall construct a $\delta$-chain
under assumptions $(H_{1})$, $(H_{2})$ and $(H_{3})$.
\begin{prop}
\label{constdeltachain} Fix $\tilde{Y}\in]0,Y[$. Assume that $V$
satisfies $(H_{V})$, that $W$ satisfies $(H_{1})$, $(H_{2})$ and
$(H_{3})$ and that $J$ satisfies $(H_{J}^{0})$. Then, there exist
a complex neighborhood $\mathcal{V}$ of $J$, two real numbers
$(A_{g},A_{d})\in\mathbb{R}^{2}$, a domain $K\subset S_{\tilde{Y}}$ and
two real sequences $\{\tau_{n}^{1}\}_{n\in\mathbb{N}}$,
$\{\tau_{n}^{2}\}_{n\in\mathbb{N}}$ such that:
\begin{itemize}\item for any $E\in\mathcal{V}$, there exists a
continuous branch $\kappa$ on $\{\varphi\in S_{\tilde{Y}}\ ;\
\mbox{Re }\varphi\in(-\infty,A_{g}]\}$ (resp. on $\{\varphi\in
S_{\tilde{Y}}\ ;\ \mbox{Re }\varphi\in[A_{d},\infty)\}$),
\item for any $E\in\mathcal{V}$, $\{K+\tau_{n}^{1}\}_{n\in\mathbb{N}}$
(resp. $\{K+\tau_{n}^{2}\}_{n\in\mathbb{N}}$) is a $\delta$-chain for
$\kappa$, $E$ and $\{\mbox{Re }\varphi\in(-\infty,A_{g}]\}$ (resp.
$\{\mbox{Re }\varphi\in[A_{d},+\infty)$)).
\end{itemize}
\end{prop}
The rest of the section \ref{deltcha} is devoted to the proof of
Proposition \ref{constdeltachain}. This proof is based on
elementary geometrical arguments. We prove the construction for
$\mbox{Re }\varphi\in(-\infty,A_{g}]$.
\subsubsection{Construction of $\delta$-strictly canonical
straight-lines} We have defined the canonical lines in section
\ref{lc} and described them in terms of the vector $t(\varphi)$.\\
We set $\alpha=\frac{1}{2}\inf\limits_{E\in J}\mbox{Im } k(E)$ and
$m=2\sup\limits_{E\in J}|\mbox{Re } k(E)|$.\\
Since the mapping $(E,\varphi)\mapsto E-W(\varphi)$ is continuous
and since $W(\varphi)$ goes to zero when $\mbox{Re }\varphi$ goes to
infinity, there exist a complex neighborhood $\mathcal{V}$ of $J$
and a real number $A_{g}$ such that:
$$\forall E\in\mathcal{V},\quad\forall\varphi\in(-\infty,A_{g}],\quad \mbox{Re } k(E-W(\varphi))\in[-m,m],\quad \mbox{Im } k(E-W(\varphi))>\alpha$$
We set $B_{g}=(-\infty,A_{g}]+i[-\tilde{Y},\tilde{Y}]$. The
canonical curves for $\mbox{Re }\varphi$ in the neighborhood of $-\infty$
are described by:
\begin{lem}
\label{geom} There exists $\theta_{0}\in]0,\pi/2[$ such that, if
$\gamma$ is a smooth curve in $B_{g}$ satisfying:
\begin{equation}
\label{geoma} \forall\varphi\in\gamma,\quad
\arg[t(\varphi)]\in]\theta_{0},\pi/2-\theta_{0}[.
\end{equation}
then, $\gamma$ is a canonical line for $\kappa$.
\end{lem}
\begin{dem}
For $\arg(u)=\theta$ and
$\cot\theta\in]-\frac{m-\delta}{\alpha},\frac{\pi+m-\delta}{\alpha}[$,
we have:
$$\mbox{Im }(\overline{(\kappa-\delta)}u)>0\quad\textrm{et}\quad\mbox{Im }(\overline{(\pi-\kappa+\delta)}u)>0$$
Consequently, $\cot\theta_{0}=\frac{m-\delta}{\alpha}$ implies
that (\ref{geoma}) is satisfied.
\end{dem}
\subsubsection{The fundamental domain $K$}
Let $\xi_{1}=-i\tilde{Y}$ and $\xi_{2}=i\tilde{Y}$. We denote by
$K$ the lozenge bounded by the straight lines containing $\xi_{1}$
and $\xi_{2}$ whose guiding vectors have the affixes
$e^{i\theta_{0}}$ and $e^{i(\pi-\theta_{0})}$.\\
We set $[-u_{0},u_{0}]=K\cap\{y=0\}$. $K$ is shown in figure
\ref{element}. Fix $x$ such that $K+x\subset B_{g}$; we shall show
that $K+x$ is a $\delta$-strictly canonical domain. According to
Lemma \ref{geom}, it suffices to write $K$ as an union of smooth
curves satisfying (\ref{geoma}).\\
For any $u\in K$, we consider a vertical segment
$[\overline{\xi},\xi]$ containing $u$ and included in $K$ (see
figure \ref{element}). The broken line
$[\xi_{1},\overline{\xi}]\cup[\overline{\xi},\xi]\cup[\xi,\xi_{2}]$
satisfies (\ref{geoma}). The relation (\ref{geoma}) is stable
under small $C^{1}$-perturbation; we slightly deform the line
$[\xi_{1},\overline{\xi}]\cup[\overline{\xi},\xi]\cup[\xi,\xi_{2}]$
to get a smooth curve which satisfies (\ref{geoma}).\\
Consequently, $K$ satisfies the following properties:\\
- $K\cap S_{\tilde{Y}}$ contains a rectangle of width $4\eta>0$.\\
- $l((K-n\eta)\cap(K-(n+1)\eta),\tilde{Y})>\eta$.\\
- $K$ is the union of curves $\gamma$ such that $\gamma-n\eta$ is
$\delta$-strictly canonical for any sufficiently large $n$.
\subsubsection{Conclusion}
To finish the proof, it suffices to adapt the proof of Lemma
\ref{lemcontfin} in section 5.9 of \cite{FK1}, by using
Proposition \ref{constdeltachain} and Proposition
\ref{infwkbprop}. The convergence of the series of general term
$\frac{1}{1+|\tau_{n}|^{s}}$ replaces the compactness. We do not
give the details.
\input{fig10}
\section{Transmission coefficient. Equation for eigenvalues}
\label{calcmattransf}\label{anares} In Theorem \ref{jostthm}, we
have constructed two functions $h_{-}^{g}$ and $h_{+}^{d}$. We
have defined the transmission coefficient
$d(E,\varphi,\varepsilon)$. We choose $m_{g}=-0+i0$ and
$m_{d}=0+i0$.\\
In Proposition \ref{bcki}, we have
introduced a consistent basis $(f_{i},f_{i}^{*})$ near the cross.
To compute $d(E,\varphi,\varepsilon)$, we shall project the
functions
$h_{-}^{g}$ and $h_{+}^{d}$ onto the basis $(f_{i},f_{i}^{*})$.\\
\subsection{Preliminaries}
\subsubsection{Introduction. Notations}
Fix $\widetilde{Y}<Y$ and $E_{0}\in J$. We have described in
section \ref{wkbconst} the complex momentum $\kappa$ and the
related geometric objects. We recall that we consider the case
\eqref{premcassc}. We use the notations introduced in section
\ref{wkbconst}. The branch points are called $\varphi_{r}^{\pm}$
and $\varphi_{i},\ \overline{\varphi_{i}}$. We
have described the Stokes lines in section \ref{stline}.\\
We have described in sections \ref{compmom} and \ref{poss} the
different branches $\kappa_{i}$, $\kappa_{g}$ and $\kappa_{d}$.
The branch $\kappa_{g}$, resp. $\kappa_{d}$, is defined and
continuous on the domain $\{\varphi\in S_{Y};\
\mbox{Re }\varphi<\varphi_{r}^{-}\}$, resp.
$\{\mbox{Re }\varphi>\varphi_{r}^{+}\}$. The branch $\kappa_{i}$ is
defined and continuous on a neighborhood of the cross. The domain
$(E-W)\left(\{\varphi\in S_{Y};\
\mbox{Re }\varphi<\varphi_{r}^{-}\}\right)$ is a simply connected domain
which intersects with real axis in only one gap. Thus, we can fix
a determination $k_{g}$ of the quasi-momentum such that:
$$k_{g}(E-W(\varphi))=\kappa_{g}(\varphi).$$
Similarly, we fix the branches $k_{i}$ and $k_{d}$ of the
quasi-momentum such that :
$$ k_{i}(E-W(\varphi))=\kappa_{i}(\varphi),\quad
k_{d}(E-W(\varphi))=\kappa_{d}(\varphi).$$ Finally, we set:
$$ q_{i}(\varphi)=\sqrt{k'_{i}(E-W(\varphi))},\quad q_{g}(\varphi)=\sqrt{k'_{g}(E-W(\varphi))}\quad q_{d}(\varphi)=\sqrt{k'_{d}(E-W(\varphi))}.$$
Let $\varphi_{g}\in\mathbb{R}$ such that $\varphi_{g}<\varphi_{r}^{-}$ and
such that the interval $[\varphi_{g},\varphi_{r}^{-}]$ does not
contain any pole of $\omega_{\pm}$. We define the path
$\gamma_{g}$ in the complex plane by:
$$\gamma_{g}=[-0+i0,\varphi_{g}+i0]\cup[\varphi_{g}-i0,-0-i0].$$
Similarly, fix $\varphi_{d}\in\mathbb{R}$ such that
$\varphi_{d}>\varphi_{r}^{+}$ and such that the interval
$[\varphi_{r}^{+},\varphi_{d}]$ does not contain any pole of
$\omega_{\pm}$. We define the path $\gamma_{d}$ in the complex
plane by:
$$\gamma_{d}=[0+i0,\varphi_{d}+i0]\cup[\varphi_{d}-i0,0-i0].$$
In the following section, we explain the choice of the
determinations $q_{i}$, $q_{g}$ and $q_{d}$.
\subsubsection{The determination $q$}
\label{deco} We recall that there exists a real number
$\sigma_{i}\in\{-1,1\}$ such that:
\begin{equation}
\label{sigmareli} \frac{q_{i}^{*}}{q_{i}}=\sigma_{i}.
\end{equation}
We refer to section \ref{precsigma}.\\
The Wronskian satisfies $w(f_{i},(f_{i})^{*})=\sigma_{i}(w_{0}k_{i}')(E-W(0)).$\\
The number $\sigma_{i}$ depends on the sign of $k'$ along the band
$B$:
\begin{itemize}
\item If the band $B$ can be written $[E_{4p+1},E_{4p+2}]$, then $k'>0$
on $B$ and $\sigma_{i}=1$.
\item If the band $B$ can be written $[E_{4p+3},E_{4p+4}]$, then $k'<0$
on $B$ and $\sigma_{i}=-1$.
\end{itemize}
We fix the branch $q_{g}$ such that $q_{g}=q_{i}$ in $S_{-}$ and
such that $q_{g}$ is analytically continued in $\{\varphi\in
S_{Y};\ \mbox{Re }\varphi<\varphi_{r}^{-}\}$. According to relation
\eqref{sigmareli}, the branch $q_{g}$ satisfies:
\begin{equation}
\label{sigmarelg} q_{g}^{*}=i\sigma_{i}q_{g}
\end{equation}
Similarly, we fix $q_{d}$ such that $q_{d}=q_{i}$ in
$\overline{S_{+}}$ and such that $q_{d}$ is analytically continued
in $\{\varphi\in S_{Y};\ \mbox{Re }\varphi>\varphi_{r}^{+}\}$. The branch
$q_{d}$ satisfies:
\begin{equation}
\label{sigmareld} q_{d}^{*}=i\sigma_{i}q_{d}
\end{equation}
According to equations \eqref{sigmarelg} and \eqref{sigmareld}, we
have also:
$$\sigma_{g}=\sigma_{i}\quad;\quad\sigma_{d}=\sigma_{i}.$$
We denote by $\widetilde{\Psi_{\pm}}^{g}(x,\mathcal{E})$,
$\widetilde{\Psi_{\pm}}^{i}(x,\mathcal{E})$ and
$\widetilde{\Psi_{\pm}}^{d}(x,\mathcal{E})$ the Bloch solutions described
in section \ref{bloch}. We set: {\small
$$\Psi_{\pm}^{i}(x,\varphi,E)=\widetilde{\Psi_{\pm}}^{i}(x,E-W(\varphi))\
;\
\Psi_{\pm}^{g}(x,\varphi,E)=\widetilde{\Psi_{\pm}}^{g}(x,E-W(\varphi))\
;\
\Psi_{\pm}^{d}(x,\varphi,E)=\widetilde{\Psi_{\pm}}^{d}(x,E-W(\varphi)).$$}
We define the functions $\omega_{\pm}^{g}$, $\omega_{\pm}^{i}$ and
$\omega_{\pm}^{d}$ associated by \eqref{omega} to the branches
$k_{g}$, $k_{i}$ and $k_{d}$.
\subsubsection{Ideas of the method}
The computation is similar to this done in \cite{FK2, FK1, FK4}.
It is based on some elementary principles that we outline now.
\begin{enumerate}
\item Periodicity.\\
The consistency condition \eqref{coh} implies that the Wronskians
are $\varepsilon$-periodic in $\varphi$. To get a total control of
the Wronskians in a horizontal strip, we only need to control them
in some vertical sub-strip of width $\varepsilon$.
\item Analyticity.\\
Since the functions $(\varphi,E)\mapsto
f_{-}^{g}(x,\varphi,E,\varepsilon)$, $(\varphi,E)\mapsto
f_{+}^{d}(x,\varphi,E,\varepsilon)$, $(\varphi,E)\mapsto
f_{\pm}^{i}(x,\varphi,E,\varepsilon)$ are analytic on
$S_{\tilde{Y}}\times \mathcal{U}$, their Wronskians are analytic
in $(\varphi,E)\in S_{\tilde{Y}}\times \mathcal{U}$. This allows
us to expand them into exponentially converging series. \\
Let $w(\varphi,E,\varepsilon)$ be an analytic function in
$(\varphi,E)$ which is $\varepsilon$-periodic in $\varphi$. We
set:
$$
w(\varphi,E,\varepsilon)=\sum\limits_{k\in\mathbb{Z}}w_{k}(E,\varepsilon)e^{\frac{2i\pi\varphi}{\varepsilon}}$$
The Cauchy formula gives an estimate of the Fourier coefficients:
\begin{equation}
\label{fourgen}
w_{k}(E,\varepsilon)=\frac{1}{\varepsilon}\int_{\varphi_{0}}^{\varphi_{0}+\varepsilon}w(\varphi,E,\varepsilon)e^{-\frac{2i
k\pi\varphi}{\varepsilon}}d\varphi,\quad\forall
k\in\mathbb{N},\quad\forall\varphi_{0}\in S_{\tilde{Y}}.
\end{equation}
By moving $\mbox{Im }\varphi_{0}$ in $[-\tilde{Y},\tilde{Y}]$, we get a
control of positive and negative coefficients.
\end{enumerate}
\subsection{Asymptotic expansion of $d(\varphi,E,\varepsilon)$}
In this section, we shall establish the following result.
\begin{prop}
\label{colina} For any $E_{0}$ in $J$, there exist a complex
neighborhood $\mathcal{U}_{0}$ of $E_{0}$ and two functions
$(\varphi,E,\varepsilon)\mapsto b_{g}^{-}(\varphi,E,\varepsilon)$
and $(\varphi,E,\varepsilon)\mapsto
b_{d}^{+}(\varphi,E,\varepsilon)$ such that:
\begin{itemize}
\item The coefficient $d$ defined in \eqref{transmcoeff} can be written:
\begin{equation}
d(\varphi,E,\varepsilon)=i\sigma_{i}w(f_{i},(f_{i})^{*})[b_{g}^{-}(b_{d}^{+})^{*}-(b_{g}^{-})^{*}b_{d}^{+}].
\end{equation}
\item The functions $(\varphi,E)\mapsto b_{g}^{-}(\varphi,E,\varepsilon)$ and $(\varphi,E)\mapsto
b_{d}^{+}(\varphi,E,\varepsilon)$ are analytic on $S_{Y}\times
\mathcal{U}_{0}$.
\item The functions $\varphi\mapsto b_{g}^{-}(\varphi,E,\varepsilon)$ and $\varphi\mapsto
b_{d}^{+}(\varphi,E,\varepsilon)$ are $\varepsilon$-periodic and
admit the following Fourier asymptotic expansion, when
$\varepsilon\rightarrow 0$:
\begin{equation}
\label{bscoeffaaa}
b_{g}^{-}(\varphi,E,\varepsilon)=\sum\limits_{k\in\mathbb{Z}}(b_{g}^{-})_{k}(E,\varepsilon)e^{\frac{2ik\pi\varphi}{\varepsilon}},
\end{equation}
with
\begin{equation}
\label{bscoeffaa}
(b_{g}^{-})_{0}(E,\varepsilon)=\sigma_{i}e^{-\frac{i}{\varepsilon}\int_{0}^{\varphi_{r}^{-}}\kappa_{i}}e^{\frac{1}{2}\int_{0}^{\varphi_{r}^{-}}(\omega_{+}^{i}-\omega_{-}^{i})}[1+o(1)],
\end{equation}
and \begin{equation} \label{bscoeffbb}\forall k \neq 0,\quad
|(b_{g}^{-})_{k}(E,\varepsilon)|<C
e^{-\alpha/\varepsilon}e^{\frac{-2|k|\pi Y_{0}}{\varepsilon}},
\end{equation}
\begin{equation}
\label{bscoeffbbb}
b_{d}^{+}(\varphi,E,\varepsilon)=\sum\limits_{k\in\mathbb{Z}}(b_{d}^{+})_{k}(E,\varepsilon)e^{\frac{2ik\pi\varphi}{\varepsilon}},
\end{equation}
with
\begin{equation} \label{bscoeffcc}(b_{d}^{+})_{0}(E,\varepsilon)=i\sigma_{i}e^{\frac{i}{\varepsilon}\int_{0}^{\varphi_{r}^{+}}\kappa_{i}}e^{\frac{1}{2}\int_{0}^{\varphi_{r}^{+}}(\omega_{+}^{i}-\omega_{-}^{i})}[1+o(1)],\end{equation}
\begin{equation} \label{bscoeffdd}(b_{d}^{+})_{1}(E,\varepsilon)=-i\sigma_{i}e^{\frac{i}{\varepsilon}\int_{\varphi_{r}^{+}}^{0}\kappa_{i}}e^{\frac{2i}{\varepsilon}\int_{0}^{\overline{\varphi_{i}}}(\kappa_{i}-\pi)}e^{\frac{1}{2}\int_{\varphi_{r}^{+}}^{0}\omega_{+}^{i}-\omega_{-}^{i}}e^{\int_{0}^{\overline{\varphi_{i}}}(\omega_{+}^{i}-\omega_{-}^{i})}[1+o(1)],\end{equation} et
\begin{equation} \label{bscoeffee}\forall k>1,\quad |(b_{d}^{+})_{k}(\varphi,E,\varepsilon)|<C |(b_{d}^{+})_{1}(E,\varepsilon)|e^{-\alpha/\varepsilon}e^{\frac{-2|k-1|\pi Y_{0}}{\varepsilon}},\end{equation}
\begin{equation} \label{bscoeffff}\forall k<0,\quad |(b_{d}^{+})_{k}(\varphi,E,\varepsilon)|<C e^{-\alpha/\varepsilon}e^{\frac{-2|k|\pi Y_{0}}{\varepsilon}},\end{equation}
\end{itemize}
\end{prop}
The rest of the section is devoted to the proof of Proposition
\ref{colina}.\\
Fix $E_{0}\in J$. According to the choice of $\kappa_{g}$ and
$\kappa_{d}$ (sections \ref{compmom} and \ref{poss}), there exist
two analytic functions $\alpha_{g}(E)$ and $\alpha_{d}(E)$ such
that:
$$(\alpha_{g}h_{-}^{g})^{*}=i\sigma_{g}\alpha_{g}h_{-}^{g},$$
$$(\alpha_{d}h_{+}^{d})^{*}=i\sigma_{d}\alpha_{d}h_{+}^{d}.$$
Now, we use the function $f_{i}$ constructed in Proposition
\ref{bcki}. There exists a neighborhood $\mathcal{U}_{0}$ of
$E_{0}$ such that we can write:
$$\alpha_{g}h_{-}^{g}=-i\sigma_{g}(b_{g}^{-})^{*}f_{i}+b_{g}^{-}(f_{i})^{*},$$
$$\alpha_{d}h_{+}^{d}=-i\sigma_{d}(b_{d}^{+})^{*}f_{i}+b_{d}^{+}(f_{i})^{*}.$$
The coefficients $\alpha_{g}$ and $\alpha_{d}$ are defined in
equations \eqref{renormconstg} and \eqref{renormconstd}. We
compute:
$$\int_{\gamma_{g}}\omega_{-}^{g}=\int_{0}^{\varphi_{r}^{-}}(\omega_{+}^{i}-\omega_{-}^{i}),$$
$$\int_{\gamma_{d}}\omega_{+}^{d}=\int_{0}^{\varphi_{r}^{+}}(\omega_{+}^{i}-\omega_{-}^{i}).$$
This leads to:
$$\alpha_{g}(E)=e^{-\frac{i}{\varepsilon}\int_{0}^{\varphi_{r}^{-}}\kappa_{i}}e^{\frac{1}{2}\int_{0}^{\varphi_{r}^{-}}(\omega_{+}^{i}-\omega_{-}^{i})},$$
$$\alpha_{d}(E)=e^{\frac{i}{\varepsilon}\int_{0}^{\varphi_{r}^{+}}\kappa_{i}}e^{\frac{1}{2}\int_{0}^{\varphi_{r}^{+}}(\omega_{+}^{i}-\omega_{-}^{i})}.$$
The coefficients $b_{g}^{-}$ and $b_{d}^{+}$ satisfy:
$$b_{g}^{-}=\alpha_{g}a_{g}^{-},\quad
b_{d}^{+}=\alpha_{d}a_{d}^{+},$$ where the coefficients
$a_{g}^{-}$ and $a_{d}^{+}$ are given by:
\begin{equation}
\label{asag}
a_{g}^{-}=\frac{w(f_{i},h_{-}^{g})}{w(f_{i},f_{i}^{*})},
\end{equation}
and:
\begin{equation}
\label{asad}
a_{d}^{+}=\frac{w(f_{i},h_{+}^{d})}{w(f_{i},f_{i}^{*})}.
\end{equation}
We compute, for $E\in\mathcal{U}_{0}$:
$$d(\varphi,E,\varepsilon)=w(\alpha_{g}h_{-}^{g},\alpha_{d}h_{+}^{d})$$
$$=\left[b_{g}^{-}(b_{d}^{+})^{*}-b_{d}^{+}(b_{g}^{-})^{*}\right]i\sigma_{i}w(f_{i},(f_{i})^{*}).$$
\subsubsection{Continuation diagram of $f_{i}$}
First, we describe the asymptotic behavior of the function $f_{i}$
in some domains of the complex plane.
\begin{lem}
\label{contdiag} We suppose that the assumptions of Proposition
\ref{bcki} are satisfied. Fix $\tilde{Y}<Y$. Fix
$\varphi_{g}<\varphi_{r}^{-}$ and $\varphi_{d}>\varphi_{r}^{+}$.
There exists $y_{0}\in]0,\mbox{Im }\varphi_{i}[$ such that the function
$f_{i}$ has the following asymptotic behavior:
\begin{itemize} \item For $\varphi\in\{\varphi\in S_{\tilde{Y}};\
\mbox{Re }\varphi\in[\varphi_{g},\varphi_{r}^{-}]\}$, $f_{i}$ has the
standard asymptotic behavior:
$$f_{i}=q_{g}e^{\frac{i}{\varepsilon}\int_{0}^{\varphi_{r}^{-}}\kappa_{i}}e^{\frac{i}{\varepsilon}\int_{\varphi_{r}^{-}}^{\varphi}\kappa_{g}}e^{\int_{0}^{\varphi_{r}^{-}}\omega_{+}^{i}}e^{\int_{\varphi_{r}^{-}}^{\varphi}\omega_{+}^{g}}\left(\Psi_{+}^{g}+o(1)\right).$$
\item For $\varphi\in\{\varphi\in S_{\tilde{Y}};\
\mbox{Re }\varphi\in[\varphi_{r}^{+},\varphi_{d}];\ \mbox{Im }\varphi>-y_{0}\}$,
$f_{i}$ has the standard asymptotic behavior:
$$f_{i}=iq_{d}e^{\frac{i}{\varepsilon}\int_{0}^{\varphi_{r}^{+}}\kappa_{i}}e^{-\frac{i}{\varepsilon}\int_{\varphi_{r}^{+}}^{\varphi}\kappa_{d}}e^{\int_{0}^{\varphi_{r}^{+}}\omega_{+}^{i}}e^{\int_{\varphi_{r}^{+}}^{\varphi}\omega_{-}^{d}}\left(\Psi_{-}^{d}+o(1)\right).$$
\item For $\varphi\in\{\varphi\in S_{\tilde{Y}};\
\mbox{Re }\varphi\in[\varphi_{r}^{+},\varphi_{d}];\ \mbox{Im }\varphi<-y_{0}\}$,
$f_{i}$ has the standard asymptotic behavior:
$$f_{i}=-iq_{d}e^{\frac{i}{\varepsilon}\int_{0}^{\varphi_{i}^{-}}\kappa_{i}}e^{\frac{i}{\varepsilon}\int_{\varphi_{i}^{-}}^{\varphi}(2\pi-\kappa_{d})}e^{\int_{0}^{\varphi_{i}^{-}}\omega_{+}^{i}}e^{\int_{\varphi_{i}^{-}}^{\varphi}\omega_{-}^{d}}\left(\Psi_{-}^{d}+o(1)\right).$$
\end{itemize}
\end{lem}
\begin{dem}
This lemma is similar to the continuation diagram presented in
section 6 of \cite{FK4}. Thus, we give only the main ideas of the
study and refer to this paper for the details. The continuation
diagram is represented in figure \ref{contdiagfig}. In this
figure, the straight arrows indicate the use of continuation lemma
(Lemma \ref{lemcontfin}), the circular arrows the use of the
Stokes lemma (Lemma \ref{stoklemma}) and the hatched zones the use
of the Adjacent Canonical Domain Principle (Lemma \ref{adjdom}).
To complete the proof, it remains to explain the connections
between the different objects of the WKB method.
\begin{itemize}
\item According to the definitions given in section \ref{compmom},
the branches $\kappa_{i}$ and $\kappa_{g}$ are equal in $S_{-}$
and, for all $\varphi\in S_{-}$, we have:
\begin{equation}
\label{bda}
\kappa_{i}(\varphi+0)=\kappa_{g}(\varphi-0),\quad\Psi_{\pm}^{i}(\varphi+0)=\Psi_{\pm}^{g}(\varphi-0),\quad
\omega_{\pm}^{i}(\varphi+0)=\omega_{\pm}^{g}(\varphi-0).
\end{equation}
Besides, it remains to link $q_{i}$ and $q_{g}$. Section
\ref{deco} implies:
$$\forall\varphi\in S_{-},\quad q_{i}(\varphi)=q_{g}(\varphi).$$
\item Similarly, we have, for all $\varphi\in \overline{S_{+}}$:
\begin{equation}
\label{bdd} \kappa_{i}(\varphi-0)=-\kappa_{d}(\varphi+0),
\quad\Psi_{\pm}^{i}(x,\varphi-0)=\Psi_{\mp}^{d}(x,\varphi+0),
\end{equation}
\begin{equation*}
\omega_{\pm}^{i}(\varphi-0)=\omega_{\mp}^{d}(\varphi+0),\quad
q_{d}(\varphi+0)=-i\ q_{i}(\varphi-0)
\end{equation*}
\item We study finally the link between $\kappa_{i}$ and $\kappa_{d}$ along the Stokes line $\bar{c}$ beginning at $\overline{\varphi_{i}}$.
We consider the quasi-momenta $k_{d}$ and $k_{i}$ associated to
$\kappa_{d}$ and $\kappa_{i}$. Equation \eqref{kl} for $k_{d}$ and
$k_{i}$, on either side of $[E_{2},E_{3}]$, implies that
$\kappa_{d}$ and $\kappa_{i}$ satisfy the following relations, for
$\varphi\in c$,
\begin{equation}
\label{bdf}
\kappa_{d}(\varphi+0)=2\pi-\kappa_{i}(\varphi-0),\quad\Psi_{\pm}^{d}(x,\varphi+0)=\Psi_{\mp}^{i}(x,\varphi-0)
\end{equation}
\begin{equation*}
\omega_{\pm}^{d}(\varphi+0)=\omega_{\mp}^{i}(\varphi-0)\quad
q_{d}(\varphi+0)=iq_{i}(\varphi-0)
\end{equation*}
\end{itemize}
\end{dem}
\input{fig11}
\subsubsection{Computation of $b_{g}^{-}$ and $b_{d}^{+}$}
Now, we compute the coefficients $b_{g}^{-}$ and $b_{d}^{+}$ given
by \eqref{asag} and \eqref{asad}.\\
According to Theorem \ref{infcontle}, we know that the asymptotic
behavior of the function $h_{-}^{g}$ remains valid in the domain
$\{\varphi\in S_{Y};\
\mbox{Re }(\varphi)\in[\varphi_{g},\varphi_{r}^{-}]\}$. Lemma
\ref{contdiag} gives the asymptotic behavior of $f_{i}$ in this
domain and we get :
\begin{equation} \forall \varphi\in S_{Y},\quad
a_{g}^{-}(\varphi,E,\varepsilon)=\sigma_{i}[1+o(1)].
\end{equation}
Fix $Y_{0}\in]0,Y[$. In the strip $S_{Y_{0}}$, we write:
\begin{equation}
a_{g}^{-}(\varphi,E,\varepsilon)=\sum\limits_{n\in
\mathbb{Z}}\alpha_{n}e^{\frac{2i\pi n\varphi}{\varepsilon}}
\end{equation}
The coefficients $\alpha_{n}$ satisfy:
\begin{equation}
\label{four}
\alpha_{n}=\frac{1}{\varepsilon}\int_{\varphi_{0}}^{\varphi_{0}+\varepsilon}a_{g}^{-}(\varphi,E,\varepsilon)e^{-2i\pi
n\frac{\varphi}{\varepsilon}}d\varphi,\quad \forall
n\in\mathbb{N},\quad\forall\varphi_{0}\in\{-Y_{0}\leq\mbox{Im }\varphi\leq
Y_{0}\}.
\end{equation}
Fix $n>0$. We estimate $|\alpha_{n}|$. We use formula (\ref{four})
for $\mbox{Im }\varphi_{0}=-(Y-\delta) $, and we get:
$$ |\alpha_{n}|\leq C e^{-2\pi
n(Y-\delta)/\varepsilon}e^{\frac{C\delta}{\varepsilon}}.$$ We
treat similarly the case $n<0$ with $\mbox{Im }\varphi_{0}=(Y-\delta)$ and
we obtain:
$$ |\alpha_{n}|\leq C e^{2\pi
n(Y-\delta)/\varepsilon}e^{\frac{C\delta}{\varepsilon}}.$$
Besides, we have:
$$\alpha_{0}=\sigma_{i}[1+o(1)].$$
We fix $\delta<\frac{2\pi(Y-Y_{0})}{C+2\pi}$. For a constant $C$
such that $\alpha<2\pi(Y-Y_{0})-\delta(C+2\pi)$, we obtain the
estimates \eqref{bscoeffaa} and \eqref{bscoeffbb}.\\
The arguments for the coefficients $a_{d}^{+}$ and $b_{d}^{+}$ are
similar.
\subsection{Proof of Lemma \ref{phaseactint}}
\label{phaseactintdem} Now, we want to express the coefficient $d$
in a more understandable form. We begin with proving Lemma
\ref{phaseactint}. We recall that we denote by $\varphi_{r}^{\pm}$
and $\varphi_{i},\ \overline{\varphi_{i}}$ the branch points of
the complex momentum, and by $E_{r}$ and $E_{i}$ the related ends
of $\sigma(H_{0})$. We shall prove the lemma in the case
\eqref{premcassc}. Let $\kappa_{i}$ be the branch
described in section \ref{poss}. $\kappa_{i}$ satisfies \eqref{kappai}.\\
We shall prove Lemma \ref{phaseactint} for the branch
$\widetilde{\kappa_{i}}=\kappa_{i}$.
\begin{itemize}
\item First, we express $\Phi$, $S$ and $\Phi_{d}$ as integrals of
the complex momentum along complex paths. Let $\gamma$ be an
oriented curve, we call $\gamma^{\dagger}$ the curve oriented in
the opposite direction. Fix $\varphi_{d}\in\mathbb{R}$ and
$\varphi_{g}\in\mathbb{R}$ such that:
$$\varphi_{d}>\varphi_{r}^{+}\ ;\
\varphi_{g}<\varphi_{r}^{-}.$$ We define the complex paths
$\gamma_{\Phi}$, $\gamma_{S}$ and $\gamma_{g,d}$:
$$\gamma_{\Phi}=[\varphi_{r}^{-}+i0,\varphi_{r}^{+}+i0]\cup[\varphi_{r}^{+}-i0,\varphi_{r}^{-}-i0],$$
$$\gamma_{S}=(\sigma+0)\cup(\sigma^{\dagger}-0),$$
$$\gamma_{g,d}=[\varphi_{g}+i0,0+i0]\cup(\sigma_{+}-0)\cup(\sigma_{+}^{\dagger}+0)\cup[0+i0,\varphi_{d}+i0].$$
These paths are represented in figure \ref{uscoeff}. We have the
following result:
\begin{lem}
\label{contour} The coefficients $\Phi$, $\Phi_{d}$ and $S$ can be
written:
$$\Phi=\frac{1}{2}\oint_{\gamma_{\Phi}}\kappa(u)du,$$
$$S=\frac{1}{2i}\oint_{\gamma_{S}}\kappa(u)du,$$
$$\Phi_{d}=\frac{1}{2}\left(\int_{\gamma_{g,d}}(\kappa(u)-\pi)du+\int_{\overline{\gamma_{g,d}}}(\widetilde{\kappa}(u)-\pi)du\right)+\pi(\varphi_{g}-\varphi_{d}).$$
where $\kappa=\kappa_{i}$ in $S_{-}$ and $\kappa$ is analytically
continued along each path; $\widetilde{\kappa}=\kappa_{i}$ in
$\overline{S_{-}}$ and $\widetilde{\kappa}$ is analytically
continued along $\overline{\gamma_{g,d}}$.
\end{lem}
\begin{dem}
\begin{itemize}
\item First, let us justify the fact that integrals along $\gamma_{\Phi}$ and
$\gamma_{S}$ can be considered along closed curves. It suffices to
show that $\kappa$ can be analytically continued along
$\gamma_{\Phi}$ and $\gamma_{S}$.\\
We consider the curve $\gamma_{\Phi}$. We have taken the cut of
$\gamma_{\Phi}$ in $\varphi_{r}^{-}$. We show that $\kappa$ has
the same values on each side of the cut. $\kappa=\kappa_{i}$ on
$[\varphi_{r}^{-}+i0,\varphi_{r}^{+}+i0]$, since $\kappa$ is
continuous to the right of $\varphi_{r}^{+}$, we obtain that
$\kappa=-\kappa_{i}$ on $[\varphi_{r}^{-}-i0,\varphi_{r}^{+}-i0]$.
In addition,
$\kappa(\varphi_{r}^{-}+i0)=0=\kappa(\varphi_{r}^{-}-i0)$, which
proves that the integral can be taken on the closed curve $\gamma_{\Phi}$ .\\
The arguments for $\gamma_{S}$ are similar.
\item We compute:
$$\frac{1}{2}\oint_{\gamma_{\Phi}}\kappa(u)du=\int_{\varphi_{r}^{-}}^{\varphi_{r}^{+}}\kappa_{i}(u)du=\Phi(E).$$
Similarly, for the coefficient $S(E)$,
$$\frac{1}{2i}\oint_{\gamma_{S}}\kappa(u)du=\frac{1}{i}[\int_{\sigma_{+}}(\pi-\kappa_{i}(u))du+\int_{\sigma_{-}}(\pi-\kappa_{i}(u))du].$$
\item It remains to study $\Phi_{d}$. We introduce the branch
$\kappa_{i}$ and we cut $\gamma_{g,d}$ in elementary segments:
$$\int_{\gamma_{g,d}}(\kappa(u)-\pi)du+\int_{\overline{\gamma_{g,d}}}(\widetilde{\kappa}(u)-\pi)du$$
$$=2\int_{\varphi_{g}}^{0}(\kappa_{i}(u)-\pi)du+2\int_{\sigma_{+}}(\kappa_{i}(u)-\pi)du+\int_{\varphi_{d}}^{0}(\kappa_{i}(u)-\pi)du-2\int_{\sigma_{-}}(\kappa_{i}(u)-\pi)du$$
$$=2\int_{\varphi_{r}^{-}}^{0}(\kappa_{i}(u)-\pi)du+2\int_{\sigma_{+}}(\kappa_{i}(u)-\pi)du+\int_{\varphi_{r}^{+}}^{0}(\kappa_{i}(u)-\pi)du-2\int_{\sigma_{-}}(\kappa_{i}(u)-\pi)du-2\pi(\varphi_{g}-\varphi_{r}^{-})+2\pi(\varphi_{d}-\varphi_{r}^{+})$$
$$=2\Phi_{d}(E)+2\pi(\varphi_{d}-\varphi_{g})$$
\end{itemize}
This ends the proof of Lemma \ref{contour}.
\end{dem}
\item We use Lemma \ref{contour} to prove the analyticity of $\Phi$, $S$ and $\Phi_{d}$.\\
First, we consider $\Phi$. We can deform $\gamma_{\Phi}$ to a
closed curve going around $[\varphi_{r}^{-},\varphi_{r}^{+}]$ and
staying at a nonzero distance from this interval. Besides,
$\kappa$ is analytic in $E$ on the integration contour when $E$ is
close enough to $J$. The analysis of the coefficient $S$ is done
in the same way. To prove that $\Phi_{d}$ is analytic, we deform
the curves $\gamma_{g,d}$ and $\overline{\gamma_{g,d}}$ to stay at
a nonzero distance of the cross.
\item Fix $E\in J$. On the interval $[\varphi_{r}^{-},\varphi_{r}^{+}]$, the
branch $\kappa_{i}$ satisfies $\kappa_{i}\in[0,\pi]$. Thus, the
function $\Phi(E)$ is real positive on $J$.\\
Now, we give a simplified expression of $S$:
\begin{equation}
\label{exps}
S(E)=-i\left[\int_{\sigma_{+}}(\pi-\kappa_{i}(u))du+\int_{\sigma_{-}}(\pi-\kappa_{i}(u))du\right]=2\mbox{Im }\int_{\sigma_{+}}(\pi-\kappa_{i}(u))du
\end{equation}
On $\sigma$, the branch $\kappa_{i}$ satisfies
$\kappa_{i}\in[0,\pi]$. According to \eqref{action} and
\eqref{exps}, we obtain that $0< S(E)\leq
2\pi\mbox{Im }\varphi_{i}(E)$.\\
Finally, we have
$\int_{\sigma_{-}}(\kappa_{i}(u)-\pi)du=-\overline{\int_{\sigma_{+}}(\kappa_{i}(u)-\pi)du}$.
Consequently, the coefficient $\Phi_{d}(E)$ is real.
\item Now, we compute $S'$ and $\Phi'$ on $J$. Let $k$ be the
branch of the Bloch momentum continuous through $[E_{r},E_{i}]$,
then $\kappa(\varphi)=k(E-W(\varphi))$ and
$$\Phi'(E)=\int_{\varphi_{r}^{-}}^{\varphi_{r}^{+}}k'(E-W(u))du+k(E-W(\varphi_{r}^{+}))-k(E-W(\varphi_{r}^{-}))=\int_{\varphi_{r}^{-}}^{\varphi_{r}^{+}}k'(E-W(u))du.$$
We recall that $k$ has some branch points of square root type at
the ends of spectral bands (see section \ref{qm2}); consequently,
the integral
$\int_{\varphi_{r}^{-}}^{\varphi_{r}^{+}}k'(E-W(u))du$ is
convergent. In the interval $[E_{r},E_{i}]$, $k'(\mathcal{E})>0$ and
$(E_{i}-E_{r})\Phi'$ takes positive values on $J$.\\
The analysis of $S'$ is similar.
\item We complete this section with the following formulas:
\begin{equation}
\label{phida}
\Phi_{d}(E)+iS(E)=\int_{\varphi_{r}^{-}}^{0}\kappa_{i}(u)du-2\int_{\sigma_{-}}(\kappa_{i}-\pi)(u)du+\int_{\varphi_{r}^{+}}^{0}\kappa_{i}(u)du
\end{equation}
\begin{equation}
\label{phidb}
-\Phi_{d}(E)+iS(E)=-\int_{\varphi_{r}^{-}}^{0}\kappa_{i}(u)du-2\int_{\sigma_{+}}(\kappa_{i}-\pi)(u)du-\int_{\varphi_{r}^{+}}^{0}\kappa_{i}(u)du
\end{equation}
When $\kappa(\varphi_{r}^{-})=\pi$, the proof is analogous for the
branch $\widetilde{\kappa_{i}}=2\pi-\kappa_{i}$.\\
\end{itemize}
\input{fig12}
\subsubsection{Further computations}
\label{comp} We recall that the functions $\omega_{+}^{i}$ and
$\omega_{-}^{i}$ are defined in \eqref{omega}. We consider the
integrals of $\omega_{+}^{i}$ and $\omega_{-}^{i}$ along some
paths of the complex plane. We have the following relations:
\begin{lem}
\label{coeffut} The integrals of $\omega_{+}^{i}$ and
$\omega_{-}^{i}$ satisfy:
\begin{equation}
\label{omegaa} \forall E\in
J,\quad\int_{[\varphi_{r}^{-},\varphi_{r}^{+}]}\omega_{+}^{i}(u,E)du=0,\quad\int_{[\varphi_{r}^{-},\varphi_{r}^{+}]}\omega_{-}^{i}(u,E)du=0
\end{equation}
\begin{equation}
\label{omegab} \forall E\in
J,\quad\int_{\sigma}\omega_{+}^{i}(u,E)du=0,\quad\int_{\sigma}\omega_{-}^{i}(u,E)du=0\end{equation}
There exists a real number $\rho$ such that:
\begin{equation}
\label{omegac} \forall E\in
J,\quad\int_{[\varphi_{r}^{+},0]\cup\sigma_{+}}(\omega_{+}^{i}(u,E)-\omega_{-}^{i}(u,E))du-\int_{\sigma_{-}\cup[0,\varphi_{r}^{-}]}(\omega_{+}^{i}(u,E)-\omega_{-}^{i}(u,E))du=i\rho
\end{equation}
\end{lem}
\begin{dem}
We consider the case \eqref{premcassc}.
\begin{itemize}
\item We first prove \eqref{omegaa}. According to \eqref{omega},
we compute:
$$\int_{[\varphi_{r}^{-},\varphi_{r}^{+}]}\omega_{+}^{i}(u,E)du=-\int_{[\varphi_{r}^{-},\varphi_{r}^{+}]}g_{+}^{i}(E-W(u))W'(u)du=\int_{E-W([\varphi_{r}^{-},\varphi_{r}^{+}])}g_{+}^{i}(e)de=0$$
Indeed, for $E\in J$, the subset
$E-W([\varphi_{r}^{-},\varphi_{r}^{+}])$ is a complex path of
energies connecting $E_{r}$ to $E_{r}$ and containing
$(E-W(0))\in]E_{1},E_{2}[)$. We have shown this path in figure
\ref{imagchem}A. Particularly,
$E-W([\varphi_{r}^{-},\varphi_{r}^{+}])$ is a closed path and does
not surround any pole of the meromorphic function $g_{+}^{i}$.
Consequently, the integral is zero. We prove similarly that
$$\int_{[\varphi_{r}^{-},\varphi_{r}^{+}]}\omega_{-}^{i}(u,E)du=0.$$
\item We consider now \eqref{omegab}. We write:
$$\int_{\sigma}\omega_{+}^{i}(u,E)du=-\int_{E-W(\sigma)}g_{+}^{i}(e)de$$
The image of the path $\sigma$ is shown in figure \ref{imagchem}B.
We deal with $\omega_{-}^{i}$ similarly.
\item Finally, we compute:
$$\int_{[\varphi_{r}^{+},0]\cup\sigma_{+}}(\omega_{+}^{i}(u,E)-\omega_{-}^{i}(u,E))du-\int_{\sigma_{-}\cup[0,\varphi_{r}^{-}]}(\omega_{+}^{i}(u,E)-\omega_{-}^{i}(u,E))du$$ $$=\int_{E-W([\varphi_{r}^{+},0]\cup\sigma_{+})}(g_{+}^{i}(e)-g_{-}^{i}(e))de-\int_{E-W(\sigma_{-}\cup[0,\varphi_{r}^{-}])}(g_{+}^{i}(e)-g_{-}^{i}(e))de $$
The images $E-W([\varphi_{r}^{+},0]\cup\sigma_{+})$ and
$E-W(\sigma_{-}\cup[0,\varphi_{r}^{-}])$ are two paths of energies
connecting $E_{r}$ to $E_{i}$ (see figure \ref{imagchem}C). By
analyticity of $(g_{+}^{i}-g_{-}^{i})$ in the domain
$\mbox{Re }(e)\in]E_{r},E_{i}[$ , we obtain that:
\begin{equation}
\label{coeffrho}
\int_{[\varphi_{r}^{+},0]\cup\sigma_{+}}(\omega_{+}^{i}(u,E)-\omega_{-}^{i}(u,E))du-\int_{\sigma_{-}\cup[0,\varphi_{r}^{-}]}(\omega_{+}^{i}(u,E)-\omega_{-}^{i}(u,E))du=2\int_{E_{r}}^{E_{i}}(g_{+}^{i}-g_{-}^{i})(e)de
\end{equation}
It remains to show that this coefficient is purely imaginary. To
do that, we point out that $(g_{-}^{i})^{*}=g_{+}^{i}$, according
to \eqref{symband}. Equation \eqref{coeffrho} becomes:
$$\int_{[\varphi_{r}^{+},0]\cup\sigma_{+}}(\omega_{+}^{i}-\omega_{-}^{i})(u,E)du-\int_{\sigma_{-}\cup[0,\varphi_{r}^{-}]}(\omega_{+}^{i}-\omega_{-}^{i})(u,E)du$$ $$=2\left(\int_{E_{r}}^{E_{i}}g_{+}^{i}(e)de-\int_{E_{r}}^{E_{i}}(g_{+}^{i})^{*}(e)de\right)=2\left(\int_{E_{r}}^{E_{i}}g_{+}^{i}(e)de-\overline{\int_{E_{r}}^{E_{i}}g_{+}^{i}(e)de}\right)$$
\end{itemize}
This ends the proof of Lemma \ref{coeffut}.
\end{dem}
\input{fig13}
\subsection{Equation for the eigenvalues}
\label{demeigeneq} The following result gives a characterization
of the eigenvalues of $H_{\varphi,\varepsilon}$.
\begin{prop} \label{eigeneq2}
We assume that $(H_{V})$, $(H_{W,r})$, $(H_{W,g })$ and $(H_{J})$
are satisfied.\\
There exist $\varepsilon_{0}>0$, a neighborhood
$\mathcal{V}=\overline{\mathcal{V}}$ of $J$, two functions
$(E,\varepsilon)\mapsto\widetilde{\Phi}(E,\varepsilon)$ and
$(E,\varepsilon)\mapsto\widetilde{\Phi_{d}}(E,\varepsilon)$
defined on $\mathcal{V}\times]0,\varepsilon_{0}[$ and two
functions $(\varphi,E,\varepsilon)\mapsto
F(\varphi,E,\varepsilon)$ and $(\varphi,E,\varepsilon)\mapsto
R_{2}(\varphi,E,\varepsilon)$ defined on
$\mathbb{R}\times\mathcal{V}\times]0,\varepsilon_{0}[$ such that:
\begin{enumerate}\item $E$ is an eigenvalue of $H_{\varphi,\varepsilon}$ if and only if:
$$ F(\varphi,E,\varepsilon)=0$$
\item The function $F$ satisfies:
$$\forall\varphi\in \mathbb{R},\ \forall E\in\mathcal{V},\ \forall\varepsilon\in]0,\varepsilon_{0}[,\quad F^{*}(\varphi,E,\varepsilon)=\overline{F(\overline{\varphi},\overline{E},\varepsilon)}=F(\varphi,E,\varepsilon).$$
\item The function $\varphi\mapsto F(\varphi,E,\varepsilon)$ is $\varepsilon$-periodic and its Fourier expansion is written:
\begin{equation}
\label{decsfoud}
F(\varphi,E,\varepsilon)=\cos\left(\frac{\widetilde{\Phi}(E)}{\varepsilon}\right)+e^{-S(E)/\varepsilon}\cos\left(\frac{\widetilde{\Phi_{d}}(E)}{\varepsilon}+\frac{2\pi\varphi}{\varepsilon}+\rho\right)+e^{-S(E)/\varepsilon}R_{2}(\varphi,E,\varepsilon)
\end{equation}
\item The functions $\widetilde{\Phi}$, $\widetilde{\Phi_{d}}$
satisfy the following properties for any
$\varepsilon\in]0,\varepsilon_{0}[$:
\begin{itemize}\item $E\mapsto\widetilde{\Phi}(E,\varepsilon)$ and
$E\mapsto\widetilde{\Phi_{d}}(E,\varepsilon)$ are analytic on
$\mathcal{V}$.
\item $\widetilde{\Phi}(E,\varepsilon)=\Phi(E)+o(\varepsilon)$ and
$\widetilde{\Phi_{d}}(E,\varepsilon)=\Phi_{d}(E)+o(\varepsilon)$
uniformly for $E\in\mathcal{V}$.
\end{itemize}
\item For any $\varepsilon\in]0,\varepsilon_{0}[$, the function $(\varphi,E)\mapsto
R_{2}(\varphi,E,\varepsilon)$ is analytic on
$\mathbb{R}\times\mathcal{V}$. Besides, there exists a constant $\alpha>0$
such that,for all $\varepsilon\in]0,\varepsilon_{0}[$, and all $E$
in $\mathcal{V}$, the function $R_{2}$ satisfies the following
properties:
$$ \int_{0}^{\varepsilon}R_{2}(u,E,\varepsilon)du=0,\quad\int_{0}^{\varepsilon}R_{2}(u,E,\varepsilon)e^{\frac{ 2i\pi u}{\varepsilon}}du=0,\quad\int_{0}^{\varepsilon}R_{2}(u,E,\varepsilon)e^{\frac{-2i\pi u}{\varepsilon}}du=0,$$
$$\sup\limits_{\varphi\in\mathbb{R},E\in\mathcal{V}}|R_{2}(\varphi,E,\varepsilon)|\leq e^{-\frac{\alpha}{\varepsilon}}$$
\end{enumerate}
The functions $\Phi$, $\Phi_{d}$, $S$ are defined in Lemma
\ref{phaseactint}. $\rho$ is a real number defined in
\eqref{omegac}.
\end{prop}
Now, we prove Proposition \ref{eigeneq2}.
\begin{itemize}\item Now, it suffices to compute the Fourier
expansion of:
$$b_{g}^{-}(b_{d}^{+})^{*}(\varphi,E,\varepsilon)=\sum\limits_{n\in\mathbb{Z}}\gamma_{n}(E,\varepsilon)e^{\frac{2i
n\pi\varphi }{\varepsilon}}.$$ By using the asymptotic expansion
of the coefficients $b_{g}^{-}$ and $b_{d}^{+}$ given in Lemma
\ref{colina}, we prove that:
$$\gamma_{0}=-ie^{-\frac{i}{\varepsilon}\int_{\varphi_{r}^{-}}^{\varphi_{r}^{+}}\kappa_{i}}
[1+o(1)].$$
$$\gamma_{1}=+ie^{-\frac{i}{\varepsilon}(\int_{\varphi_{r}^{+}}^{0}\kappa_{i}+\int_{\varphi_{r}^{-}}^{0}\kappa_{i})}e^{\frac{2i}{\varepsilon}\int_{0}^{\overline{\varphi_{i}}}(\kappa_{i}-\pi)}e^{\int_{0}^{\varphi_{r}^{+}}(\omega_{+}^{i}-\omega_{-}^{i})}e^{\int_{0}^{\varphi_{i}}(\omega_{-}^{i}-\omega_{+}^{i})}[1+o(1)].$$
$$\left|\sum\limits_{n\in\mathbb{Z}\backslash\{0,1\}}\gamma_{n}e^{\frac{2i
n\pi\varphi
}{\varepsilon}}\right|=O(e^{-\alpha/\varepsilon})\quad\textrm{
pour }\varphi\in S_{Y_{0}}.$$ Actually,
$$\gamma_{0}=\alpha_{0}\beta_{0}+\sum\limits_{n\neq 0}\alpha_{n}\beta_{-n}=-i
e^{-\frac{i}{\varepsilon}\int_{\varphi_{r}^{-}}^{\varphi_{r}^{+}}\kappa_{i}}e^{\frac{1}{2}\left[\int_{0}^{\varphi_{r}^{-}}(\omega_{+}^{i}-\omega_{-}^{i})+\int_{0}^{\varphi_{r}^{-}}(\omega_{-}^{i}-\omega_{+}^{i})\right]}[1+o(1)].$$
According to \eqref{omegaa}, we simplify:
$$\left[\int_{0}^{\varphi_{r}^{-}}(\omega_{+}^{i}-\omega_{-}^{i})+\int_{0}^{\varphi_{r}^{-}}(\omega_{-}^{i}-\omega_{+}^{i})\right]=0.$$
According to \eqref{phi},
$\int_{\varphi_{r}^{-}}^{\varphi_{r}^{+}}\kappa_{i}=\Phi(E)$.
Consequently,
$$\gamma_{0}=-i e^{\frac{i\Phi(E)}{\varepsilon}}[1+o(1)].$$ We compute:
$$\gamma_{1}=\alpha_{0}\beta_{1}+\sum\limits_{n\neq 1}\alpha_{n}\beta_{1-n}$$
We start with computing $\alpha_{0}\beta_{1}$. To do that, we
deduce from equation \eqref{phida} that:
$$\int_{\varphi_{r}^{+}}^{0}\kappa_{i}(\varphi)d\varphi+\int_{\varphi_{r}^{-}}^{0}\kappa_{i}(\varphi)d\varphi-\int_{\sigma_{-}}(\kappa_{i}(\varphi)-\pi)d\varphi=\Phi_{d}(E)+iS(E).$$
$$\alpha_{0}\beta_{1}=ie^{-\frac{i}{\varepsilon}(\int_{\varphi_{r}^{+}}^{0}\kappa_{i}+\int_{\varphi_{r}^{-}}^{0}\kappa_{i})}e^{\frac{2i}{\varepsilon}\int_{0}^{\overline{\varphi_{i}}}(\kappa_{i}-\pi)}e^{\int_{0}^{\varphi_{r}^{+}}\omega_{+}^{i}+\int_{\varphi_{r}^{-}}^{0}\omega_{-}^{i}-\int_{\sigma_{+}}(\omega_{+}^{i}-\omega_{-}^{i})}[1+o(1)]+O(e^{\frac{-\alpha}{\varepsilon}})e^{-S(E)/\varepsilon}.$$
Equation \eqref{phida} leads to:
$$\int_{\varphi_{r}^{+}}^{0}\kappa_{i}(\varphi)d\varphi+\int_{\varphi_{r}^{-}}^{0}\kappa_{i}(\varphi)d\varphi-\int_{\sigma_{-}}(\kappa_{i}(\varphi)-\pi)d\varphi=\Phi_{d}(E)+iS(E).$$
Besides, according to Lemma \ref{coeffut}, we have:
$$\int_{0}^{\varphi_{r}^{+}}\omega_{+}^{i}+\int_{\varphi_{r}^{-}}^{0}\omega_{-}^{i}-\int_{\sigma_{+}}(\omega_{+}^{i}-\omega_{-}^{i})=i\rho.$$
and:
$$\alpha_{0}\beta_{1}=ie^{-S/\varepsilon}e^{-i\Phi_{d}/\varepsilon}e^{i\rho}[1+o(1)].$$
Since $S(E)\leq 2\pi\mbox{Im }\varphi_{i}(E)$, we estimate the remainder
in the expansion:
$$|\sum\limits_{n\neq
0}\alpha_{n}\beta_{-n}|=o(e^{-S/\varepsilon}).$$ Finally, for
$p\neq 0,1$, we estimate:
$$\gamma_{p}=\sum\limits_{n\in\mathbb{Z}}\alpha_{n}\beta_{p-n}.$$
For $p>1$, we have:
$$|\gamma_{p}|=e^{-S/\varepsilon}e^{-\alpha/\varepsilon}O(e^{-\frac{2\pi Y_{0}(p-1)}{\varepsilon}}).$$
Similarly, we estimate for $p<0$,
$$|\gamma_{p}|=e^{-S/\varepsilon}e^{-\alpha/\varepsilon}O(e^{-\frac{2\pi Y_{0}(|p|-1)}{\varepsilon}}).$$
\item Now, we consider $\varphi\in\mathbb{R}$. We compute the Fourier asymptotic expansion of the coefficient $d(E,\varphi,\varepsilon)$
in a neighborhood $\mathcal{U}_{0}$ of $E_{0}$:
$$d(\varphi,E,\varepsilon)=iw(f_{i},\sigma_{i}(f_{i})^{*})\left(\lambda_{0}(E,\varepsilon)+\sum\limits_{n\in\mathbb{N}^{*}}(\lambda_{n}(E,\varepsilon)e^{\frac{2i n\pi\varphi}{\varepsilon}}+(\lambda_{n})^{*}(E,\varepsilon)e^{\frac{-2i n\pi\varphi}{\varepsilon}})\right)$$
$$=i(w_{0}k'_{i})(E-W(0))\sum\limits_{n\in\mathbb{N}}u_{n}(\varphi,E,\varepsilon).$$
where
$u_{n}(\varphi,E,\varepsilon)=\lambda_{n}(E,\varepsilon)e^{\frac{2i
n\pi\varphi}{\varepsilon}}+(\lambda_{n})^{*}(E,\varepsilon)e^{\frac{-2i
n\pi\varphi}{\varepsilon}}$, pour $n\in\mathbb{N}^{*}$, et $u_{0}(\varphi,E,\varepsilon)=\lambda_{0}(E,\varepsilon)$.\\
We have:
$$u_{0}(\varphi,E,\varepsilon)=\gamma_{0}(E,\varepsilon)-\gamma_{0}^{*}(E,\varepsilon)=-ie^{i\frac{\Phi}{\varepsilon}}g(E,\varepsilon)-ie^{-i\frac{\Phi}{\varepsilon}}g^{*}(E,\varepsilon).$$
where $g(E,\varepsilon)=1+o(1)$.\\
We define
$g(E,\varepsilon)=r_{g}(E,\varepsilon)e^{i\theta_{g}(E,\varepsilon)}$
where the functions $E\mapsto r_{g}(E,\varepsilon)$ and $E\mapsto
\theta_{g}(E,\varepsilon)$ are analytic and satisfy $$
r_{g}^{*}=r_{g},\quad r_{g}=1+o(1)\quad
\theta_{g}^{*}=\theta_{g},\quad\theta_{g}=o(1).$$ We simplify:
$$u_{0}(\varphi,E,\varepsilon)=-i r_{g}(E,\varepsilon)\cos\left(\frac{\Phi(E)}{\varepsilon}+\theta_{g}(E,\varepsilon)\right).$$
Similarly, we compute:
$$u_{1}(\varphi,E,\varepsilon)=i
r_{h}(E,\varepsilon)e^{-S(E)/\varepsilon}\cos\left(\frac{\Phi_{d}+2\pi\varphi}{\varepsilon}+\rho+\theta_{h}(E,\varepsilon)\right).$$
where the functions $E\mapsto r_{h}(E,\varepsilon)$ and $E\mapsto
\theta_{h}(E,\varepsilon)$ are analytic and satisfy $$
r_{h}^{*}=r_{h},\quad r_{h}=1+o(1)\quad
\theta_{h}^{*}=\theta_{h},\quad\theta_{h}=o(1).$$ In addition, we
have the following estimate of the remainder:
$$\left|\sum\limits_{p\geq 2}u_{p}(\varphi,E,\varepsilon)\right|\leq
Ce^{\frac{-S(E)}{\varepsilon}}e^{\frac{-\alpha}{\varepsilon}}\quad\textrm{
pour }\varphi\in \mathbb{R}.$$
\item We have proved that, for $E$
in a neighborhood of $E_{0}$, the Fourier expansion of
$d(E,\varphi,\varepsilon)$ can be written:
\begin{equation}
\label{decsfou}
\frac{d(\varphi,E,\varepsilon)}{i(w_{0}k'_{i})(E-W(0))}=-i[1+o(1)]\cos\left(\frac{\Phi(E)}{\varepsilon}+o(1)\right)
\end{equation}
$$+i[1+o(1)]e^{\frac{-S(E)}{\varepsilon}}\cos\left(\frac{\Phi_{d}+2\pi\varphi}{\varepsilon}+\rho+o(1)\right)+e^{\frac{-S(E)}{\varepsilon}}O(e^{\frac{-\alpha}{\varepsilon}}).$$
The compactness of $J$ implies that there exists a finite number
of intervals $\{J_{k}\}_{k\in\{1\cdots p \}}$ such that:
\begin{enumerate}
\item $J\subset\bigcup\limits_{k\in\{1\cdots p \}}J_{k}$ \item
For any $k\in\{1,\cdots ,p-1 \}$, the intervals $J_{k}$ and
$J_{k+1}$ overlap.\item For any $k\in\{1,\cdots, p \}$, there
exists a complex neighborhood $\mathcal{U}_{k}$ of $J_{k}$ such
that the expansion (\ref{decsfou}) is satisfied on
$\mathcal{U}_{k}$.
\end{enumerate}
We shall prove that we can define some functions
$\widetilde{\Phi}$ and $\widetilde{\Phi}_{d}$ on the whole
neighborhood $\mathcal{V}=\bigcup\limits_{k\in\{1\cdots p
\}}\mathcal{U}_{k}$. To do that, we shall ``stick'' the expansions obtained on each interval.\\
The coefficient $u_{0}$ is written:
$$\forall E\in J_{k},\quad u_{0}(E,\varepsilon)=r_{0,k}(E,\varepsilon)\cos\left(\frac{\Phi(E)}{\varepsilon}+\theta_{0,k}(E,\varepsilon)\right)$$
$$\forall E\in J_{k+1},\quad u_{0}(E,\varepsilon)=r_{0,k+1}(E,\varepsilon)\cos\left(\frac{\Phi(E)}{\varepsilon}+\theta_{0,k+1}(E,\varepsilon)\right)$$
where $r_{0,k}(E,\varepsilon)=1+o(1)$ and
$\theta_{0,k}(E,\varepsilon)=o(1)$ (resp.
$r_{0,k+1}(E,\varepsilon)=1+o(1)$ and
$\theta_{0,k+1}(E,\varepsilon)=o(1)$) for $E\in J_{k}$ (resp.
$E\in J_{k+1}$).We get that:
$$r_{0,k}(E,\varepsilon)=r_{0,k+1}(E,\varepsilon)=r_{0}(E,\varepsilon)\textrm{ et } \theta_{0,k}(E,\varepsilon)=\theta_{0,k+1}(E,\varepsilon)=\theta_{0}(E,\varepsilon)\textrm{ for
}E\in J_{k}\cap J_{k+1}$$ The function $\widetilde{\Phi}$ defined
by its restrictions to each $\mathcal{U}_{k}$ is analytic on $\mathcal{V}$.\\
The case of $\widetilde{\Phi_{d}}$ is treated similarly.
\end{itemize}
Defining
$$F(\varphi,E,\varepsilon)=\frac{d(\varphi,E,\varepsilon)}{i(w_{0}k'_{i})(E-W(0))r_{0}(E,\varepsilon)},$$
we finish the proof of Proposition \ref{eigeneq2}.
\subsection{Localization of the eigenvalues}
In this section, we deduce Theorem \ref{eigenloc} from Proposition \ref{eigeneq2}. \\
We solve equation $F(\varphi,E,\varepsilon)=0$, where $F$ is
described in (\ref{decsfoud}).
\subsubsection{Energy levels $E^{(l)}(\varepsilon)$}
\label{koe} For $E\in \mathcal{V}$, we start with solving:
\begin{equation}
\label{phasemodif}
\cos\frac{\widetilde{\Phi}(E,\varepsilon)}{\varepsilon}=0
\end{equation}
$E\mapsto\widetilde{\Phi}(E,\varepsilon)$ is a real analytic
function. For a sufficiently small $\varepsilon_{0}$, by Lemma
\ref{phaseactint}, there exists a constant $m>0$ such that:
\begin{equation} \label{phasemodifdiff}
\forall
E\in\mathcal{V},\quad\forall\varepsilon\in]0,\varepsilon_{0}[,\quad
|\widetilde{\Phi}'(E,\varepsilon)|\geq m
\end{equation}
Consequently, equation \eqref{phasemodif} has a finite number of
zeros in $J$. We denote them by $E^{(l)}(\varepsilon)$, for
$l\in\{L_{-}(\varepsilon),\dots,L_{+}(\varepsilon)\}$. They are
given by:
\begin{equation}
\label{zeroa}
\frac{\widetilde{\Phi}(E^{(l)}(\varepsilon),\varepsilon)}{\varepsilon}=l\pi+\frac{\pi}{2},\quad\quad\forall
l\in\{L_{-}(\varepsilon),\ldots,L_{+}(\varepsilon)\}.
\end{equation}
and satisfy:
\begin{equation}
\label{ecart}
E^{(l+1)}(\varepsilon)-E^{(l)}(\varepsilon)=\frac{1}{\widetilde{\Phi}'(E^{(l)}(\varepsilon))}\pi\varepsilon+o(\varepsilon).
\end{equation}
The distances between two consecutive zeros are of order
$\varepsilon$. Precisely, by combining \eqref{phasemodifdiff} with
\eqref{ecart}, we obtain that there exists a constant $c>0$ such
that:
\begin{equation}
\label{ecarta}
\frac{1}{c}\varepsilon<|E^{(l+1)}(\varepsilon)-E^{(l)}(\varepsilon)|<c\varepsilon,\quad
\forall l\in \{L_{-}(\varepsilon),\ldots,L_{+}(\varepsilon)-1\}
\end{equation}
First, we prove that the zeros of $F$ are in an exponentially
small neighborhood of the points $E^{(l)}(\varepsilon)$.
\subsubsection{First order approximation}
We give a first order approximation of the zeros of $F$.\\
We set
$$a_{0}(E,\varepsilon)=\cos\frac{\widetilde{\Phi}(E,\varepsilon)}{\varepsilon}.$$
We can assume that the neighborhood $\mathcal{V}$ is sufficiently
small and such that, for any $E\in\mathcal{V}$,
$$\mbox{Re }(S(E))>\beta>0.$$ Then, there exists a positive constant $A$
such that
$$|F(\varphi,E,\varepsilon)-a_{0}(E,\varepsilon)|<Ae^{-\beta/\varepsilon}.$$
In addition, we have the following inequality:
\begin{equation}
\label{cosinus} \exists C>0/\quad
\left|\cos\frac{\widetilde{\Phi}(E,\varepsilon)}{\varepsilon}\right|\geq\frac{C}{\varepsilon}d(E,\bigcup\limits_{l\in\{L_{-},\cdots,L_{+}\}}E^{(l)}(\varepsilon)).
\end{equation}
Actually, there exists a constant $c>0$ such that:
$$|\cos\theta|\geq c d(\theta,\pi\mathbb{Z}+\pi/2).$$ By using \eqref{phasemodifdiff}, we obtain the relation:
$$|\widetilde{\Phi}(E,\varepsilon)-\widetilde{\Phi}(E^{(l)}(\varepsilon),\varepsilon)|\geq m |E-E^{(l)}(\varepsilon)|$$
and finally:
$$\left|\cos\frac{\widetilde{\Phi}(E,\varepsilon)}{\varepsilon}\right|\geq\frac{C}{\varepsilon}d\left(E,\bigcup\limits_{l\in\{L_{-},\cdots,L_{+}\}}E^{(l)}(\varepsilon)\right).$$
For $z_{0}\in\mathbb{C}$ and $r>0$, we define $$D(z_{0},r)=\{z\in\mathbb{C}\ ;\
|z-z_{0}|<r\}.$$ Inequality (\ref{cosinus}) implies that there are
no zeros of $F$ outside exponentially small neighborhoods of the
points $E^{(l)}(\varepsilon)$. Precisely, there exists a positive
constant $D$ such that, if $r\geq D\varepsilon
e^{-\beta/\varepsilon}$, then for any $E\in
\partial D(E^{(l)}(\varepsilon),r)$, we have:
$$|F(\varphi,E,\varepsilon)-a_{0}(E,\varepsilon)|<|a_{0}(E,\varepsilon)|.$$
Rouch{\'e}'s Theorem implies that, for any $l$, $F$ has exactly
one zero $E_{l}(\varphi,\varepsilon)$, in each neighborhood
$D(E^{(l)}(\varepsilon),D\varepsilon e^{-\beta/\varepsilon})$ of
$E^{(l)}(\varepsilon)$. The relation $F=F^{*}$ allows us to
recover that the eigenvalues are real. Indeed, if $F(E)=0$, $\overline{E}$ is also a zero of $F$. By uniqueness, we obtain that $E=\overline{E}$.\\
We set:
$$E_{l}(\varphi,\varepsilon)=E^{(l)}(\varepsilon)+\varepsilon\lambda_{l}(\varphi,\varepsilon).$$
We know that $\lambda_{l}(\varphi,\varepsilon)$ is exponentially
small. Now, we compute its asymptotic behavior.
\subsubsection{Second order approximation}
We define:
$$a_{1}(\varphi,E,\varepsilon)=F(\varphi,E,\varepsilon)-a_{0}(E,\varepsilon).$$
We write
$$e^{-S(E_{l}(\varphi,\varepsilon))/\varepsilon}=e^{-S(E^{(l)}(\varepsilon))/\varepsilon}(1+O(\lambda_{l}(\varphi,\varepsilon))).$$
Similarly, with the help of the modified phase
$\widetilde{\Phi_{d}}$, we obtain the expansion:
$$\cos\left(\frac{\widetilde{\Phi_{d}}(E_{l}(\varphi,\varepsilon))+2\pi\varphi+\rho\varepsilon}{\varepsilon}\right)=\cos\left(\frac{\widetilde{\Phi_{d}}(E^{(l)}(\varepsilon))+2\pi\varphi+\rho\varepsilon}{\varepsilon}\right)+O(e^{-\beta/\varepsilon})$$
The expansion of $a_{1}$ can be written:
$$a_{1}(\varphi,E_{l}(\varphi,\varepsilon),\varepsilon)=a_{1}(\varphi,E^{(l)}(\varepsilon),\varepsilon)(1+r(\varphi,E^{(l)}(\varepsilon),\varepsilon)).$$
Moreover, we use the first order Taylor's expansion of the
function $E\mapsto a_{0}(E,\varepsilon)$:
$$a_{0}(E_{l}(\varphi,\varepsilon),\varepsilon)=(-1)^{l+1}\widetilde{\Phi}'(E^{(l)}(\varepsilon),\varepsilon)\lambda_{l}(\varphi,\varepsilon)(1+r(\varphi,E^{(l)},\varepsilon))=(-1)^{l+1}\Phi'(E^{(l)})\lambda_{l}(\varphi,\varepsilon)(1+o(1)).$$
By combining these computations, we finally obtain:
$$\lambda_{l}(\varphi,\varepsilon)=\frac{(-1)^{l+1}}{\Phi'(E^{(l)}(\varepsilon))}e^{-S(E^{(l)}(\varepsilon))/\varepsilon}\left(\cos\left(\frac{\widetilde{\Phi_{d}}(E^{(l)}(\varepsilon),\varepsilon)+2\pi\varphi}{\varepsilon}+\rho\right)+o(1)\right).$$
\subsection{Application to the trace formula} \label{trform2} In
\cite{Di1}, the author proves the existence of an asymptotic
expansion of ${\mathrm tr} [f(H_{\varphi,\varepsilon})]$, for $f\in
C_{0}^{\infty}$, when $\textrm{Supp }f$ is disjoint from the bands
of $H_{0}$; in addition, he computes explicitly the first and
second
terms of this expansion.\\
Corollary \ref{cordim} allows us to recover these terms.
\subsubsection{}
Let $J$ be an interval satisfying $(H_{J})$. Particularly, $J$ is
such that $J\cap(\sigma_{ac}\cup\sigma_{sc})=\emptyset$. For $f\in
C_{0}^{\infty}$, with $\textrm{Supp }f\subset J$, we compute:
$${\mathrm tr}[f(H_{\varphi,\varepsilon})]=\sum\limits_{l\in\{L_{-}(\varepsilon),...,L_{+}(\varepsilon)\}}f(E_{l}(\varphi,\varepsilon)).$$
Let $\beta>0$ be such that $S(E)>\beta$ for any $E\in J$;
according to Theorem \ref{eigenloc}, we know that there exists a
constant $C>0$ such that:
$$\forall u\in[0,\varepsilon],\quad\left|{\mathrm tr}[f(H_{\varphi,\varepsilon})]-{\mathrm tr}[f(H_{u,\varepsilon})]\right|<C\sum\limits_{l\in\{L_{-}(\varepsilon),...,L_{+}(\varepsilon)\}}\varepsilon e^{-\beta/\varepsilon}.$$
By integrating with respect to $u$, we obtain that:
$$\textrm{tr
}[f(H_{\varphi,\varepsilon})]=\frac{1}{\varepsilon}\int_{0}^{\varepsilon}\textrm{tr
}[f(H_{u,\varepsilon})]du+O(e^{-\beta/\varepsilon}).$$ According
to Theorem \ref{eigenloc}, we know that there exists a constant
$C$ such that
$$\forall u\in[0,\varepsilon],\quad\left|{\mathrm tr}
[f(H_{u,\varepsilon})]-\sum\limits_{l\in\{L_{-}(\varepsilon),...,L_{+}(\varepsilon)\}}f(E^{(l)}(\varepsilon))\right|<C
e^{-\beta/\varepsilon}$$ By integration, we obtain:
\begin{equation}
\label{formtrace}
\frac{1}{\varepsilon}\int_{0}^{\varepsilon}\textrm{tr
}[f(H_{u,\varepsilon})]du=\sum\limits_{l\in\{L_{-}(\varepsilon),...,L_{+}(\varepsilon)\}}f(E^{(l)}(\varepsilon))+O(e^{-\beta/\varepsilon})
\end{equation}
Now, we estimate:
$$\sum\limits_{l\in\{L_{-}(\varepsilon),...,L_{+}(\varepsilon)\}}f(E^{(l)}(\varepsilon))=\sum\limits_{l\in\{L_{-}(\varepsilon),...,L_{+}(\varepsilon)\}}f\circ\widetilde{\Phi}^{-1}(\varepsilon(l\pi+\pi/2))$$
\subsubsection{} Now, we compute this last term.
\begin{lem}
\label{trace} Let $f$ be a function in $C_{0}^{\infty}$ such that
$\textrm{Supp }f\subset J$. The trace of $H_{\varphi,\varepsilon}$
has the following asymptotic behavior:
$$\int_{0}^{\varepsilon}{\mathrm tr}
[f(H_{u,\varepsilon})]du=\frac{1}{\pi}\int_{J}
f(\mathcal{E})\widetilde{\Phi}'(\mathcal{E},\varepsilon)d\mathcal{E}+O(\varepsilon^{\infty})$$
\end{lem}
\begin{dem}
The proof of this Lemma is based on elementary results of real
analysis.
\begin{itemize} \item We apply the Poisson formula
to the function $f\circ\widetilde{\Phi}^{-1}\in C_{0}^{\infty}$:
$$\varepsilon\sum\limits_{l\in\mathbb{Z}}f\circ\widetilde{\Phi}^{-1}(\varepsilon(l\pi+\pi/2))=2\sum\limits_{n\in\mathbb{Z}}(-1)^{n}\widehat{(f\circ\widetilde{\Phi}^{-1})}\left(\frac{2n}{\varepsilon}\right).$$
Besides, the Fourier transform of $f\circ\widetilde{\Phi}^{-1}$
satisfies the estimates:
$$\forall\nu>1,\quad\exists\ C_{\nu}>0,\textrm{ such that }\left|\widehat{(f\circ\widetilde{\Phi}^{-1})}\left(\frac{2n}{\varepsilon}\right)\right|\leq
C_{\nu}\frac{\varepsilon^{\nu}}{n^{\nu}}.$$ Actually, since
$f\circ\widetilde{\Phi}^{-1}$ is $C^{\nu}$,
$|\xi^{\nu}\widehat{f\circ\widetilde{\Phi}^{-1}}(\xi)|$ is bounded.\\
This leads to:
$$\varepsilon\sum\limits_{p\in\mathbb{Z}}f\circ\widetilde{\Phi}^{-1}(\varepsilon(p\pi+\pi/2))=2\widehat{(f\circ\widetilde{\Phi}^{-1})}(0)+O(\varepsilon^{\infty}).$$
\item It remains to prove that:
$$2\widehat{(f\circ\widetilde{\Phi}^{-1})}(0)=\frac{1}{\pi}\int
f\circ\widetilde{\Phi}^{-1}(u)du.$$ With the substitution
$u=\widetilde{\Phi}(\mathcal{E})$, we obtain that:
$$2\widehat{(f\circ\widetilde{\Phi}^{-1})}(0)=\frac{1}{\pi}\int_{J}
f(\mathcal{E})\widetilde{\Phi}'(\mathcal{E},\varepsilon)d\mathcal{E}$$ This completes the
proof of Lemma \ref{trace}.
\end{itemize}
\end{dem}
\subsubsection{Conclusion}
To get an asymptotic expansion of the trace at any order, it
suffices to know an asymptotic expansion of the modified phase at
any order. Our computations are not accurate enough, but we know
that $\widetilde{\Phi}'(\mathcal{E})=\Phi'(E)+o(\varepsilon)$, hence:
$$\frac{1}{\pi}\int_{J} f(\mathcal{E})\widetilde{\Phi}'(\mathcal{E})d\mathcal{E}=\frac{1}{\pi}\int_{J}
f(\mathcal{E})\Phi'(\mathcal{E})d\mathcal{E}+o(\varepsilon).$$ To transform the right member
of previous equality, we do the substitution
$(\kappa,u)\mapsto(E(\kappa)+W(u),u)$, which implies:
$$\frac{1}{\pi}\int_{J}
f(\mathcal{E})\Phi'(\mathcal{E})d\mathcal{E}=\frac{1}{2\pi}\int_{[-\pi,\pi]}\int_{\varphi_{r}^{-}}^{\varphi_{r}^{+}}f(E(\kappa)+W(u))d\kappa
du$$ We finally obtain:
$$\int_{0}^{\varepsilon}\textrm{tr
}[f(H_{u,\varepsilon})]du=\frac{1}{2\pi}\int_{[-\pi,\pi]}\int_{\varphi_{r}^{-}}^{\varphi_{r}^{+}}f(E(\kappa)+W(u))d\kappa
du+o(\varepsilon)$$ This ends the proof of Corollary \ref{cordim}.
\subsection{Asymptotic behavior of the eigenvalues}
Now, we give a second application of Theorem \ref{eigenloc} for
the computation of the asymptotic behavior of the eigenvalues of
$H_{\varphi,\varepsilon}$. Such a computation is outlined in
\cite{CDS}, in the case $V=0$. We obtain an explicit result at
first order.\\
Under the assumptions of Theorem \ref{eigenloc}, $E_{r}$ is the
only end of $\sigma(H_{0})$ belonging to $(E-W)(\mathbb{R})$. We define:
\begin{equation}
\label{rnp} d_{p}(E_{n})=\lim\limits_{\stackrel{E\rightarrow
E_{n},}{ E\in[E_{n},E_{p}]}}\frac{k(E)-k(E_{n})}{\sqrt{E-E_{n}}}
\end{equation}
\begin{cor}
Let $H_{\varphi, \varepsilon}$ verify the assumptions of Theorem
\ref{eigenloc}. The eigenvalues $E^{(l)}(\varphi,\varepsilon)$ of
$H_{\varphi,\varepsilon}$ have the following asymptotic behavior:
$$E^{(l)}(\varphi,\varepsilon)=\widetilde{\Phi}^{-1}(\varepsilon(l\pi+\pi/2))+0(\varepsilon^{\infty}).$$
Particularly, $E^{(l)}(\varphi,\varepsilon)$ has the following
Taylor expansion at first order in $\varepsilon$ :
$$E^{(l)}(\varphi,\varepsilon)=E_{r}+W(0)+\sqrt{\frac{W"(0)}{2}}\frac{1}{d_{i}(E_{r})}(2l+1)\varepsilon+o(\varepsilon),$$
where $d_{i}(E_{r})$ is defined by \eqref{rnp}.
\end{cor}
\begin{dem}
The first equality is obvious. It suffices to give an expansion of
$\widetilde{\Phi}^{-1}(\varepsilon(l\pi+\pi/2))$. To do that, we
compute an expansion at first order of:
$$\Phi(E_{r}+W(0)+\alpha)=\int_{\varphi_{r}^{-}(E_{r}+W_{-}+\alpha)}^{\varphi_{r}^{+}(E_{r}+W_{-}+\alpha)}k(E_{r}+W(0)+\alpha-W(u))du.$$
The mapping $W$ is a bijection from $[0,\varphi_{r}^{+}]$ to
$[W(0),E_{r}]$. By the substitution $\alpha v=W(0)+\alpha-W(u)$,
we get that:
$$\int_{0}^{\varphi_{r}^{+}(E_{r}+W(0)+\alpha)}=\alpha\int_{0}^{1}\frac{k(E_{r}+\alpha v)}{W'\circ W^{-1}(W(0)+\alpha(1-v))}dv.$$
But, $\lim\limits_{\alpha\rightarrow 0}\frac{k(E_{r}+\alpha
v)}{W'\circ W^{-1}(W_{-}+\alpha(1-v))}=\frac{d_{i}(E_{r})}{\sqrt{2
W"(0)}}\frac{\sqrt{v}}{\sqrt{1-v}}$.\\
Similarly, on $[\varphi_{r}^{-},0]$, we have:
$$\Phi(E_{r}+W(0)+\alpha)=d_{i}(E_{r})\frac{\pi}{2}\sqrt{\frac{2}{W"(0)}}\alpha[1+o(1)].$$
Consequently, by inverting the expansion of $\widetilde{\Phi}$ in
the neighborhood of $E_{r}+W(0)$, we prove the result.
\end{dem}\\
We point out that, as in \ref{trform2}, a more accurate asymptotic
expansion of $\widetilde{\Phi}$ would give a better result on the
eigenvalues.
\bibliographystyle{plain}
|
{
"timestamp": "2005-03-12T19:58:38",
"yymm": "0503",
"arxiv_id": "math-ph/0503031",
"language": "en",
"url": "https://arxiv.org/abs/math-ph/0503031"
}
|
\section{Introduction}
As is well known,
\cite{UhlOrn}, \cite{Chandra}, \cite{Bal},
the ensemble average of the stochastic evolutions in {velocity space}
of a Brownian test particle\footnote{For the beginnings of the theory of
Brownian motion, see the collection of Einstein's papers
with commentary \cite{Einstein}.}
of unit mass,
immersed in a drifting uniform heat bath of fixed temperature $T$ and
constant drift velocity $\uV$, is governed by the Fokker--Planck equation
with prescribed constant coefficients of diffusion and (linear) friction,
\begin{equation}
\partial_t f (\vV;t)
=
\partial_{\vV}\cdot\Big({T}\partial_{\vV}f(\vV;t)
+
\big(\vV - \uV\big) f(\vV;t)\Big).
\label{FPbrown}
\end{equation}
Here, $f(\, .\, ;t):\mathbb{R}^3\to\mathbb{R}_+$ is the ensemble's probability density
function on velocity space at time $t\in \mathbb{R}_+$, and an overall constant
has been absorbed in the time scale.
Of course, we could also shift $\vV$ to obtain $\uV=\mathbf{0}$, then
rescale $\vV$, $t$, and $f$ to obtain $T = 1$;
however, for pedagogical purposes we refrain from doing so.
The solution $f (\vV;t)$ of \refeq{FPbrown} is given by
$f (\vV;t) = \int_{\mathbb{R}^3}G_t(\wV,\vV|\uV;T)f_0(\wV)\mathrm{d}^3\wV$,
where $f_0(\vV)\equiv f (\vV;0)$ and
\begin{equation}
G_t(\wV,\vV|\uV;T)
=
\left({2\pi T(1-e^{-2t})}\right)^{-\frac{3}{2}}
\exp\left( -\frac{1}{2 T}\frac{|\vV-\uV -\wV e^{-t}|^2}{1-e^{-2t}} \right)
\label{OUkernel}
\end{equation}
is the Green function for \refeq{FPbrown}, see \cite{UhlOrn}, \cite{Chandra}.
In its standard form, i.e. with
$ T = 1$ and $\uV=\mathbf{0}$, \refeq{OUkernel} is known as the (Mehler)
kernel of the adjoint Ornstein-Uhlenbeck semigroup
(a.k.a. Fokker--Planck semigroup).
Over the years, the Ornstein-Uhlenbeck semigroup and its adjoint
have come to play an important
role in several branches of probability theory \cite{Hsu} related,
in some form, to Brownian motions.
The fact that the explicitly known kernel \refeq{OUkernel}
of the Fokker--Planck semigroup readily lends itself to analytical
estimates has led to useful applications also outside the realm
of probability theory.
In particular, in recent years the Fokker--Planck semigroup has found
applications in kinetic theory, the subfield of transport theory
which is concerned with the approach to equilibrium and the response to
driving external forces of individual continuum systems not in
local thermal equilibrium; see, for instance, the review \cite{Vil}.
However, the linear Fokker--Planck equation itself, \refeq{FPbrown},
usually is not thought of as a {kinetic} equation for
the particle density function on velocity space of an
\textit{individual, isolated} space-homogeneous system of particles in
some compact domain, which perform a microscopic autonomous dynamics
that may be deterministic or stochastic but should satisfy the usual
conservation laws of mass (particle number), energy and,
depending on the shape of the domain in physical space
and its boundary conditions, also momentum and angular momentum.
Evidently the very meaning of $f$ and the parameters $\uV$ and $T$ in
\refeq{FPbrown} voids this interpretation.
Yet, with a re-interpretation of $f$, $\uV$ and $T$ it \emph{is}
possible to assign to \refeq{FPbrown} a kinetic meaning.
Incidentally, the first result showing that at least a partial
re-interpretation of \refeq{FPbrown} in this direction is
possible can be found in a paper by Villani \cite{Vil98} who, in his study
of the space-homogeneous Landau equation for the weak deflection (i.e. Landau)
limit of a gas of particles with Maxwellian molecular interactions, discovered
that for isotropic velocity distribution functions $f$ (and only for these)
the Landau equation is identical to \refeq{FPbrown}, with parameters
$\uV=\vect{0}$ and $T$ matched to guarantee energy conservation.
For general non-isotropic data the Landau equation for Maxwell
molecules is identical to a more complicated equation than \refeq{FPbrown}.
To pave the ground for a complete re-interpretation of \refeq{FPbrown},
which requires re-assigning the meaning of $f$, $\uV$ and $T$, we first
note that by the linearity of \refeq{FPbrown} we can scale $f$ to any
positive normalization we want.
We now introduce the following functionals of~$f$,
\noindent
the ``mass of $f$''
\begin{equation}
m(f) = \int_{\mathbb{R}^3} f(\vV;t)\mathrm{d}^3\vV\,,
\label{mOFf}
\end{equation}
the ``momentum of $f$''
\begin{equation}
\pV(f) = \int_{\mathbb{R}^3} \vV f(\vV;t)\mathrm{d}^3\vV\,,
\label{pOFf}
\end{equation}
and the ``energy of $f$''
\begin{equation}
e(f) = \int_{\mathbb{R}^3} \frac{1}{2}|\vV|^2 f(\vV;t)\mathrm{d}^3\vV\,.
\label{eOFf}
\end{equation}
The ``angular momentum of $f$'' for a space-homogeneous $f(\vV;t)$
is simply $\jV(f) = \xV_{\mathrm{CM}}\times\pV(f)$, with $\xV_{\mathrm{CM}}$
the center of mass of the system, but this does not add any further insight
and hence will not be considered explicitly.
The functionals \refeq{mOFf}, \refeq{pOFf}, and \refeq{eOFf}
inherit some time dependence from
the solution $f(\,.\,;t)$ of \refeq{FPbrown}, but to find
this dependence explicitly it is not necessary to solve for $f$ first.
Indeed, it is an elementary exercise in integration by parts
to extract from \refeq{FPbrown} the following linear
evolution equations with constant coefficients for $m$, $\pV$, and $e$,
\begin{equation}
\dot{m} = 0\,,
\label{mDOT}
\end{equation}
\begin{equation}
\dot{\pV} = m\uV - \pV\,,
\label{pDOT}
\end{equation}
\begin{equation}
\dot{e} = 3 T - 2e + \uV\cdot\pV\,,
\label{eDOT}
\end{equation}
\newpage
\noindent
which, beside the conservation of mass, i.e. $m(f) = m(f_0)$,
describe the exponentially fast convergence to a stationary state
$\pV(f) \leadsto m(f_0)\uV$ and
$e(f)\leadsto \frac{3}{2} T+ \frac{1}{2}m(f_0)|\uV|^2$.
While all this is of course quite trivial and well known,
the relevant fact to realize here is that whenever the
energy and the momentum of the initial $f_0$ equal these asymptotically
stationary values, viz. if $\pV(f_0) = m(f_0)\uV$ and
$e(f_0) = \frac{3}{2} T+ \frac{1}{2}m(f_0)|\uV|^2$, then beside
mass $m$, also energy $e$ and momentum $\pV$ will be conserved.
Conservation of mass, energy, and momentum for such a large subset
of initial data $f_0$ does not yet mean that we may already think
of the linear
equation \refeq{FPbrown} as a kinetic equation, which should
conserve mass, energy, and (depending on the shape of the domain
in physical space and its boundary conditions) also momentum
for \emph{all}
initial data, no matter what their mass, energy and momentum are;
moreover, a genuine kinetic equation for particles with (pair or higher order)
interactions must express the time derivative of $f$ in terms of an at
least\footnote{The Boltzmann, the Landau, and the Vlasov kinetic
equations have bilinear ``interaction operators,''
the Balescu--Lenard--Guernsey equation has a higher order
nonlinearity which reduces to the bilinear format in the
long wavelength regime.}
bilinear operator in $f$.
However, with the help of \refeq{mOFf}, \refeq{pOFf} and \refeq{eOFf}
we now replace $T$ and $\uV$ in \refeq{FPbrown} to obtain just such a
kinetic equation.
Indeed, consider the {\emph{a priori}} nonlinear Fokker--Planck equation
\begin{equation}
\partial_t f (\vV;t)
=
\partial_{\vV}\cdot\Big(\frac{1}{3}\big(2e(f)m(f)- |\pV(f)|^2\big)
\partial_{\vV}f(\vV;t)
+
\big(m(f)\vV - \pV(f)\big) f(\vV;t)\Big),
\label{FPkin}
\end{equation}
where $f(\, .\, ;t):\mathbb{R}^3\to\mathbb{R}_+$ now is a particle density function
on velocity space at time $t\in \mathbb{R}_+$.
The right-hand side of \refeq{FPkin} is a sum of a bilinear and a trilinear
operator acting on $f$ which now guarantees conservation of mass, momentum,
and energy for \emph{all} initial data $f_0\geq 0$, as verified by repeating
the easy exercise in elementary integrations by parts using
\refeq{FPkin} to find
$\dot{m} = 0$ as well as $\dot{\pV} = m\pV - \pV m = \mathbf{0}$ and
$\dot{e} = 2em - |\pV|^2 m - 2em + |\pV|^2m = 0$.
Of course, \emph{after this fact} of mass, momentum, and energy conservations
the {\emph{a priori}} nonlinear equation \refeq{FPkin}
in effect becomes just a completely and explicitly solvable
linear\footnote{In this sense \refeq{FPkin} is
``almost nonlinear,'' or ``essentially linear,''
depending on one's viewpoint.}
Fokker--Planck equation \refeq{FPbrown}, only now with parameters
$\uV$ and $T$ which are not prescribed but
determined through the initial data $f_0$, viz.
$\uV = \pV(f_0)/m(f_0)\equiv \uV_0$ and
$\frac{3}{2} T = e(f_0) - |\pV(f_0)|^2/2m(f_0)\equiv \varepsilon_0$;
we also set $m(f_0)\equiv m_0$ and $e(f_0) = e_0$.
Accordingly, \refeq{FPkin}
inherits from \refeq{FPbrown} the feature that,
as $t\to\infty$, its solutions $f$ converge pointwise
exponentially fast to the Maxwellian equilibrium state
\begin{equation}
f_{\mathrm{M}}(\vV) =
m_0 \left(\frac{3}{4\pi\varepsilon_0}\right)^{\frac{3}{2}}
\exp\left( -\frac{3|\vV -\uV_0|^2}{4\varepsilon_0} \right),
\end{equation}
with monotonically increasing relative entropy
\begin{equation}
S(f|f_{\mathrm{M}})
=
- \int_{\mathbb{R}^3} f(\vV;t)\ln \frac{f(\vV;t)}{f_{\mathrm{M}}(\vV)}\mathrm{d}^3\vV
\end{equation}
which in fact approaches its maximum value $0$ exponentially fast.
Since (\ref{FPkin}) displays all the
familiar features of a kinetic equation (formal nonlinearity;
conservation laws of mass, momentum, energy; an $H$-Theorem;
approach to equilibrium; Maxwellian equilibrium states), at this point
we may legitimately contemplate \refeq{FPkin} as a kinetic equation of
some spatially homogeneous, isolated system of $N$ interacting particles
in a compact spatial domain compatible with momentum conservation (e.g. a
rectangle with periodic boundary conditions).
In the remainder of this paper we show explicitly how \refeq{FPkin}
arises from the Kolmogorov equation\footnote{In the physics literature,
the Kolmogorov equation for an $N$-particle Markov process is
traditionally called ``master equation".}
for the adjoint evolution of an underlying $N$-particle Markov process
in the limit $N\to\infty$.
We use the strategy originally introduced by Kac \cite{Kac} in 1956
in the context of his work on a caricature of the Boltzmann equation;
for important recent work on Kac's original program,
see \cite{CarLoss}.
As Kac realized, the crucial property that needs
to be established in order to validate the $N\to\infty$ limit is
what he called ``propagation of chaos," which loosely speaking
means that if the particle velocities are uncorrelated at $t=0$,
they remain uncorrelated at later times;
this can be rigorously true only on the continuum scale
in the limit $N\to\infty$.
Interestingly enough, by adding some suitable lower order terms to the
putatively simplest $N$-particle Markov process that leads to the
(kinetic) Fokker--Planck equation in the limit $N\to\infty$, the corresponding
Kolmogorov equation for an ensemble of such isolated $N$-particle systems
can be simplified to be just the diffusion equation on the $3N-4$-dimensional
manifold (a sphere) of constant energy and momentum.
Since therefore both the finite-$N$ and the infinite-$N$ equations are
exactly solvable, the kinetic limit $N\to\infty$ can be carried out explicitly
and studied in great detail.
For this reason we actually defer the discussion of the underlying
$N$-particle process to Appendix Ab while in the main part of our paper
we analyse the diffusion equation on $\mathbb{S}^{3N-4}_{\sqrt{2N\varepsilon_0}}$ and derive
from it the kinetic Fokker--Planck equation on $\mathbb{R}^3$.
Technically, we apply the Laplace--Beltrami operator to a probability
density on $\mathbb{S}^{3N-4}_{\sqrt{2N\varepsilon_0}}$ and then integrate out $N-n$
velocities over their constrained domain of accessibility.
Taking next the limit $N\to\infty$ yields a Fokker--Planck
operator acting on the $n$-th marginal density on $\mathbb{R}^{3n}$.
Thus we obtain a linear Fokker--Planck hierarchy of equations indexed by $n$.
Using the Hewitt--Savage decomposition theorem, the hierarchy is seen to
be generated by the single, {\emph{a priori}}
nonlinear kinetic Fokker--Planck equation
\refeq{FPkin} which in view of the conservation laws is equivalent to the
essentially linear Fokker--Planck equation \refeq{FPbrown}
with constant parameters which are determined by the initial data.
Experts in probability theory may have noticed a similarity between the first
part of our program and what has been called the ``Poincar\'e limit''
\cite{Bak}; in fact, our approach is ``dual'' to Bakry's approach.
More specifically, Bakry \cite{Bak} has shown that the action of the
Laplace--Beltrami operator for $\mathbb{S}^N_{\sqrt{N}}\hookrightarrow \mathbb{R}^{N+1}$
on a probability density function over a ``radial'' coordinate
axis of $\mathbb{S}^N_{\sqrt{N}}$ becomes identical, in the limit $N\to\infty$,
to the action of the Ornstein--Uhlenbeck operator on the same density viewed as
a function over $\mathbb{R}$. Obviously, whenever the ``radial" function is
obtained by taking the marginal of a probability density over
$\mathbb{S}^N_{\sqrt{N}}$, i.e. by integrating out the $N-1$ Cartesian
coordinates of the embedding space which are perpendicular to a
fixed ``radial'' direction, the Ornstein--Uhlenbeck operator acts on the
limiting marginal density as $N\to\infty$.
This relationship between the operators is reflected at the spectral level
by the convergence of the whole structure of orthogonal eigenfunctions
of the Laplacian on $\mathbb{S}^N_{\sqrt{N}}$ (hyper-spherical harmonics) to the
orthogonal eigenfunctions of the Ornstein--Uhlenbeck operator on $\mathbb{R}$
(Hermite polynomials multiplied by the square root of their Gaussian
weight function); one of the earliest works is \cite{Mehler}, while
more recent works on the Poincar\'e limit, containing
interesting connections with the theory of Markov semigroups,
are \cite{Bak} and \cite{BakMaz}.
Our procedure is ``dual'' to Bakry's approach in the sense that we integrate
out subsets of the Cartesian variables of the embedding space \emph{after}
having applied the Laplace--Beltrami operator to a probability density on the
high-dimensional sphere, thereby obtaining the \emph{adjoint}
Ornstein--Uhlenbeck operator acting on the respective marginals;
in addition, while Bakry considers only mass and energy conservation, we
consider conservation of mass, energy, and momentum.
Incidentally, our work is not inspired by Bakry's works on the
Poincar\'e limit, nor by Villani's discovery about the isotropic
evolution of the space-homogeneous Landau equation, about both of
which we learned only after our own findings.
Rather, our study of the diffusion equations on the $3N-C$-dimensional spheres
of constant energy ($C=1$), respectively energy and momentum ($C=4$), which
began in \cite{KieLan04}, was originally conceived of as a
\emph{technically simpler primer} for our investigation (also in
\cite{KieLan04}) of the Balescu--Prigogine master equation for
Landau's kinetic equation.
And while the present paper is also a technical continuation of
\cite{KieLan04}, in the sense that here we supply various calculations
that we had announced in \cite{KieLan04}, the main purpose of the present
paper is to amplify the conceptual spin-off of our technical investigations,
the \emph{new physical interpretation} of one of the simplest and best known
linear transport equations as an (almost nonlinear) kinetic equation.
As should be clear from our discussion in this introduction,
this kinetic theory interpretation of the prototype Fokker--Planck
equation may have been suspected by others long ago, yet we have not
been able to find the whole story in the literature.
In what follows, for the sake of simplicity we set $m_0 =1$,
and accordingly\footnote{Setting $m_0=1$ means we should now
speak of the energy per particle $e_0$, the
thermal energy per particle $\varepsilon_0$,
and the momentum per particle $\pV_0 (=\uV_0)$.}
obtain $\pV(f_0) \equiv \uV_0$ and
$e(f_0) - |\pV(f_0)|^2/2 = e_0 - |\uV_0|^2/2 \equiv \varepsilon_0$.
With these simplifications \refeq{FPkin} now becomes
\begin{equation}
\partial_t f (\vV;t)
=
\partial_{\vV}\cdot\Big(\frac{2}{3}\varepsilon_0
\partial_{\vV}f(\vV;t)
+
\big(\vV - \uV_0\big) f(\vV;t)\Big).
\label{FPkinSIMPLE}
\end{equation}
While \refeq{FPkinSIMPLE} is essentially a linear PDE,
it should just be kept in mind that $\varepsilon_0$ and $\uV_0$
are functionals of $f$ which are determined by the initial data
$f_0$ and not chosen
independently.\footnote{The identification of \refeq{FPkin} with
\refeq{FPkinSIMPLE} is valid only for isolated systems
that can freely translate. If a driving external force
field $\mathbf{F}$ is applied, then $e(f)$ and $\pV(f)$
are no longer constant and \refeq{FPkin} -- with the addition
of the forcing term $-\mathbf{F}\cdot\partial_\vV f$ to its r.h.s.
-- is the relevant equation.}
We next shall derive \refeq{FPkinSIMPLE} from the diffusion equation equation
on $\mathbb{S}^{3N-4}_{\sqrt{2N\varepsilon_0}}$ in the spirit of Kac's program.
\section{The Finite-$N$ Ensembles}
Consider an infinite ensemble of i.i.d. random vectors
$\{\vVN_\alpha\}_{\alpha =1}^\infty$ where each
$\vVN =(\vV_1,...,\vV_N)\in \mathbb{R}^{3N}$ represents a possible
micro-state of an individual system of $N$ particles with velocities
$\vV_i=(v_{i1},v_{i2},v_{i3})\in\mathbb{R}^3$ and particle positions assumed
to be uniformly distributed over a periodic box; hence, particle positions
will not be considered explicitly.
Each $\vVN$ takes values in the $3N-4$-dimensional manifold of constant
energy $e_0$ and momentum $\uV_0$,
\begin{equation}
\mathbb{M}^{3N-4}_{\uV_0,e_0}
=
\Big\{\vVN\; :\;\sum_{k=1}^N \vV_k=N\mathbf{u}_0, \;
\sum_{k=1}^N\frac{1}{2} \abs{\vV_k}^2=Ne_0,
\; e_0 > \frac{1}{2}|\uV_0|^2 \Big\}.
\end{equation}
The manifold $\mathbb{M}^{3N-4}_{\uV_0,e_0}$ is identical to a
$3N-4$-dimensional sphere of radius $\sqrt{2N\varepsilon_0}$
(where $\varepsilon_0$ appears above \refeq{FPkinSIMPLE}),
centered at
$\uVN = (\uV_0,...,\uV_0)$ and embedded in the $3(N-1)$-dimensional
affine linear subspace of $\mathbb{R}^{3N}$ given by $\uVN + \mathbb{L}^{3N-3}$, where
$\mathbb{L}^{3N-3} \equiv \mathbb{R}^{3N}\cap\big\{\vVN\in\mathbb{R}^{3N}:
\sum_{k=1}^N \vV_k= \mathbf{0}\big\}$
is the space of velocities in any center-of-mass frame.
The ensemble at time $\tau$ is characterized by a probability density
$F^{(N)}(\vVN;\tau)$ on $\mathbb{M}^{3N-4}_{\uV_0,e_0}$, the evolution of which
is determined by the diffusion equation
\begin{equation}
\partial_\tau F^{(N)}(\vVN;\tau)
=
\Delta_{\mathbb{M}^{3N-4}_{\uV_0,e_0}}\,F^{(N)}(\vVN;\tau),
\label{heat}
\end{equation}
where $\Delta_{\mathbb{M}^{3N-4}_{\uV_0,e_0}}$ is the
Laplace--Beltrami operator on $\mathbb{M}^{3N-4}_{\uV_0,e_0}$.
Since all particles are of the same kind, we consider only solutions to
(\ref{heat}) which are invariant under the symmetric
group $S_N$ applied to the $N$ components in $\mathbb{R}^3$ of $\vVN$.
Clearly, permutation symmetry is preserved by the evolution.\footnote{In
what follows, for the sake of notational simplicity we will not
enforce this symmetry explicitly, but the reader should be aware
that (for instance) all the eigenfunctions that appear below in
the solution for $F^{(N)}$ can be easily symmetrized.}
We will show that the diffusion equation (\ref{heat}), here viewed as
a master equation, leads precisely to the essentially linear
Fokker--Planck equation \refeq{FPkinSIMPLE}
in the sense of Kac's program:
(a) the Fokker--Planck equation \refeq{FPkinSIMPLE}
arises as the $N\to\infty$ limit of the equation for the
first marginal of $F^{(N)}(\vVN;\tau)$ derived from
(\ref{heat}), and (b) propagation of chaos holds.
In this section we prepare the ground by discussing the finite-$N$
equation \refeq{heat}.
The limit $N\to\infty$ is carried out in the next section, while
propagation of chaos is discussed in the final section.
For the sake of completeness, we begin by listing some general
facts about the diffusion equation.
We note that the Laplacian $\Delta_{\mathbb{M}^{3N-4}_{\uV_0,e_0}}$
is a positive semi-definite, essentially self-adjoint operator
on the dense domain
$\mathfrak{C}^\infty (\mathbb{M}^{3N-4}_{\uV_0,e_0})
\subset\mathfrak{L}^2 (\mathbb{M}^{3N-4}_{\uV_0,e_0})$,
thus it has a unique self-adjoint extension with domain
$\mathfrak{H}^2(\mathbb{M}^{3N-4}_{\uV_0,e_0})$.
Its self-adjoint extension is the generator of a non-expansive semigroup
on $\mathfrak{L}^2 (\mathbb{M}^{3N-4}_{\uV_0,e_0})$ which is strictly contracting on
the $\mathfrak{L}^2$ orthogonal complement of the constant functions.
Thus, we may ask that the initial condition
$\lim_{t\downarrow 0}F^{(N)}(\,.\,;\tau)
= F_0^{(N)}(\,.\,)\in\mathfrak{L}^2 (\mathbb{M}^{3N-4}_{\uV_0,e_0})$
(which implies $F_0^{(N)}\in \mathfrak{L}^1 (\mathbb{M}^{3N-4}_{\uV_0,e_0})$).
Yet, as is well-known, the diffusion semigroup is so strongly regularizing
that we may even take
$F_0^{(N)}(\,.\,)\in\mathfrak{M}_{+,1} (\mathbb{M}^{3N-4}_{\uV_0,e_0})$,
a probability measure, and obtain
$F^{(N)}(\,.\,;\tau) \in \mathfrak{C}^\infty(\mathbb{M}^{3N-4}_{\uV_0,e_0})$
for all $\tau>0$.
In fact, the solutions of \refeq{heat} can be computed quite
explicitly in terms of an eigenfunction expansion.
Since via translation by $\uVN$ (choosing a center-of-mass frame)
and scaling by $\sqrt{2N\varepsilon_0}$ (choosing a convenient unit of energy)
the manifold $\mathbb{M}^{3N-4}_{\uV_0,e_0}$ can be identified with the
unit sphere centered at the origin of the linear subspace
$\mathbb{L}^{3N-3}\subset\mathbb{R}^{3N}$, the complete spectrum of
$\Delta_{\mathbb{M}^{3N-4}_{\uV_0,e_0}}$ and an orthogonal basis
of eigenfunctions can be obtained from the well-known eigenvalues and
eigenfunctions for the Laplacian on the unit sphere
$\mathbb{S}^{3N-4}\hookrightarrow\mathbb{R}^{3N-3}$.
Of course, in our case the
embedding is $\mathbb{S}^{3N-4}\hookrightarrow\mathbb{L}^{3N-3}$ with
$\mathbb{L}^{3N-3}$ isomorphic by a rotation to standard $\mathbb{R}^{3N-3}$.
Thus we start from $\mathbb{M}^{3N-4}_{\uV_0,e_0}$ and
we first carry out a rotation in $\mathbb{R}^{3N}$ that transforms $\vVN$ to
$\wVN=\mathcal{U}\vVN$ in such a way that $\mathbb{L}^{3N-3}$ is mapped to the
$3N-3$-dimensional linear subspace $\big\{\wVN\ :\ \wV_N=\mathbf{0}\big\}$.
Obviously, $\mathcal{U}^T$ is the linear transformation that diagonalizes
the projection operator onto $\mathbb{L}^{3N-3}$.
A complete orthonormal set of eigenvectors
for such a projection is readily calculated and leads to
\begin{eqnarray}
\wV_1
&=&
\sqrt{\frac{N-1}{N}}\left[\vV_1-\frac{1}{N-1}\sum_{i=2}^N\vV_i\right]
\nonumber\\
&\vdots&\nonumber\\
\wV_n&=&\sqrt{\frac{N-n}{N-n+1}}\left[\vV_n-\frac{1}{N-n}\sum_{i=n+1}^N\vV_i
\right]\nonumber\\
&\vdots&\nonumber\\
\wV_{N-1}\!\!\!\!\!\!&=&\frac{1}{\sqrt{2}}[\vV_{N-1}-\vV_N]\nonumber\\
\wV_N&=&\frac{1}{\sqrt{N}}\sum_{i=1}^N\vV_i
\label{rotation}
\end{eqnarray}
It is easily checked that the matrix associated with this transformation
is indeed orthogonal, and that $\wV_N$ vanishes whenever $\vVN\in\mathbb{L}^{3N-3}$.
More generally, the
affine subspace $\uVN+\mathbb{L}^{3N-3}$ is mapped to the linear manifold
$\big\{\wVN\ :\ \wV_N=\sqrt{N}\uV_0\big\}$
and $\mathbb{M}^{3N-4}_{\uV_0,e_0}$ is mapped to
\begin{equation}
\bigg\{\wVN\; :\; \wV_N=\sqrt{N}\uV_0,\quad
\sum_{i=1}^{N-1}\abs{\wV_i}^2=2Ne_0-N\abs{\uV_0}^2=2N\varepsilon_0
\bigg\}
\end{equation}
which implies that the truncated vector $(\wV_1,\dots,\wV_{N-1})$ belongs
to the sphere $\mathbb{S}_{\sqrt{2N\varepsilon_0}}^{3N-4}\hookrightarrow\mathbb{R}^{3N-3}$
(in $\wV_k$-coordinates).
Thus, the transform $\mathcal{U}$ allows one to analyse
the $N$-particle system with energy and momentum conservation
(``periodic box" setup) in terms of an $(N-1)$-particle system
with only energy conservation (a ``container with reflecting walls"
setup).\footnote{The gas in such a container was discussed in our
earlier work \cite{KieLan04}, but without detailed calculations.
Our calculations with the $\wV$ variables here now supply the
relevant details.}
For future reference, we also observe that for $n$ fixed and $N\to\infty$
the effect of $\mathcal{U}$ reduces to a translation of each of the $n$
velocities by $\uV_0$, in the following sense.
Consider a consistent hierarchy of vectors of increasing size $N$, in
which lower-$N$ vectors can be obtained from the higher-N ones by truncation
(i.e. projection).
Suppose that the vectors belong to $\uVN+\mathbb{L}^{3N-3}$ for all $N$, apply
the transformation in (\ref{rotation}) and look at the $n$-th component.
Since $\sum_{i=n+1}^N\vV_i=N\uV_0-\sum_{i=1}^n\vV_i$,
where $\sum_{i=1}^n\vV_i$ is independent of $N$, we find
\begin{equation}
\lim_{N\to\infty}\wV_n=\vV_n-\uV_0.
\label{wntovn}
\end{equation}
We now recall that the Laplacian is invariant under Euclidean
transformations.
Thus, under our orthogonal transformation $\mathcal{U}$,
the Laplacian $\Delta_{\mathbb{M}^{3N-4}_{\uV_0,e_0}}$ becomes the Laplacian on
$\mathbb{S}_{\sqrt{2N\varepsilon_0}}^{3N-4}$ in $\mathbb{R}^{3N-3}$, the space of
truncated vectors $(\wV_1,\dots,\wV_{N-1})$ (which will also be denoted
by $\wVN$, at the price of abusing the notation).
Since $\Delta_{\mathbb{S}_{\sqrt{2N\varepsilon_0}}^{3N-4}}=
\frac{1}{2N\varepsilon_0}\Delta_{\mathbb{S}^{3N-4}}$, and
the Laplacian on the unit sphere $\mathbb{S}^{3N-4}$ has spectrum
$j(j + 3N -5)$, $j=0,1,\dots$,
the spectrum of $\Delta_{\mathbb{M}^{3N-4}_{\uV_0,e_0}}$ is
\begin{equation}
\lambda^{(j)}_{\mathbb{M}^{3N-4}_{\uV_0,e_0}}=
\frac{j(j + 3N -5)}{2N\varepsilon_0},\qquad j=0,1,\dots\ .
\medskip
\label{eigenvaluesSN}
\end{equation}
The eigenspace on $\mathbb{S}^{3N-4}$ for the $j$-th eigenvalue has dimension
\begin{equation}
\mathcal{N}(j,3N-3)=\frac{(3N-5+2j)(3N-6+j)!}{j!(3N-5)!}
\end{equation}
and is spanned by an orthogonal basis of
hyper-spherical harmonics\footnote{The hyper-spherical harmonics
on $\mathbb{S}^n$ are restrictions to $\mathbb{S}^n\subset \mathbb{R}^{n+1}$ of
homogeneous harmonic polynomials in $\mathbb{R}^{n+1}$.
For $j>0$ the restriction has to be non-constant, since
$\widetilde{Y}_{0,1}\equiv\ const.$.}
on $\mathbb{S}^{3N-4}\subset \mathbb{R}^{3N-3}$ of order $j$, here
denoted $\widetilde{Y}_{j,\ell}(\omV;3N-3)$,
with $\ell\in\mathbb{D}_j=\{1,\dots,\mathcal{N}(j,3N-3)\}$
and with $\omV\in\mathbb{S}^{3N-4}$.
The indexing of our $\widetilde{Y}_{j,\ell}(\omV;3N-3)$ follows the
convention of \cite{Mul} for his $Y_{j,\ell}$ and differs from what
might have been anticipated from the familiar convention for spherical
harmonics on $\mathbb{S}^2$.
Our reason for using tildes atop the function symbols is to remind
the reader that we will use a normalization of the
$\widetilde{Y}_{j,\ell}(\omV;3N-3)$
which conveniently suits our purposes and does not seem to agree with any
of the existing conventions, such as in \cite{Mul} or for the spherical
harmonics on $\mathbb{S}^2$.
Our convention is motivated by the analysis of the large
$N$ behavior of the eigenfunctions, carried out in Appendix B.
Hence, the eigenspace of $\Delta_{\mathbb{M}^{3N-4}_{\uV_0,e_0}}$
associated with the $j$-th eigenvalue in (\ref{eigenvaluesSN})
is spanned by the eigenfunctions
$\widetilde{Y}_{j,\ell}\left({\wVN}/{\sqrt{2N\varepsilon_0}};3N-3\right)$,
$\ell\in\mathbb{D}_j$,
where $\wVN$ is given by (\ref{rotation}) for $n=1,\dots,N-1$.
To shorten the notation we introduce
\begin{equation}
G_{j,\ell}^{(N)}(\vVN)
\equiv \abs{\mathbb{M}^{3N-4}_{\uV_0,e_0}}^{-1}
\widetilde{Y}_{j,\ell}\left({\wVN}/{\sqrt{2N\varepsilon_0}}\,;3N-3\right);
\label{eigenfunctions}
\end{equation}
here, the factor $\abs{\mathbb{M}^{3N-4}_{\uV_0,e_0}}^{-1}$ is introduced
for later convenience.
In terms of the eigenfunctions $G_{j,\ell}^{(N)}(\vVN)$, the solution
to equation \refeq{heat} is simply given by the generalized Fourier series
\begin{equation}
F^{(N)}(\vVN;\tau)
=
\abs{\mathbb{M}^{3N-4}_{\uV_0,e_0}}^{-1}
+
\sum_{j\in\mathbb{N}} \sum_{\ell\in\mathbb{D}_j}
F_{j,\ell}^{(N)}
G_{j,\ell}^{(N)}(\vVN)\,
e^{- \textstyle{\frac{j(j +3N -5)}{2N\varepsilon_0}}\tau}
\label{FNevolution}
\end{equation}
with Fourier coefficients $F_{j,\ell}^{(N)}$ given by
\begin{equation}
F_{j,\ell}^{(N)}
=
\frac{\langle F^{(N)}_0|G_{j,\ell}^{(N)}\rangle}
{\langle G^{(N)}_{j,\ell}|G_{j,\ell}^{(N)}\rangle}
\label{FourierCOEFF}
\end{equation}
where $\langle\,.\,|\,.\,\rangle$ denotes the inner product in
$\mathfrak{L}^2(\mathbb{M}^{3N-4}_{\uV_0,e_0})$.
Notice, though, that the numerator
$\langle F^{(N)}_0|G_{j,\ell}^{(N)}\rangle$
can be extended to mean the canonical pairing of the
$G_{j,\ell}^{(N)}$s with an element of their dual space,
which allows us to take $F^{(N)}_0$ to be a measure.
In particular, we may take $F^{(N)}_0$ to be the Dirac measure concentrated
at any particular point of $\mathbb{M}^{3N-4}_{\uV_0,e_0}$.
The formula \refeq{FNevolution} then describes the fundamental solution
of the diffusion equation \refeq{heat}.
In any event, whatever $F^{(N)}_0$, \refeq{FNevolution} makes it evident
that when $\tau\to\infty$ the ensemble probability density function
on $\mathbb{M}^{3N-4}_{\uV_0,e_0}$ decays exponentially fast to the
uniform probability density $\abs{\mathbb{M}^{3N-4}_{\uV_0,e_0}}^{-1}=
\abs{\mathbb{S}_{\sqrt{2N\varepsilon_0}}^{3N-4}}^{-1}=
F_{0,1}^{(N)} G_{0,1}^{(N)}(\vVN)$, which is the constant eigenfunction
corresponding to the smallest non-degenerate eigenvalue $0$ of the Laplacian.
\section{Evolution of the Marginals}
To study the limit $N\to\infty$ for the time-evolution of the
ensemble measure, we need to consider the hierarchy of $n$-velocity
marginal distributions
\begin{equation}
F^{(n|N)}(\vV_1,\dots,\vV_n;\tau)
\equiv
\int_{\Omega^{3(N-n)-4}_{\mathbf{u}_0,e_0}}
F^{(N)}(\vVN;\tau)\, d\vV_{n+1}\dots d\vV_N
\end{equation}
where $\Omega^{3(N-n)-4}_{\mathbf{u}_0,e_0}$ is
given by all the $(\vV_{n+1},\dots ,\vV_N)$ such that
\begin{equation}
\sum_{i=n+1}^N\!\! \vV_k=N\uV_0-\sum_{i=1}^n\vV_k,
\quad \sum_{i=n+1}^N \abs{\vV_k}^2=2Ne_0-
\sum_{i=1}^n \abs{\vV_k}^2
\end{equation}
and $F^{(n|N)}$ has domain $\{(\vV_1,\dots,\vV_n):
\sum_{k=1}^n |\vV_k-\uV_0|^2\leq {4(N-n)\varepsilon_0}\}\subset\mathbb{R}^{3n}$.
The evolution equation for the $n$-th marginal
$F^{(n|N)}(\vV_1,\dots,\vV_n;\tau)$ is obtained by integrating
\refeq{heat} over $(\vV_{n+1},\dots,\vV_N)\in \mathbb{R}^{3N-3n}$,
using the representation of the Laplace--Beltrami operator given in
(\ref{heat1}) of Appendix Aa.
Then, a straightforward calculation
(previously presented in \cite{KieLan04}) shows that $F^{(n|N)}$ satisfies
\begin{eqnarray}
\partial_\tau F^{(n|N)}\!\!&=&\!\!
\sum_{i=1}^{n}
\frac{\partial}{\partial\vV_i}\cdot\frac{\partial F^{(n|N)}}{\partial\vV_i}-
\frac{1}{N}\sum_{k=1}^3\sum_{i,j=1}^{n}
\frac{\partial^2 F^{(n|N)}}{\partial v_{ik}\partial v_{jk}}
\nonumber\\
&&
-\frac{1}{2N\varepsilon_0}
\sum_{i,j=1}^{n}\frac{\partial}{\partial\vV_i}\cdot
\left((\vV_i-\uV_0)\,(\vV_j-\uV_0)\cdot
\frac{\partial F^{(n|N)}}{\partial\vV_j}\right)
\nonumber\\
&&
+\frac{3(N-n)}{2\varepsilon_0N}\sum_{i=1}^{n}
\frac{\partial}{\partial\vV_i}\cdot\Big((\vV_i-\uV_0) F^{(n|N)}\Big).
\label{nDIFFhierarchyEQ}
\end{eqnarray}
Clearly, to obtain the solutions of these equations it is advisable
to integrate the series solution for $F^{(N)}(\vVN;\tau)$,
(\ref{FNevolution}).
For this purpose, it will be convenient to calculate the marginals
in terms of the rotated variables $\wVN$.
Changing the integration variables\footnote{Note that
(\ref{rotation}) defines a one-to-one linear map
with determinant $\sqrt{\frac{N}{N-n}}$ between
$(\vV_{n+1},\dots\vV_N)$ and $(\wV_{n+1},\dots\wV_{N-1},\vect{z}_N)$,
where $\vect{z}_N\equiv\wV_N-\frac{1}{\sqrt{N}}\sum_{i=1}^n\vV_i$.}
gives
\begin{equation}
F^{(n|N)}(\vV_1,\dots,\vV_n;\tau)
=
{\textstyle{\sqrt{\frac{N}{N-n}}}}
\int F^{(N)}(\vVN;\tau)\,d\wV_{n+1}\dots d\wV_{N-1}
\label{marginW}
\end{equation}
where the integral is over
$\mathbb{S}^{3(N-n)-4}_{\sqrt{2N\varepsilon_0-\sum_{i=1}^n\abs{\wV}_i^2}}$,
and we abused the notation $F^{(N)}(\vVN;\tau)$ by applying it to what is now
regarded as a function of $(\vV_1,...,\vV_n,\wV_{n+1},...,\wV_{N-1})$.
To obtain the series solution for $F^{(n|N)}(\vVN;\tau)$ we need to
express (\ref{FNevolution}) in the variables
$(\vV_1,...,\vV_n,\wV_{n+1},...,\wV_{N-1})$ and then integrate
term by term in the spirit of \refeq{marginW}.
To accomplish this we need to choose explicitly a basis of
spherical harmonics $\widetilde{Y}_{j,\ell}$ on $\mathbb{S}^{3N-4}$.
It is convenient to do this
in an iterative fashion, by assuming that a basis is known for the
spherical harmonics with one independent variable less,
here $\widetilde{Y}_{k,m}(\omV_{3N-5};3N-4)$ with
$\omV_{3N-5}\in\mathbb{S}^{3N-5}$.
Then, the desired basis is obtained \cite{Mul}
by taking all the elements in the given lower-dimensional basis and
multiplying them by associated Legendre functions of the ``extra"
variable.
In our case the $(3N-3$)-th variable will be $w_{11}/\sqrt{\scriptstyle{{2N\vareps_0}}}$, the first
component of $\wVN/\sqrt{\scriptstyle{{2N\vareps_0}}}$, and $\omV_{3N-5}$ will be
a unit vector in the space of the remaining $3N-4$ components,
denoted by $(\wVN)_{3N-4}/\sqrt{\scriptstyle{{2N\vareps_0}}}$; thus,
\begin{equation}
\widetilde{Y}_{j,\ell}\left(\frac{\wVN}{\sqrt{\scriptstyle{{2N\vareps_0}}}};3N-3\right)
=
\widetilde{Y}_{k,m}\left(\frac{(\wVN)_{3N-4}}{\sqrt{\scriptstyle{{2N\vareps_0}}}};3N-4\right)
\,
\widetilde{P}_j^k\left(\frac{w_{11}}{\sqrt{\scriptstyle{{2N\vareps_0}}}};3N-3\right)
\label{basis1}
\end{equation}
where $k=0,1,\dots, j$, $m=1,\dots,\mathcal{N}(k,3N-4)$ and each choice
of the pair $k,m$ is associated with a value of the degeneracy index
$\ell$ for the basis $\widetilde{Y}_{j,\ell}$; moreover,
$\widetilde{P}_j^k$ is an associated Legendre
function \cite{Mul}, suitably normalized (see Appendix B).
By repeating this process $3n$ times, we write out the eigenfunctions
in the form
\begin{eqnarray}
&&\hskip-.9truecm
\widetilde{Y}_{j,\ell}\left(\frac{\wVN}{\sqrt{\scriptstyle{{2N\vareps_0}}}};3N-3\right)
=
\widetilde{Y}_{k_{3n},m}\left(\frac{(\wVN)_{3N-3n-3}}{\sqrt{\scriptstyle{{2N\vareps_0}}}};3N-3n-3\right)
\times\phantom{spaaace}
\\[0.4cm]
&\times&\!\!\!\!\!\!
\widetilde{P}_{k_{3n-1}}^{k_{3n}}\!\!
\left(\frac{w_{n3}}{\sqrt{\scriptstyle{{2N\vareps_0}}}};3N-3n-2\right)
\widetilde{P}_{k_1}^{k_2}\!\left(\frac{w_{12}}{\sqrt{\scriptstyle{{2N\vareps_0}}}};3N-4\right)
\cdots
\widetilde{P}_j^{k_1}\!\left(\frac{w_{11}}{\sqrt{\scriptstyle{{2N\vareps_0}}}};3N-3\right)
\nonumber
\label{basis2}
\end{eqnarray}
where $0\leq k_{3n}\leq \dots\leq k_1\leq j$ and
$m=1,\dots,\mathcal{N}(k_{3n},3N-3n-3)$.
Now let $g_{j,\ell}^{(n|N)}(\vV_1,\dots,\vV_n)$ denote the
$n$-th ``marginal'' of $G_{j,\ell}^{(N)}(\vVN)$
(as for $F^{(N)}$ in (\ref{marginW})), and
set $N^*\equiv N - n -1$.
We find
\begin{eqnarray}
g_{j,\ell}^{(n|N)}(\vV_1,\dots,\vV_n)
=
\abs{\mathbb{M}^{3N-4}_{\uV_0,e_0}}^{-1}
\int \widetilde{Y}_{k_{3n},m}\left(\frac{(\wVN)_{3N^*}}
{\sqrt{\scriptstyle{{2N\vareps_0}}}};3N^*\right)
d\wV_{n+1}\dots d\wV_{N-1}
\nonumber\\[0.4cm]
\times
{\textstyle{\sqrt{\frac{N}{N-n}}}}
\widetilde{P}_{k_{3n-1}}^{k_{3n}}\!\!
\left(\frac{w_{n3}}{\sqrt{\scriptstyle{{2N\vareps_0}}}};3N\!-\!3n\!-\!2\right)\!\!\!
\cdots
\widetilde{P}_j^{k_1}\!\left(\frac{w_{11}}{\sqrt{\scriptstyle{{2N\vareps_0}}}};3N\!-\!3\right)
\end{eqnarray}
where the integral is over the same domain as in \refeq{marginW}.
The integral of $\widetilde{Y}_{k_{3n},m}$ is non-zero if and only
if $k_{3n}=0$ and $m=1$, and the integrals over $\widetilde{Y}_{0,1}$
are determined only up to the overall factor $\widetilde{Y}_{0,1}$,
which we may choose to be unity without loss of generality.
Accordingly, $g_{j,\ell}^{(n|N)}\equiv 0$ unless
$\ell\in\widetilde\mathbb{D}_j\subset\mathbb{D}_j$, where
$\widetilde\mathbb{D}_j$ contains the indices of the basis functions that
``descend'' from the uniform harmonic in $\mathbb{R}^{3N-3n-3}$.
For such $\ell$'s the integrated eigenfunctions then become
\begin{eqnarray}
g_{j,\ell}^{(n|N)}(\vV_1,\dots,\vV_n)
=
\widetilde{P}_j^{k_1}\!\left(\frac{w_{11}}{\sqrt{\scriptstyle{{2N\vareps_0}}}};3N\!-\!3\right)
\cdots
\widetilde{P}_{k_{3n-1}}^{0}\!\!\left(\frac{w_{n3}}{\sqrt{\scriptstyle{{2N\vareps_0}}}},
3N\!-\!3n\!-\!2\right)
\nonumber\\[0.4cm]
\times
\sqrt{\frac{N}{N-n}}\,
\frac{\abs{\mathbb{S}^{3(N-n)-4}}}
{\abs{\mathbb{S}^{3N-4}}}
\frac{1}{{\sqrt{\scriptstyle{{2N\vareps_0}}}}^{3n}}\,
\Big(1-\frac{1}{\sqrt{\scriptstyle{{2N\vareps_0}}}}\sum_{i=1}^n|\wV_i|^2\Big)^{\frac{3(N-n)-4}{2}}.
\label{eigmarg2}
\end{eqnarray}
The series for the $n$-th marginal $F^{(n|N)}(\,.\,;\tau)$ (the integrated
\refeq{FNevolution}) is a series in the functions \refeq{eigmarg2}, viz.
\begin{equation}
F^{(n|N)}(\vV_1,\dots,\vV_n;\tau)
=
\sum_{j\in \mathbb{N}\cup\{0\}} \sum_{\ell\in\widetilde\mathbb{D}_j}
F_{j,\ell}^{(N)}
g_{j,\ell}^{(n|N)}(\vV_1,\dots,\vV_n)\,
e^{- \textstyle{\frac{j(j +3N -5)}{2N\varepsilon_0}}\tau}.
\label{FnNevolution}
\end{equation}
\section{The Limit $N\to\infty$}
We are now ready to take the infinitely many particles limit.
First of all, we observe that the evolution equation for
the marginal velocity densities
$f^{(n)}(\vV_1,\dots,\vV_n;\tau)\equiv
\lim_{N\to\infty}F^{(n|N)}(\vV_1,\dots,\vV_n;\tau)$
which obtains in the formal limit $N\to\infty$ from \refeq{nDIFFhierarchyEQ}
is the essentially linear Fokker--Planck equation in $\mathbb{R}^{3n}$,
\begin{equation}
\partial_\tau f^{(n)}
=
\sum_{i=1}^{n}
\frac{\partial}{\partial\vV_i}\cdot\Big(\frac{\partial f^{(n)}}
{\partial\vV_i}+\frac{3}{2\varepsilon_0}(\vV_i-\uV_0)\,f^{(n)}\Big).
\label{nDIFFhierarchyEQlim}
\end{equation}
We now show that the series expansion for the time-evolved
finite-$N$ marginals $F^{(n|N)}(\,.\,;\tau)$ converge under
natural conditions to solutions of these equations.
Beginning with the spectrum of $\Delta_{\mathbb{M}^{3N-4}_{\uV_0,e_0}}$,
we note that the limit $N\to\infty$ yields
\begin{equation}
\lim_{N\to\infty}
\Bigl\{
\lambda_{\mathbb{M}^{3N-4}_{\uV_0,e_0}}^{(j)}
\Bigr\}_{j=0}^\infty
=
\Big\{\textstyle{
\frac{3j}{2\varepsilon_0}}
\Big\}_{j=0}^\infty.
\label{LIMspectrumNEw}
\end{equation}
Thus, the limit spectrum is discrete.
In particular, there is a spectral
gap separating the origin from the rest of the spectrum.
As a result, the time evolution of the limit $N\to\infty$
continues to approach a stationary state exponentially
fast when $\tau\to\infty$.
Coming to the eigenfunctions, the expression
on the second line in (\ref{eigmarg2}) contains the
$n$-velocity marginal distribution of the uniform density
$\abs{\mathbb{M}^{3N-4}_{\uV_0,e_0}}^{-1}$ (the $j=0$ case).
As is well-known at least since the time of Boltzmann,
this distribution converges pointwise when $N\to\infty$
to the $n$-velocity drifting Maxwellian on $\mathbb{R}^{3n}$,
\begin{equation}
f_{\mathrm{M}}^{\otimes{n}}(\vV_1,...,\vV_n)
=
\left({\frac{3}{4\pi\varepsilon_0}}\right)^{\frac{3n}{2}}
\prod_{i=1}^n
\exp\left( -{\textstyle{\frac{3}{4\varepsilon_0}}}|\vV_i-\uV_0|^2 \right)
\label{nMaxwellian}
\end{equation}
(recall (\ref{wntovn})).
In terms of eigenfunctions this means that the ``projection'' onto
$\mathbb{R}^{3n}$ of the $j=0$ eigenfunction of the Laplace--Beltrami operator
on $\mathbb{S}^{3N-4}_{\sqrt{2N\varepsilon_0}}$ converges pointwise (in fact,
even uniformly) to the $j=0$ eigenfunction of the linear Fokker--Planck
operator in $\mathbb{R}^{3n}$, appearing in the r.h.s. of
(\ref{nDIFFhierarchyEQlim}).
The connection between the eigenfunctions generalizes to the cases
$j\neq 0$; cf. \cite{BakMaz} for the special case $\uV_0=\vect{0}$.
The asymptotic behavior for $N\to\infty$ of
the associated Legendre functions in (\ref{eigmarg2}),
which is discussed in Appendix B, together with (\ref{wntovn}),
yields that
$g_{j,\ell}^{(n)}(\vV_1,\dots,\vV_n) \equiv
\lim_{N\to\infty}g_{j,\ell}^{(n|N)}(\vV_1,\dots,\vV_n)$
exists pointwise for all $(\vV_1,\dots,\vV_n)\in\mathbb{R}^{3n}$, with
\setlength{\arraycolsep}{0.5mm}
\begin{eqnarray}
\!\!\!\!\!\!
g_{j,\ell}^{(n)}(\vV_1,\dots,\vV_n)
&=&
\frac{(-1)^j}{2^{j/2}}
H_{j-k_1}\!\left({\textstyle{\sqrt{\frac{3}{4\varepsilon_0}}}}
(v_{11}\!-u_1)\!\right)
\cdots
H_{k_{3n-1}}\!\left({\textstyle{\sqrt{\frac{3}{4\varepsilon_0}}}}
(v_{n3}\!-u_3)
\!\right)
\label{eigmarg3}
\nonumber\\
&&\times
\left({\frac{3}{4\pi\varepsilon_0}}\right)^{\frac{3n}{2}}\,
\prod_{i=1}^n
\exp\left( -{\textstyle{\frac{3}{4\varepsilon_0}}}|\vV_i-\uV_0|^2 \right)
\nonumber\\
&\equiv&\frac{(-1)^j}{2^{j/2}}\!
\left({\textstyle{\frac{3}{4\pi\varepsilon_0}}}\right)^{\!\!\frac{3n}{2}}\!
\prod_{i=1}^n
e^{-\frac{3}{4\varepsilon_0}|\vV_i-\uV_0|^2}
\prod_{l=1}^3
H_{m_{i\cdot l}}\!\left({\textstyle{\sqrt{\frac{3}{4\varepsilon_0}}}}
(v_{il}\!-u_l)\!\right)
\label{nEIGlim}
\end{eqnarray}
for all $\ell\in\widetilde\mathbb{D}_j$,
where $H_m(x)$ is the Hermite polynomial of degree $m$ on $\mathbb{R}$, and
we defined $m_1=j-k_1,m_2=k_1-k_2,\dots, m_{3n}=k_{3n-1}$. In terms of
the $m_i$'s, the index set $\widetilde\mathbb{D}_j$ counts all the choices of
integers
$0\leq m_1,\dots,m_{3n}\leq j$ such that $\sum_{i=1}^{3n}m_i=j$.
For $n=1$ one readily recognizes the well-known eigenfunctions
\cite{Risk} for the linear Fokker--Planck operator in $\mathbb{R}^3$,
viz. r.h.s.(\ref{FPkinSIMPLE}) with constant
$\varepsilon_0$ and $\uV_0$, easily calculated by separation of variables.
In fact, what we have recovered are precisely the eigenfunctions for the
linear Fokker--Planck operator in $\mathbb{R}^{3n}$, see
(\ref{nDIFFhierarchyEQlim}).
Now assume that one can choose sequences of initial conditions
$F^{(N)}_0$ such that, for each fixed $j$ and $\ell$, the Fourier coefficients
$F_{j,\ell}^{(N)}$ converge to a limit $F_{j,\ell}$ \emph{such that} each
initial $n$-velocity marginal density, $n\in\mathbb{N}$, converges in
$(\mathfrak{L}^2\cap\mathfrak{L}^1)(\mathbb{R}^{3n})$ to
\begin{equation}
f^{(n)}(\vV_1,\dots,\vV_n;0)
=
f_{\mathrm{M}}^{\otimes{n}}(\vV_1,...,\vV_n)
+
\sum_{j\in\mathbb{N}} \sum_{\ell\in\widetilde\mathbb{D}_j}
F_{j,\ell}
g_{j,\ell}^{(n)}(\vV_1,\dots,\vV_n);
\label{fnNULL}
\end{equation}
it then follows that the subsequent evolution of the
$n$-velocity marginal densities is given by
\begin{equation}
f^{(n)}(\vV_1,\dots,\vV_n;\tau)
=
f_{\mathrm{M}}^{\otimes{n}}(\vV_1,...,\vV_n)
+
\sum_{j\in\mathbb{N}} \sum_{\ell\in\widetilde\mathbb{D}_j}
F_{j,\ell}
g_{j,\ell}^{(n)}(\vV_1,\dots,\vV_n)
e^{- \textstyle{\frac{3j}{2\varepsilon_0}}\tau}.
\label{fnSOL}
\end{equation}
Formula \refeq{fnSOL} describes an exponentially fast approach
to equilibrium in the ensemble of infinite systems.
The $f^{(n)}(\,.\,;\tau)\in(\mathfrak{L}^2\cap\mathfrak{L}^1)(\mathbb{R}^{3n})$, and
in addition they automatically satisfy
\begin{eqnarray}
\int_{\mathbb{R}^{3n}} f^{(n)}(\vV_1,\dots,\vV_n;\tau)
\, d\vV_1\dots d\vV_n
\!&=&\!
1
\label{fnMASS}
\\
\int_{\mathbb{R}^{3n}}(\vV_1+\dots+\vV_n) f^{(n)}(\vV_1,\dots,\vV_n;\tau)
\, d\vV_1\dots d\vV_n
\!&=&\!
n\uV_0
\label{fnMOMENTUM}
\\
\int_{\mathbb{R}^{3n}} \frac12
(|\vV_1|^2+\dots +|\vV_n|^2)
f^{(n)}(\vV_1,\dots,\vV_n;\tau)\, d\vV_1\dots d\vV_n
\!&=&\!
n e_0
\label{fnENERGY}
\end{eqnarray}
for all $\tau\geq 0$ (recall that $e_0 = \varepsilon_0 + |\uV|_0^2/2$).
In fact, \refeq{fnSOL} solves \refeq{nDIFFhierarchyEQlim}, which now
implies that $f^{(n)}(\,.\,;\tau)$ can also be expressed through
integration of the initial data against the $n$-fold tensor product
of \refeq{OUkernel}.
The upshot is that $f^{(n)}(\,.\,;\tau)\in \mathfrak{S}(\mathbb{R}^{3n})\ \forall\tau>0$
(Schwartz space).
To vindicate these conclusions, for us it remains to show that
the infinitely many constraints on each $F_{j,\ell}$ implied by
\refeq{fnNULL}, viz.
\begin{equation}
F_{j,\ell}
=
\frac{\langle f^{(n)}_0|g_{j,\ell}^{(n)}\rangle}
{\langle g^{(n)}_{j,\ell}|g_{j,\ell}^{(n)}\rangle}\qquad \forall n\in\mathbb{N},
\label{FjlCONSTRAINTS}
\end{equation}
where $\langle\,.\,|\,.\,\rangle$ now
means inner product in $\mathfrak{L}^2(\mathbb{R}^{3n})$, do not impose impossible
consistency requirements.
To show this, recall that the $f^{(n)}_0$ by definition satisfy
\begin{equation}
\int_{\mathbb{R}^3} f_{0}^{(n+1)}(\vV_1,\dots,\vV_{n+1}) d\vV_{n+1}
=
f_{0}^{(n)}(\vV_1,\dots,\vV_{n}),
\label{fMARG}
\end{equation}
which in view of \refeq{fnNULL} implies
that the hierarchy of the $g_{j,\ell}^{(n)}$ must satisfy
\begin{equation}
\int_{\mathbb{R}^3} g_{j,\ell}^{(n+1)}(\vV_1,\dots,\vV_{n+1}) d\vV_{n+1}
=
g_{j,\ell}^{(n)}(\vV_1,\dots,\vV_{n})\prod_{i=1}^3 \delta_{k_{3(n+1)-i},0},
\label{gMARG}
\end{equation}
which is readily verified by explicit integration of
\refeq{nEIGlim}.
Thus, the constraints \refeq{FjlCONSTRAINTS} are automatically consistent,
and this vindicates our initial assumption.
\section{Propagation of Chaos}
Setting $n=1$ in \refeq{nDIFFhierarchyEQlim}, and
changing the time scale by setting $\tau = \frac{2}{3}\varepsilon_0t$,
we recover (\ref{FPkinSIMPLE}), with $f^{(1)}$ in place of $f$.
However, (\ref{FPkinSIMPLE}) (or \refeq{FPkin} for
that matter) cannot be said to have been shown to be a kinetic
equation yet.
Note that propagation of chaos has not entered the
derivation of \refeq{nDIFFhierarchyEQlim}.
In fact, (\ref{nDIFFhierarchyEQlim}) for
$n=1,2,\dots$ constitutes a ``Fokker--Planck hierarchy'' analogous to the
the well-known Boltzmann, Landau and Vlasov hierarchies which arise in the
validation of kinetic theory \cite{SpoBOOK,CIPbook} using ensembles.
In our case the hierarchy has the very simplifying feature
that the $n$-th equation in the hierarchy is decoupled from the
equation for the $n+1$-th marginal.
Since all the hierarchies used in the validation of kinetic theory
are by construction
\emph{linear}\footnote{More precisely, they are only essentially linear,
for the parameters $\varepsilon_0$ and $\uV_0$, which also enter
any of the other hierarchies whenever they describe ensembles
of systems conserving mass, momentum, and energy, are all
tied up with the initial conditions.}
in the ``vector''
of the $f^{(n)}$, whenever one has a decoupling hierarchy one
obtains closed linear equations for the $f^{(n)}$.
In particular, our equation \refeq{nDIFFhierarchyEQlim}
with $n=1$ is already a closed linear equation for $f^{(1)}$.
However, at this point, any $f^{(n)}$ is still in general
an ensemble superposition of states; in particular,
$f^{(1)}$ still describes a statistical ensemble of pure states
$f$ with same mass, momentum, and energy.
By ignoring this fact one can mislead oneself into thinking that
\refeq{nDIFFhierarchyEQlim} with $n=1$ and $f^{(1)}$ in place of $f$
is already the kinetic equation we sought.
The final step in extracting \refeq{FPkinSIMPLE} as kinetic equation
for the pure states involves the Hewitt--Savage \cite{HewSav}
decomposition theorem.
This theorem says that in the continuum limit any $f^{(n)}$
is a unique convex linear superposition of extremal (i.e. pure) $n$ particle
states, and that these pure states are products of $n$ identical
one-particle functions $f$ evaluated at $n$ generally
different velocities.
Each of the $f$ in the support of the superposition measure
represents the velocity density function of an actual individual member
of the infinite statistical ensemble of infinitely-many-particles
systems.
In formulas, at $\tau =0$ the initial data for $f^{(n)}$ read
\begin{equation}
f^{(n)}(\vV_1,\dots,\vV_n;0)
=
\langle f_0^{\otimes{n}}(\vV_1,...,\vV_n)\rangle,
\label{HWinitially}
\end{equation}
where $\langle\,.\,\rangle$ is the Hewitt--Savage \cite{HewSav}
ensemble decomposition measure on the space of initial velocity
density functions $f_0$ of {individual physical systems}
with same mass $m(f_0)(=1)$, momentum $\pV(f_0) = \uV_0$ and
energy $e(f_0) = e_0 = \varepsilon_0 + |\uV_0|^2/2$.
To extend this representation to $\tau>0$, let $U_\tau^{(n)}$ denote the
one-parameter evolution semigroup for \refeq{nDIFFhierarchyEQlim}, i.e.
$f^{(n)}(\vV_1,\dots,\vV_n;\tau)=U_\tau^{(n)}f^{(n)}_0(\vV_1,\dots,\vV_n)$.
Noting now that the Hewitt--Savage measure is of course invariant
under the evolution, and that by the linearity of \refeq{nDIFFhierarchyEQlim}
it commutes with the linear operator $U_\tau^{(n)}$ for all $\tau\geq 0$, it
follows that at later times $\tau >0$ the $n$ point density of the ensemble
is given by
\begin{equation}
f^{(n)}(\vV_1,\dots,\vV_n;\tau)
=
\langle U_\tau^{(n)}f^{\otimes{n}}_0(\vV_1,...,\vV_n)\rangle.
\end{equation}
This so far simply states that, if the ensemble is initially a statistical
mixture of pure states (product states), then at later times it is a
statistical mixture of time-evolved initially pure states.
Next we note that by inspection of \refeq{nDIFFhierarchyEQlim} it follows
that
\begin{equation}
U_\tau^{(n)}f^{\otimes{n}}_0(\vV_1,...,\vV_n)
=
(U_\tau^{(1)}f_0)^{\otimes{n}}(\vV_1,...,\vV_n),
\end{equation}
viz. pure states evolve into pure states.
Every factor $f(\vV_k;\tau)= U_\tau^{(1)}f_0(\vV_k)$
solves \refeq{FPkinSIMPLE} with $\tau = \frac{2\varepsilon_0}{3}t$, obeying
the desired conservation laws.
At last one can legitimately say that (\ref{FPkinSIMPLE})
has been derived as a full-fledged kinetic equation valid for almost
every (w.r.t. $\langle\,.\,\rangle$) individual member of the limiting
ensemble.
\section{Summary and Outlook}
In summary, the diffusion equation on $\mathbb{M}^{3N-4}_{\uV_0,e_0}$ can be
interpreted as the simplest ``master equation'' for an underlying
$N$-body Markov process with single-particle and pair terms.
The $N\to\infty$ limit for the marginal densities of solutions to
the diffusion equation is well-defined and can be carried out explicitly.
After invoking the Hewitt--Savage decomposition, the limit $N\to\infty$
is seen to produce solutions of the ``kinetic Fokker--Planck equation''
describing individual isolated systems conserving mass, momentum, and energy.
The Fokker--Planck equation \refeq{FPkin} is exactly solvable and
displays correctly the qualitative behavior of a typical kinetic
equation.
In this sense, (\ref{FPkin}) really can be regarded
as the simplest example of a kinetic equation
of the ``diffusive" type, in the same family as, for instance, the much
more complex Landau and Balescu-Lenard-Guernsey equations.
Our work raises many new questions.
1) In particular, in Appendix Ab we have only written down the generator
for the adjoint process of the underlying $N$-particle Markov process;
hence, what is the explicit characterization of this process?
2) A derivation of a kinetic equation \`a la Kac is an intermediate step
towards a full validation from some deterministic (Hamiltonian) microscopic
model, which is in general a very difficult program, see the rigorous
derivations of kinetic equations in \cite{SpoBOOK,CIPbook}.
The substitute Markov process is usually chosen to preserve some of the
essential features of the deterministic dynamics which (formally) leads
to the same kinetic equation.
Here we have only identified a stochastic model which leads to \refeq{FPkin}.
Villani's work \cite{Vil98} suggests that a deterministic model may exist
which in the kinetic regime leads to \refeq{FPkin}.
Can one indentify this model?
3) In this paper, we conveniently assumed that the Fourier coefficients
ensure convergence of the marginal density functions in $\mathfrak{L}^2\cap\mathfrak{L}^1$
and subsequently upgraded the regularity to Schwartz functions.
What are the explicit conditions on the Fourier coefficients
of the initial functions on $\mathbb{M}^{3N-4}_{\uV_0,e_0}$ which ensure
convergence in $\mathfrak{L}^2\cap\mathfrak{L}^1$, in Schwartz space, in some topology
for measures?
4)
Since the PDEs in our finite-$N$ Fokker--Planck hierarchy
are already self-contained for each $n$ (viz., they do not involve
the usual coupling to $f^{(n+1)}$), the finite-$N$ corrections to the
limiting evolutions can be studied in great detail; hence, for instance,
how do the explicit corrections to propagation of chaos look?
5) We already mentioned in a footnote that the kinetic Fokker--Planck
equation can easily be generalized to situations where the system is
exposed to some external driving force by adding a forcing term.
Can one derive this equation from some suitable ensemble of driven systems?
Under which conditions do there exist stationary
non-equilibrium states, and what are their stability properties?
6) Finally, our derivation is only valid for the space-homogeneous
Fokker--Planck equation without driving force term; hence, can one
extend our derivation to obtain the space-inhomogeneous generalization
of the kinetic Fokker--Planck equation, first without and then with
driving force term?
These are many interesting questions which should be answered in
future works.
\medskip
\noindent
\textbf{Acknowledgment}
We thank the referees for drawing our attention to \cite{BakMaz} and
for their constructive criticisms. Thanks go also to Michael Loss
for pointing out Mehler's paper \cite{Mehler}.
Kiessling was supported by NSF Grant DMS-0103808.
Lancellotti was supported by NSF Grant DMS-0318532.
\newpage
\section*{Appendix}
\subsection*{A. Two useful representations of the Laplacian on spheres}
\subsubsection*{Aa. Extrinsic representation in divergence form}
For the purpose of obtaining equations for the marginals
by integrating (\ref{heat}),
it is advantageous to express the Laplacian on the right-hand side
in terms of the projection operator
$P_{\mathbb{M}^{3N-4}_{\uV_0,e_0}}$ from $\mathbb{R}^{3N}$
to the fibers of the tangent bundle of the embedded manifold
$\mathbb{M}^{3N-4}_{\uV_0,e_0}$. It is easy to verify \cite{KieLan04} that
\begin{equation}
\Delta_{\mathbb{M}^{3N-4}_{\uV_0,e_0}} F^{(N)}
=
\nabla\cdot[P_{\mathbb{M}^{3N-4}_{\uV_0,e_0}}\nabla F^{(N)}]
\label{LapBel2}
\end{equation}
In order to have an explicit expression for
$P_{\mathbb{M}^{3N-4}_{\uV_0,e_0}}$
we introduce an orthogonal basis for the orthogonal complement
of the tangent space to $\mathbb{M}^{3N-4}_{\uV_0,e_0}$
at $\vVN\in\mathbb{M}^{3N-4}_{\uV_0,e_0}\subset \mathbb{R}^{3N}$.
Clearly, such orthogonal complement is spanned by the four vectors $\vVN$ and
$\eVN_\sigma=(\eV_{\sigma},\dots,\eV_{\sigma})$, $\sigma=1,2,3$,
where the $\eV_{\sigma}$ are the standard unit vectors in $\mathbb{R}^3$.
The vectors $\eVN_\sigma$ are orthogonal to each other but not to
$\vVN$; projecting away the non-orthogonal component of $\vVN$ yields
\begin{equation}
\biggl(
\mathbf{I}_{3N} -\frac{1}{N}\sum_{\sigma=1}^3 \eVN_\sigma\otimes \eVN_\sigma
\biggr)
\cdot\vVN
=
\vVN - \uVN.
\end{equation}
The vectors $\{\vVN-\uVN, \eVN_1, \eVN_2, \eVN_3\}$ form the desired
orthogonal basis; their magnitudes are $\abs{\eVN_\sigma}=\sqrt{N}$ and
$\abs{\vVN-\uVN}=\sqrt{2N\varepsilon_0}$.
Finally, (\ref{LapBel2}) becomes
\begin{equation}
\!\!\!\!\Delta_{\mathbb{M}^{3N-4}_{\uV_0,e_0}} F^{(N)}
=
{\partial_\vVN}\cdot
\!\left[\!\left(\!\mathbf{I}_{3N}-
\frac{1}{N} \sum_{\sigma=1}^3\eVN_\sigma\otimes\eVN_\sigma-
\frac{1}{2N\varepsilon_0}
(\vVN\!-\!\uVN)\otimes(\vVN\!-\!\uVN)\!\right)\!{\partial_\vVN F^{(N)}}
\!\right]\!
\label{heat1}
\end{equation}
\subsubsection*{Ab. Representation for the $N$-Body Markov Process}
In the main part of this paper we started from the diffusion equation on
the manifold $\mathbb{M}^{3N-4}_{\uV_0,e_0}$ of $N$-body systems with same
energy (per particle) $e_0$ and momentum (per particle) $\uV_0$, then
took the limit $N\to\infty$, obtaining the kinetic Fokker--Planck
equation \refeq{FPkinSIMPLE}, which rewrites into \refeq{FPkin}
in view of the conservation laws.
The Laplace--Beltrami operator on $\mathbb{M}^{3N-4}_{\uV_0,e_0}$
is the generator of the adjoint semigroup of the
underlying stochastic Markov process that rules the microscopic
dynamics of an individual $N$-body system.
Here we show that this generator can be written as a
sum of single particle and two-particle operators, thus characterizing
the Markov process as a mixture of individual stochastic motions and
stochastic binary interactions.
Moreover, we show that the binary
particle operators are the only ones that do not vanish in the $N\to\infty$
limit.
This means that the kinetic Fokker--Planck equation can also
be derived in terms of an $N$-body stochastic process with
purely binary interactions, which is more satisfactory
from a physical point of view.
Recall that in section 2 we explained that
$\mathbb{M}^{3N-4}_{\uV_0,e_0}$ can be identified with the sphere
$\mathbb{S}^{3N-4}_{\sqrt{2N\varepsilon_0}}$ centered at the origin
of $\mathbb{L}^{3N-3}$ (which itself is an affine linear subspace
of the space of all velocities, $\mathbb{R}^{3N}$).
Recall that
$\Delta_{\mathbb{S}^{3N-4}_{\sqrt{2N\varepsilon_0}}}
= \frac{1}{2N\varepsilon_0}\Delta_{\mathbb{S}^{3N-4}}$.
Note the well-known representation
\begin{equation}
\Delta_{\mathbb{S}^{3N-4}}
=
\!\!\!\!\!\!\!\!{\sum_{\qquad 1\leq k < l\leq 3(N-1)}}\!\!
\Big(
w_{k} \partial_{w_{l}} -w_{l} \partial_{w_{k}}
\Big)^2,
\label{LAPopDECOMPOSED}
\end{equation}
where ${w_{k}}$ is the $k$-th Cartesian component of
$\wVN\in\mathbb{S}^{3N-4}\subset\mathbb{R}^{3(N-1)}$
(note that in section 2 we used $\wVN\in\mathbb{S}^{3N-4}_{\sqrt{2N\varepsilon_0}}$,
but note furthermore that the r.h.s. of \refeq{LAPopDECOMPOSED} is invariant
under $\wVN\to\lambda\wVN$).
Grouping the components of $\wVN$ into blocks of vectors $\wV_k\in\mathbb{R}^3$,
$k=1,...,N-1$, the r.h.s. of \refeq{LAPopDECOMPOSED} can be recast as
\begin{eqnarray}
\Delta_{\mathbb{S}^{3N-4}}
=
&&\!\!\!\!
\Big. \sum_{k=1}^{N-1}
\sum_{\stackrel{l=1}{l\ne k}}^{N-1}
\Big(3\wVk\cdot\partial_{\wV_k}+\abs{\wVk}^2\partial_{\wV_l}\cdot\partial_{\wV_l}
-
\big(
\wVk\cdot\partial_{\wV_k}
\big)
\big(\wVl\cdot\partial_{\wV_l}
\big)
\Big)
\nonumber\\
&&-
\sum_{k=1}^{N-1}
\big(
\wVk\times\partial_{\wV_k}
\big)^2
\Big.
,
\label{MASTERopDECOMPOSED}
\end{eqnarray}
containing one-body terms as well as binary terms.
Note however that the first term in the binary sum is
effectively a sum of two-body terms in disguise, which
scale with factor $N-2$ and thus survive in the limit $N\to\infty$,
while the true one-body sum (second line) drops out in that limit.
This implies that the kinetic Fokker--Planck equation \refeq{FPkinSIMPLE}
can be derived from a master equation on $\mathbb{M}^{3N-4}_{\uV_0,e_0}$
which contains \emph{only} the binary terms (first line)
in \refeq{MASTERopDECOMPOSED}.
This in turn implies that \refeq{FPkinSIMPLE} is the kinetic equation
for an underlying system of $N$ particles with stochastic pair interactions.
\subsection*{B. High-Dimension Asymptotics of Associated Legendre Functions}
In \refeq{eigmarg2} the associated Legendre functions of degree
$s=0,1,2,...$ and order $r=0,...,s$ in $q$ dimensions occur.
They are defined on the interval $[-1,1]$ and given by
\begin{equation}
\widetilde{P}_s^r(t;q)
=
\sqrt{q}^{s+r}
\frac{s!}{2^r}\,\Gamma\left(\frac{q-1}{2}\right)
\sum_{l=0}^{\intgpart}
\left(-\frac{1}{4}\right)^l
\frac{(1-t^2)^{l+\frac{r}{2}}\, t^{s-r-2l}}
{l!\,(s-r-2l)!\,\Gamma\left(l+r+\frac{q-1}{2}\right)}
\label{LEGENDREf}
\end{equation}
which differ from the $P_s^r(t;q)$ in \cite{Mul} in their normalization.
In our investigation, $q = 3N - p$ and $t = \frac{w}{\sqrt{\scriptstyle{{2N\vareps_0}}}}$,
and we are interested in the limit $N\to\infty$.
The familiar asymptotics of Euler's Gamma function gives us
\begin{equation}
\frac{\Gamma\left(x\right)}
{\Gamma\left(a+x\right)}
=
x^{-a} + O\left(x^{-(a+1)}\right).
\end{equation}
for $x\gg 1$.
Applying this asymptotics with $2x = q-1=3N-p-1$ and $a=l+r$ to
\refeq{LEGENDREf},
we find that given $p\in\mathbb{N}$ and $w\in \mathbb{R}$ (which implies
$N> \max\{p/3,w^2/(2\varepsilon_0)\}$), when $N\gg 1$ we have
\begin{eqnarray}
\widetilde{P}_s^r\left(\frac{w}{\sqrt{\scriptstyle{{2N\vareps_0}}}};3N\!-\!p\right)
=&&\!\!
\sqrt{2}^{r-s}
\sum_{l=0}^{\intgpart}
(-1)^l \frac{s!}{l!\,(s-r-2l)!}
\left(\sqrt{{\textstyle{\frac{3}{\varepsilon_0}}}}\,w\right)^{s-r-2l}
\nonumber
\\
&&\!\!
+\; O\!\left(\frac{1}{\sqrt{N}} \right).
\end{eqnarray}
By comparing with the formula for the Hermite polynomial of degree $k$
on $\mathbb{R}$,
\begin{equation}
H_{k}(x)
=
\sum_{l=0}^{\big\lfloor\!\frac{{\scriptstyle k}}{2}\!\big\rfloor}
(-1)^{l+k}\frac{ s!}{l!\,(k-2l)!}(2x)^{k-2l},
\end{equation}
we see that, given $p\in\mathbb{N}$ and $w\in \mathbb{R}$, we have
\begin{eqnarray}
\widetilde{P}_s^r\left(\frac{w}{\sqrt{\scriptstyle{{2N\vareps_0}}}};3N-p\right)
=
\left(-\sqrt{2}\right)^{r-s}\,
H_{s-r}\left(\sqrt{{\textstyle{\frac{3}{4\varepsilon_0}}}}w\right)
+\; O\!\left(\frac{1}{\sqrt{N}} \right)
\end{eqnarray}
when $N\gg 1$.
Hence, for all fixed $p$ we now find that pointwise for any $w\in\mathbb{R}$,
\begin{equation}
\lim_{N\to\infty}
\widetilde{P}_s^r\left(\frac{w}{\sqrt{\scriptstyle{{2N\vareps_0}}}};3N-p\right)
=
\left(-\sqrt{2}\right)^{r-s}\,
H_{s-r}\left(\sqrt{{\textstyle{\frac{3}{4\varepsilon_0}}}}w\right)
\end{equation}
where again it is understood that
$N> \max\{p/3,w^2/(2\varepsilon_0)\}$ in the expression under the
limit in the left-hand side.
Equation (\ref{eigmarg3}) in the main text follows.
\newpage
|
{
"timestamp": "2005-11-03T23:18:24",
"yymm": "0503",
"arxiv_id": "math-ph/0503073",
"language": "en",
"url": "https://arxiv.org/abs/math-ph/0503073"
}
|
\section{Introduction}
In the last years, several theoretical approaches have predicted strong medium
effects on the pion pion interaction in the scalar isoscalar ($\sigma$) channel.
In Ref. \cite{Hatsuda:1999kd}, Hatsuda et al. studied the $\sigma$ propagator
in the linear $\sigma$ model and found an enhanced and narrow spectral
function near the $2\pi$ threshold caused by the partial restoration of the
chiral symmetry, where $m_\sigma$ would approach $m_\pi$. The same
conclusions were reached using the nonlinear chiral Lagrangians in Ref.
\cite{Jido:2000bw}.
Similar results, with large enhancements in the $\pi\pi$ amplitude around the
$2\pi$ threshold, have been found in a quite different approach by studying the
$s-$wave, $I=0$ $\pi\pi$ correlations in nuclear matter
\cite{Schuck:1988jn,Rapp:1996ir,Aouissat:1995sx,Chiang:1998di}. In these cases
the modifications of the $\sigma$ channel are induced by the strong $p-$wave
coupling of the pions to the particle-hole ($ph$) and $\Delta$-hole ($\Delta
h$) nuclear excitations. It was pointed out in
\cite{Aouissat:2000ss,Davesne:2000qj} that this attractive $\sigma$ selfenergy
induced by the $\pi$ renormalization in the nuclear medium could be
complementary to additional $s$-wave renormalizations of the kind discussed in
\cite{Hatsuda:1999kd,Jido:2000bw} calling for even larger effects.
On the experimental side, there are also several results showing strong medium
effects in the $\sigma$ channel at low invariant masses in the $A(\pi,2\pi)$
\cite{Bonutti:1996ij,Bonutti:1998zw,Camerini:1993ac,bonutti,Starostin:2000cb}
and $A(\gamma,2\pi)$ \cite{Messchendorp:2002au} reactions. At the moment, the
cleanest signal probably corresponds to the $A(\gamma,2\pi^0)$ reaction, which
shows large density effects that had been predicted in both shape and size in
Ref. \cite{Roca:2002vd}, using a model for the $\pi\pi$ final state interaction
along the lines of the present work. Note, however, that a part of the spectrum
modification could be due to quasielastic collisions of the pion
\cite{Muhlich:2004zj}.
Our aim in this paper is to study the $\pi\pi$ scattering in the scalar
isoscalar ($\sigma$) channel at finite densities in the context of the model
developed in
\cite{Dobado:1990qm,Dobado:1993ha,Oller:1998ng,Oller:1999hw,Oller:1999zr,Oller:1997ti}.
These works, which provide an economical and successful description of a
wide range of hadronic phenomenology, use as input the lowest orders of the
Lagrangian of Chiral Perturbation Theory ($\chi PT$) \cite{Gasser:1985ux} and calculate
meson meson scattering in a coupled channels unitary way.
Some nuclear medium effects, namely the $p-$wave coupling of the pions to the
particle hole ($ph$) and Delta hole ($\Delta h$) excitations, were
implemented in this framework in Refs. \cite{Chiang:1998di,Oset:2000ev}. As in
other approaches, large medium effects were found as reflected in the imaginary
part of the $\pi \pi$ scattering amplitude which showed a clear shift of
strength towards low energies as the density increases.
Although this model was able to predict the size of the medium effects on the
$(\gamma, 2\pi)$ reaction \cite{Roca:2002vd}, it was pointed out that some
probably large contributions related to nucleon tadpole diagrams
\cite{Jido:2000bw} and some vertex corrections \cite{Meissner:2001gz} were
missing. In this work, we will include those pieces and analize its influence in
the $\pi\pi$ scattering amplitude at finite nuclear densities.
In the next section we present, for the sake of completeness, a brief
description of the model used for the $\pi \pi$ interaction both in vacuum and
in a dense medium, which is already published elsewhere \cite{Chiang:1998di,Oset:2000ev}.
In Section 3 we consider further contributions to the $\pi\pi$ interaction in
the nuclear medium, associated to higher order terms in the chiral Lagrangian
than those included in Refs. \cite{Chiang:1998di,Oset:2000ev}, and some baryonic
vertex corrections advocated in Ref. \cite{Meissner:2001gz}.
\section{$\pi \pi$ interaction}
In this section we summarize the method of Ref. \cite{Oller:1997ti}
for $\pi \pi$ interaction in vacuum and Refs. \cite{Chiang:1998di,Oset:2000ev}
for the nuclear medium effects. Additional information on this and related
approaches for different spin isospin channels can be found in Refs.
\cite{Oller:1997ti,Oller:1998ng,Oller:1999hw,Nieves:2000bx}.
\subsection{Vacuum}
The basic idea is to solve a Bethe Salpeter (BS) equation, which guarantees
unitarity, matching the low energy results to $\chi PT$ predictions. We
consider two coupled channels, $\pi \pi$ and $K \bar{K}$ and neglect the $\eta
\eta $ channel which is not relevant at the low energies we are interested
in.
The BS equation is given by
\begin{equation}
\label{eq:BS}
T=V+VGT.
\end{equation}
Eq. (\ref{eq:BS}) is a matrix integral equation which
involves the two mesons one loop divergent integral (see Fig.~\ref{fig:BSF}),
where $V$ and $T$ appear off shell. However, for this channel both functions
can be factorized on shell out of the integral. The remaining off shell part
can be absorbed by a renormalization of the coupling constants
as it was shown in Refs. \cite{Oller:1997ti,Nieves:1999hp}.
Thus, the BS equation becomes purely algebraic and the
$VGT$ term, originally inside the loop integral, becomes then the product
of $V$, $G$ and $T$, with $V$ and $T$ the on shell amplitudes independent
of the integration variables, and $G$ given by the expression
\begin{equation}
G_{ii}(P) = i \int \frac{d^4 q}{(2 \pi)^4}
\frac{1}{q^2 - m_{1i}^2 + i \epsilon} \; \;
\frac{1}{(P - q)^2 - m_{2i}^2 + i \epsilon}
\end{equation}
where $P$ is the momentum of the meson meson system. This
integral is regularized with a cut-off ($\Lambda$) adjusted to optimize the fit
to the $\pi\pi$ phase shifts ($\Lambda=1.03$ GeV).
\begin{figure}
\begin{center}
\epsfig {figure=fig_1.eps,width=12.cm}
\caption{Diagrammatic representation of the Bethe Salpeter equation.}
\label{fig:BSF}
\end{center}
\end{figure}
The potential $V$ appearing in the BS equation is taken from
the lowest order chiral Lagrangian
\begin{equation}
{\cal L}_2 = \frac{1}{12 f^2} \langle (\partial_\mu \Phi \Phi - \Phi \partial_\mu
\Phi)^2 + M \Phi^4 \, \rangle
\end{equation}
\noindent
where the symbol $\langle \rangle$ indicates the trace in flavour space,
$f$ is the pion decay constant and $\Phi$, $M$ are the pseudoscalar meson
and mass $SU(3)$ matrices.
This model reproduces well phase shifts and inelasticities up to about
1.2 GeV. The $\sigma$ and $f_0 (980)$ resonances appear as poles of the
scattering amplitude in $L=0$, $I=0$.
The coupling of channels is essential to produce the
$f_0 (980)$ resonance, while the $\sigma$ pole is little
affected by the coupling of the pions to $K \bar{K}$
\cite{Oller:1997ti}.
\subsection{\label{sec:nucmed}The nuclear medium}
As we are mainly interested in the low energy region, which is not very
sensitive to the kaon channels, we will only consider the nuclear medium
effects on the pions. The main changes of the pion propagation in the
nuclear medium come from the $p-$wave selfenergy, produced basically
by the coupling of pions to particle-hole ($ph$) and Delta-hole
($\Delta h$) excitations. For a pion of momentum $q$ it is given by
\begin{equation}
\label{eq:self}
\Pi(q)= {{\left({D+F}\over{2f}\right)^2 \vec q\,^2 U(q)}
\over
{1-\left({D+F}\over{2f}\right)^2 g' U(q)}}
\end{equation}
with $g'=0.7$ the Landau-Migdal parameter, $U(q)$ the Lindhard function and
$(D+F)=1.257$. The expressions for the Lindhard functions are taken
from Ref. \cite{Oset:1990ey}.
Thus, the in-medium BS equation will include the
diagrams of Fig. \ref{fig:BSF2} where the solid line bubbles represent
the $ph$ and $\Delta h$ excitations.
\begin{figure}[htb]
\begin{center}
\epsfig{height=2.2cm,width=12.1cm,angle=0, figure=fig_2.eps}
\caption{Terms of the meson meson scattering amplitude accounting for
$ph$ and $\Delta h$ excitation.}
\label{fig:BSF2}
\end{center}
\end{figure}
In fact, as it was shown in \cite{Chanfray:1999nn}, the contact terms with
the $ph$ ($\Delta h$) excitations of diagrams (b-d) cancel the
off-shell contribution from the meson meson vertices in the term of Fig.
\ref{fig:BSF2}(a).
Hence, we just need to calculate the diagrams of the free type
(Fig. \ref{fig:BSF}) and those of Fig. \ref{fig:BSF2}(a) with the amplitudes
factorized on shell. Therefore, at first order in the baryon density, we are left
with simple meson propagator corrections which can be readily incorporated by
changing the meson vacuum propagators by the in medium ones.
The $\pi\pi$ scattering amplitude obtained using this model
exhibits a strong shift towards low energies. In Fig. \ref{fig:MED1}, we show
the imaginary part of this amplitude for several densities. Quite similar
results have been found using different models
\cite{Aouissat:1995sx} and it has been suggested
that this accumulation of strength, close to the pion threshold, could reflect
a shift of the $\sigma$ pole which would approach the mass of the pion.
\begin{figure}[htb]
\begin{center}
\epsfig{width=12.1cm, figure=fig_3.eps}
\caption{Imaginary part of the $\pi\pi$ scattering amplitude at several
densities.}
\label{fig:MED1}
\end{center}
\end{figure}
Other pion selfenergy contributions related to $2ph$ excitations, and thus
proportional to $\rho^2$, can be incorporated in the pion propagator. As we are
most interested in the region of low energies we can take as estimation the
corresponding piece of the optical potentials obtained from pionic atoms data,
following the procedure of Ref. \cite{Chiang:1998di} and substituting in Eq.
(\ref{eq:self})
\begin{equation}
\left({D+F}\over{2f}\right)^2 U(q)
\;\longrightarrow\;
\left({D+F}\over{2f}\right)^2 U(q) -4\pi C_0^* \rho ^2
\label{eq:self2}
\end{equation}
with $\rho$ the nuclear density and $C_0^*=(0.105+i 0.096) m_\pi^{-6}$.
Its effects are small except at large densities as can be appreciated by
comparing Fig. \ref{fig:MED1}, with Fig. 7 of Ref. \cite{Chiang:1998di}
where this piece is included.
\section{Further contributions}
\subsection{\label{sec:tadpoles}Higher order tadpole and related terms}
The chiral Lagrangian generates tadpole terms that could contribute to the
pion selfenergy and also in the form of vertex corrections as in Fig.
\ref{fig:NUCTAD}. At the lowest order these terms vanish in isospin symmetric
nuclear matter \cite{Oset:2000ev}. However, at next order there are terms which
provide some contribution. The complete structure of the higher order
Lagrangian adapted to the $\pi N$ system can be seen in \cite{Bernard:1995dp}.
The medium corrections associated to these new Lagrangian terms in the $\pi$ nucleus
interaction were studied in \cite{Thorsson:1995rj} and interpreted in terms of
changes of the time and space components of $f$ and changes of the pion mass
in the medium. Further developments in this direction are done in
\cite{Meissner:2001gz}.
The repercussion of these terms in $\pi\pi$ scattering in the nuclear medium
has been considered in \cite{Jido:2000bw} and we follow here the same steps. We
start from the second order $\pi N$ Lagrangian relevant for the isoscalar
sector
\begin{eqnarray}
\label{LpiN2}
{\cal L}_{\pi N}^{(2)} & = & c_3 \bar{N} (u_{\mu} u^{\mu}) N
+ (c_2 - {g_A^2 \over 8 m_N}) \bar{N}({\rm v}_{\mu} u^{\mu})^2 N \nonumber \\
& & + c_1 \bar{N}N {\rm Tr} (U^{\dagger} \chi + \chi^{\dagger} U)
+ \cdot \cdot \cdot ,
\end{eqnarray}
where $u_{\mu} = i u^{\dagger} \partial_{\mu} U u^{\dagger}$, with $U =
u^2 = {\rm exp}(i \tau^a \phi^a /f)$ in the $SU(2)$ formalism used
there, ${\rm v}_{\mu}$ is the four velocity of the nucleon, $g_A$ the axial
charge of the nucleon and $\chi = {\rm diag}(m_{\pi}^2,m_{\pi}^2)$.
The pion nucleon amplitude obtained from the
Lagrangian in Eq. (\ref{LpiN2}) is
\begin{eqnarray}
\label{TpiNfromLpiN2}
t_{\pi N} &=& \frac{4 c_1}{f^2} m_{\pi}^2 - \frac{2 c_2}{f^2} (q^0)^2
- \frac{2 c_3}{f^2} q^2
\nonumber \\
&=& ( \frac{4 c_1}{f^2} m_{\pi}^2 - \frac{2 c_2}{f^2} \omega(q)^2
- \frac{2 c_3}{f^2} m_{\pi}^2 )
\nonumber \\
& & - \frac{2 c_2 + 2 c_3}{f^2} (q^2-m_{\pi}^2)
= t_{\pi N}^{on} + t_{\pi N}^{off} \,\,\, ,
\end{eqnarray}
where in the last part of the equation we have separated what we call the
on-shell part and the off-shell part of the amplitude (term with
$(q^2-m_{\pi}^2)$). This $s-$wave $\pi N$ interaction produces a modification of
the pion propagator which we shall consider later in the solution of the Bethe
Salpeter equation in the medium.
In \cite{Jido:2000bw}
and \cite{Thorsson:1995rj} the medium effects are recast at
the mean field level in terms of a medium Lagrangian given by
\begin{eqnarray}
\label{mean-f}
\langle {\cal L} \rangle
& = & ( {f^2 \over 4} + {c_3 \over 2} \rho)\ {\rm Tr}
[\partial_{\mu} U \partial^{\mu} U^{\dagger}] \nonumber \\
& & + \ \ ( {c_2 \over 2} - {g_A^2 \over 16 m_N})\ \rho \ {\rm Tr}
[\partial_0 U \partial_0 U^{\dagger}] \nonumber \\
& & + \ \
({ f^2 \over 4} + {c_1 \over 2} \rho)
\ {\rm Tr} (U^{\dagger} \chi + \chi^{\dagger} U) \,\,\, .
\end{eqnarray}
The different corrections to the $\pi\pi$ scattering
amplitude coming from the
$\partial_{\mu} U \partial^{\mu} U^{\dagger}$, $\partial_0 U \partial^0
U^{\dagger}$ terms and the mass term in Eq.
(\ref{mean-f}) ($c_3$, $c_2$ and $c_1$ terms) are given by
\begin{eqnarray}
\label{correc_derivative}
\delta t_{\pi\pi}^{(t)}
&=&- \frac{1}{f^2} \lbrace \frac{2 c_3}{f^2} \rho (s-\frac{4}{3}m_{\pi}^2)
+ \frac{2 c_2}{f^2} \rho (s-\frac{1}{3} \sum_i \omega_i(q)^2) +
\frac{c_1}{f^2}\rho\frac{5}{6} m_{\pi}^2 \rbrace
\nonumber \\
& &+ \frac{1}{f^2} \lbrace \frac{2 c_3}{f^2} \rho \frac{1}{3} \sum_i
(q_i^2-m_{\pi}^2) + \frac{2 c_2}{f^2} \rho \frac{1}{3} \sum_i
(q_i^2-m_{\pi}^2) \rbrace \,\,\, ,
\end{eqnarray}
where we have also separated the on-shell part from the off-shell part. These
are the corrections coming from the many body tadpole diagram of Fig.
\ref{fig:NUCTAD},
which are included in the $\rho$ dependent terms of Eq. (\ref{mean-f}).
Note that in the chiral unitary approach that we follow, the external legs are
placed on shell ($q_i^2=m_{\pi}^2$). This is the case even when the diagrams
appear in loops, as in Fig. \ref{fig:NUCTADLOOP}, since the underlying physics
is the use of a dispersion relation using the $N/D$ method
\cite{Oller:1998zr,Oller:2000fj}
which determines the diagram contribution in terms of its imaginary part. In
the case of Fig. \ref{fig:NUCTADLOOP} the cut corresponds to two free pions on
shell, like in the vacuum. Hence, we shall use only the on shell part of the
correction of Eq. (\ref{correc_derivative}).
\begin{figure}[htb]
\begin{center}
\epsfig{width=6cm,figure=fig_4.eps}
\caption{Nucleon tadpole term correction to the $\pi\pi$ interaction.}
\label{fig:NUCTAD}
\end{center}
\end{figure}
\begin{figure}[htb]
\begin{center}
\epsfig{width=8cm,figure=fig_5.eps}
\caption{$\pi\pi$ rescattering diagram with tadpole vertex correction showing
the $\pi\pi$ cut.}
\label{fig:NUCTADLOOP}
\end{center}
\end{figure}
As mentioned before, at the same time, when solving the Bethe-Salpeter equation,
we have also to take into account the $s-$wave selfenergy
insertion from the Lagrangian of Eq. (\ref{LpiN2}) in the pion propagators as
depicted in Fig. \ref{fig:NUCTADLOOPSERIES}.
This is easily accounted for, at lowest order in $\rho$, adding to each pion
propagator, $D_{\pi}$, the correction $D_{\pi} t_{\pi N} \rho D_{\pi}$. A
technically simple way to account for that is to add to the
scalar isoscalar $\pi\pi$ vertex from ${\cal L}_2$, $t_{\pi\pi}$, the correction
\begin{equation}
\label{swaveinsertion}
\delta t_{\pi\pi}^{(s)} =
(t_{\pi N}^{on}+t_{\pi N}^{off}) \rho \frac{1}{q^2-m_{\pi}^2} t_{\pi\pi}
\end{equation}
for the two pion propagator lines to the left of the $\pi\pi$ vertex.
Now the separation of the on-shell and off-shell parts of $t_{\pi N}$ is most
useful since the pion propagator in Eq.
(\ref{swaveinsertion}) is cancelled out by the $(q^2-m_{\pi}^2)$ factor of the
off-shell part of $t_{\pi N}$.
Thus we have
\begin{equation}
\label{swaveinsertion2}
\delta t_{\pi\pi}^{(s)} = t_{\pi N}^{on} \, \rho \, \frac{1}{q^2-m_{\pi}^2}
t_{\pi\pi} - \frac{2 c_2 + 2 c_3}{f^2} \, \rho \, t_{\pi\pi} \,\,\, .
\end{equation}
This means that with the Lagrangian used, on top of the corrections in the loops
from the $s-$wave (on-shell) pion selfenergy, we have an additional correction
(second term of Eq. (\ref{swaveinsertion2})) of the same topology as
the tadpole term considered before.
Considering the pion
selfenergy insertion in either of the two pion propagators, we obtain for this
\begin{equation}
\label{swaveinsertion3}
\delta t_{\pi\pi}^{(st)} = -\frac{4c_2 + 4c_3}{f^2} \, \rho \, t_{\pi\pi} \,\,\, .
\end{equation}
Thus, we are left with the usual contribution in the pion loops of the ordinary
on-shell $s-$wave pion selfenergy, plus the tadpole correction of Eq.
(\ref{correc_derivative}), plus the tadpole equivalent of Eq.
(\ref{swaveinsertion3}).
\begin{figure}[htb]
\begin{center}
\epsfig{width=10cm,figure=fig_6.eps}
\caption{Nucleon tadpole correction in the pion propagator.}
\label{fig:NUCTADLOOPSERIES}
\end{center}
\end{figure}
There are still further contributions belonging to the same family. Indeed, the
$t_{\pi\pi}$ amplitude in the scalar isoscalar channel,
\begin{equation}
\label{tpipiatrhozero}
t_{\pi\pi} = -\frac{1}{f^2} (s-\frac{m_{\pi}^2}{2} -
\frac{1}{3}\sum_i(q_i^2-m_{\pi}^2)) \,\,\,,
\end{equation}
is also split in on- and off-shell parts. In \cite{Chiang:1998di,Chanfray:nn} it
was shown that the off-shell pieces could be removed from the loop calculations
for both the free pion case and the pion with a $p-$wave selfenergy.
However, the diagrams in Fig. \ref{fig:NUCTADLOOPSERIES}(a) have one free pion and a
pion with a $s-$wave medium selfenergy insertion, hence the imaginary part of the
two-pion loop is not the same
as in the mentioned cases.
It is again easy to take into
account this correction and we have, from the $s-$wave selfenergy insertions in
the pion propagators
\begin{equation}
\label{swaveinsertion4}
\delta t_{\pi\pi}^{(so)} = t_{\pi N}\rho \frac{1}{q^2-m_{\pi}^2} \frac{1}{3 f^2} (q^2-m_{\pi}^2) \equiv
\frac{1}{3 f^2} t_{\pi N}\rho
\end{equation}
for each pion line.
Next we separate the on-shell and off-shell parts of $t_{\pi N}$. For the
on-shell part we get
\begin{equation}
\label{swaveinsertion5}
\frac{1}{3 f^2} (\frac{4 c_1}{f^2}m_{\pi}^2-\frac{2 c_2}{f^2}\omega(q)^2-
\frac{2 c_3}{f^2}m_{\pi}^2) \rho \,\,\, ,
\end{equation}
which compared at threshold
to the free $t_{\pi\pi}$ amplitude, $t_{\pi\pi}= -\frac{1}{f^2} \frac{7}{2}
m_{\pi}^2$, gives
\begin{equation}
\label{swaveinsertion6}
\frac{\delta t_{\pi\pi}}{t_{\pi\pi}} \simeq \frac{1}{21 f^2}
(8 c_1 - 4 c_2 - 4 c_3) \rho \,\,\, ,
\end{equation}
which with respect to Eq. (\ref{swaveinsertion3}) gets a reduction of a factor
21, plus an extra reduction from the near on-shell cancellation of the isoscalar
$t_{\pi N}$. Hence, this correction is negligible and we take advantage of this
large reduction factor $21$ to also neglect the part involving simultaneously
the off-shell parts of $t_{\pi\pi}$ and $t_{\pi N}$.
In order to proceed we have to decide upon the $c_i$ coefficients to be used. It
is well known that the Lagrangian of Eq. (\ref{LpiN2}) leads to a part of
$p-$wave pion selfenergy \cite{Kirchbach:1996xy}, but we are explicitly taking a
$p-$wave selfenergy insertion
accounting for $ph$ and $\Delta h$ excitations. There is a work
which uses the same Lagrangian of Eq. (\ref{LpiN2}), and
in addition takes into account explicitly the $\Delta$ degrees of freedom
\cite{Fettes:2000bb}. Thus, we stick to the values of the $c_i$ coefficients
obtained there from two fits, with and without using the $\sigma$ term as a
constraint, shown in Table \ref{ci}. For comparison, the values of the
coefficients $c_i$ without
including the $\Delta$ are of the order of $c_1=-1.53$ GeV$^{-1}$, $c_2=3.22$
GeV$^{-1}$ and $c_3=-6.20$ GeV$^{-1}$ \cite{Fettes:1998ud}.
\begin{table}[ht]
\begin{center}
\begin{tabular}{|l|l|l|}
coef.& set I (GeV$^{-1}$) & set II (GeV$^{-1}$) \\
\hline
$c_1$ & $-0.35$ & $-0.32$ \\
\hline
$c_2$ & $-1.49$ & $-1.59$ \\
\hline
$c_3$ & $0.93$ & $1.15$ \\
\hline
\end{tabular}
\caption{\footnotesize{$c_i$ coefficients from Ref. \cite{Fettes:1998ud}.}}
\label{ci}
\end{center}
\end{table}
As stressed in \cite{Fettes:1998ud} the values of the coefficient $c_i$ with the
explicit contribution of the $\Delta$ are of natural order, while those obtained
without its consideration are too large and a source of problems in chiral
perturbative calculations \cite{Epelbaum:2003gr}. But in our case, as pointed
above, the choice is mandatory.
We can estimate the size of the correction of Eq. (\ref{correc_derivative})
at pion threshold, and taking advantage of the
reduction factor $1/3$ in the term $\frac{1}{3} \omega_i^2(q)$ in front of $s \simeq 4
m_{\pi}^2$, we approximate $\omega_i(q) \simeq m_{\pi}$. So we get
\begin{equation}
\label{delta}
\frac{\delta t_{\pi\pi}}{t_{\pi\pi}} = \frac{32}{21 f^2}(c_2 + c_3) \rho
+ \frac{10}{21 f^2}c_1 \rho \,\,\, ,
\end{equation}
which for the two sets of parameters of Table \ref{ci} gives
\begin{eqnarray}
\label{deltanum}
\frac{\delta t_{\pi\pi}}{t_{\pi\pi}}
&=& -0.154 \, \rho / \rho_0 \,\,\, \textrm{(set I)} \nonumber \\
&=& -0.124 \, \rho / \rho_0 \,\,\, \textrm{(set II)} \,\,\,.
\end{eqnarray}
Let us note that the correction is negative, reducing effectively the strength
of the $\pi\pi \to \pi\pi$ vertex in the medium. Note that should we have used
the values of $c_i$ without explicit $\Delta$ we would obtain a value for the
ratio of Eq. (\ref{delta}) of $-0.80 \rho / \rho_0$, certainly too large, but
also negative.
Next we consider the contribution from Eq. (\ref{swaveinsertion3}). This
correction has opposite sign to the former one. When adding the two corrections
we find, again taking the threshold for comparison,
\begin{equation}
\label{deltatotal}
\frac{\delta t_{\pi\pi}}{t_{\pi\pi}} = -\frac{52}{21 f^2}(c_2 + c_3) \rho
+ \frac{10}{21 f^2} c_1 \rho \,\,\, ,
\end{equation}
which for the two sets of values of Table \ref{ci} gives
\begin{eqnarray}
\label{deltatotalnum}
\frac{\delta t_{\pi\pi}}{t_{\pi\pi}}
&=& 0.18 \, \rho / \rho_0 \,\,\, \textrm{(set I)} \nonumber \\
&=& 0.14 \, \rho / \rho_0 \,\,\, \textrm{(set II)} \,\,\,.
\end{eqnarray}
We can see that the sign of the correction is now reversed and, altogether, we
find now an effective increase of the $\pi\pi$ vertex in the medium by a
moderate amount.
Apart from the vertex corrections, we need to include the effect of the
on-shell $s-$wave pion selfenergy in the pion propagators in the loops,
produced by the nucleon tadpole diagram. Since we have a broad range of pion
energies in the loop, we have used the $t \rho$ approximation for the $s-$wave
pion selfenergy and the amplitude $t$ has been taken from the experimental fit
to data \cite{Arndt:1995bj}. This is a more realistic approach than to take the
expression from the model used here which gives a too large $s-$wave scattering
amplitude at high energies, and in any case produces a minor effect.
The considered corrections are included in the $\pi\pi$ amplitude by modifying
the kernel of the BS equation with the on-shell part of Eq.
(\ref{correc_derivative}) and Eq. (\ref{swaveinsertion3}), namely
\begin{equation}
\label{BStad}
T = \frac{V+\delta t_{\pi\pi}^{(t)\,on}}
{1-(V+\delta t_{\pi\pi}^{(t)\,on}+\delta t_{\pi\pi}^{(st)})\,G}
\,\,\, ,
\end{equation}
and modifying the pion propagators in the calculation of the two-pion loop
function, $G$, as explained in Section \ref{sec:nucmed}.
\subsection{\label{sec:newmech}Vertex corrections from baryonic loops}
In the previous sections we have considered relevant medium effects, according
to the pion nucleus phenomenology, which describe correctly the pion in the
medium in a wide range of energies. These mechanisms lead to $s-$ and $p-$ wave
pion selfenergies in the propagators of the BS equation and some associated
vertex corrections. In Ref. \cite{Meissner:2001gz}, other vertex corrections to
the $\pi\pi$ amplitude which could provide some effect at low energies, where
the leading $p-$wave pion selfenergy is not so strong, were studied.
\begin{figure}
\begin{center}
\includegraphics[width=10cm,height=5cm]{fig_7.eps}
\caption{\label{new_mech}(a) $ph$ bubble exchange in the $t$ channel; (b) Box
diagram.}
\end{center}
\end{figure}
The mechanisms considered in \cite{Meissner:2001gz} relevant for $\pi\pi$ $s-$wave
scattering, modifying the kernel of the Bethe
Salpeter equation in the $\pi \pi$ interaction, are shown in Fig. \ref{new_mech}.
The $\pi \pi$ isoscalar contribution in the $s$ channel for the $p h$ excitation
in the $t$ channel in Fig. \ref{new_mech}(a) is given, with the unitary normalization, by
\begin{equation}
-i t = - \left( \frac{1}{4 f^2} \right) ^2 (p^0_1+k^{0}_1)(p^0_2+k^0_2) U(q) =
\tilde{t} \, U(q)
\label{textra}
\end{equation}
where $U(q)$ is the ordinary Lindhard
function for $ph$ excitation, including a factor 2 of
isospin (see Appendix of \cite{Oset:1990ey}). The second equation in Eq.
(\ref{textra}) defines $\tilde{t}$.
We have neglected the isoscalar $\pi N$ amplitude
in Eq. (\ref{textra}) since it is very small compared with the isovector one
\cite{Schroder:uq}.
The magnitude of $t$ was shown in \cite{Meissner:2001gz} to be comparable
to the $s-$wave $V$ of the lowest order chiral Lagrangian at densities of the
order of the nuclear density. Yet, there are some observations to be made:
First,
at pion threshold the diagram of Fig. \ref{new_mech}(a) is proportional to
$U(q^0=0,\vec{q}=\vec{0})$. This quantity is evaluated in \cite{Meissner:2001gz} using the
ordinary limit of the Lindhard function at $q^0=0$ and $|\vec{q}| \to 0$, which
is finite and larger in size than for any finite value of $|\vec{q}|$. This
limit is however quite different from the value of the response
function at $\vec{q}=\vec{0}$ in finite nuclei which is strictly zero,
as already noted in \cite{Meissner:2001gz,Oset:xm,Oset:sm}.
We take into account the fact that the isovector $\pi N$ amplitude reflects
the exchange of a $\rho$ in the $t$ channel \cite{Ericson:gk} and multiply
$(4f^2)^{-2}$ by a factor reflecting the two $\rho$ propagators,
$F(q)=(M_{\rho}^2/(M_{\rho}^2+\vec{q}\,^2))^2$.
\begin{figure}
\begin{center}
\includegraphics[width=10cm,height=5cm]{fig_8.eps}
\caption{\label{oneloopBS}Loop contributions to the Bethe Salpeter equation at first
order in the nuclear density, including the $t-$channel $ph$ excitation.}
\end{center}
\end{figure}
In order to estimate the importance of this contribution as compared to the
$p-$wave pion selfenergy insertions, we have evaluated the diagrams (a-e) in
Fig. \ref{oneloopBS}. Details of the calculation are given in Appendix II.
The results are shown in Fig. \ref{res:oneloopBS} for the
imaginary part of the resulting amplitude. We find that the contribution of
diagrams (d,e) is smaller than the changes produced by the insertion of the
$p-$wave pion selfenergy in the pion propagators. Similar results are found for
the real part of the amplitude.
\begin{figure}
\begin{center}
\includegraphics[width=0.7\textwidth]{fig_9.eps}
\caption{\label{res:oneloopBS}Imaginary part of the $\pi\pi$ amplitude from the
terms in Fig. \ref{oneloopBS}, as indicated in the legend. The
calculation for the dashed and dotted lines is done for $\rho=\rho_0/2$.}
\end{center}
\end{figure}
The consideration of this mechanism in the BS equation proceeds by adding the
tree level term and modifying the kernel in the loop terms with an effective
potential $\delta V$ defined as
\begin{equation}
\frac{\delta V(s,\rho)}{V} = \frac{{\cal F}(s,\rho)}{V G V} \,\,\, ,
\label{Veff}
\end{equation}
where ${\cal F}(s,\rho)$ is the amplitude corresponding to diagram (b) and $VGV$
gives the amplitude of diagram (a) in Fig. \ref{oneloopBS}.
In this sense, by
substituting $V$ by $V+\delta V$ in the $\pi\pi$ vertex, the loop function of
diagram in Fig. \ref{oneloopBS}(a) would account correctly for all the diagrams
(a-e) at the first order in the nuclear density.
One of the reasons for the small size of this contribution is that the
Lindhard function behaves
roughly as $q^{-2}$ for large values of $q$ and we should expect a large
cancellation of this piece in the loops. This would be in contrast with the
$ph$ excitations leading to the $p-$wave $\pi$ selfenergy in Fig.
\ref{oneloopBS}(b,c), since
there one has the combination $\vec{q}\,^2 U(q)$ and a priori this type of $ph$
excitation should be more important, as it is indeed the case. Thus, the
$t-$channel $ph$ exchange mechanism leads to a sizeable correction to the tree
level $\pi\pi$ scattering amplitude and a small vertex correction in the
calculation of the loops appearing in the unitarization procedure\footnote{This
mechanism would play an even smaller role in the position of the $\sigma$ pole
\cite{VicenteVacas:2002se},
which is determined by the vanishing of the denominator of the BS solution,
where the tree level term does not appear.}.
Next we consider the box diagram of Fig. \ref{new_mech}(b). This term was found
to be smaller in strength than the $ph$ exchange in \cite{Meissner:2001gz},
particularly at small energies, where the $p-$wave character of the vertices
made the contribution negligible.
The consideration of this mechanism at the pion loop level, necessary to include
it in the BS equation,
makes its contribution small since, apart from the reduction of the box diagram
for large values of $q$, there is a further cancellation of terms as we show in
Appendix III. Similar analytic treatments are done in
\cite{Herrmann:1993za,Cabrera:2000dx}. For all these reasons this contribution
should be even smaller than the one previously evaluated and one can safely
neglect it for practical purposes.
\begin{figure}
\begin{center}
\includegraphics[width=0.7\textwidth]{fig_10.eps}
\caption{\label{fig:tadeffect}Real and imaginary parts of $T$, as obtained in
Eq. (\ref{BStad}), for the two sets of parameters in Table \ref{ci} (set I,
dash-dotted line, set II, dashed line) and $\rho=\rho_0/2$. The solid line
corresponds to the result of the model in Sec. \ref{sec:nucmed} and the dotted
line is the result in vacuum.}
\end{center}
\end{figure}
\section{Results}
We solve the BS equation including the corrections discussed in Sec.
\ref{sec:tadpoles}, as they appear in Eq. (\ref{BStad}). The results are shown in
Fig. \ref{fig:tadeffect}. The new terms considered modify little the
results from Ref. \cite{Chiang:1998di}. In comparison, the imaginary part of the
$\pi\pi$ amplitude exhibits a small increase of strength at low invariant
energies whereas the real part decreases over all the calculated range of
energies. Altogether, the basic effect of the nuclear medium, as found in Ref.
\cite{Chiang:1998di}, is a strong depletion of the interaction at energies
around $500$~MeV, where the vacuum $\sigma$ pole is found, and some accumulation
of strength close to the $2\pi$ threshold, as it can be seen in Fig.
\ref{fig:tadeffectT2}, where the squared modulus of the amplitude is
depicted.
\begin{figure}
\begin{center}
\includegraphics[width=0.7\textwidth]{fig_11.eps}
\caption{\label{fig:tadeffectT2}Squared modulus of $T$. Lines as in Fig.
\ref{fig:tadeffect}.}
\end{center}
\end{figure}
The contribution of the terms discussed in Sec. \ref{sec:newmech} is shown in
Fig. \ref{fig:newmech} for the imaginary part of the $\pi\pi$ amplitude. We find
a strong reduction of the amplitude at energies close to the $2\pi$ threshold,
basically produced by the repulsive tree level term in Fig. \ref{new_mech}(a). At
these energies the amplitude stays closer to the vacuum case. A similar
reduction of the nuclear medium effects as compared to the results of Ref.
\cite{Chiang:1998di} is found in the real part of the amplitude.
\begin{figure}
\begin{center}
\includegraphics[width=0.7\textwidth]{fig_12.eps}
\caption{\label{fig:newmech}Imaginary part of $T$ including the mechanism
described in Sec. \ref{sec:newmech} at $\rho=\rho_0/2$ (dashed line). The solid line
corresponds to the result of the model in Sec. \ref{sec:nucmed} and the dotted
line is the result in vacuum.}
\end{center}
\end{figure}
Finally, we have included together the contributions of the tadpole terms,
Section \ref{sec:tadpoles}, and the $t-$channel $ph$ exchange, Section
\ref{sec:newmech}, in the BS equation, and the results are depicted in Fig.
\ref{fig:all} for the real and imaginary parts of the $\pi\pi$ amplitude. We
observe, compared to the model of Ref. \cite{Chiang:1998di} in which the basic
medium effect is due to the $p-$wave pion selfenergy, a considerable reduction
of strength close to the two pion threshold. The global effect in both
calculations is still a sizable depletion of the interaction at higher energies
and a certain accumulation of strength below the $\sigma$ pole position in
vacuum which, as suggested in \cite{VicenteVacas:2002se}, could be reflecting a
change in the $\sigma$ pole position to lower energies as a function of the
nuclear density.
\begin{figure}
\begin{center}
\includegraphics[width=0.7\textwidth]{fig_13.eps}
\caption{\label{fig:all}Real and imaginary parts of $T$ including the
mechanisms described in Secs. \ref{sec:tadpoles} and \ref{sec:newmech} at
$\rho=\rho_0/2$, using set I (dashed line). The solid line corresponds to the result of the
model in Sec. \ref{sec:nucmed} and the dotted line is the result in vacuum.}
\end{center}
\end{figure}
\section{Conclusions}
In summary, we have considered in this work the contribution of some new terms
to the $\pi\pi$ interaction in the scalar isoscalar channel at finite
densities, starting from a previous work \cite{Chiang:1998di,Oset:2000ev} in
which only medium effects associated to the $p-$wave pion selfenergy had been
accounted for.
Tadpole insertions, sometimes advocated as a possible source of a large
attraction, have been shown to affect little the $\pi\pi$ amplitude once the
Bethe Salpeter equation is solved. This is partly due to certain cancellations
which take place between vertices and internal pion propagator insertions.
We have also taken into account new terms in the driving kernel of the Bethe
Salpeter equation, which have been found important in a study based on a
chiral power counting in the many body problem. We could see that these new
terms, although large at tree level, when appearing inside loops were not as
important as one could guess from their comparison with the lowest order chiral
$\pi \pi$ amplitude in the case that all pions are on shell. As a consequence,
their consideration barely changed the results for the $\pi\pi$ interaction in
the medium.
Altogether, the final results are quite similar to those obtained previously in
\cite{Chiang:1998di,Oset:2000ev}, namely a strong reduction of the interaction
at energies around 400~MeV and beyond, and some increase of strength around the
2$\pi$ threshold.
This confirms the leading role of the strong $p-$wave pion selfenergy in the
medium modification of the $\pi\pi$ interaction in the scalar isoscalar channel.
These results are also satisfactory because a prediction on the $(\gamma,2\pi)$
reaction in nuclei \cite{Roca:2002vd} based on the previous calculation
\cite{Chiang:1998di,Oset:2000ev} of the two-pion final state interaction has
been later confirmed by experimental data \cite{Messchendorp:2002au}. The much
larger medium effects obtained at threshold energies in other approaches are
incompatible with the observed effect in the $(\gamma,2\pi)$ reaction.
\section*{Acknowledgements}
This work is partly supported by DGICYT contract no.
BFM2003-00856. D.~Cabrera acknowledges financial support from MEC.
\section*{Appendix I}
We quote in this section the Lindhard function, with an energy gap $\Delta$,
separated into the direct and crossed contributions, $U=U_d+U_c$. From
\cite{Oset:sm} we have
\begin{equation}
U_d(q^0,\vec{q},\Delta;\rho) = 4 \int \frac{d^3 p}{(2\pi)^3}
\frac{n(\vec{p}) \lbrack 1 - n(\vec{p}+\vec{q}) \rbrack}
{q^0 + \varepsilon (\vec{p}) - \varepsilon (\vec{p}+\vec{q}) - \Delta + i \epsilon}
\label{U_dgap}
\end{equation}
and $U_c(q^0,\vec{q},\Delta;\rho) \equiv U_d(-q^0,\vec{q},\Delta;\rho)$. In the
following we shall use the definitions
\begin{eqnarray}
x = \frac{q}{k_{F}} \,\,\, , \,\,\, \nu = \frac{2Mq^0}{k_F^2} \nonumber \\
\delta = \frac{2M\Delta}{k_F^2} \,\,\, , \,\,\, \rho=\frac{2}{3\pi^2}k_F^3
\label{defs}
\end{eqnarray}
with $M$ the mass of the nucleon, $k_F$ the Fermi momentum and $q \equiv
|\vec{q}|$. Once the integration in Eq. (\ref{U_dgap}) is done, the real
part of $U_d$ reads, for $x \leq 2$,
\begin{eqnarray}
\textrm{Re} \, U_d(q^0,\vec{q},\Delta;\rho) = - \frac{2Mk_F}{\pi^2} \frac{1}{2x}
\bigg \lbrace \frac{x}{2} - \frac{\nu - \delta}{4}
+ \frac{\nu - \delta}{2} \ln \bigg | \frac{\nu-\delta+x^2-2x}{\nu-\delta}\bigg |
\nonumber \\
+ \frac{1}{2}\left[ 1-\frac{1}{4} \left( \frac{\nu-\delta}{x}-x \right) ^2
\right] \ln \bigg | \frac{\nu-\delta-x^2-2x}{\nu-\delta+x^2-2x} \bigg | \bigg
\rbrace
\label{Reless2}
\end{eqnarray}
and, for $x > 2$,
\begin{eqnarray}
\textrm{Re} \, U_d(q^0,\vec{q},\Delta;\rho) = - \frac{2Mk_F}{\pi^2}\frac{1}{2x}
\bigg \lbrace \frac{-\nu+\delta+x^2}{2x} \nonumber \\
+ \frac{1}{2}\left[ 1 - \frac{1}{4} \left( \frac{\nu-\delta}{x}-x\right)^2
\right] \ln
\bigg | \frac{\nu-\delta-x^2-2x}{\nu-\delta-x^2+2x} \bigg | \bigg \rbrace
\,\,\, .
\label{Regt2}
\end{eqnarray}
The imaginary part of $U_d$ is given by
Im $U_d(q^0,\vec{q},\Delta;\rho)=$ Im $\tilde{U}(q^0-\Delta,\vec{q};\rho)
\Theta(q^0-\Delta)$, with
\begin{eqnarray}
\textrm{Im}\, \tilde{U}(q^0,\vec{q};\rho)= -\frac{3}{4} \pi \rho \frac{M}{q k_F}
\lbrack (1-z^2)\Theta(1-|z|) - (1-z'\,^2)\Theta(1-|z'|) \rbrack \frac{q^0}{|q^0|}
\,\,\, ,
\label{Utilda}
\end{eqnarray}
where $\Theta$ is the Heaviside step function and the $z$, $z'$ variables are
defined as
\begin{equation}
z=\frac{M}{q k_F}\left[ q^0-\frac{q^2}{2M}\right] \,\,\, , \,\,\,
z'=\frac{M}{q k_F}\left[ -q^0-\frac{q^2}{2M}\right] \,\,\, .
\end{equation}
\section*{Appendix II}
\begin{figure}
\begin{center}
\includegraphics[width=6cm]{fig_14.eps}
\caption{\label{loop2}Loop contribution of the $ph$ exchange in the $t$
channel.}
\end{center}
\end{figure}
The amplitude corresponding to the diagram in Fig. \ref{loop2} is given by
\begin{equation}
-i T = \int \frac{d^4q}{(2\pi)^4} (-i \tilde{t}) \frac{i}{(q+p)^2 - m_{\pi}^2 +
i\epsilon} \frac{i}{(q-p')^2 - m_{\pi}^2 + i\epsilon} (-i V(s)) \, i U(q) \,\,\,
.
\label{new_T}
\end{equation}
In order to perform the integral it is most useful to separate $U(q)$ into the
direct and crossed parts, $U(q)=U_d(q)+U_c(q)$, given their different analytical
structure.
\begin{figure}
\begin{center}
\includegraphics[width=13cm]{fig_15.eps}
\caption{\label{analyt}Analytical structure of the integrand in Eq.
(\ref{new_T}). The poles of the pion propagators are represented by '$x$' and
'$o$' symbols, and the dotted lines correspond to the analytical cuts of the
Lindhard function. The arrows indicate the circuit used for the integration of
each term.}
\end{center}
\end{figure}
In Fig. \ref{analyt} we depict the pole and cut structure for the different
terms and the path followed for the integration in the complex plane. The poles
are located at
\begin{eqnarray}
q^0 = -p^0 + \omega(\vec{p}+\vec{q}) - i\epsilon \,\,\, , \,\,\,
q^0 = -p^0 - \omega(\vec{p}+\vec{q}) + i\epsilon \nonumber \\
q^0 = p^0 + \omega(\vec{p}+\vec{q}) - i\epsilon \,\,\, , \,\,\,
q^0 = p^0 - \omega(\vec{p}+\vec{q}) + i\epsilon \,\,\, .
\label{poles}
\end{eqnarray}
The integration over the $q^0$ variable is done by closing the contour in the
complex plane in the upper half plane for the $U_d$ part and in the lower half
plane for the $U_c$ part.
The result of the integration is
\begin{eqnarray}
T = - \left( \frac{1}{4f^2}\right)^2 (2p^0)^2 V(s)
\int \frac{d^3q}{(2\pi)^3} \frac{1}{4\omega^2} \bigg \lbrace
\frac{U_c(p^0+\omega,\vec{q})}{p^0+\omega} \nonumber \\
-\frac{U_d(p^0-\omega,\vec{q})}{p^0-\omega+i\epsilon}
+ \frac{U_d(p^0-\omega,\vec{q})-U_c(p^0+\omega,\vec{q})}{p^0} \bigg \rbrace \bigg (
\frac{M_{\rho}^2}{M_{\rho}^2+\vec{q}\,^2} \bigg ) ^2 \,\,\, ,
\end{eqnarray}
where $\omega \equiv \omega(\vec{p}+\vec{q})$ and we have explicitly written the
$\rho$ meson exchange form factor arising from each $\pi\pi NN$ vertex.
Let us note that we have factorized the $\pi N \to \pi N$ vertex on shell. This
is done in analogy to what is done in \cite{Oset:1997it} where one shows that the
off shell part can be cast into a renormalization of the lowest order diagram
(no meson loop in this case). An alternative justification using dispersion
relations, which require only the on shell information, is given in
\cite{Oller:2000fj}.
\section*{Appendix III}
\begin{figure}
\begin{center}
\includegraphics[width=5cm]{fig_16.eps}
\caption{\label{box}Box diagram with two of the pions as a part of a loop.}
\end{center}
\end{figure}
We evaluate the loop function of Fig. \ref{box} containing the box
diagram of Fig. \ref{new_mech}(b)
plus all the different time orderings,
which we can see in Fig. \ref{box_orderings}. In all the diagrams the internal nucleon
lines are particle lines. This means we are taking only the terms
of order $\rho$, which are obtained when the external lines
are folded to give a single hole line in Fig. \ref{box_orderings}.
\begin{figure}
\begin{center}
\includegraphics[width=7cm]{fig_17.eps}
\caption{\label{box_orderings}Set of different time orderings of diagram in Fig.
\ref{box}. The initial and final nucleon lines correspond to a hole
propagator.}
\end{center}
\end{figure}
{\bf Diagrams (a), (b)}. For diagrams (a), (b) in Fig. \ref{box_orderings} for
$\vec{P}=\vec{p}_1+\vec{p}_2=\vec{0}$ the intermediate nucleon line after the
two pion vertices has the same momentum as the hole line (belonging to the
Fermi sea) and hence they both vanish.
Next we observe some strong
cancellations in the other diagrams. The set of the two meson propagators, which
is common to all of them, can be
written as
\begin{equation}
\frac{1}{2 P^0 \omega} \bigg \lbrace
\frac{1}{q^0-\omega+i\epsilon} \; \frac{1}{q^0+P^0+\omega-i\epsilon}
- \frac{1}{q^0+\omega-i\epsilon} \; \frac{1}{q^0+P^0-\omega+i\epsilon}
\bigg \rbrace \,\,\, ,
\label{props}
\end{equation}
with $\omega=\omega(\vec{q})$.
{\bf Diagram (c)}. The diagram (c) contains three nucleon
propagators. By making a heavy baryon approximation and neglecting the kinetic
energy of the nucleons we find for the product
\begin{equation}
\frac{1}{-p_1^0} \; \frac{1}{q^0+p_2^0+i\epsilon} \; \frac{1}{q^0+i\epsilon} \,\,\, .
\label{approx_c}
\end{equation}
Hence, multiplying this by the pion propagators and closing the contour on the
upper half plane to perform the $q^0$ integration, we find that the integral is
\begin{equation}
A \frac{1}{p_1^0} \bigg \lbrace -\frac{1}{p_1^0+\omega}\; \frac{1}{P^0+\omega}
+ \frac{1}{\omega} \; \frac{1}{p_2^0-\omega+i\epsilon} \bigg \rbrace \,\,\ .
\label{integ_c}
\end{equation}
There, the first term, which comes from the negative energy components of the
mesons, is small and has no imaginary part. The second term can lead to an
imaginary part and a more sizeable real part from the principal value.
{\bf Diagram (e)}. The set of nucleon propagators in the heavy baryon
approximation is now
\begin{equation}
\frac{1}{p_2^0} \; \frac{1}{p_2^0+q^0+i\epsilon} \; \frac{1}{-p_1^0}
\label{approx_e}
\end{equation}
and hence by closing the contour in the upper half of the complex $q^0$
plane we find for the $q^0$ integration
\begin{equation}
A \frac{1}{p_1^0}\bigg \lbrace \frac{1}{p_2^0} \; \frac{1}{p_1^0+\omega}
- \frac{1}{p_2^0} \; \frac{1}{p_2^0-\omega+i\epsilon} \bigg \rbrace \,\,\, ,
\label{integ_e}
\end{equation}
with the same $A$ as in Eq. (\ref{integ_c}).
The first term is again small, coming from the negative energy components of the
pions, and has opposite sign to the first term from diagram (c). The second term
above is the same but with opposite sign to the second term of diagram (c) at
$\omega=p_2^0$, which is the singular point. Hence there are strong
cancellations in the principal part of the integral and the imaginary part from
this source vanishes.
{\bf Diagram (d)}. Repeating the same arguments as above we find now
\begin{equation}
-A \frac{1}{\omega}\bigg \lbrace \frac{1}{p_1^0+\omega} \; \frac{1}{P^0+\omega}
+ \frac{1}{P^0-\omega+i\epsilon} \; \frac{1}{p_2^0-\omega+i\epsilon} \bigg \rbrace \,\,\, .
\label{integ_d}
\end{equation}
{\bf Diagram (f)}. For this diagram we find
\begin{equation}
A \frac{1}{p_1^0}\bigg \lbrace \frac{1}{\omega} \; \frac{1}{p_2^0+\omega}
- \frac{1}{P^0-\omega+i\epsilon} \; \frac{1}{p_1^0-\omega+i\epsilon} \bigg \rbrace \,\,\, .
\label{integ_f}
\end{equation}
Once again the first two terms from (d), (f), coming from the negative energy
part of the pion propagators, give a small contribution and partly cancel, and
the second terms which provide an imaginary part and a larger real part from the
principal value, also show cancellations. Indeed for $p_1^0=p_2^0=\omega$ the
imaginary parts corresponding to the poles $p_1^0=p_2^0=\omega$ cancel and the
real parts from the principal value would also largely cancel. At the
$P^0=\omega$ pole the cancellation would only be partial.
We thus see that when considering all the time orderings for the coupling of the
two pions and the loop with the two pion propagators there are large
cancellations of terms. In addition we have the $(p_i / M)^2$ factor of the
$p-$wave couplings for the initial pions, which make this contribution small at
small momenta of the pions. We have looked at strong cancellations of terms in
the heavy baryon approximation, which holds for small values of momenta. At
large momenta
we must note that we have two extra nucleon propagators which bring two
extra powers of $q$ in the denominator, with respect to the
ordinary Lindhard function. This makes up for the two extra $p-$wave vertices, and
hence we have a similar behaviour altogether as the one of Fig. \ref{loop2}
which lead to small contributions when evaluated into the loop. All these
elements discussed above would render this piece far smaller than the ones
of Fig. \ref{oneloopBS}(d,e) and, given the smallness of the effects found there, this
can also be neglected.
|
{
"timestamp": "2005-03-04T17:55:35",
"yymm": "0503",
"arxiv_id": "nucl-th/0503014",
"language": "en",
"url": "https://arxiv.org/abs/nucl-th/0503014"
}
|
\section{Introduction}
\newtheorem{Definition}{Definition}
\newtheorem{Lemma}{Lemma}
\newtheorem{Theorem}{Theorem}
\newtheorem{Proposition}{Proposition}
\newtheorem{Corollary}{Corollary}
Let $\Omega$ be a bounded homogeneous domain in $\mbox{\Bbb
C}^{n}.$ The class of all holomorphic functions with domain
$\Omega$ will be denoted by $H(\Omega).$ Let $\phi $ be a
holomorphic self-map of $\Omega,$ the composition operator
$C_{\phi}$ induced by $\phi$ is defined by
$$(C_{\phi}f)(z)=f(\phi(z)),$$ for $z$ in $\Omega$ and $f\in
H(\Omega)$.
Let $K(z,z)$ be the Bergman kernel function of $\Omega$, the
Bergman metric $H_{z}(u,u)$ in $\Omega$ is defined by
$$H_{z}(u,u)=\displaystyle\frac{1}{2}
\sum\limits^{n}_{j,k=1}
\displaystyle\frac{\partial^{2}\log K(z,z)}{\partial z_{j}
\partial {\overline {z}}_{k}}u_{j}{\overline u}_{k},$$
where $z\in\Omega$
and $u=(u_{1},\ldots,u_{n})\in \mbox{\Bbb C}^{n}.$
Following Timoney [1], we say that $f\in H(\Omega)$ is in the
Bloch space ${\cal B}(\Omega),$ if
$$\|f\|_{{\cal B}(\Omega)}=\sup\limits_{z\in \Omega}Q_{f}(z)<\infty,$$
where
\begin{equation}Q_{f}(z)=\sup\left\{\displaystyle\frac
{|\bigtriangledown f(z)u|}{H^{\frac{1}{2}}_{z}(u,u)}: u\in
\mbox{\Bbb C}^{n}-\{0\}\right\},\label{1}\end{equation} and
$\bigtriangledown f(z) =\left(\frac{\partial f(z)}{\partial
z_{1}}, \ldots, \frac{\partial f(z)}{\partial z_{n}} \right),
\bigtriangledown f(z)u =\sum\limits^{n}_{l=1}\frac{\partial
f(z)}{\partial z_{l}}u_{l}.$
The little Bloch space ${\cal B}_0(\Omega)$ is the closure in the
Banach space ${\cal B}(\Omega)$ of the polynomial functions.
Let $\partial\Omega$ denote the boundary of $\Omega$. Following
Timoney [2], for $\Omega=B_n$ the unit ball of $\mbox{\Bbb C}^n$,
${\cal B}_0(B_n)=\left\{f\in{\cal B}(B_n): Q_f(z)\to 0,
\mbox{as}\hspace*{2mm}z\to\partial B_n \right\};$ for $\Omega=\cal
D$ the bounded symmetric domain other than the ball $B_n$,
$\left\{ f\in{\cal B}({\cal D}): Q_f(z)\to 0,
\mbox{as}\hspace*{2mm} z\to\partial{\cal D}\right\}$ is the set of
constant functions on $\cal D.$ So if $\cal D$ is a bounded
symmetric domain other than the ball, we denote the ${\cal
B}_{0*}({\cal D})= \left\{f\in{\cal B}({\cal D}): Q_f(z)\to 0,
\mbox{as}\hspace*{2mm} z\to\partial^*{\cal D}\right\}$ and call it
little star Bloch space, here $\partial^*{\cal D}$ means the
distinguished boundary of $\cal D$. The unit ball is the only
bounded symmetric domain $\cal D$ with the property that
$\partial^*{\cal D}=\partial{\cal D}.$
Let $U^n$ be the unit polydisc of $\mbox{\Bbb C}^n$. Timoney [1]
shows that $f\in{\cal B}(U^n)$ if and only if
$$\|f\|_1=|f(0)|+\sup\limits_{z\in U^n}\sum\limits^n_{k=1}
\left|\frac{\partial f} {\partial z_k}(z)\right|(1-
|z_k|^2)<+\infty,$$ where $f\in H(U^n).$
This definition was the starting point for introducing the
$p$-Bloch spaces.
Let $p>0,$ a function $f\in H(U^n)$ is said to belong to the
$p$-Bloch space ${\cal B}^p(U^n)$ if
$$\|f\|_p=|f(0)|+\sup\limits_{z\in U^n}\sum\limits^n_{k=1}
\left|\frac{\partial f} {\partial z_k}(z)\right|\left(1-
|z_k|^2\right)^p<+\infty.$$ It is easy to show that ${\cal
B}^p(U^n)$ is a Banach space with the norm $\|\cdot\|_p.$
Just like Timoney [2], if
$$\lim_{z\to\partial U^n}\sum\limits^n_{k=1}\left|\frac{\partial f}{\partial
z_k}(z)\right|(1- |z_k|^2)^p=0,$$ it is easy to show that $f$ must
be a constant. Indeed, for fixed $z_1\in U,$
$\displaystyle\frac{\partial f}{\partial z_1}(z)(1-|z_1|^2)^p$ is
a holomorphic function in $z'=(z_2,\cdots,z_n)\in U^{n-1}$. If
$z\to\partial U^n$, then $z'\to\partial U^{n-1},$ which implies
that
$$\lim\limits_{z'\to\partial U^{n-1}}\left|\frac{\partial f} {\partial
z_1}(z)\right|\left(1- |z_1|^2\right)^p=0.$$ Hence,
$\frac{\partial f} {\partial z_1}(z)\left(1-
|z_1|^2\right)^p\equiv 0$ for every $z'\in \partial U^{n-1},$ and
for each $z_1\in U,$ and consequently $\frac{\partial f} {\partial
z_1}(z)=0$ for every $z\in U^n.$ Similarly, we can obtain that
$\frac{\partial f} {\partial z_j}(z)=0$ for every $z_j\in U^n$ and
each $j\in\{2,\cdots,n\},$ therefore $f\equiv const .$
So, there is no sense to introduce the corresponding little
$p$-Bolch space in this way. We will say that the little $p$-Bolch
space ${\cal B}_0^p(U^n)$ is the closure of the polynomials in the
$p$-Bolch space. If $f\in H(U^n)$ and
$$\sup\limits_{z\in \partial^*U^n}\sum\limits^n_{k=1}
\left|\displaystyle\frac{\partial f} {\partial
z_k}(z)\right|\left(1- |z_k|^2\right)^p=0,$$ we say $f$ belongs to
little star $p$-Bolch space ${\cal B}_{0*}^p(U^n).$ Using the same
methods as that of Theorem 4.14 in reference [2], we can show
that ${\cal B}^p_{0}(U^n)$ is a proper subspace of ${\cal
B}^p_{0*}(U^n)$ and ${\cal B}^p_{0*}(U^n)$ is a non-separable
closed subspace of ${\cal B}^p(U^n).$
Let $\phi $ be a holomorphic self-map of $U^n,$ the composition
operator $C_{\phi}$ induced by $\phi$ is defined by
$(C_{\phi}f)(z)=f(\phi(z))$ for $z$ in $U^n$ and $f\in H(U^n)$.
For the unit disc $U\subset\mbox{\Bbb C},$ Madigan and Matheson
[3] proved that $C_{\phi}$ is always bounded on ${\cal B}(U)$ and
bounded on ${\cal B}_0(U)$ if and only if $\phi\in{\cal B}_0(U).$
They also gave the sufficient and necessary conditions that
$C_{\phi}$ is compact on ${\cal B}(U)$ or ${\cal B}_0(U).$ More
recently, [4,5,7] gave some sufficient and necessary conditions
for $C_{\phi}$ to be compact on the Bloch spaces in polydisc.
We recall that the essential norm of a continuous linear operator $T$ is
the distance from $T$ to the compact operators, that is,
\begin{equation}\|T\|_e=\inf\{\|T-K\|: K \mbox{ is compact}\}.\label{2}\end{equation}
Notice that $\|T\|_e=0$ if and only if $T$ is compact, so that estimates
on $\|T\|_e$ lead to conditions for $T$ to be compact.
In this paper, we give some estimates of the essential norms of
bounded composition operators $C_{\phi}$ between ${\cal B}^p(U^n)$
(${\cal B}^p_{0}(U^n)$ or ${\cal B}^p_{0*}(U^n)$ ) and ${\cal
B}^q(U^n)$ (${\cal B}^q_{0}(U^n)$ or ${\cal B}^q_{0*}(U^n)$). As
their consequences, some necessary and sufficient conditions for
the bounded composition operators $C_{\phi}$ to be compact from
${\cal B}^p(U^n)$ (${\cal B}^p_{0}(U^n)$ or ${\cal B}^p_{0*}(U^n)$
) into ${\cal B}^q(U^n)$ (${\cal B}^q_{0}(U^n)$ or ${\cal
B}^q_{0*}(U^n)$) are obtained.
The fundamental ideals of the proof are those used by J. H.
Shpairo [8] to obtain the essential norm of a composition operator
on Hilbert spaces of analytic functions (Hardy and weighted
Bergman spaces) in terms of natural counting functions associated
with $\phi$. This paper generalizes the result on the Bloch space
in [10] to the Bloch-type space in polydisk.
Throughout the remainder of this paper $C$ will denote a positive
constant, the exact value of which will vary from one appearance
to the next.
Our main results are the following:
\begin{Theorem} Let $\phi=(\phi_1, \phi_2, \cdots, \phi_n)$ be a holomorphic
self-map of $U^n$ and $\|C_{\phi}\|_e$ the essential norm of a
bounded composition operator $C_{\phi}:$ ${\cal B}^p(U^n)$ (
${\cal B}^p_{0}(U^n)$ or ${\cal B}^p_{0*}(U^n)$)$\rightarrow$ $
{\cal B}^q(U^n)$ ( ${\cal B}^q_{0}(U^n)$ or ${\cal
B}^q_{0*}(U^n)$) , then
\begin{eqnarray}&&\displaystyle\frac{1}{n}\lim\limits_{\delta\to 0}
\sup\limits_{dist(\phi(z),\partial U^n)<\delta}\sum\limits^n_{k,l=1}
\left|\displaystyle\frac{\partial \phi_{l}}{\partial z_k}(z)\right|
\displaystyle\frac{(1-|z_k|^2)^q}{(1-|\phi_l(z)|^2)^p}\nonumber\\
&&\leq\|C_{\phi}\|_e \leq 2\lim\limits_{\delta\to 0}
\sup\limits_{dist(\phi(z),\partial
U^n)<\delta}\sum\limits^n_{k,l=1}
\left|\displaystyle\frac{\partial \phi_{l}}{\partial
z_k}(z)\right|
\displaystyle\frac{(1-|z_k|^2)^q}{(1-|\phi_l(z)|^2)^p}.\label{3}\end{eqnarray}
\end{Theorem}
By Theorem 1 and the fact that $C_{\phi}:$ ${\cal B}^p(U^n)$ (or
${\cal B}^p_{0}(U^n)$ or ${\cal B}^p_{0*}(U^n)$)$\rightarrow$ $
{\cal B}^q(U^n)$ (or ${\cal B}^q_{0}(U^n)$ or ${\cal
B}^q_{0*}(U^n)$) is compact if and only if $\|C_{\phi}\|_e=0$, we
obtain Theorem 2 at once.
\begin{Theorem}\hspace{2mm}Let $\phi=(\phi_1, \ldots, \phi_n)$ be a
holomorphic self-map of $U^{n}.$ Then the bounded composition
operator $C_{\phi}:$ ${\cal B}^p(U^n)$ (${\cal B}^p_{0}(U^n)$ or
${\cal B}^p_{0*}(U^n)$)$\rightarrow$ $ {\cal B}^q(U^n)$ (${\cal
B}^q_{0}(U^n)$ or ${\cal B}^q_{0*}(U^n)$) is compact if and only
if for any $\varepsilon>0,$ there exists a $\delta$ with
$0<\delta<1,$ such that
\begin{equation}\sup\limits_{dist(\phi(z),\partial U^n)<\delta}
\sum\limits^n_{k,l=1} \left|\displaystyle\frac{\partial
\phi_{l}}{\partial z_k}(z)\right|
\displaystyle\frac{(1-|z_k|^2)^q}{(1-|\phi_l(z)|^2)^p}<\varepsilon.\label{4}\end{equation}
\end{Theorem}
When $n=1,$ on ${\cal B}(U)$ we obtain Theorem 2 in [3]. Since
$\partial U=\partial^* U,$ ${\cal B}_0(U)={\cal B}_{0*}(U),$ we
can also obtain Theorem 1 in [3].
By Theorem 2 and Lemmas 3, 4 and 5 in next part, we can get the
following three Corollaries.
\begin{Corollary} Let $\phi=(\phi_1, \ldots, \phi_n)$ be a
holomorphic self-map of $U^{n}.$ Then\\ $C_{\phi}:{\cal
B}^p(U^{n})$(${\cal B}^p_{0}(U^n)$ or ${\cal
B}^p_{0*}(U^n)$)$\rightarrow{\cal B}^q(U^{n})$ is compact if and
only if
$$\sum\limits^n_{k,l=1} \left|\displaystyle\frac{\partial
\phi_{l}}{\partial z_k}(z)\right|
\displaystyle\frac{(1-|z_k|^2)^q}{(1-|\phi_l(z)|^2)^p}\leq C$$ for
all $z\in U^n$ and (\ref{4}) holds.\end{Corollary}
{\bf Proof}\hspace*{4mm} By Lemma 3 in next part, we know
$C_{\phi}:{\cal B}^p(U^{n})$(${\cal B}^p_{0}(U^n)$ or ${\cal
B}^p_{0*}(U^n)$)$\rightarrow{\cal B}^q(U^{n})$ is bounded. It
follows from Theorem 2 that $C_{\phi}:{\cal B}^p(U^{n})$(${\cal
B}^p_{0}(U^n)$ or ${\cal B}^p_{0*}(U^n)$)$\rightarrow{\cal
B}^q(U^{n})$ is compact.
Conversely, if $C_{\phi}:{\cal B}^p(U^{n})$(${\cal B}^p_{0}(U^n)$
or ${\cal B}^p_{0*}(U^n)$)$\rightarrow{\cal B}^q(U^{n})$ is
compact, it is clear that $C_{\phi}:{\cal B}^p(U^{n})$(${\cal
B}^p_{0}(U^n)$ or ${\cal B}^p_{0*}(U^n)$)$\rightarrow{\cal
B}^q(U^{n})$ is bounded, by Theorem 2, (\ref{4}) holds.
\begin{Corollary} Let $\phi=(\phi_1, \ldots, \phi_n)$ be a
holomorphic self-map of $U^{n}.$ Then \\
$C_{\phi}:$${\cal B}^p_{0*}(U^{n})$(${\cal
B}^p_{0}(U^n)$)$\rightarrow{\cal B}^q_{0*}(U^{n})$ is compact if
and only if $\phi_l\in {\cal B}^q_{0*}(U^n)$ for every
$l=1,2,\cdots, n$ and (\ref{4}) holds.\end{Corollary}
{\bf Proof}\hspace*{4mm} Note that Lemma 4 in next part, similar
to the proof of Corollary 1, the Corollary follows.
\begin{Corollary} Let $\phi=(\phi_1, \ldots, \phi_n)$ be a
holomorphic self-map of $U^{n}.$ Then \\
$C_{\phi}:$${\cal B}^p_{0}(U^{n})\rightarrow{\cal B}^q_{0}(U^{n})$
is compact if and only if $\phi_l\in {\cal B}^q_{0}(U^n)$ for
every $l=1,2,\cdots, n$ and (\ref{4}) holds.\end{Corollary}
{\bf Proof}\hspace*{4mm} Note that Lemma 5 in next part, similar
to the proof of Corollary 1, the Corollary follows.
\section{Some Lemmas}
In order to prove Theorem 1, we need some Lemmas.
\begin{Lemma} Let $f\in{\cal B}^p(U^n),$ then
(1) If $0\leq p<1,$ then $\|f(z)|\leq
|f(0)|+\displaystyle\frac{n}{1-p}\|f\|_p;$
(2) If $p=1,$ then $|f(z)|\leq \left(1+\displaystyle\frac{1}{n\ln
2}\right)\left(\sum\limits^n_{k=1}\ln
\displaystyle\frac{2}{1-|z_k|^2}\right)\|f\|_p.$
(3) If $p>1,$ then $|f(z)|\leq
\left(\displaystyle\frac{1}{n}+\displaystyle\frac{2^{p-1}}{p-1}\right)
\sum\limits^n_{k=1}\displaystyle\frac{1}{(1-|z_k|^2)^{p-1}}\|f\|_p.$
\end{Lemma}
{\bf Proof}\hspace{2mm} This Lemma can be proved by some integral
estimates (if necessary, the proof can be omitted).
By the definition of $\|.\|_{p}$,
$$|f(0)|\leq \|f\|_{p},\hspace*{4mm}\left|\displaystyle\frac{\partial f(z)
}{\partial z_l}\right|\leq
\displaystyle\frac{\|f\|_{p}}{(1-|z_l|^2)^p}
\hspace*{4mm}(l\in\{1,2,\cdots,n\})$$ and
\begin{eqnarray*}&&f(z)-f(0)=
\int^1_0 \displaystyle\frac{d f(tz)}{d
t}dt=\sum\limits^n_{l=1}\int^1_0 z_l\displaystyle\frac{\partial
f}{\partial\zeta_l}(tz)dt,\end{eqnarray*}
So\begin{eqnarray}&&|f(z)|\leq
|f(0)|+\sum\limits^n_{l=1}|z_l|\int^1_0\displaystyle\frac{\|f\|_p}{\left(1-t^2|z_l|^2\right)^p}dt
\nonumber\\
&&\leq
\|f\|_p+\|f\|_p\sum\limits^n_{l=1}\int^{|z_l|}_0\displaystyle\frac{1}{\left(1-t^2\right)^p}dt.\label{5}
\end{eqnarray}
If $p=1,$
\begin{equation}\int^{|z_l|}_0\displaystyle\frac{1}{\left(1-t^2\right)^p}dt=\displaystyle\frac{1}{2}
\ln\displaystyle\frac{1+|z_l|}{1-|z_l|}\leq
\displaystyle\frac{1}{2}
\ln\displaystyle\frac{4}{1-|z_l|^2}.\label{6}\end{equation} It is
clear that $\ln\displaystyle\frac{4}{1-|z_l|^2}>\ln 4=2\ln 2,$
so\begin{equation}1\leq \displaystyle\frac{1}{2\ln 2}
\ln\displaystyle\frac{4}{1-|z_l|^2}\leq
\displaystyle\frac{1}{2n\ln 2}
\sum\limits^n_{l=1}\ln\displaystyle\frac{4}{1-|z_l|^2}.\label{7}\end{equation}
Combining (\ref{5}),(\ref{6}) and (\ref{7}), we get
$$|f(z)|\leq \left(\displaystyle\frac{1}{2}+\displaystyle\frac{1}{2n\ln
2}\right)\left(\sum\limits^n_{l=1}\ln\displaystyle\frac{4}{1-|z_l|^2}\right)\|f\|_p.$$
If $p\neq 1,$
\begin{eqnarray}&&\int^{|z_l|}_0\displaystyle\frac{1}{\left(1-t^2\right)^p}dt
=\int^{|z_l|}_0\displaystyle\frac{1}{(1-t)^p}\cdot
\displaystyle\frac{1}{(1+t)^p}dt\nonumber\\
&&\leq
\int^{|z_l|}_0\displaystyle\frac{1}{(1-t)^p}dt=\displaystyle\frac{1-(1-|z_l|)^{-p+1}}{1-p}.\label{8}
\end{eqnarray}
If $0<p<1,$ (\ref{8}) gives that
$\int^{|z_l|}_0\displaystyle\frac{1}{\left(1-t^2\right)^p}dt\leq\displaystyle\frac{1}{1-p},$
it follows from (\ref{5}) that
$|f(z)|\leq\left(1+\displaystyle\frac{n}{1-p}\right)\|f\|_p.$
If $p>1,$ (\ref{8}) gives that
$$\int^{|z_l|}_0\displaystyle\frac{1}{\left(1-t^2\right)^p}dt\leq \displaystyle\frac{1-(1-|z_l|^{p-1})}{(p-1)(1-|z_l|)^{p-1}}
\leq\displaystyle\frac{2^{p-1}}{(p-1)(1-|z_l^2|)^{p-1}},$$ it
follows from (\ref{5}) that
\begin{eqnarray*}|f(z)|&\leq&\|f\|_p+\displaystyle\frac{2^{p-1}}{p-1}\left(\sum\limits^n_{l=1}
\displaystyle\frac{1}{(1-|z_l|^2)^{p-1}}\right)\|f\|_p\\
&\leq&
\left(\displaystyle\frac{1}{n}+\displaystyle\frac{2^{p-1}}{p-1}\right)\left(\sum\limits^n_{l=1}
\displaystyle\frac{1}{(1-|z_l|^2)^{p-1}}\right)
\|f\|_{p}.\end{eqnarray*}Now the Lemma is proved.
\begin{Lemma} Set $$f_w(z)=\int_0^{z_l}\frac {dt}{(1-\bar wt)^p},$$ where $w\in
U.$ Then $f\in {\cal B}^p_0(U^n)\subset{\cal
B}^p_{0*}(U^n)\subset {\cal B}^p(U^n).$ \end{Lemma}
{\bf Proof}\hspace{2mm}Since
$$\displaystyle\frac{\partial
f_w}{\partial z_l} =\left(1-\overline wt\right)^{-p},
\hspace*{4mm} \displaystyle\frac{\partial f_w}{\partial z_i}=0,
\hspace*{4mm}(i\neq l),$$ it follows that
$$|f(0)|+\sum\limits^n_{k=1}\left|\displaystyle\frac{\partial f_w}{\partial z_k}(z)\right|(1-|z_k|^2)^p
=\displaystyle\frac{(1-|z_l|^2)^p}{|1-\overline
wt|^p}\leq(1+|z_l|^2)^p\leq 2^p.$$ Hence $f_w\in {\cal B}^p(U^n).$
Now we prove that $f_w\in{\cal B}^p_0(U^n).$ Using the asymptotic
formula
$$(1-\bar w t)^{-p}=\sum\limits^{+\infty}_{k=0}\frac{p(p+1)\cdots
(p+k-1)}{k!}(\bar w)^kt^k,$$ we obtain
$$f_w(z)=\sum\limits^{+\infty}_{k=0}\frac{p(p+1)\cdots
(p+k-1)}{k!}(\bar w)^k\int^{z_l}_0 t^kdt.$$ Denote
$P_n(z)=\sum\limits^{n}_{k=0}\frac{p(p+1)\cdots (p+k-1)}{k!}(\bar
w)^k\int^{z_l}_0t^kdt,$ it is easy to see that
$$f_w(z)-P_n(z)=\sum\limits^{+\infty}_{k=n+1}\frac{p(p+1)\cdots (p+k-1)}{k!}(\bar w)^k\int^{z_l}_0t^k dt,$$
$$\left|\frac{\partial (f_w-P_n)}{\partial z_l}\right|\leq
\sum\limits^{+\infty}_{k=n+1}\frac{p(p+1)\cdots (p+k-1)}{k!}|w|^k
\to 0, \mbox{as}\ \ n\to\infty,$$
\begin{eqnarray*}\|f_w-P_n\|_p&=&|f_w(0)-P_n(0)|+\sup\limits_{z\in
U^n}\left|\frac{\partial (f_w-P_n)}{\partial
z_l}\right|(1-|z_l|^2)^p \\
&\leq& \sup\limits_{z\in U^n}\left|\frac{\partial
(f_w-P_n)}{\partial z_l}\right|\to 0, \end{eqnarray*} it shows
that $f_w\in {\cal B}^p_0(U^n).$ So $f\in {\cal
B}^p_0(U^n)\subset{\cal B}^p_{0*}(U^n)\subset {\cal B}^p(U^n).$
\begin{Lemma} Let $\phi=(\phi_1, \ldots, \phi_n)$ be a holomorphic self-map
of $U^n$, $p,q>0.$ Then $C_{\phi}: {\cal B}^p(U^n) ({\cal
B}^p_0(U^n)$ or ${\cal B}^p_{0*}(U^n))\longrightarrow {\cal
B}^q(U^n) $ is bounded if and only if there exists a constant $C$
such that
\begin{equation}\sum\limits^n_{k,l=1}\left|\displaystyle\frac{\partial \phi_{l}}
{\partial
z_k}(z)\right|\displaystyle\frac{\left(1-|z_k|^2\right)^q}
{\left(1-|\phi_l(z)|^2\right)^p}\leq C ,\label{9}\end{equation}for
all $z\in U^n.$\end{Lemma}
{\bf Proof}\hspace*{4mm}First assume that condition (\ref{9})
holds. Let $f\in{\cal B}^p(U^n)({\cal B}^p_0(U^n)$ or ${\cal
B}^p_{0*}(U^n)),$ by Lemma 1, we know the evaluation at $\phi(0)$
is a bounded linear functional on ${\cal B}^p(U^n),$ so
$|f(\phi(0))|\leq C\|f\|_p.$
On the other hand we have
\begin{eqnarray}&&\sum\limits^n_{k=1}\left|\frac{\partial
\left(C_{\phi}f(z)\right)} {\partial z_k}\right| (1-|z_k|^2)^q
=\sum\limits^n_{k=1}\left|\sum\limits^n_{l=1}\frac{\partial
f}{\partial \phi_l}(\phi(z))
\frac{\partial \phi_l}{\partial z_k}(z)\right|(1-|z_k|^2)^q\nonumber\\
&&\leq \sum\limits^n_{k, l=1}\left|\frac{\partial f}{\partial
\phi_l}(\phi(z))\frac{\partial
\phi_l}{\partial z_k}(z)\right|(1-|z_k|^2)^q\nonumber\\
&&\leq \sum\limits^n_{l=1}\left|\frac{\partial f}{\partial
\phi_l}(\phi(z))\right|\left(1-|\phi_l(z)|^2\right)^p
\sum\limits^n_{k,l=1}\left|\frac{\partial \phi_{l}} {\partial
z_k}(z)\right|\frac{\left(1-|z_k|^2\right)^q}
{\left(1-|\phi_l(z)|^2\right)^p}\label{10}\\
&&\leq\|f\|_p\sum\limits^n_{k,l=1}\left|\frac{\partial \phi_l}
{\partial z_k}(z)\right|
\frac{\left(1-|z_k|^2\right)^q}{\left(1-|\phi_l(z)|^2\right)^p}.\label{11}
\end{eqnarray}
From (\ref{11}) it follows that
$$\sum\limits^n_{k=1}\left|\frac{\partial
\left(C_{\phi}f(z)\right)} {\partial z_k}\right| (1-|z_k|^2)^q\leq
C\|f\|_p.$$ So $C_{\phi}: {\cal B}^p(U^n)\to {\cal B}^q(U^n) $ is
bounded.
For the converse, assume that $C_{\phi}: {\cal B}^p(U^n) ({\cal
B}^p_0(U^n)$ or ${\cal B}^p_{0*}(U^n))\longrightarrow {\cal
B}^q(U^n) $ is bounded, with
\begin{equation}\|C_{\phi}f\|_q\leq C\|f\|_p\label{12}\end{equation}
for all $f\in{\cal B}^p(U^n)({\cal B}^p_0(U^n)$ or ${\cal
B}^p_{0*}(U^n)).$
For fixed $l (1\leq l\leq n),$ we will make use of a family of
test functions $\{f_{w}: w\in\mbox{\Bbb C}, |w|<1\}$ in ${\cal
B}(U^n)$ defined as follows: If $p>0$, let
$$f_w(z)=\int^{z_l}_0\left(1-\overline wz_l\right)^{-p}dt.$$
It follows from Lemma 2 that $$f_w\in {\cal
B}^p_0(U^n)\subset{\cal B}^p_{0*}(U^n)\subset {\cal B}^p(U^n).$$
For $z\in U^n,$ it follows from (\ref{12}) that
\begin{equation}\sum\limits^n_{k=1}\left|\sum\limits^n_{l=1}\displaystyle\frac{\partial
f_w(\phi(z))}{\partial \phi_l} \displaystyle\frac{\partial
\phi_l}{\partial z_k}(z)\right|(1-|z_k|^2)^q\leq
C.\label{13}\end{equation}
Let $w=\phi_l(z),$ then
$$\sum\limits^n_{k=1}\left|\displaystyle\frac{\partial
\phi_{l}} {\partial z_k}(z)\right|
\displaystyle\frac{\left(1-|z_k|^2\right)^q}{\left(1-|\phi_l(z)|^2\right)^p}
\leq C.$$ Now the proof of Lemma 3 is completed.
\begin{Lemma} Let $\phi=(\phi_1, \phi_2, \cdots, \phi_n)$ be a holomorphic
self-map of $U^n.$ Then \\
$C_{\phi}:{\cal B}^p_{0*}(U^{n}) ({\cal B}^p_0(U^n))
\rightarrow{\cal B}^q_{0*}(U^{n})$ is bounded if and only if
$\phi_l\in{\cal B}^q_{0*}(U^n)$ for every $l=1,2,\cdots, n$ and
(\ref{9}) holds.\end{Lemma}
{\bf Proof}\hspace*{4mm}If $C_{\phi}:{\cal B}^p_{0*}(U^{n}) ({\cal
B}^p_0(U^n)) \rightarrow {\cal B}^q_{0*}(U^{n})$ is bounded , it
is clear that, for every $l=1,2,\cdots, n$, $f_l(z)=z_l\in{\cal
B}^p_0(U^n)\subset{\cal B}^q_{0*}(U^n),$ so
$C_{\phi}f_l=\phi_l\in{\cal B}^q_{0*}(U^n).$ In the proof of Lemma
3, note that the test functions $f_w\in {\cal
B}^p_0(U^n)\subset{\cal B}^p_{0*}(U^n),$ we know $(\ref{9})$
holds.
In order to prove the Converse, we first prove that if
$\phi_l\in{\cal B}^q_{0*}(U^n)$ for every $l=1,2,\cdots,n.,$ then
$f\circ\phi\in{\cal B}^q_{0*}(U^n)$ for any $f\in{\cal
B}^p_{0*}(U^n).$
Without loss of generality, we prove this result when $n=2.$
For any sequence $\{z^j=(z^j_1, z^j_2)\}\subset U^n$ with
$z^j\to\partial^* U^n$ as $j\to\infty,$ then $$|z^j_1|\to 1,
|z^j_2|\to 1.$$ Since $|\phi_1(z^j)|<1$ and $|\phi_2(z^j)|<1,$
there exists a subsequence $\{z^{j_s}\}$ in $\{z^j\}$ such that
$$|\phi_1(z^{j_s})|\to \rho_1, |\phi_2(z^{j_s})|\to\rho_2,$$ as
$s\to\infty .$
It is clear that $0\leq\rho_1, \rho_2\leq 1.$
\begin{eqnarray}&&\left|\displaystyle\frac{\partial(f\circ\phi)}
{\partial z_k}(z^{j_s})\right|(1-|z^{j_s}_k|^2)^q\nonumber\\
&&\leq\left| \displaystyle\frac{\partial f}{\partial
w_1}(\phi(z^{j_s}))\right| \left|\displaystyle\frac
{\partial\phi_1}{\partial
z_k}(z^{j_s})\right|(1-|z^{j_s}_k|^2)^q+\left|
\displaystyle\frac{\partial f}{\partial w_2}(\phi(z^{j_s}))\right|
\left|\displaystyle\frac
{\partial\phi_2}{\partial z_k}(z^{j_s})\right|(1-|z^{j_s}_k|^2)^q\nonumber\\
&&=\left| \displaystyle\frac{\partial f}{\partial
w_1}(\phi(z^{j_s})) \right|(1-|\phi_1(z^{j_s})|^2)^p
\left|\displaystyle\frac {\partial\phi_1}{\partial
z_k}(z^{j_s})\right|\displaystyle\frac{(1-|z^{j_s}_k|^2)^q}
{(1-|\phi_1(z^{j_s})|^2)^p}\nonumber\\
&&+\left| \displaystyle\frac{\partial f}{\partial
w_2}(\phi(z^{j_s}))\right| (1-|\phi_2(z^{j_s})|^2)^p
\left|\displaystyle\frac {\partial\phi_2}{\partial
z_k}(z^{j_s})\right|\displaystyle\frac{(1-|z^{j_s}_k|^2)^q}
{(1-|\phi_2(z^{j_s})|^2)^p},\label{14}\end{eqnarray} $k=1,2.$
Now we prove the left of $(\ref{14})\to 0$ as $s\to\infty$
according to four cases.
Case 1. If $\rho_1<1$ and $\rho_2<1.$ It is clear that there exist
$r_1$ and $r_2$ such that $\rho_1<r_1<1$ and $\rho_2<r_2<1,$ so as
$j$ is large enough, $|\phi_1(z^{j_s})|\leq r_1$ and
$|\phi_2(z^{j_s})|\leq r_2.$
By $\phi_1, \phi_2\in{\cal B}^q_{0*}(U^n)$ and (\ref{14}), we get
\begin{eqnarray*}\left|\displaystyle\frac{\partial(f\circ\phi)}
{\partial z_k}(z^{j_s})\right|(1-|z^{j_s}_k|^2)^q &\leq& \|f\|_p
\displaystyle\frac{1}{(1-r_1^2)^p}\left|\displaystyle\frac
{\partial\phi_1}{\partial z_k}(z^{j_s})\right|(1-|z^{j_s}_k|^2)^q\\
&&+\|f\|_p\displaystyle\frac{1}{(1-r_2^2)^p}\left|\displaystyle\frac
{\partial\phi_2}{\partial z_k}(z^{j_s})\right|(1-|z^{j_s}_k|^2)^q\\
& &\to 0\end{eqnarray*} as $s\to\infty.$
Case 2. If $\rho_1=1$ and $\rho_2=1.$ Then
$\phi(z^{j_s})\to\partial^*U^n,$ by (\ref{9}) and $f\in{\cal
B}^p_{0*}(U^n)$, (\ref{14}) gives that
\begin{eqnarray*}&&C\left|\displaystyle\frac{\partial(f\circ\phi)} {\partial
z_k}(z^{j_s})\right|(1-|z^{j_s}_k|^2)^q\\
&&C\leq \left| \displaystyle\frac{\partial f}{\partial
w_1}(\phi(z^{j_s})) \right|(1-|\phi_1(z^{j_s})|^2)^p +\left|
\displaystyle\frac{\partial f}{\partial w_2}(\phi(z^{j_s}))\right|
(1-|\phi_2(z^{j_s})|^2)^p\to 0\end{eqnarray*} as $s\to\infty.$
Case 3. If $\rho_1<1$ and $\rho_2=1.$ Similar to Case 1, we can
prove that
\begin{eqnarray}&&\left|
\displaystyle\frac{\partial f}{\partial w_1}(\phi(z^{j_s}))
\right|(1-|\phi_1(z^{j_s})|^2)^p\left|\displaystyle\frac
{\partial\phi_1}{\partial
z_k}(z^{j_s})\right|\displaystyle\frac{(1-|z^{j_s}_k|^2)^q}
{(1-|\phi_1(z^{j_s})|^2)^p}\nonumber\\
&&\leq\|f\|_p\displaystyle\frac{1}{(1-r_1^2)^p}
\left|\displaystyle\frac {\partial\phi_1}{\partial
z_k}(z^{j_s})\right|\displaystyle\frac{(1-|z^{j_s}_k|^2)^q}
{(1-|\phi_1(z^{j_s})|^2)^p} \to 0 \label{15}\end{eqnarray} as
$s\to\infty.$
On the other hand, for fixed $s,$ let $w^{j_s}_2=\phi_2(z^{j_s}),$
then $|w^{j_s}_2|<1.$ Denote
$$F(w_1)=\displaystyle\frac{\partial f}{\partial w_2}(w_1, w^{j_s}_2).$$
It is clear that $F(w_1)$ is holomorphic on $|w_1|<1,$ choose
$R_{j_s}\to 1$ with $r_1\leq R_{j_s}<1.$ $|\phi_1(z^{j_s})|\leq
r_1,$ so
$$|F(\phi_1(z^{j_s}))|\leq\max\limits_{|w_1|\leq r_1}|F(w_1)|
\leq\max\limits_{|w_1|\leq R_{j_s}}|F(w_1)|=
\max\limits_{|w_1|=R_{j_s}}|F(w_1)|=|F(w^{j_s}_1)|,$$ where
$|w^{j_s}_1|=R_{j_s}\to 1.$ This means that
$\left|\displaystyle\frac{\partial f}{\partial
w_2}(\phi_1(z^{j_s}), \phi_2(z^{j_s})) \right|\leq
\left|\displaystyle\frac{\partial f}{\partial w_2}(w^{j_s}_1,
w^{j_s}_2) \right|. $ Since $|w^{j_s}_1|\to 1,
|w^{j_s}_2|\to\rho_2=1$ and $f\in{\cal B}^p_{0*}(U^n),$
$$\left|\displaystyle\frac{\partial f}{\partial w_2}(w^{j_s}_1, w^{j_s}_2)\right|
(1-|w^{j_s}_2|^2)^p\to 0$$ as $s\to\infty,$ so by (\ref{9}),
\begin{eqnarray}&&\left|
\displaystyle\frac{\partial f}{\partial w_2}(\phi(z^{j_s}))\right|
(1-|\phi_2(z^{j_s})|^2) ^p\left|\displaystyle\frac
{\partial\phi_2}{\partial
z_k}(z^{j_s})\right|\displaystyle\frac{(1-|z^{j_s}_k|^2)^q}
{(1-|\phi_2(z^{j_s})|^2)^p}\nonumber\\
&&\leq C\left| \displaystyle\frac{\partial f}{\partial
w_2}(w^{j_s}_1, w^{j_s}_2)\right| (1-|w^{j_s}_2|^2)^p\to
0\label{16}\end{eqnarray} as $s\to\infty.$
By (\ref{15}) and (\ref{16}), (\ref{14}) gives
$$\left|\displaystyle\frac{\partial(f\circ\phi)}
{\partial z_k}(z^{j_s})\right|(1-|z^{j_s}_k|^2)^q\to 0,$$ as
$s\to\infty.$
Case 4. If $\rho_1=1$ and $\rho_2<1.$ Similar to Case 3, we can
prove $$\left|\displaystyle\frac{\partial(f\circ\phi)} {\partial
z_k}(z^{j_s})\right|(1-|z^{j_s}_k|^2)^q\to 0,$$ as $s\to\infty.$
Combining Case 1, Case 2, Case 3 and Case 4, we know there exists
a subsequence $\{z^{j_s}\}$ in $\{z^j\}$ such that
$$\left|\displaystyle\frac{\partial(f\circ\phi)} {\partial
z_k}(z^{j_s})\right|(1-|z^{j_s}_k|^2)^q\to 0,$$ as $s\to\infty$
for $k=1,2.$ We claim that
$$\left|\displaystyle\frac{\partial(f\circ\phi)} {\partial
z_k}(z^j)\right|(1-|z^j_k|^2)^q\to 0,$$ as $j\to\infty.$ In fact,
if it fails, then there exists a subsequence $\{z^{j_s}\}$ such
that
\begin{equation}\left|\displaystyle\frac{\partial(f\circ\phi)}
{\partial z_k}(z^{j_s})\right|(1-|z^{j_s}_k|^2)^q\to
\varepsilon>0\label{17}\end{equation} for $k=1$ or $2$. But from
the above discussion, we can find a subsequence in $\{z^{j_s}\}$
we still write $\{z^{j_s}\}$ with
$$\left|\displaystyle\frac{\partial(f\circ\phi)} {\partial
z_k}(z^{j_s})\right|(1-|z^{j_s}_k|^2)^q\to 0,$$ it contradicts
with (\ref{17}).
So for any sequence $\{z^j\}\subset U^n$ with $z^j\to\partial^*
U^n$ as $j\to\infty,$ we have
$$\left|\displaystyle\frac{\partial(f\circ\phi)} {\partial
z_k}(z^{j})\right|(1-|z^{j}_k|^2)^q\to 0$$ for $k=1,2.$ By
(\ref{9}) and Lemma 3, it is clear that $f\circ\phi\in{\cal
B}^q(U^n),$ so $f\circ\phi\in{\cal B}^q_{0*}(U^n).$
For any $f\in {\cal B}^p_0(U^n)).$ Since ${\cal
B}^p_0(U^n))\subset {\cal B}^p_{0*}(U^n)),$ then
$f\circ\phi\in{\cal B}^q_{0*}(U^n).$
By closed graph theorem we known that $$C_{\phi}:{\cal
B}^p_{0*}(U^{n}) ({\cal B}^p_0(U^n)) \rightarrow{\cal
B}^q_{0*}(U^{n})$$ is bounded. This ends the proof of Lemma 4.
{\bf Remark 1}\hspace*{4mm}For the case $C_{\phi}:{\cal
B}^p(U^n)\to{\cal B}^q_{0*}(U^n)$, the necessity is also true, but
we can't guaranty that the sufficiency is true because we can't
sure that $C_{\phi}f\in{\cal B}^q_{0*}(U^n)$ for all $f\in{\cal
B}^p(U^n$.
\begin{Lemma} Let $\phi=(\phi_1, \phi_2, \cdots, \phi_n)$ be a holomorphic
self-map of $U^n.$ Then $$C_{\phi}:{\cal B}^p_0(U^n)
\rightarrow{\cal B}^q_0(U^{n})$$ is bounded if and only if if and
only if $\phi^{\gamma}\in {\cal B}_0^q(U^n)$ for every multi-index
$\gamma$, and (\ref{9}) holds.\end{Lemma}
{\bf Proof} \hspace{2mm}Sufficiency. From (\ref{9}) and by Theorem
1 we know that $C_{\phi}:{\cal B}^p(U^n)\to{\cal B}^q(U^n)$ is
bounded, in particular
$$\|C_\phi f\|_q\leq \|C_\phi\|_{{\cal B}^p(U^n)\to
{\cal B}^q(U^n)}\|f\|_p,\quad \mbox{for all}\; f\in{\cal
B}_0^p(U^n).$$ The boundedness of $C_{\phi}: {\cal B}_0^p(U^n)\to
{\cal B}_0^q(U^n)$ directly follows, if we prove $C_\phi f\in{\cal
B}_0^q(U^n)$ whenever $f\in {\cal B}_0^p(U^n).$ So, let $f\in
{\cal B}_0^p(U^n).$ By the definition of ${\cal B}_0^p(U^n)$ it
follows that for every $\varepsilon>0$ there is a polynomial
$p_\varepsilon$ such that $\|f-p_\varepsilon\|_p<\varepsilon.$
Hence \begin{equation}\|C_\phi f-C_\phi p_\varepsilon\|_q\leq
\|C_\phi\|_{{\cal B}^p(U^n)\to {\cal
B}^q(U^n)}\|f-p_\varepsilon\|_p<\varepsilon \|C_\phi\|_{{\cal
B}^p(U^n)\to {\cal B}^q(U^n)}.\label{a}\end{equation} Since
$\phi^{\gamma}\in{\cal B}_0^q(U^n)$ for every multi-index
$\gamma,$ we obtain $C_\phi p_\varepsilon\in{\cal B}_0^q(U^n).$
From this and (\ref{a}) the result follows.
If $C_{\phi}:{\cal B}_0^p(U^n)\to{\cal B}_0^q(U^n)$ is bounded,
then (\ref{9}) can be proved as in Lemma 3, since the test
functions appearing there belong to ${\cal B}_0^p(U^n).$ Since the
polynomials $z^\gamma\in {\cal B}_0^p(U^n)$ for every multi-index
$\gamma,$ we get $C_\phi z^\gamma\in {\cal B}_0^q(U^n),$ as
desired.
{\bf Remark 2}\hspace*{4mm}For the case $C_{\phi}:{\cal
B}^p(U^n)\;\; ({\cal B}^p_{0*}(U^n))\to{\cal B}^q_{0}(U^n)$,
similar to Remark 1, the necessity is also true, but we can't
guaranty that the sufficiency is true.
\begin{Lemma} If $\{f_k\}$ is a bounded sequence in
${\cal B}^p(U^n)$, then there exists a subsequence $\{f_{k_l}\}$
of $\{f_k\}$ which converges uniformly on compact subsets of $U^n$
to a holomorphic function $f\in{\cal B}^p(U^n)$.
\end{Lemma}
{\bf Proof}\hspace{2mm} Let $\{f_k\}$ be a bounded sequence in
${\cal B}^p(U^n)$ with $\|f_k\|_p\leq C.$ By Lemma 1, $\{f_j\}$ is
uniformly bounded on compact subsets of $U^n$ and hence normal by
Montel's theorem. Hence we may extract subsequence $\{f_{j_k}\}$
which converges uniformly on compact subsects of $U^n$ to a
holomorphic function $f$. It follows that
$\displaystyle\frac{\partial f_{j_k}}{\partial
z_l}\to\displaystyle\frac{\partial f}{\partial z_l}$ for each
$l\in\{1,2,\cdots,n\}$, so
$$\sum\limits^n_{l=1}\left|\displaystyle\frac{\partial f}{\partial z_l}\right|(1-|z_l|^2)^p=
\lim\limits_{k\to\infty}\sum\limits^n_{l=1}\left|\displaystyle\frac{\partial
f_{j_k}}{\partial z_l}\right|(1-|z_l|^2)^p=\leq
\sup\limits_{k}\|f_{j_k}\|_p\leq C,$$ which implies $f\in{\cal
B}^p(U^n)$. The Lemma is proved.
\begin{Lemma} Let $\Omega$ be a domain in $\mbox{\Bbb C}^n,$
$f\in H(\Omega).$ If a compact set $K$ and its neighborhood $G$
satisfy $K\subset G\subset\subset \Omega$ and $\rho=dist(K,
\partial G)>0,$ then
$$\sup\limits_{z\in K}\left|\displaystyle\frac{\partial f}{\partial z_j}(z)
\right|\leq\displaystyle\frac{\sqrt{n}}{\rho}\sup\limits_{z\in G}|f(z)|.$$
\end{Lemma}
{\bf Proof}\hspace{2mm}Since $\rho=dist(K, \partial G)>0,$ for any $a\in K,$
the polydisc $$P_a=\left\{(z_1, \cdots, z_n)\in\mbox{\Bbb C}^n: |z_j-a_j|
<\displaystyle\frac{\rho}{\sqrt{n}}, j=1,\cdots,n\right\}$$
is contained in $G.$ By Cauchy's inequality,
$$\left|\displaystyle\frac{\partial f}{\partial z_j}(a)
\right|\leq\displaystyle\frac{\sqrt{n}}{\rho}
\sup\limits_{z\in\partial^* P_a}|f(z)|\leq
\displaystyle\frac{\sqrt{n}}{\rho}\sup\limits_{z\in G}|f(z)|.$$
Taking the supremum for $a$ over $K$ gives the desired inequality.
\section{The Proof of Theorem 1}
Now we turn to the proof of Theorem 1.
The lower estimate. It is clear that $\{m^{p-1}z^m_1\}\subset{\cal
B}^p_0(U^n) \subset{\cal B}_{0*}(U^n) \subset{\cal B}(U^n)$ for
$m=1,2,\cdots,$ and this sequence converges to zero uniformly on
compact subsets of the unit polydisc $U^n.$
\begin{equation}\|m^{p-1}z^m_1\|_p =\sup\limits_{z\in U^n}
(1-|z_1|^2)^p|m^pz^{m-1}_1|.\label{18}\end{equation} Let
$p(x)=m^p(1-x^2)^px^{m-1},$ then
$$p'(x)=-m^px^{m-2}(1-x^2)^{p-1}\left[(2p+m-1)x^2-(m-1)\right],$$ so
$p'(x)\leq 0$ for $x\in
\left[\sqrt{\displaystyle\frac{m-1}{2p+m-1}},1\right],$ and
$p'(x)\geq 0$ for $x\in
\left[0,\sqrt{\displaystyle\frac{m-1}{2p+m-1}}\right].$
That is, $p(x)$ is a decreasing function for $x\in
\left[\sqrt{\displaystyle\frac{m-1}{2p+m-1}},1\right]$ and $p(x)$
is a increasing function for $x\in
\left[0,\sqrt{\displaystyle\frac{m-1}{2p+m-1}}\right].$ Hence
$$\max\limits_{x\in
[0,1]}p(x)=p\left(\sqrt{\displaystyle\frac{m-1}{2p+m-1}}\right).$$
It follows from (\ref{18}) that
$$\|m^{p-1}z^m_1\|_p
=p\left(\sqrt{\displaystyle\frac{m-1}{2p+m-1}}\right)=\left(\displaystyle\frac{2p}{2p+m-1}\right)^pm^p
\left(\displaystyle\frac{m-1}{2p+m-1}\right)^{\frac{m-1}{2}}
\to\left(\displaystyle\frac{2p}{e}\right)^p,$$ as $m\to\infty.$
Therefore, the sequence $\{m^{p-1}z^m_1\}_{m\geq 2}$ is bounded
away from zero. Now we consider the normalized sequence
$\{f_m=\displaystyle\frac{m^{p-1}z^m_1}{\|m^{p-1}z^m_1\|_p}\}$
which also tends to zero uniformly on compact subsets of $U^n.$
For each $m\geq 2,$ we define $$A_m=\{z=(z_1, \ldots, z_n)\in U^n:
r_m\leq |z_1|\leq r_{m+1}\},$$ where
$r_m=\sqrt{\displaystyle\frac{m-1}{2p+m-1}}.$ So
\begin{eqnarray*}&&\min\limits_{A_m}
\sum\limits^n_{l=1} \left\{\left|\displaystyle\frac{\partial f_m}
{\partial z_l}(z)\right|(1-|z_l|^2)^p\right\}
=\min\limits_{A_m}\left|\displaystyle\frac{\partial f_m}
{\partial z_1}(1-|z_1|^2)^p\right|\\
&&=\displaystyle\frac{
(1-r^2_{m+1})^p|m^pr^{m-1}_{m+1}|}{\|m^{p-1}z^m_1\|_p}
=\left(\displaystyle\frac{2p+m-1}{2p+m}\right)
\left(\displaystyle\frac{m(2p+m-1)}{(m-1)(2p+m)}\right)^{\frac{m-1}{2}}=c_m.
\end{eqnarray*}
It is easy to show that $c_m$ tends to 1 as $m\to\infty$. For the
moment fix any compact operator $K:{\cal B}^p(U^n){\cal B}^p(U^n)
({\cal B}^p_0(U^n)$ or ${\cal B}^p_{0*}(U^n))\longrightarrow {\cal
B}^q(U^n)$ $({\cal B}^q_0(U^n)$ or ${\cal B}^q_{0*}(U^n)).$
The
uniform convergence on compact subsets of the sequence $\{f_m\}$
to zero and the compactness of $K$ imply that $\|Kf_m\|_q\to 0.$
It is easy to show that if a bounded sequence that is contained in
${\cal B}^p_{0*}(U^n)$ converges uniformly on compact subsets of
$U^n,$ then it also converges weakly to zero in ${\cal
B}^p_{0*}(U^n)$ as well as in ${\cal B}^p(U^n).$ Since
$\|f_m\|_p=1$, we have
\begin{eqnarray*}&&\|C_{\phi}-K\|\geq
\limsup\limits_{m}\|(C_{\phi}-K)f_m\|_q\nonumber\\
&&\geq\limsup\limits_{m}\left(\|C_{\phi}f_m\|_q -\|Kf_m\|_q\right)
=\limsup\limits_{m}\|C_{\phi}f_m\|_q\nonumber\\
&&\geq \limsup\limits_{m}\sup\limits_{z\in U^n}\sum\limits^n_{k=1}
\left\{\left|\displaystyle\frac{\partial(f_m\circ\phi)}{\partial
z_k}\right| (1-|z_k|^2)^q
\right\}\nonumber\\
&&=\limsup\limits_{m}\sup\limits_{z\in U^n}\sum\limits^n_{k=1}
\left|\displaystyle\frac{\partial f_m}{\partial
w_1}(\phi(z))\right| \left|\displaystyle\frac
{\partial\phi_1}{\partial z_k}(z)\right|(1-|z_k|^2)^q\nonumber\\
&&=\limsup\limits_{m}\sup\limits_{z\in U^n}\sum\limits^n_{k=1}
\left|\displaystyle\frac{\partial\phi_1}{\partial z_k}(z)\right|
\displaystyle\frac{(1-|z_k|^2)^q}{(1-|\phi_1(z)|^2)^p}\left|\displaystyle\frac
{\partial f_m}{\partial w_1}(\phi(z))\right|(1-|\phi_1(z)|^2)^p\nonumber\\
&&\geq\limsup\limits_{m}\sup\limits_{\phi(z)\in A_m}
\sum\limits^n_{k=1}\left|
\displaystyle\frac{\partial\phi_1}{\partial
z_k}(z)\right|\displaystyle\frac
{(1-|z_k|^2)^q}{(1-|\phi_1(z)|^2)^p}\left|\displaystyle\frac
{\partial f_m}{\partial w_1}(\phi(z))\right|(1-|\phi_1(z)|^2)^p\nonumber\\
&&\geq\limsup\limits_{m}\sup\limits_{\phi(z)\in
A_m}\sum\limits^n_{k=1}\left|
\displaystyle\frac{\partial\phi_1}{\partial z_k}(z)\right|
\displaystyle\frac
{(1-|z_k|^2)^q}{(1-|\phi_1(z)|^2)^p}\nonumber\\
& &\times\liminf\limits_{m}\min\limits_{\phi(z)\in A_m}
\left|\displaystyle\frac{\partial f_m}
{\partial w_1}(\phi(z))\right|(1-|\phi_1(z)|^2)^p\nonumber\\
&&\geq\limsup\limits_{m} \sup\limits_{\phi(z)\in
A_m}\sum\limits^n_{k=1}\left|
\displaystyle\frac{\partial\phi_1}{\partial
z_k}(z)\right|\displaystyle\frac
{(1-|z_k|^2)^q}{(1-|\phi_1(z)|^2)^p}\liminf\limits_m c_m\nonumber\\
&&\geq\limsup\limits_{m} \sup\limits_{\phi(z)\in
A_m}\sum\limits^n_{k=1}\left|
\displaystyle\frac{\partial\phi_1}{\partial
z_k}(z)\right|\displaystyle\frac
{(1-|z_k|^2)^q}{(1-|\phi_1(z)|^2)^p}.
\end{eqnarray*}
So \begin{eqnarray}\|C_{\phi}\|_e
&=&\inf\{\|C_{\phi}-K\|: K \mbox{ is compact}\}
\nonumber\\
&\geq&\limsup\limits_{m} \sup\limits_{\phi(z)\in
A_m}\sum\limits^n_{k=1}\left|
\displaystyle\frac{\partial\phi_1}{\partial
z_k}(z)\right|\displaystyle\frac
{(1-|z_k|^2)^q}{(1-|\phi_1(z)|^2)^p}.\label{19}
\end{eqnarray}
For each $l=1,2,\cdots, n, $ define
\begin{equation}a_l=\lim\limits_{\delta\to 0}
\sup\limits_{dist(\phi(z), \partial U^n)<\delta}
\sum\limits^n_{k=1}\left|
\displaystyle\frac{\partial\phi_l}{\partial
z_k}(z)\right|\displaystyle\frac
{(1-|z_k|^2)^q}{(1-|\phi_l(z)|^2)^p}.\label{20}\end{equation} For
any $\varepsilon>0,$ (\ref{20}) shows that there exists a
$\delta_0$ with $0<\delta_0<1,$ such that
\begin{equation}\sum\limits^n_{k=1}\left|
\displaystyle\frac{\partial\phi_l}{\partial
z_k}(z)\right|\displaystyle\frac
{(1-|z_k|^2)^q}{(1-|\phi_l(z)|^2)^p}>a_l-\varepsilon,\label{21}\end{equation}
whenever $dist(\phi(z),\partial U^n)<\delta_0$ and
$l=1,2,\cdots,n.$ Since $r_m\to 1$ as $m\to\infty,$ so as $m$ is
large enough, $r_m>1-\delta_0.$ If $\phi(z)\in A_m,$ $r_m\leq
|\phi_1(z)|\leq r_{m+1},$ so
$1-r_{m+1}<1-|\phi_1(z)|<1-r_m<\delta_0,$ $dist(\phi_1(z),\partial
U)<\delta_0.$ There exists $w_1$ with $|w_1|=1$ such that
$dist(\phi_1(z),w_1)=dist(\phi_1(z),\partial U)<\delta_0.$ Let
$w=(w_1, \phi_2(z),\ldots, \phi_n(z)),$ $w\in\partial U^n$, then
$$dist(\phi(z),\partial
U)\leq dist(\phi(z), w)=dist(\phi_1(z),w_1)<\delta_0.$$ By
(\ref{21}), (\ref{19}) implies that
$$\|C_{\phi}\|_e\geq a_1-\varepsilon.$$
Similarly, if we choose
$g_m(z)=\displaystyle\frac{m^{p-1}z^{m}_l}{\|m^{p-1}z^m_l\|}$, we
have
$$\|C_{\phi}\|_e\geq a_l-\varepsilon,$$
for every $l=2\cdots,n.$ So
\begin{eqnarray*}\|C_{\phi}\|_e&\geq&\displaystyle\frac{1}{n}
\sum\limits^n_{l=1}
(a_l-\varepsilon)\\
&=&\displaystyle\frac{1}{n}\sum\limits^n_{l=1}
(\lim\limits_{\delta\to 0} \sup\limits_{dist(\phi(z), \partial
U^n)<\delta} \sum\limits^n_{k=1}\left|
\displaystyle\frac{\partial\phi_l}{\partial
z_k}(z)\right|\displaystyle\frac
{(1-|z_k|^2)^q}{(1-|\phi_l(z)|^2)^p}-\varepsilon)\\
&\geq&\displaystyle\frac{1}{n} \lim\limits_{\delta\to 0}
\sup\limits_{dist(\phi(z), \partial U^n)<\delta}
\sum\limits^n_{k,l=1}
\left|\displaystyle\frac{\partial\phi_l}{\partial z_k}(z)\right|
\displaystyle\frac{(1-|z_k|^2)^q}{(1-|\phi_l(z)|^2)^p}-
\varepsilon.\end{eqnarray*} Let $\varepsilon\to 0,$ the low
estimate follows.
The upper estimate. To obtain the upper estimate we first prove
the following proposition.
\begin{Proposition} Let $\phi=(\phi_1, \ldots, \phi_n)$ a holomorphic self-map of $U^{n}.$ The operators
$K_m$ ($m\geq 2$) as follows:
$$K_mf(z)=f(\displaystyle\frac{m-1}{m}z),$$ for
$f\in H(U^n).$ Then the operators $K_m$ have the following
properties:
(i)\hspace*{2mm} For any $f\in H(U^n),$ $K_mf\in {\cal
B}^p_0(U^n)\subset {\cal B}^p_{0*}(U^n)\subset {\cal B}^p(U^n).$
(ii)\hspace*{2mm} If $C_{\phi}:$ ${\cal B}^p(U^n)$ ( ${\cal
B}^p_{0}(U^n)$ or ${\cal B}^p_{0*}(U^n)$)$\rightarrow$ $ {\cal
B}^q(U^n)$ ( ${\cal B}^q_{0}(U^n)$ or ${\cal B}^q_{0*}(U^n)$) is
bounded, then $C_{\phi}K_mf\in{\cal B}^q(U^n)$ (${\cal
B}^q_{0}(U^n)$ or ${\cal B}^q_{0*}(U^n)$) for all $f\in H(U^n).$
(iii) \hspace*{2mm}For fixed $m$, the operator $K_m$ is compact on
${\cal B}^p(U^n)$ (${\cal B}^p_{0}(U^n)$ or ${\cal
B}^p_{0*}(U^n)$).
(iv)\hspace*{2mm} If $C_{\phi}:$ ${\cal B}^p(U^n)$ ( ${\cal
B}^p_{0}(U^n)$ or ${\cal B}^p_{0*}(U^n)$)$\rightarrow$ $ {\cal
B}^q(U^n)$ ( ${\cal B}^q_{0}(U^n)$ or ${\cal B}^q_{0*}(U^n)$) is
bounded, then $C_{\phi}K_mf\in{\cal B}^q(U^n)$ (${\cal
B}^q_{0}(U^n)$ or ${\cal B}^q_{0*}(U^n))$ is compact.
(v) \hspace*{2mm} $\|I-K_m\|\leq 2.$
(vi) \hspace*{2mm}$(I-K_m)f$ converges uniformly to zero on
compact subset of $U^n$.\end{Proposition}
{\bf Proof}\hspace*{2mm} (i)\hspace*{2mm} Let $f\in H(U^n),$
$r_m=\displaystyle\frac{m-1}{m}, (0<r_m<1)$ and
$f_m(z)=K_mf(z)=f(r_mz).$ First note that
\begin{eqnarray}\|f_m\|_p&=&|f(0)|+\sup\limits_{z\in U^n}\sum\limits^n_{k=1}
r_m\left|\frac{\partial f} {\partial z_k}(r_mz)\right|\left(1- |z_k|^2\right)^p\nonumber\\
&\leq&|f(0)|+\sup\limits_{z\in U^n}\sum\limits^n_{k=1}
\left|\frac{\partial f} {\partial z_k}(r_mz)\right|\left(1-
|r_mz_k|^2\right)^p\leq\|f\|_p.\label{b}\end{eqnarray}
On the other hand, $f_m\in H(\frac{1}{r_m}U^n).$
$0<\displaystyle\frac{2}{1+r_m}<\displaystyle\frac{1}{r_m},$
$\displaystyle\frac{2}{1+r_m}\overline{U^n}\subset
\displaystyle\frac{1}{r_m}U^n.$ which implies that for fixed $m,$
and $\varepsilon=\displaystyle\frac{1}{j}, j=1,2,\cdots,$ there is
a polynomial $P^{(j)}_m$ such that
$$\sup_{z\in \frac{2}{1+r_m}\overline{U^n}}|f_m(z)-P^{(j)}_m(z)|<(1-r_m)^2\displaystyle\frac{1}{j}.$$
Let $K=\overline{U^n},$ $G=\displaystyle\frac{2}{1+r_m}U^n,$
$\Omega=\displaystyle\frac{1}{r_m}U^n,$ then $K\subset
G\subset\subset\Omega$, and $\rho=dist(K,\partial
G)=\displaystyle\frac{1-r_m}{1+r_m}>0$, so $\forall w\in U^n$,
$k\in\{1,\cdots,n\}$, it follows from Lemma 7 that
\begin{eqnarray*}
&&\Big|\frac{\partial (f_m-P_m^{(j)})}{\partial w_k}(w)\Big|\leq
\sup_{w\in K}
\Big|\frac{\partial (f_m-P_m^{(j)})}{\partial w_k}(w)\Big|\\[6pt]
&\leq& \frac{\sqrt{n}(1+r_m)}{1-r_m}\sup_{w\in G}|f_m(w)-P_m^{(j)}(w)|\\[6pt]
&\leq& \frac{\sqrt{n}(1+r_m)}{1-r_m}(1-r_m^2)\frac{1}{j}\leq
4\sqrt{n}\frac{1}{j}.
\end{eqnarray*}
Therefore
$$
\sum_{k=1}^n\Big|\frac{\partial (f_m-P_m^{(j)})}{\partial
w_k}(w)\Big|(1-|w_k|^p)^p \leq 4n\sqrt{n}\frac{1}{j}\to 0
$$
as $j\to \infty.$ that is,
$$
||f_m-P_m^{(j)}||_{{\cal B}^p}=|f_m(0)-P_m^{(j)}(0)|+\sup_{w\in
U^n}\sum_{k=1}^n\Big|\frac{\partial (f_m-P_m^{(j)})}{\partial
w_k}(w)\Big|(1-|w_k|^p)^p\to 0.
$$
$P_m^{(j)}(w)\in {\cal B}^p_0(U^n)$ implies that $f_m\in{\cal
B}^p_0(U^n)$.
(ii)\hspace*{2mm} By (i), as desired.
(iii) \hspace{2mm} For any sequence $\{f_{j}\}\subset{\cal
B}^p(U^n)$ (${\cal B}^p_{0}(U^n)$ or ${\cal B}^p_{0*}(U^n)$) with
$\|f_j\|_p\leq M,$ by (i), $\{K_mf_{j}\}\in {\cal B}^p_{0}(U^n).$
By Lemma 6, there is a subsequence $\{f_{j_s}\}$ of $\{f_j\}$
which converges uniformly on compact subsets of $U^n$ to a
holomorphic function $f\in{\cal B}^p(U^n)$ and $\|f\|_p\leq M.$
$\left\{\displaystyle\frac{\partial f_{j_s}}{\partial z_i}
\right\}, i=1,2,\cdots,n,$ also converges uniformly on compact
subsets of $U^n$ to the holomorphic function
$\displaystyle\frac{\partial f} {\partial z_i}.$ So as $s$ is
large enough, for any $w\in E=\{\frac{m-1}{m}z: z\in
\overline{U^n}\}\subset U^n$
\begin{equation}\left|\displaystyle\frac
{\partial (f_{j_s}-f)}{\partial
w_l}(w)\right|<\varepsilon,\label{23}\end{equation} for every
$l=1,2,\cdots,n.$ So
\begin{eqnarray}&&\left\|K_mf_{j_s}-K_mf\right\|_p
=\left\|f_{j_s}(\displaystyle\frac{m-1}{m}z)-
f(\displaystyle\frac{m-1}{m}z)\right\|_p\nonumber\\
&&=\sup\limits_{z\in U^n}\sum\limits^n_{k=1}
\left\{\left|\displaystyle\frac{\partial \left[(f_{j_s}-f)
(\displaystyle\frac{m-1}{m}z)\right]} {\partial z_k}\right|
(1-|z_k|^2)^p
\right\}+|f_{j_s}(0)-f(0)|\nonumber\\
&&\leq\sup\limits_{z\in
U^n}\sum\limits^n_{k=1}\sum\limits^n_{l=1}\left|
\displaystyle\frac{\partial (f_{j_s}-f)}{\partial w_l}
(\displaystyle\frac{m-1}{m}z)\right|\displaystyle\frac{m-1}{m}+|f_{j_s}(0)-f(0)|
\nonumber\\
&&\leq n\sup\limits_{w\in
E_1}\displaystyle\frac{m-1}{m}\sum\limits^n_{l=1}
\left|\displaystyle\frac {\partial (f_{j_s}-f)}{\partial
w_l}(w)\right|+|f_{j_s}(0)-f(0)|\to 0,\label{24}
\end{eqnarray}
as $s\to\infty.$ This shows that $\{K_mf_{j_s}\}$ converges to
$g=K_mf\in{\cal B}^p_{0}(U^n)\subset {\cal B}^p_{0*}(U^n)\subset
{\cal B}^p(U^n).$ So $K_m$ is compact on ${\cal B}^p(U^n)$(${\cal
B}^p_0(U^n)$ or ${\cal B}^p_{0*}(U^n)).$
(iv)\hspace*{2mm} By (i) and (iii), the result is obvious.
(v)\hspace*{2mm}In fact, for any $f\in{\cal B}^p(U^n)({\cal
B}^p_{0}(U^n)$ or ${\cal B}^p_{0*}(U^n))$, note that
$(I-K_m)f(0)=0$, so
\begin{eqnarray*}&&\|(I-K_m)f\|_p
=\sup\limits_{z\in U^n}\sum\limits^n_{k=1}
\left|\displaystyle\frac{\partial (I-K_m)f}{\partial z_k}(z)
\right|(1-|z_k|^2)\\
&&=n\sup\limits_{z\in U^n}\max\limits_{1\leq k\leq n}
\left|\displaystyle\frac{\partial f}{\partial z_k}(z)
-(1-\frac{1}{m}) \displaystyle\frac{\partial f}{\partial
z_k}((1-\frac{1}{m})z)
\right|(1-|z_k|^2)^p\\
&&\leq\sup\limits_{z\in U^n}\sum\limits^n_{k=1}
\left|\displaystyle\frac{\partial f}{\partial z_k}(z)\right|(1-|z_k|^2)^p\\
&&+(1-\frac{1}{m})\sup\limits_{z\in
U^n}\sum\limits^n_{k=1}\left|\displaystyle\frac{\partial
f}{\partial z_k}((1-\frac{1}{m})z)
\right|(1-|(1-\frac{1}{m})z_k|^2)^p\\
&&\leq \|f\|_p+\|f\|_p=2\|f\|_p,
\end{eqnarray*}
so $\|I-K_m\|\leq 2.$
(vi)\hspace*{2mm} For any compact subset $E\subset U^n$, $\exists
r,$ $0<r<1$ such that $E\subset rU^n\subset \subset U^n$. For
$\forall z\in E$,
\begin{eqnarray*}
|(I-K_m)f(z)|&=&|f(z)-f_m(z)|=|f(z)-f(r_mz)|\\
&=&\left|\int_{r_m}^1\frac{d}{dt}(f(tz))dt\right|=\left|\int_{r_m}^1\sum_{k=1}^n
\frac{\partial f}{\partial w_k}(tz)\cdot z_kdt\right|\\
&\leq& \sum_{k=1}^n\int_{r_m}^1\left|\frac{\partial f}{\partial
w_k}(tz)\right|dt.
\end{eqnarray*}
$t\in[r_m,1]$, $\forall z\in U^n,\hspace*{4mm}
|tz_k|=t|z_k|<|z_k|<r,$\hspace*{4mm} so
$\displaystyle\frac{\partial f}{\partial w_k}(w)$ is bounded in
$r\overline{U^n}$, i.e., $\forall z\in E,$\hspace*{4mm}
$\left|\displaystyle\frac{\partial f} {\partial
w_k}(tz)\right|\leq M$. So
$$
|(I-K_m)f(z)|\leq nM(1-r_m)\to 0$$ as $m\to \infty$, the results
follows.
Now return to the upper estimate. For the convenience, we denote
$\|f\|=\|f\|_p.$
\begin{eqnarray}&&\|C_{\phi}\|_e \leq\|C_{\phi}-C_{\phi}K_m\|
=\|C_{\phi}(I-K_m)\|
=\sup\limits_{\|f\|=1}\|C_{\phi}(I-K_m)f\|_q\nonumber\\
&&=\sup\limits_{\|f\|=1}\left(\sup\limits_{z\in
U^n}\sum\limits^n_{k=1} \left\{\left|\displaystyle\frac{\partial
(I-K_m)(f\circ\phi)} {\partial
z_k}\right|(1-|z_k|^2)^q\right\}+\left|(I-K_m)f(\phi(0))\right|\right)
\nonumber\\
&&\leq\sup\limits_{\|f\|=1}\sup\limits_{z\in
U^n}\sum\limits^n_{k=1}
\sum\limits^n_{l=1}\left|\displaystyle\frac{\partial (I-K_m)f}
{\partial w_l}(\phi(z))\right| \left|\displaystyle\frac
{\partial\phi_l}{\partial z_k}(z)\right|(1-|z_k|^2)^q\nonumber\\
&& +\sup\limits_{\|f\|=1}
\left|f(\phi(0))-f(\frac{m-1}{m}\phi(0))\right|\nonumber\\
&&\leq \sup\limits_{\|f\|=1}\sup\limits_{z\in U^n}
\sum\limits^n_{k, l=1}\left|
\displaystyle\frac{\partial\phi_l}{\partial z_k}(z)\right|
\displaystyle\frac {(1-|z_k|^2)^q}{(1-|\phi_l(z)|^2)^p}
\left|\displaystyle\frac{\partial (I-K_m)f}{\partial w_l}(\phi(z))
\right|(1-|\phi_l(z)|^2)^p\nonumber\\
&& +\sup\limits_{\|f\|=1}
\left|f(\phi(0))-f(\frac{m-1}{m}\phi(0))\right|.\label{26}\end{eqnarray}
Denote $G_1=\{z\in U^n: dist(\phi(z), \partial U^n)<\delta\},$
$G_2=\{z\in U^n: dist(\phi(z), \partial U^n)\geq\delta\},$
$G=\{w\in U^n: dist(w, \partial U^n)\geq\delta\}$, where $G$ is a
compact subset of $\mbox{\Bbb C}^n.$
Then by Lemma 3, Lemma 4 and Lemma 5, condition (9) holds, so
\begin{eqnarray}\|C_{\phi}\|_e
&\leq& \sup\limits_{\|f\|=1}\sup\limits_{z\in G_1}
\sum\limits^n_{k, l=1}\left|
\displaystyle\frac{\partial\phi_l}{\partial
z_k}(z)\right|\displaystyle\frac
{(1-|z_k|^2)^q}{(1-|\phi_l(z)|^2)^p} \left|\displaystyle\frac
{\partial (I-K_m)f}{\partial
w_l}(\phi(z))\right|(1-|\phi_l(z)|^2)^q \nonumber\\
&&+C\sup\limits_{\|f\|=1}\sup\limits_{z\in G_2}
\sum\limits^n_{l=1}(1-|\phi_l(z)|^2)^p
\left|\displaystyle\frac{\partial (I-K_m)f}{\partial
w_l}(\phi(z))\right|\nonumber\\
&& +\sup\limits_{\|f\|=1}
\left|f(\phi(0))-f(\frac{m-1}{m}\phi(0))\right|
\nonumber\\
&\leq& \|I-K_m\|\sup\limits_{z\in G_1} \sum\limits^n_{k,
l=1}\left| \displaystyle\frac{\partial\phi_l}{\partial
z_k}(z)\right|\displaystyle\frac
{(1-|z_k|^2)^q}{(1-|\phi_l(z)|^2)^p}\nonumber\\
&&+C\sup\limits_{\|f\|=1}\sup\limits_{z\in
G_2}\sum\limits^n_{l=1}(1-|\phi_l(z)|^2)^p
\left|\displaystyle\frac{\partial (I-K_m)f}{\partial w_l}(\phi(z))\right|\nonumber\\
&& +\sup\limits_{\|f\|=1}
\left|f(\phi(0))-f(\frac{m-1}{m}\phi(0))\right|\nonumber\\
&\leq& 2\sup\limits_{z\in G_1} \sum\limits^n_{k, l=1}\left|
\displaystyle\frac{\partial\phi_l}{\partial
z_k}(z)\right|\displaystyle\frac
{(1-|z_k|^2)^q}{(1-|\phi_l(z)|^2)^p}\nonumber\\
&&+C\sup\limits_{\|f\|=1}\sup\limits_{z\in G_2}
\sum\limits^n_{l=1}(1-|\phi_l(z)|^2)^p
\left|\displaystyle\frac{\partial (I-K_m)f}{\partial
w_l}(\phi(z))\right|\nonumber\\
&& +\sup\limits_{\|f\|=1}
\left|f(\phi(0))-f(\frac{m-1}{m}\phi(0))\right|.\label{27}
\end{eqnarray}
Denote the second term and third term of the right hand side of
(\ref{27}) by $I_1$ and $I_2$. Then Theorem 1 is proved if we can
prove
$$\lim\limits_{m\to\infty}I_1=0\hspace*{4mm} \mbox{and}\hspace*{4mm} \lim\limits_{m\to\infty}I_2=0.$$
To do this, let $z\in G_2$ and $w=\phi(z),$ then $w\in G$
\begin{eqnarray}I_1&\leq&C\sup\limits_{\|f\|=1}
\sup\limits_{w\in G}\sum\limits^n_{l=1}(1-|w_l|^2)^p
\left|\displaystyle\frac{\partial f}{\partial w_l}(w)-
(1-\frac{1}{m})\displaystyle\frac{\partial f}{\partial w_l}
((1-\frac{1}{m})w)\right|\nonumber\\
&\leq& C\sup\limits_{\|f\|=1} \sup\limits_{w\in
G}\sum\limits^n_{l=1}(1-|w_l|^2)^p
\left|\displaystyle\frac{\partial f}{\partial w_l}(w)-
\displaystyle\frac{\partial f}{\partial w_l}
((1-\frac{1}{m})w)\right|\nonumber\\
& &+\displaystyle\frac{C}{m}\sup\limits_{\|f\|=1}
\sup\limits_{w\in G}\sum\limits^n_{l=1}(1-|w_l|^2)^p
\left|\displaystyle\frac{\partial f}{\partial w_l}
((1-\frac{1}{m})w)\right|\nonumber\\
&\leq& C\sup\limits_{\|f\|=1} \sup\limits_{w\in
G}\sum\limits^n_{l=1}(1-|w_l|^2)^p
\left|\displaystyle\frac{\partial f}{\partial w_l}(w)-
\displaystyle\frac{\partial f}{\partial w_l}
((1-\frac{1}{m})w)\right|+\displaystyle\frac{C}{m}.\label{28}
\end{eqnarray}
Let $w=(w_1,w_2,\cdots,w_{n-1},w_n),$ for $m$ large enough, we have
\begin{eqnarray}&&\left|\displaystyle\frac{\partial f}{\partial w_l}(w)-
\displaystyle\frac{\partial f}{\partial w_l}((1-\frac{1}{m})w)
\right|\nonumber\\
&&\leq\sum\limits^n_{j=1}\left|\displaystyle\frac{\partial f}
{\partial w_l}\left((1-\frac{1}{m})w_1,\cdots,
(1-\frac{1}{m})w_{j-1},
w_j,\cdots, w_n\right)\right.\nonumber\\
& &-\left.\displaystyle\frac{\partial f}{\partial w_l}
\left((1-\frac{1}{m})w_1,\cdots,(1-\frac{1}{m})
w_j,w_{j+1},\cdots,w_n\right)\right|
\nonumber\\
&&=\sum\limits^n_{j=1}\left|\int^{w_j}_{(1-\frac{1}{m})w_j}
\displaystyle\frac{\partial^2 f}{\partial w_l\partial w_j}
\left((1-\frac{1}{m})w_1,\cdots,
(1-\frac{1}{m})w_{j-1},\zeta,
w_{j+1},\cdots, w_n\right)d\zeta\right|\nonumber\\
&&\leq\frac{1}{m}\sum\limits^n_{j=1} \sup\limits_{w\in
G}\left|\displaystyle\frac {\partial^2 f}{\partial w_l\partial
w_j}(w)\right|.\label{29}\end{eqnarray} Denote $G_3=\left\{w\in
U^n:dist(w,\partial U^n)> \displaystyle\frac{\delta}{2}\right\},$
then $G\subset G_3\subset\subset U^n.$
Since $dist(G, \partial G_3)=\displaystyle\frac{\delta}{2},$ then
by Lemma 7, (\ref{29}) gives
\begin{equation}\left|\displaystyle\frac{\partial f}{\partial w_l}(w)-
\displaystyle\frac{\partial f}{\partial w_l}
((1-\frac{1}{m})w)\right| \leq\displaystyle\frac{2n\sqrt{n}}
{m\delta}\max\limits_{z\in G_3} \left|\displaystyle\frac{\partial
f}{\partial w_l}(w)\right|.\label{30}\end{equation} On the other
hand, on the unit ball of ${\cal B}^p(U^n)$, we have
$$\sup\limits_{z\in G_3}(1-|w_l|^2)^p\left|\displaystyle\frac{\partial f}
{\partial w_l}(w)\right|=\sup\limits_{dist(w,\partial
U^n)>\frac{\delta}{2}}
(1-|w_l|^2)^p\left|\displaystyle\frac{\partial f} {\partial
w_l}(w)\right|\leq \|f\|_p=1,$$ namely
\begin{equation}\sup\limits_{z\in G_3}\left|\displaystyle\frac{\partial f}
{\partial w_l}(w)\right|
\leq\displaystyle\frac{1}{1-\left(\frac{\delta}{2}\right)^2}=
\displaystyle\frac{4}{4-\delta^2}.\label{31}\end{equation}
Combining (\ref{28}), (\ref{30}) and (\ref{31}), imply
$$I_1\leq\displaystyle\frac{2n\sqrt{n}C}{m\delta}
\displaystyle\frac{4}{4-\delta^2}+\displaystyle\frac{C}{m}$$ and
$\lim\limits_{m\to\infty}I_1=0.$
Now we can prove $\lim\limits_{m\to\infty}I_2=0$. In fact,
\begin{eqnarray*}&&f(\phi(0))-f(\frac{m-1}{m}\phi(0))=
\int^{1}_{\frac{m-1}{m}}\displaystyle\frac{d f(t\phi(0))}{d
t}dt=\sum\limits^n_{l=1}\int^{1}_{\frac{m-1}{m}}
\phi_l(0)\displaystyle\frac{\partial
f}{\partial\zeta_l}(t\phi(0))dt.\end{eqnarray*} By Lemma 1, it
follows that for any compact subset $K\subset U^n$, $|f(z)|\leq
C_K \|f\|_p=C_K.$ Let $K=\{z\in U^n: |z_i|\leq|\phi_i(0)|\},$
So\begin{eqnarray*}&&|f(\phi(0))-f(\frac{m-1}{m}\phi(0))|\leq
\sum\limits^n_{l=1}|\phi_l(0)|\int^{1}_{\frac{m-1}{m}}C_K dt \leq
nC_K(1-\frac{m-1}{m})=\frac{nC_K}{m},
\end{eqnarray*} so $I_2\leq \frac{nC_K}{m}\to 0.$
Thus let first $m\to\infty,$ then $\delta\to 0$ in (\ref{27}), we
get the upper estimate of $\|C_{\phi}\|_e$:
$$\|C_{\phi}\|_e\leq 2\lim\limits_{\delta\to 0}
\sup\limits_{dist(\phi(z),\partial
U^n)<\delta}\sum\limits^n_{k,l=1}
\left|\displaystyle\frac{\partial \phi_{l}}{\partial
z_k}(z)\right|
\displaystyle\frac{(1-|z_k|^2)^q}{(1-|\phi_l(z)|^2)^p}.$$ Now the
proof of Theorem 1 is finished.
|
{
"timestamp": "2005-12-27T16:13:13",
"yymm": "0503",
"arxiv_id": "math/0503723",
"language": "en",
"url": "https://arxiv.org/abs/math/0503723"
}
|
\section{Introduction}
Quantum billiards, that is, closed compact domains in the two-dimensional
Euclidean plane, are the
simplest model of a quantum system corresponding to physical instances
such as quantum dots
or microstructures. The statistical properties of the quantum energy
levels of such systems have
been investigated, and it turns out that the statistical quantum
behaviour can be related to the
classical properties of the system. It is believed that systems
whose classical motion is chaotic
have energy levels behaving like eigenvalues of random matrix ensembles \cite{BohGiaSch84},
whereas the energy levels of systems whose classical motion is integrable are Poisson distributed,
i.e. they behave like independent uniformly distributed random variables \cite{BerTab77a}. Both
numerical evidence and some analytical results support these conjectures
\cite{AndAlt95, AgaAndAlt95, Mar98}.\\
Among systems which are classically neither chaotic nor integrable, some systems have been found to
display an eigenvalue statistics which is intermediate between the Poisson and the
Random matrix distribution. The characteristics of such intermediate statistics are
\cite{BogGerSch99} level repulsion, exponential decrease of the nearest-neighbour
spacing distribution at infinity and linear asymptotic behaviour of the number variance
(which is related to a non-vanishing form factor at small arguments).
The form factor at the origin is equal to $1$ for classically integrable systems, to $0$
for chaotic systems, and it is found numerically to take values between 0 and 1 for intermediate
statistics, the case $\overline{K_2(0)}=1/2$ corresponding to semi-Poisson statistics \cite{BogGerSch99}.
Numerous quantum systems have been found to display numerically intermediate
statistics: for example, pseudo-integrable systems such as
rational polygonal billiards (polygons in which all angles are commensurate
with $\pi$) \cite{CasPro99}, or quantum maps \cite{GirMarOke04}.\\
An analytical approach to the study of level statistics is the
semiclassical trace
formula, which gives an expansion of the density of
energy levels as a sum over periodic orbits \cite{BalBlo72, Gut89}, or families of
periodic orbits in the case of integrable systems \cite{BerTab76}. For
diffractive systems, the trace formula can be modified to include diffractive orbits
contributions \cite{Sie99, BogPavSch00}. It can be argued however
(see \cite{BogGirSch01}
for a discussion) that only the periodic orbits contribute to the semiclassical
form factor at small arguments, $\overline{K_2(0)}$. The calculation of this quantity
therefore only
requires to find the
periodic orbits and the areas occupied by the pencils of periodic orbits
in a given system. Unfortunately, this is not
a simple task. For instance is is not known whether any acute triangle has a periodic
orbit. In the case of rational polygonal billiards, it has been shown \cite{Mas90} that
the number ${\mathcal N}(L)$ of periodic orbits of length less than $L$ is quadratically bounded,
namely there exist $c_1$ and $c_2$ such that $c_1L^2\leq {\mathcal N}(L)\leq c_2L^2$, but even for
general rational polygonal billiards exact asymptotics is not known.
There exist however certain specific rational polygonal billiards for which more precise
statements are known. For instance for Veech billiards \cite{Vee89, Vor96}, a special class
of rational polygonal billiards (whose stabilizer is a discrete cofinite subgroup of $SL(2, \mathbb{R})$),
precise asymptotics for ${\mathcal N}(L)$ is known, and in \cite{BogGirSch01} it was possible to
calculate analytically the form factor at the origin for triangular Veech billiards.\\
This paper presents the calculation of the semiclassical form factor at the origin
for a billiard which does not have this special Veech property, the barrier billiard. The barrier billiard
is one of the simplest pseudo-integrable billiards. It was introduced by Hannay and McCraw
\cite{HanMcc90} and consists of a rectangle $[0,a]\times[0,b]$ containing a barrier
described by the segment $\{\epsilon_0 a\}\times[0,\alpha b]$ with $0\leq\epsilon_0, \alpha<1$
(see Figure \ref{billard} left). It is a rational polygonal billiard with six angles equal to $\pi/2$
and one angle equal to $2\pi$. It is therefore a pseudo-integrable billiard
\cite{BerRic81},
and the movement in phase space takes place on a surface of genus 2.
When the height of the barrier is such that $\alpha\in\mathbb{Q}$ then the barrier billiard
is a Veech billiard. But when $\alpha$ is irrational the billiard loses
this property.
Nevertheless, from results obtained in \cite{EskMasSch01},
it is still possible to work out the distribution of the
periodic orbits in this latter case, and thus calculate analytically
the semiclassical form
factor at the origin,
provided the position of the barrier is a rational number with
respect to the size of the side: $\epsilon_0=p/q$ with $p,q\in\mathbb{N}$ coprime.
We will first devise a method to obtain a complete
characterization of the periodic orbit pencils in
the non-Veech barrier billiard (Section \ref{section3}). We then
rigourously derive asymptotics for each family of periodic orbit pencils
(Section \ref{onpeutremplacer}), then use this result to calculate the
semiclassical form factor at small arguments
(Section \ref{calculff}). Previously obtained
analytical results show that the semi-classical form factor at the origin
takes non-universal values between 0 and 1. For Veech triangular
billiards with angles $(\pi/2, \pi/n, \pi/2-\pi/n)$,
the value $K_2(0)=\frac{1}{3}(n+\epsilon(n))/(n-2)$ with
$\epsilon(n)=0,2$ or $6$ was found \cite{BogGirSch01}.
For a rectangular billiard perturbed by an
Aharonov-Bohm flux line, we obtained
$K_2(0)=1-\kappa\overline{\alpha}+4\overline{\alpha}^2$
where $\overline{\alpha}\in[0,1/2[$ is the strength of the magnetic flux
and $\kappa$ a rational depending on the position of the flux line in the
billiard (for irrational positions, $\kappa=3$) \cite{BogGirSch01}.
For a circular billiard perturbed by an Aharonov-Bohm flux line,
a similar result $K_2(0)=1-\kappa\overline{\alpha}(1-\overline{\alpha})$,
with $\kappa\in[0,2]$ an explicit function of the position of the flux,
was derived \cite{TheseGir02}.
In the case of the barrier billiard, we obtain $K_2(0)=1/2+1/q$.
This value depends on the position of the barrier inside the rectangle,
which reflects the fact that the structure and the properties of periodic
orbits strongly depend on it. This analytical expression for $K_2(0)$ extends previous
results to the case of non-Veech polygonal billiards.
\section{Periodic orbits in the barrier billiard}
\label{section3}
The aim of this section is to characterize periodic orbits
in a barrier billiard. We first begin by the simple case of a
rectangular billiard.
\subsection{Periodic orbits in the rectangular billiard}
\label{casrectangle}
Let us consider a rectangle of area ${\mathcal A}=a\times b$ with Dirichlet
boundary conditions. It is easy to work out the density of the lengths
of periodic orbits. Any orbit in the rectangle can
be unfolded into a straight line in a torus (a rectangle with periodic
boundary conditions) of size $2a\times 2b$; a periodic orbit is
therefore defined by two integers $M$ and $N$ and has length
\begin{equation}
\label{lprectangle}
l_p=\sqrt{(2 M a)^2+(2 N b)^2}.
\end{equation}
If we restrict ourselves to $(M,N)$ in the upper right quadrant,
each family of periodic orbits occupies an area $4{\mathcal A}$ ($2{\mathcal A}$
for the orbit itself, $2{\mathcal A}$ for its time-reverse). The number
${\mathcal N}(l)$ of pencils of length less than $l$ is just the number of
lattice points $(2Ma, 2Nb)$ within a (quarter of a) disk of radius $l$.
It has the asymptotic expression ${\mathcal N}(l)\sim\pi l^2/16{\mathcal A}$.
The corresponding density of periodic orbits is the derivative of ${\mathcal N}(l)$:
\begin{equation}
\label{rhol}
\rho(l)\sim\frac{\pi l}{8{\mathcal A}}.
\end{equation}
The density of primitive periodic orbits is given by (see e.g. \cite{BogGirSch01})
\begin{equation}
\label{densiterectangle}
\rho_{pp}(l)\sim\frac{3 l}{4\pi{\mathcal A}}.
\end{equation}
We want to obtain a similar result for the barrier billiard. In the rest
of this section we investigate the periodic orbits of the barrier billiard,
and Section \ref{onpeutremplacer} leads to Equation \eqref{rhoppf} which
gives the density of primitive periodic orbits for the barrier billiard.
\subsection{The translation surface}
\label{transsurf}
Instead of studying directly the barrier billiard itself, we will consider the equivalent
problem of studying the translation surface associated to this billiard \cite{GutJud00}.
\begin{figure}[ht]
\begin{center}
\epsfig{file=fig1.eps,width=11cm}
\end{center}
\caption{The barrier billiard and its translation surface}
\label{billard}
\end{figure}
A construction due to Zemlyakov and Katok \cite{ZemKat76} shows that the translation surface
associated to a generic rational polygonal billiard is obtained by unfolding the polygon
with respect to each of its sides, which means gluing to the initial polygon its images by reflexion
with respect to each of its sides and repeating the operation. If the angles of the polygon are
$\alpha_i=\pi m_i/n_i$ and $N$ is the least common multiple of the $n_i$, then $2N$ copies of the initial billiard
are needed. Here all the angles are multiples of $\pi/2$, therefore only 4 copies are needed,
and the translation surface $S$ obtained by this construction is represented
in Figure \ref{billard} (right). In this surface, all opposite sides are identified.
Any trajectory in the barrier billiard can be unfolded to a straight line on the translation surface.
The surface $S$ is of genus 2: there are two singular angles of measure $4\pi$
that we will represent respectively by $z_1$ (a dot in Figure \ref{billard}) and $z_2$ (a cross
in Figure \ref{billard}). The two singularities are traditionally
called saddles \cite{EskMasZor03} and a geodesic joining them is called a saddle-connexion.
\subsection{Periodic orbits in the barrier billiard}
\label{pobb}
In this subsection, our aim is to describe qualitatively the periodic
orbits in the barrier billiard in a given direction.
On translation surfaces the periodic orbits occur in pencils, or
cylinders, of periodic orbits of
same length. These cylinders are bounded by saddle-connexions and are characterized by their length
and their height.
Let us consider a 'rational direction' on the translation surface $S$:
\begin{equation}
\label{vecteurs}
{\bf v}=(2 M a/q, 2 N b),
\end{equation}
with $M$ and $N$ two coprime positive integers.
The length of the vector ${\bf v}$ is
\begin{equation}
\label{lgbarr}
l_p=\sqrt{(2a M/q)^2+( 2 b N)^2}.
\end{equation}
Let us label by the integers $k=0$, 1,..., $q-1$ the positions on the
translation surface such that the barrier on the ''left'' of the
translation surface in Figure \ref{billard} be at position $p-1$
and the barrier on the ''right'' in Figure \ref{billard} be at position
$q-p$ (see Figure \ref{unfolded}).
Since the opposite sides on the translation surface are identified, then
when a trajectory hits the barrier at position $p-1$ it
reappears at position $q-p$, and vice-versa. The translation by vector
${\bf v}$ induces a permutation
$\sigma_{{\bf v}}$ of the positions $\{0, 1,..., q-1\}$.
Let us define
\begin{equation}
w_1=\min\left\{k\in{\mathbb{N}}; \sigma^{k}_{{\bf v}}(p-1)\in\{p-1, q-p\}\right\}
\end{equation}
and in the same way
\begin{equation}
w_2=\min\left\{k\in{\mathbb{N}}; \sigma^{k}_{{\bf v}}(q-p)\in\{p-1, q-p\}\right\}.
\end{equation}
A translation by the vector $w_1 {\bf v}$ takes $z_1$ to itself and defines
a saddle-connexion of length $w_1 l_p$. The second saddle-connexion
joining $z_1$ to itself starts at position $q-p$ and its length
is $w_2 l_p$.\\
Figure \ref{unfolded} shows, as an example, the two saddle-connexions
going from $z_1$ to itself in the
direction $(9,2)$ for $p/q=1/3$. The translation by the vector ${\bf v}$ induces
the permutation $(012)\mapsto (021)$.
One of the saddle-connexions goes from the position 0 to itself
and has a length $l_p$; the other goes from position 2 to itself
via position 1 and has a length $2l_p$.
In any direction, there are always two saddle-connexions going from
$z_1$ to itself, and, in the same way, two from $z_2$ to itself.
These four saddle-connexions form the boundary of three cylinders of
periodic orbits (see Figure \ref{3cylindres}).
\begin{figure}[ht]
\begin{center}
\epsfig{file=fig2.eps,width=13cm}
\end{center}
\caption{Starting from points of abscissa 0, 1 or 2 ( for $q=3$) in the direction $(M=9,N=2)$,
one arrives at 0, 2 or 1: there are two saddle-connexions $0\to 0$ and $2\to 1\to 2$.}
\label{unfolded}
\end{figure}
The lengths of these cylinders are necessarily of the form $w_1 l_p$, $w_2 l_p$
and $(w_1+w_2)l_p$, with $w_i\in\mathbb{N}$, and their heights $(2b/M)h_i$ are such that $h_1+h_3\in\mathbb{Z}$,
$h_2+h_3\in\mathbb{Z}$ and $h_3-\sigma \delta_2\in\mathbb{Z}$ for some $\sigma=\pm 1$ and $\delta_2=\{M\alpha\}$,
the fractional part of $M\alpha$. For instance in Figure
\ref{3cylindres}, there is one cylinder immediately above the saddle-connexion $0\to 0$, one immediately
above the saddle-connexion $2\to 1\to 2$, and the third cylinder is below both.
\begin{figure}[ht]
\begin{center}
\epsfig{file=fig3.eps,width=11cm}
\end{center}
\caption{Three cylinders of periodic orbits bounded by four saddle-connexions in the case $M=4, N=1$.}
\label{3cylindres}
\end{figure}
The results of this section can be summed up as follows. We set
\begin{eqnarray}
s_1=h_1+h_3\nonumber\\
s_2=h_2+h_3,
\end{eqnarray}
so that $s_1$ and $s_2$ are integers. Then in each direction ${\bf v}$ defined by
$(M,N)$ with $M$ and $N$ coprime, there are three cylinders of periodic orbits
of lengths $w_i l_p$ and heights $(2b/M)h_i$ with $i=1,2,3$.
The cylinders can be described
by the following five characteristic numbers:
\begin{itemize}
\item[-] the integers $w_1$ and $w_2$ (giving the lengths of the two short cylinders and the
length $(w_1+w_2)l_p$ of the long cylinder)
\item[-] the real number $h_3$ (giving the height $(2b/M)h_3$ of the long cylinder)
\item[-] the integers $s_1$ and $s_2$ (giving the heights $(2b/M)(s_1-h_3)$
and\\
$(2b/M)(s_2-h_3)$
of the short cylinders).
\end{itemize}
Note that by definition of the $s_i$ we need to have $0<h_3<\min(s_1,s_2)$.
Also note that the condition that the sum of the areas of the cylinders
be $4{\mathcal A}$ can be expressed as $s_1 w_1+s_2 w_2=q$.
\section{Asymptotics for the periodic orbit lengths}
\label{onpeutremplacer}
Let us define ${\mathcal F}$ as the set of all 4-uples
$(w_1,w_2,s_1,s_2)\in {(\mathbb{N}^{*})}^4$
such that $(s_1, s_2)$ are coprime and $s_1 w_1+s_2 w_2=q$. We say that a
direction ${\bf v}$ belongs to the family $f\in{\mathcal F}$ if the three
cylinders in the direction ${\bf v}$
have the characteristic numbers $w_1,w_2,s_1,s_2$.
The goal of this section is to calculate, for a fixed family $f\in{\mathcal F}$ and a
fixed interval $I\subset[0,\min(s_1,s_2)[$, the asymptotics
for the number ${\mathcal N}^{(q)}_{f,I}(l)$ of directions ${\bf v}$ belonging to
the family $f$, such that $||{\bf v}||<l$ and such that the height $h_3$
of the third cylinder in the direction ${\bf v}$ belongs to the interval $I$.
\subsection{Counting periodic orbits}
The asymptotics for the number ${\mathcal N}^{(q)}(l)$ of cylinders of length less than $l$
have been calculated in \cite{EskMasSch01}.
These asymptotics are obtained by applying a Siegel-Veech formula
to the space ${\mathcal M}_q(1,1)$ of $q$-fold coverings of the torus with two branch points and
area 1. If $V(S)$ is the set of vectors associated with cylinders of periodic orbits on a
'stable' $q$-fold torus cover $S$, then it is shown that there is a constant $\kappa(S)$
depending only on the connected component ${\mathcal M}(S)$ of ${\mathcal M}_q(1,1)$ containing $S$,
such that
\begin{equation}
\label{quadraticasymptotics1}
|V(S)\cap B(T)|\sim\pi\kappa T^2,
\end{equation}
where $|.|$ denotes the cardinal of a set and $B(T)$ is the ball of radius $T$ centered at the
origin. The constant $\kappa$ is given by the following Siegel-Veech formula:
for any continuous compactly supported $\varphi:{\mathbb{R}}^2\to\mathbb{R}$,
\begin{equation}
\label{quadraticasymptotics2}
\frac{1}{\tilde{\mu}({\mathcal M}(S))}\int_{{\mathcal M}(S)}\hat{\varphi}\ d\tilde{\mu}
=\kappa\int_{{\mathbb{R}}^2}\varphi,
\end{equation}
where $\hat{\varphi}$ is the Siegel-Veech transform of $\varphi$ defined by
\begin{equation}
\label{quadraticasymptotics3}
\hat{\varphi}(S)=\sum_{v\in V(S)}\varphi(v)
\end{equation}
and $\tilde{\mu}$ is the measure on ${\mathcal M}_q(1,1)$ (Theorem 2.4 of \cite{EskMasSch01}).
This Theorem applies to the translation surface constructed from the barrier billiard,
provided $S$ be a stable $q$-fold torus cover, which is true only if the height $\alpha$ of the
barrier is irrational. It is shown that in this case ${\mathcal M}(S)$ is the set
${\mathcal P}_q(1,1)\subset{\mathcal M}_q(1,1)$ of primitive torus covers.
The following asymptotics are then obtained (here we have a factor 1/16
differing from the factor in \cite{EskMasSch01} because of our conventions
for the counting of the time-reverse partner of a periodic orbit):
\begin{equation}
\label{eskin1}
{\mathcal N}^{(q)}(l)\sim c\frac{\pi l^2}{16{\mathcal A}}.
\end{equation}
The constant $c$ is given by
\begin{equation}
\label{eskin2}
c=\frac{q}{N_q}\sum_{r|q}\mu(r)
\hspace{-.5cm}\sum_{\genfrac{}{}{0pt}{}{(s_1,s_2)=1}{s_1 u_1+s_2 u_2=q/r}}\hspace{-.5cm}
u_1 u_2 (u_1+u_2) \min(s_1,s_2)\left(\frac{1}{u_1^2}+\frac{1}{u_2^2}+\frac{1}{(u_1+u_2)^2}\right)
\end{equation}
(the gcd of $s, s'$ will be noted either $\gcd(s, s')$ or simply $(s,s')$), and
\begin{equation}
\label{nqp}
N_q=\sum_{r|q}\mu(r)r^2\hspace{-.5cm}\sum_{\genfrac{}{}{0pt}{}{(s_1,s_2)=1}{s_1 u_1+s_2 u_2=q/r}}
\hspace{-.5cm}u_1 u_2 (u_1+u_2) \min(s_1,s_2)
\end{equation}
(Proposition 4.14 of \cite{EskMasSch01}).
The constant $N_q$ is the number of primitive covers of degree $q$ of a surface of
genus 2 with 2 branch points.
\subsection{Siegel-Veech formula}
The proof leading to Equation \eqref{eskin1} can be adapted to any subset of $V(S)$
provided it varies linearly under $SL(2, \mathbb{R})$ action, i.e. provided the subset
verifies $\forall g\in SL(2, \mathbb{R})$, $V(g S)=g V(S)$
(see Section 2 of \cite{EskMas01} for more detail).
To obtain the asymptotics for a fixed pair $F=(f,I)$ with $f=(w_1,w_2,s_1,s_2)\in{\mathcal F}$
and $I$ an interval, $I\subset[0,\min(s_1,s_2)[$, let us define
$V_{F}(S)$ the set of vectors ${\bf v}\in{\mathbb{R}}^2$ defined by \eqref{vecteurs},
such that the triple of cylinders in the direction ${\bf v}$ belongs to
the family $f$, with $h_3\in I$.
Then along the same lines of the proof of Theorem 2.4 in \cite{EskMasSch01}, one
can show that when the height $\alpha$ of the barrier is irrational, the
translation surface $S$ of the barrier billiard is a stable $q$-fold torus cover and
\begin{equation}
\label{vb}
|V_F(S)\cap B(T)|\sim\pi\kappa_F T^2,
\end{equation}
where the constant $\kappa_F$ is given by the Siegel-Veech formula
\begin{equation}
\label{svf2}
\frac{1}{\tilde{\mu}({\mathcal P}_q(1,1))}\int_{{\mathcal P}_q(1,1)}\hat{\varphi_F}\ d\tilde{\mu}
=\kappa_F\int_{{\mathbb{R}}^2}\varphi_F,
\end{equation}
with $\hat{\varphi_F}$ the Siegel-Veech transform
\begin{equation}
\label{svfF}
\hat{\varphi_F}(S)=\sum_{v\in V_F(S)}\varphi(v),
\end{equation}
for some continuous compactly supported $\varphi:{\mathbb{R}}^2\to\mathbb{R}$.
\subsection{Asymptotics for a family of periodic orbits}
Following the steps leading from the Siegel-Veech formula
\eqref{quadraticasymptotics1}-\eqref{quadraticasymptotics3}
to the asymptotics \eqref{eskin1}-\eqref{eskin2} in \cite{EskMasSch01},
we can now derive asymptotics for
the number of cylinders in each family $(f, I)$. Recall that
${\mathcal N}^{(q)}_{f,I}(l)$ is the number of directions ${\bf v}$ belonging
to a family characterized by the numbers $f=(w_1, w_2, s_1, s_2)$,
with a height $h_3\in I$, and such that $l_p$ given by Equation
\eqref{lgbarr} is less than $l$. (Note that ${\mathcal N}^{(q)}_{f,I}(l)$
is a number of directions and not a number of cylinders.)
Let $\rho_{f, I}(l)$ be the corresponding density.
According to Equation \eqref{vb}, ${\mathcal N}^{(q)}_{f,I}(l)$ is proportional
to $l^2$; we define the constant $c_{f,I}$ by
\begin{equation}
\label{eskin1f}
{\mathcal N}^{(q)}_{f,I}(l)\sim c_{f,I}\frac{\pi l^2}{16{\mathcal A}}.
\end{equation}
The proof leading to the asymptotics for ${\mathcal N}^{(q)}_{f,I}(l)$ is
essentially the same as the proof in \cite{EskMasSch01}, section 4.4,
provided we replace the counting functions of the cylinders
in \cite{EskMasSch01} by counting functions of directions in which
the cylinders belong to the family $(f, I)$ we are interested in.
We take $\varphi$ to be the characteristic function of a disc of radius
$\epsilon$ in ${\mathbb{R}}^2$. Therefore its Siegel-Veech transform $\hat{\varphi}$,
as defined by \eqref{svfF},
counts the number of directions on $S$ in which the cylinders
belong to family $(f,I)$ and such that $l_p<\epsilon$.
For $\epsilon$ small enough, the Siegel-Veech formula \eqref{svf2} is
equivalent to
\begin{equation}
\label{svf3}
\pi\epsilon^2 c_{f, I}=\zeta(2)\frac{1}{\tilde{\mu}({\mathcal P}_q(1,1))}
\int_{{\mathcal P}_q(1,1)}\tilde{\varphi_F}\ d\tilde{\mu},
\end{equation}
where $\zeta$ is the Riemann Zeta function and
\begin{equation}
\tilde{\varphi}_{f, I}(S)=\left\{
\begin{array}{cl}
1&\textrm{ \ \ \ if the cylinders in the horizontal direction belong}\cr
&\textrm{to the family $f$, if $h_3\in I$ and if}\ ||{\bf v}||<\epsilon\cr
0&\textrm{ \ \ \ otherwise.}
\end{array}
\right.
\end{equation}
We define $\chi_{f, I}:{\mathbb{R}}^2\mapsto\mathbb{R}$ by
\begin{equation}
\chi_{f, I}(v)=\left\{
\begin{array}{cl}
1&\textrm{ \ \ \ if the cylinders in the horizontal direction belong}\cr
&\textrm{to the family $f$, if $h_3\in I$ and if}\ ||{\bf v}||<\epsilon\sqrt{q}\cr
0&\textrm{ \ \ \ otherwise.}
\end{array}
\right.
\end{equation}
Following \cite{EskMasSch01}, we parametrize ${\mathcal P}_q(1,1)$ and perform
the integration in \eqref{svf3}. Part of it can be related to the integral
over $\chi_{f, I}$, which is $\int_{{\mathbb{R}}^2}\chi_{f, I}(v)\ dv=\pi\epsilon^2 q$.
The integration yields
\begin{equation}
\int_{{\mathcal P}_q(1,1)}\tilde{\varphi_F}\ d\tilde{\mu}=
\frac{\pi\epsilon^2 q}{q\zeta(2)}\sum_{r|(w_1, w_2)}
\frac{\mu(r)}{r}w_1 w_2 (w_1+w_2)|I|.
\end{equation}
From \cite{EskMasSch01} we get $\tilde{\mu}({\mathcal P}_q(1,1))=N_q/q$, with
$N_q$ given by \eqref{nqp}. Equation \eqref{svf3} finally gives
\begin{equation}
\label{eskin2f}
c_{f,I}=\frac{q}{N_q}\sum_{r|(w_1, w_2)}\frac{\mu(r)}{r}w_1 w_2 (w_1+w_2)|I|,
\end{equation}
where $|I|$ is the length of the interval $I$.
Equation \eqref{eskin2f} shows that $c_{f,I}$ depends on $I$ only
through its length.
Is is therefore convenient to introduce the density of
directions $p=(M,N)$ corresponding to a
family $f$ and such that $l_p<l$ and $h_3=h$:
\begin{equation}
\label{eskin1fh}
{\mathcal N}^{(q)}_{f,h}(l)\sim c_{f,h}\frac{\pi l^2}{16{\mathcal A}},
\end{equation}
with $c_{f,h}$ given by
\begin{equation}
\label{eskin2fh}
c_{f,h}=\frac{q}{N_q}\sum_{r|q}\frac{\mu(r)}{r}w_1 w_2 (w_1+w_2)\theta_f(h)
\end{equation}
for any family $f=(w_1, w_2, s_1, s_2)$ of ${\mathcal F}$ and $h\in \mathbb{R}$.
The function $\theta_f$ is the characteristic function of the interval
$[0,\min(s_1,s_2)[$.
The density of primitive periodic orbit lengths for the family
$f\in{\mathcal F}$ and $h\in\mathbb{R}$
is
\begin{equation}
\label{rhoppf}
\rho_{pp,f, h}(l)\sim c_{f,h} \frac{3 l}{4\pi{\mathcal A}}.
\end{equation}
It is easy to verify that the expression \eqref{eskin1fh} of
${\mathcal N}^{(q)}_{f,h}(l)$
is consistent with the total number ${\mathcal N}^{(q)}(l)$ of pencils of
periodic orbits with length less than $l$. This comes from the fact
that any pencil of periodic orbits contributing to
${\mathcal N}^{(q)}(l)$ belongs to a certain family $f$ and has a length
$w_i l_{p}\leq l$, which implies that $l_{p}\leq l/w_i$. Therefore
\begin{eqnarray}
{\mathcal N}^{(q)}(l)&=&\sum_{f\in{\mathcal F}}\int dh\sum_{i=1}^{3}
{\mathcal N}^{(q)}_{f,h}(l/w_i)\nonumber\\
&\sim&\sum_{f\in{\mathcal F}}\int dh\ c_{f,h}\frac{\pi l^2}{16{\mathcal A}}
\left(\frac{1}{w_1^2}+\frac{1}{w_2^2}+\frac{1}{(w_1+w_2)^2}\right).
\end{eqnarray}
Using Equation \eqref{eskin2fh}, we obtain, after integration over $h$,
\begin{eqnarray}
{\mathcal N}^{(q)}(l)&\sim&\frac{\pi l^2}{16{\mathcal A}}
\frac{q}{N_q}\sum_{\genfrac{}{}{0pt}{}{(s_1,s_2)=1}{s_1 w_1+s_2 w_2=q}}
\sum_{r|(w_1,w_2)}\frac{\mu(r)}{r} w_1 w_2 (w_1+w_2)\nonumber\\
&&\times\min(s_1,s_2)
\left(\frac{1}{w_1^2}+\frac{1}{w_2^2}+\frac{1}{(w_1+w_2)^2}\right).
\end{eqnarray}
Making the substitution $w_i=r u_i$ and inverting the two sums, we get
exactly the expression given by Equations (\ref{eskin1}) and (\ref{eskin2}).
\section{Calculation of the form factor at $\tau=0$}
\label{calculff}
\subsection{Definitions}
\label{section2}
The spectrum $\{E_n, n\in\mathbb{N}\}$ of a quantum billiard can be described
by the density
\begin{equation}
d(E)\equiv\sum_n \delta(E-E_n).
\end{equation}
The two-point correlation form factor is defined as the Fourier
transform of the two-point correlation function of the density of states:
\begin{equation}
\label{formfactor}
K_2(\tau)=\int_{-\infty}^{\infty}\frac{d\epsilon}{\bar{d}}\langle
d(E+\epsilon/2)d(E-\epsilon/2)\rangle_{\textrm{c}}
e^{2 i \pi \bar{d} \tau\epsilon}.
\end{equation}
Here the product of the densities is averaged over an energy window of width
$\Delta E\gg 1/\bar{d}$ centered around $E=k^2$ and such that
$\Delta E \ll E$. If ${\mathcal A}$ is the area of the billiard,
$\bar{d}={\mathcal A}/4\pi$ is the non-oscillating part of the density of states.
The subscript c means that one only considers the connected part of the
correlation function. It can be argued that in the case of
pseudo-integrable systems, the leading term of the
semiclassical expansion of $K_2(\tau)$ at small argument ($\tau\to 0$)
is given in the diagonal approximation by the contribution of periodic orbits
only: $K_2(\tau)=K^{\textrm{diag}}(\tau)+O(\tau)$, with
\begin{equation}
K^{\textrm{diag}}(\tau)=\frac{1}{8\pi^2\bar{d}}
\sum_{p} \frac{|S_{p}|^2}{l_p}
\delta (l_{p}-4\pi k \bar{d} \tau)
\label{k0}
\end{equation}
(see \cite{BogGirSch01} for the derivation of this expression, based
on heuristic arguments). The sum is
performed over all pencils of periodic orbits $p$ of length $l_p$.
In general, there can be several pencils having exactly the same
length: in Equation \eqref{k0}, $S_p$ is the sum of the areas
occupied by all pencils having, when (possibly) multiply repeated,
a length $l_p$.
The aim of the present section is to calculate the
semiclassical form factor at small arguments \eqref{k0},
using the result (\ref{rhoppf}) for the distribution of pencils of
periodic orbits in the barrier billiard. Let us take a $C^{\infty}$,
compactly supported test function and integrate the distribution $K^{\textrm{diag}}(\tau)$
over $\tau$. If the density of periodic pencils depends linearly on $l$
(as is the case for the barrier billiard or the rectangular billiard), the
integration over families of periodic orbits yields
$K^{\textrm{diag}}(\tau)=\lambda\Theta(\tau)$, where $\lambda$ is a constant
and $\Theta$ the Heaviside step function. In such a case, we define
$\overline{K_2(0)}=\lambda$.
As an introduction, we first
deal with the simpler case of a rectangular billiard.
\subsection{Rectangular billiard}
In the case of the rectangular billiard, discussed in section
\ref{casrectangle}, the periodic orbits have lengths $n l_{pp}$,
where $l_{pp}$ is given by \eqref{lprectangle} with $(M,N)$ coprime,
and $n\in\mathbb{N}$ is the repetition number. The area of each pencil
of primitive periodic orbits $pp$ is $A_{pp}=4{\mathcal A}$.
When the sides $a$ and $b$ of the rectangle are incommensurable, there
is only one pencil of length $l_{pp}$ and therefore in Equation (\ref{k0})
$S_p=4{\mathcal A}$. Equation (\ref{k0}) becomes
\begin{equation}
K^{\textrm{diag}}(\tau)=\frac{1}{8\pi^2\bar{d}}
\sum_{pp}\sum_{n}\frac{|4{\mathcal A}|^2}{n^2 l_{pp}}
\delta (l_{pp}-4\pi k \bar{d} \tau/n)
\end{equation}
hence (using the fact that $\bar{d}={\mathcal A}/4\pi$ and turning the sum over
$pp=(M,N)$ with $M$ and $N$ coprime into an integral over $l$ with density
$\rho_{pp}(l)$)
\begin{equation}
\label{k2rec}
K^{\textrm{diag}}(\tau)=\frac{8{\mathcal A}}{\pi}\sum_n
\int_{0}^{\infty}dl\ \frac{1}{n^2 l}\rho_{pp}(l)
\delta (l-4\pi k \bar{d} \tau/n)
\end{equation}
The density $\rho_{pp}(l)$ of periodic orbits is given by Equation
(\ref{densiterectangle}) and yields $K^{\textrm{diag}}(\tau)=1$, as expected for
integrable systems.
\subsection{Barrier billiard}
In the case of the barrier billiard, the periodic orbits
have a length of the form $n w l_p$ with $l_p$ given by \eqref{lgbarr}: here
the primitive length is $w l_p$ and $n$ is the repetition number.
Two pencils of periodic orbits $p$ and $p'$ have the same length provided
there exist repetition numbers $n$ and $n'$ such that
$n w l_{p}=n' w' l_{p'}$. When $a$ and $b$ are incommensurable,
this implies $p=p'$, i.e. two pencils can
have same length only if they are in the
same direction. For a given direction $(M,N)$ with $M$ and $N$
coprime (which will now be labeled by $p$),
there are three cylinders of area ${\mathcal A}_i$ and
length $w_i l_{p}$, $1\leq i \leq 3$, and therefore
$w,w'$ belong to the set $\{w_1, w_2, w_1+w_2\}$. Equation (\ref{k0}) becomes
\begin{equation}
\label{dpo2barr}
K^{\textrm{diag}}(\tau)=\frac{1}{8\pi^2\bar{d}}
\sum_{p}\sum_{n}\frac{|S_{p,n}|^2}{n l_p}
\delta (n l_{p}-4\pi k \bar{d} \tau)
\end{equation}
where $l_p$ is given by \eqref{lgbarr} and $S_{p,n}$ is the sum over
the ${\mathcal A}_i$ corresponding to a $w_i$ which divides $n$:
\begin{equation}
\label{sppn}
S_{p,n}=\sum_{i=1}^{3}{\mathcal A}_i \delta_{w_i|n},
\end{equation}
with $\delta_{r|t}=1$ if $r$ divides $t$, 0 otherwise.
Each area ${\mathcal A}_i$ is equal to $(2b/M)h_i\times (w_i l_{p})\cos\varphi_{p}$ ($\varphi_{p}$
is the angle between the orbit and the horizontal). This can be rewritten as
\begin{equation}
\label{area}
{\mathcal A}_i=\frac{4{\mathcal A}}{q}h_i w_i
\end{equation}
(note that since $\sum_ih_iw_i=s_1w_1+s_2 w_2=q$, one has $\sum_i{\mathcal A}_i=4{\mathcal A}$, i.e. the total area
of the translation surface, as expected). Therefore $S_{p,n}$ only depends of the five numbers
$f=(w_1, w_2,s_1,s_2)$ and $h_3$, and can be rewritten:
\begin{equation}
S_{f,h_3,n}=\frac{4{\mathcal A}}{q}\left[(s_1-h_3)w_1\delta_{w_1|n}+(s_2-h_3)w_2\delta_{w_2|n}
+h_3(w_1+w_2)\delta_{(w_1+w_2)|n}\right].
\end{equation}
The sum (\ref{dpo2barr}) over all periodic orbits can be
partitioned into sums running over primitive pencils of periodic orbits $p(f,h)$ belonging to
a family $f$ with a height of the long cylinder in $[h, h+dh[$; \eqref{dpo2barr}
becomes
\begin{equation}
K^{\textrm{diag}}(\tau)=\sum_{f\in{\mathcal F}}\int dh\sum_{p(f,h)}\sum_{n}
\frac{|S_{p,n}|^2}{8\pi^2 n^2 l_{p} \bar{d}}
\delta(l_{p}-\frac{4\pi k \bar{d} \tau}{n}).
\end{equation}
Each of the sums corresponding to a family $f$
can be replaced, as in (\ref{k2rec}), by an integral with density
$\rho_{pp,f,h}(l)$, and
$S_{p,n}$ by $S_{f,h,n}$:
\begin{equation}
K^{\textrm{diag}}(\tau)=\sum_{f\in{\mathcal F}}\int dh\sum_{n}
\frac{|S_{f,h,n}|^2}{8\pi^2 n^2\bar{d}}
\int_{0}^{\infty}dl\ \frac{\rho_{pp,f,h}(l)}{l}
\delta(l-\frac{4\pi k \bar{d} \tau}{n}).
\end{equation}
Replacing the density $\rho_{pp,f,h}$ by its expression (\ref{rhoppf}), the
integration over $l$ becomes straightforward and yields $K^{\textrm{diag}}(\tau)=\overline{K_2(0)}\Theta(\tau)$,
where
\begin{eqnarray}
\overline{K_2(0)}=\frac{1}{q^2}\frac{6}{\pi^2}\sum_{n}\frac{1}{n^2}\sum_{f\in{\mathcal F}}\int dh
\left[(s_1-h)w_1\delta_{w_1|n}\right.\\
\nonumber
+\left.(s_2-h)w_2\delta_{w_2|n}
+h(w_1+w_2)\delta_{(w_1+w_2)|n}\right]^2 c_{f,h}
\end{eqnarray}
(we have used the fact that $\bar{d}={\mathcal A}/4\pi$). Expanding the square, we can perform the
summation over $n$, using the identity
\begin{equation}
\sum_{n=1}^{\infty}\frac{\delta_{w_1|n}\delta_{w_2|n}}{n^2}=\frac{\pi^2}{6}
\frac{\gcd(w_1,w_2)^2}{w_1^2 w_2^2}.
\end{equation}
The form factor can therefore be written, after simplifications using the fact that
$\gcd(w_1,w_1+w_2)=\gcd(w_2,w_1+w_2)=\gcd(w_1,w_2)$, as
\begin{eqnarray}
\overline{K_2(0)}&=&\frac{1}{q^2}\sum_{f\in{\mathcal F}}\int dh\ c_{f,h}
\left[3h^2-2\left(s_1+s_2+q\frac{\gcd(w_1,w_2)^2}{w_1 w_2 (w_1+w_2)}\right)h\right.\nonumber\\
&+&\left.s_1^2+s_2^2+2 s_1 s_2\frac{\gcd(w_1,w_2)^2}{w_1 w_2}\right].
\end{eqnarray}
Replacing the weight $c_{f,h}$ by its expression (\ref{eskin2fh}), we can easily perform the
integration over $h$, which consists of terms of the form
\begin{equation}
\int_{0}^{\min(s_1, s_2)}dh\ h^{\nu}=\frac{\min(s_1,s_2)^{\nu+1}}{\nu+1}
\end{equation}
for $\nu=0,1,2$. The form factor becomes
\begin{eqnarray}
\overline{K_2(0)}&=&\frac{1}{q N_q}\sum_{\genfrac{}{}{0pt}{}{(s_1,s_2)=1}{s_1 w_1+s_2 w_2=q}}
\sum_{r|(w_1,w_2)}\frac{\mu(r)}{r} w_1 w_2 (w_1+w_2)
\left[\min(s_1,s_2)^3\right.\nonumber\\
&-&\left(s_1+s_2+q\frac{\gcd(w_1,w_2)^2}{w_1 w_2 (w_1+w_2)}\right)\min(s_1,s_2)^2\nonumber\\
&+&\left.\left(s_1^2+s_2^2+2 s_1 s_2\frac{\gcd(w_1,w_2)^2}{w_1 w_2}\right)\min(s_1,s_2)\right].
\end{eqnarray}
This sum can be evaluated with some cumbersome arithmetic manipulations; the calculation is given in
the Appendix, and the final result is unexpectedly simple:
\begin{equation}
\label{resultatfinal}
\overline{K_2(0)}=\frac{1}{2}+\frac{1}{q}.
\end{equation}
There are several comments to make concerning this value. First, it is close
to the result corresponding to semi-Poisson statistics $\overline{K_2(0)}=1/2$
\cite{BogGirSch01}. This result is not
valid for $q=2$, since in that case there is an additional symmetry in the billiard,
with respect to the barrier, and the spectrum has to be desymmetrized. The calculation in
this case has been done in \cite{Wie02} for a height of the barrier equal to $b/2$ (half the height of
the rectangle), and yields $\overline{K_2(0)}=1/2$.
The calculation for $q=2$ and a barrier with any height has been done in \cite{TheseGir02} using
a different method, and also yields $\overline{K_2(0)}=1/2$.
The result (\ref{resultatfinal}) is similar to previously obtained results
\cite{BogGirSch01} for rational polygonal billiards having the Veech property.
For instance for triangular billiards with angles $(\pi/2, \pi/n, \pi/2-\pi/n)$
the form factor at the origin was found to be between $1/3$ and $3/5$ \cite{BogGirSch01}.
Here the form factor lies between $1/2$ and $5/6$, which again is close to the
semi-Poisson result.
\section*{Acknowledgments}
The author thanks Professor Alex Eskin for helpful discussions.
The funding of the Leverhulme trust and the Department of Physics of
the University of Bristol, where most of this work has been done,
are gratefully acknowledged for their support. The funding of
post-doctoral CNRS fellowship and the theoretical physics laboratory
of the University of Toulouse have made the completion of this work
possible.
\section*{Appendix}
In this appendix, we want to evalute the quantity
\begin{equation}
\label{depart}
\overline{K_2(0)}=\frac{1}{q N_q}\sum_{\genfrac{}{}{0pt}{}{(s_1,s_2)=1}{s_1 w_1+s_2 w_2=q}}
\sum_{r|(w_1,w_2)}\frac{\mu(r)}{r}f(s_1, s_2, w_1, w_2,q),
\end{equation}
where
\begin{eqnarray}
\label{homogeneite}
f(s_1, s_2, w_1, w_2,q)&=& w_1 w_2 (w_1+w_2)\left[\min(s_1,s_2)^3\right.\nonumber\\
&-&\left.\left(s_1+s_2+q\frac{\gcd(w_1,w_2)^2}{w_1 w_2 (w_1+w_2)}\right)\min(s_1,s_2)^2\right.\nonumber\\
&+&\left.\left(s_1^2+s_2^2+2 s_1 s_2\frac{\gcd(w_1,w_2)^2}{w_1 w_2}\right)\min(s_1,s_2)\right].
\end{eqnarray}
The function $f$ is homogeneous, in the sense that it verifies
\begin{equation}
f(s_1, s_2,\lambda w_1,\lambda w_2, \lambda q)=f(\lambda s_1,\lambda s_2,w_1, w_2,\lambda q)
=\lambda^3 f( s_1, s_2,w_1,w_2,q).
\end{equation}
In (\ref{depart}), the first sum goes over all integers $w_i\geq 1$ and $s_i\geq 1$, $i=1,2$,
verifying $s_1 w_1+s_2 w_2=q$ and $\gcd(s_1,s_2)=1$. The number $N_q$ is given by (\ref{nqp}).
The first step is to exchange the sum over $(s_i, w_i)$ and the sum over $r$ in (\ref{depart}), and
substitute $w_i=r u_i$: using the homogeneity of $f$, we get
\begin{equation}
\label{stari}
\overline{K_2(0)}=\frac{1}{q N_q}\sum_{r|q}\mu(r)r^2
\sum_{\genfrac{}{}{0pt}{}{(s_1,s_2)=1}{s_1 u_1+s_2 u_2=q/r}}f(s_1, s_2, u_1, u_2, \frac{q}{r}),
\end{equation}
To get rid of the co-primality condition on $(s_1, s_2)$
we use the exclusion-inclusion principle, which for any function $\varphi$ gives
\begin{equation}
\label{excluinclu}
\sum_{(s,s')=1}\varphi(s,s')=\sum_{s, s'=1}^{\infty}\sum_{t=1}^{\infty}\mu(t)\varphi(t s, t s').
\end{equation}
This allows to rewrite the form factor as
\begin{eqnarray}
\overline{K_2(0)}&=&\frac{1}{q N_q}\sum_{r|q}\mu(r)r^2\sum_{t=1}^{\infty}\mu(t)
\hspace{-.5cm}\sum_{t s_1 u_1+t s_2 u_2=q/r}f(t s_1, t s_2,u_1, u_2, \frac{q}{r})\hspace{-.5cm}\\
&=&\frac{1}{q N_q}\sum_{r|q}\mu(r)r^2\sum_{t|q}\mu(t)t^3
\hspace{-.5cm}\sum_{s_1 u_1+s_2 u_2=q/(rt)}f(s_1, s_2, u_1, u_2, \frac{q}{r t})\hspace{-.5cm}\nonumber
\end{eqnarray}
Here the sum over $t$ from 1 to $\infty$ has been replaced by a sum over $t|q$ since for all the other
values of $t$ there is no value of $(s_1, s_2,u_1, u_2)$ fulfilling the condition $t s_1 u_1+t s_2 u_2=q/r$.
Again, the homogeneity of $f$ (Equation (\ref{homogeneite})) has been used. Setting $d=r t$
we get
\begin{eqnarray}
\label{starf}
\overline{K_2(0)}=\frac{1}{q N_q}\sum_{d|q}\left(\sum_{t|d}\mu(t)\mu(\frac{d}{t}) d^2 t\right)
\sum_{s_1 u_1+s_2 u_2=q/d}f(s_1, s_2, u_1, u_2, \frac{q}{d}).\nonumber
\end{eqnarray}
We need to evaluate
\begin{equation}
\label{k0total}
\overline{K_2(0)}=\frac{1}{q N_q}\sum_{d|q}\left(\sum_{t|d}\mu(t)\mu(\frac{d}{t}) d^2 t\right)(G_{q/d}+H_{q/d}),
\end{equation}
where
\begin{eqnarray}
G_n&\equiv&\hspace{-.5cm} \sum_{s_1 u_1+s_2 u_2=n}\hspace{-.5cm}u_1 u_2 (u_1+u_2)\left[\min(s_1,s_2)^3\right.\\
&-&\left.\left(s_1+s_2\right)\min(s_1,s_2)^2+\left(s_1^2+s_2^2+2 s_1 s_2\right)\min(s_1,s_2)
\right]\nonumber
\end{eqnarray}
and
\begin{eqnarray}
H_n&\equiv&\hspace{-.5cm} \sum_{s_1 u_1+s_2 u_2=n}\hspace{-.5cm}u_1 u_2 (u_1+u_2)
\left[-q\frac{\gcd(u_1,u_2)^2}{u_1 u_2 (u_1+u_2)}\min(s_1,s_2)^2\right.\nonumber\\
&&\hspace{3cm}+\left.2 s_1 s_2\frac{\gcd(u_1,u_2)^2}{u_1 u_2}\min(s_1,s_2)\right].
\end{eqnarray}
The quantities $G_n$ and $H_n$ will be evaluated separately.
This evaluation will require the use of a theorem proved in
\cite{HuaOuSpeWil02}:\\
{\bf Theorem.} Let $f:\mathbb{Z}$$^{4}\rightarrow \mathbb{C}$ such that
\begin{equation}
f(a,b,x,y)-f(x,y,a,b)=f(-a, -b, x,y)-f(x,y,-a,-b)
\end{equation}
for all integers $a,b,x$ and $y$. Then for $n\in\mathbb{N}$, $n\geq 1$,
\begin{eqnarray}
\label{theoreme}
\sum_{\genfrac{}{}{0pt}{}{a,b,x,y\geq 1}{a x+b y=n}}
\left[f(a,b,x,-y)-f(a,-b,x,y)+f(a,a-b,x+y,y)\right.\nonumber\\
-\left. f(a,a+b,y-x,y)+f(b-a, b, x, x+y)-f(a+b, b, x, x-y)\right]\nonumber\\
=\sum_{d|n}\sum_{x=1}^{d-1}\left[
f(0, \frac{n}{d}, x, d)+f(\frac{n}{d},0,d, x)+f(\frac{n}{d},\frac{n}{d},d-x,-x)\right.\nonumber\\
-\left. f(x, x-d, \frac{n}{d},\frac{n}{d})-f(x,d,0,\frac{n}{d})-f(d,x,\frac{n}{d},0)\right].
\end{eqnarray}
\subsection*{a. Evaluation of $G_n$}
We can immediately point out that the identity
\begin{equation}
\min(s_1,s_2)^2-(s_1+s_2)\min(s_1,s_2)=-s_1 s_2,
\end{equation}
valid for any integers $s_1$ and $s_2$, allows to simplify $G_n$. We now need to evaluate
the sum
\begin{equation}
G_n=\hspace{-.5cm}\sum_{s_1 u_1+s_2 u_2=n}\hspace{-.5cm}
u_1 u_2 (u_1+u_2)\min(s_1,s_2)\left(s_1^2+s_2^2-s_1 s_2\right).
\end{equation}
for any integer $n$. Writing $\min(a,b)=\frac{1}{2}(a+b-|a-b|)$, we have
\begin{eqnarray}
\label{gn2sommes}
G_n&=&\frac{1}{2}\sum_{s_1 u_1+s_2 u_2=n}\hspace{-.5cm}u_1 u_2\left[
s_1^3 u_2+s_2^3 u_1-\frac{1}{3}(s_1^3 u_1+s_2^3 u_2)\right.\\
&-&\left.\vphantom{\frac{1}{2}}(u_1+u_2)(s_1^2+s_2^2-s_1 s_2)|u_1-u_2|\right]
+\frac{2}{3}\hspace{-.5cm}\sum_{s_1 u_1+s_2 u_2=n}\hspace{-.5cm}u_1 u_2(s_1^3 u_1+s_2^3 u_2).\nonumber
\end{eqnarray}
The first sum in (\ref{gn2sommes}) can be evaluated by applying Theorem (\ref{theoreme}) to the function
\begin{equation}
f(a,b,x,y)=\frac{1}{3}\left(x y-\frac{|x y|}{2}\right)\left|(a-b)(x-y)\right|(a^2+b^2-a b)
\end{equation}
and is equal to
\begin{equation}
\frac{n^2 (n-1)}{18}\sum_{d|n}d.
\end{equation}
The second sum in (\ref{gn2sommes}) can be evaluated by applying Theorem (\ref{theoreme}) to the function
$f(a,b,x,y)=b^2y^4-b^2 x y^3$ (see \cite{HuaOuSpeWil02}). It gives
\begin{equation}
\frac{4}{3}\sum_{a x+b y=n}\hspace{-.3cm}a^3 x^2 y
=\frac{n^2}{18}\sum_{d|n}\left(3d^3+(1-4n)d\right).
\end{equation}
Finally we get
\begin{equation}
\label{gn}
G_n=\frac{n^2}{6}\sum_{d|n}\left(d^3-n d\right).
\end{equation}
If we now evaluate the quantity
\begin{eqnarray}
\sum_{s_1 u_1+s_2 u_2=n}\hspace{-.5cm}
u_1 u_2 (u_1+u_2)\min(s_1,s_2)=\frac{1}{2}\sum_{s_1 u_1+s_2 u_2=n}\hspace{-.5cm}u_1 u_2
\left[s_1 u_2+s_2 u_1\vphantom{\frac{1}{2}}\right.\nonumber\\
\left.-\frac{1}{3}(s_1 u_1+s_2 u_2)
-(u_1+u_2)|u_1-u_2|\right]
+\frac{2n}{3}\hspace{-.5cm}\sum_{s_1 u_1+s_2 u_2=n}\hspace{-.5cm}u_1 u_2,
\end{eqnarray}
the first sum is given by Theorem (\ref{theoreme}) applied to the function
\begin{equation}
f(a,b,x,y)=\frac{1}{3}\left(x y-\frac{|x y|}{2}\right)\left|(a-b)(x-y)\right|
\end{equation}
and the second one is given by Theorem (\ref{theoreme}) applied to the
function\\
$f(a,b,x,y)=n x y/3$; altogether, this gives
\begin{equation}
\sum_{s_1 u_1+s_2 u_2=n}\hspace{-.5cm}
u_1 u_2 (u_1+u_2)\min(s_1,s_2)=\frac{n}{3}\sum_{d|n}\left(d^3-n d\right).
\end{equation}
Together with Equation (\ref{gn}) we get
\begin{equation}
G_{n}=\frac{n}{2}\sum_{s_1 u_1+s_2 u_2=n}\hspace{-.5cm}u_1 u_2 (u_1+u_2)\min(s_1,s_2).
\end{equation}
\subsection*{b. Evaluation of $H_n$}
We want to evaluate
\begin{eqnarray}
\label{departH}
H_n&=&\hspace{-.5cm}\sum_{s_1 u_1+s_2 u_2=n}\hspace{-.5cm} u_1 u_2 (u_1+u_2)\min(s_1,s_2)\hspace{4cm}\nonumber\\
&&\hspace{2cm}\left(-n\frac{\min(s_1,s_2)}{ u_1 u_2 (u_1+u_2)}+\frac{2 s_1 s_2}{u_1 u_2}\right)
\gcd(u_1, u_2)^2
\end{eqnarray}
for any integer $n$. Summing over all the possible values $r$ of the $\gcd$ of $u_1$ and $u_2$, and
substituting $u_i=r v_i$, we have
\begin{equation}
H_n=\sum_{r|n}\hspace{-.2cm}
\sum_{\genfrac{}{}{0pt}{}{(v_1,v_2)=1}{s_1 v_1+s_2 v_2=n/r}}\hspace{-.2cm}
r^3 v_1 v_2 (v_1+v_2)\min(s_1,s_2)
\left(-\frac{n}{r}\frac{\min(s_1,s_2)}{v_1 v_2 (v_1+v_2)}+\frac{2 s_1 s_2}{v_1 v_2}\right).
\end{equation}
Then, as before, the co-primality condition can be expressed by a sum over $t$ (see
Equation (\ref{excluinclu})). Restricting the sum over $t$ as before,
we get
\begin{eqnarray}
H_n&=&\sum_{r|n}\sum_{t|n}\mu(t)r^3 t \hspace{-.5cm}\sum_{s_1 v_1+s_2 v_2=n/(r t)}\hspace{-.5cm}
v_1 v_2 (v_1+v_2)\min(s_1,s_2)\hspace{2cm}\nonumber\\
&&\hspace{5cm}\left(-\frac{n}{r t}\frac{\min(s_1,s_2)}{v_1 v_2 (v_1+v_2)}+\frac{2 s_1 s_2}{v_1 v_2}\right).
\end{eqnarray}
Setting $d=rt$ we get
\begin{eqnarray}
\label{arriveeH}
H_n&=&\sum_{d|n}\left(\sum_{t|d}\mu(t)\frac{d^3}{t^2}\right)\sum_{s_1 v_1+s_2 v_2=n/d}\hspace{-.5cm}
v_1 v_2 (v_1+v_2)\min(s_1,s_2)\hspace{2cm}\nonumber\\
&&\hspace{5cm}\left(-\frac{n}{d}\frac{\min(s_1,s_2)}{v_1 v_2 (v_1+v_2)}+\frac{2 s_1 s_2}{v_1 v_2}\right).
\end{eqnarray}
Let us now evaluate, for any integer $m$, the quantity
\begin{eqnarray}
K_m&=&\sum_{s_1 v_1+s_2 v_2=m}v_1 v_2 (v_1+v_2)\min(s_1,s_2)
\left(-m\frac{\min(s_1,s_2)}{v_1 v_2 (v_1+v_2)}+\frac{2 s_1 s_2}{v_1 v_2}\right)\nonumber\\
&=&-m\hspace{-.5cm}\sum_{s_1 v_1+s_2 v_2=m}\hspace{-.5cm}\min(s_1,s_2)^2
+2\hspace{-.5cm}\sum_{s_1 v_1+s_2 v_2=m}\hspace{-.5cm}s_1 s_2 (v_1+v_2)\min(s_1,s_2).
\end{eqnarray}
Let
\begin{eqnarray}
L_m&=&\hspace{-.5cm}\sum_{s_1 v_1+s_2 v_2=m}\hspace{-.5cm}(2 s_1 s_2-v_1 v_2)(v_1+v_2)\min(s_1,s_2)\\
&=&\hspace{-.5cm}\sum_{s_1 v_1+s_2 v_2=m}\hspace{-.5cm}
(2 v_1 v_2(s_1+s_2)\min(v_1,v_2)-v_1 v_2(v_1+v_2)\min(s_1,s_2))\nonumber
\end{eqnarray}
after exchanging $(s_1, s_2)$ and $(v_1, v_2)$ in the first half of the right member.
Writing $\min(a,b)=\frac{1}{2}(a+b-|a-b|)$ and applying Theorem (\ref{theoreme}) to
the function
\begin{equation}
f(a,b,x,y)=-\frac{1}{2}\left|a b (a-b)(x-y)\right|
\end{equation}
one gets
\begin{equation}
L_m=m\sum_{d|m}\sum_{x=1}^{d-1}x(d-x).
\end{equation}
Applying Theorem (\ref{theoreme}) to
the function
\begin{equation}
f(a,b,x,y)=\frac{a b-\left|a b\right|}{2}
\end{equation}
one gets
\begin{equation}
\sum_{s_1 v_1+s_2 v_2=m}\hspace{-.5cm}\min(s_1,s_2)^2=\sum_{d|m}\sum_{x=1}^{d-1}x(d-x).
\end{equation}
This proves that
\begin{equation}
K_m=\hspace{-.5cm}\sum_{s_1 v_1+s_2 v_2=m}\hspace{-.5cm}v_1 v_2(v_1+v_2)\min(s_1,s_2)
\end{equation}
and therefore
\begin{equation}
H_n=\sum_{d|n}\left(\sum_{t|d}\mu(t)\frac{d^3}{t^2}\right)
\sum_{s_1 v_1+s_2 v_2=n/d}\hspace{-.5cm}v_1 v_2 (v_1+v_2)\min(s_1,s_2).
\end{equation}
\subsection*{c. Calculation of $\overline{K_2(0)}$}
The evaluation of (\ref{k0total}) will require to introduce the functions
\begin{equation}
f(n)=\frac{\mu(n)}{n}\ \ \ \ \ \ \ \textrm{and}\ \ \ \ g(n)=\sum_{d|n}\frac{\mu(d)}{d^2}
\end{equation}
For $f_1$ and $f_2$ two arithmetic functions, the Dirichlet convolution is defined by
\begin{equation}
f_1*f_2(n)=\sum_{d|n}f_1(d)f_2(\frac{n}{d}).
\end{equation}
Replacing the expressions found for $G_{q/d}$ and $H_{q/d}$ in Equation (\ref{k0total})
we get
\begin{eqnarray}
\label{deuxtermes}
\overline{K_2(0)}&=&\frac{1}{q N_q}\sum_{d|q}(f*\mu)(d)d^3
\left\{\sum_{s_1 u_1+s_2 u_2=q/d}\hspace{-.3cm}\left(\frac{q}{2d}\right)u_1 u_2 (u_1+u_2)\min(s_1,s_2)
\right.\nonumber\\
&+&\left.\sum_{d'|q/d}d'^3g(d')
\hspace{-.5cm}\sum_{s_1 u_1+s_2 u_2=q/dd'}\hspace{-.5cm}u_1 u_2 (u_1+u_2)\min(s_1,s_2)\right\}.
\end{eqnarray}
Rewriting the constant $N_q$ given by (\ref{nqp}), using the inclusion-exclusion principle
and following the steps from Equations (\ref{stari}) to (\ref{starf}), we get
\begin{equation}
N_q=\sum_{d|q}(f*\mu)(d)d^2
\hspace{-.5cm}\sum_{s_1 u_1+s_2 u_2=q/d}\hspace{-.5cm}u_1 u_2 (u_1+u_2)\min(s_1,s_2).
\end{equation}
We see that the first
term in (\ref{deuxtermes}) is equal to 1/2. If we set $\delta=d d'$, the second term gives
\begin{equation}
\label{2emeterm}
\frac{1}{q N_q}\sum_{\delta|q}(\frac{\delta}{d})^3\left(\sum_{d|\delta}(f*\mu)(d)g(\frac{\delta}{d})\right)
\sum_{s_1 u_1+s_2 u_2=q/\delta}\hspace{-.5cm}u_1 u_2 (u_1+u_2)\min(s_1,s_2).
\end{equation}
But
\begin{equation}
\sum_{d|\delta}(f*\mu)(d)g(\frac{\delta}{d})=[(f*\mu)*g](\delta)=[f*(\mu*g)](\delta)
\end{equation}
by associativity of Dirichlet convolution, and
\begin{equation}
(\mu*g)(\delta)=\frac{\mu(\delta)}{\delta^2}
\end{equation}
by Moebius inversion formula. Finally the term (\ref{2emeterm}) simplifies to $1/q$,
which completes the proof.
|
{
"timestamp": "2005-03-31T14:55:13",
"yymm": "0503",
"arxiv_id": "nlin/0503067",
"language": "en",
"url": "https://arxiv.org/abs/nlin/0503067"
}
|
\section{Introduction}
Quantum entanglement and quantum nonlocality are two striking
aspects of quantum mechanics. They are introduced by Einstein,
Podolsky, and Rosen (EPR) in their famous paper \cite{epr}. The
relationship between them has been paid much attention. They play
an essential role in the modern understanding of quantum
phenomena, and quantum information transmission and processing.
Bell proposed a remarkable inequality imposed by a local hidden
variable theory \cite{bel}, which enables a quantitative test on
quantum nonlocality. The quantum nonlocality test can be performed
on an entangled system composed of two coherent systems. This
entangled system can be used as a quantum entangled channel for
quantum information transfer. Numerous theoretical studies and
experimental demonstrations have been carried out to understand
nonlocal properties of quantum states. Various versions of Bell's
inequality \cite{chs,ch} are proposed. Gisin and Peres found pairs
of observable whose correlations violate Bell's inequality for a
discrete $N$-dimensional entangled state \cite{gis} . Banaszek and
W\'{o}dkiewicz studied Bell's inequality for continuous-variable
states in terms of Wigner representation in phase space based upon
parity measurement and displacement operation. Recently, Chen {\it
et al.} studied Bell's inequality of continuous-variable states
\cite{chen1} using their newly defined pseudospin Bell operators,
and they showed that the EPR state can maximally violate Bell's
inequality in their framework.
Recently, $N$-photon entangled states and their superposition
states are paid much attention. They are widely used to realize
quantum lithography \cite{kuan,kok,bjo}, super-resolving phase
measurements \cite{mit}, and quantum teleportation \cite{coc}. In
this letter, we study quantum entanglement and quantum nonlocality
of the following $N$-photon entangled states and their
superpositions
\begin{eqnarray}
\label{e1}
|\psi_{N m}\rangle && =\mathcal{N}_{m}[\cos\gamma|N-m\rangle_{1}|m\rangle_{2} \nonumber \\
&&+e^{i\theta_{m}}\sin\gamma|m\rangle_{1}|N-m\rangle_{2}],
\end{eqnarray}
where $m$ takes its values from zero to $N$, the normalization
factor is given by $\mathcal{N}_{m}^{-2}=1+\cos\theta_{m}\sin
2\gamma \delta_{N,2m}$ with $\gamma$ and $\theta_{m}$ being an
entanglement angle and a relative phase, respectively. We will
calculate the von Neumann entropy and study the Bell's inequality
for quantum states $|\psi_{N m}\rangle$ and their superposition
states. We will show that quantum superpositions for the two modes
may increase the amount of entanglement, and $N$-photon entangled
states can maximally violate Bell's inequality. This letter is
organized as follows. In Sec. II, we study quantum entanglement of
$N$-photon entangled states and their superposition states through
analyzing their von Neumann entropy. Quantum nonlocality of
$N$-photon entangled states and their superposition states is
investigated through discussing the violation of the Bell's
inequality in the pseudospin Bell-operator formalism developed by
Chen {\it et al.} \cite{chen1} in Sec. III. The last section is
devoted to summary and conclusion.
\section{Quantum entanglement for $N$-photon entangled states}
In this section, we study properties of quantum entanglement of
the $N$-photon entangled states given by Eq. (\ref{e1}). In order
to this, we consider the nontrivial case of $N\neq 2m$, in which
the normalization constant $\mathcal{N}_{m}=1$. The degree of the
entanglement can be described by the von Neumann entropy defined
by
\begin{eqnarray}
\label{e2}
E(\rho_{1})=-Tr_{1}(\rho_{1}log\rho_{1}),
\end{eqnarray}
where $\rho_{1}$ is the reduced density operator of the first
mode.
For the $N$-photon entangled states defined by Eq. (\ref{e1}) we
find the von Neumann entropy to be
\begin{eqnarray}
\label{e3}
E_{1}&=&-\cos^{2}\gamma\log\cos^2\gamma -
\sin^{2}\gamma\log\sin^2\gamma,
\end{eqnarray}
which indicates that quantum entanglement of the $N$-photon
entangled states given by Eq. (\ref{e1}) is independent of the
superposition phases $\theta_m$ and the total photon number of the
two modes $N$, and changes periodically with respect to the
entanglement angle $\gamma$. In particular, the amount of
entanglement reaches the maximal value of $E_{1}=1$ when the
entanglement angle takes values by $\gamma=k\pi + \pi/4$ with $k$
being an integer. In Fig. 1 we plot the change of the von Neumann
entropy with respect to the entanglement angle.
\begin{figure}[htp]
\center
\includegraphics[width=3.3in,height=2.1in]{fig1.eps}
\caption{ The von Neumann entropy of the $N$-photon entangled
state is plotted against the entanglement angle $\gamma$.}
\end{figure}
We then study quantum entanglement of superposition states based
on the $N$-photon entangled states given by Eq. (\ref{e1}).
Firstly, we consider a two-state superposition state defined by
\begin{eqnarray}
\label{e4}
|\Psi_{2}\rangle=\frac{1}{\sqrt{2}}(|\psi_{30}\rangle +
|\psi_{31}\rangle),
\end{eqnarray}
which leads to the von Neumann entropy
\begin{eqnarray}
\label{e5}
E_{2}=1-\cos^{2}\gamma\log\cos^2\gamma -
\sin^{2}\gamma\log\sin^2\gamma,
\end{eqnarray}
which implies that the amount of entanglement of the two-component
superposition state only depends on the entanglement angle
$\gamma$ of its basis. Especially, comparing Eq. (\ref{e5}) with
(\ref{e5}) we find that the difference of the amount of
entanglement between the two-component superposition state
(\ref{e4}) and the basis state (\ref{e1}) is a positive constant,
i.e., $E_2-E_1=1$. This indicates that the entanglement amount of
the superposition state is always larger than that of the basis
state for an arbitrary entanglement angle $\gamma$. In other
words, starting with basis states defined by (\ref{e1}) one can
construct quantum superposition states with larger amount of
entanglement than that of the basis states. Hence, we may conclude
that quantum superpositions for the two modes may increase the
amount of entanglement.
In order to further demonstrate the above idea of quantum
superpositions increasing quantum entanglement, in what follows we
take into account a multi-component quantum superposition state
consisting of $N$ bases with a fixed photon number $N$, but with
different distributions $m$,
\begin{eqnarray}
\label{e6}
|\psi_{N}\rangle&=&A\sum_{m=0}^{N}\alpha_{m}|\psi_{Nm}\rangle,
\end{eqnarray}
where $A$ is a normalization factor. It is straightforward to
express the $N$-component superposition state as the following
number-sum Bell state,
\begin{eqnarray}
\label{e7}
|\Psi_{N}\rangle&=&\sum_{m=0}^{N}d_{m}|N-m\rangle_{1}|m\rangle_{2},
\end{eqnarray}
which is the eigenstate of the number-sum Bell operators
$\hat{N}=\hat{N}_1+\hat{N}_2$ with $\hat{N}_i$ being the number
operators of the respective modes. The coefficients in Eq.
(\ref{e7}) are given by
\begin{eqnarray}
\label{e8}
d_{m}=A(\alpha_{m}\mathcal{N}_{m}\cos\gamma+
\alpha_{N-m}\mathcal{N}_{N-m}e^{i\theta_{N-m}}\sin\gamma),
\end{eqnarray}
where the normalization of the superposition state (\ref{e8})
implies that $d_{m}$ satisfy the condition
$\sum_{m=0}^{N}|d_{m}|^{2}=1$.
In order to calculate the von Neumann entropy of the superposition
state (\ref{e7}), we need the reduced density operator
\begin{eqnarray}
\label{e9}
\rho_{1}&=&\sum_{m=0}^{N}|d_{m}|^{2}|N-m\rangle_{1}\langle
N-m|,
\end{eqnarray}
which leads to the following von Neumann entropy
\begin{eqnarray}
\label{e10} E_{N}&=&-\sum_{m=0}^{N}|d_{m}|^{2}\log|d_{m}|^{2}.
\end{eqnarray}
\begin{figure}[htp]
\center
\includegraphics[width=3.5in,height=2.1in]{fig2.eps}
\caption{ The von Neumann entropy of multi-component
superposition states based on $|\psi_{Nm}\rangle$ is plotted
against the entanglement angle $\gamma$ for $N=1, 2, 3$, and $4$,
respectively.}
\end{figure}
For the sake of simplicity and without the loss of generality, we
consider the situation of equal-weight superposition in which we
choose the superposition coefficients $\alpha_{m}=1/\sqrt{N+1}$
and the relative phases $\theta_{m}=2\pi m/N$. In this case, we
have
\begin{eqnarray}
\label{e11}
|d_{m}|^{2}=A^2|\mathcal{N}_{m}|^{2}\{1+\cos[2\pi(N-m)/N]\sin2\gamma\},
\end{eqnarray}
where the normalization constant is given by
\begin{eqnarray}
\label{e12}
A^{-2}=\sum_{m=0}^{N}|\mathcal{N}_{m}|^{2}\{1+\cos[2\pi(N-m)/N]\sin2\gamma\}.
\end{eqnarray}
Then the von Neumann entropy of the superposition state can be
directly obtained through substituting (\ref{e11}) and (\ref{e12})
into (\ref{e10}). From Eqs. (\ref{e10})-(\ref{e12}) we can find
the maximal value of the von Neumann entropy of the superposition
state to be $E_{N,max}=\log_2(N+1)$ when $\gamma=k\pi/2$ with $k$
being an arbitrary integer. This implies that the maximal amount
of entanglement of the $N$-component superposition state only
depends on the total photon number $N$, and increases with
increasing the total photon number $N$. In order to clearly see
the influence of the number of components $N$ and the entanglement
angle $\gamma $, in Fig. 2 we plot the von Neumann entropy of the
superposition states when $N=1, 2, 3$, and $4$, respectively. From
Fig. 2 we can see that the entanglement amount increases with
increasing the number of components, i.e., the photon number $N$,
and changes periodically with respect to the entanglement angle
$\gamma $.
\section{Quantum nonlocality for $N$-photon entangled states}
In this section, we study quantum nonlocality of $N$-photon entangled states and
their superposition states through discussing the violation of the
Bell's inequality in the pseudospin Bell-operator formalism
developed by Chen and coworkers \cite{chen1}. Let us begin with a
brief review of the pseudospin-operator formalism \cite{chen1}.
For a single-mode boson field, the pseudospin operators can be
defined in terms of project operators in a Fock space in the
following form
\begin{eqnarray}
\label{e13} S_{z}&=&\sum_{n=0}^{\infty}[| 2n+1 \rangle\langle 2n+1
|-|2n\rangle\langle 2n
|],\nonumber\\
S_{-}&=&\sum_{n=0}^{\infty}[| 2n \rangle\langle 2n+1
|]=(S_{j+})^{\dag},
\end{eqnarray}
where $ |n\rangle$ are the usual Fock states of the boson mode.
The operator $S_{z}=-(-1)^{\hat{N}}$ with $\hat{N}$ being the
number operator and $(-1)^{N} $ being the parity operator, $S_{+}$
and $S_{-}$ being the ``parity-flip" operators. They satisfy the
commutation relations of the $su(2)$ Lie algebra
\begin{eqnarray}
\label{e14}
[S_{z},S_{ \pm}]=\pm 2S_{\pm},\hspace{0.3cm}
[S_{+},S_{-}]&=S_{z}.
\end{eqnarray}
For an arbitrary vector living on the surface of a unit sphere
$\vec{a}$ = ($\sin\theta_{a}\cos\varphi_{a},
\sin\theta_{a}\sin\varphi_{a}, \cos\theta_{a}$), we have the
following dot product
\begin{eqnarray}
\label{e15}
\vec{a}\cdot\vec{S}=S_{z}\cos\theta_{a}+\sin\theta_{a}(e^{i\varphi_{a}}S_{-}+e^{-i\varphi_{a}}S_{+}).
\end{eqnarray}
Then for a two-mode boson field, the Bell operator due to
Clauser, Horne, Shimony, and Holt (CHSH) \cite{cla} can be defined
by
\begin{eqnarray}
\label{e16} B&=& (\vec{a}\cdot\vec{S_{1}}) \otimes
(\vec{b}\cdot\vec{S_{2}})+(\vec{a}\cdot\vec{S_{1}})\otimes
(\vec{b'}\cdot\vec{S_{2}}) \nonumber \\
&&+(\vec{a'}\cdot\vec{S_{1}})\otimes
(\vec{b}\cdot\vec{S_{2}})-(\vec{a'}\cdot\vec{S_{1}})\otimes
(\vec{b'}\cdot\vec{S_{2}}),
\end{eqnarray}
where $ \vec{a'}, \vec{b}$, and $\vec{b'}$ are three unit vectors
similarly defined as $\vec{a}$, $\vec{S_{1}}$ and $\vec{S_{2}}$
are defined as in Eq. (\ref{e13}).
As well known, local hidden variable theories impose the Bell-CHSH
inequality $|\langle B \rangle|\leq 2$ where $\langle B \rangle$
is the mean value of the Bell operator with respect to a given
quantum state. However, in the quantum theory it is found that
$|\langle B \rangle|\leq 2\sqrt{2}$, which implies that the
Bell-CHSH inequality is violated. In particular, when $|\langle B
\rangle|= 2\sqrt{2}$ for a given quantum state, we say that the
Bell-CHSH inequality is maximally violated by the quantum state.
Quantum nonlocality of a quantum state can be described by the
violation of the Bell-CHSH inequality. The expectation value of
the Bell operator with respect to a quantum state $|\psi \rangle$
can be expressed in terms of the correlation functions as
\begin{eqnarray}
\label{e17}\langle B
\rangle&=&E(\theta_{a},\theta_{b})+E(\theta_{a},\theta_{b'})+E(\theta_{a'},\theta_{b})\nonumber
\\&&-E(\theta_{a'},\theta_{b'}),
\end{eqnarray}
where the correlation functions are defined by
\begin{equation}
\label{e18} E(\theta_{a},\theta_{b})=\langle
\psi|S_{\theta_{a}}^{(1)}\otimes S_{\theta_{b}}^{(2)}|\psi\rangle,
\end{equation}
with
\begin{equation}
\label{e19}
S_{\theta_{a}}^{(j)}=S_{jz}\cos\theta_{a}+S_{jx}\sin\theta_{a}.
\end{equation}
We now investigate quantum nonlocality of the $N$-photon
entangled state given by Eq. (\ref{e1}). For this multi-photon
entangled state we find the correlation function to be
\begin{eqnarray}
\label{e20}
E_{Nm}(\theta_{a},\theta_{b})&=&\mathcal{N}_{m}^{2}\{[(-1)^{N}+K(\theta_{m},\gamma)\delta_{N,2m}]\nonumber\\
&&\times\cos\theta_{a}\cos\theta_{b}
+\delta_{N,2m\pm 1}\nonumber\\
&&\times K(\theta_{m},\gamma)\sin\theta_{a}\sin\theta_{b}\},
\end{eqnarray}
where we have introduced the following effective state parameter
\begin{eqnarray}
\label{e21}K(\theta_{m},\gamma)=\cos\theta_{m}\sin 2\gamma,
\end{eqnarray}
which describes the effect of the basis state defined by
(\ref{e1}) on the correlation functions.
Making use of the correlation function (\ref{e20}), from Eq.
(\ref{e17}) we can get the expectation value of the Bell operator
for arbitrary values of all azimuthal angles $\theta_{a}$,
$\theta_{b}$, $\theta_{a'}$ and $\theta_{b'}$. As a concrete
example, we consider the situation of $\theta_{a}=0$,
$\theta_{a'}=\pi/2$ and $\theta_{b}=-\theta_{b'}$. In this case,
from Eqs. (\ref{e17}), (\ref{e20}) and (\ref{e21}) we can obtain
the expectation value of the Bell operator given by
\begin{eqnarray}
\label{e22}\langle B
\rangle&=&2\mathcal{N}_{m}^{2}\{[(-1)^{N}+K(\theta_{m},\gamma)\delta_{N,2m}]\cos\theta_{b}\nonumber\\
&&+\delta_{N,2m\pm 1}K(\theta_{m},\gamma)\sin\theta_{b}\}.
\end{eqnarray}
In order to observe quantum nonlocality of the $N$-photon
entangled state (\ref{e1}), we consider two different cases of
$N=2m$ and $N=2m \pm 1$, respectively. When $N=2m$, the quantum
state given by Eq. (\ref{e1}) is disentangled, and reduces to
\begin{eqnarray}
\label{e23}|\psi_{N
m}\rangle=\mathcal{N}_{m}(\cos\gamma+e^{i\theta_{m}}\sin\gamma)|m\rangle_{1}|m\rangle_{2}.
\end{eqnarray}
Making use of Eqs. (\ref{e17}), (\ref{e18}) and (\ref{e20}), we
obtain the expectation value of the Bell operator with respect to
the disentangled state (\ref{e23}) $\langle B
\rangle=2\cos\theta_{b}$, which means that $|\langle B
\rangle|\leq 2$. Hence, the unentangled state (\ref{e23}) cannot
produce a violation of Bell's inequality.
On the other hand, when $N=2m\pm 1$, the expectation value of the
Bell operator with respect to the state (\ref{e1}) is given by
\begin{eqnarray}
\label{e24}\langle B
\rangle=2[K(\theta_{m},\gamma)\sin\theta_{b}-\cos\theta_{b}],
\end{eqnarray}
which indicates that for a fixed entanglement angle $\gamma$ and a
fixed relative phase $\theta_m$, when $\theta_{b}=-\tan^{-1}K$,
the expectation value of the Bell operator $\langle B \rangle$
reaches its maximum given by
\begin{eqnarray}
\label{e25} \langle B
\rangle_{max}=2\sqrt{1+K^{2}(\theta_{m},\gamma)},
\end{eqnarray}
which implies that $\langle B \rangle_{max}\geq 2$ due to
$|K(\theta_{m},\gamma)|\leq1.$ Thus, the $N$-photon entangled
state always violates the Bell's inequality if
$K(\theta_{m},\gamma)\neq 0$. From Eqs. (\ref{e21})
and(\ref{e24}) we can see that the degree of violation of the
Bell's inequalities depends upon both the entangling angle
$\gamma$ and the relative phase $\theta_{m}$, and it changes
periodically with respect to both $\gamma$ and $\theta_{m}$. In
particular, we note that the relative phase $\theta_{m}$ seriously
affects the mean value of the Bell operator although it dos not
affect quantum entanglement of the quantum state given by
(\ref{e1}). It is easy to find that when $|\psi_{N m}\rangle$
reaches maximal entanglement i.e., $\gamma=\pi/4$, and
$\theta_{m}=0$, we have $K(\theta_{m},\gamma)=1$. Under these
conditions, we can reach the maximal violation of Bell's
inequality with $ \langle B \rangle_{max}=2\sqrt{2}$.
In Fig. 3 we plot the degree of the violation of the Bell's
inequality for $N$-photon entangled state against $\gamma$ and
$\theta_{m}$. From Fig. 3 we can see that the mean value of the
Bell operator with respect to the multi-photon entangled state (1)
changes periodically with both of the entanglement angle and the
relative phase, and the Bell's inequality is always violated for
arbitrary values of the entanglement angle and the relative phase
except $\gamma=k\pi/2$ and $\theta=(2k+1)\pi/2$ with $k$ is an
integer.
\begin{figure}[htp]
\center
\includegraphics[width=3.3in,height=3.1in]{fig3.eps}
\caption{The degree of the maximal violation of the Bell's
inequality for $N$-photon entangled state $\langle B
\rangle_{max}$ is plotted against the entanglement angle $\gamma$
and the relative phase $\theta_{m}$. }
\end{figure}
Finally, we consider quantum nonlocality of quantum
superposition states which is formed using the multi-photon
entangled state (\ref{e1}). In order to calculate the expectation
value of the Bell operator with respect to the superposition
states based on entangled states given by (\ref{e1}), we need the
following correlation function
\begin{eqnarray}
\label{e26} E_{m m'}=\langle\psi_{N m}|S_{\theta_{a}}^{(1)}\otimes
S_{\theta_{b}}^{(2)}|\psi_{N m'}\rangle,
\end{eqnarray}
which is given by the following expression
\begin{eqnarray}
\label{e27}E_{m
m'}&=&\mathcal{N}_{m}\mathcal{N}_{m'}^{*}\cos\theta_{a}\cos\theta_{b}\nonumber
\\
&&\times \left \{(-1)^{N}\delta_{m,m'}\delta_{N,m+m'}\cos\gamma
\sin\gamma \right.\nonumber\\
& &\times [\cos^{2}\gamma+e^{i (\theta_{m'}-\theta_{m})} \sin^{2}\gamma] \nonumber\\
&& + (-1)^{m+m'}(e^{i \theta_{m'}} +e^{-i\theta_{m}})\}\nonumber\\
&& + \mathcal{N}_{m}\mathcal{N}^*_{m'}\sin\theta_{a}\sin\theta_{b}
\nonumber \\
&& \times \left\{[\cos^{2}\gamma +
e^{i(\theta_{m'}-\theta_{m})}\sin^{2}\gamma]\delta_{m,m'\pm 1}
\right. \nonumber\\
&& \left. + \cos \gamma \sin
\gamma(e^{i\theta_{m'}}+e^{-i\theta_{m}})\delta_{N,m+m'\pm
1}\right \}.
\end{eqnarray}
As a simple example of analyzing quantum nonlocality of
superposition states, we consider the following two-component
superposition state,
\begin{eqnarray}
\label{e28}|\psi_{3}\rangle=C(\alpha_{0}|\psi_{30}\rangle+\alpha_{1}|\psi_{31}\rangle),
\end{eqnarray}
where $C$ is the normalization constant. Making use of Eq.
(\ref{e27}), we can obtain the expectation value of the Bell
operator for this superposition state
\begin{eqnarray}
\label{e29}\langle B \rangle&=&-2\cos\theta_{b} +
4\sin\theta_{b}(|\alpha_{0}|^{2}+|\alpha_{1}|^{2})^{-1}\nonumber
\\&&\times Re[\alpha_{0}^{*}\alpha_{1}(\cos^{2}\gamma +e^{i
(\theta_{1}-\theta_{0})} \sin^{2}\gamma)],
\end{eqnarray}
which indicates that $\langle B \rangle$ depends upon not only the
superposition coefficients $\alpha_{0},\alpha_{1}$, the azimuthal
angles $\theta_{a}, \theta_{b}$, but also upon the state
parameters of the basis states given by Eq. (\ref{e1}),
$\theta_{0}, \theta_{1}$ and $\gamma$.
In order to observe the maximal violation of the Bell inequality,
we can choose $\alpha_{0}=\alpha_{1}=1$, the Bell function
$\langle B \rangle$ given by Eq. (\ref{e29}) becomes
\begin{eqnarray}
\label{e30}\langle B \rangle&=&2\sin\theta_{b}[\cos^{2}\gamma
+\cos(\theta_{1}-\theta_{0})\sin^{2}\gamma]\nonumber\\
&&-2\cos\theta_{b}.
\end{eqnarray}
Making use of Eq. (\ref{e30}), we can show that the maximal
violation of the Bell inequality can be reached by controlling the
azimuthal angle $\theta_{b}$, the state parameters of the basis
states given by Eq. (1), $\theta_{0}, \theta_{1}$ and $\gamma$. In
fact, when $\theta_{1}-\theta_{0}=2k\pi$ with $k$ being an
arbitrary integer, we can arrive at the following Bell function
\begin{eqnarray}
\label{e31}\langle B \rangle&=2(\sin\theta_{b}-\cos\theta_{b})
\end{eqnarray}
which means that the values of the Bell function of $\langle B
\rangle$ are independent of the entangling parameter of the basis
state defined in (1), and we can reach the maximal violation of
the Bell inequality $\langle B \rangle_{max}=2\sqrt{2}$ when
$\theta_{b}=-\pi/4$.
On the other hand, when the phase difference takes
$\theta_{1}-\theta_{0}=(2k+1)\pi$ with $k$ being an arbitrary
integer, the Bell function given by Eq. (30) becomes
\begin{eqnarray}
\label{e32}\langle B
\rangle&=2[\sin\theta_{b}\cos(2\gamma)-\cos\theta_{b}],
\end{eqnarray}
from which we can see that when the entanglement angle satisfies
$\gamma=k\pi$ with $k$ being an arbitrary integer, Eq. (\ref{e32})
becomes Eq. (\ref{e31}), hence the maximal violation of the Bell
inequality is reached with $\theta_{b}=-\pi/4$. However, when the
entanglement angle satisfies $\gamma=(2k+1)\pi/2$ with $k$ being
an arbitrary integer, the Bell function (\ref{e32}) reduces to
\begin{eqnarray}
\label{e33}\langle B \rangle&=-2(\sin\theta_{b}+\cos\theta_{b}),
\end{eqnarray}
which implies that the violation of the Bell inequality reaches
its maximal value $\langle B \rangle_{max}=2\sqrt{2}$ when
$\theta_{b}=\pi/4$. Therefore, we can conclude that superposition
states based on entangled states (\ref{e1}) can provide us with
more ways of reaching the maximal violation of the Bell's
inequality.
\section{Concluding remarks}
In summary, we have studied quantum entanglement and quantum
nonlocality of $N$-photon entangled states for the two modes
defined in Eq. (1) and their superpositions through investigating
the von Neumann entropy and the violation of the Bell's inequality
in the pseudospin Bell-operator formalism. For the multi-photon
entangled states defined in Eq. (1) we have indicated that the von
Neumann entropy is independent of the relative phase $\theta_{m}$.
Hence, quantum entanglement of these states only depends on the
entanglement angle. However, quantum nonlocality of these quantum
states exhibits different dependence upon the state parameters. We
have shown that both of the entanglement angle and the relative
phase seriously affect the violation of the Bell's inequality. And
we have indicated that under certain conditions the maximal
violation of the Bell's inequality can be reached.
It is worthwhile to mention that multi-component superposition
states made from $N$-photon entangled states for the two modes
defined in Eq. (1) exhibit some interesting characteristics on
their quantum entanglement and quantum nonlocality. Firstly, we
have found that these multi-component superposition states have
larger amount of entanglement than that of the basis states
defined by (1). Hence, quantum superpositions for the two modes
can increase the amount of entanglement. This indicates the
possibility of obtaining entangled states with a larger amount of
entanglement starting from entangled states with a smaller amount
of entanglement. Secondly, we have found that for these quantum
superposition states there are more ways to make corresponding
Bell's inequality reach the maximal violation, and revealed that
quantum nonlocality can be controlled and manipulated by adjusting
the state parameters of $|\psi_{N m}\rangle$, superposition
coefficients, and the azimuthal angles of the Bell operator. We
hope that these results obtained in present paper would find their
applications in quantum information processing
\cite{nie,che,lu,zhou2} and the test of quantum nonlocality
\cite{jeo,pit}.
\acknowledgments This work was supported in part the National
Fundamental Research Program (2001CB309310), the National Natural
Science Foundation of China under Grant Nos. 90203018,
10325523,10347128 and 10075018, the foundation of the Education
Ministry of China, and the Educational Committee of Human Province
under Grant Nos. 200248 and 02A026.
|
{
"timestamp": "2005-03-10T02:32:56",
"yymm": "0503",
"arxiv_id": "quant-ph/0503099",
"language": "en",
"url": "https://arxiv.org/abs/quant-ph/0503099"
}
|
\section{Introduction}
Consider the problem of finding a zero of a function $\varphi : \mathbb{R}
\to \mathbb{R}$. If there are several zeros, it is required to find at
least one of them. It is supposed that the function can be
measured at any point, with some random error. The standard
algorithm of stochastic approximation consists in calculating
successive approximations of the required value, $x_0$,\, $x_1$,\,
$x_2, \ldots$, according to the rule
\begin{equation}\label{eqal1}
x_{t} = x_{t-1} - \gamma_{t-1} y_t, \quad t=1,\ 2,\ldots,
\end{equation}
where
\begin{equation}\label{eqal2}
y_t = \varphi(x_{t-1}) + \xi_t \hspace{26mm}
\end{equation}
is the value of $\varphi$ measured at $x_{t-1}$,\, $\xi_t$ is the
measurement error; \,$\gamma_0$,\, $\gamma_1$,\, $\gamma_2, \ldots$ is
the sequence of step sizes of the algorithm. Usually it is assumed
that the step sizes are positive real numbers satisfying the
relations $\sum \gamma_t = \infty$,\, $\sum \gamma_t^2 < \infty$.
Then, under some additional assumptions on $\varphi$ and $\xi_t$,
the algorithm a.s. converges to a zero point of $\varphi$ (see,
e.g., \cite{b007,b001}). In practice, however, the convergence rate of
this algorithm may prove to be unsatisfactory, therefore, when
solving practical tasks, various modifications of the algorithm
are used. There are widely utilized heuristical algorithms using
random, rather than deterministic, step size, which is corrected
in the course of the algorithm, according to the current data
\cite{a005,a011,a100,a006}. In particular, there is used the idea that prescribes to
decrease the step size if the sequence of increments $x_{t} -
x_{t-1}$ changes the sign often enough, indicating that the
current value $x_t$ is close to the set of zeros of $\varphi$, and
hence, the measurement error $\xi_t$ of the function is big enough
with respect to the function itself $\varphi(x_{t-1})$.
Alternatively, one should increase the step size, or leave it
unchanged. So, Kesten in the theoretical work \cite{a008}
considered an algorithm using (\ref{eqal1}), (\ref{eqal2}), and
the rule of modification of $\gamma_t$:
\begin{equation}\label{eqK}
\gamma_t = \gamma(s_t), \ \ \ \ \ \ s_t = \left\{
\begin{array}{lll}
s_{t-1} & \textrm{ if } & y_{t-1} y_t > 0\\
s_{t-1} + 1 & \textrm{ if } & y_{t-1} y_t \le 0,
\end{array}
\right.\quad t=2,3,\ldots.
\end{equation}
where $s_0 = 0$,\, $s_1 = 1$;\, $\gamma(0)$,\, $\gamma(1)$,\,
$\gamma(2), \ldots$ is a sequence of positive numbers satisfying the
relations $\sum \gamma(m) = \infty$,\, $\sum \gamma^2(m) < \infty$.
Thus, the step size cannot increase in the course of algorithm; it
can only decrease or remain unchanged. It is supposed that there
is a unique zero of $\varphi$. Kesten proved that $x_t$ a.s.
converges to this zero point. A multidimensional version of this
algorithm is considered in \cite{a003}.
There are also heuristical procedures (in particular, in
artificial neural networks), where at each moment $t$ the step
size is multiplied by a positive constant less than 1, if the
measurement data indicate that $x_t$ is close enough to the zero
set of $\varphi$, and by a constant more than 1, elsewhere
\cite{a067,a001,a100,a101}. This kind of rules ensure
sufficiently high convergence rate, however the step size
converges like a geometric progression, therefore $\sum \gamma_t <
\infty$, which means that the limit of $\{ x_t \}$ need not be a
zero point of $\varphi$, but instead, the sequence may "get stuck"
on its way to the set of zeros of $\varphi$. Nevertheless, such a
procedure may be justified if it gives a value close enough to one
of the zeros of $\varphi$.
In the present paper, a stochastic approximation algorithm
utilizing this rule of step size modification is considered.
Namely, the rule (\ref{eqal1}), (\ref{eqal2}), jointly with the
following rule
\begin{eqnarray}
\gamma_t = \left\{
\begin{array}{lll}
\min\{u\, \gamma_{t-1},\, \bar{\mathrm{g}}\} & \textrm{ if } & y_{t-1} y_t > 0,\\
d\, \gamma_{t-1} & \textrm{ if } & y_{t-1} y_t \le 0,
\end{array}
\right.\quad t=2,3,\ldots. \label{eqal3}
\end{eqnarray}
is used. Here $0 < d < 1 < u$,\, $0 < \gamma_0$,\, $\gamma_1 \le
\bar{\mathrm{g}}$,\, $\bar{\mathrm{g}}$ is a positive constant. Let us point out the
main differences between (\ref{eqal3}) and Kesten's rule
(\ref{eqK}). First, according to (\ref{eqal3}), $\gamma_t$ can both
decrease and increase. Second, in Kesten's algorithm one always
has $\sum \gamma_t = \infty$. On the other hand, it looks likely
that in the case of convergence of the algorithm (\ref{eqal1}),
(\ref{eqal2}), (\ref{eqal3}), $\gamma_t$ converges like a geometric
progression (this conjecture will be justified in the section 3),
therefore the limit of algorithm may not be a zero point of
$\varphi$.
Suppose that $\{ \xi_t \}$ is a sequence of i.i.d.r.v. with zero
mean, besides $\mathrm{P} (\xi_t > 0) = \mathrm{P} (\xi_t < 0)$. Under some
additional assumptions on $\varphi$, $\xi_t$, and $\bar{\mathrm{g}}$, stated
below, the process defined by (\ref{eqal1}), (\ref{eqal2}),
(\ref{eqal3}) a.s. diverges if $ud > 1$, and converges if $ud <
1$, moreover the limit of $\{ x_t \}$ belongs to $\mathcal{U} (\frac{\ln
u}{-\ln d})$. Here $\mathcal{U}(\lambda)$,\, $0 < \lambda < 1$, is a monotone
decreasing family of sets of real numbers, besides every set
$\mathcal{U}(\lambda)$ contains the set $\mathrm{Z}$ of zeros of $\varphi$, and
$\partial (\mathcal{U}(\lambda), \mathrm{Z}) \to 0$ as $\lambda \to 1^-$. (Here by
definition $\partial (A,B) = \sup_{x\in A} \inf_{y\in B} |x-y|$
for any two sets of real numbers $A$ and $B$.)
This statement is a consequence of the main
theorem, which will be stated in section 2 and proved in section
3. Thus, by adjusting the parameters $u$ and $d$ (for example,
fixing $u$ and letting $d \to 1/u - 0$), one can reach necessary
precision of the algorithm; higher precision is obtained at the
expense of lower convergence rate.
\section{Definition of the algorithm and statement of the main result}
Consider the algorithm given by (\ref{eqal1}), (\ref{eqal2}),
(\ref{eqal3}). The rule (\ref{eqal3}) means that at each instant
$t$, step size is multiplied by $u$ or by $d$, if the result of
multiplication is less than $\bar{\mathrm{g}}$; otherwise, step size is set
to be $\bar{\mathrm{g}}$. Thus, the maximal possible value of step size
equals $\bar{\mathrm{g}}$.
The rule (\ref{eqal3}) can be written in the form
\begin{equation}\label{eqal4n}
\begin{array}{l@{ = }l}
\ln \tilde \gamma_t & \ln \gamma_{t-1}
+ \ln u \cdot \;\mathbb{I}(y_{t-1} y_t>0) + \ln d \cdot \;\mathbb{I}(y_{t-1} y_t \le 0), \\
\ln \gamma_t & \min\{ \ln \tilde \gamma_t, \ln \bar{\mathrm{g}}\}.
\end{array}
\end{equation}
Let us take the following assumptions:
\begin{description}
\item [A1] Denote ${\cal F}_t$, $t = 0,1,2,\ldots$ the $\sigma$-algebra
generated by $x_i$, $\gamma_i$, and $\xi_i$, $0\le i \le t$; then
$\xi_{t+1}$ does not depend on ${\cal F}_t$.
\item [A2] The values $\xi_t$ are identically distributed, with
zero mean and finite variance: $\mathrm{E} \xi_t = 0$,\, $\mathrm{Var} \xi_t =: S
< +\infty$.
\item [A3] (a) There exists $L >0$ such that for any
interval
$I \subset [-L,\, L]$, $\mathrm{P}(\xi_1 \in I)>0$;\\
\hspace*{-2mm}(b) $\mathrm{P}(\xi_1 = 0) = 0$.
\item [A4] $\varphi \in \mathbb{C}^1(\mathbb{R})$ and $\sup_x |\varphi'(x)| =: M <
\infty$.
\item [A5] $\bar{\mathrm{g}} < 2/M$.
\item [A6] There exists $R>0$ such that
\begin{itemize}
\item[(a)] $x \varphi(x) > 0$ as $|x| \ge R$, and \item[(b)]
$\displaystyle \inf_{|x|\ge R} \varphi^2(x) > \frac{\bar{\mathrm{g}} M
S}{2-\bar{\mathrm{g}} M}$.
\end{itemize}
\end{description}
\begin{remark
From \A{A4} and \A{A6}\,(a) it follows that the set $\mathrm{Z}$ is
non-empty and is contained in $(-R,\, R)$.
\end{remark}
\begin{remark
Note that assumptions \A{A4}--\A{A6} guarantee convergence of the
deterministic counterpart of algorithm (\ref{eqal1}),
(\ref{eqal2}), (\ref{eqal3}) (that is, of the algorithm with
$\xi_t \equiv 0$). Moreover, under these conditions, any
deterministic algorithm $x_t = x_{t-1} - \gamma_{t-1}
\varphi(x_{t-1})$ converges, whatever the sequence $\{\gamma_t\}$
satisfying $\gamma_t \le \bar{\mathrm{g}}$.
\end{remark}
Introduce the functions:
\begin{equation} \label{ast1}
k_+(z) := \lim_{\epsilon\to0^+} \sup\{ \mathrm{P}( (\varphi_1+\xi_1)(\varphi_2+\xi_2) >
0),\
|\varphi_1 - z| < \epsilon,\ |\varphi_2 - z| < \epsilon\},
\end{equation}
\begin{equation} \label{ast2}
k_-(z) := \lim_{\epsilon\to0^+} \inf\{ \mathrm{P}( (\varphi_1+\xi_1)(\varphi_2+\xi_2) >
0),\
|\varphi_1 - z| < \epsilon,\ |\varphi_2 - z| < \epsilon\};
\end{equation}
one has $k_+(z) \ge 1/2$,\, $0 \le k_{\pm}(z) \le 1$,\,
$\lim_{z\to\infty} k_{\pm}(z)=1$.
Further, define the sets of real numbers
\begin{equation}\label{ast3}
V_{\pm}^{(a)} := \{ x : k_\pm(\varphi(x)) < a \}, \quad
V_{\pm}^{[a]} := \{ x : k_\pm(\varphi(x)) \le a \};
\end{equation}
obviously, $V_+^{(a)} \subset V_-^{(a)}$,\, $V_\pm^{(a)} \subset
V_\pm^{[a]}$ for any $a$.
Note that $V_+^{(a)}$ is open. Indeed, let $x \in V_+^{(a)}$, then
there exists $\epsilon > 0$ such that
$$
\sup\{ \mathrm{P}( (\varphi_1+\xi_1)(\varphi_2+\xi_2) > 0),\
|\varphi_1 - \varphi(x)| < \epsilon,\ |\varphi_2 - \varphi(x)| <
\epsilon\} =: c < a.
$$
Then for $x'$ close enough to $x$ one has $|\varphi(x')-
\varphi(x)| < \varepsilon/2$, hence
$$
\sup\{ \mathrm{P}( (\varphi_1+\xi_1)(\varphi_2+\xi_2) > 0),\
|\varphi_1 - \varphi(x')| < \epsilon/2,\ |\varphi_2 - \varphi(x')| <
\epsilon/2 \} \le c < a.
$$
This implies that $k_+(\varphi(x')) < a$, hence $x' \in
V_+^{(a)}$.
Denote also
\begin{equation}\label{eq5}
\mathrm{k} := \frac{\ln(1/d)}{\ln(u/d)}.
\end{equation}
Denote by $\mathrm{Z}$ the set of zeros of $\varphi$, i.e., $\mathrm{Z}:=\{ x :
\varphi(x)=0 \}$. Suppose that $x\in V_+^{(\mathrm{k})}$,\,
$x_{t-2} \in (x-\epsilon,\, x+\epsilon) \subset
V_+^{(\mathrm{k})}$, and $\gamma_{t-2} < \epsilon$, where
$\epsilon$ is a small positive number. Then, with a probability
close to 1,\, $x_{t-1}$ also belongs to a small (possibly larger)
neighborhood of $x$ contained in $V_+^{(\mathrm{k})}$, and
taking into account (\ref{ast1}) and (\ref{ast3}), one gets
\[
\begin{array}{l}
\mathrm{P}( y_{t-1} y_t > 0 \,{\Big |}\, |x_{t-2} - x| < \epsilon, \gamma_{t-2} < \epsilon ) =\\
=\mathrm{P}( (\varphi(x_{t-2}) + \xi_{t-1}) (\varphi(x_{t-1}) + \xi_{t}) > 0
\,{\Big |}\, |x_{t-2}-x|<\epsilon, \gamma_{t-2} < \epsilon) <
\mathrm{k}.
\end{array}
\]
Then, using (\ref{eqal4n}) and (\ref{eq5}), one obtains
\[
\begin{array}{l}
\mathrm{E}[ \ln \gamma_t-\ln \gamma_{t-1} \,{\Big |}\, |x_{t-2}-x| < \epsilon, \gamma_{t-2} < \epsilon ]
\le \\
\ln u \cdot \mathrm{P}(y_{t-1} y_t > 0 \,{\Big |}\, |x_{t-2}-x| < \epsilon, \gamma_{t-2} < \epsilon ) +
\ln d \cdot \mathrm{P}(y_{t-1} y_t \le 0 \,{\Big |}\, |x_{t-2} - x| < \epsilon, \gamma_{t-2} < \epsilon ) \\
< \ln u \cdot \mathrm{k} + \ln d \cdot (1-\mathrm{k}) = 0.
\end{array}
\]
Thus, in a sense, the set $V_+^{(\mathrm{k})}$ can be regarded
to be a \textit{domain of decrease of step size}: if several
consecutive values of $x_t$ belong to $V_+^{(\mathrm{k})}$ and
are close enough to each other, and if the first term of the
sequence of corresponding step sizes $\gamma_t$ is small enough,
then the sequence of their mean values $E \gamma_t$ decreases.
Now, suppose that $x \in \mathbb{R} \setminus V_-^{[\mathrm{k}]}$,\,
$x_{t-2} \in (x-\epsilon,\, x+\epsilon) \subset \mathbb{R} \setminus
V_-^{[\mathrm{k}]}$, and that $\gamma_{t-2} < \epsilon$.
Analogously, for $\epsilon$ small enough, one has
\[
\mathrm{P}( y_{t-1} y_t > 0 \,{\Big |}\, |x_{t-2} - x| < \epsilon, \gamma_{t-2} < \epsilon ) > \mathrm{k},
\]
and then, using again (\ref{eqal4n}) and (\ref{eq5}) and taking
into account that for $\epsilon < \bar{\mathrm{g}}/u^2$,\, $\tilde\gamma_t =
\gamma_t$, one obtains
\[
\begin{array}{l}
\mathrm{E}[ \ln \gamma_t - \ln \gamma_{t-1} \,{\Big |}\, |x_{t-2} - x| < \epsilon, \gamma_{t-2} < \epsilon] = \\
\ln u \cdot \mathrm{P}( y_{t-1} y_t > 0 \,{\Big |}\, |x_{t-2} - x | < \epsilon, \gamma_{t-2} < \epsilon])
+ \ln d \cdot \mathrm{P}( y_{t-1} y_t \le 0 \,{\Big |}\, |x_{t-2} - x| < \epsilon, \gamma_{t-2} < \epsilon ]) \\
> \ln u \cdot \mathrm{k} + \ln d \cdot (1-\mathrm{k}) = 0.
\end{array}
\]
Thus, the set $\mathbb{R} \setminus V_-^{[\mathrm{k}]}$ can be regarded
as a \textit{domain of increase of step size}: if several
consecutive values of $x_t$ belong to $\mathbb{R}\setminus
V_-^{[\mathrm{k}]}$ and are close enough to each other, and if
the first of the corresponding values of $\gamma_t$ is small
enough, then the sequence of their mean values $E \gamma_t$
increases.
Note that if $\mathrm{k} > k_+(0)$ then, by virtue of
(\ref{ast3}), $\mathrm{Z} \subset V_+^{(\mathrm{k})}$, that is, all the
zeros of $\varphi$ belong to the region of decrease of step size.
On the other hand, if $\mathrm{k} < \inf_z k_-(z)$ then
$V_-^{[\mathrm{k}]} = \emptyset$, which means that the region of
increase of step size coincides with $\mathbb{R}$.
It seems likely that in the first case the algorithm can converge,
and in the second one, cannot. This conjecture is confirmed by the
following theorem, which is the main result of the paper.
\vspace{2mm}
\textbf{Theorem} \textit{ Let the assumptions \A{A1}--\A{A6} be
satisfied; consider the process $\{ x_t,\ \gamma_t \}$ defined by
(\ref{eqal1}), (\ref{eqal2}), (\ref{eqal3}). Recall that
$\mathrm{k} = \frac{\ln(1/d)}{\ln(u/d)}$.
Then\\
(a) If $\mathrm{k} > k_+(0)$ then $\{x_t\}$ a.s. converges to a
point from $V_-^{[\mathrm{k}]}$.\\
(b) If $\mathrm{k} < \inf_z k_-(z)$ then $\{ x_t \}$ a.s.
diverges. }
\vspace{2mm}
Suppose that $\mathrm{P} (\xi_1 = x) = 0$ for any real $x$ and that $\mathrm{P}
(\xi_1 > 0) = \mathrm{P} (\xi_1 < 0)$. Then the function $k(\cdot) :=
k_+(\cdot)$ coincides with $k_-(\cdot)$, is continuous, and is
given by
$$
k(z) = \mathrm{P} ((z + \xi_1)(z + \xi_2) > 0);
$$
$z = 0$ is the unique minimum of $k(\cdot)$, and $k(0) = \inf_z
k(z) = 1/2$. After a simple algebra, one can rewrite the
hypotheses of theorem in the form (a) $ud < 1$, (b) $ud > 1$.
Denote $\mathcal{U}(\lambda) := V^{[\frac{1}{1+\lambda}]} = \{ x :\,
k(\varphi(x)) \le \frac{1}{1 + \lambda} \}$; \, $\mathcal{U}(\lambda)$,\, $1 <
\lambda < 1$ is a monotone decreasing family of sets containing $\mathrm{Z}$
and tending to $\mathrm{Z}$ as $\lambda \to 1^-$.
Thus, one comes to \vspace{2mm}
\textbf{Corollary} \textit{ Let, in addition to assumptions
\A{A1}--\A{A6}, $\mathrm{P} (\xi_1 = x) = 0$ for any $x \in \mathbb{R}$, and $\mathrm{P}
(\xi_1 > 0) = \mathrm{P} (\xi_1 < 0) = 1/2$. Consider the process defined
by (\ref{eqal1}), (\ref{eqal2}), (\ref{eqal3}). Then there exists
a monotone decreasing family of sets $\mathcal{U}(\lambda)$,\, $0 < \lambda <
1$ such that $\mathcal{U}(\lambda) \supset
\mathrm{Z}$,\, $\partial (\mathcal{U}(\lambda), \mathrm{Z}) \to 0$ as $\lambda \to 1^-$, and\\
(a) if $ud < 1$ then $\{x_t\}$ a.s. converges to a point from
$\mathcal{U} (\frac{\ln u}{-\ln d})$;\\
(b) if $ud > 1$ then $\{x_t\}$ a.s. diverges. }
\begin{remark
Theorem does not give any information about behavior of the
algorithm for the values $u$,\, $d$ such that
$$
\inf\nolimits_z k_-(z) \le \frac{\ln(1/d)}{\ln(u/d)} \le k_+(0).
$$
In particular, under the hypotheses of corollary, the case $ud =
1$ remains unexplored. These issues will be addressed elsewhere.
\end{remark}
\section{Proof of theorem}
First we prove 10 auxiliary lemmas, and then, basing on them, we
prove theorem.
Here all statements about random variables are supposed
to be true almost surely.
In the sequel, we shall mainly designate random values by Greek
letters, and real numbers and functions from $\mathbb{R}$ to $\mathbb{R}$, by
Latin ones; the letters $t$,\, $i$,\, $j$,\, $s$ will denote
integer non-negative numbers. The function $\varphi$ and the random
values $x_t$,\, $y_t$ are exceptions; also, traditional notation
$\epsilon$,\, $\delta$ for small positive numbers will be used.
\begin{lemma}
If $\sum_t \gamma_t < \infty$ then the sequence $\{ x_t \}$ converges.
\end{lemma}
\textit{Proof.} Note that without loss of generality one can
assume that $x_0$ is bounded. Indeed, replacing $x_0$ by $\tilde
x_0 = x_0 \cdot \;\mathbb{I}(|x_0|<X)$ changes the process only with
probability $\mathrm{P}(|x_0|>X)$. By taking $X$ large enough, one can
make this probability arbitrarily small.
Let $C>0$; define the stopping time
$\tau_C = \inf \{ t : \sum_{i=0}^t \gamma_i > C\}$ and introduce the new
process $x_t^C$, $\gamma_t^C$ by
\[
\begin{array}{l}
x_t^C = x_t, \quad \gamma_t^C=\gamma_t \textrm{ as } t < \tau_c, \textrm{ and } \\
x_t^C = x_{\tau_C}, \quad \gamma_t^C=0 \textrm{ as } t \ge
\tau_c.
\end{array}
\]
First, let us prove that the sequence $\{x_t^C\}$ is bounded.
Designate $M_R := \sup_{|x|\ge R} \frac{\varphi(x)}{x}$; from
\A{A4} it follows that $M_R<\infty$. One has
\begin{equation}\label{eq9}
|x_t^C| \le |x_{t-1}^C - \gamma_{t-1}^C \varphi(x^C_{t-1})| + \gamma_{t-1}^C |\xi_t|.
\end{equation}
Using that $\gamma_{t-1}^C \le C$ and $|\varphi(x_{t-1})^C| \le |\varphi(0)| + M |x_{t-1}^C|$,
one obtains
\begin{equation}\label{eq10}
|x_t^C| \le |x_{t-1}^C|(1 + CM) + \gamma_{t-1}^C(|\varphi(0)|+|\xi_t|).
\end{equation}
If $\gamma_{t-1}^C \le 2/M_R$, an even more precise estimate for
$x_t^C$ can be obtained. We shall distinguish between two cases:
(i) $|x_{t-1}|\le R$ and (ii) $|x_{t-1}^C| > R$.
In case (i), designating $\bar{b} := \sup_{|x|\le R}
|\varphi(x)|$, one has
\begin{equation}\label{eq11}
|x_{t-1}^C - \gamma_{t-1}^C \varphi(x_{t-1}^C)| \le |x_{t-1}^C| + \gamma_{t-1}^C \bar{b}.
\end{equation}
In the case (ii) one has
\[
0 \le \gamma_{t-1}^C \frac{\varphi(x_{t-1}^C)}{x_{t-1}^C} \le \frac{2}{M_R} M_R = 2,
\]
hence
\begin{equation}\label{eq12}
|x_{t-1}^C - \gamma_{t-1}^C\varphi(x_{t-1}^C)| \le |x_{t-1}^C|.
\end{equation}
Thus, in both cases (i) and (ii), from (\ref{eq9}), (\ref{eq11}), and (\ref{eq12}) one gets
\begin{equation}\label{eq13}
|x_t^C| \le |x_{t-1}^C| + \gamma_{t-1}^C ( \bar{b} + |\xi_t| ).
\end{equation}
The overall number of values of $t$ such that $\gamma_{t-1}^C \le 2/M_R$
is less than $CM_R/2$; therefore, using (\ref{eq10}) and (\ref{eq13}), one concludes that
\begin{equation}\label{eq14}
|x_t^C| \le \left( |x_0| + \sum_{i=1}^t
\gamma_{i-1}^C(\bar{b}+|\varphi(0)|+|\xi_i|) \right) \cdot
(1+CM)^{CM_R/2}.
\end{equation}
Denote $c_0 := \bar{b} + |\varphi(0)|+\mathrm{E}|\xi_1|$ and $\zeta_t :=
|\xi_t| - \mathrm{E}|\xi_t|$; using that $\sum_1^\infty \gamma_{i-1}^C \le
C$ one gets
\begin{equation}\label{eq15}
|x_t^C| \le \left( |x_0| + C\, c_0 + \sum_{i=1}^t \gamma_{i-1}^C
\zeta_i \right)\cdot(1+CM)^{CM_R/2}.
\end{equation}
Using that $\sum_1^\infty \mathrm{E}(\gamma_{t-1}^C \zeta_t)^2 = \mathrm{E}
\zeta_1^2 \cdot \sum_1^\infty \mathrm{E}(\gamma_{t-1}^C)^2 < \infty$, one
obtains that the martingale $\sum_1^t \gamma_{i-1}^C \zeta_i$ is
bounded; the value $x_0$ is also bounded, so, by (\ref{eq15}), one
concludes that the sequence $\{x_t^C\}$ is bounded.
Now, let us show that $\{x_t^C\}$ converges. From the definition of $x_t^C$ and
$\gamma_t^C$ it follows that
\[
x_t^C = x_0 - \sum_1^t \gamma_{i-1}^C \varphi(x_{i-1}^C) - \sum_1^t \gamma_{i-1}^C \xi_i.
\]
Using that the sequence $\{ \varphi(x_{i-1}^C)\}$ is bounded and that $\sum_1^\infty \gamma_{i-1}^C \le C$,
one gets that the series $\sum_1^\infty \gamma_{i-1}^C \varphi(x_{i-1}^C)$ converges. Further, one has
\[
\sum_1^\infty \mathrm{E}(\gamma_{t-1}^C \xi_t)^2 = S\cdot \sum_1^\infty
\mathrm{E}(\gamma_{t-1}^C)^2 < \infty,
\]
hence the martingale $\sum_1^t \gamma_{i-1}^C \xi_i$ converges. This implies
that $\{x_t^C\}$ also converges.
Define the events $A_C = \{ \sum_t \gamma_t \le C\}$ and $A_\infty
= \{ \sum_t \gamma_t < \infty \}$. One has $A_\infty = \cup_C
A_C$. If $\sum_t \gamma_t \le C$ then $x_t^C=x_t$ for any $t$;
this means that $\;\mathbb{I}(A_C)\cdot(x_t^C - x_t)=0$ for any $t$ and $C$.
The sequence $\{ \;\mathbb{I}(A_C) x_t^C\}$ converges, therefore the
sequence $\{\;\mathbb{I}(A_C)x_t\}$ also converges, and passing to the limit
$C\to\infty$ one obtains that $\{ \;\mathbb{I}(A_\infty) x_t\}$ converges.
This means exactly that if $\sum_t \gamma_t < \infty$ then $\{ x_t
\}$ converges. \hfill$\Box$
\begin{lemma}
If $\lim_{t\to\infty} x_t = x$ then $x \in V_-^{[\mathrm{k}]}$.
\end{lemma}
\textit{Proof.} Note that, using \A{A3}\,(a), it is easy to show
that there exists $\delta_0 >0$ such that $\mathrm{P}(\xi_1 \not \in
[x-L/2,\, x+L/2]) > \delta_0$, whatever $x \in
\mathbb{R}$.
Next, for any $x\not \in V_-^{([\mathrm{k}])}$ there exist
$w(x)>0$ and $0 < \epsilon(x) < L/4$ such that the
following holds: for any two random variables $\phi_1$ and
$\phi_2$ satisfying the relations $|\phi_l - \varphi(x)| \le
\epsilon(x)$,\, $l=1,2$ one has
\[
\mathrm{P}( (\phi_1+\xi_1)(\phi_2+\xi_2) >0 ) > \frac{\ln(1/d)+w(x)}{\ln u + \ln(1/d)}.
\]
Choose a countable set of intervals $U_i =
(\varphi(x_i)-\epsilon(x_i),\ \varphi(x_i)+\epsilon(x_i))$
covering the set $\varphi(\mathbb{R}\setminus V_-^{[\mathrm{k}]})$, and
denote $w_i := w(x_i)$. Fix $i$ and $s\in\{0,\, 1,\, 2,\ldots\}$,
and define the auxiliary process $x_t^{(is)}$, $\gamma_t^{(is)}$
by formulas:
\vspace{1mm}
if $t<s$ then $x_t^{(is)} = x_t$, \ and \ if $t \ge s$ then
\begin{eqnarray}\label{eq16}
x_t^{(is)} = \left\{
\begin{array}{ll}
x_{t-1}^{(is)} - \gamma_{t-1}^{(is)}\, y_t^{(is)} &
\textrm{ if } \ \varphi(x_{t-1}^{(is)} - \gamma_{t-1}^{(is)}\, y_t^{(is)}) \in U_i,\\
x_i & \textrm{ elsewhere};
\end{array}
\right.
\end{eqnarray}
\begin{equation}\label{eq17}
y_t^{(is)} = \varphi(x_{t-1}^{(is)}) + \xi_t, \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \hspace{35mm}
\end{equation}
\begin{eqnarray}\label{eq18}
\gamma_t^{(is)} = \left\{
\begin{array}{l@{\textrm{ if }}l}
\min\{u\gamma_{t-1}^{(is)},\, \bar{\mathrm{g}}\} \ \ \ & \ y_{t-1}^{(is)}\, y_t^{(is)} > 0,\\
d \gamma_{t-1}^{(is)} \ \ \ & \ y_{t-1}^{(is)}\, y_t^{(is)} \le 0.
\end{array}
\right. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\end{eqnarray}
So, as $t \ge s$,\, $\varphi(x_t^{(is)})$ is forced to be
contained in $U_i$.
For $t\ge s+2$, using that $y_{t-1}^{(is)} =
\varphi(x_{t-2}^{(is)})+\xi_{t-1}$,\, $y_t^{(is)} =
\varphi(x_{t-1}^{(is)}) + \xi_t$,\, $\varphi(x_{t-2}^{(is)}) \in
U_i$, one obtains that
\[
\mathrm{P}( y_{t-1}^{(is)}\, y_t^{(is)} > 0) > \frac{\ln(1/d) + w_i}{\ln u + \ln(1/d)}
\]
and
\[
\mathrm{P}( y_{t-1}^{(is)}\, y_t^{(is)} \le 0 ) < \frac{\ln u - w_i}{\ln u + \ln (1/d)},
\]
hence
$$
\displaystyle \mathrm{E}[\ln u\cdot \;\mathbb{I}(y_{t-1}^{(is)}\, y_t^{(is)}>0)\,
+\, \ln d\cdot \;\mathbb{I}(y_{t-1}^{(is)}\, y_t^{(is)}\le 0) ] >
$$
$$
\displaystyle > \ln u \cdot \frac{\ln(1/d) + w_i}{\ln u +
\ln(1/d)}\ +\ \ln d \cdot \frac{\ln u - w_i}{\ln u + \ln(1/d)} =
w_i.
$$
Consider variables $\phi_1=f_1(\xi_1, \xi_2)$ and $\phi_2 =
f_2(\xi_1,\xi_2)$ providing a solution of the (deterministic)
minimization problem:
\[
(\phi_1 + \xi_1)(\phi_2+\xi_2) \to \min,
\]
subject to
\[
\begin{array}{l}
|\phi_1 - \varphi(x_i)| \le \epsilon(x_i) \\
|\phi_2 - \varphi(x_i)| \le \epsilon(x_i),\\
\end{array}
\]
and denote $Y_{t-1}^1 = f_1(\xi_{t-1},\xi_t) + \xi_{t-1}$,
$Y_{t}^2 = f_2(\xi_{t-1},\xi_t) + \xi_{t}$, $\eta_t = \ln u \cdot
\;\mathbb{I}(Y_{t-1}^1 Y_{t-1}^2 > 0) + \ln d \cdot \;\mathbb{I}( Y_{t-1}^1 Y_{t-1}^2
\le 0)$.
One has
(i) $\eta_t \le \ln u \cdot \;\mathbb{I}( y_{t-1}^{(is)}\, y_t^{(is)} > 0)
+ \ln d \cdot \;\mathbb{I}( y_{t-1}^{(is)}\, y_t^{(is)} \le 0)$;
(ii) $\eta_t$ are identically distributed, and $\mathrm{E} \eta_t \ge
w_i$;
(iii) the set of random variables $\{\eta_t,\ t \textrm{ even},\ t
\ge s+2 \}$ as well as the set $\{\eta_t,\ t \textrm{ odd},\ t \ge
s+2 \}$, are mutually independent.
From (ii)--(iii) it follows that almost surely $\sum_t \eta_t =
+\infty$, and from (i) it follows that
\[
\sum_t [\ln u \cdot \;\mathbb{I}(y_{t-1}^{(is)}\, y_t^{(is)} > 0 ) + \ln d
\cdot \;\mathbb{I}( y_{t-1}^{(is)}\, y_t^{(is)} \le 0)]= +\infty,
\]
so, by virtue of (\ref{eq18}), $\gamma^{(is)}$ does not go to zero.
Thus, there exists a random value $\chi > 0$ such that for
infinitely many values of $t$,\, $\gamma_t^{(is)} \ge \chi$.
Define a sequence of stopping times $\tau_0$, $\tau_1$, $\tau_2,
\ldots$ inductively, letting $\tau_0=0$ and $\tau_j = \inf\{ t >
\tau_{j-1} : \gamma_t^{(is)} \ge \chi\}$ for $j\ge 1$. The events
$B_j = \{ |\xi_{\tau_j+1} + \varphi(x_i)|>L/2\}$ happen
with probability more that $\delta_0$ (recall the remark done in
the beginning of proof), and every event $B_j$, $j\ge 2$ does not
depend on the set of events $\{ B_1, \ldots, B_{j-1} \}$.
Therefore, for infinitely many values of $j$, $B_j$, takes place,
i.e., $|\xi_{\tau_j+1} + \varphi(x_i)| > L/2$, and hence,
taking into account that $|y_{\tau_j+1}| \ge |\xi_{\tau_j+1} +
\varphi(x_i)| - |\varphi(x_{\tau_j}) - \varphi(x_i)|$ and
$|\varphi(x_{\tau_j})-\varphi(x_i)| < \epsilon(x_i) <
L/4$, for these values of $j$ one has $|y_{\tau_j+1}| \ge
L/4$. Thus, one concludes that
\begin{equation}\label{eqast}
\textrm{for infinitely many values of }j, \ \ |\gamma_{\tau_j}
y_{\tau_j + 1}| \ge \chi\, L/4.
\end{equation}
Suppose that $x_t$ converges to a point from $\mathbb{R}\setminus
V_-^{[\mathrm{k}]}$, then for some $i$ and $s$ one has $x_t \in
U_i$ as $t \ge s$, hence the process $x_t^{(is)}$,
$\gamma_t^{(is)}$ coincides with $x_t$,\, $\gamma_t$, and
therefore $\gamma_t\, y_{t+1} \to 0$ as $t \to \infty$. The last
relation contradicts (\ref{eqast}), thus Lemma 2 is proved.
\hfill$\Box$
\begin{lemma}
Let $\sum_t \gamma_t = \infty$. Then for any open set ${\cal O}$
containing $\mathrm{Z}$ there exists a positive constant
$g=g({\cal O})$ such that either (i) for some $t$,\,
$x_t\in{\cal O}$, or (ii) for some $t$, $|x_t|<R$ and $\gamma_t >
g$.
\end{lemma}
\textit{Proof.} Designate by $f$ the primitive of $\varphi$ such that $\inf_x f(x) = 0$.
Define the stopping time
\[
\tau = \tau({\cal O},g) :=
\inf \{ t : \textrm{ either
(i) } x_t \in {\cal O},
\textrm{ or (ii) } |x_t|<R \textrm{ and } \gamma_t \ge g\}.
\]
The value of $g \in (0,\bar{\mathrm{g}})$ will be specified below.
Consider the sequence $\mathrm{E}_t = \mathrm{E}[ f(x_t) \;\mathbb{I}(t<\tau)]$. Introducing
shorthand notation $f(x_t) =: f_t$,\, $\;\mathbb{I}(t<\tau)=: I_t$,\,
$f'(x_t)=:f_t'=\varphi_t$, and using that $I_t \le I_{t-1}$, one
gets
\begin{equation}\label{eq20}
E_t - E_{t-1} = \mathrm{E}[f_t \;\mathbb{I}_t - f_{t-1} \;\mathbb{I}_{t-1}]\, \le\, \mathrm{E}[(f_t - f_{t-1}) \;\mathbb{I}_{t-1}].
\end{equation}
Next, we utilize the Taylor decomposition
$$
f_t = f(x_{t-1} - \gamma_{t-1} y_t) = f_{t-1} - f'_{t-1}\,
\gamma_{t-1} y_t + \frac 12\, f''(x')\, \gamma_{t-1}^2 y_t^2,
$$
$x'$ being some point between $x_{t-1}$ and $x_t$. Substituting
$y_t = \varphi_{t-1}+\xi_t$ and recalling that
$f'_{t-1}=\varphi_{t-1}$ and $f''(x')=\varphi'(x') \le M $, one
obtains
\begin{equation}\label{eq21}
f_t - f_{t-1} \le -\gamma_{t-1}\, \varphi_{t-1} (\varphi_{t-1} +
\xi_t) + {M\over 2}\, \gamma_{t-1}^2\, (\varphi_{t-1} + \xi_t)^2.
\end{equation}
Using (\ref{eq20}) and (\ref{eq21}) and taking into account that
each of the values $\gamma_{t-1}$,\, $\varphi_{t-1}$,\, $\;\mathbb{I}_{t-1}$
is mutually independent with $\xi_t$ (see \A{A1}), one gets
\begin{equation}\label{eq23}
\begin{array}{l}
E_t - E_{t-1} \le \mathrm{E}[
(-\gamma_{t-1}\, \varphi_{t-1}^2 -
\gamma_{t-1}\, \varphi_{t-1} \xi_t +
{M\over 2} \gamma_{t-1}^2\, \varphi_{t-1}^2 +
M\gamma_{t-1}^2\, \varphi_{t-1} \xi_t +
{M\over 2} \gamma_{t-1}^2\, \xi_t^2) \;\mathbb{I}_{t-1}] =\\
= \mathrm{E}[ (-\varphi_{t-1}^2 + \frac M2 \gamma_{t-1}\, \varphi_{t-1}^2 + {M\over 2} \gamma_{t-1} S) \gamma_{t-1} \;\mathbb{I}_{t-1}] = \\
= \mathrm{E}[ (-\varphi_{t-1}^2(1-M\gamma_{t-1}/2) + M\gamma_{t-1} S/2) \gamma_{t-1} \;\mathbb{I}_{t-1}].
\end{array}
\end{equation}
If $\;\mathbb{I}_{t-1}=1$ then~ either~ (i)~ $x_{t-1} \in [-R,R]\setminus{\cal O}$
and $\gamma_{t-1} < g$,~~ or~ (ii)~ $|x_{t-1}| \ge R$.
In the case (i) one has
\begin{equation}\label{eq23.1}
-\varphi_{t-1}^2(1-M\gamma_{t-1}/2) + M\gamma_{t-1}S/2 \le -c_0(1-Mg/2) + M g S/2 =: -c'_g,
\end{equation}
where $c_0 := \inf\{ |\varphi(x)| : x\in[-R,R]\setminus{\cal O} \}$;
obviously, $c_0>0$. Let us fix a $g \in (0,\bar{\mathrm{g}})$ such that
$c'_g>0$.
In the case (ii), designating $b_0 := \inf_{|x|\ge R}
\varphi^2(x)$, one has
\begin{equation}\label{eq23.2}
-\varphi_{t-1}^2(1-M\gamma_{t-1}/2) + M\gamma_{t-1} S/2 \le
-b_0(1-M\bar{\mathrm{g}}/2)+M\bar{\mathrm{g}} S/2 =: -c''.
\end{equation}
Using \A{A6}, one gets that $c''>0$.
Denote $c=\min\{c'_g,c''\}$. The relations (\ref{eq23.1}) and
(\ref{eq23.2}) imply that if $\;\mathbb{I}_{t-1}=1$ then
$-\varphi_{t-1}^2(1-M\gamma_{t-1}/2) + M\gamma_{t-1} S/2 \le -c <
0$, hence, by virtue of (\ref{eq23}),
\begin{equation}\label{eq24}
E_t - E_{t-1} \le -c \cdot \mathrm{E}[\gamma_{t-1} \;\mathbb{I}_{t-1}].
\end{equation}
Summing up both sides of (\ref{eq24}) over $t=1,\ldots,s$ and
denoting $\;\mathbb{I}_\infty = \;\mathbb{I}(\tau=\infty)=\min_t \;\mathbb{I}_t$, one obtains
\[
\mathrm{E}_s - \mathrm{E}_0 \le -c \cdot \mathrm{E} \left[\sum_{i=0}^{s-1} \gamma_i \cdot \;\mathbb{I}_\infty
\right].
\]
One has $\mathrm{E}_s \ge 0$, and $x_0$ is bounded, hence $E_0 < \infty$.
Thus, for arbitrary $s$
\[
\mathrm{E} \left[\sum_{i=0}^{s-1} \gamma_i \cdot \;\mathbb{I}_\infty \right] \le \frac{\mathrm{E}_0}{c} < \infty.
\]
This implies that a.s. either $\sum_0^\infty \gamma_i < \infty$,
or $\tau = \infty$. Lemma 3 is proved. \hfill$\Box$
\vspace{2mm}
Denote $c_1 := 1- M \bar{\mathrm{g}}/2$. Recall that $f$ is the primitive of
$\varphi$ such that $\inf_x f(x) = 0$; the assumption \A{A6}
implies that $\lim_{x\to\pm \infty} f(x) = +\infty$. Denote $H:=
\sup_{|x|\le R} f(x)$. Denote also~ $c_3:= \bar{\mathrm{g}} \cdot \sup\{
|\varphi(x)| : f(x) \le H\} + 1$,~ $z^{l} := \inf \{ x : f(x) \le
H \} - c_3$,\ $z^{r} := \sup\{ x : f(x) \le H\} + c_3$,~ $c_2 :=
\inf\{ |\varphi(x)| : x \in [z^{l},\, z^{r}] \setminus{\cal O}\}$,~ and~
$\mathrm{K} := \sup\{ |\varphi(x)| : x\in[z^{l},\, z^{r}]\}$.~
Obviously, $c_1
> 0$ and $\mathrm{K}\ge c_2 > 0$.
Fix an open set ${\cal O}$ containing $\mathrm{Z}$. Let $g > 0$,\, $0 < w <
1$. We shall say that a (finite or infinite) deterministic
sequence $\{z_0, z_1, z_2, \ldots \}$ is $(g,\,
w)$-admissible if $|z_0|\le R$ and there exist deterministic
sequences $\{q_t\},$ $\{h_t\}$ such that
1) $|h_t| \le w$;
2) if $\{ z_0,z_1,\ldots,z_t\}
\subset[z^{l},\,z^{r}]\setminus{\cal O}$~ then~ $g d^2 \le q_s
\le \bar{\mathrm{g}}$,~ $s=0,1,\ldots,t$;
3) $z_t = z_{t-1} - q_{t-1}\, \varphi(z_{t-1})-h_t$,~
$t=1,2,\ldots$.
\begin{proposition}
There exists constants $t_0$ and $w$ such that any $(g,\,
w)$-admissible sequence $\{z_t,\ t=0,\, 1,\ldots,t_0\}$ has
non-empty intersection with ${\cal O}$.
\end{proposition}
\textit{Proof.} Let $w:= \min\{ 1,\, g d^2 c_2^2
c_1/(2\mathrm{K})\}$. Designate $\tilde{t}=\inf\{ t:z_t\in{\cal O}\}$;\,
$\tilde{t}$ takes values from $\{0,\, 1, \ldots, t_0,\,
+\infty\}$. We shall use shorthand notation $f_t := f(z_t)$,
$f'_t= \varphi_t := \varphi(z_t)$. One has
\begin{equation}\label{eq25}
f_t = f(z_{t-1}-q_{t-1} \varphi_{t-1} - h_t) = f(z_{t-1} - q_{t-1}
\varphi_{t-1}) - f'(\tilde z).h_t,
\end{equation}
where $\tilde z$ is a point between $z_{t-1} - q_{t-1}
\varphi_{t-1}$ and $z_{t-1} - q_{t-1} \varphi_{t-1} -h_t$.
Next, one has
\begin{equation}\label{eq26}
f(z_{t-1}-q_{t-1} \varphi_{t-1}) = f_{t-1} - f'_{t-1} q_{t-1}
\varphi_{t-1} + \frac 12 f''(\hat z)\, q_{t-1}^2 \varphi_{t-1}^2,
\end{equation}
where $\hat z$ is a point between $z_{t-1}$ and $z_{t-1}-q_{t-1}
\varphi_{t-1}$.
We are going to prove by induction that
\begin{equation}\label{eq29}
\textrm{if } 0\le s \le \tilde{t} \ \textrm{ then } \ f_s \le H-s\cdot g d^2 c_2^2 c_1 / 2.
\end{equation}
For $s=0$,~ (\ref{eq29}) follows from the condition $|z_0|\le R$
and the definition of $H$. Now, let $1 \le t \le \tilde{t}$;~
suppose that formula (\ref{eq29}) is true for $0 \le s \le t-1$
and prove it for $s = t$. For $0 \le s \le t-1$, one has $f(z_s)
\le H$,\, $z_s \not \in {\cal O}$, therefore $z_s \in [z^{l},\, z^{r}]
\setminus {\cal O}$; hence, by virtue of 2), $g d^2 \le q_s \le
\bar{\mathrm{g}}$ for $0 \le s \le t-1$.
One has $f(z_{t-1}) \le H$,\, $|q_{t-1} \varphi_{t-1}| \le \bar{\mathrm{g}}
\cdot \sup\{ |\varphi(x)| : f(x) \le H\}$, and $|h_t| \le w \le
1$, hence $|q_{t-1} \varphi_{t-1}| \le c_3$,\, $|q_{t-1}
\varphi_{t-1} + h_t| \le c_3$, and so, $z_{t-1} - q_{t-1}
\varphi_{t-1} \in [z^{l},\, z^{r}]$,\,
$z_{t-1}-q_{t-1}\varphi_{t-1} -h_t \in [z^{l},\, z^{r}]$, thus
$\tilde z$ also belongs to $[z^{l},\, z^{r}]$. This implies that
$|\varphi(\tilde z)| = |f'(\tilde z)| \le \mathrm{K}$. Then, combining
(\ref{eq25}) and (\ref{eq26}) and using that $|h_t| \le w$ and
$|f''(\hat z)| = |\varphi'(\hat z)|\le M$, one obtains
\begin{equation}\label{eq27}
f_t \le f_{t-1} - q_{t-1} \varphi^2_{t-1}(1-{1\over 2} q_{t-1} M) + w\mathrm{K}.
\end{equation}
One has $z_{t-1} \in [z^{l},\, z^{r}] \setminus {\cal O}$, hence
$|\varphi(z_{t-1})| = |\varphi_{t-1}| \ge c_2$. Using also that
$q_{t-1} \ge g d^2$,\, $1-{1 \over 2} q_{t-1} M \ge c_1$, and
$w\mathrm{K} \le g d^2 c_2^2 c_1/2$, one gets from (\ref{eq27})
that
\[
f_t \le f_{t-1} - g d^2 c_2^2 c_1 / 2,
\]
and using the induction hypothesis, one concludes that
\[
f_t \le H - t \cdot g d^2 c_2^2 c_1 /2.
\]
Formula (\ref{eq29}) is proved.
Let $t_0 := \lfloor 2H/(g d^2 c_2^2 c_1) \rfloor + 1$; here
$\lfloor z \rfloor$ stands for the integral part of $z$. Then,
taking into account that $f_s \ge 0$, from (\ref{eq29}) one
concludes that $\tilde{t} < t_0$, thus Proposition 1 is proved.
\hfill$\Box$.
\begin{proposition}
If $\gamma_{t-1} < 1/(3M)$,\, $|\xi_t|<c_2$,\, $|\xi_{t+1}|<
c_2$,\, $x_{t-1}$ and $x_t$ belong to $[z^{l},\, z^{r}] \setminus
{\cal O}$,~ then $\gamma_{t+1} \ge \gamma_t$.
\end{proposition}
\textit{Proof.} Using notation $\varphi_t := \varphi(x_t)$, one gets
$$
\varphi_t = \varphi(x_{t-1} - \gamma_{t-1}(\varphi_{t-1}+\xi_t)) =
\varphi_{t-1} - \varphi'(\tilde x) \cdot
\gamma_{t-1}(\varphi_{t-1}+\xi_t),
$$
where $\tilde x$ is a point between $x_{t-1}$ and $x_t$.
Therefore,
\[
\varphi_{t-1} \varphi_t = \varphi^2_{t-1} \cdot[ 1-
\varphi'(\tilde x) \gamma_{t-1} \cdot(1+\xi_t / \varphi_{t-1})].
\]
Using that $|\varphi'(\tilde x)| \le M$,\, $\gamma_{t-1} <
1/(3M)$,\, $|\xi_t| < c_2$,\, $|\varphi_{t-1}| \ge c_2$, one
obtains $1-\varphi'(\tilde x)\, \gamma_{t-1} \cdot (1 + \xi_t /
\varphi_{t-1}) \ge 1/3$, hence $\varphi_{t-1} \varphi_t >0$.
Further, using that $|\xi_t| < c_2$,\, $|\xi_{t+1}|<c_2$,\,
$|\varphi_{t-1}| \ge c_2$,\, $|\varphi_t| \ge c_2$, one gets
\[
y_{t}\, y_{t+1} = \varphi_{t-1} \varphi_t
\cdot(1+\xi_t/\varphi_{t-1})(1+\xi_{t+1}/\varphi_t) > 0.
\]
This implies that $\gamma_{t+1} = \min\{ u \gamma_t, \bar{\mathrm{g}}\} \ge
\gamma_t$. \hfill$\Box$
\begin{lemma
For any open set ${\cal O}$, containing $\mathrm{Z}$, and any $g > 0$~
there exists $\delta = \delta({\cal O}, g) > 0$ such that
\[
\text{if } \ |x_0| \le R, \ \gamma_0 \ge g \ \text{ then
} \ \ \mathrm{P}(\textrm{for some } t, \ x_t \in {\cal O}) \ge \delta.
\]
\end{lemma}
\textit{Proof.} Without loss of generality suppose that $g <
1/(3M)$. Define the event
\[
A := \{ |\xi_i| < \min\{ c_2,\, w/\bar{\mathrm{g}} \}, \ i=1,2,\ldots,t_0\},
\]
where $w$ and $t_0$ are the same as in the proof of Proposition
1:~ $w = \min\{ 1,\, g d^2 c_2^2 c_1/(2\mathrm{K})\}$,\ $t_0 =
\lfloor 2H/(g d^2 c_2^2 c_1) \rfloor + 1$.
Denote
\[
\delta := P(A) = ( \mathrm{P}( |\xi_1| < \min\{ c_2,\, w/\bar{\mathrm{g}}\}))^{t_0};
\]
by virtue of \A{A3}\,(a), $\delta > 0$. Let us show that for any
elementary event $\omega \in A$, the sequence $\{ z_t =
x_t(\omega),\ t=0, 1, \ldots, t_0\}$ is $(g,\,
w)$-admissible.
One has $|z_0|=|x_0(\omega)| < R$. Further, one has $z_t = z_{t-1}
- q_{t-1} \varphi(z_{t-1})- h_t$, with $q_{t-1} =
\gamma_{t-1}(\omega)$,\, $h_t = \gamma_{t-1}(\omega)\,
\xi_t(\omega)$, and using that $\gamma_{t-1}(\omega) \le \bar{\mathrm{g}}$
and $|\xi_t(\omega)| < \omega / \bar{\mathrm{g}}$, one gets $|h_t| \le w$.
Thus, conditions 1) and 3) are verified.
Now, let $\{z_0, z_1, \ldots,z_t\} \subset [z^{l},\, z^{r}]
\setminus {\cal O}$,~ $t \le t_0$. Let $s_0 \in \{ 0,1,2,\ldots,t\}$ be
the minimal value such that $q_{s_0} = \min\{ q_0, q_1, \ldots,
q_t \}$. If $s_0=0$ then $\min\{ q_0, q_1,\ldots, q_t \} = q_0 =
\gamma_0(\omega)\ge g \ge g d^2$.
If $s_0=1$ then
$\min\{ q_0, q_1, \ldots, q_t\} = q_1 = \gamma_1(\omega) \ge
g d \ge g d^2$. If $s_0 \ge 2$ then
$\gamma_{s_0-2}(\omega) \ge 1/(3M)$; otherwise, using that
$|\xi_{s_0-1}| < c_2$,\, $|\xi_{s_0}| < c_2$,\,
$x_{s_0-2}(\omega)$ and $x_{s_0-1}(\omega)$ belong to $[z^{l},\,
z^{r}] \setminus {\cal O}$, and applying Proposition 2, one would
conclude that $\gamma_{s_0}(\omega) \ge \gamma_{s_0-1} (\omega)$,
which contradicts the definition of $s_0$.
Thus, $\gamma_{s_0}(\omega) \ge 1/(3M) \cdot d^2 \ge g d^2 $,
and therefore, $\min\{ q_0, q_1, \ldots, q_t \} =
\gamma_{s_0}(\omega) \ge g d^2$. So, the condition 2) is also
verified.
Now, applying Proposition 1 to the $(g,\, w)$-admissible
sequence $\{z_t\}$, one concludes that there exists a non-negative
$\tau \le t_0$ such that $z_{\tau} = x_\tau(\omega) \in {\cal O}$. This
implies that
\[
\mathrm{P}(\textrm{for some } t, \ x_t \in {\cal O}) \ge \mathrm{P}(A) = \delta.
\]
\hfill$\Box$
\begin{lemma
If $\sum_t \gamma_t = \infty$ then for any open set ${\cal O}$
containing $\mathrm{Z}$ there exists $t$ such that $x_t \in {\cal O}$.
\end{lemma}
\textit{Proof.} Let us fix an open set ${\cal O} \supset \mathrm{Z}$, and denote
$\delta = \delta({\cal O}, g({\cal O}))$. Combining Lemma 3 and Lemma 4,
one concludes that for any ${\cal O} \supset \mathrm{Z}$ there exists $\delta >
0$ such that whatever the initial conditions $x_0$, $\gamma_0$,
$\gamma_1$,
\[
\mathrm{P}(\textrm{for some } t, \ x_t \in {\cal O} \,{\Big |}\, \sum_t \gamma_t =
\infty) > \delta.
\]
Then one can choose a measurable integer-valued function
$n(\cdot,\cdot,\cdot)$ defined on $\mathbb{R} \times (0,\bar{\mathrm{g}}] \times
(0,\bar{\mathrm{g}}]$ such that for $\nu=n(x_0,\gamma_0,\gamma_1)$ one will
have
\[
\mathrm{P}(\textrm{for some } t\le \nu, \ x_t \in {\cal O} \,{\Big |}\, \sum_t \gamma_t=\infty) > \delta/2
\]
Designate
\[
\bar p = \sup \mathrm{P}(\textrm{for all } t, \ x_t \not \in {\cal O} \,{\Big |}\,
\sum_t \gamma_t = \infty),
\]
the supremum being taken over all the initial conditions $x_0$, $\gamma_0$, $\gamma_1$.
Fix $x_0$, $\gamma_0$, $\gamma_1$, then
\begin{equation}\label{eq30}
\begin{array}{l}
\mathrm{P}(\textrm{for all } t, \ x_t \not \in {\cal O} \,{\Big |}\, \sum_t \gamma_t = \infty ) =\\
= \mathrm{P}(\textrm{for all } t > \nu, \ x_t \not \in {\cal O} \,{\Big |}\,
\textrm{for all } t \le \nu, \ x_t \not \in {\cal O} \textrm{ and } \sum_t \gamma_t =\infty) \cdot \\
\cdot \mathrm{P}(\textrm{for all } t \le \nu, \ x_t \not \in {\cal O}\, | \sum_t
\gamma_t = \infty) \le \bar p\, (1-\delta/2).
\end{array}
\end{equation}
Taking supremum of the left hand side of (\ref{eq30}) over all
$(x_0, \gamma_0, \gamma_1) \in \mathbb{R} \times (0,\bar{\mathrm{g}}] \times
(0,\bar{\mathrm{g}}]$, one obtains $\bar p \le \bar p\, (1- \delta/2)$,
hence $\bar p = 0$. Lemma~5 is proved. \hfill$\Box$.
\vspace{2mm}
Denote ${\cal O}_* = \{ x: |\varphi(x)| < L/2 \}$.
\begin{lemma
For any open bounded sets $\mathcal{O}$,\, $\mathcal{O}_1$
such that $\bar{\mathcal{O}} \subset \mathcal{O}_1 \subset {\cal O}_*$
and for any $w > 0$ there exists $\delta = \delta({\cal O}, {\cal O}_1, w)> 0$
such that
$$
\text{if } \ x_0 \in \mathcal{O} \text{ then } \ \mathrm{P}(\textrm{for
some } n,\ x_{n} \in {\cal O}_1 \text{ and } \gamma_{n} < w) \ge \delta.
$$
\end{lemma}
\textit{Proof.} Denote $n = \lfloor \frac{\ln\bar{\mathrm{g}} - \ln w}{\ln
(1/d)} \rfloor + 2$. Denote also
$$
\varepsilon = \min \left\{ \frac{L}{2},\ \frac{
\partial (\mathcal{O},\, \mathbb{R} \setminus \mathcal{O}_1)}{n \bar{\mathrm{g}}}
\right\},
$$
where $\partial (A,B) := \sup_{x\in A} \inf_{y\in B} |x-y|$ for
arbitrary sets of real numbers $A$,\, $B$. Using assumption
\A{A3}\,(a), one obtains that there exists $\delta_1 > 0$ such
that for any $x \in {\cal O}_1$ and for any integer $t$,
$$
\mathrm{P} \left( (-1)^{t-1} \varphi(x) < (-1)^t \xi_1 < (-1)^{t-1}
\varphi(x) + \varepsilon \right) \ge \delta_1.
$$
This implies that if $x_0 \in {\cal O}$ then
$$
\mathrm{P} (0 < (-1)^t y_t < \varepsilon,\ \text{dist}(x_{t-1},\, {\cal O}) < (t
- 1) \bar{\mathrm{g}} \varepsilon,\ t = 1,\, 2,\ldots, n+1) \ge
\delta_1^{n+1}.
$$
Denoting $\delta = \delta_1^{n+1}$, one concludes that the
following statements (i) and (ii) hold with probability at least
$\delta$:
(i) dist$(x_n,\, {\cal O}) < n \bar{\mathrm{g}} \varepsilon \le$ dist$({\cal O},\,
\mathbb{R} \setminus {\cal O}_1)$, hence $x_n \in {\cal O}_1$;
(ii) as $t = 2,\, 3,\ldots, n+1$, one has $y_{t-1} y_t < 0$, hence
$\gamma_t = d \gamma_{t-1}$, therefore
$\gamma_{n} = d^{n-1} \gamma_1 \le d^{n-1} \bar{\mathrm{g}} < w$.\\
Lemma~6 is proved. \hfill$\Box$
\begin{lemma
If $\sum_t \gamma_t = \infty$,\ ${\cal O}$ is an open set containing
$\mathrm{Z}$, and $w > 0$ then for some $t$,~ $x_{t-1} \in {\cal O}$ and
$\gamma_t < w$.
\end{lemma}
\textit{Proof.} Without loss of generality, suppose that ${\cal O}$ is
bounded and ${\cal O} \subset {\cal O}_*$. Choose an open set ${\cal O}_1$ such that
$\mathrm{Z} \subset {\cal O}_1$,\, $\bar{\cal O}_1 \subset {\cal O}$;~ applying Lemmas 5 and
6, one gets that for $\delta = \delta({\cal O}_1, {\cal O}, w)$ and for
arbitrary initial conditions,
$$
\mathrm{P} (\textrm{for some } t,\ x_{t} \in {\cal O} \textrm{ and } \gamma_t <
w)
> \delta.
$$
Repeating the argument of Lemma 5, one concludes that there
exists $t$ such that $x_{t} \in {\cal O}$ and $\gamma_t < w$.
\hfill$\Box$
\vspace{2mm}
From now on we suppose that $\mathrm{k} > k_+(0)$. Choose $k'$
such that $k_+(0) < k' < \mathrm{k}$; using \A{A3}\,(b), one
obtains that for some $\varepsilon_0 > 0$,\, $\mathrm{P} ( \xi_1 \xi_2 >
0, \text{ or } |\xi_1| < \varepsilon_0, \text{ or } |\xi_2| <
\varepsilon_0) \le k'$. Denote ${\cal O}_0 = \{ x:\, |\varphi(x)| <
\varepsilon_0 \}$ and $\tau = \inf \{ t: \ x_t \not\in {\cal O}_0 \}$.
Without loss of generality, suppose that ${\cal O}_0$ is bounded.
\begin{lemma
Suppose that $\mathrm{k} > k_+(0)$,~ then there exist a constant
$b
> 0$ and a monotone decreasing function $p(\cdot)$ such that
$\lim_{a\to+\infty} p(a) = 0$ and
$$
\text{if } \ \gamma_0 < w \text{ then } \ \mathrm{P} (\ln\gamma_t < \ln v
- bt \text{ for all } t < \tau ) > 1 - p(v/w).
$$
\end{lemma}
\textit{Proof.} Define the sequences $\{ \rho_t \}$ and $\{
\sigma_t \}$ by
\begin{eqnarray*}
\rho_t &=& \ln u \cdot \;\mathbb{I}(\xi_{t-1} \xi_t > 0, \text{ or }
|\xi_{t-1}|
< \varepsilon_0, \text{ or } |\xi_t| < \varepsilon_0) +\\
&+& \ln d \cdot \;\mathbb{I}(\xi_{t-1} \xi_t \le 0 \ \, \& \, \ |\xi_{t-1}| \ge
\varepsilon_0 \ \, \& \ \, |\xi_t| \ge \varepsilon_0),
\end{eqnarray*}
$$
\sigma_t = \ln w + \sum_{i=1}^t \rho_i.
$$
Using (\ref{eqal4n}) and definition of $\tau$, one obtains that
for all $t < \tau$,\, $\gamma_t \le \sigma_t$. The variables
$\rho_t$ are identically distributed, take the values $\ln u$ and
$\ln d$, and
\begin{eqnarray*}
E\rho_t &=& \ln u \cdot \mathrm{P}(\xi_{t-1} \xi_t > 0, \text{ or }
|\xi_{t-1}| < \varepsilon_0, \text{ or } |\xi_t| < \varepsilon_0) +\\
&+& \ln d \cdot \mathrm{P}(\xi_{t-1} \xi_t \le 0 \ \, \& \, \ |\xi_{t-1}| \ge
\varepsilon_0 \ \, \& \ \, |\xi_t| \ge \varepsilon_0) \le\\
&\le& \ln u \cdot k' + \ln d \cdot (1 - k') < \ln u \cdot
\mathrm{k} + \ln d \cdot (1-\mathrm{k}) = 0.
\end{eqnarray*}
Moreover, the variables in the set $\{ \rho_t, \ t \text{ even}
\}$, as well as the variables in the set $\{ \rho_t, \ t \text{
odd} \}$, are independent.
Denote $b = -E\rho_t/2$. One has
$$
\mathrm{P} (\ln\gamma_t < \ln v - b t \ \text{ for all } t < \tau ) \ge \mathrm{P}
(\sigma_t < \ln v - b t \ \text{ for all } t) =
$$
$$
= \mathrm{P} (\sum_{i=1}^t (\rho_i + 2b) < \ln v - \ln w + b t \ \text{
for all } t ) \ge 1 - p(v/w),
$$
where $p(a) = p_1(a) + p_2(a)$,
$$
p_1(a) = \mathrm{P} \left( {\sum_{1\le i\le t}}' (\rho_i + 2 b) \ge \frac
{\ln a}2 + \frac b2\, t \ \text{ for all } t \right),
$$
$$
p_2(a) = \mathrm{P} \left( {\sum_{1\le i\le t}}'' (\rho_i + 2 b) \ge \frac
{\ln a}2 + \frac b2\, t \ \text{ for all } t \right);
$$
the sum $\sum'$ ($\sum''$) is taken over the even (odd) values of
$i$. Both $\sum'$ and $\sum''$ are sums of i.i.d.r.v. with zero
mean, hence both $p_1(a)$ and $p_2(a)$ tend to zero as $a \to
+\infty$. Lemma 8 is proved.
\hfill$\Box$
Define the stopping times $\tau_v = \inf \{ t: \ x_t \not\in {\cal O}_0
\text{ or } \ln\gamma_t \ge \ln v - bt \}$. Recall that $f$ is the
primitive of $\varphi$ such that $\inf_x f(x) = 0$. Fix an open
set ${\cal O}'$ such that $\mathrm{Z} \subset {\cal O}' \subset {\cal O}_0$ and
$\sup_{x\in{\cal O}'} f(x) < \inf_{x\not\in{\cal O}_0} f(x)$, and denote
$\delta = \inf_{x\not\in{\cal O}_0} f(x) - \sup_{x\in{\cal O}'} f(x)$.
\begin{lemma
Let $\mathrm{k} > k_+(0)$,\ $x_0 \in {\cal O}'$, and $\gamma_0 < w$,
then
$$
\mathrm{P} (\tau_v < \infty ) \le K\, v^2 + p(v/w);
$$
here $K$ is a positive constant, and $p(\cdot)$ satisfies the
statement of lemma 8.
\end{lemma}
\textit{Proof.} We shall use shorthand notation of Lemma 3: $f_t
:= f(x_t)$ and $\varphi_t := \varphi(x_t)$. According to
(\ref{eq21}), one has
$$
f_t - f_{t-1} \le -\gamma_{t-1} \varphi_{t-1} (\varphi_{t-1} +
\xi_t) + {M\over 2}\, \gamma_{t-1}^2(\varphi_{t-1} + \xi_t)^2 \le
$$
$$
\le -\gamma_{t-1} \varphi_{t-1} \xi_t + M \gamma_{t-1}^2
(\varphi_{t-1}^2 + \xi_t^2).
$$
This implies that $f_t - f_1 \le Q_t' + Q_t''$, with
$$
Q_t' = \big| \sum_{i=2}^{t} \gamma_{i-1} \varphi_{i-1} \xi_i
\big|, \ \ \ \ \ Q_t'' = M \sum_{i=2}^{t} \, \gamma_{i-1}^2
(\varphi^{2}_{i-1} + \xi_i^2).
$$
Using Lemma 8, one gets
$$
\mathrm{P} (\tau_v < \infty)\, \le\, p(v/w) + P' + P'',
$$
where
$$
P' = \mathrm{P} (Q'_{\tau_v} \ge {\delta}/{2}) \ \ \text{ and } \ \
P'' = \mathrm{P} (Q''_{\tau_v} \ge {\delta}/{2}).
$$
According to the Chebyshev inequality,
$$
P'\, \le\, \frac{4}{\delta^2}\, EQ'^2_{\tau_v} =
\frac{4}{\delta^2} \sum_{i,j=1}^{\infty} E_{ij},
$$
where
$$
E_{ij}\, =\, E \left[ \gamma_{i-1} \varphi_{i-1} \xi_i\, \;\mathbb{I}(i-1 <
\tau_v) \cdot \gamma_{j-1} \varphi_{j-1} \xi_j\, \;\mathbb{I}(j-1 < \tau_v)
\right].
$$
Using that the values $\gamma_i$,\, $\varphi_i$,\, $\xi_i$, and
$\;\mathbb{I} (i < \tau_v)$ are $\mathcal{F}_{i}$-measurable, and using
assumptions \A{A1} and \A{A2}, one obtains that for $i \ne j$,\,
$E_{ij} = 0$, and for $i = j$,
$$
E_{ii} = E \left[ \gamma_{i-1}^2 \varphi_{i-1}^2 \;\mathbb{I}(i-1 < \tau_v)
\cdot \xi_i^2 \right] \le v^2 e^{-2bi} \sup_{x\in{\cal O}_0}
\varphi^2(x) \cdot S.
$$
Therefore,
$$
P'\, \le\, \frac{4}{\delta^2} \sum_{i=2}^\infty E_{ii} \le
\frac{4v^2 S}{\delta^2}\ \frac{e^{-4b}}{1 - e^{-2b}}\
\sup_{x\in{\cal O}_0} \varphi^2(x).
$$
Similarly,
$$
P''\, \le\, \frac{2}{\delta}\, EQ_{\tau_v}'' =
\frac{2M}{\delta} \sum_{i=2}^\infty E \left[ \gamma^2_{i-1}
(\varphi^{2}_{i-1} + \xi_i^2) \;\mathbb{I}(i-1 < \tau_v) \right] \le
$$
$$
\le \frac{2M v^2}{\delta} \sum_{i=2}^\infty e^{-2bi} \left(
\sup_{x\in{\cal O}_0} \varphi^2(x) + S \right) = \frac{2M v^2}{\delta}\
\frac{e^{-4b}}{1 - e^{-2b}} \left( \sup_{x\in{\cal O}_0} \varphi^2(x) +
S \right).
$$
Taking
$$
K\, =\, \left[ \frac{4S}{\delta^2} \sup_{x\in{\cal O}_0} \varphi^2(x)\,
+\, \frac{2M}{\delta} \left( \sup_{x\in{\cal O}_0} \varphi^2(x) + S
\right) \right] \frac{e^{-4b}}{1 - e^{-2b}},
$$
one gets that $P' + P'' \le K\, v^2$. Lemma 9 is proved.
\hfill$\Box$
\begin{lemma
If $\mathrm{k} > k_+(0)$ then $\sum_t \gamma_t < \infty$.
\end{lemma}
\textit{Proof.} From the definition of $\tau_v$ one easily sees
that if $\tau_v = \infty$ for some $v > 0$, then $\sum_t \gamma_t
< \infty$. This implies that for any $v > 0$
\begin{equation}\label{1point}
\mathrm{P} \left(\sum \gamma_t\, =\, \infty \right) \le \mathrm{P}(\tau_v =
\infty).
\end{equation}
Further, by virtue of Lemma 9, if $x_0 \in {\cal O}'$ and $\gamma_0 < w$
then
\begin{equation}\label{2points}
\mathrm{P} (\tau_{\sqrt{w}}\, <\, \infty)\, \le\, Kw + p(1/\sqrt{w}).
\end{equation}
Combining (\ref{1point}) and (\ref{2points}), one gets that for
any $w > 0$
\begin{equation}\label{3points}
\mathrm{P} \left(\sum \gamma_t = \infty\ |\ x_0 \in {\cal O}' \text{ and }
\gamma_0 < w\right)\, \le\, Kw\, +\, p(1/\sqrt{w}).
\end{equation}
Define the event $\mathcal{A}_w = \{ \text{ for some } t,\ x_t \in {\cal O}'
\text{ and } \gamma_t < w \}$, then by virtue of (\ref{3points}),
\begin{equation}\label{101}
\mathrm{P} \left(\sum \gamma_t = \infty\ \big|\ \mathcal{A}_w \right) \le Kw +
p(1/\sqrt{w}).
\end{equation}
Denote by $\bar\mathcal{A}_w$ the complementary event, $\bar\mathcal{A}_w = \{
\text{ for any } t,\ x_t \not\in {\cal O}' \text{ or } \gamma_t \ge w
\}$. By virtue of Lemma 7,
\begin{equation}\label{102}
\mathrm{P} \left(\sum \gamma_t = \infty \ \, \& \, \ \bar\mathcal{A}_w \right) =
0.
\end{equation}
Using (\ref{101}) and (\ref{102}), one gets
\begin{eqnarray*}
\mathrm{P} \left( \sum \gamma_t = \infty \right) = \mathrm{P} \left( \sum
\gamma_t = \infty \, \ \& \ \, \mathcal{A}_w \right) + \mathrm{P} \left( \sum
\gamma_t = \infty \ \, \& \ \, \bar\mathcal{A}_w \right) \le
\end{eqnarray*}
$$
\le (Kw + p(1/\sqrt{w})) \cdot \mathrm{P} (\mathcal{A}_w).
$$
Taking into account that $w$ can be chosen arbitrarily small and
that $Kw + p(1/\sqrt{w}) \to 0$ as $w \to 0^+$, one concludes that
$\mathrm{P} \left( \sum_t \gamma_t = \infty \right) = 0$.
\hfill$\Box$
\vspace{2mm}
Now, we are in a position to prove the theorem. Suppose that
$\mathrm{k} < \inf_z k_-(z)$, then $V_-^{[\mathrm{k}]} =
\emptyset$, and by Lemma 2, $\{ x_t \}$ diverges. So, the
statement (b) of Theorem is proved.
On the other hand, according to Lemma 10, if $\mathrm{k} >
k_+(0)$ then $\sum_t \gamma_t < \infty$, and by Lemmas 1 and 2,
the sequence $\{ x_t \}$ converges to a point from
$V_-^{[\mathrm{k}]}$. Thus, the statement (a) of theorem is also
established.
\section*{Acknowledgements}
This work was partially supported by the R\&D Unit CEOC (Center
for Research in Optimization and Control). The second author (PC)
also gratefully acknowledges the financial support by the
Portuguese program PRODEP `Medida 5 - Ac\c c\~ao 5.3 - Forma\c
c\~ao Avan\c cada de Docentes do Ensino Superior - Concurso nr.
2/5.3/PRODEP/2001'.
|
{
"timestamp": "2005-03-21T19:44:42",
"yymm": "0503",
"arxiv_id": "math/0503434",
"language": "en",
"url": "https://arxiv.org/abs/math/0503434"
}
|
\section{Introduction}
\subsection{}
This work arose from an attempt to understand the results of the
paper ~\cite{gl} of A.~Givental and Y.-P.~Lee where the authors
perform some computations related to ``quantum $K$-theory" of flag
varieties (as well as some results from ~\cite{neok} related to 5d
$SU(n)$-gauge theory compactified on a circle) in the framework of
representation theory. Similar approach to quantum cohomology of
flag varieties (and to partition functions of 4d gauge theory) is
discussed in ~\cite{b} and ~\cite{be}.
In \cite{gl} the authors consider the moduli spaces $\fQ_{\ul{d}}$
introduced by G.~Laumon in ~\cite{la1}, ~\cite{la2}. These are
certain closures of the moduli spaces of based maps of degree
$\ul{d}$ from $\BP^1$ to the flag variety $\CB$ of
$\mathfrak{sl}_n$.
A Cartan torus $T$ of $SL_n$ acts on $\fQ_{\ul{d}}$. The
multiplicative group $\BC^*$ of dilations of $\BP^1$ (loop
rotations) also acts on $\fQ_{\ul{d}}$. The formal character of
the (infinite dimensional) $T\times\BC^*$-module
$R\Gamma(\fQ_{\ul{d}},\CO_{\ul{d}})$ turns out to be a rational
function on $T\times\BC^*$.
One may form a certain generating function $\fJ$ of these rational
functions for all degrees $\ul{d}$. Computing the function $\fJ$
presumably should give rise to a computation of the
$SL_n$-equivariant quantum $K$-theory ring of $\CB$ (which to the
best of the authors' knowledge has not yet been defined in the
literature).
A.~Givental and Y.-P.~Lee prove that $\fJ$ satisfies a certain
$v$-difference version of the quantum Toda lattice equations (here
$v$ stands for the tautological character of $\BC^*$). Moreover,
they suggest another way to construct solutions of the
$v$-difference Toda system: as the Shapovalov scalar product of
the Whittaker vectors in the universal Verma module for the
quantum group $U_v(\mathfrak{sl}_n)$. The latter construction was
worked out independently in ~\cite{e}, ~\cite{s2}.
\subsection{}
The principal goal of the present paper is to identify these two
constructions of solutions of the $v$-difference Toda system.
Namely, we prove that the natural correspondences between the
moduli spaces $\fQ_{\ul{d}}$ (for the degrees differing by a
simple root) give rise to the action of the standard generators of
$U_v(\mathfrak{sl}_n)$ on the localized equivariant $K$-theory
$\oplus_{\ul{d}}\ul{K}^{T\times\BC^*} (\fQ_{\ul{d}})$. Here the
localization is taken with respect to the
$K^{T\times\BC^*}(\cdot)=\BC[T\times\BC^*]$, that is, we tensor
everything with the fraction field of $\BC[T\times\BC^*]$. This is
needed since the above correspondences are not proper, but the
subspaces of their $T\times\BC^*$-fixed points are proper (in
fact, they are finite), so their action is well defined only in
the localized equivariant $K$-theory. This way we get a
$U_v(\mathfrak{sl}_n)$-module, and we identify it with the
universal Verma module $M$. We also compute in geometric terms the
Shapovalov scalar product on $M$, and the Whittaker vectors. It
turns out that the generating function for the Shapovalov scalar
product of the Whittaker vectors is a simple modification of the
Givental-Lee generating function $\fJ$. Thus we reprove the Main
Theorem of Givental-Lee.
\subsection{}
There is a similar generating function $J$ for equivariant
integrals of the unit cohomology classes of $\fQ_{\ul{d}}$ which
controls the $T$-equivariant quantum cohomology of $\CB$. It
satisfies the quantum Toda lattice differential system, as proved
originally by A.~Givental and B.~Kim.
For the simple Lie algebras $\fg$ other than $\mathfrak{sl}_n$
there is no analogue of the Laumon moduli spaces $\fQ_{\ul{d}}$
but there is Drinfeld's moduli space of Quasimaps
$\CZ_{\ul{d}}(\fg)$. It also exists for the case of affine Lie
algebras, under the name of Uhlenbeck compactification. In the
$\mathfrak{sl}_n$ case $\fQ_{\ul{d}}$ is a small resolution of
$\CZ_{\ul{d}}(\mathfrak{sl}_n)$. In the affine
$\widehat{\mathfrak{sl}}_n$ case
$\CZ_{\ul{d}}(\widehat{\mathfrak{sl}}_n)$ possesses a semismall
resolution of singularities: the moduli space $\CP_{\ul{d}}$ of
torsion free parabolic sheaves on $\BP^1\times\BP^1$ endowed with
some additional structures. Thus in the affine case we can define
an analog of the function $J$ which we denote by $J_{\aff}$ (this
is discussed in \cite{b}).
The generating function $J$ (for any simple $G$) is known to
satisfy the quantum (differential) Toda equations (cf. \cite{gk}
and \cite{kim}).
In the work ~\cite{b}, the generating function $J$ for equivariant
integrals of the unit cohomology classes of $\CZ_{\ul{d}}(\fg)$
was proved to satisfy the quantum Toda lattice by constructing the
action of the Langlands dual Lie algebra $\check\fg$ in the
equivariant Intersection Cohomology of the Drinfeld
compactifications. Also in the affine case the function $J_{\aff}$
was shown to satisfy some non-stationary analog of ``the most
basic" (quadratic) Toda equation. Thus ~\cite{b} offered a
representation theoretic explanation of the Givental-Kim results
as well as generalized them to the affine case. And the present
work is a multiplicative analogue of ~\cite{b} in the simplest
case of $\mathfrak{sl}_n$.
\subsection{}
It would be extremely interesting to extend our work to other
simple and affine Lie algebras. It would require something like an
equivariant ``IC $K$-theory'' of $\CZ_{\ul{d}}(\fg)$ which is not
defined at the moment. In case of $\mathfrak{sl}_n$ the IC
cohomology of $\CZ_{\ul{d}}(\mathfrak{sl}_n)$ coincides with the
cohomology of the small resolution $\fQ_{\ul{d}}$, while in the
affine case of $\widehat{\mathfrak{sl}}_n$ the IC cohomology of
$\CZ_{\ul{d}}(\widehat{\mathfrak{sl}}_n)$ is a direct summand in
the cohomology of the semismall resolution $\CP_{\ul{d}}$.
Accordingly, one might look for the correct ``IC $K$-theory'' of
$\CZ_{\ul{d}}(\widehat{\mathfrak{sl}}_n)$ as an appropriate direct
summand of the usual $K$-theory of $\CP_{\ul{d}}$.
This is sketched in the Section ~\ref{p}. Namely, similarly to the
case of Laumon spaces, the quantum affine group
$U_v(\widehat{\mathfrak{sl}}_n)$ acts by the natural
correspondences on the direct sum of localized equivariant
$K$-groups $\oplus_{\ul{d}}\ul{K}^{T\times\BC^*\times\BC^*}
(\CP_{\ul{d}})$. However, this module looks more like the
universal Verma module for $U_v(\widehat{\mathfrak{gl}}_n)$, and
we have to specify a certain submodule isomorphic to the universal
Verma module for $U_v(\widehat{\mathfrak{sl}}_n)$. Then we
construct geometrically the Shapovalov scalar product, and the
Whittaker vectors. It turns out that the Shapovalov scalar product
of the Whittaker vectors can be expressed via the formal
characters of the global sections
$R\Gamma(\CP_{\ul{d}},\CO_{\ul{d}})$ as in the case of
$\mathfrak{sl}_n$. However, we were unable to derive any
$v$-difference equation for the affine version of the generating
function $\fJ$.
\subsection{Acknowledgments}
M.F. is obliged to V.~Schechtman, A.~Stoyanovsky, B.~Feigin,
E.~Vasserot, and R.~Bezrukavnikov who, ever since the appearance
of ~\cite{fk}, urged him to consider its equivariant $K$-theory
analogue. While trying to guess the correct formulae in the low
ranks, we profited strongly from the computational help of
V.~Dotsenko, V.~Golyshev, A.~Kuznetsov. We are also grateful to
P.~Etingof and A.~Joseph for very useful explanations; to
M.~Kashiwara for bringing the reference ~\cite{nz} to our
attention, and to the referee for the valuable comments. We would
like to thank the Weizmann Institute and RIMS, Kyoto, as well as
the University of Chicago, for the hospitality and support.
M.F. was partially supported by the CRDF award RM1-2545-MO-03.
A.B. was partially supported by the NSF grant DMS-0300271.
\section{Laumon spaces and quantum groups}
\subsection{}
We recall the setup of ~\cite{fk}. Let $\bC$ be a smooth
projective curve of genus zero. We fix a coordinate $z$ on $\bC$,
and consider the action of $\BC^*$ on $\bC$ such that
$v(z)=v^{-2}z$. We have $\bC^{\BC^*}=\{0,\infty\}$.
We consider an $n$-dimensional vector space $W$ with a basis
$w_1,\ldots,w_n$. This defines a Cartan torus $T\subset
G=SL_n\subset Aut(W)$. We also consider its $2^{n-1}$-fold cover,
the bigger torus $\widetilde{T}$, acting on $W$ as follows: for
$\widetilde{T}\ni\ul{t}=(t_1,\ldots,t_n)$ we have
$\ul{t}(w_i)=t_i^2w_i$. We denote by $\CB$ the flag variety of
$G$.
\subsection{}
Given an $(n-1)$-tuple of nonnegative integers
$\ul{d}=(d_1,\ldots,d_{n-1})$, we consider the Laumon's
quasiflags' space $\CQ_{\ul{d}}$, see ~\cite{la2}, ~4.2. It is the
moduli space of flags of locally free subsheaves
$$0\subset\CW_1\subset\ldots\subset\CW_{n-1}\subset\CW=W\otimes\CO_\bC$$
such that $\on{rank}(\CW_k)=k$, and $\deg(\CW_k)=-d_k$.
It is known to be a smooth projective variety of dimension
$2d_1+\ldots+2d_{n-1}+\dim\CB$, see ~\cite{la1}, ~2.10.
\subsection{} We consider the following locally closed subvariety
$\fQ_{\ul{d}}\subset\CQ_{\ul{d}}$ (quasiflags based at
$\infty\in\bC$) formed by the flags
$$0\subset\CW_1\subset\ldots\subset\CW_{n-1}\subset\CW=W\otimes\CO_\bC$$
such that $\CW_i\subset\CW$ is a vector subbundle in a
neighbourhood of $\infty\in\bC$, and the fiber of $\CW_i$ at
$\infty$ equals the span $\langle w_1,\ldots,w_i\rangle\subset W$.
It is known to be a smooth quasiprojective variety of dimension
$2d_1+\ldots+2d_{n-1}$.
\subsection{}
\label{fixed points} The group $G\times\BC^*$ acts naturally on
$\CQ_{\ul{d}}$, and the group $\widetilde{T}\times\BC^*$ acts
naturally on $\fQ_{\ul{d}}$. The set of fixed points of
$\widetilde{T}\times\BC^*$ on $\fQ_{\ul{d}}$ is finite; we recall
its description from ~\cite{fk}, ~2.11.
Let $\widetilde{\ul{d}}$ be a collection of nonnegative integers
$(d_{ij}),\ i\geq j$, such that $d_i=\sum_{j=1}^id_{ij}$, and for
$i\geq k\geq j$ we have $d_{kj}\geq d_{ij}$. Abusing notation we
denote by $\widetilde{\ul{d}}$ the corresponding
$\widetilde{T}\times\BC^*$-fixed point in $\fQ_{\ul{d}}$:
$\CW_1=\CO_\bC(-d_{11}\cdot0)w_1,$
$\CW_2=\CO_\bC(-d_{21}\cdot0)w_1\oplus\CO_\bC(-d_{22}\cdot0)w_2,$
$\ldots\ \ldots\ \ldots\ ,$
$\CW_{n-1}=\CO_\bC(-d_{n-1,1}\cdot0)w_1\oplus\CO_\bC(-d_{n-1,2}\cdot0)w_2
\oplus\ldots\oplus\CO_\bC(-d_{n-1,n-1}\cdot0)w_{n-1}.$
\subsection{}
For $i\in\{1,\ldots,n-1\}$, and $\ul{d}=(d_1,\ldots,d_{n-1})$, we
set $\ul{d}+i:=(d_1,\ldots,d_i+1,\ldots,d_{n-1})$. We have a
correspondence $\CE_{\ul{d},i}\subset\CQ_{\ul{d}}\times
\CQ_{\ul{d}+i}$ formed by the pairs $(\CW_\bullet,\CW'_\bullet)$
such that for $j\ne i$ we have $\CW_j=\CW'_j$, and
$\CW'_i\subset\CW_i$, see ~\cite{fk}, ~3.1. In other words,
$\CE_{\ul{d},i}$ is the moduli space of flags of locally free
sheaves
$$0\subset\CW_1\subset\ldots\CW_{i-1}\subset\CW'_i\subset\CW_i\subset
\CW_{i+1}\ldots\subset\CW_{n-1}\subset\CW$$ such that
$\on{rank}(\CW_k)=k$, and $\deg(\CW_k)=-d_k$, while
$\on{rank}(\CW'_i)=i$, and $\deg(\CW'_i)=-d_i-1$.
According to ~\cite{la1}, ~2.10, $\CE_{\ul{d},i}$ is a smooth
projective algebraic variety of dimension
$2d_1+\ldots+2d_{n-1}+\dim\CB+1$.
We denote by $\bp$ (resp. $\bq$) the natural projection
$\CE_{\ul{d},i}\to\CQ_{\ul{d}}$ (resp.
$\CE_{\ul{d},i}\to\CQ_{\ul{d}+i}$). We also have a map $\br:\
\CE_{\ul{d},i}\to\bC,$
$$(0\subset\CW_1\subset\ldots\CW_{i-1}\subset\CW'_i\subset\CW_i\subset
\CW_{i+1}\ldots\subset\CW_{n-1}\subset\CW)\mapsto\on{supp}(\CW_i/\CW'_i).$$
The correspondence $\CE_{\ul{d},i}$ comes equipped with a natural
line bundle $\CL_i$ whose fiber at a point
$$(0\subset\CW_1\subset\ldots\CW_{i-1}\subset\CW'_i\subset\CW_i\subset
\CW_{i+1}\ldots\subset\CW_{n-1}\subset\CW)$$ equals
$\Gamma(\bC,\CW_i/\CW'_i)$.
Finally, we have a transposed correspondence
$^\sT\CE_{\ul{d},i}\subset \CQ_{\ul{d}+i}\times\CQ_{\ul{d}}$.
\subsection{}
Restricting to $\fQ_{\ul{d}}\subset\CQ_{\ul{d}}$ we obtain the
correspondence
$\fE_{\ul{d},i}\subset\fQ_{\ul{d}}\times\fQ_{\ul{d}+i}$ together
with line bundle $\fL_i$ and the natural maps $\bp:\
\fE_{\ul{d},i}\to\fQ_{\ul{d}},\ \bq:\
\fE_{\ul{d},i}\to\fQ_{\ul{d}+i},\ \br:\
\fE_{\ul{d},i}\to\bC-\infty$. We also have a transposed
correspondence $^\sT\fE_{\ul{d},i}\subset
\fQ_{\ul{d}+i}\times\fQ_{\ul{d}}$. It is a smooth quasiprojective
variety of dimension $2d_1+\ldots+2d_{n-1}+1$.
\subsection{}
We denote by ${}'M$ the direct sum of equivariant (complexified)
$K$-groups:
${}'M=\oplus_{\ul{d}}K^{\widetilde{T}\times\BC^*}(\fQ_{\ul{d}})$.
It is a module over
$K^{\widetilde{T}\times\BC^*}(pt)=\BC[\widetilde{T}\times\BC^*]=
\BC[t_1,\ldots,t_n,v\ :\ t_1\cdots t_n=1]$.
We define $M=\ {}'M\otimes_{K^{\widetilde{T}\times\BC^*}(pt)}
\on{Frac}(K^{\widetilde{T}\times\BC^*}(pt))$.
We have an evident grading $M=\oplus_{\ul{d}}M_{\ul{d}},\
M_{\ul{d}}=K^{\widetilde{T}\times\BC^*}(\fQ_{\ul{d}})
\otimes_{K^{\widetilde{T}\times\BC^*}(pt)}
\on{Frac}(K^{\widetilde{T}\times\BC^*}(pt))$.
\subsection{}
\label{operators} The grading and the correspondences
$^\sT\fE_{\ul{d},i},\fE_{\ul{d},i}$ give rise to the following
operators on $M$ (note that though $\bp$ is not proper, $\bp_*$ is
well defined on the localized equivariant $K$-theory due to the
finiteness of the fixed point sets):
$K_i=t_{i+1}t_i^{-1}v^{2d_i-d_{i-1}-d_{i+1}+1}:\ M_{\ul{d}}\to
M_{\ul{d}}$;
$L_i=t_1^{-1}\cdots t_i^{-1}v^{d_i+\frac{1}{2}i(n-i)}:\
M_{\ul{d}}\to M_{\ul{d}}$;
$f_i=\bp_*\bq^*:\ M_{\ul{d}}\to M_{\ul{d}-i}$;
$F_i=t_{i+1}^it_i^{-i}v^{2id_i-id_{i-1}-id_{i+1}-i}\bp_*\bq^*:\
M_{\ul{d}}\to M_{\ul{d}-i}$;
$e_i=-t_{i+1}^{-1}t_i^{-1}v^{d_{i+1}-d_{i-1}}\bq_*(\fL_i\otimes\bp^*):\
M_{\ul{d}}\to M_{\ul{d}+i}$,
$E_i=-t_{i+1}^{-i-1}t_i^{i-1}v^{(i-1)d_{i-1}+(i+1)d_{i+1}-2id_i-i}
\bq_*(\fL_i\otimes\bp^*):\ M_{\ul{d}}\to M_{\ul{d}+i}$.
\subsection{}
We recall the notations and results of ~\cite{s} in the special
case of quantum group of $SL_n$ type.
$U$ is the $\BC[v,v^{-1}]$-algebra with generators
$E_i,L_i^{\pm1},K_i^{\pm1},F_i,\ 1\leq i\leq n-1$, subject to the
following relations:
\begin{equation}
\label{och} L_iL_j=L_jL_i,\ K_1=L_1^2L_2^{-1},\
K_i=L_{i-1}^{-1}L_i^2L_{i+1}^{-1},\ K_{n-1}=L_{n-2}^{-1}L_{n-1}^2
\end{equation}
\begin{equation}
\label{ochev} L_iE_jL_i^{-1}=v^{\delta_{i,j}}E_j,\
L_iF_jL_i^{-1}=v^{-\delta_{i,j}}F_j
\end{equation}
\begin{equation}
\label{ochevidno}
E_iF_j-F_jE_i=\delta_{i,j}\frac{K_i-K_i^{-1}}{v-v^{-1}}
\end{equation}
\begin{equation}
\label{Serre1} |i-j|>1\ \Longrightarrow\
E_iE_j-E_jE_i=0=F_iF_j-F_jF_i
\end{equation}
\begin{equation}
\label{Serre2} |i-j|=1\ \Longrightarrow\
E_i^2E_j-(v+v^{-1})E_iE_jE_i+E_jE_i^2=0=
F_i^2F_j-(v+v^{-1})F_iF_jF_i+F_jF_i^2
\end{equation}
Sevostyanov considers elements $e_i,f_i\in U$ depending on a
choice of $(n-1)\times(n-1)$-matrices $n_{ij},\ c_{ij}$. We make
the following choice:
\begin{equation}
\label{vybor} n_{i,i}=-2i;\ n_{i,i+1}=n_{i,i-1}=i,
\end{equation}
otherwise $n_{ij}=0$.
\begin{equation}
\label{Rossii} i<n-1\ \Longrightarrow\ c_{i,i+1}=-1,\ c_{i+1,i}=1,
\end{equation}
otherwise $c_{ij}=0$. In other words, $c_{ij}=n_{ij}-n_{ji}$.
Then we have
\begin{equation}
\label{Seva} f_i:=L_{i-1}^iL_i^{-2i}L_{i+1}^iF_i=K_i^{-i}F_i,\
e_i:=E_iL_{i-1}^{-i}L_i^{2i}L_{i+1}^{-i}=E_iK_i^i.
\end{equation}
Clearly, the algebra $U$ is generated by
$e_i,L_i^{\pm1},K_i^{\pm1},f_i,\ 1\leq i\leq n-1$, and the
relations ~(\ref{ochev})--~(\ref{Serre2}) above are equivalent to
the relations ~(\ref{ochev'})--~(\ref{Serre2'}) below.
\begin{equation}
\label{ochev'} L_ie_jL_i^{-1}=v^{\delta_{i,j}}e_j,\
L_if_jL_i^{-1}=v^{-\delta_{i,j}}f_j
\end{equation}
\begin{equation}
\label{ochevidno'}
e_if_j-v^{c_{ij}}f_je_i=\delta_{i,j}\frac{K_i-K_i^{-1}}{v-v^{-1}}
\end{equation}
\begin{equation}
\label{Serre1'} |i-j|>1\ \Longrightarrow\
e_ie_j-e_je_i=0=f_if_j-f_jf_i
\end{equation}
\begin{equation}
\label{Serre2'} |i-j|=1\ \Longrightarrow\
e_i^2e_j-v^{c_{ij}}(v+v^{-1})e_ie_je_i+v^{2c_{ij}}e_je_i^2=0=
f_i^2f_j-v^{c_{ij}}(v+v^{-1})f_if_jf_i+v^{2c_{ij}}f_jf_i^2
\end{equation}
\subsection{Remark}
The elements $f_i$ of the subalgebra $U_{\leq0}$ generated by
$F_1,\ldots,F_{n-1},K_1,\ldots,K_{n-1}$ were introduced by
C.~M.~Ringel in ~\cite{r}. They are the natural generators of the
Hall algebra of the $A_{n-1}$-quiver with the set of vertices
$1,\ldots,n-1$, and orientation $i\longrightarrow i+1$. More
generally, Ringel's construction works for an arbitrary
orientation of an $ADE$ quiver, and produces Sevostyanov's
generators $f_i$ (in the simply laced case). It can be seen easily
that the set of Sevostyanov's matrices $c_{ij}$ (parametrizing the
choices of his ``Coxeter realizations'') is in a natural bijection
with the set of orientations of the corresponding quiver.
\subsection{}
We are finally able to formulate our main theorem. Recall the
operators $E_i,e_i,L_i^{\pm1},K_i^{\pm1},F_i,f_i$ on $M$ defined
in ~\ref{operators}.
\begin{thm}
\label{main} The operators $E_i,L_i^{\pm1},K_i^{\pm1},F_i,\ 1\leq
i\leq n-1$, on $M$ satisfy the relations
~(\ref{och})--~(\ref{Serre2}). Equivalently, the operators
$e_i,L_i^{\pm1},K_i^{\pm1},f_i,\ 1\leq i\leq n-1$, on $M$ satisfy
the relations ~(\ref{och}), ~(\ref{ochev'})--~(\ref{Serre2'}).
\end{thm}
The relations ~(\ref{och}) and ~(\ref{ochev}) are evident. The
relation ~(\ref{ochevidno}) for $i\ne j$ follows from a
transversality property formulated in the next subsection.
\subsection{}
We consider the subvarieties $\bp_{12}^{-1}(\fE_{\ul{d},i})$ and
$\bp_{23}^{-1}(\ ^\sT\fE_{\ul{d}+i-j,j})$ in
$\fQ_{\ul{d}}\times\fQ_{\ul{d}+i}\times\fQ_{\ul{d}+i-j}$.
Similarly, we consider the subvarieties $\bp_{12}^{-1}(\
^\sT\fE_{\ul{d}-j,j})$ and $\bp_{23}^{-1}(\fE_{\ul{d}-j,i})$ in
$\fQ_{\ul{d}}\times\fQ_{\ul{d}-j}\times\fQ_{\ul{d}+i-j}$.
\begin{lem}
\label{trans} For $i\ne j$ the intersection (a)
$\bp_{12}^{-1}(\fE_{\ul{d},i})\cap\bp_{23}^{-1}(\
^\sT\fE_{\ul{d}+i-j,j})$ in
$\fQ_{\ul{d}}\times\fQ_{\ul{d}+i}\times\fQ_{\ul{d}+i-j}$ (resp.
(b) $\bp_{12}^{-1}(\
^\sT\fE_{\ul{d}-j,j})\cap\bp_{23}^{-1}(\fE_{\ul{d}-j,i})$ in
$\fQ_{\ul{d}}\times\fQ_{\ul{d}-j}\times\fQ_{\ul{d}+i-j}$) is
transversal.
(c) $\bp_{12}^{-1}(\fE_{\ul{d},i})\cap\bp_{23}^{-1}(\
^\sT\fE_{\ul{d}+i-j,j}) \simeq \bp_{12}^{-1}(\
^\sT\fE_{\ul{d}-j,j})\cap\bp_{23}^{-1}(\fE_{\ul{d}-j,i})$.
\end{lem}
\begin{proof}
We prove (a). By definition, $\bp_{12}^{-1}(\fE_{\ul{d},i})$ is
the moduli space of pairs of flags
$$(0\subset\CW'_1=\CW_1\subset\CW'_2=\CW_2\subset
\ldots\subset\CW'_i\subset\CW_i
\subset\ldots\subset\CW'_{n-1}=\CW_{n-1}\subset\CW,$$
$$0\subset\CW'''_1\subset\CW'''_2\subset\ldots\subset\CW'''_{n-1}\subset\CW)$$
of prescribed ranks and degrees, while $\bp_{23}^{-1}(\
^\sT\fE_{\ul{d}+i-j,j})$ is the moduli space of pairs of flags
$$(0\subset\CW_1\subset\ldots\subset\CW_{n-1}\subset\CW,$$
$$0\subset\CW'_1=\CW'''_1\subset\CW'_2=\CW'''_2\subset
\ldots\subset\CW'_j\subset
\CW'''_j\subset\ldots\subset\CW'_{n-1}=\CW'''_{n-1}\subset\CW)$$
of prescribed ranks and degrees.
Their intersection is the moduli space of flags (say, $i<j$)
$$0\subset\CW'_1=\CW_1=\CW'''_1\subset\ldots\subset\CW'_i=\CW'''_i\subset\CW_i
\subset\ldots\subset\CW'_j=$$
$$=\CW_j\subset\CW'''_j\subset\ldots\subset
\CW'_{n-1}=\CW_{n-1}=\CW'''_{n-1}\subset\CW$$ of prescribed ranks
and degrees which is smooth according to ~\cite{la1}, ~2.10. This
implies that at any closed point of the scheme-theoretic
intersection $\bp_{12}^{-1}(\fE_{\ul{d},i})\cap\bp_{23}^{-1}(\
^\sT\fE_{\ul{d}+i-j,j})$ the Zariski tangent space to
$\bp_{12}^{-1}(\fE_{\ul{d},i})\cap\bp_{23}^{-1}(\
^\sT\fE_{\ul{d}+i-j,j})$ is the intersection of tangent spaces to
$\bp_{12}^{-1}(\fE_{\ul{d},i})$ and $\bp_{23}^{-1}(\
^\sT\fE_{\ul{d}+i-j,j})$. Comparing the dimensions we conclude
that the sum of tangent spaces to $\bp_{12}^{-1}(\fE_{\ul{d},i})$
and $\bp_{23}^{-1}(\ ^\sT\fE_{\ul{d}+i-j,j})$ must coincide with
the tangent space to
$\fQ_{\ul{d}}\times\fQ_{\ul{d}+i}\times\fQ_{\ul{d}+i-j}$. Hence
the intersection is transversal. This completes the proof of (a).
In (b) we prove similarly that $\bp_{12}^{-1}(\
^\sT\fE_{\ul{d}-j,j})\cap\bp_{23}^{-1}(\fE_{\ul{d}-j,i})$ is the
moduli space of flags (say, $i<j$)
$$0\subset\CW_1=\CW'''_1=\CW''_1\subset\ldots\subset\CW'''_i\subset\CW_i
=\CW''_i \subset\ldots\subset\CW_j\subset$$
$$\subset\CW'''_j=\CW''_j\subset \ldots\subset
\CW_{n-1}=\CW'''_{n-1}=\CW''_{n-1}\subset\CW$$ of prescribed ranks
and degrees which is smooth according to ~\cite{la1}, ~2.10. Hence
the intersection is transversal by the same argument as in the
proof of (a). This completes the proof of (b).
Part (c) was proved in ~\cite{fk}, ~3.6. We just recall that the
mutually inverse isomorphisms send a triple
$(\CW_\bullet,\CW'_\bullet,\CW'''_\bullet)$ to
$(\CW_\bullet,\CW''_\bullet,\CW'''_\bullet)$ where
$\CW''_\bullet:=\CW_\bullet+\CW'''_\bullet$, and a triple
$(\CW_\bullet,\CW''_\bullet,\CW'''_\bullet)$ to
$(\CW_\bullet,\CW'_\bullet,\CW'''_\bullet)$ where
$\CW'_\bullet:=\CW_\bullet\cap\CW'''_\bullet$.
\end{proof}
\subsection{}
We return to the proof of relation ~(\ref{ochevidno}) for $i\ne
j$. The composition $E_iF_j$ is given by the action of
correspondence
$$f(\ul{t})g(v)\bp_{13*}(\bp_{12}^*\fL_i
\stackrel{L}{\otimes}_{\CO_{\fQ_{\ul{d}}\times\fQ_{\ul{d}+i}\times
\fQ_{\ul{d}+i-j}}}\bp_{23}^*\CO_{^\sT\fE_{\ul{d}+i-j,j}})$$ where
$f$ (resp. $g$) is a certain monomial in $\ul{t}$ (resp. $v$).
Because of the transversality in ~\ref{trans}(a), $\bp_{12}^*\fL_i
\stackrel{L}{\otimes}_{\CO_{\fQ_{\ul{d}}\times\fQ_{\ul{d}+i}\times
\fQ_{\ul{d}+i-j}}}\bp_{23}^*\CO_{^\sT\fE_{\ul{d}+i-j,j}}$ is a
line bundle $\fL_{i,j}$ on
$\bp_{12}^{-1}(\fE_{\ul{d},i})\cap\bp_{23}^{-1}(\
^\sT\fE_{\ul{d}+i-j,j})$ whose fiber at a point
$(\CW_\bullet,\CW'_\bullet,\CW'''_\bullet)$ is equal to
$\Gamma(\bC,\CW_i/\CW'''_i)$.
Similarly, due to the transversality in ~\ref{trans}(b), the
composition $F_jE_i$ is given by the action of correspondence
$$f'(\ul{t})g'(v)\bp_{13*}(\fL'_{i,j})$$
where $f'$ (resp. $g'$) is a certain monomial in $\ul{t}$ (resp.
$v$), and $\fL'_{i,j}$ is a line bundle on $\bp_{12}^{-1}(\
^\sT\fE_{\ul{d}-j,j})\cap\bp_{23}^{-1}(\fE_{\ul{d}-j,i})$ whose
fiber at a point $(\CW_\bullet,\CW''_\bullet,\CW'''_\bullet)$ is
equal to $\Gamma(\bC,\CW_i/\CW'''_i)$.
Now the isomorphism in ~\ref{trans}(c) clearly takes $\fL_{i,j}$
to $\fL'_{i,j}$, and a routine check shows that
$f(\ul{t})g(v)=f'(\ul{t})g'(v)$. This completes the proof of the
relations ~(\ref{ochevidno}) for $i\ne j$.
\subsection{}
\label{matrix} To prove the relation ~(\ref{ochevidno}) for $i=j$
we use the localization to the fixed points.
According to the Thomason localization theorem (see e.g.
~\cite{cg}), restriction to the $\widetilde{T}\times\BC^*$-fixed
point set induces an isomorphism
$$K^{\widetilde{T}\times\BC^*}(\fQ_{\ul{d}})
\otimes_{K^{\widetilde{T}\times\BC^*}(pt)}
\on{Frac}(K^{\widetilde{T}\times\BC^*}(pt))\to
K^{\widetilde{T}\times\BC^*}(\fQ_{\ul{d}}^{\widetilde{T}\times\BC^*})
\otimes_{K^{\widetilde{T}\times\BC^*}(pt)}
\on{Frac}(K^{\widetilde{T}\times\BC^*}(pt))$$ (resp.
$$K^{\widetilde{T}\times\BC^*}(\fE_{\ul{d},i})
\otimes_{K^{\widetilde{T}\times\BC^*}(pt)}
\on{Frac}(K^{\widetilde{T}\times\BC^*}(pt))\to
K^{\widetilde{T}\times\BC^*}(\fE_{\ul{d},i}^{\widetilde{T}\times\BC^*})
\otimes_{K^{\widetilde{T}\times\BC^*}(pt)}
\on{Frac}(K^{\widetilde{T}\times\BC^*}(pt)))$$
The classes of the structure sheaves $[\widetilde{\ul{d}}]$ of the
$\widetilde{T}\times\BC^*$-fixed points $\widetilde{\ul{d}}$ (see
~\ref{fixed points}) form a basis in
$\oplus_{\ul{d}}K^{\widetilde{T}\times\BC^*}
(\fQ_{\ul{d}}^{\widetilde{T}\times\BC^*})
\otimes_{K^{\widetilde{T}\times\BC^*}(pt)}\on{Frac}
(K^{\widetilde{T}\times\BC^*}(pt))$. In order to compute the
matrix coefficients of $E_i,F_i$ in this basis, we have to know
the character of the $\widetilde{T}\times\BC^*$-action in the
tangent spaces $\CT_{\widetilde{\ul{d}}}\fQ_{\ul{d}}$ and also in
the tangent spaces to the fixed points in the correspondences.
This is the subject of the following Proposition.
\subsection{}
Note that a point $(\widetilde{\ul{d}},\widetilde{\ul{d}}{}')$
lies in the correspondence $\fE_{\ul{d},i}$ if and only if
$d_{k,j}=d'_{k,j}$ with a single exception $d'_{i,j}=d_{i,j}+1$
for certain $j\leq i$.
\begin{prop}
\label{zanudstvo} a) The character $\chi_{\widetilde{\ul{d}}}$ of
$\widetilde{T}\times\BC^*$ in the tangent space
$\CT_{\widetilde{\ul{d}}}\fQ_{\ul{d}}$ equals
$$\sum_{1\leq k,j\leq n-1}t_k^2t_j^{-2}\left(\sum_{l=0}^{d_{k-1,j}}v^{2l}-
\sum_{l=d_{k-1,j}-d_{k,k}+1}^{d_{k-1,j}}v^{2l}+ \sum_{n-1\geq
i\geq k,j}\sum_{l=d_{i,j}-d_{i,k}+1}^{d_{i,j}-d_{i+1,k}}v^{2l}
\right)-\sum_{1\leq j<k\leq n-1}t_k^2t_j^{-2}$$
where we set $d_{n,k}=0$.
b) The character
$\chi_{(\widetilde{\ul{d}},\widetilde{\ul{d}}{}')}$ of
$\widetilde{T}\times\BC^*$ in the tangent space
$\CT_{(\widetilde{\ul{d}},\widetilde{\ul{d}}{}')}\fE_{\ul{d},i}$
equals
$$\chi_{\widetilde{\ul{d}}}+
\sum_{k\leq i}t_j^2t_k^{-2}v^{2d'_{i,k}-2d_{i,j}}- \sum_{k\leq
i-1}t_j^2t_k^{-2}v^{2d_{i-1,k}-2d_{i,j}}$$
if $d'_{i,j}=d_{i,j}+1$ for certain $j\leq i$.
c) The character
$\lambda_{(\widetilde{\ul{d}},\widetilde{\ul{d}}{}')}$ of
$\widetilde{T}\times\BC^*$ in the fiber of $\fL_i$ at the point
$(\widetilde{\ul{d}},\widetilde{\ul{d}}{}')$ equals
$t_j^2v^{-2d_{i,j}}$ if $d'_{i,j}=d_{i,j}+1$.
\end{prop}
\begin{proof}
Let $\CQ$ be the moduli space of flags of locally free subsheaves
$$0\subset\CW_1\subset\CW_2\subset\ldots\subset\CW_r\subset\CW$$ of fixed
ranks. Then the tangent space $\CT_{\CW_\bullet}\CQ$ equals the
kernel of
$$\sum_{1\leq l<r}p_{l-1}^*\otimes\on{Id}-\on{Id}\otimes q_l:\
\oplus_l\on{Hom}(\CW_l,\CW/\CW_l)\twoheadrightarrow
\oplus_l\on{Hom}(\CW_l,\CW/\CW_{l+1})$$ where $p_l:\
\CW_l\hookrightarrow\CW_{l+1};\ q_l:\
\CW/\CW_l\twoheadrightarrow\CW/\CW_{l+1}$ (see e.g. ~\cite{gl},
~3.2).
The parts a), b) follow easily. The part c) is obvious.
\end{proof}
\subsection{}
Let us denote by
$S\chi_{\widetilde{\ul{d}}}=\Lambda^{-1}\chi_{\widetilde{\ul{d}}}$
(resp. $S\chi_{(\widetilde{\ul{d}},\widetilde{\ul{d}}{}')}=
\Lambda^{-1}\chi_{(\widetilde{\ul{d}},\widetilde{\ul{d}}{}')}$)
the character of $\widetilde{T}\times\BC^*$ in the symmetric
algebra $\on{Sym}^\bullet\CT_{\widetilde{\ul{d}}}\fQ_{\ul{d}}$
(resp.
$\on{Sym}^\bullet\CT_{(\widetilde{\ul{d}},\widetilde{\ul{d}}{}')}
\fE_{\ul{d},i}$). It is the inverse of the character of the
corresponding exterior algebra, thus it lies in the fraction field
$\on{Frac}(K^{\widetilde{T}\times\BC^*}(pt))$.
According to the Bott-Lefschetz fixed point formula, the matrix
coefficient
$\bp_*\bq^*_{[\widetilde{\ul{d}}{}',\widetilde{\ul{d}}]}$ of
$\bp_*\bq^*:\ M_{\ul{d}'}\to M_{\ul{d}}$ with respect to the basis
elements $[\widetilde{\ul{d}}]\in K^{\widetilde{T}\times\BC^*}
(\fQ_{\ul{d}}),\ [\widetilde{\ul{d}}{}']\in
K^{\widetilde{T}\times\BC^*} (\fQ_{\ul{d}'})$ (see ~\ref{matrix})
equals $S\chi_{(\widetilde{\ul{d}},\widetilde{\ul{d}}{}')}/
S\chi_{\widetilde{\ul{d}}{}'}$. Similarly, the matrix coefficient
$\bq_*(\fL_i\otimes\bp^*)_{[\widetilde{\ul{d}},\widetilde{\ul{d}}{}']}$
of $\bq_*(\fL_i\otimes\bp^*):\ M_{\ul{d}}\to M_{\ul{d}'}$ equals
$\lambda_{(\widetilde{\ul{d}},\widetilde{\ul{d}}{}')}
S\chi_{(\widetilde{\ul{d}},\widetilde{\ul{d}}{}')}/
S\chi_{\widetilde{\ul{d}}}$.
Hence, the matrix coefficient
$E_{i[\widetilde{\ul{d}},\widetilde{\ul{d}}{}']}$ of $E_i:\
M_{\ul{d}}\to M_{\ul{d}'}$ equals
$$-t_{i+1}^{-i-1}t_i^{i-1}v^{(i-1)d_{i-1}+(i+1)d_{i+1}-2id_i-i}
\lambda_{(\widetilde{\ul{d}},\widetilde{\ul{d}}{}')}
S\chi_{(\widetilde{\ul{d}},\widetilde{\ul{d}}{}')}/
S\chi_{\widetilde{\ul{d}}}.$$ And the matrix coefficient
$F_{i[\widetilde{\ul{d}},\widetilde{\ul{d}}{}']}$ of $F_i:\
M_{\ul{d}}\to M_{\ul{d}'}$ equals
$t_{i+1}^it_i^{-i}v^{2id_i-id_{i-1}-id_{i+1}-i}
S\chi_{(\widetilde{\ul{d}}{}',\widetilde{\ul{d}})}/
S\chi_{\widetilde{\ul{d}}}$.
Thus, Proposition ~\ref{zanudstvo} admits the following Corollary.
\begin{cor}
\label{coefficients}
$$E_{i[\widetilde{\ul{d}},\widetilde{\ul{d}}{}']}=
-t_{i+1}^{-i-1}t_i^{i-1}v^{(i-1)d_{i-1}+(i+1)d_{i+1}-2id_i-i}
t_j^2v^{-2d_{i,j}}\times$$
$$(1-v^2)^{-1}\prod_{j\ne k\leq i}(1-t_j^2t_k^{-2}v^{2d_{i,k}-2d_{i,j}})^{-1}
\prod_{k\leq i-1}(1-t_j^2t_k^{-2}v^{2d_{i-1,k}-2d_{i,j}})$$ if
$d'_{i,j}=d_{i,j}+1$ for certain $j\leq i$;
$$F_{i[\widetilde{\ul{d}},\widetilde{\ul{d}}{}']}=
t_{i+1}^it_i^{-i}v^{2id_i-id_{i-1}-id_{i+1}-i}\times$$
$$(1-v^2)^{-1}\prod_{j\ne k\leq i}(1-t_k^2t_j^{-2}v^{2d_{i,j}-2d_{i,k}})^{-1}
\prod_{k\leq i+1}(1-t_k^2t_j^{-2}v^{2d_{i,j}-2d_{i+1,k}})$$ if
$d'_{i,j}=d_{i,j}-1$ for certain $j\leq i$;
All the other matrix coefficients of $E_i,F_i$ vanish.
\end{cor}
Now the relation ~(\ref{ochevidno}) boils down to the following
identity.
\begin{prop}
\label{mrak}
$$\frac{t_it_{i+1}^{-1}v^{d_{i-1}-2d_i+d_{i+1}-1}-
t_i^{-1}t_{i+1}v^{-d_{i-1}+2d_i-d_{i+1}+1}}{v-v^{-1}}
(1-v^2)^2v^{d_{i-1}-d_{i+1}}t_it_{i+1}=$$
$$\sum_{j\leq i}t_j^2v^{-2d_{i,j}+2}
(1-t_i^2t_j^{-2}v^{2d_{i,j}-2d_{i+1,i}})
(1-t_{i+1}^2t_j^{-2}v^{2d_{i,j}-2d_{i+1,i+1}})\times$$
$$\times\prod_{k\leq i}^{k\ne j}(1-t_k^2t_j^{-2}v^{2d_{i,j}-2d_{i,k}})^{-1}
(1-t_j^2t_k^{-2}v^{2d_{i,k}-2d_{i,j}+2})^{-1}\times$$
$$\times\prod_{k\leq i-1}(1-t_k^2t_j^{-2}v^{2d_{i,j}-2d_{i+1,k}})
(1-t_j^2t_k^{-2}v^{2d_{i-1,k}-2d_{i,j}+2})-$$
$$-\sum_{j\leq i}t_j^2v^{-2d_{i,j}}
(1-t_i^2t_j^{-2}v^{2d_{i,j}-2d_{i+1,i}+2})
(1-t_{i+1}^2t_j^{-2}v^{2d_{i,j}-2d_{i+1,i+1}+2})\times$$
$$\times\prod_{k\leq i}^{k\ne j}(1-t_k^2t_j^{-2}v^{2d_{i,j}-2d_{i,k}+2})^{-1}
(1-t_j^2t_k^{-2}v^{2d_{i,k}-2d_{i,j}})^{-1}\times$$
$$\times\prod_{k\leq i-1}(1-t_k^2t_j^{-2}v^{2d_{i,j}-2d_{i+1,k}+2})
(1-t_j^2t_k^{-2}v^{2d_{i-1,k}-2d_{i,j}}).$$
\end{prop}
\begin{proof} We introduce the new variables $q:=v^2;\ s_j:=t_j^2v^{-2d_{ij}},\
1\leq j\leq i;\ r_k:=t_k^2v^{-2d_{i+1,k}},\ 1\leq k\leq i+1;\
p_k:=t_k^2v^{-2d_{i-1,k}},\ 1\leq k\leq i-1$. Then the LHS of
~\ref{mrak} equals
$$(1-q)\left(q\prod_{k=1}^{i+1}r_k\prod_{j=1}^is_j^{-1}-
\prod_{j=1}^is_j\prod_{k=1}^{i-1}p_k^{-1}\right)$$ while the RHS
of ~\ref{mrak} equals
$$\prod_{j=1}^is_j\prod_{k=1}^{i-1}p_k^{-1}
\left(q\sum_{j\leq i}s_j^{-2}\prod_{k=1}^{i+1}(s_j-r_k)
\prod_{k=1}^{i-1}(p_k-qs_j) \prod_{k\leq i}^{k\ne
j}(s_j-s_k)^{-1}(s_k-qs_j)^{-1}\right.-$$
$$\left.\sum_{j\leq i}s_j^{-2}\prod_{k=1}^{i+1}(s_j-qr_k)
\prod_{k=1}^{i-1}(p_k-s_j) \prod_{k\leq i}^{k\ne
j}(s_j-qs_k)^{-1}(s_k-s_j)^{-1}\right)$$ Dividing both the LHS and
the RHS by $\prod_{j=1}^is_j\prod_{k=1}^{i-1}p_k^{-1}$ we arrive
at
$$(1-q)(q\prod_{j=1}^is_j^{-2}\prod_{k=1}^{i-1}p_k\prod_{k=1}^{i+1}r_k-1)=$$
$$q\sum_{j\leq i}s_j^{-2}\prod_{k=1}^{i+1}(s_j-r_k)
\prod_{k=1}^{i-1}(p_k-qs_j) \prod_{k\leq i}^{k\ne
j}(s_j-s_k)^{-1}(s_k-qs_j)^{-1}-$$
$$\sum_{j\leq i}s_j^{-2}\prod_{k=1}^{i+1}(s_j-qr_k)
\prod_{k=1}^{i-1}(p_k-s_j) \prod_{k\leq i}^{k\ne
j}(s_j-qs_k)^{-1}(s_k-s_j)^{-1}.$$ If we subtract the LHS from the
RHS we obtain a rational expression in $s_j$ of degree 0, that is,
the degree of numerator is not bigger than the degree of
denominator. We see easily that as $s_j$ tends to $\infty$, the
difference of the RHS and the LHS tends to 0. The possible poles
of the difference can occur at $s_j=0,\ s_j=s_k,\ s_j=qs_k,\
s_j=q^{-1}s_k$. We see easily that the principal parts of the
difference at these points vanish. We conclude that the difference
is identically 0. This completes the proof of the Proposition.
\end{proof}
\subsection{}
\label{import} To finish the proof of relation ~(\ref{ochevidno})
we note that the commutator correspondence $E_iF_i-F_iE_i$ is
concentrated on the diagonal of $\fQ_{\ul{d}}\times\fQ_{\ul{d}}$.
This is proved exactly as in Lemma ~\ref{trans}. In other words,
$E_iF_i-F_iE_i$ is given by tensor product
$?\mapsto?\stackrel{L}{\otimes}X_i$ for certain $X_i\in
M_{\ul{d}}$. This means that in the basis $[\widetilde{\ul{d}}]$
the operator $E_iF_i-F_iE_i$ is diagonal. Now the Proposition
~\ref{mrak} computes the matrix coefficient
$(E_iF_i-F_iE_i)_{[\widetilde{\ul{d}},\widetilde{\ul{d}}]}$ and
proves that it equals
$\frac{K_i-K_i^{-1}}{v-v^{-1}}|_{M_{\ul{d}}}$. This completes the
proof of the relation ~(\ref{ochevidno}).
\subsection{}
\label{alternate} Alternatively, the relation ~(\ref{ochevidno})
follows from the next Conjecture. We consider a 2-dimensional
vector space with a basis $\fw_1,\fw_2$. Let $\fT$ be a torus
acting on $\fw_1$ (resp. $\fw_2$) via a character $\tau_1^2$
(resp. $\tau_2^2$). Let $\fZ_{\fd_1,\fd_2}$ be the moduli stack of
flags of coherent sheaves $\fW_1\subset\fW_2$ on $\bC$ locally
free at $\infty\in\bC$, equipped with a trivialization
$\fW_1|_{\infty}=\langle\fw_1\rangle,\
\fW_2|_{\infty}=\langle\fw_1,\fw_2\rangle$, and such that
$\deg\fW_1=-\fd_1,\ \deg\fW_2/\fW_1=-\fd_2$. We have a natural
correspondence
$\fE_{\fd_1}\subset\fZ_{\fd_1,\fd_2}\times\fZ_{\fd_1+1,\fd_2-1}$
formed by the pairs $(\fW_1,\fW_2;\fW'_1,\fW'_2)$ such that
$\fW'_1\subset\fW_1\subset\fW_2=\fW'_2$. The projection
$\fE_{\fd_1}\to\fZ_{\fd_1,\fd_2}$ (resp.
$\fE_{\fd_1}\to\fZ_{\fd'_1,\fd_2}$) is denoted by $\bp$ (resp.
$\bq$). Finally, $\fE_{\fd_1}$ is equipped with the line bundle
$\fL_{\fd_1}$ whose fiber at the point
$(\fW_1,\fW_2;\fW'_1,\fW'_2)$ equals $\Gamma(\bC,\fW_1/\fW'_1)$.
The stack $\fZ_{\fd_1,\fd_2}$ is smooth, and acted upon by
$\fT\times\BC^*$. So it makes sense to consider the operators
$$f:=\bp_*\bq^*:$$ $$K^{\fT\times\BC^*}(\fZ_{\fd_1,\fd_2})
\otimes_{K^{\fT\times\BC^*}(pt)}\on{Frac}(K^{\fT\times\BC^*}(pt))\to
K^{\fT\times\BC^*}(\fZ_{\fd_1-1,\fd_2+1})
\otimes_{K^{\fT\times\BC^*}(pt)}\on{Frac}(K^{\fT\times\BC^*}(pt)),$$
$$e:=-\tau_1^{-1}\tau_2^{-1}v^{\fd_1+\fd_2}
\bq_*(\fL_{\fd_1}\otimes\bp^*):$$
$$K^{\fT\times\BC^*}(\fZ_{\fd_1,\fd_2})
\otimes_{K^{\fT\times\BC^*}(pt)}\on{Frac}(K^{\fT\times\BC^*}(pt))\to
K^{\fT\times\BC^*}(\fZ_{\fd_1+1,\fd_2-1})
\otimes_{K^{\fT\times\BC^*}(pt)}\on{Frac}(K^{\fT\times\BC^*}(pt)),$$
$$K=\tau_1^{-1}\tau_2v^{\fd_1-\fd_2+1}:$$
$$K^{\fT\times\BC^*}(\fZ_{\fd_1,\fd_2})
\otimes_{K^{\fT\times\BC^*}(pt)}\on{Frac}(K^{\fT\times\BC^*}(pt))\to
K^{\fT\times\BC^*}(\fZ_{\fd_1,\fd_2})
\otimes_{K^{\fT\times\BC^*}(pt)}\on{Frac}(K^{\fT\times\BC^*}(pt))$$
\begin{conj}
\label{inteligent} $ef-fe=\frac{K-K^{-1}}{v-v^{-1}}$.
\end{conj}
\subsection{}
\label{derivation} To derive the relation ~(\ref{ochevidno}), or
equivalently, ~(\ref{ochevidno'}) for $j=i$ from Conjecture
~\ref{inteligent} we consider the map
$$\fz_{\ul{d}}:\ \fQ_{\ul{d}}\to\fZ_{d_i-d_{i-1},d_{i+1}-d_i},\
\CW_\bullet\mapsto(\CW_i/\CW_{i-1},\CW_{i+1}/\CW_{i-1}).$$ Then we
have
$$(\fQ_{\ul{d}}\times\fZ_{d_i-d_{i-1}+1,d_{i+1}-d_i-1})
\times_{\fZ_{d_i-d_{i-1},d_{i+1}-d_i}\times\fZ_{d_i-d_{i-1}+1,d_{i+1}-d_i-1}}
\fE_{d_i-d_{i-1}}=\fE_{\ul{d},i}\subset\fQ_{\ul{d}}\times\fQ_{\ul{d}+i}.$$
We also have the natural maps
$$\fe_{\ul{d},i}:\ \fE_{\ul{d},i}\to\fE_{d_i-d_{i-1}},$$
$$^\sT\fe_{\ul{d},i}:\ {}^\sT\fE_{\ul{d},i}\to\ {}^\sT\fE_{d_i-d_{i-1}},$$
$$\fh_{\ul{d},i}:\ \fE_{\ul{d}-i,i}\circ\ {}^\sT\fE_{\ul{d}-i,i}\to
\fE_{d_i-d_{i-1}-1}\circ\ {}^\sT\fE_{d_i-d_{i-1}-1},$$
$$'\fh_{\ul{d},i}:\ {}^\sT\fE_{\ul{d},i}\circ\fE_{\ul{d},i}\to\
{}^\sT\fE_{d_i-d_{i-1}}\circ\fE_{d_i-d_{i-1}}.$$ We may consider
$e_i$ (resp. $f_i,e,f$) as an element of
$K^{\widetilde{T}\times\BC^*}(\fE_{\ul{d},i})$ (resp.
$K^{\widetilde{T}\times\BC^*}(\ {}^\sT\fE_{\ul{d},i}),\\
K^{\widetilde{T}\times\BC^*}(\fE_{d_i-d_{i-1}}),\
K^{\widetilde{T}\times\BC^*}(\ {}^\sT\fE_{d_i-d_{i-1}})$). We
evidently have
$$\fe_{\ul{d},i}^*e=e_i,\ {}^\sT\fe_{\ul{d},i}^*f=f_i.$$
Moreover, according to ~\cite{n}, ~8.2 (Restriction of the
convolution to submanifolds), we have
\begin{equation}
\label{nak} \fh_{\ul{d},i}^*(e*f)=e_i*f_i,\
'\fh_{\ul{d},i}^*(f*e)=f_i*e_i.
\end{equation}
We already know from the argument in ~\ref{import} that the
correspondence $e_i*f_i-f_i*e_i$ acts as tensor multiplication
with a certain class $X_i\in M_{\ul{d}}$. Similarly, the
correspondence $e*f-f*e$ acts in
$K^{\widetilde{T}\times\BC^*}(\fZ_{d_i-d_{i-1},d_{i+1}-d_i})
\otimes_{K^{\widetilde{T}\times\BC^*}(pt)}
\on{Frac}(K^{\widetilde{T}\times\BC^*}(pt))$ as tensor
multiplication with a certain class $\fX\in
K^{\widetilde{T}\times\BC^*}(\fZ_{d_i-d_{i-1},d_{i+1}-d_i})
\otimes_{K^{\widetilde{T}\times\BC^*}(pt)}
\on{Frac}(K^{\widetilde{T}\times\BC^*}(pt))$ By ~(\ref{nak}) we
must have $X_i=\fz_{\ul{d}}^*\fX$. Thus the relation
~(\ref{ochevidno'}) for $j=i$ follows from Conjecture
~\ref{inteligent}.
\subsection{}
\label{univerma} To complete the proof of Theorem ~\ref{main} it
remains to check the relations ~(\ref{Serre1}), ~(\ref{Serre2}).
To this end we consider the algebra $\widetilde{U}$ given by the
generators $E_i,L_i^{\pm1},K_i^{\pm1},F_i,\ 1\leq i\leq n-1$, and
the relations ~(\ref{och})--~(\ref{ochevidno}). Thus, $U$ is the
quotient of $\widetilde{U}$ by the Serre relations.
We extend the scalars to
$\on{Frac}(\BC[L_1^{\pm1},\ldots,L_{n-1}^{\pm1}])$: we set
$$U'=U\otimes_\BC
\on{Frac}(\BC[L_1^{\pm1},\ldots,L_{n-1}^{\pm1}]),\
\widetilde{U}{}'=\widetilde{U}\otimes_\BC
\on{Frac}(\BC[L_1^{\pm1},\ldots,L_{n-1}^{\pm1}])$$ Note that
$\widetilde{U}{}'$ acts in $M$, so $U'$ acts in the quotient
$\overline{M}$ of $M$ by the two-sided ideal $\CI$ in
$\widetilde{U}{}'$ generated by the Serre relations. So it
suffices to check that $\overline{M}=M$, or equivalently, $\CI
M=0$.
Now $M$ has the size of the universal Verma module over $U'$ which
is an irreducible $U'$- (and $\widetilde{U}{}'$-) module. In
effect, a bijection between the set $\{[\widetilde{\ul{d}}]\}$,
and the set of Kostant partitions for $\mathfrak{sl}_n$ is defined
e.g. in ~\cite{fk}, 2.11. Hence we only have to check that $\CI
M\ne M$. But any element $x\in\CI$ of principal grading degree 0
annihilates the lowest weight vector $[(0,\ldots,0)]$ of $M$ since
we may shift the generators $e_i$ in the expression of $x$ to the
right.
This completes the proof of the Serre relations in $M$ along with
the proof of Theorem ~\ref{main}.
\subsection{Remark}
\label{joseph} (A.~Joseph) We have constructed a basis
$\{[\widetilde{\ul{d}}]\}$ in the universal Verma module $M$ over
$U$. Though we can not identify it with any known type of basis,
the parametrization of this basis coincides with the polyhedral
realization of the crystal base of $U^+_v(\mathfrak{sl}_n)$
corresponding to the reduced expression in the Weyl group of
$SL_n$:
$$w_0=s_{n-1}s_{n-2}\ldots s_1s_{n-1}s_{n-2}\ldots s_2\ldots s_{n-1}s_{n-2}
s_{n-1}$$ (see ~\cite{nz}).
\subsection{}
\label{Shapovalov} Recall that the universal Verma module $M$ over
$U$ is equipped with the symmetric Shapovalov form $(,)$ with
values in $\on{Frac}(\BC[\widetilde{T}\times\BC^*])$. It is
characterized by the properties
(a) $([\widetilde{\ul{d}}{}_0],[\widetilde{\ul{d}}{}_0])=1$ where
$[\widetilde{\ul{d}}{}_0]=[(0,\ldots,0)]$ is the lowest weight
vector;
(b) $(E_ix,y)=(x,F_iy)\ \ \forall\ x,y\in M$.
\medskip
We will write down a geometric expression for the Shapovalov form.
Evidently, the different weight spaces of $M$ are orthogonal with
respect to the Shapovalov form. We consider the line bundle $\D_i$
on $\fQ_{\ul{d}}$ whose fiber at the point $(\CW_\bullet)$ equals
$\det R\Gamma(\bC,\CW_i)$. We also define the line bundle
$\D:=\bigotimes_{i=1}^{n-1}\D_i$.
\begin{prop}
\label{Shapoval} For $\CG_1,\CG_2\in M_{\ul{d}}$ we have
$$(\CG_1,\CG_2)=(-1)^{\sum_{i=1}^{n-1}d_i}
v^{\sum_{i=1}^{n-1}2id_i^2-\sum_{i=2}^{n-1}(2i-1)d_id_{i-1}}
\prod_{i=1}^nt_i^{(2i-1)(d_{i-1}-d_i)}[R\Gamma(\fQ_{\ul{d}},
\CG_1\otimes\CG_2\otimes\D)]$$
\end{prop}
\begin{proof}
Since $\det R\Gamma$ is multiplicative in short exact sequences,
we have an equality of line bundles on the correspondence
$\fE_{\ul{d},i}:\ \bp^*\D=\bq^*\D\otimes\fL_i$. Now the projection
formula shows that the operators $\bp_*\bq^*$ and
$\bq_*(\fL_i\otimes\bp^*)$ are adjoint with respect to the pairing
$\CG_1,\CG_2\mapsto
R\Gamma(\fQ_{\ul{d}},\CG_1\otimes\CG_2\otimes\D)$. Finally, it is
easy to see that the $v,t$-factor takes care of the scaling
coefficients of our $E_i,F_i$.
\end{proof}
\subsection{}
\label{conjugate} While the operators $E_i,F_i$ are conjugate to
each other with respect to the Shapovalov form, the operators
$e_i,f_i$ are not. In fact, obviously, $e_i^*=K_i^{2i}f_i$. It is
known that a completion of the universal Verma module $M$ contains
a unique vector $\fk=\sum_{\ul{d}}\fk_{\ul{d}}$ (resp.
$\fw=\sum_{\ul{d}}\fw_{\ul{d}}$) such that
$\fk_{(0,\ldots,0)}=\fw_{(0,\ldots,0)}=[(0,\ldots,0)]$, and
$f_i\fk=(1-v^2)^{-1}\fk$ (resp. $e_i^*\fw=(1-v^2)^{-1}\fw$) for
any $i$ (the {\em Whittaker vectors}).
The following proposition gives a geometric construction of the
Whittaker vectors $\fk,\fw\in M$.
\begin{prop}
\label{hlop} a) $\fk_{\ul{d}}=[\CO_{\ul{d}}]$ (the class of the
structure sheaf of $\fQ_{\ul{d}}$);
b) $\fw_{\ul{d}}=
v^{\sum_{i=1}^{n-1}(1-2i)d_i^2-\sum_{i=2}^{n-1}(2-2i)d_id_{i-1}-
\sum_{i=1}^{n-1}d_i}
\prod_{i=1}^nt_i^{(2-2i)(d_{i-1}-d_i)}[\D^{-1}_{\ul{d}}]$.
\end{prop}
\begin{proof}
a) We have $\bq^*\CO_{\ul{d}+i}=\CO_{\fE_{\ul{d},i}}$.
Furthermore, since $\bp\times\br:\
\fE_{\ul{d},i}\to\fQ_{\ul{d}}\times(\bC-\infty)$ is proper and
birational, and both the source and the target are smooth, we have
$(\bp\times\br)_*[\CO_{\fE_{\ul{d},i}}]=
[\CO_{\ul{d}}]\boxtimes[\CO_{\bC-\infty}]$. In effect,
$(\bp\times\br)_*\CO_{\fE_{\ul{d},i}}=
\CO_{\ul{d}}\boxtimes\CO_{\bC-\infty}$, and the higher direct
images $R^{>0}(\bp\times\br)_*\CO_{\fE_{\ul{d},i}}$ vanish.
Finally, $pr_*[\CO_{\ul{d}}\boxtimes\CO_{\bC-\infty}]=
(1-v^2)^{-1}[\CO_{\ul{d}}]$ where $pr:\
\fQ_{\ul{d}}\times(\bC-\infty)\to \fQ_{\ul{d}}$ is the projection
to the first factor.
b) Recall that $e_i^*=K_i^{2i}f_i$. Thus we have to check that
$f_i[\D^{-1}_{\ul{d}+i}]=t_i^2v^{2d_{i-1}-2d_i}(1-v^2)^{-1}[\D^{-1}_{\ul{d}}]$.
Furthermore, recall that on $\fE_{\ul{d},i}$ we have a canonical
isomorphism
$\bq^*\D^{-1}_{\ul{d}+i}=\fL_i\otimes\bp^*\D^{-1}_{\ul{d}}$. By
the projection formula we are reduced to
\begin{equation}
\label{zhelaem}
\bp_*[\fL_i]=t_i^2v^{2d_{i-1}-2d_i}(1-v^2)^{-1}[\CO_{\ul{d}}]
\end{equation}
This can be calculated in the basis $[\widetilde{\ul{d}}]$ where
we already know the matrix coefficients of our operators (see
Corollary ~\ref{coefficients}). More precisely, by the
Bott-Lefschetz fixed point formula, we have to check
$$\sum_{j\leq i}t_j^2v^{-2d_{ij}}(1-v^2)^{-1}
\prod_{j\ne k\leq i}(1-t_j^2t_k^{-2}v^{2d_{i,k}-2d_{i,j}})^{-1}
\prod_{k\leq i-1}(1-t_j^2t_k^{-2}v^{2d_{i-1,k}-2d_{i,j}})=$$
$$=t_i^2v^{2d_{i-1}-2d_i}(1-v^2)^{-1}$$
Recall the change of variables we used in the proof of Proposition
~\ref{mrak}: $s_j:=t_j^2v^{-2d_{ij}},\ 1\leq j\leq i;\
p_k:=t_k^2v^{-2d_{i-1,k}},\ 1\leq k\leq i-1$. Then we have to
prove
$$\sum_{j\leq i}s_j\prod_{k\leq i}^{k\ne j}(1-s_js_k^{-1})^{-1}
\prod_{k\leq i-1}(1-s_jp_k^{-1})=s_1\cdots s_ip_1^{-1}\cdots
p_{i-1}^{-1}$$ This follows immediately from the well known
identity
$$\sum_{j\leq i}\prod_{k\leq i-1}(p_k-s_j)
\prod_{k\leq i}^{k\ne j}(s_k-s_j)^{-1}=1.$$ This completes the
proof of the Proposition.
\end{proof}
\begin{cor}
\label{vot tebe} The Shapovalov scalar product of the Whittaker
vectors equals
$(\fk_{\ul{d}},\fw_{\ul{d}})=(-1)^{\sum_{i=1}^{n-1}d_i}
v^{\sum_{i=1}^{n-1}d_i^2-\sum_{i=2}^{n-1}d_id_{i-1}-\sum_{i=1}^{n-1}d_i}
\prod_{i=1}^nt_i^{d_{i-1}-d_i}[R\Gamma(\fQ_{\ul{d}},\CO_{\ul{d}})]$.
\end{cor}
\subsection{}
\label{generating} According to the works ~\cite{e}, ~\cite{s2},
the appropriate generating function of the Shapovalov scalar
product of the Whittaker vectors satisfies a $v$-deformed
($v$-difference) version of the quantum Toda lattice equations.
Let us recall the required notations and results.
We introduce the formal variables $\sz_1,\ldots,\sz_n$, and we set
$\sQ_i=\exp(\sz_i-\sz_{i+1}),\ i=1,\ldots,n-1$. We set
$\hbar=\log(v)$, so that $v=\exp(\hbar)$. We introduce the shift
operators $\sT_i,\ i=1,\ldots,n$, acting on the space of functions
of $\sz_1,\ldots,\sz_n$ invariant with respect to the simultaneous
translations
$f(\sz_1,\ldots,\sz_n)=f(\sz_1+\sz,\ldots,\sz_n+\sz)$. Namely, we
set $\sT_if(\sz_1,\ldots,\sz_n)=
f(\sz_1,\ldots,\sz_i+\hbar,\ldots,\sz_n)$.
We define the following $v$-difference operators:
\begin{equation}
\label{Etingof} {\mathfrak
S}:=\sum_{j=1}^n\sT_j^2+v^{-2}\sum_{i=1}^{n-1}\sQ_i\sT_i\sT_{i+1}
\end{equation}
\begin{equation}
\label{Givental} {\mathfrak
G}:=\sT_1^2+\sT_2^2(1-\sQ_1)+\ldots+\sT_n^2(1-\sQ_{n-1})
\end{equation}
We also consider the following generating functions:
\begin{equation}
\label{etingof} {\mathfrak
I}:=\prod_{i=1}^{n-1}\sQ_i^{\frac{-\log(t_1\cdots t_i)}{\hbar}}
\sum_{\ul{d}}(\fk_{\ul{d}},\fw_{\ul{d}})\sQ_1^{d_1}\cdots\sQ_{n-1}^{d_{n-1}}
\end{equation}
\begin{equation}
\label{givental} {\mathfrak
J}:=\prod_{i=1}^{n-1}\sQ_i^{\frac{-\log(t_1\cdots t_i)}{\hbar}}
\sum_{\ul{d}}[R\Gamma(\fQ_{\ul{d}},
\CO_{\ul{d}})]\sQ_1^{d_1}\cdots\sQ_{n-1}^{d_{n-1}}
\end{equation}
Then according to the last formula of ~\cite{s2} (or equivalently,
the formula ~(5.7) of ~\cite{e}), we have
\begin{equation}
\label{etisev} {\mathfrak S}{\mathfrak
I}=\left(\sum_{i=1}^nt_i^2\right){\mathfrak I}
\end{equation}
In effect, the seeming discrepancy between the formula
~(\ref{Etingof}) above, and the formula ~(5.7) of ~\cite{e} is
explained by the fact that (a) our $v$ corresponds to $q$ of
~\cite{e}; (b) our Whittaker vectors have eigenvalue
$(1-v^2)^{-1}$, whereas the Whittaker vectors of ~\cite{e} have
eigenvalue 1, which takes care of the factor $(q-q^{-1})^2$ in the
second summand of the formula ~(5.7) of ~\cite{e}.
Now the argument of ~\cite{e}, section ~6 (see the formula ~(6.5))
together with Corollary ~\ref{vot tebe}, establishes
\begin{equation}
\label{talgiven} {\mathfrak G}{\mathfrak
J}=\left(\sum_{i=1}^nt_i^2\right){\mathfrak J}
\end{equation}
thus reproving the Main Theorem ~2 of ~\cite{gl}.
\section{Parabolic sheaves and affine quantum groups}
In this section we want to generalize the previous results to the
affine setting. \label{p}
\subsection{Parabolic sheaves}
We recall the setup of ~\cite{fgk}. Let $\bX$ be another smooth
projective curve of genus zero. We fix a coordinate $x$ on $\bX$,
and consider the action of $\BC^*$ on $\bX$ such that
$u(x)=u^{-2}x$. We have $\bX^{\BC^*}=\{0_\bX,\infty_\bX\}$. Let
$\bS$ denote the product surface $\bC\times\bX$. Let $\bD_\infty$
denote the divisor $\bC\times\infty_\bX\cup\infty_\bC\times\bX$.
Let $\bD_0$ denote the divisor $\bC\times0_\bX$.
Given an $n$-tuple of nonnegative integers
$\ul{d}=(d_0,\ldots,d_{n-1})$, we say that a {\em parabolic sheaf}
$\CF_\bullet$ of degree $\ul{d}$ is an infinite flag of torsion
free coherent sheaves of rank $n$ on $\bS:\
\ldots\subset\CF_{-1}\subset\CF_0\subset\CF_1\subset\ldots$ such
that:
(a) $\CF_{k+n}=\CF_k(\bD_0)$ for any $k$;
(b) $ch_1(\CF_k)=k[\bD_0]$ for any $k$: the first Chern classes
are proportional to the fundamental class of $\bD_0$;
(c) $ch_2(\CF_k)=d_i$ for $i\equiv k\pmod{n}$;
(d) $\CF_0$ is locally free at $\bD_\infty$ and trivialized at
$\bD_\infty:\ \CF_0|_{\bD_\infty}=W\otimes\CO_{\bD_\infty}$;
(e) For $-n\leq k\leq0$ the sheaf $\CF_k$ is locally free at
$\bD_\infty$, and the quotient sheaves $\CF_k/\CF_{-n},\
\CF_0/\CF_k$ (both supported at $\bD_0=\bC\times0_\bX\subset\bS$)
are both locally free at the point $\infty_\bC\times0_\bX$;
moreover, the local sections of $\CF_k|_{\infty_\bC\times \bX}$
are those sections of $\CF_0|_{\infty_\bC\times
\bX}=W\otimes\CO_\bX$ which take value in $\langle
w_1,\ldots,w_{n-k}\rangle\subset W$ at $\infty_\bX\in \bX$.
\medskip
According to ~\cite{fgk}, ~3.5, the fine moduli space
$\CP_{\ul{d}}$ of degree $\ul{d}$ parabolic sheaves exists and is
a smooth connected quasiprojective variety of dimension
$2d_0+\ldots+2d_{n-1}$.
The group $\widetilde{T}\times\BC^*\times\BC^*$ acts naturally on
$\CP_{\ul{d}}$, and its fixed point set is finite.
\subsection{Correspondences}
If the collections $\ul{d}$ and $\ul{d}'$ differ at the only place
$i\in I:=\BZ/n\BZ$, and $d'_i=d_i+1$, then we consider a
correspondence
$\sE_{\ul{d},i}\subset\CP_{\ul{d}}\times\CP_{\ul{d}'}$ formed by
the pairs $(\CF_\bullet,\CF'_\bullet)$ such that for $j\not\equiv
i\pmod{n}$ we have $\CF_j=\CF'_j$, and for $j\equiv i\pmod{n}$ we
have $\CF'_j\subset\CF_j$.
It is a smooth quasiprojective algebraic variety of dimension
$2\sum_{i\in I}d_i+1$. In effect, the argument of ~\cite{fgk},
~Lemma~3.3, reduces this statement to the corresponding fact about
Laumon correspondences (see ~\cite{la1}, ~2.10).
We denote by $\bp$ (resp. $\bq$) the natural projection
$\sE_{\ul{d},i}\to\CP_{\ul{d}}$ (resp.
$\sE_{\ul{d},i}\to\CP_{\ul{d}'}$). For $j\equiv i\pmod{n}$ the
correspondence $\sE_{\ul{d},i}$ is equipped with a natural line
bundle $\sL_j$ whose fiber at $(\CF_\bullet,\CF'_\bullet)$ equals
$\Gamma(\bC,\CF_{j-n}/\CF'_{j-n})$. Finally, we have a transposed
correspondence
$^\sT\sE_{\ul{d},i}\subset\CP_{\ul{d}'}\times\CP_{\ul{d}}$.
\subsection{}
We denote by ${}'\CM$ the direct sum of equivariant (complexified)
$K$-groups:
${}'\CM=\oplus_{\ul{d}}K^{\widetilde{T}\times\BC^*\times\BC^*}(\CP_{\ul{d}})$.
It is a module over $K^{\widetilde{T}\times\BC^*\times\BC^*}(pt)
=\BC[\widetilde{T}\times\BC^*\times\BC^*]= \BC[t_1,\ldots,t_n,v,u\
:\ t_1\cdots t_n=1]$. We define $\CM=\
{}'\CM\otimes_{K^{\widetilde{T}\times\BC^*\times\BC^*}(pt)}
\on{Frac}(K^{\widetilde{T}\times\BC^*\times\BC^*}(pt))$.
We have an evident grading $\CM=\oplus_{\ul{d}}\CM_{\ul{d}},\
\CM_{\ul{d}}=K^{\widetilde{T}\times\BC^*\times\BC^*}(\CP_{\ul{d}})
\otimes_{K^{\widetilde{T}\times\BC^*\times\BC^*}(pt)}
\on{Frac}(K^{\widetilde{T}\times\BC^*\times\BC^*}(pt))$.
\subsection{}
\label{operators'} The grading and the correspondences
$^\sT\sE_{\ul{d},i},\sE_{\ul{d},i}$ give rise to the following
operators on $\CM$ (note that though $\bp$ is not proper, $\bp_*$
is well defined on the localized equivariant $K$-theory due to the
finiteness of the fixed point sets):
$K_i=t_{i+1}t_i^{-1}u^{\delta_{0,i}}v^{2d_i-d_{i-1}-d_{i+1}+1}:\
\CM_{\ul{d}}\to\CM_{\ul{d}}$,
$C=uv^n$,
For $i=0,\ldots,n-1$ we define $L_i=t_1^{-1}\cdots
t_i^{-1}v^{d_i+\frac{1}{2}i(n-i)}:\ M_{\ul{d}}\to M_{\ul{d}}$
(that is, $L_0=v^{d_0}$),
$f_i=\bp_*\bq^*:\ \CM_{\ul{d}}\to\CM_{\ul{d}-i}$;
For $n>2$ and $i=0,\ldots,n-1$ we define
$F_i=t_{i+1}^{-1}v^{d_{i+1}-d_i+\frac{n+1}{2}-i}\bp_*\bq^*:\
\CM_{\ul{d}}\to\CM_{\ul{d}-i}$,
For $n=2$ we define $F_i=f_i$,
$e_i=-t_i^{-1}t_{i+1}^{-1}u^{\delta_{0,i}}v^{d_{i+1}-d_{i-1}}
\bq_*(\sL_i\otimes\bp^*):\ \CM_{\ul{d}}\to\CM_{\ul{d}+i}$,
For $n>2$ and $i=0,\ldots,n-1$ we define
$E_i=-t_i^{-1}u^{\delta_{0,i}}v^{d_i-d_{i-1}+\frac{1-n}{2}+i}
\bq_*(\sL_i\otimes\bp^*):\ \CM_{\ul{d}}\to\CM_{\ul{d}+i}$,
For $n=2$ we define $E_i=e_i$.
\subsection{Sevostyanov's form of affine quantum $SL_n$}
Let $I$ denote the set $\BZ/n\BZ$ of residue classes modulo $n$.
$\CU$ is the $\BC[v,v^{-1}]$-algebra with generators
$E_i,L_i^{\pm1},K_i^{\pm1},C^{\pm1},F_i,\ i\in\BZ/n\BZ$, subject
to the following relations:
\begin{equation}
\label{ev} L_iL_j=L_jL_i,\ K_i=L_i^2L_{i+1}L_{i-1}C^{\delta_{i,0}}
\end{equation}
\begin{equation}
\label{evid} L_jE_iL_j^{-1} =v^{\delta_{i,j}}E_i,\ L_jF_iL_j^{-1}
=v^{-\delta_{i,j}}F_i,\
C\hphantom{m}\on{is}\hphantom{m}\on{central}
\end{equation}
\begin{equation}
\label{evident}
E_iF_j-F_jE_i=\delta_{i,j}\frac{K_i-K_i^{-1}}{v-v^{-1}}
\end{equation}
\begin{equation}
\label{Ser1} |i-j|>1\ \Longrightarrow\
E_iE_j-E_jE_i=0=F_iF_j-F_jF_i
\end{equation}
\begin{equation}
\label{Ser2} n>2\ \&\ |i-j|=1\ \Longrightarrow\
E_i^2E_j-(v+v^{-1})E_iE_jE_i+E_jE_i^2=0=
F_i^2F_j-(v+v^{-1})F_iF_jF_i+F_jF_i^2
\end{equation}
\begin{equation}
\label{Ser22} n=2\ \&\ |i-j|=1\ \Longrightarrow\
E_i^3E_j-(v^2+1+v^{-2})E_i^2E_jE_i+
(v^2+1+v^{-2})E_iE_jE_i^2-E_jE_i^3=0
\end{equation}
\begin{equation}
\label{Ser22F} n=2\ \&\ |i-j|=1\ \Longrightarrow\
F_i^3F_j-(v^2+1+v^{-2})F_i^2F_jF_i+
(v^2+1+v^{-2})F_iF_jF_i^2-F_jF_i^3=0
\end{equation}
For $n>2$ we also consider elements $e_i,f_i\in\CU$ depending on
the following choice of $n\times n$-matrices $n_{ij},\ c_{ij}$
(cf. ~\cite{s}, ~Remark ~3):
\begin{equation}
\label{choice} n_{i,i}=1,\ n_{i,i+1}=-1,\ n_{i+1,i}=0,
\end{equation}
otherwise $n_{ij}=0$.
\begin{equation}
\label{Russia} c_{i,i+1}=-1,\ c_{i+1,i}=1,
\end{equation}
otherwise $c_{ij}=0$.
Then we set
\begin{equation}
\label{Sevo} f_i:=L_iL_{i+1}^{-1}F_i,\ e_i:=E_iL_i^{-1}L_{i+1}.
\end{equation}
Clearly, the algebra $\CU$ is generated by
$e_i,L_i^{\pm1},K_i^{\pm1}, C^{\pm1},f_i,\ i\in\BZ/n\BZ$, and the
relations ~(\ref{evid})--~(\ref{Ser2}) above are equivalent to the
relations ~(\ref{evid'})--~(\ref{Ser2'}) below.
\begin{equation}
\label{evid'} L_je_iL_j^{-1} =v^{\delta_{i,j}}e_i,\ L_jf_iL_j^{-1}
=v^{-\delta_{i,j}}f_i,\
C\hphantom{m}\on{is}\hphantom{m}\on{central}
\end{equation}
\begin{equation}
\label{evident'}
e_if_j-v^{c_{ij}}f_je_i=\delta_{i,j}\frac{K_i-K_i^{-1}}{v-v^{-1}}
\end{equation}
\begin{equation}
\label{Ser1'} |i-j|>1\ \Longrightarrow\
e_ie_j-e_je_i=0=f_if_j-f_jf_i
\end{equation}
\begin{equation}
\label{Ser2'} |i-j|=1\ \Longrightarrow\
e_i^2e_j-v^{c_{ij}}(v+v^{-1})e_ie_je_i+v^{2c_{ij}}e_je_i^2=0=
f_i^2f_j-v^{c_{ij}}(v+v^{-1})f_if_jf_i+v^{2c_{ij}}f_jf_i^2
\end{equation}
\subsection{} The following
is an affine analogue of Theorem ~\ref{main}. Recall the operators
$E_i,e_i,K_i^{\pm1},L_i^{\pm1},C^{\pm1},F_i,f_i,\ i\in I$, on
$\CM$ defined in ~\ref{operators'}.
\begin{conj}
\label{main'} The operators
$E_i,K_i^{\pm1},L_i^{\pm1},C^{\pm1},F_i,\ i\in I$, on $\CM$
satisfy the relations ~(\ref{ev})--~(\ref{Ser22F}). Equivalently,
if $n>2$, the operators $e_i,K_i^{\pm1},L_i^{\pm1},C,f_i,\ i\in
I$, satisfy the relations ~(\ref{ev}),
~(\ref{evid'})--~(\ref{Ser2'}).
\end{conj}
\subsection{}
\label{long} We can prove Conjecture ~\ref{main'} for $n>2$ . Let
us sketch this proof. It is parallel to the proof of Theorem
~\ref{main}. In effect, the relation ~(\ref{evident'}) for $i\ne
j$ follows from the transversality statement absolutely similar to
Lemma ~\ref{trans}. More precisely, the argument of ~\cite{fgk}
~(Lemma ~3.3), reduces the required smoothness to that proved in
Lemma ~\ref{trans}.
The relation ~(\ref{evident'}) for $j=i$ follows from Conjecture
~\ref{inteligent} by the argument of ~\ref{derivation}. Since we
can not prove Conjecture ~\ref{inteligent} at the moment, we will
derive the relation ~(\ref{evident'}) for $j=i$ from its weaker
but accessible form.
To this end we consider the following closed substack
$\fZ'_{\fd_1,\fd_2}\subset\fZ_{\fd_1,\fd_2}$. Recall that a
coherent sheaf $\fW_1$ (resp. $\fW_2$) contains the maximal
torsion subsheaf $\fW_1^{tors}$ (resp. $\fW_2^{tors}$) with the
locally free quotient sheaf $\fW_1^{free}$ (resp. $\fW_2^{free}$).
Moreover, we have $\fW_1\simeq\fW_1^{tors}\oplus\fW_1^{free}$
(resp. $\fW_2\simeq\fW_2^{tors}\oplus\fW_2^{free}$). The closed
substack $\fZ'_{\fd_1,\fd_2}\subset\fZ_{\fd_1,\fd_2}$ classifies
the flags of coherent sheaves (with trivialization at
$\infty\in\bC$) $\fW_1\subset\fW_2$ such that
$\deg\fW_1^{free}\leq0\geq\deg\fW_2^{free}$. We define
$K^{\fT\times\BC^*}(\fZ'_{\fd_1,\fd_2})$ as the $K$-group of
$\fT\times\BC^*$-equivariant coherent sheaves on the smooth stack
$\fZ_{\fd_1,\fd_2}$ supported on the closed substack
$\fZ'_{\fd_1,\fd_2}$. Note that for any
$\ul{d}=(d_1,\ldots,d_{n-1})$ the map $\fz_{\ul{d}}:\
\fQ_{\ul{d}}\to\fZ_{d_i-d_{i-1},d_{i+1}-d_i}$ factors through the
same named map into the closed substack
$\fZ'_{d_i-d_{i-1},d_{i+1}-d_i}$. Similarly, for any
$\ul{d}=(d_0,d_1,\ldots,d_{n-1})$ the map
$$\fz\fz_{\ul{d}}:\ \CP_{\ul{d}}\to\fZ_{d_i-d_{i-1},d_{i+1}-d_i},\
\CF_\bullet\mapsto(\CF_i/\CF_{i-1},\CF_{i+1}/\CF_{i-1})$$ factors
through the same named map into the closed substack
$\fZ'_{d_i-d_{i-1},d_{i+1}-d_i}$.
Let $(\fW_1\subset\fW_2)$ be a $\fT\times\BC^*$-fixed point of
$\fZ'_{\fd_1,\fd_2}$. Let $\iota_{(\fW_1\subset\fW_2)}$ denote its
locally closed embedding into $\fZ'_{\fd_1,\fd_2}$. Let
$Aut_{(\fW_1\subset\fW_2)}$ stand for its automorphisms' group.
One can easily check the following
\begin{lem}
\label{easily} There exists $n,\ i,\ 1\leq i\leq n-1,\
\ul{d}=(d_1,\ldots,d_n)$, and a fixed point
$\widetilde{\ul{d}}\in\fQ_{\ul{d}}^{\widetilde{T}\times\BC^*}$
such that
(a) $\fz_{\ul{d}}(\widetilde{\ul{d}})=(\fW_1\subset\fW_2)$;
(b) $\fz_{\ul{d}}(\widetilde{T}\times\BC^*)$ is a maximal torus of
$Aut_{(\fW_1\subset\fW_2)}$.
\end{lem}
\subsection{}
\label{birka} One way to prove Conjecture ~\ref{inteligent} would
be to reverse the argument of ~\ref{derivation} and derive it from
the relations ~(\ref{ochevidno'}) for all $n,i$. In effect, we
must compute (notations of ~\ref{derivation}) $\fX\in
K^{\fT\times\BC^*}(\fZ_{\fd_1,\fd_2})
\otimes_{K^{\fT\times\BC^*}(pt)}
\on{Frac}(K^{\fT\times\BC^*}(pt))$ while we know
$\fz_{\ul{d}}^*\fX$ for all $n,i,\ul{d}$ such that
$d_i-d_{i-1}=\fd_1,\ d_{i+1}-d_i=\fd_2$ (also, the homomorphism of
tori $\widetilde{T}_n\to\fT$ acts on the characters as
$\tau_1=t_i,\ \tau_2=t_{i+1}$).
Let us denote by $\fY\in K^{\fT\times\BC^*}(\fZ'_{\fd_1,\fd_2})
\otimes_{K^{\fT\times\BC^*}(pt)}
\on{Frac}(K^{\fT\times\BC^*}(pt))$ the restriction of $\fX$ to
$\fZ'_{\fd_1,\fd_2}$.
The Lemma ~\ref{easily} implies that the kernel $Ker_1$ of the
direct product of inverse images
$$\prod_{n,i,\ul{d}}\fz_{\ul{d}}^*:\ K^{\fT\times\BC^*}(\fZ'_{\fd_1,\fd_2})
\otimes_{K^{\fT\times\BC^*}(pt)}\on{Frac}(K^{\fT\times\BC^*}(pt))\to
\prod_{n,i,\ul{d}}M_{\ul{d}}$$ coincides with the kernel $Ker_2$
of the direct product of restrictions
$$\prod_{(\fW_1\subset\fW_2)\in(\fZ'_{\fd_1,\fd_2})^{\fT\times\BC^*}}
\iota^*_{(\fW_1\subset\fW_2)}:\
K^{\fT\times\BC^*}(\fZ'_{\fd_1,\fd_2})
\otimes_{K^{\fT\times\BC^*}(pt)}\on{Frac}(K^{\fT\times\BC^*}(pt))\to$$
$$\to\prod_{(\fW_1\subset\fW_2)\in(\fZ'_{\fd_1,\fd_2})^{\fT\times\BC^*}}
K^{\fT\times\BC^*\times Aut_{(\fW_1\subset\fW_2)}}(pt)
\otimes_{K^{\fT\times\BC^*}(pt)}\on{Frac}(K^{\fT\times\BC^*}(pt))$$
It follows that for any $n,\ i,\ 0\leq i\leq n-1,\
\ul{d}=(d_0,\ldots,d_n)$, such that $\fd_1=d_i-d_{i-1},\
\fd_2=d_{i+1}-d_i$, the kernel $Ker_1=Ker_2$ is contained in the
kernel $Ker_3$ of the inverse image
$$\fz\fz_{\ul{d}}^*:\ K^{\fT\times\BC^*}(\fZ'_{\fd_1,\fd_2})
\otimes_{K^{\fT\times\BC^*}(pt)}\on{Frac}(K^{\fT\times\BC^*}(pt))\to
\CM_{\ul{d}}$$
By the argument of ~\ref{derivation} we know that
$\fY=\frac{K-K^{-1}}{v-v^{-1}}$ modulo $Ker_1$, and hence the same
holds modulo $Ker_3$. The argument of {\em loc. cit.} then shows
that the relation ~(\ref{evident'}) for $j=i$ holds in $\CM$.
\subsection{}
\label{longer} It remains to check the Serre relations. The
relations for negative generators follow from the relations for
positive generators because they are adjoint with respect to the
nondegenerate Shapovalov form, see ~\ref{Shapo} below. So it
suffices to consider the relations ~(\ref{Ser1}), ~(\ref{Ser2})
between $E_i,E_j,\ i\ne j$. It is here that we need the assumption
$n>2$ for technical reasons. Namely, for $n>2$ we can find $k\in
I$ such that $i\ne k\ne j$.
We consider an $n$-dimensional vector space with a basis
$\fw_1,\ldots,\fw_n$, and a torus $\fT$ acting on $\fw_l$ by the
character $\tau^2_l$. Let $\fZ_n$ be the moduli stack of flags of
coherent sheaves $\fW_1\subset\ldots\subset\fW_n$ on $\bC$ locally
free at $\infty\in\bC$, equipped with compatible trivializations
$\fW_l|_\infty=\langle\fw_1,\ldots,\fw_l\rangle$. Note that
$\fZ_n$ has connected components numbered by the degrees of
$\fW_l$, which for $n=2$ coincide with the stacks
$\fZ_{\fd_1,\fd_2}$. Absolutely similarly to ~\ref{alternate} we
introduce the correspondences between various connected
components, which give rise to the operators
$E^\fZ_1,\ldots,E^\fZ_{n-1}$ on the localized equivariant
$K$-theory of $\fZ_n$.
As in ~\ref{long} above, we have a closed substack
$\fZ'_n\subset\fZ_n$ classifying the flags such that
$\deg\fW_l^{free}\leq0,\ 1\leq l\leq n$.
We have a map
$$\fz\fz_k:\ \CP_{\ul{d}}\to\fZ_n,\ (\CF_\bullet)\mapsto
(\CF_{k+1}/\CF_k\subset\ldots\subset\CF_{k+n}/\CF_k)$$ factoring
through the same named map $\CP_{\ul{d}}\to\fZ'_n$. For any $N\geq
n$, and $m$ such that $0\leq m\leq N-n$, and
$\ul{d}=(d_1,\ldots,d_N)$, we also have a map
$$\fz_{m,\ul{d}}:\ \fQ_{\ul{d}}\to\fZ_n,\
(\CW_\bullet)\mapsto(\CW_{m+1}/\CW_m\subset\ldots\subset\CW_{m+n}/\CW_m)$$
factoring through the same named map $\fQ_{\ul{d}}\to\fZ'_n$.
Now the argument of ~\ref{derivation} shows that the Serre
relation between $E_i,E_j$ would follow from the Serre relation
between $E^\fZ_{i'},E^\fZ_{j'}$ for certain $i',j'$. Though we
cannot establish the latter relations, the argument of
~\ref{birka} shows that they hold modulo the subspace $Ker_1$
(because we already know the Serre relations for $\mathfrak{sl}_N$
with arbitrary $N$), and also shows that this suffices to derive
the former relations.
This completes the proof of the Serre relations for $n>2$. Thus,
Conjecture ~\ref{main'} is proved for $n>2$.
\subsection{}
Similarly to ~\ref{Shapoval}, we will write down a geometric
expression for a Shapovalov form on $\CM$, that is a symmetric
$\operatorname{Frac}(\BC[\widetilde{T}\times\BC^*\times\BC^*])$-valued
bilinear form on $\CM$ such that $(E_im_1,m_2)=(m_1,F_im_2)$ for
any $i\in I$, and $m_1,m_2\in\CM$. The different weight spaces of
$\CM$ will be orthogonal with respect to this geometric Shapovalov
form. For $i=0,\ldots,n-1$, we consider the line bundle $\D_i$ on
$\CP_{\ul{d}}$ whose fiber at the point $(\CF_\bullet)$ equals
$\det R\Gamma(\bS,\CF_{i-n})$. We also define the line bundle
$\D_{\ul{d}}:=\bigotimes_{i=0}^{n-1}\D_i$. For
$\CG_1,\CG_2\in\CM_{\ul{d}}$, we set
\begin{equation}
\label{Chapo} (\CG_1,\CG_2):=(-1)^{\sum_{i=0}^{n-1}d_i}
v^{-\sum_{i=0}^{n-1}d_i^2+\sum_{i=0}^{n-1}d_id_{i+1}
+\sum_{i=0}^{n-1}(n-2i)d_i}u^{-d_0}
\prod_{i=0}^{n-1}t_i^{d_i-d_{i-1}}R\Gamma(\CP_{\ul{d}},
\CG_1\otimes\CG_2\otimes\D_{\ul{d}})
\end{equation}
Clearly, the form $(,)$ is nondegenerate, since the classes of the
structure sheaves of the
$\widetilde{T}\times\BC^*\times\BC^*$-fixed points form an
orthogonal basis of $\CM$.
The following proposition is proved exactly as ~\ref{Shapoval}.
\begin{prop}
\label{Shapo} For $i\in I,\ \CG_1\in\CM_{\ul{d}},\
\CG_2\in\CM_{\ul{d}+i}$ we have
$(E_i\CG_1,\CG_2)=(\CG_1,F_i\CG_2)$.
\end{prop}
\subsection{}
We define a formal sum in a completion of $\CM$ as follows:
$\fn=\sum_{\ul{d}}\fn_{\ul{d}}:=\sum_{\ul{d}}[\CO_{\ul{d}}]
=\sum_{\ul{d}}[\CO_{\CP_{\ul{d}}}]$. We also consider the
following formal sum: $\fu=\sum_{\ul{d}}\fu_{\ul{d}}$ where
\begin{equation}
\fu_{\ul{d}}=v^{2\sum_{i=0}^{n-1}d_i^2-2\sum_{i=0}^{n-1}d_id_{i+1}
-\sum_{i=0}^{n-1}(n-2i+1)d_i}u^{2d_0}
\prod_{i=1}^nt_i^{2d_{i-1}-2d_i}[\D^{-1}_{\ul{d}}]
\end{equation}
\begin{prop}
\label{vykusi} a) $\fn$ is a common eigenvector of the operators
$f_i$ with the eigenvalue $(1-v^2)^{-1}$;
b) $\fu$ is a common eigenvector of the operators $e_i^*$ with the
eigenvalue $(1-v^2)^{-1}$.
\end{prop}
\begin{proof} a) is proved exactly as Proposition ~\ref{hlop} ~(a).
To check b) we argue as in the proof of Proposition ~\ref{hlop}
~(b), and reduce it to
\begin{equation}
\label{hotim}
\bp_*[\sL_i]=t_i^2u^{-2\delta_{0,i}}v^{2d_{i-1}-2d_i}
(1-v^2)^{-1}[\CO_{\ul{d}}]
\end{equation}
To verify this we recall the setup of ~\ref{alternate}, and claim
that in the notations of {\em loc. cit.} we have
\begin{equation}
\label{imeem} \bp_*[\fL_{\fd_1}]=\tau_1^2v^{-2\fd_1}(1-v^2)^{-1}
[\CO_{\fZ_{\fd_1,\fd_2}}]
\end{equation}
In effect, ~(\ref{imeem}) is deduced from ~(\ref{zhelaem}) by the
argument of ~\ref{birka}. Finally, ~(\ref{hotim}) is deduced from
~(\ref{imeem}) by the argument of ~\ref{derivation}.
The Proposition is proved.
\end{proof}
\begin{cor}
\label{vot te} The Shapovalov scalar product of the Whittaker
vectors equals
$(\fn_{\ul{d}},\fu_{\ul{d}})=(-1)^{\sum_{i=0}^{n-1}d_i}
v^{\sum_{i=0}^{n-1}d_i^2-\sum_{i=0}^{n-1}d_id_{i-1}-\sum_{i=0}^{n-1}d_i}
u^{d_0}\prod_{i=1}^nt_i^{d_{i-1}-d_i}R\Gamma(\CP_{\ul{d}},\CO_{\ul{d}})$.
\end{cor}
\subsection{}
\label{new} We define $\CM'\subset\CM$ as a minimal
$\CU$-submodule containing the lowest weight vector
$[0,\ldots,0]$. The relations ~(\ref{evident'}) show that $\CM'$
is generated from $[0,\ldots,0]$ by the action of operators $e_i,\
i\in I$. Clearly, $\CM'$ is isomorphic to a universal Verma module
over $\CU$.
\begin{conj}
\label{nakos} The class of the structure sheaf $[\CO_{\ul{d}}]$
lies in $\CM'_{\ul{d}}$.
\end{conj}
In what follows we shall assume the validity of Conjecture
~\ref{main'} (as was explained above this is actually not an
assumption for $n>2$).
\begin{prop}
\label{nakosi} The class of $[\D^{-1}_{\ul{d}}]$ lies in
$\CM'_{\ul{d}}$.
\end{prop}
\begin{proof} We have $\CM=\CM'\oplus\CM''$ where $\CM''$ is the
orthogonal complement of $\CM'$ in $\CM$ with respect to the
Shapovalov form. We have to prove that $[\D^{-1}_{\ul{d}}]$ is
orthogonal to $\CM''$. Let $A\in\CM''_{\ul{d}}$. Suppose $A=e_iB$
for some $i\in I$ and $B\in\CM''_{\ul{d}-i}$. Then
$(A,[\D^{-1}_{\ul{d}}])=(B,e_i^*[\D^{-1}_{\ul{d}}])$. Thus up to
(an invertible) monomial in $t,u,v$ we have
$(A,[\D^{-1}_{\ul{d}}])=(B,[D^{-1}_{\ul{d}-i}])$. Hence, arguing
by induction in $\ul{d}$ we may assume that $A\in\CM''_{\ul{d}}$
is orthogonal to the image of any $e_i$. Then $e_i^*A=0$ or,
equivalently, $f_iA=0$ for any $i\in I$. Up to (an invertible)
monomial in $t,u,v$ we have
$(A,[\D^{-1}_{\ul{d}}])=R\Gamma(\CP_{\ul{d}},A)$. Thus we are
reduced to the following claim for $\ul{d}\ne(0,\ldots,0)$:
\begin{equation}
\label{ponjatno} f_iA=0\ \forall\ i\in I\ \Longrightarrow\
R\Gamma(\CP_{\ul{d}},A)=0.
\end{equation}
We will derive ~(\ref{ponjatno}) from the corresponding claim in
the equivariant (complexified) Borel-Moore homology
$H^{\widetilde{T}\times\BC^*\times\BC^*}_{BM}(\CP_{\ul{d}})$. Let
$\on{Td}_{\CP_{\ul{d}}}$ denote the equivariant Todd class in the
completion of the equivariant cohomology. Let also $\on{ch}_*$
denote the homological Chern character map from the equivariant
$K$-theory to the completion of the equivariant Borel-Moore
homology (see e.g. ~\cite{cg}). We define
$$a:=\on{Td}_{\CP_{\ul{d}}}\cup\on{ch}_*A\in
\widehat{H}{}^{\widetilde{T}\times\BC^*\times\BC^*}_{BM}(\CP_{\ul{d}})$$
By the bivariant Riemann-Roch Theorem (see e.g. ~\cite{cg},
~5.11.11) we have
$\on{ch}_*(f_iA)=\on{ch}_*(\bp_*\bq^*A)=\bp_*\bq^*a$ where in the
RHS $\bp_*$ and $\bq^*$ refer to the operations in the (localized
and completed) equivariant Borel-Moore homology. We also have
$R\Gamma(\CP_{\ul{d}},A)=\int_{\CP_{\ul{d}}}a$. Since $\on{ch}_*$
is injective, and the operation
$?\mapsto\on{Td}_{\CP_{\ul{d}}}\cup?$ is invertible, the claim
~(\ref{ponjatno}) follows from the corresponding claim in the
equivariant Borel-Moore homology
$\widehat{H}{}^{\widetilde{T}\times\BC^*\times\BC^*}_{BM}(\CP_{\ul{d}})$:
\begin{equation}
\label{clear} {\mathfrak f}_ia=0\ \forall\ i\in I\
\Longrightarrow\ \int_{\CP_{\ul{d}}}a=0.
\end{equation}
Here ${\mathfrak f}_i=\bp_*\bq^*$ is a part of the action of the
affine Lie algebra $\widehat{\mathfrak{sl}}_n$ on ${\mathfrak
M}:=\oplus_{\ul{d}}
\widehat{\ul{H}}{}^{\widetilde{T}\times\BC^*\times\BC^*}_{BM}(\CP_{\ul{d}})$
(localized and completed equivariant Borel-Moore homology). The
positive generators act as ${\mathfrak e}_i=-\bq_*\bp^*$. This can
be checked along the lines of ~\ref{long}--\ref{longer} but
simpler.
Reversing the argument in the beginning of the proof, we see that
~(\ref{clear}) is equivalent to the statement that the fundamental
cycle $[\CP_{\ul{d}}]\in
\widehat{\ul{H}}{}^{\widetilde{T}\times\BC^*\times\BC^*}_{BM}(\CP_{\ul{d}})$
is contained in the subspace ${\mathfrak M}'$ of ${\mathfrak M}$
generated by the action of ${\mathfrak e}_i,\ i\in I$, from
$[\CP_{(0,\ldots,0)}]$.
Recall the semismall resolution morphism $\pi_{\ul{d}}:\
\CP_{\ul{d}}\to\frP_{\ul{d}}$ to the Uhlenbeck flag space, see
~\cite{fgk}. By the Decomposition Theorem of
Beilinson-Bernstein-Deligne-Gabber, the direct sum of (localized
and completed) equivariant Intersection Homology $'{\mathfrak
M}:=\oplus_{\ul{d}}
\widehat{\ul{IH}}{}^{\widetilde{T}\times\BC^*\times\BC^*}(\frP_{\ul{d}})$
is a direct summand of $\mathfrak M$.
Now ~\cite{b} defines the action of $\widehat{\mathfrak{sl}}_n$ on
$'{\mathfrak M}$, and one can check that the action of ~\cite{b}
is the restriction of the above $\widehat{\mathfrak{sl}}_n$-action
on $\mathfrak M$. It follows that $'{\mathfrak M}={\mathfrak M}'$.
Finally, it is proved in ~\cite{b} that $[\CP_{\ul{d}}]\in\
'{\mathfrak M}$.
This completes the proof of the Proposition.
\end{proof}
\subsection{}
We conclude that $\fu$ is the unique Whittaker vector in the
completion of the Verma module $\CM'$ with the lowest weight
component $\fu_{(0,\ldots,0)}=[(0,\ldots,0)]$ (the common
eigenvector of $e_i^*,\ i\in I$, with the eigenvalue
$(1-v^2)^{-1}$).
Let $\fn'\in\widehat{\CM}{}'$ be the unique common eigenvector of
$f_i,\ i\in I$, with the eigenvalue $(1-v^2)^{-1}$ and with the
lowest weight component $\fn'_{(0,\ldots,0)}=[(0,\ldots,0)]$. Then
$\fn'$ is the orthogonal projection of $\fn$ onto
$\widehat{\CM}{}'$ along $\widehat{\CM}{}''$. Hence the Corollary
~\ref{vot te} yields the following
\begin{cor}
\label{last} One has
$$(\fn'_{\ul{d}},\fu_{\ul{d}})=(-1)^{\sum_{i=0}^{n-1}d_i}
v^{\sum_{i=0}^{n-1}d_i^2-\sum_{i=0}^{n-1}d_id_{i-1}-\sum_{i=0}^{n-1}d_i}
u^{d_0}\prod_{i=1}^nt_i^{d_{i-1}-d_i}[R\Gamma(\CP_{\ul{d}},\CO_{\ul{d}})].$$
\end{cor}
\subsection{Some further remarks}
The next natural step would be to study the generating function of all
$[R\Gamma(\CP_{\ul{d}},\CO_{\ul{d}})]$'s in a way similar to
subsection ~\ref{generating}; let us denote this function by
$\fJ_{\aff}$. The cohomology (as opposed to $K$-theory) analogue
of this is performed in \cite{b} and \cite{be}. In particular, in
\cite{b} it is shown that such a function is an eigen-function of
a certain linear differential operator of 2nd order (the
``non-stationary analogue" of the quadratic affine Toda
hamiltonian). This fact is used in \cite{be} in order to show that
certain asymptotic of this function is given by the {\it
Seiberg-Witten prepotential} of the corresponding classical affine
Toda system. This agrees well with the results of \cite{neok}
about a similar asymptotic of the partition function of N=2
supersymmetric gauge theory in 4 dimensions.
Unfortunately, in the present ($K$-theoretic) case we can't derive
any good equation for the the function $\fJ_{\aff}$. Thus we do
not know how to generalize the results of \cite{be} to this case.
One can probably show that the results of \cite{neok} on 5d gauge
theory imply that a similar asymptotic (when the classical affine
Toda lattice is replaced by the classical affine relativistic
Toda) is valid for the function $\fJ_{\aff}$, but we do not know
how to derive it from Corollary ~\ref{last}.
|
{
"timestamp": "2005-06-13T11:04:42",
"yymm": "0503",
"arxiv_id": "math/0503456",
"language": "en",
"url": "https://arxiv.org/abs/math/0503456"
}
|
\section{Introduction}
Recently, Szmytkowski \cite{Szmytkowski} proposed a new formulation
of the eigenchannel method for quantum scattering from Hermitian
short-range potentials, different from that presented by Danos and
Greiner \cite{Danos}. Some ideas leading to this method were drawn
from works on electromagnetism theory by Garbacz \cite{Garbacz1} and
Harrington and Mautz \cite{Harrington1}. This method was further
extended to the case of zero-range potentials for Schr\"odinger
particles by Szmytkowski and Gruchowski \cite{Szmytkowski2} and then
for Dirac particles by Szmytkowski \cite{Szmytkowski3} (see also
\cite{Szmytkowski5}).
On the other hand, it is the well-known fact that separable
potentials, since they provide analytical solutions to the
Lippmann-Schwinger equations \cite{LippSchw}, have found
applications in many branches of physics, both in the
non-relativistic and relativistic cases \cite{Zast}. (It should be
noted that much larger effort has been devoted to the separable
potentials in the non-relativistic regime.) Especially, their
utility was confirmed in nuclear physics by successful use for
describing nucleon-nucleon interactions \cite{NN}. Moreover,
methods allowing one to approximate an arbitrary non-local
potential by a separable one are known \cite{metody}.
In view of what has been said above, it seems interesting to pose
the question: {\it how does the new method apply to quantum
scattering from non-local separable potentials?} Partially, the
answer was given by the author by applying the method to quantum
scattering of Schr\"odinger particles from separable potentials
\cite{moja}. In the present contribution, we extend considerations
from \cite{moja} to the case of Dirac particles.
This paper is organized as follows. In Section 2 some facts and
notions from the theory of potential scattering of Dirac particles
(see \cite{Thaller}) are provided.
In Section 3 we concentrate on the special class of non-local
potentials, namely, separable potentials. In this context,
expressions for the bispinor as well as matrix scattering
amplitudes are provided. Section 4 contains main ideas and
results. Here, we define {\it eigenchannel vectors}, directly
related to eigenchannels, as solutions to a certain weighted
eigenproblem. Moreover, we introduce eigenphase-shifts, relating
them to eigenvalues of this spectral problem. Within this
approach, we also calculate expressions for the scattering
amplitude and the average total cross section. In Section 5,
scattering from a rank one delta-like separable potential is
discussed as an illustrative example. The paper ends with two
appendices.
\section{Quantum scattering of Dirac particles from non-local potentials}
\label{SecII}
Let us assume that a free Dirac particle of energy $E$ (with
$|E|>mc^{2}$) described by the following monochromatic plane wave
\begin{equation}\label{I.1}
\phi_{i}(\wektor{r})\equiv\braket{\wektor{r}}{\wektor{k}_{i}\chi_{i}}=
U_{i}(\wektor{k}_{i})e^{i\wektor{k}_{i}\cdot\wektor{r}},
\end{equation}
where
\begin{equation}\label{I.2}
U_{i}(\wektor{k}_{i})=\frac{1}{\sqrt{1+\varepsilon^{2}}} \left(
\begin{array}{c}
\chi_{i} \\*[0.2ex]
\displaystyle\varepsilon\wektor{\sigma}\cdot\wersor{k}_{i}\,\chi_{i}
\end{array}
\right),
\end{equation}
\begin{equation}
\varepsilon=\sqrt{\frac{E-mc^{2}}{E+mc^{2}}}
\end{equation}
is being scattered from a non-local potential given by a kernel
$\mathsf{V}(\wektor{r},\wektor{r}')$, which in general may be a
$4\times 4$ matrix. In the above equation, $\chi_{i}$ stands for a
normalized pure spin-$\frac{1}{2}$ state belonging to
$\mathbb{C}^{2}$. Orientation of the spin in $\mathbb{R}^{3}$ will
be denoted by $\wektor{\nu}_{i}$ and is related to $\chi_{i}$ by
$\wektor{\nu}_{i}=\chi^{\dagger}_{i}\wektor{\sigma}\chi_{i}$,
where $\wektor{\sigma}$ is a vector consisting of the standard
Pauli matrices, i.e.,
\begin{equation}\label{I.3}
\wektor{\sigma}= \left[ \left(
\begin{array}{cc}
0 & 1\\
1 & 0
\end{array}
\right), \left(
\begin{array}{cc}
0 & -i\\
i & 0
\end{array}
\right), \left(
\begin{array}{cc}
1 & 0\\
0 & -1
\end{array}
\right) \right].
\end{equation}
Moreover, $\wektor{p}_{i}=\hbar\wektor{k}_{i}$ is a momentum of
the incident particle and $k$ denotes the Dirac wave number and is
given by
\begin{equation}\label{DiracWaveNumber}
k=\mathrm{sgn}(E)\sqrt{\frac{E^{2}-\left(mc^{2}\right)^{2}}{c^{2}\hbar^{2}}}.
\end{equation}
Thereafter, we shall consider only Hermitian potentials, i.e.,
those with kernels obeying
$\mathsf{V}(\wektor{r},\wektor{r}')=\mathsf{V}^{\dagger}(\wektor{r}',\wektor{r})$.
For this scattering process we may write the Lippmann-Schwinger
equation \cite{LippSchw} of the form
\begin{eqnarray}\label{I.4}
&&\hspace{-1.4cm}\psi(\wektor{r})=\phi_{i}(\wektor{r})\nonumber\\
&&\hspace{-1cm}-\calkaob{r}'\calkaob{r}''\:G(E,\wektor{r},\wektor{r}')\mathsf{V}(\wektor{r}',\wektor{r}'')
\psi(\wektor{r}'').
\end{eqnarray}
Function $G(E,\wektor{r},\wektor{r}')$ appearing above is the
relativistic free--particle outgoing Green function given by
\begin{equation}\label{I.6}
G(E,\wektor{r},\wektor{r}')= \frac{1}{4\pi c^{2}\hbar^{2}}\left(
-ic\hbar\wektor{\alpha}\cdot\wektor{\nabla}+\beta mc^{2}+E\jed_{4}
\right)
\frac{e^{ik|\wektor{r}-\wektor{r}'|}}{|\wektor{r}-\wektor{r}'|},
\end{equation}
and formally is a kernel of the relativistic outgoing Green
operator defined as
\begin{equation}\label{I.7}
\hat{G}(E)=\lim_{\epsilon\downarrow
0}[\hat{\mathcal{H}}_{0}-E-i\epsilon]^{-1},
\end{equation}
with
$\hat{\mathcal{H}}_{0}=-ic\hbar\wektor{\alpha}\cdot\wektor{\nabla}+
\beta mc^{2}$ being a Dirac free--particle Hamiltonian. Here
\begin{equation}\label{I.5}
\wektor{\alpha}= \left(
\begin{array}{cc}
0 & \wektor{\sigma}\\
\wektor{\sigma} & 0
\end{array}
\right), \qquad \beta= \left(
\begin{array}{cc}
\jed_{2} & 0\\
0 & -\jed_{2}
\end{array}
\right),\qquad \jed_{2}=\left(
\begin{array}{cc}
1 & 0\\
0 & 1
\end{array}
\right)
\end{equation}
and $\jed_{4}=\jed_{2}\ot\jed_{2}$.
It is worth noticing that within the relativistic
regime the Green function (\ref{I.6}) is a $4\times 4$ matrix.
For purposes of further analysis, it is useful to introduce the
following projector:
\begin{equation}\label{I.8}
\mathcal{P}(\wektor{k})=\frac{c\hbar
\wektor{\alpha}\cdot\wektor{k}+\beta mc^{2}+E\jed_{4}}{2E},
\end{equation}
which, as one can immediately infer, may be decomposed in the
following way
\begin{eqnarray}\label{I.9}
\mathcal{P}(\wektor{k})&=&\Theta_{+}(\wektor{k})
\Theta_{+}^{\dagger}(\wektor{k})+\Theta_{-}(\wektor{k})\Theta_{-}^{\dagger}(\wektor{k})\nonumber\\
&=&\frac{1}{1+\varepsilon^{2}} \left(
\begin{array}{cc}
\mathbbm{1}_{2} & \varepsilon \wektor{\sigma}\cdot\wersor{k}\\
\varepsilon \wektor{\sigma}\cdot\wersor{k} &
\varepsilon^{2}\mathbbm{1}_{2},
\end{array}
\right)
\end{eqnarray}
with $\Theta_{\pm}(\wektor{k})$ being defined as
\begin{equation}\label{I.10}
\Theta_{\pm}(\wektor{k})= \frac{1}{\sqrt{1+\varepsilon^{2}}}\left(
\begin{array}{c}
\theta_{\pm}\\
\varepsilon\wektor{\sigma}\cdot\wersor{k}\,\theta_{\pm}
\end{array}
\right).
\end{equation}
Spinors $\theta_{\pm}$ constitute an arbitrary orthonormal basis
in $\mathbb{C}^{2}$, i.e.,
$\theta_{s}^{\dagger}\theta_{t}=\delta_{st}$ $(s,t=-,+)$ and
$\sum_{s=-}^{+}\theta_{s}\theta_{s}^{\dagger}=\jed_{2}$. What is
important for further considerations, the matrix (\ref{I.8})
possesses the obvious property that
$\mathcal{P}(\wektor{k})\Theta_{\pm}(\wektor{k})=\Theta_{\pm}(\wektor{k})$
and therefore
\begin{equation}\label{I.11}
\mathcal{P}(\wektor{k}_{i})U_{i}(\wektor{k}_{i})=U_{i}(\wektor{k}_{i}).
\end{equation}
We shall be exploiting this property in later analysis.
Considering scattering processes we usually tend to find
expressions for a scattering amplitude and various cross sections.
To this aim we need to find an asymptotic behavior of the
relativistic outgoing Green function. From Eq. (\ref{I.6}), using
the projector (\ref{I.8}), we have
\begin{equation}\label{I.12}
G(E,\wektor{r},\wektor{r}')\stackrel{r\to\infty}{\sim}
\frac{E}{2\pi c^{2}\hbar^{2}}
\mathcal{P}(\wektor{k}_{f})\frac{e^{ikr}}{r}
e^{-i\wektor{k}_{f}\cdot\wektor{r}'},
\end{equation}
where $\wektor{k}_{f}=k\wektor{r}/r$ is a wave vector of the
scattered particle. Notice that due to the fact that we deal with
elastic processes $|\wektor{k}_{i}|=|\wektor{k}_{f}|=k$. After
application of Eq. (\ref{I.12}) to Eq. (\ref{I.4}), we obtain
\begin{equation}\label{I.13}
\psi(\wektor{r})\stackrel{r\to\infty}
{\sim}\underset{r\to\infty}{\mathrm{asymp}}\, \phi_{i}(\wektor{r})
+\Amplituda\frac{e^{ikr}}{r},
\end{equation}
where $\Amplituda$ is the bispinor scattering amplitude and is
defined through the relation
\begin{eqnarray}\label{I.14}
&&\hspace{-1.5cm}\Amplituda=-\frac{E}{2\pi
c^{2}\hbar^{2}}\mathcal{P}(\wektor{k}_{f})\nonumber\\
&&\hspace{-1cm}\times\calkaob{r}'\calkaob{r}''\,e^{-i\wektor{k}_{f}\cdot\wektor{r}'}
\mathsf{V}(\wektor{r}',\wektor{r}'')\psi(\wektor{r}'')
\end{eqnarray}
and, in general, is of the form
\begin{equation}\label{I.15}
\Amplituda=\frac{1}{\sqrt{1+\varepsilon^{2}}}\left(
\begin{array}{c}
\chi_{f}\\
\varepsilon\wektor{\sigma}\cdot\wersor{k}_{f}\chi_{f}
\end{array}
\right).
\end{equation}
Here $\chi_{f}$ is a spinor transformed from the initial spinor
$\chi_{i}$ by the scattering process. Vector
$\wektor{\nu}_{f}=(\chi_{f}^{\dagger}\wektor{\sigma}\chi_{f})/(\chi_{f}^{\dagger}\chi_{f})$
responds for an orientation of the spin of the scattered particle.
Therefore let us assume that there exist a matrix, such that
$\chi_{f}=\Amplitudaaa\chi_{i}$. Then it is easy to verify that
the bispinor scattering amplitude may be written in the form
\begin{equation}\label{I.16}
\Amplituda=\Amplitudaa U_{i}(\wektor{k}_{i}),
\end{equation}
where the matrix $\Amplitudaa$ is related to $\Amplitudaaa$ by
\begin{equation}\label{I.17}
\Amplitudaa=\frac{1}{1+\varepsilon^{2}} \left(
\begin{array}{cc}
\Amplitudaaa &\quad
\varepsilon\Amplitudaaa\wektor{\sigma}\cdot\wersor{k}_{i}\\
\varepsilon\wektor{\sigma}\cdot\wersor{k}_{f}\Amplitudaaa &\quad
\varepsilon^{2}\wektor{\sigma}\cdot\wersor{k}_{f}\Amplitudaaa\wektor{\sigma}\cdot\wersor{k}_{i}
\end{array}
\right).
\end{equation}
Henceforth matrices $\Amplitudaa$ and $\Amplitudaaa$ will be
called the matrix scattering amplitudes. The differential cross
section for scattering from the direction $\wektor{k}_{i}$ and the
spin arrangement $\wektor{\nu_{i}}$ onto $\wektor{k_{f}}$ and
$\wektor{\nu_{f}}$ is defined as
\begin{equation}\label{I.18}
\frac{\mathrm{d}\sigma}{\mathrm{d}\Omega_{f}}
=\chi_{f}^{\dag}\chi_{f}=\chi_{i}^{\dagger}
\Amplitudaaa^{\dagger}\Amplitudaaa\chi_{i},
\end{equation}
Subsequently, after integration the above over all the directions
of $\wektor{k}_{f}$, we arrive at the total cross section
\begin{equation}\label{I.19}
\sigma(\wektor{k}_{i},\wektor{\nu}_{i})=
\calkapow{k}_{f}\:\chi_{f}^{\dag}\chi_{f}.
\end{equation}
Finally, averaging over all directions of incidence
$\hat{\wektor{k}}_{i}$ and the initial spin orientation
$\hat{\wektor{\nu}}_{i}$, one finds the average total cross
section
\begin{equation}\label{I.20}
\sigma_{t}(E)=
\frac{1}{(4\pi)^{2}}\calkapow{k}_{i}\calkapow{\nu}_{i}\calkapow{k}_{f}
\:\chi_{f}^{\dag}\chi_{f}.
\end{equation}
Obviously all the mentioned cross sections may be expressed in terms
of all the scattering amplitudes $\Amplituda$, $\Amplitudaa$, and
$\Amplitudaaa$.
\section{Special class of non--local separable potentials}
\setcounter{equation}{0} \noindent\indent In this section we
employ the above considerations to the special class of non--local
separable potentials. As previously mentioned, such a class of
potentials allows to find solutions to the Lippmann--Schwinger
equations in an analytical way.
Consider the following class of potential kernels:
\begin{equation}\label{II.1}
\mathsf{V}(\wektor{r},\wektor{r}')=\sum_{\mu}\omega_{\mu}
\mathsf{u}_{\mu}(\wektor{r})\mathsf{u}_{\mu}^{\dag}(\wektor{r}')
\end{equation}
where it is assumed that in general $\mu$ may denote the arbitrary
finite set of indices, i.e., $\mu=\{\mu_{1},\ldots,\mu_{k}\}$ and
all the coefficients $\omega_{\mu}$ different from zero. Functions
$\mathsf{u}_{\mu}(\wektor{r})$ are assumed to be four--element
columns.
Substitution of Eq. (\ref{II.1}) to Eq. (\ref{I.4}) leads us to
the Lippmann--Schwinger equation for the separable potentials:
\begin{eqnarray}\label{II.2}
&&\hspace{-0.2cm}\psi(\wektor{r})=\phi_{i}(\wektor{r})-\sum_{\mu}\omega_{\mu}
\calkaob{r}'\,G(E,\wektor{r},\wektor{r}')\mathsf{u}_{\mu}(\wektor{r}')
\nonumber\\
&&\hspace{1cm}\times\calkaob{r}''\,\mathsf{u}_{\mu}^{\dag}(\wektor{r}'')\psi(\wektor{r}''),
\end{eqnarray}
which may be equivalently rewritten as a set of linear algebraic
equations. Indeed, using the Dirac notation one finds
\begin{equation}\label{II.3}
\sum_{\mu} \left[ \delta_{\nu\mu}+
\avg{\mathsf{u}_{\nu}}{\hat{G}(E)}{\mathsf{u}_{\mu}} \omega_{\mu}
\right]
\braket{\mathsf{u}_{\mu}}{\psi}=\braket{\mathsf{u}_{\nu}}{\phi_{i}}.
\end{equation}
For further convenience we introduce the following notations
\begin{eqnarray}\label{II.4}
&&\big<\mathsf{\mathsf{u}}\big|\varphi\big>= \left(
\begin{array}{c}
\big<\mathsf{u}_{1}\big|\varphi\big>\\*[0.5ex]
\big<\mathsf{u}_{2}\big|\varphi\big>\\
\vdots
\end{array}
\right),\nonumber\\
&&\big<\varphi\big|\mathsf{\mathsf{u}}\big>=\big<\mathsf{u}\big|\varphi\big>^{\dagger}=
\left( \big<\varphi\big|\mathsf{u}_{1}\big>\;
\big<\varphi\big|\mathsf{u}_{2}\big>\; \dots \right).
\end{eqnarray}
Consequently, we may rewrite Eq. (\ref{II.3}) as a matrix equation
$(\mathbbm{1}+\mathsf{G}\mathsf{\Omega})
\big<\mathsf{u}\big|\psi\big>=\big<\mathsf{u}\big|\phi_{i}\big>$
or equivalently as
\begin{equation}\label{II.6}
\big<\mathsf{u}\big|\psi\big>=(\mathbbm{1}+\mathsf{G}\mathsf{\Omega})^{-1}
\big<\mathsf{u}\big|\phi_{i}\big>,
\end{equation}
with $\mathsf{G}$ being a matrix composed of the elements
$\avg{\mathsf{u}_{\nu}}{\hat{G}(E)}{\mathsf{u}_{\mu}}$ and
$\sf{\Omega}=\mathrm{diag}[\omega_{\mu}]$. Similarly, substituting
Eq. (\ref{II.1}) to Eq. (\ref{I.14}) and again using Eq.
(\ref{I.8}), we arrive at the bispinor scattering amplitude for the
separable potentials in the form
\begin{eqnarray}\label{II.7}
&&\hspace{-1.5cm}\Amplituda=\frac{-E}{2\pi
c^{2}\hbar^{2}}\mathcal{P}(\wektor{k}_{f})\sum_{\mu}\omega_{\mu}\calkaob{r}\:e^{-i\wektor{k}_{f}\cdot\wektor{r}}
\mathsf{u}_{\mu}(\wektor{r})\nonumber\\
&&\times
\calkaob{r}'\:\mathsf{u}_{\mu}^{\dag}(\wektor{r}')\psi(\wektor{r}'),
\end{eqnarray}
which, by virtue of Eqs. (\ref{I.9}) and (\ref{II.6}), reduces to
\begin{equation}\label{II.8}
\Amplituda=\frac{-E}{2\pi c^{2}\hbar^{2}}
\sum_{s=-}^{+}\Theta_{s}(\wektor{k}_{f})\braket{\wektor{k}_{f}\theta_{s}}{\mathsf{u}}
\mathsf{\Omega}\left(\mathbbm{1}+\mathsf{G}\mathsf{\Omega}\right)^{-1}
\big<\mathsf{u}\big|\phi_{i}\big>
\end{equation}
and, utilizing the fact that for all invertible matrices
$\mathsf{X}$ and $\mathsf{Y}$ the relation
$(\mathsf{X}\mathsf{Y})^{-1}=\mathsf{Y}^{-1}\mathsf{X}^{-1}$ is
satisfied, finally to
\begin{equation}\label{II.9}
\Amplituda=\frac{-E}{2\pi c^{2}\hbar^{2}}
\sum_{s=-}^{+}\Theta_{s}(\wektor{k}_{f})\braket{\wektor{k}_{f}\theta_{s}}{\mathsf{u}}
\left(\mathsf{\Omega}^{-1}+\mathsf{G}\right)^{-1}
\big<\mathsf{u}\big|\phi_{i}\big>.
\end{equation}
Subsequently, using the fact that (\ref{I.11}), we obtain the
bispinor scattering amplitude in the following form
\begin{eqnarray}\label{II.10}
&&\hspace{-1.02cm}\Amplituda=\frac{-E}{2\pi
c^{2}\hbar^{2}}\sum_{s,t=-}^{+}\Theta_{s}(\wektor{k}_{f})\braket{\wektor{k}_{f}\theta_{s}}{\mathsf{u}}\nonumber\\
&&\times \left(\mathsf{\Omega}^{-1}+\mathsf{G}\right)^{-1}
\braket{\mathsf{u}}{\wektor{k}_{i}\theta_{t}}\Theta_{t}^{\dagger}(\wektor{k}_{i})
U_{i}(\wektor{k}_{i}),
\end{eqnarray}
which, after comparison with Eq. (\ref{I.16}), gives the formulae
for the $4\times4$ matrix scattering amplitude:
\begin{eqnarray}\label{II.11}
&&\hspace{-2.02cm}\Amplitudaa=\frac{-E}{2\pi c^{2}\hbar^{2}}
\sum_{s,t=-}^{+}\Theta_{s}(\wektor{k}_{f})\braket{\wektor{k}_{f}\theta_{s}}{\mathsf{u}}\nonumber\\
&&\hspace{-1cm}\times
\left(\mathsf{\Omega}^{-1}+\mathsf{G}\right)^{-1}
\braket{\mathsf{u}}{\wektor{k}_{i}\theta_{t}}\Theta_{t}^{\dagger}(\wektor{k}_{i}),
\end{eqnarray}
and finally, after straightforward movements, for the $2\times 2$
matrix scattering amplitude as
\begin{equation}\label{II.12}
\Amplitudaaa=\frac{-E}{2\pi c^{2}\hbar^{2}}
\sum_{s,t=-}^{+}\theta_{s}
\braket{\wektor{k}_{f}\theta_{s}}{\mathsf{u}}
\left(\mathsf{\Omega}^{-1}+\mathsf{G}\right)^{-1}
\braket{\mathsf{u}}{\wektor{k}_{i}\theta_{t}}\theta_{t}^{\dagger}.
\end{equation}
\section{The eigenchannel method}\label{III}
\setcounter{equation}{0} \noindent\indent Now we are in position
to apply the eigenchannel method proposed recently by Szmytkowski
\cite{Szmytkowski} to scattering of the Dirac particles from
potentials of the form (\ref{II.1}). As we shall see below, such a
class of potentials allows us to formulate this method in a
simplified algebraic form.
We start from the decomposition of the matrix
$\mathsf{\Omega}^{-1}+\mathsf{G}_{\mathrm{D}}$ into its Hermitian
and non-Hermitian parts, i.e.,
\begin{equation}\label{III.1}
\mathsf{\Omega}^{-1}+\mathsf{G}=\mathsf{A}+i\mathsf{B},
\end{equation}
where matrices $\sf{A}$ and $\sf{B}$ are defined through relations
\begin{equation}\label{III.2}
\mathsf{A}=\mathsf{\Omega}^{-1}+\frac{1}{2}
\left(\mathsf{G}+\mathsf{G}^{\dagger}\right),\qquad
\mathsf{B}=\frac{1}{2i}\left(\mathsf{G}-
\mathsf{G}^{\dagger}\right).
\end{equation}
It is evident from these definitions that both matrices $\sf{A}$
and $\sf{B}$ are Hermitian. Moreover, utilizing the fact that
\begin{equation}\label{III.4}
\wektor{\nabla}\frac{\mathrm{e}^{ik|\wektor{r}-\wektor{r}'|}}{|\wektor{r}-\wektor{r}'|}=
\frac{\wektor{r}-\wektor{r}'}{|\wektor{r}-\wektor{r}'|}\left(\frac{ik\mathrm{e}^{ik|\wektor{r}-\wektor{r}'|}}
{|\wektor{r}-\wektor{r}'|}-\frac{\mathrm{e}^{ik|\wektor{r}-\wektor{r}'|}}{|\wektor{r}-\wektor{r}'|^{2}}\right)
\end{equation}
where $\wektor{\varrho}=\wektor{r}-\wektor{r}'$, the
straightforward calculations lead us to their matrix elements of
the form
\begin{eqnarray}\label{III.6}
&&\mathsf{A}_{\nu\mu}=
\omega^{-1}_{\nu}\delta_{\nu\mu}-\frac{k}{4\pi c^{2}\hbar^{2}}
\calkaob{r}\calkaob{r}'\mathsf{u}_{\nu}^{\dagger}(\wektor{r})\nonumber\\
&&\times\left[ ic\hbar
k\wektor{\alpha}\cdot\frac{\wektor{\varrho}}{|\wektor{\varrho}|}
y_{1}(k|\wektor{\varrho}|)+(\beta
mc^{2}+E)y_{0}(k|\wektor{\varrho}|)
\right]\mathsf{u}_{\mu}(\wektor{r}')\nonumber\\
\end{eqnarray}
and
\begin{eqnarray}\label{III.7}
&&\mathsf{B}_{\nu\mu}=\frac{k}{4\pi c^{2}\hbar^{2}}
\calkaob{r}\calkaob{r}'\mathsf{u}_{\nu}^{\dagger}(\wektor{r})\nonumber\\
&&\times\left[ic\hbar
k\wektor{\alpha}\cdot\frac{\wektor{\varrho}}{|\wektor{\varrho}|}
j_{1}(k|\wektor{\varrho}|)+(\beta
mc^{2}+E)j_{0}(k|\wektor{\varrho}|)
\right]\mathsf{u}_{\mu}(\wektor{r}'),\nonumber\\
\end{eqnarray}
where $j_{0}(z)$, $j_{1}(z)$, $y_{0}(z)$ and $y_{1}(z)$ are,
respectively, the Bessel and Neumann spherical functions
\cite{AbrStegun}. Recall that in general
$j_{0}(z)=(\sin z)/z$,
$y_{0}(z)=(\cos z)/z$
and
\begin{equation}
j_{1}(z)=-\frac{\cos z}{z}+\frac{\sin z}{z^{2}}, \quad
y_{1}(z)=-\frac{\sin z}{z}-\frac{\cos z}{z^{2}}.
\end{equation}
The main idea of the present paper, adopted from
\cite{Szmytkowski}, is to construct the following weighted
spectral problem:
\begin{equation}\label{III.8}
\mathsf{A}X_{\gamma}(E)=\lambda_{\gamma}(E)\mathsf{B}
X_{\gamma}(E),
\end{equation}
where $X_{\gamma}(E)$ and $\lambda_{\gamma}(E)$ is, respectively,
an eigenvector and an eigenvalue. Thereafter the eigenvectors
$\{X_{\gamma}(E)\}$ will be called {\it eigenchannel vectors}.
They are directly related to eigenchannels defined in
\cite{Szmytkowski} as state vectors. In fact, they constitute a
projection of eigenchannels onto subspace spanned by
$\mac{u}_{\mu}(\wektor{r})$.
Using the fact that matrices $\mathsf{A}$ and $\mathsf{B}$ are
Hermitian and, as it is proven in Appendix \ref{AppA}, the matrix
$\mathsf{B}$ is positive semi-definite, one finds that the
eigenvalues $\{\lambda_{\gamma}(E)\}$ are real, i.e.,
$\lambda_{\gamma}^{*}(E)=\lambda_{\gamma}(E)$. Moreover, the
eigenchannels associated with different eigenvalues obey the
orthogonality relation
\begin{equation}\label{III.9}
X_{\gamma'}^{\dagger}(E)\mathsf{B}X_{\gamma}(E)=0 \qquad
(\lambda_{\gamma'}(E)\neq\lambda_{\gamma}(E)).
\end{equation}
In case of degeneration of some eigenvalues one may always choose
the corresponding eigenvectors to be orthogonal according to the
above relation. Then, imposing the normalization
$X_{\gamma}^{\dagger}(E)\mathsf{B}X_{\gamma}(E)=1$, one obtains
the following orthonormality relation
\begin{equation}\label{III.10}
X_{\gamma'}^{\dagger}(E)\mathsf{B}X_{\gamma}(E)=\delta_{\gamma'\gamma}.
\end{equation}
From Eqs. (\ref{III.8}) and (\ref{III.10}) one infers that the
eigenvalues $\{\lambda_{\gamma}(E)\}$ may be related to the matrix
$\mathsf{A}$ as follows
\begin{equation}\label{III.11}
\lambda_{\gamma}(E)=X_{\gamma}^{\dagger}(E)\mathsf{A}X_{\gamma}(E).
\end{equation}
Similar reasoning may be carried out employing the matrices
$\mathsf{A}$ and $\mathsf{\Omega}^{-1}+\mathsf{G}$. Indeed, after
algebraic manipulations we arrive at
\begin{eqnarray}\label{III.12}
&&X_{\gamma'}^{\dagger}(E)\mathsf{A}X_{\gamma}(E)=
\lambda_{\gamma}(E)\delta_{\gamma'\gamma},\nonumber\\
&&\hspace{-1cm}X_{\gamma'}^{\dagger}(E)(\mathsf{\Omega}^{-1}+\mathsf{G})X_{\gamma}(E)=
[i+\lambda_{\gamma}(E)]\delta_{\gamma'\gamma},
\end{eqnarray}
and
$\lambda_{\gamma}(E)=X_{\gamma}^{\dagger}(E)(\mathsf{\Omega}^{-1}+\mathsf{G})X_{\gamma}(E)-i$.
Since the eigenchannels $\{X_{\gamma}(E)\}$ are the solutions of
the Hermitian eigenvalue problem, they may satisfy the following
closure relations
\begin{eqnarray}\label{III.14}
&\displaystyle \sum_{\gamma}X_{\gamma}(E)X_{\gamma}^{\dagger}(E)\mathsf{B}=\jed,&\nonumber\\
&\displaystyle \sum_{\gamma}\lambda_{\gamma}^{-1}(E)
X_{\gamma}(E)X_{\gamma}^{\dagger}(E)\mathsf{A}=\jed,&
\end{eqnarray}
and
\begin{equation}\label{III.15}
\sum_{\gamma}
\frac{1}{i+\lambda_{\gamma}(E)}X_{\gamma}(E)X_{\gamma}^{\dagger}(E)
(\mathsf{\Omega}^{-1}+\mathsf{G})=\jed,
\end{equation}
where $\jed$ is an identity matrix, which dimension depends on the
dimension of the matrix $\mathsf{G}$. For purposes of further
analyzes the above closure relations are assumed to hold. Below, we
employ the above reasoning to the derivation of the scattering
amplitudes. From Eq. (\ref{III.15}) one deduces that
\begin{equation}\label{III.16}
(\mathsf{\Omega}^{-1}+\mathsf{G})^{-1}=\sum_{\gamma}
\frac{1}{i+\lambda_{\gamma}(E)}
X_{\gamma}(E)X_{\gamma}^{\dagger}(E).
\end{equation}
After substitution of Eq. (\ref{III.16}) to Eq. (\ref{II.11}) and
rearranging terms, we have
\begin{eqnarray}\label{III.17}
&&\hspace{-0.5cm}\Amplitudaa=\frac{-E}{2\pi c^{2}\hbar^{2}}
\sum_{\gamma}\frac{1}{i+\lambda_{\gamma}(E)}\sum_{s=-}^{+}\Theta_{s}(\wektor{k}_{f})
\braket{\wektor{k}_{f}\theta_{s}}{\mathsf{u}}X_{\gamma}(E)\nonumber\\
&&\hspace{0.5cm}\times \sum_{t=-}^{+}X_{\gamma}^{\dagger}(E)
\braket{\mathsf{u}}{\wektor{k}_{i}\theta_{t}}\Theta_{t}^{\dagger}(\wektor{k}_{i}).
\end{eqnarray}
Let us define the following angular functions
\begin{equation}\label{III.18}
\mathcal{Y}_{\gamma}(\wektor{k})=\sqrt{\frac{Ek}{8\pi^{2}c^{2}\hbar^{2}}}
\sum_{s=-}^{+}\Theta_{s}(\wektor{k})\braket{\wektor{k}\theta_{s}}{\mathsf{u}}
X_{\gamma}(E),
\end{equation}
hereafter termed the {\it eigenchannel bispinor harmonics}. The
functions $\{\mathcal{Y}_{\gamma}(\wektor{k})\}$ are orthonormal
on the unit sphere (for proof, see Appendix \ref{AppB}), i.e.,
\begin{equation}\label{III.19}
\calkapow{k}\,\mathcal{Y}_{\gamma'}^{\dagger}(\wektor{k})\mathcal{Y}_{\gamma}(\wektor{k})
=\delta_{\gamma'\gamma}.
\end{equation}
Application of Eq. (\ref{III.18}) to Eq. (\ref{III.17}) yields
\begin{equation}\label{III.20}
\Amplitudaa=\frac{4\pi}{k}\sum_{\gamma}e^{i\delta_{\gamma}(E)}\sin\delta_{\gamma}(E)
\mathcal{Y}_{\gamma}(\wektor{k}_{f})\mathcal{Y}_{\gamma}^{\dagger}(\wektor{k}_{i}),
\end{equation}
where $\{\delta_{\gamma}(E)\}$ are called {\it eigenphase-shifts}
and are related to $\{\lambda_{\gamma}(E)\}$ according to
\begin{equation}\label{III.21}
\lambda_{\gamma}(E)=-\cot\delta_{\gamma}(E).
\end{equation}
Similar considerations may be carried out for the $2\times 2$
matrix scattering amplitude $\Amplitudaaa$. Indeed, in virtue of
Eq. (\ref{I.17}) we may rewrite Eq. (\ref{II.12}) in the form
\begin{equation}\label{III.22}
\Amplitudaaa=\frac{4\pi}{k}\sum_{\gamma}e^{i\delta_{\gamma}(E)}
\sin\delta_{\gamma}(E)\Upsilon_{\gamma}(\wektor{k}_{f})\Upsilon_{\gamma}^{\dagger}(\wektor{k}_{i}),
\end{equation}
where the angular functions $\{\Upsilon_{\gamma}(\wektor{k})\}$,
hereafter called {\it eigenchannel spinor harmonics}, are defined as
follows
\begin{equation}\label{III.23}
\Upsilon_{\gamma}(\wektor{k})=\sqrt{\frac{Ek}{8\pi^{2}c^{2}\hbar^{2}}}
\sum_{s=-}^{+}\theta_{s}\braket{\wektor{k}\theta_{s}}{\mathsf{u}}
X_{\gamma}(E).
\end{equation}
Moreover, they are orthogonal on the unit sphere (for proof, see
Appendix \ref{AppB})
\begin{equation}\label{III.24}
\calkapow{k}\,\Upsilon_{\gamma'}^{\dagger}(\wektor{k})\Upsilon_{\gamma}(\wektor{k})=\delta_{\gamma'\gamma},
\end{equation}
and, as one can verify, are related to the eigenchannel bispinor
harmonics $\{\mathcal{Y}_{\gamma}(\wektor{k})\}$ via the relation
\begin{equation}\label{III.25}
\mathcal{Y}_{\gamma}(\wektor{k})=\frac{1}{\sqrt{1+\varepsilon^{2}}}
\left(
\begin{array}{c}
\Upsilon_{\gamma}(\wektor{k})\\
\varepsilon\wektor{\sigma}\cdot\wersor{k}\Upsilon_{\gamma}(\wektor{k})
\end{array}
\right).
\end{equation}
Now we are in position to compute scattering cross-sections.
Substitution of Eq. (\ref{III.22}) to Eq. (\ref{I.18}) and
integration over all directions of scattering
$\hat{\wektor{k}}_{f}$, by virtue of relation (\ref{III.24}),
yields
\begin{equation}\label{III.26}
\sigma(\wektor{k}_{i},\wektor{\nu}_{i})=\frac{16\pi^{2}}{k^{2}}
\sum_{\gamma}\sin^{2}\delta_{\gamma}(E)\left|\chi_{i}^{\dagger}\Upsilon_{\gamma}(\wektor{k}_{i})\right|^{2}.
\end{equation}
To compute the total cross-section averaged over all arrangements
of spin of the incident particle, we have to notice that the
projector onto the pure state $\chi_{i}$ may be written as
$\chi_{i}\chi_{i}^{\dagger}=(1/2)[\jed_{2}+\wektor{\nu}_{i}\cdot\wektor{\sigma}]$
with $|\wektor{\nu}_{i}|=1$.
Therefore, substituting of the above to Eq. (\ref{III.26}) and
averaging over all directions of $\boldsymbol{\nu}_{i}$, we arrive
at
\begin{equation}\label{III.28}
\sigma(\wektor{k}_{i})=
\frac{8\pi^{2}}{k^{2}}\sum_{\gamma}\sin^{2}\delta_{\gamma}(E)\Upsilon_{\gamma}^{\dagger}(\wektor{k}_{i})
\Upsilon_{\gamma}(\wektor{k}_{i}).
\end{equation}
Finally, averaging the above scattering cross-section over all
directions of incidence $\hat{\wektor{k}}_{i}$, again by virtue of
Eq. (\ref{III.24}), we get the total cross-section in the form
\begin{equation}\label{III.29}
\sigma_{t}(E)=\frac{2\pi}{k^{2}}\sum_{\gamma}\sin^{2}\delta_{\gamma}(E).
\end{equation}
It should be emphasized that all the above considerations
respecting scattering cross-sections may be repeated using the
eigenchannel bispinor harmonics
$\{\mathcal{Y}_{\gamma}(\wektor{k})\}$ instead of the eigenchannel
spinor harmonics $\{\Upsilon_{\gamma}(\wektor{k})\}$. The
significant difference is that then the integrals over
$\hat{\wektor{k}}_{f}$ and $\hat{\wektor{k}}_{i}$ need to be
calculated using relation (\ref{III.19}) instead of
(\ref{III.24}).
\section{Example}
\setcounter{equation}{0}
We conclude our considerations providing here an illustrative
example concerning the scattering from a spherical shell of radius
$R$, centered at the origin of the coordinate system. Due to the
assumption of non-locality of potentials under consideration, we
shall simulate this process by using a potential of the form
\begin{equation}\label{Ex1}
\mac{V}(\wektor{r},\wektor{r}')=\omega
v(\wektor{r})v(\wektor{r}')\jed_{4},\qquad
v(\wektor{r})=\frac{1}{\sqrt{4\pi}}\frac{\delta(r-R)}{R^{2}},
\end{equation}
where $\omega\neq 0$. Notice that the potential defined above is
the special case of that proposed recently by de Prunel\'e
\cite{Prunele} (see also \cite{Prunele1}). Scattering of the Dirac
particles from delta-like potentials was also studied e.g. in
Refs. \cite{Dombey,Loewe}. However, in these papers the authors
considered only local potentials and not non-local ones.
At the very beginning, we need to bring the potential (\ref{Ex1}) to
the previously postulated form (\ref{II.1}). To this aim, let
$\mac{e}_{1}$ and $\mac{e}_{2}$ constitute a standard basis in
$\mathbb{C}^{2}$, i.e., $\mac{e}_{1}=(1\;0)^{T}$ and
$\mac{e}_{2}=(0\;1)^{T}$. Moreover, let
$\mac{e}_{ij}=\mac{e}_{i}\otimes\mac{e}_{j}$ and then by virtue of
the fact that
$\jed_{4}=\sum_{i,j=1}^{2}\mac{e}_{ij}\mac{e}_{ij}^{\dagger}$, we
may rewrite (\ref{Ex1}) as
\begin{equation}\label{Ex2}
\mac{V}(\wektor{r},\wektor{r}')=\omega\sum_{i,j=1}^{2}\mac{u}_{ij}(\wektor{r})\mac{u}_{ij}^{\dagger}(\wektor{r}),\qquad
\mac{u}_{ij}(\wektor{r})=v(\wektor{r})\mac{e}_{ij}.
\end{equation}
Now, we are in position to compute the matrix $\mac{G}$. Using
Eqs. (\ref{I.6}) and (\ref{Ex1}), after straightforward
integrations we have
\begin{eqnarray}\label{Ex3}
\mac{G}=ikj_{0}(kR)h_{0}^{(+)}(kR) \left(
\begin{array}{cc}
\eta_{+}\jed_{2} & 0\\
0 & \eta_{-}\jed_{2}
\end{array}
\right),
\end{eqnarray}
where $\eta_{\pm}=(E\pm mc^{2})/c^{2}\hbar^{2}$ and
$h_{0}^{(+)}(z)=j_{0}(z)+iy_{0}(z)$ is the spherical Hankel
function of the first kind. Hence, by the definitions given in Eq.
(\ref{III.2}), we find that the explicit forms of matrices
$\mac{A}$ and $\mac{B}$ are
\begin{equation}\label{Ex4}
\mac{A}=\left(
\begin{array}{cc}
[\omega^{-1}-kj_{0}(kR)y_{0}(kR)\eta_{+}]\jed_{2} &\hspace{-1.7cm} 0\\
0 & \hspace{-1.7cm}
[\omega^{-1}-kj_{0}(kR)y_{0}(kR)\eta_{-}]\jed_{2}
\end{array}
\right)
\end{equation}
and
\begin{equation}\label{Ex5}
\mac{B}=kj_{0}^{2}(kR) \left(
\begin{array}{cc}
\eta_{+}\jed_{2} & 0\\
0 & \eta_{-}\jed_{2}
\end{array}
\right).
\end{equation}
According to the method formulated in Sec. \ref{III}, we may
construct the following spectral problem
\begin{equation}\label{Ex6}
\mac{A}X_{\gamma}(E)=\lambda_{\gamma}(E)\mac{B}X_{\gamma}(E)\qquad
(\gamma=1,2,3,4),
\end{equation}
which, as one can easily verify, has two different eigenvalues
\begin{equation}\label{Ex7}
\lambda_{\pm}(E)=\frac{\omega^{-1}-kj_{0}(kR)y_{0}(kR)\eta_{\pm}}{kj_{0}^{2}(kR)\eta_{\pm}}
\end{equation}
and respective eigenvectors
\begin{eqnarray}\label{Ex8}
&\displaystyle X_{+}^{(1(2))}(E)=\frac{1}{\sqrt{k\eta_{+}}j_{0}(kR)}\,\mac{e}_{1}\otimes\mac{e}_{1(2)},&\nonumber\\
&\displaystyle X_{-}^{(1(2))}(E)=\frac{1}{\sqrt{k\eta_{-}}j_{0}(kR)}
\,\mac{e}_{2}\otimes\mac{e}_{1(2)}.&
\end{eqnarray}
Then, using Eq. (\ref{III.18}) and by virtue of the fact that
\begin{eqnarray}\label{Ex9}
&&\hspace{-1.5cm}\braket{\wektor{k}\chi}{\mac{\bf{u}}}=\sqrt{\frac{4\pi}{1+\varepsilon^{2}}}j_{0}(kR) \nonumber\\
&&\hspace{-1cm}\times \left( \chi^{\dagger}\mac{e}_{1}\;\;
\chi^{\dagger}\mac{e}_{2}\;\;\varepsilon\chi^{\dagger}\wektor{\sigma}\cdot\wersor{k}\,
\mac{e}_{1}\;\;
\varepsilon\chi^{\dagger}\wektor{\sigma}\cdot\wersor{k}\,
\mac{e}_{2}\right),
\end{eqnarray}
we arrive at the four eigenchannel bispinor harmonics
$\{\mathcal{Y}_{\gamma}(\wektor{k})\}$ in the form
\begin{equation}\label{Ex10}
\mathcal{Y}_{+}^{(1(2))}(\wektor{k})=
\frac{1}{\sqrt{4\pi(1+\varepsilon^{2})}}\, \left(
\begin{array}{c}
\mac{e}_{1(2)}\\
\varepsilon\wektor{\sigma}\cdot\wersor{k}\,\mac{e}_{1(2)}
\end{array}
\right)
\end{equation}
and
\begin{equation}
\mathcal{Y}_{-}^{(1(2))}(\wektor{k})=\frac{1}{\sqrt{4\pi(1+\varepsilon^{2})}}\,
\left(
\begin{array}{c}
\varepsilon\wektor{\sigma}\cdot\wersor{k}\,\mac{e}_{1(2)}\\
\mac{e}_{1(2)}
\end{array}
\right).
\end{equation}
Then, by virtue of Eq. (\ref{III.23}), one obtains the eigenchannel
spinor harmonics $\{\Upsilon_{\gamma}(\wektor{k})\}$ in the form
\begin{equation}\label{Ex11}
\Upsilon_{+}^{(1(2))}(\wektor{k})=
\frac{1}{\sqrt{4\pi}}\,\mac{e}_{1(2)},\quad
\Upsilon_{-}^{(1(2))}(\wektor{k})=\frac{1}{\sqrt{4\pi}}\,\wektor{\sigma}\cdot\wersor{k}\,\mac{e}_{1(2)}.
\end{equation}
The latter may be equivalently obtained combining Eqs.
(\ref{III.25}) and (\ref{Ex11}). Moreover, as one may easily
verify, functions given by Eqs. (\ref{Ex10}) and (\ref{Ex11}) are
orthonormal, respectively, in the sense (\ref{III.19}) and
(\ref{III.24}).
Before we find an expression for total cross section, we compute
the scattering amplitude. Since, as shown in Sec. \ref{SecII}, the
bispinor and both matrix scattering amplitudes are mutually
related, we restrict our considerations to the $2\times 2$
scattering amplitude. Thus, combining Eqs. (\ref{III.22}),
(\ref{Ex5}), and (\ref{Ex11}) we obtain
\begin{eqnarray}\label{Ex12}
&&\hspace{-0.9cm}\Amplitudaaa=-j_{0}^{2}(kR)\left[
\frac{\jed_{2}}{ik
j_{0}(kR)h_{0}^{(+)}(kR)+(\omega\eta_{+})^{-1}}\right.\nonumber\\
&&\left.+\frac{(\wektor{\sigma}\cdot\wersor{k}_{f})
(\wektor{\sigma}\cdot\wersor{k}_{i})}
{ikj_{0}(kR)h_{0}^{(+)}(kR)+(\omega\eta_{-})^{-1}} \right].
\end{eqnarray}
Finally, substitution of Eqs. (\ref{Ex8}) and (\ref{Ex11}) to Eq.
(\ref{III.26}) with the aid of Eq. (\ref{III.21}) yields
\begin{eqnarray}\label{Ex13}
&&\hspace{-1cm}\sigma(\wektor{k}_{i},\wektor{\nu}_{i})=\frac{4\pi}{k^{2}}
j_{0}^{4}(kR)\nonumber\\
&&\hspace{-0.5cm}\times\left\{
\frac{1}{[(k\omega \eta_{+})^{-1}-j_{0}(kR)y_{0}(kR)]^{2}+j_{0}^{4}(kR)}\right.\nonumber\\
&&\hspace{-0.5cm}+\left.\frac{1}{[(k\omega\eta_{-})^{-1}-j_{0}(kR)y_{0}(kR)]^{2}
+j_{0}^{4}(kR)}\right\}.
\end{eqnarray}
Here it is evident that
$\sigma(\wektor{k}_{i},\wektor{\nu}_{i})=\sigma(\wektor{k}_{i})=\sigma_{t}(E)$.
In order to illustrate the obtained results, the eigenphaseshifts
for two different values of $\omega$, derived from Eqs.
(\ref{III.21}) and (\ref{Ex7}), are plotted in Fig. 1 and 2.
Figures 3 and 4 present partial $\sigma_{\pm}(E)$ as well as total
$\sigma_{t}(E)$ cross sections.
It seems interesting to investigate the behavior of both eigenvalues
$\lambda_{\pm}(E)$ in the non-relativistic limit, i.e., when
$c\to\infty$. From (\ref{DiracWaveNumber}) one concludes that
\begin{equation}
\eta_{+}\stackrel{c\to\infty}{\LRA}\frac{2m}{\hbar^{2}},\qquad
\eta_{-}\stackrel{c\to\infty}{\LRA}0
\end{equation}
and therefore
\begin{equation}
\lambda_{+}(E)\stackrel{c\to\infty}{\LRA}\frac{(\hbar^{2}/2m\omega)-kj_{0}(kR)y_{0}(kR)}{kj_{0}^{2}(kR)}
\end{equation}
and
\begin{equation}
\lambda_{-}(E)\stackrel{c\to\infty}{\LRA}\mathrm{sgn}(\omega)\infty.
\end{equation}
This means that $\delta_{-}(E)\to n\pi$ $(n\in \mathbb{Z})$ in the
limit of $c\to\infty$. Therefore the cross section $\sigma_{-}(E)$
vanishes in the non-relativistic limit and in this sense it has a
purely relativistic character leading to the fact that the
resonance appearing in Fig. 4 at about $1.25 mc^{2}$ is purely
relativistic effect.
One sees that in the non-relativistic limit the cross section
(\ref{Ex13}) reduces to
\begin{eqnarray}
&&\hspace{-1cm}\sigma_{t}(E)\stackrel{c\to\infty}{\LRA}\frac{4\pi}{k^{2}}
j_{0}^{4}(kR)\nonumber\\
&&\hspace{-0.8cm}\times\left\{ \frac{1}{[(\hbar^{2}/2m
k\omega)-j_{0}(kR)y_{0}(kR)]^{2}+j_{0}^{4}(kR)}\right\}.
\end{eqnarray}
The above cross section may also be obtained using non-relativistic
formulation of the present method given in Ref. \cite{moja}.
\begin{figure}[ht]
\includegraphics[width=6cm]{fig1.eps}
\caption{Behavior of eigenphaseshifts $\delta_{+}(E)$ (solid
curve) and $\delta_{-}(E)$ (dashed curve) as functions of energy
$E$ (in units of $mc^{2}$) for $\omega=-\hbar^{3}/m^{2}c$ and
$R=\hbar/mc$. The eigenphaseshift $\delta_{+}(E)$ has been
constrained to the range $[-\pi/2,\pi/2]$. }
\end{figure}
\begin{figure}[ht]
\includegraphics[width=6cm]{fig2.eps}
\caption{Behavior of eigenphaseshifts $\delta_{+}(E)$ (solid curve)
and $\delta_{-}(E)$ (dashed curve) as functions of energy $E$ (in
units of $mc^{2}$) for $\omega=-5\hbar^{3}/m^{2}c$ and $R=\hbar/mc$.
Both eigenphaseshifts have been constrained to the range
$[-\pi/2,\pi/2]$.}
\end{figure}
\begin{figure}[ht]
\includegraphics[width=6cm]{fig3.eps}
\caption{Partial $\sigma_{+}(E)$ (dashed curve), $\sigma_{-}(E)$
(dotted curve), and total $\sigma_{t}(E)$ (solid curve) cross
sections (all in units of $R^{2}$) as functions of energy $E$ (in
units of $mc^{2}$) for $\omega=-\hbar^{3}/m^{2}c$ and
$R=\hbar/mc$.}
\end{figure}
\begin{figure}[ht]
\includegraphics[width=6cm]{fig4.eps}
\caption{Partial $\sigma_{+}(E)$ (dashed curve), $\sigma_{-}(E)$
(dotted curve), and total $\sigma_{t}(E)$ (solid curve) cross
sections (all in units of $R^{2}$) as functions of energy $E$ (in
units of $mc^{2}$) for $\omega=-5\hbar^{3}/m^{2}c$ and
$R=\hbar/mc$. }
\end{figure}
\section{Conclusions}
In this work, an application of the recently proposed eigenchannel
method \cite{Szmytkowski} to the scattering of Dirac particles from
non-local separable potentials has been presented. Application of
such a particular case of the non-local potentials reduces naturally
the general weighted eigenvalue problem stated in Ref.
\cite{Szmytkowski} to its matrix counterpart given by Eq.
(\ref{III.8}) leading to the definition of eigenchannel vectors.
Using the notion of the eigenchannel vectors the definitions of
eigenchannel spinor as well as bispinor harmonics have been given.
The latter provide us with the formulas for scattering amplitudes
similar to that well-known for central potentials generalizing
them at the same time to the case of non-local separable
potentials.
The general considerations have been extended with an illustrative
example in which the Dirac particles are scattered from non-local,
delta-like potential. In this particular case, the general
eigenvalue problem (\ref{III.8}) become just a $4\times 4$ matrix
equation and therefore is easily solvable (notice that in the case
of non-relativistic scattering it would be just a one-dimensional
problem). The eigenvalues of this problem are two-fold degenerated
and therefore give two different eigenphase-shifts from which one
has a purely relativistic character in the sense that it tends to
$n\pi$ $(n\in \mathbb{Z})$ whenever $c\to\infty$ giving no
contribution to total cross sections in non-relativistic limit.
One sees also that even such a simple example of non-local
potentials may lead to some resonances (see Fig. 4).
The next step in our considerations will be to investigate the
applicability of the new formulation of the eigenchannel method in
the case of inelastic scattering from separable potentials.
Moreover it seems also interesting to investigate the
applicability of the method to the other, more complicated
examples of separable potentials.
\section*{Acknowledgments}
I am grateful to R.~Szmytkowski for very useful discussions,
suggestions and commenting on the manuscript. Discussions with
M.~Czachor are also acknowledged.
|
{
"timestamp": "2007-11-05T18:23:44",
"yymm": "0503",
"arxiv_id": "physics/0503196",
"language": "en",
"url": "https://arxiv.org/abs/physics/0503196"
}
|
\section{\label{sec:level1}Introduction}
Recent advances in quantum information science have shown that, on one hand,
photons are ideal carriers of quantum information, and on the other hand,
atoms represent reliable and long-lived storage and processing units. In
recent years quantum light storage is one of the extensively studied tasks
of quantum optics. Basic idea of light storage is electromagnetically
induced transparency (EIT) \cite{EIT}.
Electromagnetically induced transparency is a coherent interaction process
in which a coupling laser field is used to made the optical dense media
transparent for the probe field. Since its discovery, a number of new
effects and techniques for light-matter interaction have appeared [2-6].
Most notably, from the point of view of the work presented here, particular
attention has been devoted to ultraslow light propagation and light storage
techniques [3-5].
The key concept of EIT is the dark state and population trapping \cite%
{arrimon}. The dark state is a specific coherent superposition state which
does not contain excited short-living atomic level due to\ destructive
interference between two interaction paths. The dark-state is eigenstate of
the light -- atom interaction Hamiltonian, so the atom prepared in a
dark-state can not be excited and cannot leave the dark-state if the
interaction is adiabatic (Fig.1). The population trapping via applying
strong coupling leads to the adiabatic formation of the dark-state. Since
the interaction is realized by the light pulses the infuence of nonadiabatic
corrections may become important \cite{adiabatica}, \cite{Nasha} . This
infuence has been studied e.g. in \cite{adiabatica}. In particular, the
first nonadiabatic correction connects the dark state and the bright state,
so the depletion of the bright state, because of optical pumping, can affect
the dark state. Despite of large amount of experimental and theoretical
papers concerning light storage (see \cite{revlukin}\ and citations there),
and applications in quantum information science, the influence of
decoherence level width on information carried by stored light is studied
insufficiently.
In this work we present theoretical study which discusses and explains
influence of all relaxations on probe propagation both analytically and
numerically (it is essential in especially, solid state systems \cite{solid}%
). The goal is to study comprehensively how the depletion of bright state
will affect the pulse propagation in an EIT\ media and light storage in
particular.\ By solving the coupled system of Maxwell and density matrix
equations to the first order of the nonstationary pertrubation theory with
respect to nonadiabaticity and decoherence we obtain analytical solution
which completely describes the probe pulse propagation and is consistent
with the recent light storage experiments.
The paper is organized as follows. In section II the basic equations are
written down and the probe pulse propagation equation is derived and
analyzed. In section III and Appendixes the analytical solution for
counterintuitive pulse switching order and for matched pulses are obtained
and the asymptotic solutions discussed. Section IV deals with the physical
consequences of the obtained solution, namely the necessary conditions of
the pulse storage and retrieving, also the numerical results are
demonstrated. In section V we consider the transverse relaxation of the
coherence induced in the medium. Section VI concludes the paper.
\section{\label{basic}Basic Equations}
Figure 1 shows a schematic diagram of the atomic system in the EIT basis:
media of three level atoms interacting with two laser pulses $%
E_{p}=A_{p}\cos \left( \omega _{p}t-k_{p}z+\varphi _{p}\right) $ (probe)\
and $E_{c}=A_{c}\cos \left( \omega _{c}t-k_{c}z+\varphi _{c}\right) $\emph{\
}(coupling). The probe field resonantly connects the state $|1\rangle $ to
the state $|3\rangle $ and the coupling field connects $|2\rangle $ to $%
|3\rangle $. The Hamiltonian of the system in the rotating wave
approximation is:
\begin{equation*}
H=\hbar \Delta \sigma _{33}-\hbar \Omega _{p}\sigma _{31}-\hbar \Omega
_{p}\sigma _{32}+H.c.\text{,}
\end{equation*}%
where $\Omega _{p,c}=\dfrac{A_{p,c}\mu _{3i}}{\hbar }$ ($i=1,2$) are the
respective Rabi frequencies, $\sigma _{ij}=|i\rangle \langle j|$ are the
atomic transition operators, $\Delta =\omega _{p}-\omega _{31}=$ $\omega
_{c}-\omega _{32}$ is the detuning of the pulse frequencies from the upper
level and $\mu _{3i}$ ($i=1,2$) are the dipole moments of corresponding
transitions.
We assume\ that: (i) the probe field is weak as compared to the coupling
pulse field $\Omega _{p}<<\Omega _{c}$; (ii) the interaction is adiabatic ($%
\Omega _{c}T>>1$, where $T$ is the interaction duration). Then, the atomic
density matrix equation may be written as
\begin{eqnarray}
\overset{.}{\rho }_{31} &=&-\Gamma \rho _{31}+i\Omega _{p}+i\Omega _{c}\rho
_{21}, \notag \\
\overset{.}{\rho }_{21} &=&i\Omega _{c}^{\ast }\rho _{31}, \label{bloch} \\
\rho _{11} &=&1, \notag \\
\rho _{22} &=&\rho _{33}=\rho _{32}=0, \notag
\end{eqnarray}%
where $\Gamma $ is the width of the upper level which is the sum of the
spontaneous decay and transverse relaxations rates. It is supposed that
interaction is fast enough to neglect the decoherence between metastable
levels (sec. III, IV), or to take it into account to the first order (sec.
V).
The propagation of the pulses is governed by the Maxwell equation for slowly
varying amplitudes,%
\begin{eqnarray}
\left( \frac{\partial }{\partial x^{\prime }}+\frac{1}{c}\frac{\partial }{%
\partial t^{\prime }}\right) \Omega _{p} &=&iq_{p}\rho _{31}, \label{prop}
\\
\text{ \ }\left( \frac{\partial }{\partial x^{\prime }}+\frac{1}{c}\frac{%
\partial }{\partial t^{\prime }}\right) \Omega _{p} &=&iq_{c}\rho _{32},
\notag
\end{eqnarray}%
where $q_{i}=\dfrac{2\pi \mu _{3i}\omega _{i}N}{\hbar c}$, $N$ is the atomic
number density.
System of equations (\ref{bloch}) can be reduced to one equation for $\rho
_{31}$%
\begin{equation}
\overset{..}{\rho }_{31}-\overset{.}{\rho }_{31}\frac{\overset{.}{\Omega }%
_{c}}{\Omega _{c}}+\rho _{31}\Omega _{c}^{2}+\Gamma \left( \overset{.}{\rho }%
_{31}-\rho _{31}\frac{\overset{.}{\Omega }_{c}}{\Omega _{c}}\right) =i\Omega
_{c}\overset{.}{\theta } \label{dens}
\end{equation}%
where $\theta =\dfrac{\Omega _{p}}{\Omega _{c}}$ is the common used notation
for the so called mixing angle. Influence of the first two terms in (\ref%
{dens}) can be neglected if we confine to\ only first terms with respect to
the nonadiabaticity (i.e. $\left( \Omega _{c}T\right) ^{-2}<<1$ is
neglected). Influence of the fourth term in (\ref{dens}) is essential
parameter only under the assumption%
\begin{equation}
\Gamma T>>1. \label{largeg}
\end{equation}%
The meaning of the condition (\ref{largeg})\ is obvious: under the condition
of complete adiabaticity relaxation does not affect the pulse propagation
(dark-state), but taking into account first nonadiabatic correction, has
essential influence. The relaxation can be neglected when $\Gamma T\lesssim
1 $.
Finally, by substituting the Maxwell equation (\ref{prop}) into (\ref{dens})
one gets pulse propagation equation in wave variables $x=x^{\prime }$, $%
t=t-x^{\prime }/c$,%
\begin{equation}
\frac{1}{\Gamma _{1}}\frac{\partial ^{2}\theta }{\partial x\partial t}+\frac{%
q_{p}}{\Omega _{c}^{2}}\frac{\partial \theta }{\partial t}+\frac{\partial
\theta }{\partial x}=0, \label{probe_prop}
\end{equation}
where notation $\Gamma _{1}=\dfrac{\Omega _{c}^{2}}{\Gamma }$ is used. In
this connection the coherence dynamics is governed by the following equation:%
\begin{equation}
\overset{.}{\rho }_{21}=-\Gamma _{1}\rho _{21}-\Gamma _{1}\theta
\label{coh_dyn}
\end{equation}
Thus $\Gamma _{1}$ is the coherence decay rate, or width of EIT\ resonance,
due to applied coupling. Under the condition%
\begin{equation}
\Gamma _{1}T>>1 \label{bright}
\end{equation}%
equation (\ref{coh_dyn}) has the well known quasi-stationary solution $\rho
_{21}=-\theta $ \cite{polariton}, and the equation (\ref{probe_prop}) passes
to the dark state polariton propagation equation. The condition (\ref{bright}%
) means, that width of EIT\ resonance exceeds the spectral width of the
probe.
\section{\label{solution of propag}Solution of propagation equation}
The obtained probe pulse propagation equation (\ref{probe_prop}) is solved
by the method presented in \cite{Mostowski}. Since (\ref{probe_prop}) is
linear in $\theta $ and $\Omega _{c}\left( t\right) $ is independent of $x$,
it can be solved by using the Laplace transform with respect to $x$. The
solution of (\ref{probe_prop}) for $\theta $'s Laplace image can be found
easily:%
\begin{equation}
\overset{\symbol{126}}{\theta }\left( s,t\right) =\int\limits_{-\infty }^{t}%
\frac{\overset{.}{\theta }_{0}+\Gamma _{1}\theta _{0}}{s+q_{p}/\Gamma }%
B\left( s,t,t_{1}\right) dt_{1}+c\left( s\right) B\left( s,t,-\infty \right)
\label{image_sol}
\end{equation}%
where $B\left( s,t,t_{1}\right) =\exp \left( -\dfrac{s}{s+q_{p}/\Gamma }%
\int\limits_{t_{1}}^{t}\Gamma _{1}dt^{\prime }\right) $, $c(s)$ is an
integration constant that is determined by the initial condition, $c\left(
s\right) =\overset{\symbol{126}}{\theta }\left( s,-\infty \right) $. If
pulses are switched in counterintuitive sequence (coupling turns on earlier
than the probe does) then $c\left( s\right) =0$, since $\theta \left(
z,-\infty \right) =\theta _{0}\left( -\infty \right) =0$ (see appendix A).
Space time evolution of the probe pulse is obtained by implementing the
reverse Laplace transform in (\ref{image_sol}).%
\begin{eqnarray}
\theta \left( z,t\right) &=&\int\limits_{-\infty }^{t}dt_{1}\left( \theta
_{0}\left( t\right) \Gamma _{1}+\overset{.}{\theta }_{0}\left( t\right)
\right) \times \label{solution} \\
&&\times \exp \left( -z-\alpha \left( t_{1},t\right) \right) I_{0}\left( 2%
\sqrt{z\alpha \left( t_{1},t\right) }\right) , \notag
\end{eqnarray}
where $z=\dfrac{q_{p}x}{\Gamma }$ is propagation distance normalized to
linear absorption factor and, for convenience, the notation $\alpha \left(
t_{1},t\right) =\int\limits_{t_{1}}^{t}\Gamma _{1}\left( t^{\prime }\right)
dt^{\prime }$ is used. \ By using the condition $z\alpha \left(
t_{1},t\right) >>1$ one can substitute the modified Bessel function by its
asymptote, so the solution (\ref{solution})\ reduces to the following:
\begin{eqnarray}
\theta \left( z,t\right) &=&\frac{1}{2\sqrt{\pi }}\int\limits_{-\infty
}^{t}dt_{1}\left( \theta _{0}\left( t\right) \Gamma _{1}\left( t_{1}\right) +%
\overset{.}{\theta }_{0}\left( t\right) \right) \times \label{gauss} \\
&&\times \exp \left( -\left( \sqrt{z}-\sqrt{\alpha \left( t_{1},t\right) }%
\right) ^{2}\right) \left( z\alpha \left( t_{1},t\right) \right) ^{-1/4}.
\notag
\end{eqnarray}
Depending on optical propagation distance $z$ two simple asymptotes for (\ref%
{gauss}) can be obtained (see appendix B). The first is the case where%
\begin{equation}
\frac{\Gamma _{1m}T}{\sqrt{z}}>>4\sqrt{\ln 2} \label{polariton_cond}
\end{equation}%
$\Gamma _{1m}$ is the maximal value of $\Gamma _{1}\left( t\right) $ (see
also \cite{polariton}). Under condition (\ref{polariton_cond}), solution (%
\ref{solution}) reduces to the dark-state polariton propagation solution
with correction in (\ref{bright}):%
\begin{equation}
\theta \left( z,t\right) =\theta _{0}\left( \xi \right) +\frac{1}{\Gamma
_{1}\left( \xi \right) }\overset{.}{\theta }_{0}\left( \xi \right)
\label{polariton}
\end{equation}%
where $\xi $ is the non-linear time determined by $\int\limits_{\xi
}^{t}\Omega _{c}^{2}\left( t_{1}\right) dt_{1}=q_{p}x$ \cite{Nasha}. Note,
that turning off the coupling $\Omega _{c}\left( t\right) $ does not reduce $%
\Gamma _{1}\left( \xi \right) $\ to zero, since $\xi $ retards from $t$. In
this case, as it will be shown below, the information stored in the medium
can be well retrieved (see sec. IV).
In the case of condition reversed to (\ref{polariton_cond}),%
\begin{equation}
\frac{\Gamma _{1m}T}{\sqrt{z}}<<4\sqrt{\ln 2}. \label{bluring_cond}
\end{equation}%
the\emph{\ }solution (\ref{solution}) reduces to
\begin{equation}
\theta \left( z,t\right) =R\exp \left( -\left( \sqrt{z}-\sqrt{\alpha \left(
t_{0},t\right) }\right) ^{2}\right) \left( z\alpha \left( t_{0},t\right)
\right) ^{-1/4} \label{bluring}
\end{equation}%
where $t_{0}$ is the maximal value of $\theta _{0}\left( t\right) $, $%
R=\int\limits_{-\infty }^{\infty }\Gamma _{1}\left( t^{\prime }\right)
\theta _{0}\left( t^{\prime }\right) dt^{\prime }$ and does not depend on
time. We emphasize that for propagation distances meeting the condition (\ref%
{bluring_cond}), the obtained pulse loses all the information about its
initial temporal shape, since the right hand side in (\ref{bluring}) does
not contain time dependent $\theta _{0}$.
\section{\label{calc}Discussion}
In this section the probe pulse propagation dynamics obtained from the
analytical solution (\ref{solution}) is presented. First of all we consider
the case of constant coupling field. Shape of the initial pulse is chosen to
be double-humped in order to visualize the propagation dynamics. For the
small propagation distances when the condition (\ref{polariton_cond})\ is
met influence of $\Gamma $ is negligible (Fig.2a). When the condition (\ref%
{polariton_cond}) is violated, the influence of upper level width becomes
essential as one can see from Figs. 2b,c. Thus influence of $\Gamma $ breaks
the adiabaton propagation regime.
As it was mentioned above, propagation over very long distances (\ref%
{bluring_cond}) leads to the lost of the information on the initial pulse
temporal shape. This can be seen in Fig 3, where propagation over the same
distance of two pulses with different temporal shapes but with the same
initial area is depicted. By propagating over very long distance (\ref%
{bluring_cond}) they lose any information about their initial temporal
shapes.
In Fig. 4 we show that the increase of $\Gamma _{1}$ suppresses the smearing
of the probe. This is caused by the decrease of the bright state population
and hence leads to the decrease the influence of $\Gamma $ on pulse
propagation. Note, that in the dark-state propagation regime the pulse
temporal shape does not depend on coupling field amplitude or on unstable
level width.
Summarizing presented results one can see that to minimize the pulse
smearing during its propagation one has to either increase $\Gamma _{1}$ or
decrease the propagation distance $z$. The situation changes dramatically
for the light storage and retrieving process ($\Omega _{c}\neq const$).
It is known, that pulse can be completely stored and retrieved from the
medium if the medium length and $\Gamma _{1}$ meet the condition (see for
example \cite{Nasha}):%
\begin{equation}
z\gtrsim \Gamma _{1m}T. \label{fitting}
\end{equation}
For efficient storage and retrieving the condition (\ref{polariton_cond})
also has to be met. Combining this two nonequalities one gets that to
completely store and well retrieve the light pulse, $\Gamma _{1}$ has to
meet the following condition:$\ $%
\begin{equation}
\Gamma _{1m}T>>16\ln 2>>1. \label{good_storage}
\end{equation}
Therefore, influence of the second term in (\ref{polariton}) is insufficient
when the condition (\ref{good_storage}) is met.
Storage and retrieving of the light pulse for different propagation
distances under the condition (\ref{good_storage}) is depicted in Fig 5. For
small propagation distances when the condition (\ref{fitting}) is violated
only the falling edge of the pulse is stored and can be retrieved (Fig. 5a),
because when this edge enters the medium, the leading edge emerges already.
For larger propagation distances when the condition (\ref{fitting}) is
satisfied the whole pulse can be stored and then well retrieved by turning
on the coupling field.
Let us now consider the case when the condition (\ref{good_storage}) is not
met (Fig. 6). For small propagation distances when the condition (\ref%
{polariton_cond}) is satisfied but (\ref{fitting})\ is not, only the falling
edge of the pulse can be stored and retrieved. Propagation over longer
distances brings to satisfying of (\ref{fitting})\ and violation of (\ref%
{polariton_cond}). Thus the whole pulse can be stored but the retrieved
pulse temporal shape is smeared.
We present finally comparison of the experimental results with our
analytical solution. In Fig 7a the experimental data of storage and
retrieving of light pulse from \cite{experiment} are presented. Curve in Fig
7b is plotted from our analytical solution (\ref{solution}): all parameters
correspond to the conditions of the experiment. One can see good consistency
between experimental data and our analytical solution (Note that the storage
in case of experiment () is incomplete as was discussed above).
\section{\label{sec_V}Consideration of $\protect\rho _{21}$ transverse decay}
In this sections we take into acount the quantity $\gamma T$ in first order.
This leads, instead of (\ref{bloch}), to the equations.
\begin{eqnarray*}
\overset{.}{\rho }_{31} &=&-\Gamma \rho _{31}+i\Omega _{p}+i\Omega _{c}\rho
_{21}, \\
\overset{.}{\rho }_{21} &=&-\gamma \rho _{21}+i\Omega _{c}^{\ast }\rho _{31},
\\
\rho _{11} &=&1, \\
\rho _{22} &=&\rho _{33}=\rho _{32}=0,
\end{eqnarray*}
Thus, probe pulse propagation equation is written as follows:%
\begin{equation}
\frac{1}{\Gamma _{1}}\frac{\partial ^{2}\theta }{\partial x\partial t}+\frac{%
\partial \theta }{\partial x}+\frac{q_{p}}{\Gamma _{1}\left( \Gamma +\gamma
\right) }\frac{\partial \theta }{\partial t}+\frac{q_{p}\gamma }{\Gamma
_{1}\left( \Gamma +\gamma \right) }\theta =0, \label{V_2}
\end{equation}
where $\Gamma _{1}\left( t\right) $ now is%
\begin{equation}
\Gamma _{1}\left( t\right) =\frac{\Omega _{c}^{2}+\gamma \left( \Gamma +%
\dfrac{\dot{\Omega}_{c}}{\Omega _{c}}\right) }{\Gamma +\gamma }. \label{V_3}
\end{equation}
As results from (\ref{V_3}), to completely stop the light in the medium $%
\left( \Gamma _{1}=0\right) $ one should turn off the coupling field $\Omega
_{c}$ at the rate $\Gamma $ (i.e., $\Gamma +\dfrac{\dot{\Omega}_{c}}{\Omega
_{c}}=0$).
By performing the stated above Laplace transform procedure one obtains
analytical solution of the equation (\ref{V_2}) in the form%
\begin{eqnarray}
\theta \left( z,t\right) &=&\int\limits_{-\infty }^{t}dt_{1}\left( \theta
_{0}+\Gamma _{1}\dot{\theta}_{0}\right) \times \label{V_4} \\
&&\times \exp \left( -z-\int\limits_{t_{1}}^{t}\Gamma _{1}dt^{\prime
}\right) I_{0}\left( 2\sqrt{z\alpha \left( t_{1},t\right) }\right) \notag
\end{eqnarray}%
where $\alpha \left( t_{1},t\right) =\int\limits_{t_{1}}^{t}\Gamma
_{1}\left( t^{\prime }\right) -\gamma dt^{\prime }$ and $z=\dfrac{q_{p}x}{%
\Gamma +\gamma }$.
More detailed analysis of the expression (\ref{V_2}) will be performed in a
subsequent publication.
\section{Conclusion}
We have considered the propagation, storage and retrieving of the light
pulse in EIT media by taking into account all dephasing rates. From coupled
system of Maxwell and density matrix equations we derive the probe pulse
propagation equation, which in particular case passes into the dark-state
polariton propagation equation. We find an analytical solution and analyzed
its physical consequences. We derived a simple asymptotes of the solution,
and showed strong dependence of light pulse temporal shape on optical
propagation distance in the presence of relaxations. We demonstrated that an
efficient storage of light is possible by choosing appropriate coupling
intensities and optical propagation distances. Finally, we compared our
solution with experimental data and showed that our solution is well
consistent with the recent experiments.
\begin{acknowledgments}
We are grateful to Prof. M.Fleischhauer, Prof. V.Chaltykyan and Prof.
Yu.Malakyan for helpful discussions. The work was supported by the ISTC
Grant No. \#A-1095.
\end{acknowledgments}
\bigskip
|
{
"timestamp": "2005-04-01T23:34:38",
"yymm": "0503",
"arxiv_id": "quant-ph/0503209",
"language": "en",
"url": "https://arxiv.org/abs/quant-ph/0503209"
}
|
\section{introduction}
\vskip -0.15in
Since the early realization of sub-micron atom lithography
\cite{timp}, the subject of focusing neutral atoms by use of light
fields continues to attract a great deal of attention. The basic
principle of atom lithography relies on the possibility of
concentrating the atomic flux in space utilizing a spatially
modulated atom-light interaction. In the conventional
atom-lithographic schemes, a standing wave (SW) of light is used
as a mask on atoms to concentrate the atomic flux periodically and
create desired patterns at the nanometer scale \cite{review}. The
technique has been applied to many atomic-species in one
\cite{{prentis},{mcc1},{sodium},{mcc2},{chr},{alu},{ces},{ytt},{iro}}
as well as two-dimensional \cite{twod} pattern formations. There are
two ways to focus a parallel beam of atoms by light masks in close
correspondence with conventional optics. In the thin-lens
approach, atoms are focused outside the region of light field
which happens for low intensity light beams. On the other hand,
the atoms can be focused within the light beam when its intensity
is high. This is known as thick-lens regime and is very similar to
the graded-index lens of traditional optics. The laser focusing of
atoms depends on parameters such as thickness of light beam, velocity
spread of atoms, detuning of laser frequency from the atomic transition
frequencies, etc,. Experimentally, atomic nanostructures have been reported
with sodium \cite{{timp},{sodium}}, chromium \cite{{mcc2},{chr}},
aluminium \cite{alu}, cesium \cite{ces},
ytterbium \cite{ytt}, and iron \cite{iro} atoms.
Most of the theoretical studies on atom lithography employ a particle optics
approach to laser focusing of atoms \cite{{prentis},{mcc1},{ashkin}}.
The classical trajectories of atoms in the potentials induced by light fields suffice
to study the focal properties of light lens. In the case of direct
laser-guided atom deposition, the diffraction resolution limit will be ultimately
determined by the de Broglie wavelength of atoms, and may reach several picometers
for typical atomic beams \cite{lee}. In practice, however, this limit has never
been relevant because of the surface diffusion process, the quality of the atomic
beam, and severe aberrations due to anharmonicity of the sinusoidal dipole
potential. As a result, all current atom lithography schemes suffer from a considerable
background in the deposited structures. A possible way to overcome the aberration
problem was suggested in \cite{sch}, by using nanofabricated mechanical masks
that block atoms passing far from the minima of the dipole potential. However, this
complicates considerably the setup and reduces the deposition rate. Therefore, there
is a considerable need in a pure atom optics solution for the enhanced focusing of an
atomic beam having a significant angular spread.
In paraxial approximation, the steady-state propagation of an
atomic beam through a standing light wave is closely connected to
the problem of the time-dependent lateral motion of atoms subject
to a spatially periodic potential of an optical lattice. From this
point of view, enhanced focusing of the atomic beam can be
considered as a squeezing process on atoms in the optical lattice.
In recent work \cite{mleib}, novel squeezing technique has been
introduced for atoms in a pulsed optical lattice. The approach
considered a time modulation of the SW with a series of short
laser pulses. Based on specially designed aperiodic sequence of
pulses, it has been shown that atoms can be squeezed to the minima
of the light-induced potential with reduced background level.
Oskay {\it et al.} \cite{raizen} have verified this proposal
experimentally using Cs atoms in an optical lattice. In Refs.
\cite{{mleib},{raizen}}, the atoms were loaded into
the optical lattice and the dynamics of atoms along the
direction of SW was studied as a time-dependent problem. The aim
of the present work is to extend the focusing scenario of Ref.
\cite{mleib} to the beam configuration employed for atomic
nanofabrication. We generalize the results on atomic squeezing in
the pulsed SW to a system involving the atomic-beam traversing
several layers of light masks. In particular, we will investigate
prospects for reducing spherical and chromatic aberrations in atom
focusing with double-layer light masks. High-resolution deposition
of chromium atoms will be considered as an example.
The plan of the paper is as follows. In Sec. II, the basic
framework of the problem is defined and the linear focusing of
atoms by a double-layer light mask is studied using the particle
optics approach in paraxial approximation. In Sec. III, we examine
the optimal squeezing scheme of \cite{mleib} in application to the
atomic-beam traversing two layers of light masks. The effects of
beam collimation and chromatic aberrations are considered in Sec.
IV. Here, we optimize the double lens performance and give
parameters for the minimum spot-size in the atom deposition.
Finally, in Sec. V, we summarize our main results.
\section{squeezing of atoms by multi-layer light masks - classical treatment}
\vskip -0.15in
The focusing property of a single SW light has been studied in
great details by McClelland {\it et al.} \cite{{mcc1},{mcc2}}. The
light acts like an array of cylindrical lenses for the incident
atomic beam, focusing the atoms into a grating on the substrate.
However, because of the non-parabolic nature of the light-induced
potential, the focusing of atoms is subject to spherical
aberrations giving a finite width to the deposited features
\cite{mcc1}. A doublet of light masks made from two standing light
waves may, in principle, reduce the focusing imperfections due to
a clear physical mechanism. In this configuration, the first SW
prefocuses the atoms towards the minima of the sinusoidal
potential. When the pre-focused atoms cross the second SW, they
see closely the parabolic part of the potential which should
result in a reduction of the over-all spherical aberrations.
To test this scheme, we consider the propagation of an atomic-beam
through a combination of two SWs formed by counter-propagating
laser beams. The two SWs are identical except for their
intensities and are assumed to be formed along the x-direction.
Atoms are described as two-level systems with transition frequency
$\omega_o$. We take the direction of propagation of atoms through
the SW fields along the z-direction. If the atoms move
sufficiently slow (adiabatic conditions) through the light fields,
the internal variables of atoms maintain a steady state during
propagation \cite{cohen}. In this approximation, the atoms can be
described as point-like particles moving under the influence of an
average dipole-force. The potential energy of interaction is given
by \cite{{ashkin},{conserve}}
\begin{equation}
U(x,z) = \frac{\hbar \Delta}{2}~\hbox{ln}[1 + p(x,z)]~,
\label{poten}
\end{equation}
where
\begin{equation}
p(x,z) = \frac{\gamma^2}{\gamma^2 + 4 \Delta^2}~\frac{I(x,z)}{I_s}~.
\label{pxz}
\end{equation}
In Eq. (\ref{pxz}), $\Delta$ is the detuning of the laser frequency from
the atomic resonance, $I(x,z)$ is the light intensity, $\gamma$ is
\vskip -0.2 in
\begin{figure}[t]
\epsfxsize=220pt
\centerline{
\epsfbox{nfig1.eps}
}
\end{figure}
\vskip -0.1in
\noindent
FIG. 1. Schematic representation of the laser focusing of atoms by a
double layer of Gaussian standing waves. The intensity profile shows
the Gaussian envelopes along the z-axis and the sinusoidal variations along
the x-axis.
\vskip 0.2in
\noindent
the spontaneous decay rate of excited level, and $I_s$ is the saturation
intensity associated with the atomic transition. For the arrangement of
two SW light masks (denoted by 1 and 2) with separation $S$ between them,
the net intensity profile of light is given by
\begin{eqnarray}
I(x,z) &=& \left[I_1 \exp(-2 z^2/\sigma_z^2) + I_2 \exp(-2 {(z - S)}^{2}/\sigma_z^2)
\right] \nonumber \\
&&~~~~~ \times \sin^2(k x)~.
\end{eqnarray}
Here, $\sigma_z$ is the $1/e^2$ radius and $\lambda = 2 \pi/k$ is
the wavelength of laser beams forming the SWs. We consider
Gaussian intensity profiles and ignore any y-dependence of laser
intensities as the force on atoms along the y-direction is
negligible compared to that along the direction of SW (x-axis).
$I_1$ and $I_2$ denote the maximum intensity of the standing light
waves 1 and 2, respectively. We neglect the overlap and
interference between two SWs. The intensity profile of light and
the focusing of atoms by light fields are shown schematically in
Fig. 1.
The classical trajectories of atoms in the potential
($\ref{poten}$) induced by the double-layer light masks obey the
Newton's equations of motions~:
\begin{equation}
\frac{d^2 x}{d t^2} + \frac{1}{m} \frac{\partial U(x,z)}{\partial x}
= 0~,~~~~
\frac{d^2 z}{d t^2} + \frac{1}{m} \frac{\partial U(x,z)}{\partial z} = 0~.
\end{equation}
Using the conservation of energy, we can combine the above two equations
and solve for $x$ as a function of $z$. This results in two first-order
coupled differential equations for $x(z)$, $\alpha \equiv dx(z)/dz$~~:
\begin{eqnarray}
\frac{dx(z)}{dz} &=& \alpha~~, \label{newton} \\
\frac{d\alpha(z)}{dz} &=& \frac{1 + \alpha^2}{2 (E - U)} \left(\alpha
\frac{dU}{dz} - (1 + \alpha^2) \frac{dU}{dx} \right)~~. \nonumber
\end{eqnarray}
Here, $E$ represents the total energy of the incoming atoms (the
kinetic energy in the field-free region) and $\alpha$ gives the
slope of the trajectory $x(z)$.
\vskip -0.2 in
\begin{figure}[t]
\centerline{ \epsfxsize=220 pt
\epsfbox{nfig2ab.eps}}
\centerline{ \epsfxsize=220 pt
\epsfbox{nfig2c.eps}}
\end{figure}
\vskip -0.2in
\noindent
FIG. 2. Numerically calculated trajectories of atoms for laser focusing by
a single- (a) and double-layer (b) light masks. The parameters used are
$I_1/I_s = 1000$, $I_2/I_s = 0$ (a) and $I_1/I_s = 1000$, $I_2 =
I_1$, $S = 500$ (b). All other parameters are the same as in Table
I. The solid (dashed) curve in graph (c) shows the probability
density of atoms at the focal point $z = z_f \approx 650~(700)$ of
the double (single) light lens. The region to the right of origin
$(x = 0)$ in graph (c) is zoomed and shown in the inset.
\vskip 0.2in
We first study the focal properties of the light fields, and
solve numerically Eq. ($\ref{newton}$) for an atomic beam that is
initially parallel to the z-axis. The linear focal points and
principal-plane locations can be obtained by tracing paraxial
trajectories as discussed in \cite{mcc1}. Some typical results are
shown in Fig. 2, where we present the numerical calculation of a
series of atomic trajectories entering the nodal region of both
single $(I_1 \neq 0,I_2 \equiv 0)$ and double $(I_1,I_2 \neq 0)$
light masks. Table I lists the parameters used in dimensionless
units, in which length is expressed in units of $\lambda$, and
frequency is in units $\omega_{r} \equiv \hbar k^2 /2 m $
corresponding to the recoil energy. We have considered the
intensities of light SWs to be equal in the case of double
light masks. For the other variables, the values close to the
experimental parameters of the chromium atom-deposition \cite{dirk}
are taken as an example, though the general conclusions to be drawn
should apply to other atoms. It is seen from Fig. 2, that a sharp
focal spot appears in the flux of focused atoms \cite{pflux}.
Despite the small size of the focal spot, the overall localization of
atoms in the focal plane is not very marked. Atomic background in the
focal plane indeed gets reduced with double light masks as shown in the
inset of Fig. 2(c), however this effect is not very pronounced. To
take full advantage from double-mask arrangement, we have to replace the
concept of linear focusing (useful for paraxial trajectories only)
by the notion of optimized nonlinear spatial squeezing.
\section{optimal squeezing theory - \\application to atom nanolithography}
\vskip -0.1in
We have seen that the double light lens leads to some improvement
in feature contrast in the focal plane in comparison to the single
light lens. However, even for a single SW, the best squeezing of
atoms (maximal spatial compression) is achieved not at the focal
plane, but after the linear focusing phenomenon takes place. To
characterize the spatial localization of atoms we use a convenient
figure of merit, the localization factor \cite{mleib}:
\begin{eqnarray}
L(z) &=& 1 - <\cos\left(2 k x(z,x_o)\right)> \nonumber \\
&\equiv& \frac{2}{\lambda} \int_{-\lambda/4}^{\lambda/4} dx_o
\left[1 - \cos\left(2 k x(z,x_o)\right) \right]~, \label{local}
\end{eqnarray}
where $x(z,x_o)$ is the solution of the differential equations
($\ref{newton}$) satisfying the initial condition $x\rightarrow
x_{0}$ at $z \rightarrow - \infty$. The average in Eq.
($\ref{local}$) is taken over the random initial positions of
atoms and the localization factor is measured as a function of
distance $z$ from the center $(z=0)$ of the first SW. The
localization factor equals zero for an ideally localized atomic
ensemble, and is proportional to the mean-square variation of the
x coordinate (modulo standing wave period) in the case of
well-localized distribution $(L << 1)$.
\vskip 0.1in
Ref. \cite{mleib} considered the squeezing process in the
time-domain by analyzing the action of pulsed SWs on atoms. In the
Raman-Nath approximation, this corresponds to the thin-lens regime
(in space domain) for interaction of a propagating atomic-beam
with multiple layers of light masks. According to the optimal
squeezing strategy \cite{mleib}, the time sequence of pulses
applied to the atomic system is determined by minimizing the
localization factor. To apply this procedure to the atom squeezing
by multi-layer light masks, we should minimize the localization
factor ($\ref{local}$) in the parameter space: the separations
between the light SWs, their intensities, and the relative
distance of substrate surface with respect to the layers of light
masks. This optimization can be done numerically
\vskip 0.1in
\begin{table}[htb]
TABLE I. Parameters in scaled units. Frequency is measured in recoil
units, and length in units of the optical wavelength. Energy is given
in the units of recoil energy $\hbar \omega_r$.
\vskip 0.03in
\begin{tabular}[t]{lc}
~~~~~~~~~~Parameter & Numerical value \\ \hline
Spontaneous emission rate $\gamma$ & 238 \\
Detuning $\Delta$ & 9500 \\
1/$e^2$ radius of SW $\sigma_z$ & 120 \\
Energy of the incoming atoms $E$ & 3$~\times 10^9$ \\
\end{tabular}
\end{table}
\begin{figure}[t]
\epsfxsize=220pt
\centerline{
\epsfbox{nfig3.eps}
}
\end{figure}
\vskip -0.2in
\noindent
FIG. 3. Localization factor of the atomic distribution for
squeezing by a single- (dashed curve) and double-layer (solid
curve) light masks. The parameters used are $I_1/I_s = 1000$,
$I_2/I_s = 0$ (dashed curve) and $I_1/I_s = 1000$, $I_2 = I_1$, $S
= S_m \approx 1000$ (solid curve). All other parameters are the
same as in Table I. The minimal value of $L(z)$ is 0.31 (0.42) and
it occurs at $z = z_m \approx 1450~(1300)$ for the solid (dashed)
curve. In the case of the double light mask, the point $(z_m,S_m)$
corresponds to the numerically found global minimum of the
localization factor.
\vskip 0.1in
\noindent
using the established simplex-search method. Our numerical analysis shows
that the localization factor exhibits multiple local minima even
for the simplest case of double light masks. In Fig. 3, we plot
the localization factor as a function of distance $z$ both for
single and double light masks around its global minimum
$(z_m,S_m)$. The intensities of SWs have been chosen to be equal
and satisfy the thin-lens condition of atom-light interaction
\cite{thin}. The graph shows that the localization factor gets a
sizable reduction with double light masks indicating for an
enhanced focusing of atoms. The minimum values of $L(z)$ in Fig. 3
are in conformity with the values obtained for optimal squeezing
of atoms with single and double pulses in the time-dependent
problem \cite{mleib}. We emphasize that the best squeezing
(localization) of atoms does not occur at the focal point. Figure
4 displays the spatial distribution of atoms at the point of best
localization. Instead of a single focal peak, a two-peaked spatial
distribution of atoms near the potential minima is observed in
Fig. 4. The origin of these peaks can be related to the formation
of rainbows in the wave optics and quantum mechanics, and it is
discussed in detail in \cite{{mleib},{rain}}. Moreover, on
comparing the inset of Figs. 2(c) and (4), it is seen that the
optimized separation between layers of the double light mask
results in a considerable reduction of atomic deposition in the
background. This also leads to an overall increased concentration
of atomic flux near the potential minima.
We note, that according to \cite{{mleib},{raizen}}, further
squeezing of atoms can be achieved by increasing the number of
identical SWs in the multi-layer light masks. For the best
localization, again the optimized values for the separations
between light masks should be used.
\vskip 0.8in
\begin{figure}[h]
\epsfxsize=220pt
\centerline{
\epsfbox{nfig4.eps}
}
\end{figure}
\vskip -0.5in
\noindent
FIG. 4. Probability density of atoms at the point of the best
squeezing by a single- (dashed curve) and double-layer (solid
curve) light masks. The parameters used are $I_1/I_s = 1000$,
$I_2/I_s = 0$, $z = z_m \approx 1300$ (dashed curve) and $I_1/I_s
= 1000$, $I_2 = I_1$, $S = S_m \approx 1000$, $z = z_m \approx
1450$ (solid curve). All other parameters are the same as in Table
I. The region to the right of origin $(x = 0)$ is zoomed and shown
in the inset.
\vskip -0.1in
In the above analysis, we have considered the case of equal
intensities for the light lenses and the problem has been studied
in the thin-lens \cite{firstthin} regime of atom focusing by light
masks. However, in many current atom-lithographic schemes, focusing of
atoms is generally achieved using an intense SW light. This corresponds
to the thick-lens regime of atom-light interaction. In this limit,
the focal point is within or close to the region of laser fields and
hence a detailed information on atomic motion within the light is
required for a full description \cite{mcc1}. For the chromium atoms
deposition, the focusing of atoms to the center of an intense SW
has been extensively studied both theoretically \cite{mcc1} and
experimentally \cite{mcc2}. We show here that a combination of a
thin and thick lenses can result in the enhanced localization of
atoms
\begin{figure}[h]
\epsfxsize=215pt
\centerline{
\epsfbox{nfig5.eps}
}
\end{figure}
\vskip -0.5in
\noindent
FIG. 5. Minimal localization factor (maximal squeezing) of
the atomic distribution as a function of the relative intensity
$I_r \equiv I_2/I_1$ of standing light waves in a double light
mask. The parameters used are same as in Table I with $I_1/I_s =
1500$.
\noindent
with minimal background structures. For illustration, we
consider the focusing of atoms by a doublet of light masks made of
a thin and a thick lens. We fix the intensity of the first SW
light mask to satisfy the thin-lens limit and study the best
localization of atoms that can be achieved by varying the
intensity of the second SW. A plot of the minimal value of the
localization factor versus the relative intensity of the second
light mask is shown in Fig. 5. The graph shows that the
localization factor becomes almost insensitive to the variation in
relative intensity after the intensity ratio reaches the value of
5, and it approaches a small value of $L=0.15$. This result is to
be compared with the value of $L=0.31$ for the optimal squeezing
by two thin lenses, and $L = 0.42$ achievable by a single thin lens.
Fig. 6 shows the corresponding trajectories of atoms and a plot of
atomic distribution at
\begin{figure}[h]
\centerline{ \epsfxsize=185 pt \epsfbox{nfig6a.eps}} \centerline{
\epsfxsize=215 pt \epsfbox{nfig6b.eps}}
\end{figure}
\vskip -0.25in
\noindent
FIG. 6. (a) Numerical trajectory calculation for laser focusing
by a double light mask. The parameters used for the calculation
are $I_1/I_s = 1500$, $I_2 = 25 I_1$, and $S = S_m \approx 1500$.
All other parameters are the same as in Table I. (b) Probability
density of the atomic distribution at the point $(z_m,S_m)$ of maximal
squeezing by the double light mask. The parameters used are the
same as those of (a) with $z = z_m \approx 1550$. The point
$(z_m,S_m)$ is the numerically found global minimum of the
function $L(z)$ with respect to the variables $(z,S)$. The dashed
curve in graph (b) shows the atomic distribution at the point $z =
z_m$ of the best squeezing by a single thick light lens with
parameters $I_1/I_s = 37500$, $I_2/I_s = 0$, $z_m \approx 90$. The
region to the right of origin $(x = 0)$ is zoomed and shown in the
inset.
\noindent
the point of the best localization.
Note that the optimized double light mask reduces the atomic background
by a factor of three in the midpoint $(x = 0.25)$ between two deposition
peaks (see the inset of Fig. 6). Moreover, the background in the
optimized double mask scheme is five times smaller compared to the
usual atom deposition in the focal plane (graph not shown) of a single
thick lens.
\section{parameters for optimal squeezing of a thermal atomic beam}
\vskip -0.11in
The effects that have been discussed so far assume an initially
collimated $(\alpha = 0)$ beam of atoms with fixed velocity (or
energy). However, in atom optics experiments involving thermal
atomic beams, the atoms possess a wide range of velocities along
the longitudinal (z-axis) and transverse (x-axis) directions. In
order to characterize the atom spatial squeezing under such
conditions, we need to average the localization factor Eq.
(\ref{local}) over the random initial velocities and angles of the
beam. The averaging can be done by using the normalized
probability density \cite{mcc1}
\begin{equation}
P(v,\alpha) = \frac{1}{2 \sqrt{2\pi}}~\frac{1}{\alpha_o v_o^5}~v^4 \exp\left[- \frac{v^2}{2 v_o^2}
\left(1 + \frac{\alpha^2}{\alpha_o^2}\right)\right]~, \label{aver}
\end{equation}
where $v_o$ is the root mean square speed of atoms with average energy $\bar{E} \equiv m v_o^2/2$.
In the above equation, the term proportional to $v^3 \exp(-v^2/2v_o^2)dv$ represents the thermal
flux probability of having a longitudinal velocity $v$ along the z-direction. The probability of
having a transverse velocity $v_x = \alpha v$ along the x-direction is proportional to the
Gaussian distribution $\exp(-v_x^2/2v_o^2 \alpha_o^2) dv_x$, where $\alpha_o$ is the FWHM of the
angular distribution. Using the probability density ($\ref{aver}$), the averaged localization
factor is thus given by
\begin{eqnarray}
L(z) &=& \frac{2}{\lambda} \int_{\alpha = -\infty}^{\alpha = \infty} \int_{v=0}^{v=\infty}
\int_{x_o = -\lambda/4}^{x_o = \lambda/4} P(v,\alpha) \nonumber \\
&&~~~~~\times \left[1 - \cos\left(2 k x(z,x_o)\right) \right] dx_o dv d\alpha \label{mainlocal}~.
\end{eqnarray}
Here, $x(z,x_o)$ represents the solution of differential equations ($\ref{newton}$) for varying
initial conditions $(x_o,v,\alpha)$ at $z \rightarrow -\infty$ of atoms. Note that the solution
of Eq. ($\ref{newton}$) depends on the initial conditions $(v,\alpha)$ through the energy term
$E \equiv m v^2 (1 + \alpha^2)/2$ as well.
Since the focal length of light masks depends on velocity of the
incoming atoms, the velocity spread in the atomic beam leads to
the broadening of the deposited feature size. In the particle
optics context of atom focusing, this is referred to as chromatic
aberration. In addition, the initial angular divergence $(\alpha
\neq 0)$ of the atomic beam degrades greatly the focusing of
atoms. We are interested in the extent to which the velocity and
angular spreads degrade the optimal squeezing of atoms. The best
feature contrast in the presence of aberrations is again defined
by minimizing the localization factor, Eq. (\ref{mainlocal}). We
have carried out the triple integration in Eq. (\ref{mainlocal})
\begin{figure}[h]
\centerline{ \epsfxsize=210 pt \epsfbox{nfig7ab.eps}} \centerline{
\epsfxsize=215 pt \epsfbox{nfig7c.eps}}
\end{figure}
\vskip -0.2in
\noindent
FIG. 7. Localization factor of the atomic distribution for squeezing
by a single- (a) and double-layer (b) light masks. The parameters used
are $\bar{E} = 3 \times 10^9$, $\alpha_o = 10^{-4}$, $I_1/I_s = 1500$,
and (a) $I_2/I_s = 0$, (b) $I_2 = I_1$, $S = S_m \approx 800$. All
other parameters used are the same as in Table I. The minimal value of
$L(z)$ is 0.67~[0.8] and it occurs at $z = z_m \approx 1350~[975]$
in the graph (b)~[(a)]. The dashed and solid curves in graph (c)
give the atomic distribution at the point $(z_m,S_m)$ of best
squeezing by the single- and double-layer light masks with the
parameters of (a) and (b). The region to the right of origin $(x =
0)$ in graph (c) is zoomed and shown in the inset.
\vskip 0.2in
\noindent
numerically and optimized the localization factor $L(z)$ in the
parameter space $(z,S)$ for the case of the double-layer light
masks. Figures 7 and 8 display atomic distribution at the point
of the best squeezing by thin-thin and thin-thick lenses
configurations. On comparing the results with those ones for a
single thin or thick lens, it is seen that the thin-thick lens
combination provides the smallest feature size for the atom
deposition. In the case of thin-thin lenses, the effects of
chromatic aberrations are greater because of the strong dependence
of focal length on the atomic velocity. We note that,
though the initial velocity and angular spread of thermal beam
worsen the optimal squeezing of atoms, the effects may become less
important with increasing the number of layers in the multi-layer
light masks. Further, chromatic aberrations can be greatly reduced
by employing low-temperature supersonic beams of highly collimated
atoms.
\section{summary}
\vskip -0.2in
In this paper, we presented the particle-optics analysis for atom
lithography using multiple layers of SW light
\begin{figure}[h]
\centerline{ \epsfxsize=210 pt \epsfbox{nfig8ab.eps}} \centerline{
\epsfxsize=215 pt \epsfbox{nfig8c.eps}}
\end{figure}
\vskip -0.24in
\noindent
FIG. 8. Localization factor of the atomic distribution for squeezing
by a single- (a) and double-layer (b) light masks. The parameters used
are $\bar{E} = 3 \times 10^9$, $\alpha_o = 10^{-4}$, and (a) $I_1/I_s
= 37500$, $I_2/I_s = 0$, (b) $I_1/I_s = 1500$, $I_2 = 25 I_1$, $S = S_m
\approx 1200$. All other parameters used are the same as in Table
I. The minimal value of $L(z)$ is 0.5~[0.66] and it occurs at $z =
z_m \approx 1350~[160]$ in the graph (b) [(a)]. The dashed and
solid curves in graph (c) give the atomic distribution at the
point $(z_m,S_m)$ of best squeezing by the single- and
double-layer light masks with the parameters of (a) and (b). The
region to the right of origin $(x = 0)$ in graph (c) is zoomed and
shown in the inset.
\vskip 0.2in
\noindent
masks. In
particular, we studied the spatial squeezing of atoms by a double
layer of standing light waves with particular reference to
minimizing the feature size of atom deposition. At first, linear
focusing of atoms using paraxial approximation was considered.
This showed an improvement in feature contrast at the focal plane,
but the effect was rather modest. We then applied the approach of
optimal squeezing that was suggested recently for the enhanced
localization of atoms in a pulsed SW \cite{mleib}. We showed that
this approach works effectively for atomic nanofabrication and can
considerably reduce the background in the atom deposition.
Based on the optimal squeezing approach, a new figure of merit,
the localization factor, was introduced to characterize the atomic
localization. Both, thin-thin and thin-thick lens regimes of atom
focusing were considered for monoenergetic as well as thermal
beams of atoms. The parameters for the smallest feature size were
found by minimizing the localization factor. We have shown that
using a proper choice of lens parameters, it is possible to narrow
considerably the atomic spatial distribution using the
double-layer light mask instead of the single-layer one. Finally,
we note that our model calculations neglect the effects of atomic
recoil due to spontaneous emission and the dipole force fluctuations.
These effects are generally beyond the scope of the classical particle
optics analysis and can be treated by means of a fully quantum approach.
A detailed quantum mechanical study of the optimal atomic squeezing in
application to nanofabrication will be published elsewhere.
\begin{center}
{\bf ACKNOWLEDGMENTS}
\end{center}
This work was supported by German - Israeli Foundation for Scientific
Research and Development.
|
{
"timestamp": "2005-03-22T13:20:06",
"yymm": "0503",
"arxiv_id": "quant-ph/0503181",
"language": "en",
"url": "https://arxiv.org/abs/quant-ph/0503181"
}
|
\section{Introduction and main result}
Dirichlet's theorem on diophantine approximation tells us that we can approximate any real number by rational numbers quite well, namely:
\begin{thm}\label{theorem1}
For any real $\theta$ and any positive integer $N$, there exist integers $a$ and $q$, with $1 \leq q \leq N$, such that
\[
\left| \theta - \frac{a}{q} \right| < \frac {1}{qN}.
\]
\end{thm}
Moreover, the bound $1/(qN)$ is best possible, apart from the constant factor. To see this, it suffices to consider the golden ratio $\theta = (\sqrt{5} - 1)/2$ (see \cite[\S 11.8]{HW}). During his work in \cite{C}, the first author accidentally stumbled across the following analogous question:
\begin{question}\label{q1}
For any real $\theta$ and any positive integer $N$, give an upper bound for
\[
\min_{ \substack{ a_1, a_2, q_1, q_2 \in \mathbb{Z}\\ 1 \leq q_1, q_2 \leq N}} \left| \frac{a_1}{q_1} + \frac{a_2}{q_2} - \theta \right|.
\]
\end{question}
With the golden ratio in mind, we know that the upper bound can be no better than $O\big( 1/(q_1 q_2 N^2) \big)$. So, what is the best possible upper bound? More generally,
\begin{question}\label{q2}
Let $k$ be a positive integer. For any real $\theta$ and any positive integer $N$, give an upper bound for
\[
\min_{ \substack{ a_1, \dots, a_k, q_1, \dots, q_k \in \mathbb{Z}\\ 1 \leq q_1, \dots, q_k \leq N}} \left| \frac{a_1}{q_1} + \dots + \frac{a_k}{q_k} - \theta \right|.
\]
\end{question}
To these, we have the following result:
\begin{thm}\label{theorem2}
Let $k$ be a positive integer. For any real $\theta$ and any positive integer $N$, there exist integers $a_1, \dots, a_k$, $q_1, \dots, q_k$, with $1 \le q_1, \dots q_k \le N$, such that
\[
\left| \frac {a_1}{q_1} + \dots + \frac {a_k}{q_k} - \theta \right| \ll N^{-k}.
\]
\end{thm}
The bound $N^{-k}$ is best possible in the sense that, for some $\theta$, the minimum in Question \ref{q2} can be as large as $N^{-k}$. For example, if one considers $\theta = 1/(2N^k)$,
\[
\left| \frac{a_1}{q_1} + \cdots + \frac{a_k}{q_k} - \theta \right| \ge \frac 1{2N^k}
\]
for any choice of $a_1, \dots, a_k, q_1, \dots, q_k$, with $1 \le q_1, \dots, q_k \le N$. However, one expects such pathological examples to be relatively rare, and so one may wonder if it is possible to obtain a sharper upper bound involving the $q_i$'s. For example, is it possible to replace $N^{-k}$ by $(q_1 \cdots q_k)^{-1}N^{-k}$ in Theorem \ref{theorem2}? We shall briefly address this issue in the last section.
\section{Proof of Theorem \ref{theorem2}}
\begin{lem}
\label{lemma1}
Suppose that $k \ge 1$ is an integer. There is a number $x_0(k) \ge 1$ such that
\[ \sideset{}{^*} \sum_{1 \le q_1, \dots, q_k \le x} q_1 \cdots q_k \gg x^{2k}, \] whenever $x \ge x_0(k)$. Here, $\sum^*$ denotes a summation over the $k$-tuples $q_1, \dots, q_k$ such that $(q_i, q_j) = 1$ whenever $1 \le i < j \le k$.
\end{lem}
\begin{proof}
It suffices to show that
\begin{equation}\label{1}
\sum_{ \substack{ n \le x\\ (n, m) = 1}} \phi(n)^{\alpha}n^{1 - \alpha} \gg x^2\phi(m)m^{-1},
\end{equation}
whenever $0 \le \alpha \le k - 1$, $1 \le m \le x^{k - 1}$, and $x \ge x_0(k)$. The conclusion of the lemma will then follow by successive applications of \eqref{1} with $\alpha = 0, 1, \dots, k - 1$ to the summations over $q_k, q_{k - 1}, \dots, q_1$.
We now proceed to establish \eqref{1}. We start by showing that
\begin{equation}\label{2}
\sum_{ \substack{ n \le x\\ (n, m) = 1}} \left( \frac n{\phi(n)} \right)^{\alpha} \ll x \phi(m)m^{-1}.
\end{equation}
Define the multiplicative functions
\[
f(n) = \begin{cases}
\big( n/\phi(n) \big)^{\alpha} & \text{if } (n, m) = 1, \\
0 & \text{if } (n, m) > 1,
\end{cases} \qquad g(n) = \sum_{d \mid n} f(d)\mu(n/d).
\]
Then $g(n) \ge 0$, and
\begin{align*}
\sum_{n \le x} f(n) &= \sum_{n \le x} \sum_{d \mid n} g(d) = \sum_{d \le x} g(d) \left\lfloor \frac xd \right\rfloor \le x \sum_{d \le x} g(d)d^{-1}\\
&\le x \prod_{p \le x} \sum_{\nu = 0}^{\infty} g(p^{\nu})p^{-\nu} = x \prod_{p \le x} \left( 1 - p^{-1} \right) \sum_{\nu = 0}^{\infty} f(p^{\nu})p^{-\nu} \\
&\le x \prod_{ \substack{ p \mid m\\ p \le x}} \left( 1 - p^{-1} \right) \prod_p \left( 1 + \frac {p^{\alpha} - (p - 1)^{\alpha}}{p(p - 1)^{\alpha}} \right) \\
&\le x \prod_{p \mid m} \left( 1 - p^{-1} \right) \prod_p \left( 1 + \frac {p^{\alpha} - (p - 1)^{\alpha}}{p(p - 1)^{\alpha}} \right) + O(1).
\end{align*}
The last inequality follows on noting that $m$ has at most $k - 2$ prime divisors $p > x$, and hence,
\[
\prod_{ \substack{ p \mid m\\ p > x}} \left( 1 - p^{-1} \right) = 1 + O \big( x^{-1} \big).
\]
This proves \eqref{2}. On the other hand, when $\alpha = 0$, we have
\[
\sum_{ \substack{ n \le x\\ (n, m) = 1}} n = \sum_{d \mid m} \mu(d) d \sum_{k \le x/d} k = \frac {\phi(m)}{2m} x^2 + O(x\tau(m)),
\]
whence
\begin{equation}\label{3}
\sum_{ \substack{ n \le x\\ (n, m) = 1}} n^{1/2} \ge x^{-1/2} \sum_{ \substack{ n \le x\\ (n, m) = 1}} n \gg x^{3/2}\phi(m)m^{-1}.
\end{equation}
Finally, \eqref{1} follows from \eqref{2}, \eqref{3}, and Cauchy's inequality:
\[
\sum_{ \substack{ n \le x\\ (n, m) = 1}} \phi(n)^{\alpha}n^{1 - \alpha} \ge \bigg\{ \sum_{ \substack{ n \le x\\ (n, m) = 1}} n^{1/2} \bigg\}^2 \bigg\{ \sum_{ \substack{ n \le x\\ (n, m) = 1}} \left( \frac n{\phi(n)} \right)^{\alpha} \bigg\}^{-1} \gg x^2\phi(m)m^{-1}.
\]
\end{proof}
\begin{proof}[Proof of Theorem \ref{theorem2}]
For $0 < \Delta < 1/2$, define
\[
t(x) = \max \big( 1 - |x|/\Delta, 0 \big), \qquad g(x) = \sum_{n = -\infty}^{\infty} t(x - n).
\]
The function $g$ has a Fourier expansion
\[
g(x) = \sum_{h = -\infty}^{\infty} \hat g_h e(h x), \qquad \hat g_h = \Delta \left( \frac{\sin \pi \Delta h}{\pi \Delta h} \right)^2.
\]
We consider the sum
\begin{equation}\label{4}
\mathcal S = \sideset{}{^*} \sum_{1 \le q_1, \dots, q_k \le N} \sum_{a_1 = 1}^{q_1} \dots \sum_{a_k = 1}^{q_k} g \left( \frac{a_1}{q_1} + \dots + \frac{a_k}{q_k} - \theta \right),
\end{equation}
where $\sum^*$ has the same meaning as in the Lemma. Putting in the Fourier expansion for $g$, we get
\begin{align}\label{5}
\mathcal S =& \sideset{}{^*} \sum_{1 \le q_1, \dots, q_k \le N} \sum_{a_1 = 1}^{q_1} \dots \sum_{a_k = 1}^{q_k} \sum_{h = -\infty}^{\infty} \hat g_h e \left( h \left ( \frac{a_1}{q_1} + \dots + \frac {a_k}{q_k} - \theta \right) \right) \\
=& \sideset{}{^*} \sum_{1 \le q_1, \dots, q_k \le N} \sum_{h = -\infty}^{\infty} \hat g_h e(-h \theta) \sum_{a_1 = 1}^{q_1} e \big( ha_1/ q_1 \big) \dots \sum_{a_k = 1}^{q_k} e \big( ha_k/ q_k \big) \notag\\
=& \sideset{}{^*} \sum_{1 \le q_1, \dots, q_k \le N} q_1 \cdots q_k \sum_{ \substack{ h = -\infty\\ q_1 \cdots q_k \mid h}}^{\infty} \hat g_h e(-h \theta) \notag,
\end{align}
as
\[
\sum_{a = 1}^{q} e(ha/q) = \begin{cases}
q & \text{if } q \mid h, \\
0 & \text{otherwise}.
\end{cases}
\]
If $m$ is a positive integer and $\Delta \le m^{-1}$, we have
\begin{equation}\label{6}
\begin{split}
\sum_{h \ne 0} \big| \hat g_{mh} \big| &\le 2 \bigg\{ \sum_{h = 1}^H \Delta + \frac 1{\Delta m^2} \sum_{h = H + 1}^{\infty} h^{-2} \bigg\} \\
&\le 2 \bigg( H\Delta + \frac 1{H\Delta m^2} \bigg) \le 6m^{-1},
\end{split}
\end{equation}
where $H = \big\lceil (\Delta m)^{-1} \big\rceil$; whereas if $\Delta > m^{-1}$, we have
\begin{equation}\label{7}
\sum_{h \ne 0} \big| \hat g_{mh} \big| \le \frac {2\zeta(2)}{\Delta m^2} \le 4m^{-1}.
\end{equation}
Putting \eqref{6} and \eqref{7} (with $m = q_1 \cdots q_k$) into \eqref{5}, we obtain
\[
\mathcal S = \Delta \sideset{}{^*} \sum_{1 \le q_1, \dots, q_k \le N} q_1 \cdots q_k + O \big( N^k \big),
\]
the $O$-implied constant being absolute (in fact, it is $6$). Therefore, upon choosing $\Delta = cN^{-k}$ with a sufficiently large $c > 0$, it follows from the Lemma that $\mathcal S > 0$. Hence, by \eqref{4},
\[
g\left( \frac {a_1}{q_1} + \dots + \frac {a_k}{q_k} - \theta \right) > 0
\]
for some integers $a_1, \dots, a_k, q_1, \dots, q_k$ with $1 \le q_1, \dots, q_k \le N$. Then, by the definition of $g$,
\[
\left| \frac {a_1}{q_1} + \dots + \frac {a_k}{q_k} - n - \theta \right| \le cN^{-k}
\]
for some integer $n$. This establishes the theorem.
\end{proof}
\section{Closing remarks}
We conclude this note with a short discussion of possible improvement on the bound $N^{-k}$ in Theorem \ref{theorem2}. For example, is it possible to replace $N^{-k}$ by $(q_1 \cdots q_k)^{-1} N^{-k}$? While such a result may appear to be the right generalization of Dirichlet's theorem, it is not true in general. Indeed, suppose that for any real $\theta$, there exist integers $a_1, \dots, a_k, q_1, \dots, q_k$, with $1 \le q_1, \dots, q_k \le N$, such that
\begin{equation}\label{8}
\left| \frac{a_1}{q_1} + \cdots + \frac{a_k}{q_k} - \theta \right| \le \frac C{q_1 \cdots q_kN^k}.
\end{equation}
Then
\begin{equation}\label{9}
[0, 1] \subseteq \bigcup_{q \in \mathcal D_k(N)} \bigcup_{0 \le a \le q} \left\{ \theta \in \mathbb R \; : \; |\theta - a/q| \le C/(qN^k) \right\},
\end{equation}
where $\mathcal D_k(N)$ denotes the set of least common denominators of the sums appearing on the left side of \eqref{8}. By a result of Erd\"os \cite{E}, $\mathcal D_k(N)$ has cardinality
\[
|\mathcal D_k(N)| \ll N^k(\log N)^{-c}
\]
for some constant $c = c(k) > 0$, so it follows from \eqref{9} that
\[
1 \le \sum_{q \in \mathcal D_k(N)} \sum_{0 \le a \le q} \frac {2C}{qN^k} \le 4CN^{-k}|\mathcal D_k(N)| \ll (\log N)^{-c},
\]
which is impossible when $N \to \infty$. On the other hand, one may hypothesize that the set of fractions with denominators in $\mathcal D_k(N)$ is distributed similarly to the set of all fractions $a/q$ with denominators $q \le N^k$. Under such a hypothesis, one might hope for an estimate with $|\mathcal D_k(N)|^{-1}$ in place of the term $(\log 3N)^cN^{-k}$ on the right side of \eqref{10} below, and such an estimate, if true, would be essentially best possible. However, upon observing that
\[
|\mathcal D_k(N)| \ge \sum_{q_1 \le N} \cdots \sum_{q_k \le N} d(q_1 \cdots q_k)^{-1} \ge \bigg\{ \sum_{q \le N} d(q)^{-1} \bigg\}^{k} \gg N^k(\log 3N)^{-k},
\]
we will take a more cautious approach and pose the following
\begin{question}
Let $k$ be a positive integer. Determine the least value of $c_k$ such that for any real $\theta$ and any positive integer $N$, there exist integers $a_1, \dots, a_k$, $q_1, \dots, q_k$, with $1 \le q_1, \dots, q_k \le N$, such that
\begin{equation}\label{10}
\left| \frac {a_1}{q_1} + \dots + \frac {a_k}{q_k} - \theta \right| \ll \frac {(\log 3N)^{c_k}}{q_1 \cdots q_k N^k}.
\end{equation}
\end{question}
We leave the answer to this question to the future.
\bigskip
\begin{acknowledgement}
The first author would like to thank the American Institute of Mathematics for support. The second author would like to thank Jeff Vaaler for several enlightening conversations on this and related topics.
\end{acknowledgement}
|
{
"timestamp": "2005-04-12T18:25:08",
"yymm": "0503",
"arxiv_id": "math/0503440",
"language": "en",
"url": "https://arxiv.org/abs/math/0503440"
}
|
\section{Introduction}
Let $G$ be a connected reductive group over $\ensuremath{{\mathbb R}}$, and $T$ a
maximal torus in $G$. Assume that $G$ has a discrete series of
representations. Let $A$
be the split part of $T$, and $M$ the centralizer of $A$ in $G$. It is a Levi
subgroup of $G$
containing $T$. Let $E$ be a finite-dimensional representation of
$G(\ensuremath{{\mathbb C}})$,
and consider the packet $\Pi_E$ of discrete series representations $\pi$ of
$G(\ensuremath{{\mathbb R}})$ which have the same infinitesimal and central characters as $E$.
Write $\Theta_{\pi}$ for the character of $\pi$, and put
\[ \Theta^E=(-1)^{q(G)} \sum_{\pi \in \Pi_E} \Theta_{\pi}. \]
\noindent Here $q(G)$ is half the dimension of the symmetric space
associated with $G$.
Note that $\Theta^E(\gamma)$ will not extend to all elements $\gamma \in T(\ensuremath{{\mathbb R}})$,
in particular to $\gamma=1$.
Define the number $D_M^G(\gamma)$ by
\[ D_M^G(\gamma) = \det(1-\Adj(\gamma); \Lie(G)/ \Lie(M)). \]
Then a result of Arthur and Shelstad [1] states that the
function
\[ \gamma \mapsto |D_M^G(\gamma)|^{\frac{1}{2}} \Theta^E(\gamma), \]
\noindent defined on the set of regular elements $T_{\reg}(\ensuremath{{\mathbb R}})$
extends continuously to $T(\ensuremath{{\mathbb R}})$. We denote this extension by
$\Phi_M(\gamma,\Theta^E)$. These quantities give the contribution
from the real place to the $L^2$-Lefschetz numbers of Hecke operators in
[1] and [2].
An expression for $\Phi_M(\gamma,\Theta^E)$ as essentially a sum over elements
in the Weyl group $W$ of $T$ in $G$ appears in the proof of Lemma 4.1 in
[2]. Although this expression suffices to prove the lemma, it can be
considerably refined when $\gamma$ is in the maximal elliptic subtorus
$T_e(\ensuremath{{\mathbb R}})$ of $T(\ensuremath{{\mathbb R}})$.
\bigskip
The following theorem is proved in section 4.
\bigskip
{\bf Theorem 1.} {\it If $\gamma \in T_e(\ensuremath{{\mathbb R}})$, then \it}
\[ \Phi_M(\gamma,\Theta^E)=(-1)^{q(L)}\cdot |W_L| \cdot
\sum_{\omega \in W^{LM}}
\varepsilon(\omega) \cdot
\tr(\gamma;V^M_{\omega(\lambda_B+\rho_B)-\rho_B}). \]
Here we write $L$ for the centralizer of $T_c$ in $G$, where $T_c$
is the maximal compact subtorus of $T$. Also write $W_L$
and $W_M$ for the Weyl groups of $T$ in $L$ and $M$. The latter
are subgroups of $W$ which commute and have trivial intersection.
Here $W^{LM}$ is a certain set of representatives for the cosets
$(W_L \times W_M) \backslash W$. It is defined explicitly in section 5.
We write $\varepsilon$
for the sign character of $W$. Finally by
$V^M_{\omega(\lambda_B+\rho_B)-\rho_B}$ we
denote the irreducible finite-dimensional representation of $M$,
with highest weight $\omega(\lambda_B+\rho_B)-\rho_B$, where $\lambda_B$
is the $B$-dominant highest weight of $E$.
\bigskip
In particular, we obtain the extremely simple expression,
\[ \Phi_A(1,\Theta^E)=(-1)^{q(G)} \cdot |W|, \]
\noindent in the case of a split torus $T=A$.
\bigskip
We now describe the organization of this paper.
\bigskip
In section 2, we spell out the relationship between the root
systems of $G$, $L$, and $M$. There are two distinct systems of chambers in
$X_*(A) \otimes_{\ensuremath{{\mathbb Z}}} \ensuremath{{\mathbb R}}$ obtained from these root systems which are important to
understand.
\bigskip
In section 3, we take the aforementioned lemma a step further to
express $\Phi_M(\gamma,\Theta^E)$ explicitly as a linear combination of
characters. (Actually we do the computation for any stable virtual
character $\Theta$, as it is no more difficult.) The sum over $W$ simplifies
to a sum over Kostant representatives $W^M$.
\bigskip
In section 4, in which we deal specifically with
$\Phi_M(\gamma,\Theta^E)$, we
distill out the action of $W_L$. A sum over $W^{LM}$ remains. At a key step we use
a result of section 5, the computation of an alternating sum of stable
discrete
series constants.
\bigskip
In section 5, we prove the result mentioned above, in the
context of abstract root systems. It is independent from the rest of the paper.
\bigskip
I am indebted to my advisor Robert Kottwitz for suggesting the problem and
many useful comments. I also thank Christian Kaiser for some helpful conversations.
This project was carried out during a stay at the Max-Planck-Institut
f\"{u}r Mathematik in Bonn, and I am grateful to the Institut for its
support and hospitality.
\section{$L$-chambers and $\ensuremath{{\mathcal P}}$-chambers}
Let $G$ be a connected reductive group over $\ensuremath{{\mathbb R}}$, and $T$ a
maximal torus of $G$.
Assume that $G$ has a discrete series, or equivalently, that $G$ has an
elliptic maximal torus.
\bigskip
Write $T_c$, respectively $A$, for the maximal compact,
resp. split,
subtori of $T$ with centralizers $L$, resp. $M$, in $G$.
Write $R$ for the root system of $T$ in $G$, and $R_L$,
resp. $R_M$, for the set of roots of $T$ in $L$, resp. $M$. Then $R_L$ is the
subset of $R$ consisting of real roots, and $R_M$ is the subset of imaginary roots.
Write $W_L$ and $W_M$ for the respective Weyl groups. They are commuting subgroups
of $W$ with trivial intersection. Note that $W_L$ fixes each root in $R_M$.
\bigskip
$A$ is contained as a split maximal torus in $L_{\der}$, the
derived group of $L$, and we
may identify $R_L$ with the set of roots of $A$ in $L_{\der}$.
\bigskip
Write $\ensuremath{{\mathfrak a}}_M$ for $X_*(A) \otimes_{\ensuremath{{\mathbb Z}}} \ensuremath{{\mathbb R}}$.
For any $\alpha \in R \backslash R_M$ the root hyperplane
$H_{\alpha}$ of
$X^*(T)_{\ensuremath{{\mathbb R}}}:=X_*(T) \otimes \ensuremath{{\mathbb R}}$ gives a hyperplane in $\ensuremath{{\mathfrak a}}_M$. Let us
consider two kinds of chambers in $\ensuremath{{\mathfrak a}}_M$ obtained from these.
Define $\ensuremath{{\mathcal P}}$-chambers to be those obtained by deleting from $\ensuremath{{\mathfrak a}}_M$ all
the hyperplanes $H_{\alpha}$, with $\alpha \in R \backslash R_M$.
Define $L$-chambers to be those obtained by deleting all the $H_{\alpha}$
with $\alpha \in R_L$. The latter are the Weyl chambers for $A$ in
$L_{\der}$; therefore $W_L$ acts simply transitively on them.
\bigskip
Observe that $R_L \subset (R \backslash R_M)$. Any additional hyperplanes
coming from roots in $R \backslash (R_L \cup R_M)$ divide the $L$-chambers
into $\ensuremath{{\mathcal P}}$-chambers. Thus every $\ensuremath{{\mathcal P}}$-chamber is contained in a unique $L$-chamber.
\bigskip
Write $\ensuremath{{\mathcal P}}(M)$ for the set of parabolic subgroups of $G$
admitting $M$ as a Levi
component. There is a one-to-one correspondence between $\ensuremath{{\mathcal P}}(M)$ and the set of
$\ensuremath{{\mathcal P}}$-chambers in $\ensuremath{{\mathfrak a}}_M$, obtained as follows: for $P=MN \in \ensuremath{{\mathcal P}}(M)$, the
corresponding $\ensuremath{{\mathcal P}}$-chamber is
\[ \ensuremath{{\mathfrak a}}_P^+= \{x \in \ensuremath{{\mathfrak a}}_M : \langle \alpha,x \rangle > 0 \text{, for all } \alpha \in
R_N \}, \]
\noindent where $R_N$ denotes the set of roots of $T$ in $\Lie(N)$.
\bigskip
Recall that the set of $L$-chambers is in bijection with the set
of Borel
subgroups of $L$ containing $T$, or equivalently the set of positive root systems
$R_L^+$ in the root system $R_L$.
\bigskip
Now let $C_P$ be a $\ensuremath{{\mathcal P}}$-chamber, and let $P=MN$ be the
corresponding element of
$\ensuremath{{\mathcal P}}(M)$. It is easy to see that $R_N \cap R_L$ is a positive system in
$R_L$,
and this corresponds to an $L$-chamber $C_L$. Thus we have defined a map $C_P
\mapsto C_L$ from the set of $\ensuremath{{\mathcal P}}$-chambers to the set of $L$-chambers. It is
the obvious one which associates to $C_P$ the unique $L$-chamber
containing $C_P$.
\section{A Linear Combination of Characters}
A stable virtual character is a finite $\ensuremath{{\mathbb Z}}$-linear combination
$\Theta$ of
characters $\Theta_{\pi}$ so that
\[ \Theta(\gamma)=\Theta(\gamma^{\prime}) \]
\noindent whenever $\gamma$ and $\gamma^{\prime}$ are regular, stably conjugate
elements of
$G(\ensuremath{{\mathbb R}})$.
In Lemma 4.1 of [2], it is proved that for a stable virtual
character $\Theta$ on
$G(\ensuremath{{\mathbb R}})$, the function
\[ \gamma \mapsto |D^G_M(\gamma)|^{\frac{1}{2}}\Theta(\gamma) \]
\noindent on $T_{\reg}(\ensuremath{{\mathbb R}})$ extends continuously to $T(\ensuremath{{\mathbb R}})$. A key
ingredient of the proof is the
fact that the expression at the bottom of page 497 is a linear combination
of irreducible finite-dimensional representations of $M$.
In this section we will compute explicitly the
coefficients and the representations involved, in the case where the
element $a$ appearing in the proof is equal to $1$.
\bigskip
We translate the set-up of the proof in [2] as follows.
We take $\Gamma$ to be the identity component of $T(\ensuremath{{\mathbb R}})$. The root
system $R_{\Gamma}$ is then simply $R_L$. Fix a
positive root system $R_L^+$ in $R_L$, and let $C$ be the corresponding
$L$-chamber in $\ensuremath{{\mathfrak a}}_M$.
We then choose a parabolic subgroup $P=MN$ so that $R_L \cap R_N \subseteq R_L^+$.
Note that $R_L \cap R_N$ is also a system of positive roots, so this condition is
equivalent to having $R_L \cap R_N=R_L^+$. Thus we simply require that
the $\ensuremath{{\mathcal P}}$-chamber corresponding to $P$ be contained in $C$.
\bigskip
Although at the end of our computations we will allow $\gamma$ to
be nonregular,
we choose now $\gamma$ to be a regular element of $\Gamma= T_c(\ensuremath{{\mathbb R}}) \cdot
\exp(\bar{C})$.
\bigskip
The expression is
\begin{equation}
\sum_B m(B) \frac{\Delta_P(\gamma) \cdot \lambda_B(\gamma)}{\Delta_B(\gamma)}.
\end{equation}
\noindent The sum ranges over Borels containing $T$, which correspond to
elements of
$W$.
\bigskip
\noindent Here $\lambda_B$ is the $B$-dominant highest weight of $E$,
\[ \Delta_B=\prod_{\alpha >0} (1-\alpha^{-1}) \text{, and }
\Delta_P=\prod_{\alpha \in R_N}(1-\alpha^{-1}). \]
Fix now a Borel $B$ of $G$ with $T \subseteq B \subseteq P$, for
the rest of
this paper.
\bigskip
Recall the set of Kostant representatives $W^M$
for the Weyl group $W_M$ of $M$, relative to B.
It is the set $\{ w \in W| w^{-1}R_M^+ \subset R^+ \}$.
\bigskip
If $w \in W$, write $w * B$ for $wBw^{-1}$.
\bigskip
We will use the observation that for $\omega \in W^M, (\omega *
B)_M=B_M$.
Indeed, if $\alpha \in R^+ \cap R_M$, then $\omega^{-1} \alpha \in R^+$, which implies that $\alpha \in
\omega R^+ \cap R_M$.
\bigskip
Our sum (1) breaks up as
\begin{equation}
\label{break}
\sum_{\omega \in W^M} m(\omega * B) \cdot \Delta_P(\gamma) \cdot \sum_{w_M \in W_M}
\frac{w_M (\omega\lambda_B)(\gamma)}{\Delta_{w_M \omega * B}(\gamma)}.
\end{equation}
We would prefer the denominator inside the sum to be $\Delta_{
w_M * B_M}(\gamma)$.
(Recall that $B_M=B \cap M$.) Note that $\Delta_P \cdot \Delta_{B_M} = \Delta_B$, since
$R^+$ is the disjoint union of $R_M^+$ and $R_N$.
\bigskip
So we consider the quantity
\begin{equation}
\label{quantity}
\frac{ \Delta_P \cdot \Delta_{w_M * B_M}}{\Delta_{w_M \omega * B}}=
\frac{ \Delta_B \cdot \Delta_{w_M * B_M}}{\Delta_{B_M} \cdot \Delta_{w_M \omega *
B}} .
\end{equation}
\noindent Observe that if $\BB$ is a Borel, $\Delta_{\BB}=\delta_{\BB}
\cdot
\rho_{\BB}^{-1}$,
where $\delta_{\BB}=\prod_{\alpha > 0}
(\alpha^{\frac{1}{2}}-\alpha^{-\frac{1}{2}})$ and $\rho_{\BB}$ is the usual half sum of positive
roots.
Since $\delta_{w * \BB}=\varepsilon(w)\delta_{\BB}$, we compute that
\[ \frac{\Delta_{w * \BB}}{\Delta_{\BB}}=\varepsilon(w) \cdot
(\rho_{\BB}-w\rho_{\BB}). \]
Thus \eqref{quantity} becomes
\[ \varepsilon(\omega) (w_M(\omega \rho_B-\rho_{B_M})-\rho_B+\rho_{B_M}). \]
Next observe that for $w_M \in W_M$,
\[ w_M(\rho_B-\rho_{B_M})=\rho_B-\rho_{B_M}. \]
Indeed, the roots of $R^+$ not in $R_M^+$ are in $R_N$, and are
thus normalized by
$W_M$. So the above expression simplifies to
\[ \varepsilon(\omega) \cdot w_M(\omega \rho_B-\rho_B). \]
We can therefore rewrite \eqref{break} as
\begin{equation}
\label{rewrite}
\sum_{\omega \in W^M} m(\omega * B) \cdot \varepsilon(\omega) \cdot \sum_{w_M \in W_M}
\frac{w_M (\omega(\lambda_B+\rho_B)-\rho_B)(\gamma)}{\Delta_{w_M * B_M}(\gamma)}.
\end{equation}
Since $\omega$ is a Kostant representative, the
weight
$\omega(\lambda_B+\rho_B)-\rho_B$ is positive for $B_M$,
and we may use the Weyl character formula to rewrite this as
\begin{equation}
\label{Weyl}
\sum_{\omega \in W^M} m(\omega * B) \cdot \varepsilon(\omega) \cdot \tr(\gamma; V^M_{\omega(\lambda_B+\rho_B)-\rho_B}).
\end{equation}
\noindent Here $V^M_{\omega(\lambda_B+\rho_B)-\rho_B}$ denotes the
irreducible finite-dimensional
representation of $M$ with highest weight $\omega(\lambda_B+\rho_B)-\rho_B$.
\section{A Formula for $\Phi_M(\gamma,\Theta^E)$}
To identify \eqref{Weyl} with $\Phi_M(\gamma,\Theta^E)$, we
replace $m(\omega *
B)$ with $n(\gamma,\omega * B)$ as on page 500 of [2], and multiply it by the
factor $\delta_P^{\frac{1}{2}}(\gamma)$:
\begin{equation}
\label{factor}
\delta_P^{\frac{1}{2}}(\gamma) \cdot \sum_{\omega \in W^M} n(\gamma,\omega * B) \cdot
\varepsilon(\omega) \cdot
\tr(\gamma; V^M_{\omega(\lambda_B+\rho_B)-\rho_B}).
\end{equation}
\noindent Here $\delta_P$ is the modulus character of $P$.
(We are still only considering regular $\gamma$.)
\bigskip
Write $A_G$ for the split component of the center of $G$. Let
$\lambda_0 \in X^*(A_G)$ denote the character by which $A_G$ acts on $E$.
It extends to $X^*(T)_{\ensuremath{{\mathbb R}}}$ in the usual way, and is
$W$-invariant.
\bigskip
Let $T_e$ denote the subtorus of $T$ generated by $T_c$ and $A_G$. It is
the maximal subtorus of $T$ which is elliptic in $G$.
\bigskip
Write $p_M$ for the projection from $X^*(T)_{\ensuremath{{\mathbb R}}}$ to
$X^*(A)_{\ensuremath{{\mathbb R}}}$, and note
that it is $W_L$-invariant.
The group $W_L$ fixes each root of $M$, thus it acts on $W^M$.
For every orbit of this action, there is a unique
member $\omega$ so that $p_M(\omega(\lambda_B + \rho_B-\lambda_0))$ is
dominant with respect
to $C$. We denote the set of these elements by $W^{LM}$, one element for
each orbit of $W_L$ on $W_M$.
\bigskip
If $\lambda \in X^*(T)$ and $w_L \in W_L$, then
plainly
$w_L \lambda-\lambda \in \ensuremath{{\mathfrak a}}_M^*$.
Write $(\chi_{w_L,\omega,B},\ensuremath{{\mathbb C}}_{w_L,\omega,B})$ for the
one-dimensional
representation of $M$, acting through $A$, with weight $w_L\omega(\lambda_B+\rho_B)-\omega(\lambda_B+\rho_B)$.
Note that $T_c$ and $A_G$ act trivially on
$\ensuremath{{\mathbb C}}_{w_L,\omega,B}$, thus so does $T_e$.
\bigskip
Thus we have
\[ V^M_{ w_L\omega(\lambda_B+\rho_B)-\rho_B} \cong V^M_{
\omega(\lambda_B+\rho_B)-\rho_B} \otimes \ensuremath{{\mathbb C}}_{w_L,\omega,B}. \]
\bigskip
Our formula \eqref{factor} is now (replacing $\omega \in W^M$
with $w_L
\omega$, where $\omega$ is now in $W^{LM}$):
\begin{equation}
\label{replace}
\delta_P^{\frac{1}{2}}(\gamma) \cdot \sum_{\omega \in W^{LM}} \varepsilon(\omega) \cdot
\tr(\gamma; V^M_{\omega(\lambda_B+\rho_B)-\rho_B}) \cdot
\sum_{w_L \in W_L} \varepsilon(w_L) \cdot \chi_{w_L,\omega,B}(\gamma) \cdot
n(\gamma, w_L \omega
* B),
\end{equation}
Of course we now wish to simplify the inner sum. Recall from
page 500 of [2] that
\[ n(\gamma,w_L \omega * B)= \bar{c}(x,p_M(w_L \omega \lambda_B+ w_L\omega
\rho_B - \lambda_0)),
\]
\noindent where $x$ is in the interior of $C$.
Here $\bar{c}(x,\lambda)$ is the integer-valued ``stable discrete series
constant'' on
\[ (X_*(A/A_G)_{\ensuremath{{\mathbb R}}})_{\reg} \times (X^*(A/A_G)_{\ensuremath{{\mathbb R}}})_{\reg}, \]
\noindent as defined, for instance, on page 493 of [2].
Recall that $\lambda_0 \in X^*(T)_{\ensuremath{{\mathbb R}}}$ is obtained from the character
$\lambda_0 \in X^*(A_G)$ by which $A_G$ acts on $E$, and is thus
$W$-invariant.
\bigskip
As $p_M$ commutes with $w_L$, the inner sum of
\eqref{replace} is now
\begin{equation}
\label{inner}
\sum_{w_L \in W_L} \varepsilon(w_L) \cdot \bar{c}(x,w_L \Lambda)
\cdot \chi_{w_L,\omega,B}(\gamma),
\end{equation}
\noindent where $\Lambda=p_M(\omega \lambda_B+ \omega \rho_B -
\lambda_0)$.
\bigskip
We would like to consider the limit of \eqref{inner} as $x$
approaches
$0$. Recall we can write $\gamma=\gamma_c \cdot \exp(x)$, with
$\gamma_c \in T_c(\ensuremath{{\mathbb R}})$ and $x$ in $\bar{C}$. Also recall that $\gamma$ is
still regular (not for long!). Consider the above formula with $\gamma_c$ fixed and $x$
going to $0$ along regular elements of $\bar{C}$. Fix some element $x_0$ in
the interior of $C$.
The value
\[ \bar{c}(x,w_L \Lambda)=\bar{c}(x_0,w_L \Lambda) \]
is unchanged, but $\chi_{w_L,\omega,B}(\gamma)$ approaches
$\chi_{w_L,\omega,B}(\gamma_c)=1$.
Thus \eqref{inner} converges to
\[ \sum_{w_L \in W_L} \varepsilon(w_L) \cdot \bar{c}(x_0,w_L \Lambda) \]
\noindent for some $x_0 \in C$.
\bigskip
But this is simply $(-1)^{q(L)}|W_L|$, by Proposition 1(ii) in
Section 5 below. Here we use that $\omega \in W^{LM}$. Note that $-1$ is
in the Weyl group of the root system by the argument on page 499 of [2].
\bigskip
It is easy to modify this argument to get the
same limit as $x$ approaches an element of $X_*(A_G)_{\ensuremath{{\mathbb R}}}$.
\bigskip
Finally note that $\delta_P$ is a positive character and therefore trivial
on the compact group $T_c(\ensuremath{{\mathbb R}})$. It is thus trivial on $T_e(\ensuremath{{\mathbb R}})$.
\bigskip
Now consider irregular $\gamma$. We take the limit in
\eqref{replace} and obtain our theorem:
\begin{thm}
If $\gamma \in T_e(\ensuremath{{\mathbb R}})$, then
\begin{equation}
\label{main}
\Phi_M(\gamma,\Theta^E) = (-1)^{q(L)}\cdot |W_L| \cdot \sum_{w \in
W^{LM}} \varepsilon(w) \cdot \tr(\gamma; V^M_{w(\lambda_B+\rho_B)-\rho_B}).
\end{equation}
\end{thm}
\bigskip
\noindent For the reader's convenience, we review the definition of
$W^{LM}$.
\bigskip
The definition depends on the choice of a parabolic $P=MN$ and a
Borel subgroup $B$ with $T \subseteq B \subseteq P$. The choice of $B$ gives a
set of positive roots $R^+$ for $R$ and a set of positive roots $R_M^+$
for $R_M$. It also gives $B$-dominant elements $\lambda_B$ and $\rho_B$
of $X^*(T)_{\ensuremath{{\mathbb R}}}$. The choice of $P$ determines an $L$-chamber $C$ as in
Section 2. Recall the character $\lambda_0$ determined by $A_G$ on $E$
and the projection $p_M$ from $X^*(T)_{\ensuremath{{\mathbb R}}}$ to $X^*(A)_{\ensuremath{{\mathbb R}}}$. Then
\[ W^{LM} = \{ w \in W | w^{-1}R_M^+ \subseteq R^+ \text{ and }
p_M(w(\lambda_B + \rho_B - \lambda_0)) \text{ is dominant w.r.t. } C \}.
\]
\bigskip
We now evaluate \eqref{main} for $\Phi_M$ on the extreme cases
for $T$. If $T=A$ is split, then $M=A$, $L=G$, $W^{LM}$ is trivial, but so is
$T_c$. We conclude that for $z \in A_G(\ensuremath{{\mathbb R}})$,
\[ \Phi_A(z,\Theta^E)=(-1)^{q(G)} \cdot |W| \cdot \lambda_0(z). \]
If $T$ is elliptic, then $M=G$, $L=T$, $W^{LM}$ is again
trivial, and so for $\gamma \in T$,
\[ \Phi_G(\gamma,\Theta^E)= \tr(\gamma; E). \]
\noindent Note that this agrees with the results of
Theorems 5.1 and 5.2 of [2], since
\[ \tr(\gamma^{-1}; E^*)= \tr(\gamma; E). \]
\section{The Sum of the Stable Discrete Series Constants}
Let $(X,X^*,R,\check{R})$ be a root system. Write $W$ for the Weyl group of the root
system, and $\varepsilon$ for its sign character. Assume that $R$
generates the real vector space $X$ and that $-1 \in W$.
Write $q(R)$ for $(|R^+|+\dim(X)) / 2$, as in [2].
Let $x_0$ be a regular element of $X$, and $\lambda$ a regular
element of $X^*$. Write $C_0$ for the chamber of $X$ containing
$x_0$, and $C_0^{\vee}$ for its dual chamber in $X^*$.
Recall the stable discrete series constants $\bar{c}_R(x_0,\lambda)$ from
section 3 of [2].
\begin{prop} We have the following formulas for sums of discrete
series constants:
\begin{itemize}
\item[(i)] For all such $ \lambda, \sum_{w \in W}
\bar{c}_R(wx_0,\lambda)=|W|.$
\item[(ii)] For $\lambda=\lambda_0 \in C_0^{\vee}$, we have $\sum_{w
\in W}
\varepsilon(w) \cdot \bar{c}_R(wx_0,\lambda_0)=(-1)^{q(R)}|W|.$
\end{itemize}
\noindent The same formulas hold if we sum over the $W$-orbit of
$\lambda$ rather than
that of $x_0$.
\end{prop}
We make a few comments before beginning the proof. The proof
begins by using the
``inductive'' property (4) of the discrete series constants from page 493 of
[2], to change the sum
over chambers into a sum over certain facets of $X$. In fact we consider those facets which
separate the chambers of $X$, i.e., those which span the root hyperplanes $Y$ of $X$.
\bigskip
In the course of the proof, we (mis)use the term ``facet'' only
in reference to these
particular facets, of codimension $1$. So a facet in this sense will be the common face
of two adjacent chambers.
\bigskip
The hyperplanes $Y$ have their own chambers, and we examine the
relationship between the
facets and these smaller chambers. Not every facet is equal to such a chamber, as in the
case of $B_3$ when $Y$ is the root hyperplane of a long root. The facets
in $Y$ give a $B_2$ system, but the chambers of $R_Y$ give an $A_1
\times A_1$ system.
\bigskip
Finally induction on the rank of the root system gives the
calculation.
\begin{proof}
\noindent The second formula follows from the first by applying Theorem
3.2(2) on page 494 of [2].
\bigskip
We induce on $r=\dim X$. The proposition is clear when $r=0$.
\bigskip
We associate these discrete series constants with the
various chambers and
facets of $X$, and introduce some appropriate notation.
\bigskip
Write $c(\ensuremath{{\mathcal C}})$ for $\bar{c}_R(x,\lambda)$, when $x$ is in the
interior of a chamber
$\ensuremath{{\mathcal C}}$.
\bigskip
Suppose $F$ is a facet in $X$, $y$ is in the interior of $F$,
and
$\bar{F}:=\Span(F)=Y$.
Then write $c(F)=\bar{c}_{R_Y}(y,\lambda_Y)$, notation as on page 493 of
[2].
\bigskip
Thus if $F$ is the common face of distinct chambers $\ensuremath{{\mathcal C}}$ and
$\ensuremath{{\mathcal C}}^{\prime}$, then
\[ 2c(F)=c(\ensuremath{{\mathcal C}})+c(\ensuremath{{\mathcal C}}^{\prime}). \]
Each chamber has $r$ faces, and it follows that
\begin{equation}
\label{summing}
r \cdot \sum_{\ensuremath{{\mathcal C}}} c(\ensuremath{{\mathcal C}})= 2 \sum_{F} c(F),
\end{equation}
\noindent where we are summing over all chambers and then all facets.
\bigskip
We show the right hand side of \eqref{summing} is equal to $r
\cdot |W|$ to
prove the proposition.
\bigskip
Now every facet is on some root hyperplane
$X_{\alpha}=X_{-\alpha}$, so we have
\[ 2 \sum_F c(F)= \sum_{\alpha \in R} \sum_{\bar{F}=X_{\alpha}} c(F). \]
We now work with the inner sum. There is a root system on
$X_{\alpha}$ whose set of coroots
is $\check{R} \cap X_{\alpha}$, which defines chambers $\ensuremath{{\mathcal C}}_{\alpha}$ in $X_{\alpha}$ and
constants $c_{\alpha}(\ensuremath{{\mathcal C}}_{\alpha})$. Write $W_{\alpha}$ for
the Weyl group of $X_{\alpha}$. We have
\[ \sum_{\bar{F}=X_{\alpha}} c(F)=\sum_{\ensuremath{{\mathcal C}}_{\alpha}} \sum_{F \subset
\ensuremath{{\mathcal C}}_{\alpha}} c_{\alpha}(\ensuremath{{\mathcal C}}_{\alpha})=
\sum_{\ensuremath{{\mathcal C}}_{\alpha}} \sum_{W_{\alpha} \backslash \{F \subset
X_{\alpha} \}} c_{\alpha}(\ensuremath{{\mathcal C}}_{\alpha}) = \sum_{W_{\alpha} \backslash \{ F \subset
X_{\alpha} \} } \sum_{\ensuremath{{\mathcal C}}_{\alpha}}c_{\alpha}(\ensuremath{{\mathcal C}}_{\alpha}). \]
\noindent For the first equality, note that every facet $F$ with
$\bar{F}=X_{\alpha}$ is contained in
a some chamber $\ensuremath{{\mathcal C}}_{\alpha}$.
\bigskip
The second equality follows because $W_{\alpha}$ acts transitively on the
chambers $C_{\alpha}$.
\bigskip
Write $n(\alpha)$ for the order of $W_{\alpha} \backslash \{ F
\subset X_{\alpha} \}$.
It is equal to the number of facets in a given chamber $\ensuremath{{\mathcal C}}_{\alpha}$. Then
by induction the above is merely
\[ n(\alpha) \cdot |W_{\alpha}|, \]
\noindent which is exactly the number of facets in $X_{\alpha}$. It
follows that \eqref{summing} is
simply equal to twice the total number of facets in $X$.
\bigskip
Since $W$ has $r$ orbits on the set of facets in $X$, and the
stabilizer in $W$ of any
facet has order $2$, we conclude that the total number of facets is half of $r \cdot |W|$,
as desired.
\end{proof}
|
{
"timestamp": "2005-03-24T10:50:13",
"yymm": "0503",
"arxiv_id": "math/0503524",
"language": "en",
"url": "https://arxiv.org/abs/math/0503524"
}
|
\section{Introduction}
Experiments in cooling and trapping of neutral gases have paved the way
toward a new parameter regime of ionized gases, namely the regime of
ultracold neutral plasmas (UNPs). Experimentally, UNPs are produced by
photoionizing a cloud of laser-cooled atoms collected in a magneto-optical
trap \cite{Kil99}, with temperatures down to 10 $\mu$K. By tuning the frequency
of the ionizing laser, initial electron kinetic energies of $E_{\rm{e}}/k_{\rm B} = 1
\mbox{K} - 1000 \mbox{K}$ have been achieved. The time evolution of several
quantities characterizing the state of the plasma, such as the plasma
density \cite{Kil99,Kul00,Sim04}, the rate of expansion of the plasma cloud
into the surrounding vacuum \cite{Kul00}, the energy-resolved population of
bound Rydberg states formed through recombination \cite{Kil01}, or electronic
\cite{Rob04,Van04} as well as ionic \cite{Sim04} temperature have been measured
using various plasma diagnostic methods.
Despite the low typical densities of $\approx 10^9$ cm$^{-3}$, the very low
initial temperatures suggest that these plasmas have been produced well within
the strongly coupled regime, with Coulomb coupling parameters up to
$\Gamma_{\rm{e}} = 10$ for the electrons and even $\Gamma_{\rm{i}} = 30000$ for the ions.
Thus, UNPs seem to offer a unique opportunity for a laboratory study of neutral
plasmas where, depending on the initial electronic kinetic energy, either one
component (namely the ions) or both components (ions and electrons) may be
strongly coupled. Moreover, the plasma is created in a completely
uncorrelated state, i.e.\ far away from thermodynamical equilibrium. The
relaxation of a strongly correlated system towards equilibrium is an
interesting topic in non-equilibrium thermodynamics and has been studied for
decades. The history of this problem must be traced back to the important
contributions of Klimontovich
\cite{Kli72,Kli73,Kli82}, who pointed out that kinetic energy conserving
collision integrals such as the Boltzmann, Landau and Lenard-Balescu
collision integrals are not appropriate for such a situation, and derived
non-Markovian kinetic equations taking correctly into account total energy
conservation of the system. In the following years this problem has attracted
much attention and the relaxation of nonequilibrium strongly coupled plasmas
has been studied by a variety of different methods
\cite{Wal78,Bel96,Bon98,Zwi99}.
The very low densities of UNPs make it now possible to directly
observe the dynamical development of spatial correlations, which may
serve as the first experimental check of the present understanding of the
strongly coupled plasma dynamics. Moreover, it turns out that the timescale of
the plasma expansion, the correlation time as well as the relaxation time of
the ions are almost equal. Therefore Bogoliubov's functional hypothesis,
usually used in kinetic theory, breaks down under the present conditions,
which may lead to a very interesting relaxation behavior but also causes some
difficulties in the theoretical description of these systems, since the
plasma dynamics can not be divided into different relaxation stages.
\section{Theoretical approach}
A full molecular dynamics simulation of ultracold neutral plasmas over
experimentally relevant timescales is infeasible with present-day computer
resources due to the large number of particles ($N \approx 10^5$) and the
long observation times ($t \approx 10^{-4}$ s) involved.
In order to model the evolution of UNPs, we have developed a hybrid molecular
dynamics (HMD) approach which treats electrons and ions on different levels
of sophistication, namely in a hydrodynamical approximation on the one hand
(for the electrons) and on a full molecular dynamics level on the other hand
(for the ions) \cite{PPR04}. For the electrons, it has been shown that several
heating effects, such as continuum threshold lowering \cite{Hah02}, build-up
of correlations \cite{Kuz02}, and, predominantly, three-body recombination
\cite{Rob02} rapidly increase the electronic temperature. As a consequence,
the electrons are always weakly coupled, $\Gamma_{\rm{e}} < 0.2$, over the whole
course of the system evolution. Moreover, due to the small electron-to-ion
mass ratio, the relaxation timescale of the electrons is much smaller than
that of the ions as well as the timescale of the plasma expansion. Hence, an
adiabatic approximation may safely be applied, assuming instant equilibration
of the electrons. This allows for the use of much larger timesteps than in
a full MD simulation since the electronic motion does not need to be
resolved. It is this adiabatic approximation for the electrons which makes
a molecular dynamics treatment of the ionic motion in UNPs computationally
feasible. The electronic density is determined self-consistently from the
Poisson equation. The fact that the potential well created by the ions which is
trapping the electrons has a finite depth is taken into account by using a
King-type distribution \cite{Kin66} known from simulations of globular clusters
rather than a Maxwell-Boltzmann distribution for the electron velocities, with the electronic temperature $T_{\rm{e}}$ obtained from energy
conservation. The finite well depth also leads to evaporation of a
fraction of the free electrons in the very early stage of the plasma evolution,
which is accounted for by determining the fraction of trapped electrons
from the results of \cite{Kil99}. The dynamics of the heavy particles is
described in the framework of a chemical picture, where inelastic processes,
namely three-body recombination and electron impact
ionization, excitation and deexcitation, are taken into account on the basis
of Boltzmann-type collision integrals \cite{Kli81,Kli82}, with the transition
rates taken from \cite{Man69}. Numerically, the resulting collision integrals
are evaluated using a Monte Carlo sampling as described in
\cite{Rob03,PPR04,PPR04c}. The ions and recombined atoms are then propagated
individually in a molecular dynamics simulation, taking
into account the electronic mean-field potential and the full interaction
potential of the remaining ions\footnote{In order to bring out clearly the role
of ionic correlations, it is also possible to neglect them in the HMD approach
by propagating the ions in the mean-field
potential created by all charges rather than the full ionic interaction.}.
In order to allow for larger particle numbers, the most time-consuming part
of the HMD simulation, namely the
calculation of the interionic forces, is done using a treecode procedure
originally designed for astrophysical problems \cite{Bar90}, which scales like
$N_{\rm{i}} \ln N_{\rm{i}}$ rather than $N_{\rm{i}}^2$ with the number $N_{\rm{i}}$ of ions.
As shown in several publications \cite{PPR04,PPR04a,PPR04b,PPR04c}, the HMD
approach outlined above provides a powerful method for the description of
UNPs, taking full account of ionic correlation effects. However, due to the
large numerical effort involved, it is limited to particle numbers of $N_{\rm{i}}
\approx 10^5$. While this permits a direct simulation of many, particularly
of the early, experiments, an increasing number of experiments is performed
with larger particle numbers up to $10^7$. Thus, an alternative method which
is able to treat such larger systems is desirable. Such a method is indeed
available \cite{PPR04}, based on a hydrodynamical description of both electrons
and ions similar to that introduced in \cite{Rob02,Rob03}. Starting from the
first equation of the BBGKY hierarchy, one obtains the evolution equations for
the one-particle distribution functions $f$ of the electrons and ions.
Neglecting again electron-electron as well as electron-ion correlations, and
employing the same adiabatic approximation for the electrons already used in
the HMD approach, a quasineutral approximation \cite{Dor98} permits
expressing the mean-field
electrostatic potential in terms of the ionic density, leading to a closed
equation for the ion distribution function which contains the electron
temperature as a parameter. A Gaussian ansatz for the ion distribution
function,
\begin{equation} \label{e1}
f_{\rm{i}} \propto \exp{\left(-\frac{r^2}{2\sigma^2}\right)}\exp{\left(-
\frac{m_{\rm{i}}\left({\bf{v}}-\gamma{\mathbf{r}}\right)^2}{2k_{\rm{B}}
T_{\rm{i}}}\right)} \; ,
\end{equation}
which corresponds to the initial state of the plasma cloud, is then inserted
into the evolution equations for the second moments $\langle r^2 \rangle$,
$\langle \bf{r} \bf{v} \rangle$ and $\langle v^2 \rangle$ of the ion
distribution function. In this way,
evolution equations for the width $\sigma$ of the cloud, the parameter $\gamma$
of the hydrodynamical expansion velocity $\bf{u} = \gamma{\mathbf{r}}$ and
the ionic temperature $T_{\rm{i}}$ are obtained. Ionic correlations are taken into
account in an approximate way using a local density approximation together
with a gradient expansion, reducing the description of their influence on the
plasma dynamics to the evolution of a single macroscopic quantity, namely the
correlation energy $U_{\rm{ii}}$ of a homogeneous plasma. The relaxation behavior
of $U_{\rm{ii}}$ is modeled using a correlation-time approximation \cite{Bon96}
with a correlation time equal to the inverse of the ionic plasma frequency,
$\tau_{\rm corr} = \omega_{\rm{p,i}}^{-1}$, together with an analytical expression
for the equilibrium value of $U_{\rm{ii}}$ \cite{Cha98}. Finally, inelastic
processes such as three-body recombination and electron impact ionization,
excitation and deexcitation are incorporated on the basis of rate
equations, and the influence of the recombined Rydberg atoms on the expansion
dynamics is taken into account assuming equal hydrodynamical velocities for
atoms and ions. The final set of evolution equations then reads
\begin{subequations} \label{e2}
\begin{eqnarray}
\label{e2a}
\dot{\sigma}&=&\gamma\sigma\;,\\
\label{e2b}
\dot{\gamma}&=&\frac{N_{\rm{i}}\left(k_{\rm{B}}T_{\rm{e}}+k_{\rm{B}}T_{\rm{i}}+
\frac{1}{3}U_{\rm{ii}}\right)}{\left(N_{\rm{i}}+N_{\rm{a}}\right)m_{\rm{i}}
\sigma^2}-\gamma^2\;,\\
\label{e2c}
k_{\rm{B}}\dot{T}_{\rm{i}}&=&-2\gamma k_{\rm{B}}T_{\rm{i}}-\frac{2}{3}\gamma
U_{\rm{ii}}-\frac{2}{3}\dot{U}_{\rm{ii}}\;,\\
\label{e2d}
\dot{U}_{\rm{ii}}&=&-\omega_{\rm{p,i}}\left(U_{\rm{ii}}-U_{\rm{ii}}^{\rm{(eq)}}
\right)\\
\label{e2e}
\dot{{\cal{N}}}_{\rm{a}}(n)&=&\sum_{p\neq n}\left[R_{\rm{bb}}{(p,n)}
{\cal{N}}_{\rm{a}}(p)-R_{\rm{bb}}{(n,p)}{\cal{N}}_{\rm{a}}(n)\right]
\nonumber\\&&+R_{\rm{tbr}}(n)N_{\rm{i}}-R_{\rm{ion}}{(n)}{\cal{N}}_{\rm{a}}(n)
\end{eqnarray}
and the electronic temperature is determined by energy conservation,
\begin{equation}
\label{e2f}
N_{\rm{i}}k_{\rm{B}}T_{\rm{e}}+\left[N_{\rm{i}}+N_{\rm{a}}\right]
\left[k_{\rm{B}}T_{\rm{i}}+m_{\rm{i}}\gamma^2\sigma^2\right]+\frac{2}{3}
N_{\rm{i}}U_{\rm{ii}}-\frac{2}{3}\sum_n{\cal{N}}_{\rm{a}}(n)
\frac{{\cal{R}}}{n^2}={\rm{const.}}\;,
\end{equation}
\end{subequations}
where ${\cal{N}}_{\rm{a}}(n)$ defines the population of Rydberg states,
$N_{\rm{a}}=\sum_n{\cal{N}}_{\rm{a}}(n)$ is the total number of atoms and
${\cal{R}}=13.6$eV is the Rydberg constant.
The preceeding hydrodynamical method is much more approximate than the HMD
approach, but, on the other hand, it is much simpler and quicker.
For particle numbers of $N_i \approx 10^5$, it requires about two orders
of magnitude less CPU time. Since its computational effort is independent of
the number of particles, it allows for a simulation of larger plasma clouds
corresponding to a number of current experiments. Moreover, and maybe
equally important, it provides physical insight into the plasma dynamics
since it is based on a few simple evolution equations for the macroscopic
observables characterizing the state of the plasma. As we have investigated
in detail in \cite{PPR04}, there is generally surprisingly good agreement
between the hydrodynamical simulation and the more sophisticated HMD
calculation as long as macroscopic, i.e.\ spatially averaged, quantities such
as electronic temperature, expansion velocity, ionic correlation energy etc.\
are considered.
\begin{figure}[tb]
\centerline{\psfig{figure=f1a.eps,width=6.3cm} \hfill
\psfig{figure=f1b.eps,width=6.3cm}}
\caption{\label{f1}
Electronic temperature $T_{\rm{e}}(t)$ for an expanding plasma of $40000$ Sr ions
with an initial average density of $\rho_{\rm{i}}=10^9$cm$^{-3}$ and an initial electron
kinetic energy of $20\:$K, obtained from the HMD simulation (a) and from
eqs.\ (\ref{e2}) (b), with (solid) and without (dotted) the inclusion of ionic
correlations. The inset shows the ratio of the electron temperatures obtained
from both methods.}
\end{figure}
As an example, we show in figure \ref{f1} the time evolution of
the electronic temperature for a plasma of 40000 Sr ions
with an initial average density of $10^9$cm$^{-3}$ and an initial electron
kinetic energy of $20\:$K, obtained from the HMD simulation (a) and from
eqs.\ (\ref{e2}) (b). During the whole system evolution, the agreement between
the two simulation methods is better than about 8\%, and it becomes even
better at later times. Thus, we conclude that, for the present type of
experimental setups, the hydrodynamical method outlined above, and in particular
the approximate treatment of ionic correlations, is well suited for the
description of the behavior of UNPs.
\section{Results and discussion}
\subsection{Comparison with experiments}
In fact, fig.\ \ref{f1} only shows good agreement between the two theoretical
simulation methods, without comparison with experiment. Such a comparison is
now also possible, since measurements of the electron temperature dynamics have
recently been reported in \cite{Rob04}. Fig.\ \ref{f2} shows the time evolution
of the electronic temperature for a Xenon plasma with $N_{\rm{i}}(0)=1.2\cdot10^6$,
$\rho_{\rm{i}}(0)=1.35\cdot10^9$cm$^{-3}$ and two different initial temperatures
of $T_{\rm{e}}(0)=66.67$K and $T_{\rm{e}}(0)=6.67$K.
In addition to the full hydrodynamical simulation according to equations
(\ref{e2}), fig.\ \ref{f2} also shows corresponding calculations where the
effect of inelastic electron-ion collisions, eq.\ (\ref{e2e}), is neglected
(dashed lines). (The plasmas in these experiments are too large to be simulated
using the HMD approach.)
\begin{figure}[tb]
\centerline{\psfig{figure=f2.eps,width=9cm}}
\caption{\label{f2}
Electronic temperature $T_{\rm{e}}(t)$ for a plasma of $1.2\cdot10^6$ Xenon ions with
an initial average density of $1.35\cdot10^9$cm$^{-3}$ for two different
initial electron temperatures, $T_{\rm{e}} =6.67$K (filled dots) and $T_{\rm{e}} = 66.67$K
(open dots). The lines show the hydrodynamical simulation (solid lines:
including inelastic collisions, dashed lines: without inelastic collisions),
the dots the experiment \cite{Rob04}, scaled down by 26\% (see text).}
\end{figure}
\begin{figure}[tb]
\centerline{\psfig{figure=f3a.eps,height=3.8cm} \hfill
\psfig{figure=f3b.eps,height=3.8cm}}
\caption{\label{f3}
Time evolution of the average electron density of a Xenon plasma of 500000
ions with an initial average density of $10^9$cm$^{-3}$ and an initial electron
temperature of $T_{\rm{e}}=210$K (a) and $T_{\rm{e}}=2.6$K (b). The lines show the results of
the model equations (\ref{e2}) (solid lines: including inelastic collisions,
dashed lines: without inelastic collisions) and the dots the experimental data
from \cite{Kul00}.}
\end{figure}
For the high initial temperature, there is close agreement between
the two corresponding simulations, showing that inelastic processes are almost
negligible in this case. Indeed, it is known that the high-temperature
plasma expansion is well described by the collisionless plasma dynamics, and
the hydrodynamical model is expected to accurately reproduce the plasma
dynamics in this regime. Since an overall systematic error of about $70\%$ for
the temperature measurement has been reported in \cite{Rob04}, we have
exploited this fact to calibrate the measured temperatures by scaling down
both experimental data sets by $26\%$ in order to match the high-temperature
results to our calculations. As can be seen in the figure, there is excellent
agreement between simulation and experiment also for the lower temperature.
(We stress that there is no further scaling of the low-temperature data in order
to achieve quantitative agreement, the same calibration factor as in the
high-temperature case is used.) In this case, inelastic collisions play a
decisive role for the evolution of the system. More specifically, as has been
found already in \cite{Rob02}, three-body recombination heats the plasma and
significantly changes its behavior, leading to a weakly coupled electron gas,
as discussed above in connection with the omission of
electronic correlation effects in the numerical treatment. Moreover, there
has been some discussion in the literature whether the collision rates of
\cite{Man69} would still be applicable at these ultralow temperatures, or
whether three-body recombination would be significantly altered. The close
agreement between the present simulation and the experimental data in fig.\
\ref{f2} suggests that the rates of \cite{Man69}, while ultimately diverging
$\propto T_{\rm{e}}^{-9/2}$ for $T_{\rm{e}} \to 0$, still adequately describe three-body
recombination processes in the temperature range under consideration.
As a second example, figure \ref{f3} shows the time evolution of the electronic
density for a Xenon plasma of $500000$ ions with an initial average density of
$10^9$cm$^{-3}$ and two different initial electron temperatures of
$T_{\rm{e}}(0)=210$K and $T_{\rm{e}}(0)=2.6$K \cite{Kul00}. Again, it can be seen that the
model
equations nicely reproduce the density evolution in both temperature regimes,
in agreement with \cite{Rob02} where it was shown that the low-temperature
enhancement of the expansion velocity \cite{Kul00} is caused by recombination
heating and is not due to strong-coupling effects of the electrons.
\subsection{Role of ionic correlations}
Having thus established the validity of our numerical methods for the
description of UNPs, we can now turn to a more detailed investigation of the
role of ionic correlations in these systems. It is found that, for situations
corresponding to the type of experiments \cite{Kil99,Rob04,Van04}, they
hardly influence the macroscopic expansion behavior of the plasma. This becomes
evident, e.g., in fig.\ \ref{f1}, where the ``full'' simulations as
described above (solid lines) are compared to a mean-field treatment of the
system completely neglecting correlation effects (dotted lines). The
correlation-induced heating of the ions \cite{Mur01,Ger03a,Ger03b} leads
to a slightly faster expansion of the plasma, which in turn results in a
slightly faster adiabatic cooling of the electrons \cite{PPR04}. However, the
overall effect is almost negligible.
\begin{figure}[bt]
\centerline{\psfig{figure=f4.eps,width=9cm}}
\caption{\label{f4}
Spatial density $\rho_{\rm{i}}$ (solid) of the ions, at $t=3\:\mu$s, compared to the
Gaussian profile assumed for the kinetic model (dashed). Additionally, $\rho_{\rm{i}}$
obtained from the particle simulation using the mean-field interaction only is
shown as the dotted line. Initial-state parameters are the same as in fig.\
\ref{f1}.}
\end{figure}
A closer look, on the other hand, reveals that certain aspects of the
expansion dynamics are indeed significantly affected by the strong ion-ion
interaction, as can be seen in figure \ref{f4}. There, the spatial density
of the ions is shown after $t=3\:\mu$s for the same plasma as in fig.\
\ref{f1}. A mean-field treatment of the particle interactions \cite{Rob03}
predicts that a shock front should form at the plasma edge, seen as the
sharp spike in fig.\ \ref{f4} (dotted line). Apparently, with ionic
correlations included (solid line) the peak structure is much
less pronounced than in mean-field approximation. This is due
to dissipation caused by ion-ion collisions which are fully taken into account
in the HMD simulation. As shown in \cite{Sac85}, by adding an ion viscosity
term to the hydrodynamic equations of motion, dissipation tends to
stabilize the ion density and prevents the occurrence of wavebreaking which was
found to be responsible for the diverging ion
density at the plasma edge in the case of a dissipationless plasma
expansion. Furthermore, the initial correlation heating of the ions largely
increases the thermal ion velocities, leading to a broadening of the peak
structure compared to the zero-temperature case.
Another obvious aspect where ionic correlations play a dominant role is the
behavior of the ionic temperature.
Considering the huge ionic coupling
constants suggested by the low initial ion temperatures, this temperature
turns out to be an important quantity since it directly determines the value
of $\Gamma_{\rm{i}}$. According to a mean-field treatment, the ions would
remain the (near) zero temperature fluid they are initially.
However, as has been pointed
out before, the ions are created in a completely uncorrelated non-equilibrium
state, and they quickly heat up through the build-up of correlations as the
system relaxes toward thermodynamical equilibrium. As shown in
\cite{PPR04}, even at early times the ionic velocity distribution is locally
well described by a Maxwell distribution corresponding to a (spatially) local
temperature, justifying the definition of a --- due to the spherical symmetry
of the plasma --- radius-dependent ion temperature
$T_i(r,t)$. Moreover, if the spatially averaged temperature is
identified with the ion temperature determined by the model equations
(\ref{e2}) one can find again good agreement between both approaches concerning
the timescale of the initial heating as well as the magnitude of the ion
temperature, even at later times \cite{PPR04}. However, as becomes apparent
from fig.\ \ref{f5}, the HMD simulations show temporal oscillations of the
ionic temperature, which can, of course, not be described by the linear ansatz
of the correlation-time approximation.
\begin{figure}[bt]
\centerline{\psfig{figure=f5.eps,width=9cm}}
\caption{\label{f5}
Time evolution of the density-scaled average ionic temperature for a plasma
consisting of $400000$ ions with an initial electronic Coulomb coupling
parameter of $\Gamma_{\rm{e}}(0)=0.07$.}
\end{figure}
Such temporal oscillations of the temperature during the initial relaxation
stage are known from molecular dynamics simulations of homogeneous
one-component \cite{Zwi99} and two-component \cite{Mor03} plasmas, which are
clearly caused by the strongly coupled collective ion dynamics, since they
increase in strength with increasing $\Gamma_{\rm{i}}$ and
disappear for $\Gamma_{\rm{i}}(0)<0.5$ \cite{Zwi99}.
\begin{figure}[tb]
\centerline{\psfig{figure=f6a.eps,width=6.1cm} \hfill
\psfig{figure=f6b.eps,width=6.1cm}}
\caption{\label{f6}
Time evolution of the ion temperature determined from a central sphere
with a radius of half of the plasma width $\sigma$ (a) and time dependence of
the amplitude of the corresponding oscillations (b). The initial-state
parameters are the same as in fig.\ \ref{f5}.}
\end{figure}
\begin{figure}[t]
\centerline{\psfig{figure=f7.eps,width=8cm}}
\caption{\label{f7}
Temporally and spatially resolved time evolution of the ion temperature. The
initial-state parameters are the same as in fig.\ \ref{f5}.}
\end{figure}
Despite the fact that the maximum initial coupling constant used in
\cite{Zwi99} is $\Gamma_{\rm{i}}=5$, while a value of $\Gamma_{\rm{i}}(0)\approx40000$ is
considered in the case of fig.\ \ref{f5}, the
oscillations observed in \cite{Zwi99} are much more pronounced and persist much
longer than in the present case. It becomes apparent that the rapid damping of
the temperature oscillations can be traced to the inhomogeneity of the Gaussian
density profile by looking at the central part of the plasma only, where the
ionic density is approximately constant (figure \ref{f6}).
The temperature oscillations with an oscillation period
of half of the inverse plasma frequency $\nu_0=\omega_{{\rm{p,i}}}(0)/2\pi$ defined
in the central plasma region are much more pronounced in this case, showing
an exponential decay with a characteristic damping rate of $\nu_0$ (fig.\
\ref{f6}(b)). The temporally and spatially resolved temperature evolution shown
in fig.\ \ref{f7} shows that the radially decreasing ion density leads to
local temperature oscillations with radially increasing frequencies, thereby
causing also spatial oscillations of the local ion temperature. Therefore,
the seemingly enhanced damping rate, which has also been observed in recent
experiments, is purely an effect of the averaging of these local oscillations
over the total plasma volume.
\subsection{Coulomb crystallization through laser cooling}
The above considerations show that, while not dramatically affecting the
overall expansion behavior of the plasma cloud, strong-coupling effects
play an important role in different aspects of the evolution of UNPs. Thus,
UNPs provide a prime example of laboratory realizations of strongly nonideal
plasmas. Moreover, the HMD approach developed in \cite{PPR04} is well suited
for an accurate description of these systems over experimentally relevant
timescales, allowing for direct comparison between experiment and theory.
Many interesting aspects of the relaxation behavior of these non-equilibrium
plasmas may thus be studied in great detail. However, while effects of strong
ionic coupling become apparent in UNPs,
the naively expected
regime with $\Gamma > 100$ can not be reached with
the current experimental setups. For the electrons, it is predominantly
three-body recombination which heats them by several orders of magnitude, so
that $\Gamma_{\rm{e}} < 0.2$ during the whole system evolution. The ionic component,
on the other hand, is heated by the correlation-induced heating until
$\Gamma_{\rm{i}} \approx 1$, i.e.\ just at the border of the strongly coupled regime
\cite{Mur01,Sim04}. Thus, it is the very build-up of ionic correlations one
wishes to study that eventually shuts off the process and limits the amount
of coupling achievable in these systems.
\begin{figure}[bt]
\centerline{\psfig{figure=f8.eps,height=5cm}}
\caption{\label{f8}
Radial density and a central slice of a plasma with $N_{\rm{i}}(0)=80000$,
$\Gamma_{\rm{e}}(0)=0.08$, cooled with a damping rate of
$\beta=0.2\omega_{\rm{p,i}}(0)$ at a time of $\omega_{\rm{p,i}}(0)t=216$. (For
better contrast, different cuts have been overlayed.)}
\end{figure}
As soon as the reason for this ionic heating became clear, several proposals
have been made in order to avoid or at least reduce the effect, among them
({\em i}) using fermionic atoms cooled below the Fermi temperature in the
initial state, so that the Fermi hole around each atom prevents the occurrence
of small interatomic distances \cite{Mur01}; ({\em ii}) an intermediate step
of exciting atoms into high Rydberg states, so that the interatomic spacing is
at least twice the radius of the corresponding Rydberg state \cite{Ger03a};
and ({\em iii}) the continuous laser-cooling of the plasma ions after their
initial creation, so that the correlation heating is counterbalanced by the
external cooling \cite{Kil03,PPR04a}. We have simulated the latter scenario
using the HMD method, extended to allow for the description of laser cooling,
as well as elastic electron-ion collisions which are negligible for the free
plasma expansion but not necessarily in the laser-cooled case
\cite{PPR04a,PPR04c}. Laser cooling is modeled by adding a Langevin force
\begin{equation}
{\bf{F}}_{\rm cool} =-m_i\beta{\bf{v}}+\sqrt{2\beta k_{\rm{B}}T_c m_i}
{\bm{\xi}}
\end{equation}
to the ion equation of motion, where ${\bf{v}}$ is the ion
velocity, ${\bm{\xi}}$ is a stochastic variable with $\left<{\bm{\xi}}\right>
={\bf{0}}$, $\left<{\bm{\xi}}(t){\bm{\xi}}(t+\tau)\right>=3\delta(\tau)$,
and the cooling rate $\beta$ and the
corresponding Doppler temperature $T_c$ are determined by the
properties of the cooling laser \cite{Met99}. Elastic electron-ion collisions
are taken into account on the basis of the corresponding Boltzmann collision
integral, which is again evaluated by a Monte-Carlo procedure \cite{PPR04c}.
\begin{figure}[tb]
\centerline{\psfig{figure=f9a.eps,width=3.8cm} \hfill
\psfig{figure=f9b.eps,width=3.7cm} \hfill \psfig{figure=f9c.eps,width=3.7cm}}
\caption{\label{f9}
Arrangement of the ions on the first (a), third (b) and fifth (c)
shell of the plasma of fig.\ \ref{f8}.}
\end{figure}
It is found that laser cooling leads to qualitative changes of the plasma
dynamics. In particular, it significantly decelerates the expansion of the
plasma, whose width is found to increase only as $\sigma\propto t^{1/4}$, in
contrast to freely expanding plasmas which behave as $\sigma\propto t$. It is
this drastic slow-down of the expansion which favors the development of strong
ion correlations, compared to a free plasma where the expansion
considerably disturbs the relaxation of the system. The simulations show
further that
strongly coupled expanding plasmas can indeed be created under realistic
conditions, with ionic coupling constants far above the crystallization limit
for homogeneous plasmas of $\Gamma_{\rm{i}}\approx174$ \cite{Dub99}. Here we
find, depending on the initial conditions, i.e.\ ion number
and initial electronic Coulomb coupling parameter, strong
liquid-like short-range correlations or even the onset of a radial
crystallization of the ions. This is demonstrated in fig.\ \ref{f8}, showing the
radial density and a central slice of a plasma with $N_{\rm{i}}(0)=80000$,
$\Gamma_{\rm{e}}(0)=0.08$, cooled with a damping rate of
$\beta=0.2\omega_{\rm{p,i}}(0)$, at a scaled time of
$\omega_{\rm{p,i}}(0)t=216$. The formation of concentric shells in the center of the cloud is clearly visible. As
illustrated in fig.\ \ref{f9}, beside
the radial ordering there is also significant intra-shell ordering, namely a formation of hexagonal structures on the
shells, which are, however, considerably disturbed by the curvature of the
shells.
\section{Conclusions}
In summary, we have used an HMD approach to study the behavior
of ultracold neutral plasmas on long time scales. We have shown that effects of
strong interionic
coupling are indeed visible in such systems, e.g.\ most prominently in the
relaxation behavior of the ion temperature, which is connected
with transient temporal as well as spatial oscillations.
Nevertheless, the strongly coupled regime of $\Gamma > 100$ is not
reached with the current experimental setups. We have demonstrated, however,
that additional continuous laser cooling of the ions during the plasma
evolution qualitatively changes the expansion behavior of the system and
should allow for the Coulomb crystallization of the plasma
\cite{PPR04a,PPR04c}. It will be an interesting subject for further
investigation to study in detail the dynamics of this
crystallization process, which differs from the shell structure formation
observed in trapped nonneutral plasmas \cite{Dub99} as explained in
\cite{PPR04a}. In particular, the
influence of the plasma expansion, which presumably causes the transition from
liquid-like short-range correlation to the radial ordering, deserves more
detailed studies.
Other future directions include the study of effects
induced by additional magnetic fields, or of ways to confine the plasma in
a trap.
We gratefully acknowledge many helpful discussions with J.M.\ Rost, as well as
conversations with T.C.\ Killian and F.\ Robicheaux.
|
{
"timestamp": "2005-03-01T20:13:39",
"yymm": "0503",
"arxiv_id": "physics/0503018",
"language": "en",
"url": "https://arxiv.org/abs/physics/0503018"
}
|
\section{Introduction}\label{sec:int}
A simple generalization of a closed space curve is the notion of a ribbon.
An ideal narrow ribbon in three-dimensional space is specified by the position of one edge at each point along its length, together with the unit normal vector to the ribbon at each point of this edge.
If the ribbon is a closed loop (with two faces, not one as a M{\"o}bius band), then the two edges are non-intersecting closed curves in space which may wind around each other if the looped ribbon is twisted.
Such ideal twisted ribbon loops are important in applications, for instance modelling circular duplex DNA molecules (Fuller 1971, 1978; Pohl 1980; Bauer \textit{et al.} 1980; Hoffman \textit{et al.} 2003), magnetic field lines (Moffatt \& Ricca 1992), phase singularities (Winfree \& Strogatz 1983; Dennis 2004), rotating body frames (Hannay 1998; Starostin 2002), and various aspects of geometric phase theory (Chiao \& Wu 1986; Kimball \& Frisch 2004).
A fundamental result in the geometry of twisted, closed ribbon loops is C\u{a}lug\u{a}\-rea\-nu's theorem (C\u{a}lug\u{a}reanu 1959, 1961; Moffatt \& Ricca 1992) (also referred to as White's formula (White 1969; Pohl 1980; Kauffman 2001; Eggar 2000), and the C\u{a}lug\u{a}reanu-White-Fuller theorem (Adams 1994; Hoffman \textit{et al.} 2003), which is expressed as
\begin{equation}
Lk = Tw + Wr. \label{eq:cwf}
\end{equation}
$Lk$ is the topological linking number of the two edge curves; it is the classical Gauss linking number of topology (described, for example, by Epple (1998)).
The theorem states that this topological invariant is the sum of two other terms whose proper definitions will be given later and which individually depend on geometry rather than topology: the twist $Tw$ is a measure of how much the ribbon is twisted about its own axis, and the writhe $Wr$ is a measure of non-planarity (and non-sphericity) of the axis curve.
The formula actually has a very simple interpretation in terms of `views' of the ribbon from different projection directions (this is hinted at in the discussion of Kauffman (2001)).
Such interpretations of $Lk$ and $Wr$ go back to Fuller (1971, 1978) and Pohl (1968{\em a}, {\em b}), but do not seem to have been extended to $Tw.$
This is our first result.
Our second uses this picture to construct, for any curve, a particular ribbon on it which has zero $Lk.$
This is the writhe framing ribbon, anticipated algebraically by J. H. Maddocks (private communication, unpublished notes; also see Hoffman \textit{et al.} 2003).
The topological argument can be paraphrased very simply.
Consider the two edges of a ribbon loop, for example that in figure \ref{fig:loop}.
Viewing a particular projection as in the picture, there are a number of places where one edge crosses the other.
A positive direction around the ribbon is assigned arbitrarily, so that the two edges can be given arrows.
At each crossing between the two edge curves, a sign ($\pm$) can be defined according to the the sense of rotation of the two arrows at the crossing ($+1$ for right handed, $-1$ for left handed, as shown in figure \ref{fig:loop}).
Summing the signs ($\pm 1$) of the crossings gives twice the linking number $Lk$ of the ribbon edges (the sign of $Lk$ is positive for a ribbon with planar axis and a right-handed twist, and is negative for a planar ribbon with a left-handed twist).
Since the ribbon is two-sided, the total number of crossings must be even, ensuring that $Lk$ is an integer.
\begin{figure}
\begin{center}
\includegraphics*[width=8cm]{loops.eps}
\end{center}
\caption{A projection of a twisted ribbon exhibiting the types of crossing described in the text.
The two edges of the ribbon are represented in different colours, and their linking number $Lk$ is +1.
At the left and right sides of the figure, when the ribbon is edge-on, there are `local crossings' (a `right-handed' or positive crossing on the right, a `left-handed' negative crossing on the left).
Two more positive crossings occur in the middle (where the ribbon crosses over itself); these correspond to the two crossings of the ribbon edges with themselves (i.e. a nonlocal crossing).}
\label{fig:loop}
\end{figure}
The crossings between the two edge curves naturally fall into two types: `local,' which will be associated with $Tw,$ and `nonlocal,' which will be associated with $Wr.$
Local crossings are where the ribbon is edge-on to the viewing direction: one edge of the ribbon is crossing its own other edge.
If the ribbon is arbitrarily narrow, then the two edges are indefinitely close in space as they cross in the projection.
Nonlocal type crossings between the two edges are caused by the ribbon crossing over itself.
More precisely, they occur when a single edge curve of the ribbon crosses over itself.
This does not itself count towards the linking number, but such a crossing implies that there are two crossings with the other edge close by, as in the centre of figure \ref{fig:loop}.
These two counts, the signed local crossings and signed nonlocal crossings (which come in pairs as described) add up to twice $Lk$ by definition.
These quantities are integers, but depend on the choice of projection direction (although their sum does not).
It is well known (Fuller 1971, 1978; Adams 1994; Kauffman 2001) that the number of self-crossings of the ribbon axis curve, signed appropriately, and averaged democratically over the sphere of all projection directions, equals the writhe.
Therefore, the signed sum of nonlocal crossings between opposite edges is twice the writhe.
We claim that the local crossing counterpart (that is, the average over all projection directions of the signed local crossing number) is twice $Tw.$
The C\u{a}lug\u{a}reanu theorem then follows automatically.
This is described in the following section.
Section \ref{sec:formalism} is a review of standard material formalising the notions of linking and writhe; this is used in section \ref{sec:writhe} to construct the natural writhe framing of any closed curve.
The interpretation of twist and writhe in terms of local and nonlocal crossings gives insight into a common application of the theorem, namely `supercoiling' of elastic ribbons (such as DNA, or telephone cords; see, for example, Adams (1994), Pohl (1980), Bauer \textit{et al.} (1980) and Hoffman \textit{et al.} (2003)).
Repeated local crossings (i.e. a high twist) are energetically unfavourable, whereas nonlocal crossings (represented by writhe), where the ribbon repeatedly passes over itself (`supercoiling'), are preferred elastically.
\section{Local crossings, twist, and a proof of the C\u{a}lug\u{a}reanu theorem}\label{sec:proof}
In order to describe the argument more formally, it is necessary to introduce some notation.
We will represent the edges of the narrow ribbon by two closed curves, $\mathcal{A}$ and $\mathcal{B},$ whose points are $\mathbf{a}(s)$ and $\mathbf{b}(s)$ (and where no confusion will ensue, just $s$).
$s$ is an arbitrary parameterization (giving the sense of direction along the curves), and $\rd \bullet/\rd s$ is denoted $\dot{\bullet}.$
$\mathcal{A}$ will be referred to as the {\em axis curve} and will play the primary role of the two curves.
Its unit tangent $\mathbf{t}(s)$ is proportional to $\dot{\mathbf{a}}(s).$
The other curve $\mathcal{B}$ will be regarded as derived from $\mathcal{A}$ via a {\em framing} of $\mathcal{A},$ by associating at each point $\mathbf{a}(s)$ a unit vector $\mathbf{u}(s)$ perpendicular to the tangent ($\mathbf{t}\cdot\mathbf{u} = 0$).
Then we define
\begin{equation}
\mathbf{b}(s) = \mathbf{a}(s) + \varepsilon \mathbf{u}(s) \label{eq:bdef}
\end{equation}
for $\varepsilon$ arbitrarily small, ensuring that the ribbon nowhere intersects itself.
$Tw$ may now be defined formally, as the integral around the curve $\mathcal{A}$ of the rate of rotation of $\mathbf{u}$ about $\mathbf{t}:$
\begin{equation}
Tw = \frac{1}{2\pi} \int_{\mathcal{A}} \rd s \, (\mathbf{t} \times \mathbf{u}) \cdot \dot{\mathbf{u}}. \label{eq:twdef}
\end{equation}
(We adopt the convention of writing the integrand at the end of an integration throughout.)
$Tw$ is local in the sense that it is an integral of quantities defined only by $s$ on the curve, and clearly depends on the choice of framing (ribbon).
A simple example of a framing is the {\em Frenet framing}, where $\mathbf{u}(s)$ is the direction of the normal vector $\dot{\mathbf{t}}(s)$ to the curve (where defined).
This framing plays no special role in the following.
As described in the introduction, the crossings in any projection direction can be determined to be either nonlocal (associated with writhe) or local.
Choosing an observation direction $\mathbf{o},$ there is a local crossing at $\mathbf{a}(s)$ (the ribbon appears edge-on), when $\mathbf{o}$ is linearly dependent on $\mathbf{t}(s)$ and $\mathbf{u}(s),$ i.e. there exists some $\theta$ between 0 and $\pi$ such that
\begin{equation}
\mathbf{o} = \mathbf{t} \cos \theta + \mathbf{u} \sin \theta \qquad \hbox{at a local crossing.}
\label{eq:local}
\end{equation}
We therefore define the vector
\begin{equation}
\mathbf{v}(s, \theta) = \mathbf{t}(s) \cos \theta + \mathbf{u}(s) \sin \theta
\label{eq:vdef}
\end{equation}
dependent on parameters $s,$ labelling a point of $\mathcal{A},$ and angle $\theta$ with $0 \le \theta \le \pi.$
We claim that twice $Tw$ is the average, over all projection directions $\mathbf{o},$ of the local crossing number.
For each $\mathbf{o},$ the local crossing number is defined as the number of coincidences of $\mathbf{v}$ with $\mathbf{o}$ or $-\mathbf{o}.$
$Tw$ itself (rather than its double) can therefore be determined by just counting coincidences of $\mathbf{v}$ with $\mathbf{o}$ (not $-\mathbf{o}$).
The sign of the crossing is determined by which way the tangent plane of the
ribbon at $s$ sweeps across $\mathbf{o}$ as $s$ passes through the edge-on position (which side of the ribbon is visible before the crossing and which after).
Thus the vector ($\partial_{\theta} \mathbf{v} \times \partial_s \mathbf{v}$) is either parallel or antiparallel to $\mathbf{v},$ and this decides the crossing sign: i.e. the sign of $(\partial_{\theta} \mathbf{v} \times \partial_s \mathbf{v})\cdot \mathbf{v}$ is minus the sign of the crossing.
This sign is opposite to the usual crossing number (described, for instance, in the next section), because $\theta$ increases in the opposite direction to $s$ along the ribbon.
The average over all projection directions $\mathbf{o}$ can be replaced by an integral over $s$ and $\theta,$ since the only projection directions which count are those for which there exist $s, \theta$ such that $\mathbf{o} = \mathbf{v}(s,\theta).$
This transformation of variables gives rise to a jacobian factor of $|(\partial_{\theta} \mathbf{v} \times \partial_s \mathbf{v})\cdot\mathbf{v}|$ (the modulus of the quantity whose sign gives the crossing sign).
Our claim (justified in the following), that $Tw$ is the spherical average of crossing numbers, is therefore
\begin{equation}
Tw = \frac{1}{4\pi} \int_{\mathcal{A}} \rd s \int_0^{\pi} \rd \theta \, (\partial_{\theta} \mathbf{v} \times \partial_s \mathbf{v})\cdot\mathbf{v}.
\label{eq:twdef1}
\end{equation}
The two expressions for twist, equations (\ref{eq:twdef}) and (\ref{eq:twdef1}), are equal; this can be seen by integrating $\theta$ in equation (\ref{eq:twdef1}),
\begin{multline}
\frac{1}{4\pi} \int_0^{\pi} \rd \theta \, (\partial_{\theta} \mathbf{v} \times \partial_s \mathbf{v})\cdot\mathbf{v} \\
\begin{aligned}
&= \frac{1}{4\pi} \int_0^{\pi} \rd \theta \,((\mathbf{t} \cos \theta + \mathbf{u} \sin \theta) \times (-\mathbf{t} \sin \theta + \mathbf{u} \cos \theta) ) \cdot (\dot{\mathbf{t}} \cos \theta + \dot{\mathbf{u}} \sin \theta) \\
&= \frac{1}{4\pi} \int_0^{\pi} \rd \theta \, (\cos^2 \theta + \sin^2 \theta) (\mathbf{t} \times \mathbf{u})\cdot (\dot{\mathbf{t}} \cos \theta + \dot{\mathbf{u}} \sin \theta) \\
&= \frac{1}{2\pi} (\mathbf{t} \times \mathbf{u})\cdot\dot{\mathbf{u}},
\end{aligned}
\label{eq:twder}
\end{multline}
which is the integrand of equation (\ref{eq:twdef}).
Thus the two expressions for twist, the conventional one (equation (\ref{eq:twdef})) and the local crossing count averaged over viewing directions (equation (\ref{eq:twdef1})), are the same.
The preceding analysis of $Tw$ is reminiscent of the more abstract analysis of Pohl (1980), albeit with a different interpretation.
Using the notion of direction-averaged crossing numbers, the C\u{a}lug\u{a}reanu theorem follows immediately, as we now explain.
For a sufficiently narrow ribbon, it is straightforward to decompose the crossings between different edges into local and nonlocal for each projection of the ribbon.
It is well known (Fuller (1971, 1978), Adams (1994)) that writhe $Wr$ equals the sum of signed nonlocal crossings, averaged over direction, i.e. twice the average of self-crossings of the axis curve with itself (a proof is provided in the next section).
The average of local crossings (between different edges, counting with respect to both $\mathbf{o}$ and $-\mathbf{o}$), has been shown to equal twice the twist $Tw$ defined in equation (\ref{eq:twdef}).
The sum of local plus nonlocal crossings is independent of the choice of projection direction, and is twice the linking number $Lk$ of the two curves.
So, averaging the crossings over the direction sphere, $Lk = Tw + Wr.$
Any ambiguity as to whether a crossing is local or nonlocal arises only when the projection direction $\bf{o}$ coincides with the (positive or negative) tangent $\pm \mathbf{t}.$
However, the set of such projection directions is only one-dimensional (parameterised by $s$), and so does not contribute (has zero measure) to the total two-dimensional average over the direction sphere.
\section{Formalism for writhe and linking number}\label{sec:formalism}
We include the present section, which reviews known material (e.g. Fuller 1971, 1978; Adams 1994; Kauffman 2001; Hannay 1998), to provide some formal geometrical tools for the next section, as well as providing further insight into the proof from the previous section.
The linking number $Lk$ between the curves $\mathcal{A}$ and $\mathcal{B}$ is related to the system of (normalised) {\em cross chords}
\begin{equation}
\mathbf{c}_{\mathcal{A}\mathcal{B}}(s,s') = \frac{\mathbf{a}(s) - \mathbf{b}(s')}{| \mathbf{a}(s) - \mathbf{b}(s') |}.
\label{eq:cabdef}
\end{equation}
$Lk$ is represented mathematically using Gauss's formula:
\begin{eqnarray}
Lk & =& \frac{1}{4\pi} \int_{\mathcal{A}} \rd s \int_{\mathcal{B}} \rd s'
\frac{( \dot{\mathbf{a}}(s) \times \dot{\mathbf{b}}(s') )\cdot(\mathbf{a}(s) - \mathbf{b}(s'))}{| \mathbf{a}(s) - \mathbf{b}(s') |^3} \nonumber \\
& =& \frac{1}{4\pi} \int_{\mathcal{A}} \rd s \int_{\mathcal{B}} \rd s' \, (\partial_s \mathbf{c}_{\mathcal{A}\mathcal{B}} \times \partial_{s'} \mathbf{c}_{\mathcal{A}\mathcal{B}})\cdot \mathbf{c}_{\mathcal{A}\mathcal{B}}. \label{eq:lk}
\end{eqnarray}
This bears some similarity to the formula for twist in equation (\ref{eq:twdef}).
$Lk$ is invariant with respect to reparameterization of $s$ and $s',$ and, of course, any topological deformation avoiding intersections.
The domain of integration in equation (\ref{eq:lk}), $\mathcal{A} \times \mathcal{B},$ is the cross chord manifold (secant manifold) of pairs of points on the two curves, topologically equivalent to the torus.
The mapping $\mathbf{c}_{\mathcal{A}\mathcal{B}}(s,s')$ takes this torus smoothly to the sphere of directions, with the torus `wrapping around' the sphere an integer number of times (the integer arises since the cross chord manifold has no boundary, and the mapping is smooth); the wrapping is a two-dimensional generalization of the familiar `winding number' of a circle around a circle.
This wrapping integer, the integral in equation (\ref{eq:lk}), is the linking number of the two curves (it is the degree of the mapping; see, for example, Madsen \& Tornhave (1997), Epple (1998)).
It is easy to see that this interpretation of $Lk$ agrees with that defined earlier in terms of crossings.
Choosing the observation direction $\mathbf{o},$ the crossings in the projection are precisely those chords $\mathbf{c}_{\mathcal{AB}}(s,s')$ coinciding with $\mathbf{o}$ where the sign of the scalar triple product $(\partial_s \mathbf{c}_{\mathcal{A}\mathcal{B}} \times \partial_{s'} \mathbf{c}_{\mathcal{A}\mathcal{B}})\cdot \mathbf{c}_{\mathcal{A}\mathcal{B}}$ gives the sign of the crossing.
As with $Tw,$ a crossing is only counted in the integral when $\mathbf{c}_{\mathcal{AB}}(s,s')$ is parallel to $\mathbf{o}$ (not antiparallel).
Since the cross chord manifold is closed (has no boundary), the total sum of signed crossings does not depend on the choice of $\mathbf{o}.$
\begin{figure}
\begin{center}
\includegraphics*[width=11cm]{writhes.eps}
\end{center}
\caption{Illustrating the writhe mesh construction.
a) The curve on the torus $(-0.3 \cos(2s), \cos(s)(1+0.3 \sin(2s)), \sin(s)(1+0.3 \sin(2s)).$
The points on the curve are represented by different colours on the colour wheel.
b) The writhe mesh for the curve in a).
The black lines are the tangent curves $\pm \mathbf{t}(s),$ and the coloured lines are the chord fans $\mathcal{C}_s,$ whose colours correspond to the points $s$ on the curve in a).
Note that the total writhe in this example is less than $4\pi$ (i.e. the writhe mesh does not completely cover the direction sphere).}
\label{fig:writhe}
\end{figure}
Writhe has a similar interpretation to link.
We define the chords between points of the same curve $\mathcal{A},$
\begin{equation}
\mathbf{c}_{\mathcal{A}}(s,s') = \frac{\mathbf{a}(s) - \mathbf{a}(s')}{| \mathbf{a}(s) - \mathbf{a}(s') |},
\label{eq:cadef}
\end{equation}
noting that, as $s'\to s$ from above, $\mathbf{c}_{\mathcal{A}} \to \mathbf{t}(s),$ and as $s'\to s$ from below, $\mathbf{c}_{\mathcal{A}} \to -\mathbf{t}(s).$
The writhe is the total area on the direction sphere traversed by the vector as $s$ and $s'$ are varied; this two-dimensional surface embedded on the sphere will be referred to as the {\em writhe mesh}:
\begin{eqnarray}
Wr &=& \frac{1}{4\pi} \int_{\mathcal{A}} \rd s \int_{\mathcal{A}} \rd s'
\frac{( \dot{\mathbf{a}}(s) \times \dot{\mathbf{a}}(s') )\cdot(\mathbf{a}(s) - \mathbf{a}(s'))}{| \mathbf{a}(s) - \mathbf{a}(s') |^3} \nonumber \\
&=& \frac{1}{4\pi} \int_{\mathcal{A}} \rd s \int_{\mathcal{A}} \rd s' \, (\partial_s \mathbf{c}_{\mathcal{A}} \times \partial_{s'} \mathbf{c}_{\mathcal{A}})\cdot \mathbf{c}_{\mathcal{A}}. \label{eq:wr}
\end{eqnarray}
The interpretation of crossings applies to the integral for writhe as well as link and twist; however, for writhe, the visual crossings are uniquely associated with a chord (the chord from $s$ to $s',$ and its reverse, each have separate associated viewing directions).
Thus, for writhe, each crossing is counted exactly once, and the average over projections is not divided by 2 (unlike link and twist).
(However, as shown in figure \ref{fig:loop}, at every nonlocal crossing of curve $\mathcal{A}$ with itself, there are two crossings (of the same sign) with the other curve.)
Unlike the cross chord manifold $\mathbf{c}_\mathcal{AB},$ the writhe mesh $\mathbf{c}_{\mathcal{A}}$ has a boundary: it is topologically equivalent to an annulus (i.e. a disk with a hole).
The two boundary circles map to the tangent indicatrix curves (i.e. the loops $\pm \mathbf{t}(s)$ on the direction sphere, as $s$ varies).
For each $s,$ the following locus on the sphere, referred to as the {\em chord fan}
\begin{equation}
\mathcal{C}_s = \mathbf{c}_{\mathcal{A}}(s,s') \quad \hbox{(varying $s'$ from $s$ round to $s$ again)},
\label{eq:csetdef}
\end{equation}
follows the directions of all the chords to the point $s,$ starting in the positive tangent direction $+\mathbf{t}(s),$ and ending at its antipodal point $-\mathbf{t}(s).$
Topologically, it is a `radial' line joining the two edges of the writhe mesh annulus, although on the direction sphere it may have self-intersections.
In addition to the partial covering bounded by the tangent indicatrix curves $\pm\mathbf{t}(s),$ the total writhe mesh may cover the direction sphere an integer number of times.
The signed number of crossings, as a function of viewing direction $\mathbf{o},$ changes (by $\pm2$) when $\mathbf{o}$ crosses $\pm \mathbf{t}(s).$
The writhe integral (equation (\ref{eq:wr})) is its average value over all viewing directions.
The writhe mesh construction is illustrated in figure \ref{fig:writhe}.
The approach of section \ref{sec:proof} may also naturally be interpreted topologically on the direction sphere.
The vector $\mathbf{v}(s,\theta),$ dependent on two parameters, also defines a mesh on the direction sphere, the {\em twist mesh}.
For fixed $s,$ the locus of points on the twist mesh is a semicircle,
\begin{equation}
\mathcal{S}_s = \mathbf{v}(s,\theta) \qquad \hbox{($0 \le \theta \le \pi$)}
\label{eq:ssetdef}
\end{equation}
whose endpoints are at $\pm\mathbf{t}(s)$ and whose midpoint is the framing vector $\mathbf{u}(s).$
As $s$ varies around the curve, the semicircle $\mathcal{S}_s$ sweeps out the solid angle $Tw.$
Like the writhe mesh, the twist mesh is topologically an annulus, with $\mathcal{S}_s$ the `radial' line labelled by $s;$ its boundary is again the set of tangent directions $\pm \mathbf{t}(s).$
Therefore, a topological visualization of $Tw + Wr$ is the union of the meshes for twist and writhe; each mesh has the same boundary, so the closure of the union is the join of two annuli along their boundary, topologically a torus.
This torus therefore wraps around the sphere the same number of times (has the same degree) as the cross chord manifold $\mathbf{c}_{\mathcal{AB}}.$
It is therefore possible to interpret the loop $\mathcal{L}_s$ on the manifold of cross chords $\mathcal{A}\times\mathcal{B},$ labelled by $s$ on $\mathcal{A}$ (varying $s'$ on $\mathcal{B}$), in terms of the twist semicircle $\mathcal{S}_s$ and the chord fan $\mathcal{C}_s.$
The loop $\mathcal{L}_s,$ mapped to the direction sphere, is the set of cross chords directions between fixed $\mathbf{a}(s)$ and all $\mathbf{b}(s')$ on $\mathcal{B}.$
When the ribbon is vanishingly thin, for $s'$ outside the neighbourhood of $s,$ the chords $\mathbf{c}_{\mathcal{AB}}(s,s')$ can be approximated by the chords of $\mathbf{c}_{\mathcal{A}}(s,s').$
When $s'$ is in the neighbourhood of $s,$ $\mathbf{c}_{\mathcal{AB}}(s,s')$ is approximated by $\mathbf{v}(s,\theta)$ for some $\theta$ (exact at $\theta = \pi/2$ when $s' = s$).
As $\varepsilon \to 0,$ these approximations improve, and $\mathcal{L}_s$ approaches the union of $\mathcal{C}_s$ and $\mathcal{S}_s.$
Link is therefore the area on the direction sphere swept out by the loop union of $\mathcal{C}_s$ and $\mathcal{S}_s$ for $s$ varying around the curve; since this family of loops generates the closed torus, the direction sphere is enveloped an integer number of times.
Thus, C\u{a}lug\u{a}reanu's theorem may be interpreted as a natural decomposition of the integrand in Gauss's formula (\ref{eq:lk}) in terms of the writhe mesh and twist mesh.
This is very close, in a different language, to White's proof (White 1969, also see Pohl 1968{\em a}, and particularly Pohl 1980), where objects analogous to the writhe mesh and the twist mesh appears as boundaries to a suitably regularised (blown-up) 3-manifold of chords from $\mathcal{A}$ to points on surface of the ribbon with boundary $\mathcal{A}, \mathcal{B}.$
\section{The writhe framing}\label{sec:writhe}
In this section, we use the description of $Tw$ and $Wr$ to define, for any non-self-intersecting closed curve in space, a natural framing (i.e. ribbon) whose linking number is zero.
Such a framing is useful since it can be used as a reference to determine the linking number for any other framing: if $\mathbf{u}_0(s)$ represents this zero framing, and $\mathbf{u}(s)$ any other framing with linking number $Lk,$ then
\begin{equation}
Lk = \frac{1}{2\pi} \int_{\mathcal{A}} \rd s\, \arccos \mathbf{u}\cdot\mathbf{u}_0.
\label{eq:zerolinkref}
\end{equation}
Clearly the twist of a zero framing is equal to minus the writhe by C\u{a}lug\u{a}reanu's theorem, and therefore we will refer to our natural framing as the {\em writhe framing}.
Expressing $Lk$ in terms of an integral involving the difference between two framings is reminiscent of C\u{a}lug\u{a}reanu's original proof (C\u{a}lug\u{a}reanu 1959, 1961; also see Moffatt \& Ricca 1992), in which the twist is defined as the total number of turns the framing vector makes with respect to the Frenet framing around the curve (provided the normal to the curve is everywhere defined).
The total twist is therefore the sum of this with the integrated torsion around the curve (i.e. $Tw$ of the Frenet framing).
As stated before, the Frenet framing plays no role in our construction.
It is easy to construct an unnatural zero framing by taking any framing, cutting anywhere, and rejoining with a compensating number of twists locally.
In contrast, the framing we construct here is natural in the canonical sense that, at any point $s,$ the definition of $\mathbf{u}_0(s)$ and its corresponding semicircle $\mathcal{S}_s$ (defined in equation (\ref{eq:ssetdef})) depends only on the view from $s$ of the rest of the closed curve, that is on the chord fan $\mathcal{C}_s$ (defined in equation (\ref{eq:csetdef})).
The rate of rotation of the resulting $\mathbf{u}_0$ exactly compensates the corresponding writhe integrand, that is, the integrands (with respect to $s$) of equation (\ref{eq:wr}) and the twist of the writhe framing (equation (\ref{eq:twdef})) are equal and opposite.
\begin{figure}
\begin{center}
\includegraphics*[width=12.15cm]{framings.eps}
\end{center}
\caption{Illustrating the writhe framing construction, using the same example curve as figure \ref{fig:writhe}.
a) The chord fan for $s = 0,$ is represented in red, the semicicle which bisects its area represented in pink.
The tangent curves are also represented.
b) The same as part a), and in addition the chord fan with $s = -0.4$ is shown in dark blue, and its bisecting semicircle in light blue.
No area is swept out by these closed curves as $s$ evolves.}
\label{fig:framing}
\end{figure}
This may be interpreted topologically as follows.
$\mathcal{C}_s$ and $\mathcal{S}_s$ correspond topologically to `radial' lines of their respective annular meshes; their union is a closed loop on the direction sphere.
The sum of the twist and writhe integrands, by the discussion in section \ref{sec:proof}, is the area swept out by this changing closed loop as $s$ develops; it was proved that the total area swept out by this loop is $4\pi Lk.$
The writhe framing construction of $\mathbf{u}_0$ below arranges that the loop has a constant area (say zero), and therefore the rate of area swept by this loop is zero as $s$ evolves, giving zero total area swept (which is $4\pi Lk$).
The direction of $\mathbf{u}_0(s)$ for the writhe framing is that for which the semicircle $\mathcal{S}_s$ bisects the chord fan $\mathcal{C}_s$ in the following sense.
Since $\mathcal{C}_s$ is a curve on the direction sphere with endpoints $\pm\mathbf{t}(s),$ it may be closed by an arbitrary semicircle with the same endpoints (i.e. any rotation of $\mathcal{S}_s$ about $\pm\mathbf{t}(s)$).
The total closed curve encloses some area on the sphere (mod 4$\pi$); the (unique) semicircle which gives zero area (mod $4\pi$) defines $\mathbf{u}_0.$
Since the spherical area enclosed by this closed curve is contant, the curve does not sweep out any area as it evolves (since as much leaves as enters).
An example of the writhe framing is represented in figure \ref{fig:framing}.
The rate of area swept out by this curve is the integrand with respect to $s$ in the $Lk$ expression (equation (\ref{eq:lk})), (i.e. the sum of the $Tw, Wr$ integrands).
Since it has been shown that this is zero, the writhe framing $\mathbf{u}_0$ indeed has zero linking number.
Of course, although the choice of zero area is most natural, the writhe faming vector $\mathbf{u}_0$ could be defined such that the area between the chord fan $\mathcal{C}_s$ and the twist semicircle $\mathcal{S}_s$ is any fixed value - the important feature in the construction is that the area does not change with $s.$
\begin{acknowledgements}
We are grateful to John Maddocks for originally pointing out to us the problem of the writhe framing, and useful correspondence. MRD acknowledges support from the Leverhulme Trust and the Royal Society of London.
\end{acknowledgements}
|
{
"timestamp": "2005-06-10T11:48:58",
"yymm": "0503",
"arxiv_id": "math-ph/0503012",
"language": "en",
"url": "https://arxiv.org/abs/math-ph/0503012"
}
|
\section{Introduction}
The Weizs\"acker-Bethe mass formula has played an important role
for many years.
It offers a guideline for developing modern mass formulas.
Indeed, modern mass formulas, such as the droplet mass formula (FRDM)
\cite{Moller}
(excluding the Hartree-Fock approaches \cite{Aboussir,Goriely})
largely retain the original form.
They include the pairing and symmetry energy terms, in addition to
the volume, surface and Coulomb energy terms. While the pairing energy
has been investigated by combining with the microscopic pairing correlations,
the symmetry energy has not necessarily been investigated from the point
of view of the shell model.
A recent detailed work \cite{Duflo} succeeded in obtaining a precise
mass formula. That work is based on the rigorous microscopic guidelines
given in Ref. 5), which considers the monopole field and pairing
structure providing the dominant terms of the mass formula.
The study presented in Ref. 5) considers general properties of
the shell model Hamiltonian. However, while the pairing interaction
as the origin of the pairing energy determines the structure of wave functions,
the symmetry energy is not treated symmetrically with the pairing energy.
In the present paradigm for the nuclear mass formula, the symmetry energy
is regarded as a basic concept. In the shell model, however,
there is no approach other than describing the symmetry energy in terms of
nuclear correlations. The asymmetrical treatment of the symmetry energy
and pairing energy leaves a missing link in relating mass formulas
to nuclear structure.
This paper proposes an alternative approach, in which the ``symmetry energy"
is not treated as a fundamental concept and explains the symmetry energy
in terms of certain correlations, as the pairing energy is determined
in terms of the pair correlations. We wish to understand the symmetry energy
derived from the mean field theory from the point of view of the shell model.
Our treatment begins from the $jj$ coupling shell model
based on a $Z=N$ doubly-closed shell core, and we do not discuss Strutinski's
prescription \cite{Strut}.
The purpose of this paper is not to give a new mass formula better than
modern sophisticated mass formulas, but to present a useful understanding
of the mass formula. We therefore start from an old fashioned mass formula
in order to clearly show the basic idea.
In the $jj$ coupling shell model with an effective interaction,
the energy depending on the total isospin $T$ comes from the interactions
between valence nucleons in $j$ orbits. The corresponding correlations
are not yet reduced to a mean field but determine wave functions
or structure of nuclei, in the shell model description.
We consider such correlations in even-even $N=Z$ ($A_0+m\alpha$) nuclei
that give no contribution of the symmetry energy.
Here, $A_0$ represents a doubly-closed-shell core, $\alpha$ is
a two-neutron-two-proton ($2n-2p$) quartet with $T=0$, and $m$ is an integer.
We show that the interaction energies of the $A_0+m\alpha$
nuclei characterize the binding energies of $N \approx Z$ nuclei,
excluding the bulk energy depending on mass $A$.
The strength of the correlations can be evaluated in terms of the difference
between the mass of an $A_0+m\alpha +2n$ ($A_0+m\alpha +2p$) nucleus
and the average mass
of its neighboring nuclei with $A=A_0+m\alpha$ and $A=A_0+(m+1)\alpha$.
With this indicator, Gambhir, Ring and Schuck \cite{Gamb} studied
a superfluid state of many $\alpha$ {\it particles}.
The term $\alpha$, however, represents only a $T=0$ $2n-2p$ quartet,
not the spatial $\alpha$ cluster. We call the correlations
``$T=0$ $2n-2p$ correlations" in the sense of many-body correlations.
(We use ``$\alpha$-like" as a concise term for the superfluid state.)
This paper shows that the symmetry energy is derived from the nonparticipation
of redundant nucleons in the $T=0$ $2n-2p$ correlations,
in parallel with the pairing energy derived from the nonparticipation
of an odd nucleon in the $T=1$ pair correlations.
The energy of valence nucleons is separated from the binding energy,
and the leading role of $T=0$ $2n-2p$ correlations is discussed in $\S$2.
Section 3 discusses the fundamental $T=0$ $2n-2p$ correlated structure
in $N \approx Z$ nuclei, (which is called ``$\alpha$-like superfluidity").
Section 4 explains the mass differences between even-even nuclei
in terms of multi-pair structure on the base of $\alpha$-like superfluidity.
In $\S$5, we discuss how the pairing energy should be evaluated.
Section 6 gives concluding remarks.
\section{Correlations of valence nucleons buried in the binding energy}
\subsection{Extraction of the energy of valence nucleons}
In the old fashioned mass formula, the bulk of the binding energy is
written in terms of the volume, surface and Coulomb energies as
\begin{equation}
B_{VSC}(A) = - a_V A + a_S A^{2/3} + a_C Z^2 / A^{1/3}. \label{eq:1}
\end{equation}
We can consider that these main terms basically represent a nuclear
potential in the shell model picture, while the other terms of the mass
formula are related to the shell model interactions.
It must be stressed that the symmetry energy depending on the total
isospin $T$ is attributed to the shell model interactions in this picture.
Let us estimate the interaction energy by subtracting $B_{VSC}(A)$
from the experimental binding energy $B(A)$ \cite{Audi} for $A_0+m\alpha$
nuclei with $T=0$. (Note that the sign of the binding energy $B(A)$ is
negative in this paper.)
The values $B(A)-B_{VSC}(A)$ calculated with a few mass formulas with simple
forms \cite{Duflo,Yagi,Ring,Samanta} are listed in Table \ref{table1}.
[In the third line, we used the six-parameter mass formula in Ref. 4).
The Coulomb energy term in Refs. 4) and 11) is expressed in terms of
different functions of the proton number $Z$.]
In Table \ref{table1}, we tabulate the values $B(A)-B_{VSC}(A)$
for seven $A=A_0+m\alpha$
nuclei with $T=0$, where the symmetry energy makes no contribution.
These values display variation depending on $A$.
\begin{table}[b]
\caption{The values of $B(A)-B_{VSC}(A)$ for $A_0+m\alpha$ nuclei with $T=0$,
calculated using a few mass formulas.}
\begin{center}
\begin{tabular}{c|rrrrrrr} \hline
ref. & $^{16}$O & $^{20}$Ne & $^{28}$Si & $^{40}$Ca & $^{44}$Ti & $^{56}$Ni
& $^{64}$Ge \\ \hline
\cite{Yagi} & $-5.81$ & $-2.13$ & $-4.82$ & $-2.88$ & $-1.35$ & $-7.99$ & $-4.63$ \\
\cite{Ring} & $-12.80$ & $-10.21$ & $-14.92$ & $-15.79$ & $-15.17$ & $-24.50$ & $-22.94$ \\
\cite{Duflo} & $-4.85$ & $-0.87$ & $-2.97$ & $-0.15$ & $ 1.67$ & $-4.10$ & $-0.17$ \\
\cite{Samanta} & $-7.21$ & $-3.52$ & $-6.08$ & $-3.85$ & $-2.21$ & $-8.49$ & $-4.91$ \\
\hline
\end{tabular}
\end{center}
\label{table1}
\end{table}
Table \ref{table1} shows that the mass formula \cite{Ring} fitted
for heavy nuclei is not good for $N \approx Z$ nuclei but that the other three
display parallel and interesting behavior (dips at $^{28}$Si and
$^{56}$Ni) as $A$ increases.
According to ordinary mass formulas, there remains the pairing term $\delta_P$,
which contributes to the $T=0$ even-even nuclei under consideration.
However, the deviations of the experimental binding energies from $B_{VSC}$
shown in Table \ref{table1} are much larger than the pairing effect.
The characteristic behavior of $B(A_0+m\alpha)-B_{VSC}(A_0+m\alpha)$
cannot be explained as a simple variation of $\delta_P$ depending on $A$,
like $\delta_P \propto A^p$. The existing mass formulas do not describe
the behavior. The characteristic behavior of
$B(A_0+m\alpha)-B_{VSC}(A_0+m\alpha)$
must reflect correlations stronger than the pairing correlations
from the point of view of the shell model.
This is worth investigating further.
Let us start from the old simple mass formula \cite{Yagi}
with the parameters $a_V=15.56$, $a_S=17.23$ and $a_C=0.6986$ in MeV.
It is noticed in Table \ref{table1} that the values
in the first line \cite{Yagi} resemble those in the fourth line obtained
with the mass formula \cite{Samanta}, and the mass formula \cite{Duflo}
has an elaborate form, so that the deviations
$B(A_0+m\alpha)-B_{VSC}(A_0+m\alpha)$ may be small.
The values in Table \ref{table1} indicate the insufficiency of $B_{VSC}$
for the doubly-closed-shell nuclei $^{16}$O and $^{40}$Ca, which are
the bases for the shell model calculations. We suppose that the deviations
in $^{16}$O and $^{40}$Ca require adjustments of the depth of the shell model
potential. The adjustment parameter $\delta_P$ for even-even nuclei has
the form $A^{-3/4}$ in the old convention. If we adopt the $A^{-3/4}$
adjustment for the potential depth, we can fix its parameter
so as to make the deviations $B(A_0+m\alpha)-B_{VSC}(A_0+m\alpha)$
nearly zero for $^{16}$O and $^{40}$Ca.
We assume that the main part of the mass formula corresponding to
the shell model potential is approximated by
\begin{eqnarray}
& {} & B_0(A) = B_{VSC}(A) + \delta U_{pot}(A), \\ \label{eq:2}
& {} & \delta U_{pot}(A) = - 46.4 / A^{3/4}. \label{eq:3}
\end{eqnarray}
The deviation $B(A)-B_0(A)$ could be regarded as the energy of valence
nucleons outside a doubly-closed-shell core,
\begin{equation}
E(A) = B(A) - B_0(A). \label{eq:4}
\end{equation}
The energies $E(A_0+m\alpha)$ for the even-even $N=Z$ nuclei with $T=0$
are plotted in Fig. \ref{fig1}, which displays the characteristic behavior
of the binding energies $B(A_0+m\alpha)$ mentioned above.
In the shell model calculation, the experimental energy of correlated
valence nucleons outside a doubly-closed-shell core $A_0=(N_0,Z_0)$ is
evaluated using
\begin{equation}
E_{shl}(N,Z) = B(N,Z) - B(N_0,Z_0)
- \lambda (A-A_0) - \Delta E_C(N,Z), \label{eq:5}
\end{equation}
where $\Delta E_C(N,Z)$ is a Coulomb energy correction for the valence
nucleons. For instance, the correction
$\Delta E_C(N,Z) = p(Z-Z_0)+q(Z-Z_0)(Z-Z_0-1)+r(Z-Z_0)(N-N_0)$
was used by Caurier et al. \cite{Caurier} in the shell model calculations
for $f_{7/2}$ shell nuclei. We calculated the energy $E_{shl}(N,Z)$
using the parameter values $p=7.279$, $q=0.15$, $r=-0.065$ and $\lambda =-12.45$
(in MeV) for the $pf$ shell nuclei and $p=3.54$, $q=0.20$, $r=0.0$ and
$\lambda =-11.2$ (in MeV) for the $sd$ shell nuclei. The calculated values
are indicated by the dotted curves in Fig. \ref{fig1}. We can see that
$E(A_0+m\alpha) \approx E_{shl}(A_0+m\alpha)$ in the first half of the shells,
which supports our assumption that $E(A)$ in Eq. (\ref{eq:4}) represents
the energy of valence nucleons.
The disagreement between $E(A_0+m\alpha)$ and $E_{shl}(A_0+m\alpha)$ is large
in the latter half of the shells. (Note that the disagreement is smaller
in the heavier $pf$ shell nuclei.) The hole picture
would be better for the latter half of the shells.
\begin{figure}[t]
\begin{center}
\includegraphics[width=7cm,height=7cm]{fig1.eps}
\caption{The energies of $E(A_0+m\alpha)=B(A_0+m\alpha)-B_0(A_0+m\alpha)$
and the curve connecting them, which defines the $T=0$ base level.}
\label{fig1}
\end{center}
\end{figure}
\subsection{The leading role of the $T=0$ $2n-2p$ correlations}
From the above, we find that the energy $E(A)$ defined by Eq. (\ref{eq:4})
approximately represents the total energy of valence nucleons outside
the doubly-closed-shell core $^{16}$O or $^{40}$Ca. To obtain a guide
for the discussion given in the following sections, let us consider
the main features of $E(A)$ near $^{40}$Ca,
where $E(A)$ corresponds well with the shell model
energy $E_{shl}(A)$. Figure \ref{fig2} depicts the ground state energies
$E(A)$ of $^{40}$Ca, an $A=41$ system with $T=1/2$, an $A=42$ system with $T=1$,
$^{44}$Ti and $^{48}$Cr.
(This figure is similar to the diagram for the pairing vibrations.)
The energy $E(A)$ in Eq. (\ref{eq:4}) does not
give exactly the same energy to $^{41}$Ca and $^{41}$Sc. A correction term
representing the Coulomb energy is necessary in the final stage.
In Fig. \ref{fig2},
we show the average energy of $E(^{41}{\rm Ca})$ and $E(^{41}{\rm Sc})$
for the $A=41$ system with $T=1/2$. Similarly, we show the average energy
of $E(^{42}{\rm Ca})$ and $E(^{42}{\rm Ti})$
[which is approximately equal to $E(^{42}{\rm Sc})$]
for the $A=42$ system with $T=1$.
The energy $E(A=41)$ represents
an effective single-particle energy $e_{sp}$ in the nuclear potential
represented by $B_0(A)$, whose depth is adjusted to be zero for
$^{40}$Ca. If there are no interactions between valence nucleons,
the energy of the $A=40+n_v$ system ($n_v$ being the number of valence nucleons)
is $n_v e_{sp}$. However, the real energy $E(A=42)$ lies substantially below
the line $n_v e_{sp}$ in Fig. \ref{fig2}. The difference is the pair correlation
energy. Half of the absolute pair correlation energy is called the
(three-point) odd-even mass difference $\Delta$. The value $\Delta$ is
often used as the indicator of the pair correlations. We show the odd-even mass
difference $\Delta$ at $A=41$ in Fig. \ref{fig2}. The definition of $\Delta$
given in Eq. (\ref{eq:30}) explains the geometrical relations
shown in Fig. \ref{fig2}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=7.5cm,height=7.5cm]{fig2.eps}
\caption{Schematic depiction of the pair correlations and
$T=0$ $2n-2p$ correlations and their indicators
for the nuclei $^{40}$Ca, $A=41$ with $T=1/2$, $A=42$ with
$T=1$, $^{44}$Ti and $^{48}$Cr.}
\label{fig2}
\end{center}
\end{figure}
The energy of $^{44}$Ti measured from the line $n_v e_{sp}$ is the interaction
energy of the four nucleons with $T=0$ outside the $^{40}$Ca core,
which is denoted by the dotted line in Fig.~\ref{fig2}. Figure \ref{fig2} shows
that the $T=0$ four nucleon correlations, which we call $T=0$ $2n-2p$ correlations,
experience an energy gain much larger than
that of the pair correlations $2 \Delta$.
The strong $T=0$ $2n-2p$ correlations
cause the four nucleons to form an $\alpha$-like quartet outside the $^{40}$Ca
core, as described by the core plus $\alpha$-cluster model. The study of
$\alpha$-like correlations has a long history.
\cite{Maru,Danos,Arim,Kamim,Tomo,Cauvin,Duss1,Gamb,Jensen,Apos,Duss2,Curut,Suga,Hase1,Hase1B,Merm,Hase2,Sakuda}
(Also, see other references cited in Ref. 28), especially for the core
plus $\alpha$-cluster model.)
Sometimes the indicator of the $T=0$ $2n-2p$ correlations was evaluated
after subtracting the symmetry energy from the binding energy
(for instance, see Ref. 20)). As shown below, however,
different involvements of valence nucleons in
the $T=0$ $2n-2p$ correlations result in the ordering
of the ground-state energies of $N \approx Z$ nuclei according to that of
the total isospin $T$ (which results in the symmetry energy
in the mean field theory).
If we limit the comparative study of different $T$ nuclei
by excluding the symmetry energy from the binding energy at the beginning,
we miss a substantial energy gain due to the underlying $T=0$ $2n-2p$
correlations, and we therefore cannot understand the formation
of the $\alpha$-cluster in $^{20}$Ne and $^{44}$Ti
in contrast to
the absence of the $\alpha$-cluster outside the core in $^{20}$O and $^{44}$Ca.
Let $e_s$ denote the energy of the $T=1$ correlated pair and $l$ the number
of correlated pairs. If there is no interaction between the $2n$ and $2p$
pairs, the energy of $^{44}$Ti is expected to be on the line $l e_s$.
The real energy of $^{44}$Ti lies far below the line $l e_s$
in Fig. \ref{fig2}. The difference is the interaction energy between the $2n$
and $2p$ pairs, which is denoted by the dot-dashed line in Fig. \ref{fig2}.
Half of the absolute interaction energy between the $2n$ and $2p$ pairs
is a good indicator of the $T=0$ $2n-2p$ correlations. This indicator
is called the ``ODD-EVEN mass difference for the $\alpha$-like correlations"
by Gambhir, Ring and Schuck \cite{Gamb}, where ODD and EVEN
are used for the number of pairs. We define it in Eq. (\ref{eq:13})
and Eq. (\ref{eq:14}), and we write it as $\delta M(A_0+m\alpha +2)$ and
$\delta W(A_0+m\alpha +2:T=1)$ in the respective cases.
The value $\delta M$ is indicated by the solid line
at $A=42$ in Fig. \ref{fig2}.
The definitions given in Eq. (\ref{eq:13}) and Eq. (\ref{eq:14}) explain
the geometrical relations. The total interaction energy of
the $T=0$ $2n-2p$ quartet is $-2(2 \Delta + \delta M)$.
Figure \ref{fig2} clearly shows that the ODD-EVEN mass difference $\delta M$
for the $T=0$ $2n-2p$ correlations is much larger than the odd-even
mass difference $\Delta$ for the pair correlations. The ODD-EVEN mass
difference $\delta M$ for the $A=42$ system with $T=1$, which is measured from
the $T=0$ line, reflects the symmetry energy. In other words,
the symmetry energy
in the framework of the mass formula can be explained in terms of
the $T=0$ $2n-2p$ correlations from the point of view of the shell model.
It should be noticed that the symmetry energy and pairing energy in the mass
formula are treated on the same footing here.
Because the $T=0$ $2n-2p$ correlations are very strong, we believe
that the $T=0$ $2n-2p$ quartet is approximately a good excitation mode.
Let its energy be $e_\alpha$. Then the energy of the $A=40+m \alpha$ system
is expected to be nearly $m e_\alpha$. This expectation holds roughly
for $^{48}$Cr, as shown in Fig. \ref{fig2}, where $E(^{48}{\rm Cr})$ lies
below but near the line $m e_\alpha$.
It is shown in Ref. 29) that the $^{48}$Cr nucleus is described
quite well by the $^{40}$Ca core plus two $\alpha$-cluster model.
Figure \ref{fig1} shows that the energies $E(40+m \alpha)$
are below the line $m e_\alpha$.
This result indicates the important point that the interaction between
the $T=0$ $2n-2p$ quartets is attractive, and the $T=0$ $2n-2p$
correlations are collective in systems of many quartets.
Figure \ref{fig1} shows that $^{56}$Ni in the middle of the $pf$ shell
is different from the typical doubly-closed-shell nuclei $^{16}$O and
$^{40}$Ca, but it resembles $^{28}$Si in the middle of the $sd$ shell.
Because $B_0(A_0+m\alpha)$ as a function of $m$ is monotonic
in the regions $A=16-36$ and $A=40-72$ of the $A_0+m\alpha$ nuclei,
we cannot attribute the difference between the $^{56}$Ni nucleus
and the $^{16}$O and $^{40}$Ca nuclei to special behavior of $B_0(A)$.
The difference is due to correlations of valence nucleons or a shell effect.
The rigid core of $^{16}$O and $^{40}$Ca is supported by the successful
description of $^{20}$Ne and $^{44}$Ti with the core plus $\alpha$-cluster
model. Figure \ref{fig1} suggests structure of $^{56}$Ni ($^{28}$Si)
that differs from a rigid core.
We suppose that $^{16}$O and $^{40}$Ca have rather rigid cores and
that the other $N \approx Z$ nuclei are described as systems of
correlated valence nucleons outside the respective cores,
as is done in ordinary shell model calculations.
\subsection{Examination by means of the shell model calculation}
Let us examine the above picture by carrying out shell model calculations
with a realistic effective interaction in the $pf$ shell nuclei
outside the $^{40}$Ca core.
The shell model Hamiltonian describing valence nucleons outside the core
is composed of the single-particle energy part and the effective interaction:
\begin{equation}
H = H_{sp} + H_{int}. \label{eq:H1}
\end{equation}
We adopt the Honma interaction, \cite{Honma} which accurately describes
the $pf$ shell nuclei near $^{56}$Ni,
and we consider systems in the $jj$ coupling scheme.
To compare with Fig. \ref{fig1}, we use the same parameter value
$\lambda = -12.45$ MeV, as in Eq. (\ref{eq:5}),
though a somewhat different Coulomb energy correction
is used in Ref. 30). The adopted single-particle energies are
$e(f_{7/2})=3.862$, $e(p_{3/2})=6.7707$, $e(p_{1/2})=8.313$ and
$e(f_{5/2})=11.0671$ in MeV.
It is useful to decompose the effective interaction $H_{int}$ into
the monopole part and the residual part \cite{Dufour}. The $T=0$ monopole
field defined by the following equation is especially important, because it
determines the main part of the interaction energy (expectation value
$\langle H_{int} \rangle$):
\begin{eqnarray}
& {} & H_{mp}^{T=0} = - k^0 \sum_{a \leq b} \sum_{JM}
A^\dagger_{JM00}(ab) A_{JM00}(ab), \\ \label{eq:H2}
& {} & k^0 = \frac{\sum_{ab} \overline{V}(ab:T=0)} {\sum_{ab}1} \label{eq:H3}
\end{eqnarray}
with
\begin{equation}
\overline{V}(ab:T=0) = \frac{ \sum_J (2J+1) V(abab:J,T=0) }{\sum_J (2J+1)},
\label{eq:H4}
\end{equation}
where $A^\dagger_{JMTK}(ab)$ is the creation operator of a nucleon pair
with spin $JM$ and isospin $TK$ in the single-particle orbits
($a,b$) and $V(abab:JT)$ is a diagonal two-body interaction matrix element.
Let us write the effective interaction as
\begin{equation}
H_{int} = H_{mp}^{T=0} + H_{res} \quad ( H_{res} = H_{int} - H_{mp}^{T=0}).
\label{eq:H5}
\end{equation}
The monopole field $H_{mp}^{T=0}$ is expressed exactly as
\begin{equation}
H_{mp}^{T=0} = - \frac{k^0}{2} \Big\{ \frac{\hat n_v}{2}
\big(\frac{\hat n_v}{2} +1\big)
- {\hat T}({\hat T}+1) \Big\}, \label{eq:H6}
\end{equation}
where ${\hat n_v}$ stands for the number of valence nucleons, and
${\hat T}$ stands for the total isospin.
It is well known that realistic effective interactions have large and
comparable values of the centroids $\overline{V}(ab:T=0)$.
The expression (\ref{eq:H6}) with a large average value $k^0$
(for instance, $k^0=1.44$ MeV for $^{56}$Ni) shows that the symmetry energy
comes mainly from $H_{mp}^{T=0}$ with the $T(T+1)$ term \cite{Kaneko}.
The monopole field in the form (\ref{eq:H6}) can be regarded
as an additional term to the Hartree-Fock mean field, in a sense.
However, the residual interaction $H_{res}$, which determines
the microscopic structure, contributes significantly
to the symmetry energy \cite{Kaneko}.
The symmetry energy cannot be reduced to a simple mean field
but is affected by dynamical interactions in the shell model.
\begin{figure}[b]
\begin{center}
\includegraphics[width=6.8cm,height=7.5cm]{fig3a.eps}
\includegraphics[width=6.8cm,height=7.5cm]{fig3b.eps}
\caption{Expectation values of $H_{sp}$, $H_{sp}+H_{mp}^{T=0}$ and $H$
for (a) even-even $N=Z$ nuclei with $A=40+m\alpha$ and (b)
odd-$A$ nuclei with $A=40+m\alpha+1n$.
The residual interaction energy $\langle H_{res}\rangle$
is denoted by the solid-line arrows, and the monopole contribution
$\langle H_{mp}^{T=0} \rangle$ is denoted by the dashed-line arrows.}
\label{fig3}
\end{center}
\end{figure}
We carried out numerical calculations using Mizusaki's code,
\cite{Mizusaki,Mizusaki2} which makes large-scale shell model calculations
possible by means of extrapolation.
The calculated results for the $A=40+m\alpha$ nuclei from $^{44}$Ti to
$^{64}$Ge are illustrated in Fig. \ref{fig3}(a),
where $\langle H_{sp} \rangle$, $\langle H_{sp}+H_{mp}^{T=0} \rangle$
and $\langle H \rangle$ denote their expectation values
for the ground state. The behavior of the ground-state energy
$\langle H \rangle$ corresponds well with that of the energy $E(A_0+m\alpha)$
seen in Figs.~\ref{fig1} and \ref{fig2}. Figure \ref{fig3}(a)
shows that the energy $E(A_0+m\alpha)$ represents the ground-state energies
of the even-even $N=Z$ nuclei and, moreover,
that $E(A)$ hides significant correlations
in the background. This figure supports the schematic explanation
for the large energy gains of the $A_0+m\alpha$ nuclei in Fig. \ref{fig2}.
Even if we regard the monopole field as a part of the mean field,
the residual interaction energy $\langle H_{res} \rangle$ is still large
in Fig. \ref{fig3}(a).
The residual interaction energy $\langle H_{res} \rangle$ is essential
for bringing the values of $E(A_0+m\alpha)$ ($ \approx \langle H \rangle$)
close to the zero line.
In $^{56}$Ni, for instance, $\langle H_{res} \rangle$ is approximately 20 MeV,
which overwhelms the single-particle energy gap between
$f_{7/2}$ and $p_{3/2}$. The closed-shell configuration $(f_{7/2})^{16}$
does not exceed 68\% in the wave function of the ground state, according to
Ref. 30). The new Tamm-Dancoff solution for the $J=T=0$
four-particle excitation mode indicates that the ground state of $^{56}$Ni
cannot be described within a perturbation expansion starting with
the closed-shell configuration \cite{Hase3}.
This situation is called ``$\alpha$-like superfluidity" in the next section.
The $^{56}$Ni nucleus can be regarded as a correlated state of
valence nucleons outside the $^{40}$Ca core.
Figure \ref{fig3}(a) shows the upward turn of $\langle H \rangle$
from $^{56}$Ni to $^{60}$Zn resulting from an energy loss of four additional
nucleons occupying the upper orbits beyond the semi-magic number $Z=N=28$.
The variation of $E(A_0+m\alpha)$ is therefore related to a shell effect
as well as correlations.
However, it should be noted that the position of
$\langle H(^{60}\mbox{Zn}) \rangle$ in Fig. \ref{fig3}(a) depends on
significant collapse of the $^{56}$Ni core.
Figure \ref{fig3}(b) displays the shell model results for odd-mass nuclei
with $A_0+m\alpha+1n$. This figure is very similar to Fig. \ref{fig3}(a).
The ground-state energy $\langle H \rangle$ exhibits a dip at $^{57}$Ni
in Fig. \ref{fig3}(b), like the dip at $^{56}$Ni in Fig. \ref{fig3}(a).
In the shell model results for $^{57}$Ni, the occupation probabilities
of the respective orbits indicate strong correlations of valence nucleons
and collapse of the $^{56}$Ni core, which is contrary to a simple picture
of the $^{56}$Ni core plus one neutron. The single-particle energy gap
between $f_{7/2}$ and $p_{3/2}$ is negligible as compared
with the interaction energy, though the energy loss of four additional
nucleons occupying the upper orbits ($p_{3/2}$, $p_{1/2}$, $f_{5/2}$)
causes an upward turn of $\langle H \rangle$ from $^{57}$Ni to $^{61}$Zn.
The present shell model also explains the behavior of the experimental energy
$E(A_0+m\alpha+1n)$ shown in Fig. \ref{fig4}(b).
Thus, Figs. \ref{fig3}(a) and \ref{fig3}(b) support our picture, which regards
nuclei around $^{56}$Ni as correlated states of valence nucleons
outside the $^{40}$Ca core.
It is interesting that a $A_0+m\alpha+1n$ nucleus appears to be composed of
an $A_0+m\alpha$ system and a last neutron with an effective single-particle
energy. We can assume roughly the same correlations forming a common
structure in the two nuclei with $A=A_0+m\alpha$ and $A=A_0+m\alpha+1n$.
The correlations buried in the energies $E(A)$ of the $A_0+m\alpha$
nuclei have been considered little in the framework of the mass formula.
The inconspicuous values of $E(A_0+m\alpha)$, however, hide the $T=0$
$2n-2p$ correlations which are stronger than the $T=1$ pair correlations,
in the background. The energy $E(A_0+m\alpha)$ should be considered
explicitly in the mass formulas.
Figure \ref{fig2} suggests a description of the energies $E(A)$
of $N \ne Z$ nuclei with the indicators $\delta M$ and $\Delta$
on the $T=0$ line connecting the $A_0+m\alpha$ nuclei.
We now note the importance of the $T=0$ line in Fig. \ref{fig1}.
Ignoring the variation of the $T=0$ line affects the masses of
all $N \approx Z$ nuclei.
The values of $E(A_0+m\alpha)$ are not zero, and the substantial deviations
should not be ignored.
\section{Fundamental $T=0$ $2n-2p$ correlated structure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=6.8cm,height=7.4cm]{fig4a.eps}
\includegraphics[width=6.8cm,height=7.4cm]{fig4b.eps}
\caption{Experimental energies $E(A)$ defined in Eq. (\ref{eq:4})
for (a) even-even nuclei with $N \ge Z$ and
(b) nuclei with $A_0+m\alpha$, $A_0+m\alpha+1n$ and
$A_0+m\alpha+1n1p$ around $^{56}$Ni.}
\label{fig4}
\end{center}
\end{figure}
We have seen significant correlations buried in $E(A)$ of $A_0+m\alpha$
and $A_0+m\alpha+1n$ nuclei.
What is the nature of the correlations in other $N \neq Z$ nuclei?
To answer this, let us consider Fig.~\ref{fig4}(a),
in which experimental values of $E(A)$ are plotted
for even-even nuclei with $N \ge Z$.
Even-even nuclei with $N > Z$ can be classified according to the number of
neutron pairs added to the $A_0+m\alpha$ systems,
such as $A=A_0+m\alpha +2n$, $A=A_0+m\alpha +4n$, $\cdots$,
and each series of them with increasing $m$ has the same $T$.
Figure \ref{fig4}(a) indicates a parallelism
with regard to $E(A)$ between even-even nuclei with $T>0$ and $T=0$.
More precisely, every $T$ line connecting a series of nuclei
is parallel to the $T=0$ line of the $A_0+m\alpha$ nuclei.
The same parallelism is seen in the experimental values of $E(A)$
for even-even $N \le Z$ nuclei
($A=A_0+m\alpha$, $A=A_0+m\alpha +2p$, $A=A_0+m\alpha +4p$, $\cdots$),
though they are omitted in Fig. \ref{fig4}(a) for simplicity.
The experimental energy $E(A)$ for the $T=1$, $0^+$ states of odd-odd $N=Z$
nuclei also varies in a manner parallel to $E(A_0+m\alpha)$,
as shown in Fig. \ref{fig4}(b).
Realistic shell model calculations faithfully reproduce the parallelism of
the experimental $E(A)$ in Fig. \ref{fig4}.
This parallelism is expressed in terms of the symmetry energy
in ordinary mass formulas.
It should, however, be noted that the characteristic behavior of
$E(A_0+m\alpha)$, which hides important correlations and resulting structure,
appears in the $T>0$ lines of other nuclei.
The parallel variations of $E(A)$ suggest the existence of a common structure
formed in nuclei with $A=A_0+m\alpha$, $A=A_0+m\alpha +2n(2p)$,
$A=A_0+m\alpha +4n(4p)$, etc., and also in $A_0+m\alpha +1n1p$ nuclei
with $T=1$.
It is notable in Fig. \ref{fig4}(b) that parallel variation of $E(A)$
appears also in the odd-$A$ nuclei with $A=A_0+m\alpha +1n$, which is
seen in the shell model results of Fig. \ref{fig3}. The energy difference
$E(A_0+m\alpha+1n)-E(A_0+m\alpha)$ is related to the pairing energy
in the ordinary mass formulas.
This parallelism again suggests a common structure in the two nuclei
with $A=A_0+m\alpha$ and $A=A_0+m\alpha +1n$.
This common structure is probably the $T=0$ $2n-2p$ correlated
structure of the $A_0+m\alpha$ nuclei.
\subsection{Multi-quartet structure of even-even $N=Z$ nuclei}
The study of nuclear structure has clarified the importance of
the $T=0$ $2n2p$ correlations in $N \approx Z$ nuclei.
Recall that the core plus $\alpha$ cluster (two $\alpha$ cluster)
model accurately describes $^{20}$Ne, $^{24}$Mg, $^{44}$Ti and even $^{48}$Cr.
In a simplified picture ignoring the spatial correlations of $\alpha$,
the ground states of
$A_0+m\alpha$ nuclei can be approximated in the following way
\cite{Hase1}:
\begin{equation}
|\Phi_0(A_0+m\alpha) \rangle \approx \frac{1}{\sqrt{N(A_0+m\alpha)}}
(\alpha_{J=T=0}^\dagger)^m |A_0 \rangle, \label{eq:6}
\end{equation}
where $\alpha_{J=T=0}^\dagger$ consists of a linear combination
of four valence nucleons
$(c^\dagger)^4_{J=T=0}$ determined by the Tamm-Dancoff equation
$[H, \alpha_{J=T=0}^\dagger ] \approx e_\alpha \alpha_{J=T=0}^\dagger$,
$N(A_0+m\alpha)$ is a normalization constant,
and $|A_0 \rangle$ denotes the doubly-closed-shell core.
When the Hamiltonian has only two-body interactions,
the energy $E(A_0+m\alpha)$ can be calculated as
\begin{eqnarray}
& {} & E(A_0+m\alpha) = m \frac{\langle A_0| (\alpha_0)^m (\alpha_{J=T=0}^\dagger)^{m-1}
[H, \alpha_{J=T=0}^\dagger ]|A_0 \rangle}{N(A_0+m\alpha)} \nonumber \\
& {} & + \frac{1}{2} m(m-1)
\frac{\langle A_0| (\alpha_0)^m (\alpha_{J=T=0}^\dagger)^{m-2}
[[H, \alpha_{J=T=0}^\dagger ], \alpha_{J=T=0}^\dagger ]|A_0 \rangle}{N(A_0+m\alpha)}.
\label{eq:7}
\end{eqnarray}
The microscopic calculations of Eq. (\ref{eq:7}) in Ref. 25)
approximately reproduce the experimental energies $E_{shl}(A_0+m\alpha)$
[and hence $E(A_0+m\alpha)$] for $^{44}$Ti, $^{48}$Cr, $^{52}$Fe and $^{56}$Ni.
Therefore, the $A_0+m\alpha$ nuclei have a multi-quartet structure
approximated by Eq. (\ref{eq:6}), at least up to $^{56}$Ni.
The characteristic behavior of $E(A_0+m\alpha)$ in Fig. \ref{fig1}
indicates the leading role of the multi-quartet structure.
The shell model results in $\S$2.3 allow us to imagine the multi-quartet
structure for the $A_0+m\alpha$ nuclei beyond $^{56}$Ni.
In order to get a simple formula, let us transfer the fermion equation
(\ref{eq:7}) into an interacting $\alpha$-boson model,
\begin{equation}
E_{IBM}(A_0+m\alpha) = m e_\alpha - \frac{1}{2} m(m-1) G_{\alpha \alpha},
\label{eq:8}
\end{equation}
where $e_\alpha$ is the energy of the $\alpha$-boson and $G_{\alpha \alpha}$
denotes the interaction between the $\alpha$-bosons. Because the $\alpha$-boson
is regarded as being mapped from the four correlated fermions, the interaction
strength $G_{\alpha \alpha}$ should reflect the Pauli principle.
The stable doubly-closed-shell nuclei $^{16}$O and $^{40}$Ca suggest
that the four correlated nucleons are mainly in one major shell \cite{Tomo}.
As the number of $\alpha_{J=T=0}^\dagger$ increases in the major shell,
the Pauli principle applied to $\alpha_{J=T=0}^\dagger$ must restrict
the degrees of freedom for $\alpha_{J=T=0}^\dagger$.
Let us take the Pauli principle effect into account by expressing
the interaction strength $G_{\alpha \alpha}$ in the form of a decreasing
function of $m$ (the number of $\alpha_{J=T=0}^\dagger$). The simplest way
to do this is
to approximate the decline function by a linear function of $m$, such as
\begin{equation}
G_{\alpha \alpha} = g_{\alpha \alpha} \{ 1 - C_\alpha (m-1) \}. \label{eq:9}
\end{equation}
The factor $C_\alpha$ represents something like the scale of the subspace
$\{(\alpha_{J=T=0}^\dagger)^m|A_0 \rangle \}$, depending on the shell structure.
This interacting $\alpha$-boson model can reproduce the experimental values
of $E(A_0+m\alpha)$ up to $^{56}$Ni, as shown in Fig. \ref{fig1},
where the values denoted by the open squares are obtained with the parameter
values
$e_\alpha=2.75$, $g_{\alpha \alpha}=5.3$, $C_\alpha=0.21$ in MeV
for $^{20}$Ne to $^{28}$Si and $e_\alpha=1.35$, $g_{\alpha \alpha}=3.75$,
$C_\alpha=0.18$ in MeV for $^{44}$Ti to $^{56}$Ni.
The interacting $\alpha$-boson model describes the peaks
at $A=A_0+\alpha$ ($^{20}$Ne and $^{44}$Ti) and the decline toward $^{28}$Si
and $^{56}$Ni.
The most important point here is that the interaction between the composite
quartets $\alpha_{J=T=0}^\dagger$ is attractive and quite strong.
Other composite fermion units, like Cooper pairs and vibrational phonons
with $J=2$ and $J=3$ in nuclear physics, have repulsive interactions
between them and do not actually have a boson-like property, because of
the Pauli principle. Only the $\alpha$-like quartet with $J=T=0$ has
the possibility to resemble a boson, because of a special mechanism
in couplings of spin and isospin.
The large energy gain due to the strong $T=0$ $2n-2p$ correlations
and the attractive interaction between the $\alpha$-like quartets cause
the effect of the (collective) $T=0$ $2n-2p$ correlations
on the nuclear mass to be rather inconspicuous.
The interacting $\alpha$-boson model (\ref{eq:8}) with (\ref{eq:9})
roughly reproduces the energies $E(^{32}$S$)$ and $E(^{60}$Zn$)$ given in Fig.
\ref{fig1}. The shell model calculation in $\S$2.3, however, shows
that the upward turn to $^{60}$Zn in the graph of $E(A_0+m\alpha)$ is
due to a shell effect. The formula (\ref{eq:8}) cannot be applied
to the regions $32<A<40$ and $60<A<80$, because the expression (\ref{eq:9})
is not valid there. A hole picture is probably suitable for these latter halves
of the $sd$ and $pf$ shells. Then we obtain the same type of states as in Eq.
(\ref{eq:6}) by replacing $\alpha_{J=T=0}^\dagger$ with a linear combination
of four holes $(c_h^\dagger)^4_{J=T=0}$. The corresponding boson picture,
the interacting $\alpha$-hole boson model, may give a formula similar to
(\ref{eq:8}).
It is, however, difficult to obtain a simple formula that reproduces
the variation of $E(A_0+m\alpha)$ including a shell effect in the entire region
of $N=Z$ nuclei. We therefore abandon this problem and instead adopt
the experimental values given in Fig. \ref{fig1} for the energy
$E(A_0+m\alpha)$ in this paper.
The approximation (\ref{eq:6}) is very simplified in comparison with
the realistic shell model. Adding other collective modes
of the $T=0$ $2n-2p$ correlations with $J>0$ is necessary to better reproduce
the variation of $E(A_0+m\alpha)$ for the $f_{7/2}$ shell nuclei \cite{Hase1B}.
We should consider the $T=0$ $2n-2p$ correlations as correlations of collective
$T=0$ $2n-2p$ quartets with various $J$ in an improved approximation.
In fact, although we usually imagine the multi $J$$=$$0$ pair structure
for the $T=1$ pair correlated state, a realistic shell model wave function
includes components of various $J>0$ pairs in nuclear physics.
We use the term ``$T=0$ $2n-2p$ correlations" in such a broader sense.
In the following sections, we express the $T=0$ $2n-2p$ correlated states as
\begin{equation}
|\Phi_0(A_0+m\alpha) \rangle \propto (\alpha_{T=0}^\dagger)^m |A_0 \rangle, \label{eq:10}
\end{equation}
where $\alpha_{T=0}^\dagger$ represents a quartet of $T=0$ $2n-2p$ correlated
nucleons or holes.
We formally use the expression (\ref{eq:10}) also in the hole regions,
$32<A<40$ and $60<A<80$.
\subsection{Superfluid state induced by the $T=0$ $2n-2p$ correlations}
In Fig. \ref{fig1}, the line connecting the energies $E(A_0+m\alpha)$
of the $T=0$ nuclei plays an important role in the mass formula,
because the energies of the other nuclei with $T>0$ are measured from this line.
For $E(A_0+m\alpha)$, we provisionally use the experimental values
evaluated from $B(A_0+m\alpha)-B_0(A_0+m\alpha)$, as mentioned above.
Let us extrapolate the line for $E(A_0+m\alpha)$
to nuclei with $A \neq A_0+m\alpha$ as follows:
\begin{eqnarray}
E_{T=0}(A_0+m\alpha+2) & = & (E(A_0+m\alpha) + E(A_0+m\alpha+\alpha)) /2, \nonumber \\
E_{T=0}(A_0+m\alpha+1) & = & (E(A_0+m\alpha) + E_{T=0}(A_0+m\alpha+2)) /2, \nonumber \\
E_{T=0}(A_0+m\alpha+3) & = & (E_{T=0}(A_0+m\alpha+2) + E(A_0+m\alpha+\alpha)) /2.
\label{eq:11}
\end{eqnarray}
These equations define the $T=0$ plane as the base level of energy
in the mass table. At this stage, the binding energy is written
\begin{equation}
B(A) = B_0(A) + E_{T=0}(A) + W(A). \label{eq:12}
\end{equation}
The pairing energy, symmetry energy, Wigner energy and a correction
for odd-odd nuclei are included in the residual energy $W(A)$
in Eq. (\ref{eq:12}).
The $T=0$ $2n-2p$ correlations are related to
the ``$\alpha$-like superfluidity" proposed in Ref. 7),
where, by analogy to pairing superfluidity, $\alpha$-like superfluidity
is indicated by the following mass difference corresponding to the odd-even
mass difference $\Delta$:
\begin{eqnarray}
\delta M(A_0+m\alpha+2) & = & (B(A_0+m\alpha+2n)+B(A_0+m\alpha+2p))/2 \nonumber \\
& - & (B(A_0+m\alpha)+B(A_0+m\alpha+\alpha))/2. \label{eq:13}
\end{eqnarray}
This quantity is called the ODD-EVEN mass difference in $\S$2.2.
The average energy of the $A_0+m\alpha+2n$ and $A_0+m\alpha+2p$ nuclei
is used so as to remove the Coulomb energy effect. Although the
$\alpha$-like superfluidity in heavy nuclei is discussed in Ref. 7),
we are concerned with $N \approx Z$ nuclei, for which the isospin is
a good quantum number. In place of Eq. (\ref{eq:13}), we define the
following quantity as an indicator of the $\alpha$-like superfluidity:
\begin{eqnarray}
\delta W(A_0+m\alpha+2:T=1,K) & = & W(A_0+m\alpha+2:T=1,K) \nonumber \\
& - & (W(A_0+m\alpha)+W(A_0+m\alpha+\alpha))/2, \label{eq:14}
\end{eqnarray}
where $K=1$, 0 and $-1$ correspond to the $T=1$ states of the $A_0+m\alpha+2n$,
$A_0+m\alpha+1n1p$ and $A_0+m\alpha+2p$ nuclei, respectively. The residual
energy $W(A)$ is measured from the $T=0$ plane [$E_{T=0}(A)$],
and hence we have $W(A_0+m\alpha)=W(A_0+m\alpha+\alpha)=0$.
It is thus seen that $W(A_0+m\alpha+2:T=1,K)$
is identically the ODD-EVEN mass difference for $\alpha$-like superfluidity,
\begin{equation}
\delta W(A_0+m\alpha+2:T=1,K) = W(A_0+m\alpha+2:T=1,K). \label{eq:15}
\end{equation}
The average of $W(A_0+m\alpha+2:T=1,K=1)$ and $W(A_0+m\alpha+2:T=1,K=-1)$
corresponds with $\delta M(A_0+m\alpha+2)$ in Eq. (\ref{eq:13}),
which represents the $n-p$ (mainly $T=0$) interaction energy between $2n$
and $2p$ in an $A_0+m\alpha+\alpha$ nucleus \cite{Kaneko}.
The values of $W(A_0+m\alpha+2:T=1,K)$ for the $A_0+m\alpha+2n$,
$A_0+m\alpha+1n1p$ and $A_0+m\alpha+2p$ nuclei are plotted
by the dotted curves in Fig. \ref{fig5}.
Figure \ref{fig5} shows that the ODD-EVEN mass difference for the $T=0$ $2n-2p$
correlations is larger than the odd-even mass difference $\Delta$
for the pairing superfluidity.
The shell model calculation in $\S$2.3 shows
that there is a significant contribution of
the monopole field $H_{mp}^{T=0}$ to the ODD-EVEN mass difference
[in Eq. (\ref{eq:14})]. The $H_{mp}^{T=0}$ contribution is $3k^0/2$.
For $A\approx 58$, for instance, the value is about 2.14 MeV,
while $W(A=58:T=1,K=1) \approx 3.4$ MeV.
The $H_{res}$ contribution to the ODD-EVEN mass difference is about 1.25 MeV,
which is comparable to the odd-even mass difference
$\Delta \approx 1.34$ MeV near $A=58$.
It should be noted here that the $T=1$ pair correlations of neutron and proton
pairs joining in the formation of the $T=0$ $2n-2p$ quartet are not included
in the ODD-EVEN mass difference. We can say that the strong $T=0$ $2n-2p$
correlations cause a superfluid state, like the pairing superfluid state,
as claimed by Gambhir {\it et al.}\cite{Gamb}
It is notable that the ODD-EVEN mass difference for $N \approx Z$ nuclei
is larger than that for $N>Z$ nuclei, with regard to which the term
``$\alpha$-superfluidity" was first used \cite{Gamb}.
We call the strongly correlated state an ``$\alpha$-like superfluid state".
As discussed in the previous subsection, the $\alpha$-like superfluid state
has the multi-quartet structure (\ref{eq:6}) in the $A_0+m\alpha$ nuclei,
at least up to $^{56}$Ni.
Figure \ref{fig5} displays the systematic differences among the $A_0+m\alpha+2n$,
$A_0+m\alpha+1n1p$ and $A_0+m\alpha+2p$ nuclei. This indicates that the effect
of the Coulomb interaction remains after subtracting the Coulomb energy
term $a_C Z^2/A^{1/3}$. It may be necessary for a practical mass formula
to add some correction terms in order to remove the differences
between the states with different $K$. In fact, modern mass formulas do have
such correction terms. However, we leave this problem and
employ different parameters for neutrons and protons in this paper,
where we aim to explain our basic idea.
\begin{figure}[t]
\begin{center}
\includegraphics[width=7cm,height=7cm]{fig5.eps}
\caption{The energy $W(A_0+m\alpha+2:T=1,K)$
for $A_0+m\alpha+2$ nuclei with $T=1$. This quantity represents
the ODD-EVEN mass difference for $\alpha$-like superfluidity.}
\label{fig5}
\end{center}
\end{figure}
\subsection{Bogoliubov transformation for the $\alpha$-like superfluid state}
We consider $A_0+m\alpha+2$ nuclei with $T=1$,
where structure is roughly expressed as
\begin{eqnarray}
& {} & |A=A_0+m\alpha+2:T=1,K \rangle \propto S_K^\dagger
(\alpha_{T=0}^\dagger)^m |A_0 \rangle , \label{eq:16} \\
& {} & S_K^\dagger \propto (c^\dagger c^\dagger)_{J=0,T=1,K} . \nonumber
\end{eqnarray}
The large values of $W(A_0+m\alpha+2:1K)$ allow us to regard
the state
$|\Phi_0(A_0+m\alpha) \rangle $ as an $\alpha$-like superfluid state
$(\alpha_{T=0}^\dagger)^m |A_0 \rangle$. After a kind of the Bogoliubov
transformation, the $\alpha$-like superfluid state is the vacuum state
$|0(\alpha) \rangle$ for a quasi-pair $\mbox{\boldmath $S$}_K^\dagger$,
which is transformed from $S_K^\dagger$. In this picture, the state
(\ref{eq:16}) is regarded as a quasi-pair state, like the quasi-particle
state in the BCS theory,
\begin{equation}
|A=A_0+m\alpha+2:1,K \rangle = \mbox{\boldmath $S$}_K^\dagger
|0(\alpha) \rangle . \label{eq:17}
\end{equation}
Measuring the energy $W(A)$ from the $T=0$ plane [$E_{T=0}(A)$]
defined in Eq. (\ref{eq:11}) corresponds to the above transformation
for the wave functions. The energy $W(A_0+m\alpha+2:1K)$ is the energy
of the quasi-pair $\mbox{\boldmath $S$}_K^\dagger$.
This discussion is parallel to the quasi-particle picture
concerning the pairing energy, as seen below.
The above transformation is, in fact, difficult to carry out
for the four composite fermions $\alpha_{T=0}^\dagger$.
Instead, we illustrate our plan using the interacting boson model (IBM),
as done by Gambhir {\it et al.} \cite{Gamb}.
The IBM for $N \approx Z$ nuclei is called the IBM3. The IBM3 Hamiltonian
is expressed in terms of the $s$ boson ($J=0$) and $d$ boson ($J=2$)
with $T=1$ \cite{Thompson,Hase4},
\begin{equation}
s_K^\dagger = s_{J=0,T=1,K}^\dagger , \quad
d_{MK}^\dagger = d_{2M1K}^\dagger. \label{eq:18}
\end{equation}
The $sd$ boson image of $\alpha_{J=T=0}^\dagger$ is given by
\begin{equation}
\alpha_{J=T=0}^\dagger \Rightarrow x (s^\dagger s^\dagger)_{J=T=0}
+ \sqrt{1-x^2} (d^\dagger d^\dagger)_{J=T=0}. \label{eq:19}
\end{equation}
For the $\alpha$-like superfluid state, the quasi $s$ and $d$ bosons
($\mbox{\boldmath $s$}_K$, $\mbox{\boldmath $d$}_{MK}$) are introduced
through the Bogoliubov transformation
\begin{eqnarray}
s_K^\dagger &=& U_s \mbox{\boldmath $s$}_K^\dagger
+ V_s (-)^{1-K} \mbox{\boldmath $s$}_{-K} , \nonumber \\
d_{MK}^\dagger &=& U_d \mbox{\boldmath $d$}_{MK}^\dagger
+ V_d (-)^{2-M}(-)^{1-K}\mbox{\boldmath $d$}_{-M-K}. \label{eq:20}
\end{eqnarray}
Here, we have $U_i^2 - V_i^2 =1$ ($i=s$, $d$).
We have a boson-type gap equation and can calculate the quasi-boson
energies $e_s$ and $e_d$ using an appropriate IBM3 Hamiltonian.
In this quasi-boson picture, the quasi-pair state (\ref{eq:17})
of the $A_0+m\alpha+2$ nuclei is written
\begin{equation}
|A=A_0+m\alpha+2:1,K \rangle \Rightarrow \mbox{\boldmath $s$}_K^\dagger |0(\alpha)),
\label{eq:21}
\end{equation}
where the $\alpha$-like superfluid vacuum state is replaced by that
for the quasi-bosons ($\mbox{\boldmath $s$}_K$, $\mbox{\boldmath $d$}_{MK}$).
There is a well-defined IBM3 Hamiltonian for the $f_{7/2}$ shell nuclei
\cite{Thompson}.
Using the IBM3 Hamiltonian, we evaluated the quasi-boson energy $e_s$,
which should be equal to $W(A_0+m\alpha+2:T=1)$ given in Eq. (\ref{eq:15}).
The calculated values of $e_s$ for $^{46}$Ti, $^{50}$Cr and $^{54}$Fe
are plotted in Fig. \ref{fig5}.
It is seen that the quasi-boson energies $e_s$ accurately reproduce
the experimental values of $W(A_0+m\alpha+2n)$ which is least sensitive
to the Coulomb interaction effect.
This success supports our interpretation of the $\alpha$-like superfluidity
of $A_0+m\alpha+2$ nuclei.
\section{Multi-pair structure on the base of $\alpha$-like superfluidity}
Because the picture of $\alpha$-like superfluidity is good, the $J=0$
ground states of even-even nuclei can be approximated by
\begin{eqnarray}
|A=A_0+m\alpha+2l:T=l \rangle & \propto & (S^\dagger)^l
|\Phi_0(A_0+m\alpha) \rangle , \nonumber \\
& \Rightarrow & (\mbox{\boldmath $s$}^\dagger)^l |0(\alpha)). \label{eq:22}
\end{eqnarray}
Similar wave functions are considered in the microscopic derivation
of a mass formula \cite{Zuker}. Now we have reached the second stage,
which can be compared with the first stage considering the
multi-quartet state $(\alpha_{T=0}^\dagger)^m|A_0 \rangle$.
We have another interacting boson picture for the Cooper pair,
\begin{equation}
W(A_0+m\alpha+2l:T=l) = l e_s + \frac{1}{2} l(l-1) g_{ss}. \label{eq:23}
\end{equation}
The interaction between the Cooper pairs (like-nucleon pairs)
is repulsive because of the Pauli principle.
The repulsive interaction between the quasi-$s$-bosons gives a quadratic
increase of the mass, depending on the boson number $l$.
The quasi-$s$-boson $\mbox{\boldmath $s$}_K^\dagger$ increases the isospin
of the state by 1, and the number of $\mbox{\boldmath $s$}_K^\dagger$
can be replaced with the isospin $T$ in Eq. (\ref{eq:23}).
Let us write Eq. (\ref{eq:23}) in the ordinary form
\begin{eqnarray}
W(A_0+m\alpha+2l:T=l) & = & a_{sym} T^2 + b_{Wig} T , \label{eq:24} \\
a_{sym} & = & \frac{1}{2}g_{ss}, \quad b_{Wig}=e_s-\frac{1}{2}g_{ss}.
\label{eq:25}
\end{eqnarray}
The first term here is called the symmetry energy, and the second term is called
the Wigner energy in the mass formulas. Our interacting boson picture
for the multi-pair states explains the structural origins
of the symmetry energy and Wigner energy.
\begin{figure}[b]
\begin{center}
\includegraphics[width=8cm,height=8cm]{fig6.eps}
\caption{Energies $W(m\alpha + 2ln:T=l)$ of multi-quasi-neutron-pair states
$(S_n^\dagger)^{l=T}|\Phi_0(m\alpha) \rangle$.
Experimental values (flat dots) are compared with the theoretical
values obtained with the parameters (\ref{eq:26})
(which are at the intersections of the solid and dotted curves).}
\label{fig6}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=8cm,height=5.71cm]{fig7.eps}
\caption{Energies $W(m\alpha + 2lp:T=l)$ of multi-quasi-proton-pair states
$(S_p^\dagger)^{l=T}|\Phi_0(m\alpha) \rangle$,
shown in the same manner as Fig. \ref{fig5}.
The theoretical values (at the intersections of the two curves)
are obtained with the parameters (\ref{eq:27}).}
\label{fig7}
\end{center}
\end{figure}
Figures \ref{fig6} and \ref{fig7} display the experimental energies of
the multi-quasi-pair states (\ref{eq:22}), $W(A_0+m\alpha+2ln)$ and
$W(A_0+m\alpha+2lp)$. We can fix the parameter values $e_s$ and $g_{ss}$
in the approximation (\ref{eq:23}) [$a_{sym}$ and $b_{Wig}$ in Eq.
(\ref{eq:24})] from the experimental values of $W(A_0+m\alpha+2l:T=l)$.
They can be expressed in the same form as that determined
in the microscopic mass formula \cite{Duflo}.
The parameters, which are fixed separately for neutrons and protons, are
\begin{eqnarray}
a_{sym}^{(n)} & = & 116 (1-1.52/A^{1/3})/A, \nonumber \\
b_{Wig}^{(n)} & = & 218 (1-1.52/A^{1/3})/A, \label{eq:26} \\
a_{sym}^{(p)} & = & 82 (1-1.0/A^{1/3})/A, \nonumber \\
b_{Wig}^{(p)} & = & 92 (1-1.0/A^{1/3})/A. \label{eq:27}
\end{eqnarray}
The quasi-$s$-boson energies $e_s^{(n)}$ and $e_s^{(p)}$ are plotted
by the solid and dash-dot curves in Fig. \ref{fig5} [where a curve fitted
to $W(A_0+m\alpha+1n1p)$ is also shown]. We see in Figs. \ref{fig6} and
\ref{fig7} that the approximation (\ref{eq:24}) with the parameters
(\ref{eq:26}) and (\ref{eq:27}) is very good, and hence our interacting
boson picture for Cooper pairs is also good. From the parameters
(\ref{eq:26}) and (\ref{eq:27}), the coefficient of the Wigner energy
is larger than that of the symmetry energy
($b_{Wig}^{(n)} \approx 1.88 a_{sym}^{(n)})$ for neutrons and
$b_{Wig}^{(p)} \approx a_{sym}^{(p)}$ for protons.
The symmetry energy coefficient $a_{sym}^{(n)}$ is nearly equal to
that determined in Ref. 4).
The values of these parameters depend on the manner of evaluating
the Coulomb energy.
We point out that the quasi-$s$-boson energy is, for instance,
$e_s^{(n)} \approx 6.4$ MeV for $A=20$ and $e_s^{(n)} \approx 4.3$ MeV
for $A=44$, as obtained from Fig.~\ref{fig5}. [The interaction energy between
the quasi-$s$-bosons is $g_{ss}^{(n)} \approx 5.1$ MeV for $A=20$
and $g_{ss}^{(n)} \approx 3.0$ MeV for $A=44$, from Eq. (\ref{eq:26}).]
The quasi-$s$-boson energy $e_s$ is larger than the $\alpha$-boson
energy $e_\alpha$ ($e_\alpha =1.35$ MeV for the $sd$ shell nuclei and
$e_\alpha =2.75$ MeV for the $pf$ shell nuclei). The fact that
$e_\alpha$ is much smaller than $e_s^{(n)}+e_s^{(p)}$ indicates
the very large energy gain of the $\alpha$-like quartet.
Moreover, while the $\alpha$-like quartet interaction is attractive,
the quasi-pair interaction is repulsive.
The characteristic patterns in Figs. \ref{fig6} and \ref{fig7} are
due to the repulsive interaction between the quasi-pairs
(a quasi-pair transfers isospin 1). These are contrast with
the inconspicuous effect of the multi-quartet structure on the energy
$E(m \alpha)$, shown in Fig.~\ref{fig1}.
(The $\alpha$-like quartet transfers no quantum number
other than the nucleon number.)
If there was no such great energy gain caused by the collective
$T=0$ $2n-2p$ correlations, the nuclear mass table would be different.
\section{Structure having an unpaired neutron and/or an unpaired proton}
Before ending the second stage treating the multi-pair states,
let us write our mass formula as
\begin{equation}
B(A) = B_0(A)
+ E_{T=0}(A) + w_T(A) + w_v(A). \label{eq:28}
\end{equation}
We extend the $T$-dependent energy $W(A_0+m\alpha+2l:T=l)$ in Eq. (\ref{eq:24})
to odd-$A$ nuclei and odd-odd nuclei, as we extended $E(A_0+m\alpha)$
to $E_{T=0}(A)$, expressing it as
\begin{equation}
w_T(A) = a_{sym}(A) T^2 + b_{Wig}(A) T, \label{eq:29}
\end{equation}
where we permit $T$ to be a half integer for odd-$A$ nuclei.
The last term, $w_v(A)$ in Eq. (\ref{eq:28}), represents the energy of
unpaired nucleon(s). The subscript $v$ is the seniority quantum
number. It should be noted that in the mass formula (\ref{eq:28}),
the energy $w_v(A)$ of an odd-$A$ (or odd-odd) nucleus is measured
from the base {\it curve} given by (\ref{eq:29}).
\subsection{Shifted quasi-particle energy for odd-mass nuclei}
The strength of the pairing correlations in an odd-$A$ nucleus is
usually evaluated with the odd-even mass difference. We define it
using $W(A)$ of Eq. (\ref{eq:12}) in the same form as Eq. (\ref{eq:14}),
\begin{equation}
\Delta(A=A_0+m\alpha+2l+1)
= W(A) - ( W(A-1) + W(A+1) ) /2. \label{eq:30}
\end{equation}
This relation is illustrated in Fig. \ref{fig2}.
Substituting the relation (\ref{eq:29}) for $W(A-1)$ and $W(A+1)$,
we obtain the approximate relation
\begin{equation}
\Delta(A=A_0+m\alpha+2l+1)
\approx W(A) - ( w_{T=l+1/2}(A) + a_{sym}(A)/4 ). \label{eq:31}
\end{equation}
The energy $w_{v=1}(A)$ for an odd-$A$ nucleus with $A=A_0+m\alpha+2l+1$
in Eq. (\ref{eq:28}) is given by
\begin{eqnarray}
w_{v=1}(A) & \equiv & W(A) - w_{T=l+1/2}(A) \nonumber \\
& \approx & \Delta(A) + \frac{1}{4} a_{sym}(A). \label{eq:32}
\end{eqnarray}
The energy shift $a_{sym}(A)/4$ is inevitable when we measure the energy
$w_{v=1}(A)$ from the base curve (\ref{eq:29}).
\begin{figure}[b]
\begin{center}
\includegraphics[width=6.8cm,height=7.2cm]{fig8a.eps}
\includegraphics[width=6.8cm,height=7.2cm]{fig8b.eps}
\caption{Energies $w_{v=1}(A)=d_n(A)$ for odd-$N$ nuclei and
$w_{v=1}(A)=d_p(A)$ for odd-$Z$ nuclei.}
\label{fig8}
\end{center}
\end{figure}
In the last stage, we consider $A_0+ m\alpha +2l+1$ nuclei
with $T=l+1/2$, which have the structure
\begin{equation}
|A=A_0+m\alpha+2l+1 \rangle \propto c^\dagger (S^\dagger)^l
|\Phi_0(A_0+m\alpha) \rangle . \label{eq:33}
\end{equation}
This structure is expressed approximately as a direct product of
the three modules $|\Phi_0(A_0+m\alpha) \rangle$, $(S^\dagger)^l$, and
the last odd nucleon $c^\dagger$.
We regard the multi-pair structure $(S^\dagger)^l$ as the pairing
superfluid structure as usual. After the Bogoliubov transformation,
the odd-$A$ nucleus is regarded as the one quasi-particle state,
\begin{equation}
|A=A_0+m\alpha+2l+1 \rangle = a^\dagger
|0(lS)\otimes 0(A_0+m\alpha) \rangle . \label{eq:34}
\end{equation}
In this picture, the energy $w_{T=l}(A)$ given in Eq. (\ref{eq:28}) represents
the energy of the pairing superfluid state $|0(lS) \rangle$,
which is the vacuum for the quasi-particle $a^\dagger$,
and the quantity $\Delta (A)$ in Eq. (\ref{eq:30})
can be regarded as the quasi-particle energy.
Let us rewrite the ``shifted quasi-particle energy" $w_{v=1}(A)$ as
\begin{equation}
d_n(A) = \Delta_n(A) + a_{sym}^{(n)}/4, \quad
d_p(A) = \Delta_p(A) + a_{sym}^{(p)}/4. \label{eq:35}
\end{equation}
Experimental values of $w_{v=1}(A)$ calculated with the experimental
values of $E_{T=0}(A)$ in Eq. (\ref{eq:28}) are plotted
in Fig. \ref{fig8}.
With the approximate relation (\ref{eq:32}), we can parameterize
the quantity $\Delta(A)$ in the same form ($\propto A^{-1/3}$)
as that of the microscopic mass formula \cite{Duflo}, {\it i.e.},
\begin{equation}
\Delta_n(A) = 5.18/A^{1/3}, \quad
\Delta_p(A) = 4.6/A^{1/3}. \label{eq:36}
\end{equation}
The neutron value, $\Delta_n(A)$, is equal to that given in Ref. 4),
and the proton value, $\Delta_p(A)$, is smaller than $\Delta_n(A)$.
It should be noted that $\Delta(A)$ is a measure of the $T=1$ pair
correlations and $W(A:T=1)=e_s$ is approximately a measure of
the $T=0$ $n-p$ correlations between the $T=1$ pairs \cite{Kaneko,Janecke}.
This leads to the different $A$ dependences of $\Delta(A)$ and $e_s$.
According to Ref. 39), because the symmetry energy contribution
is cancelled by the curvature contribution from a smooth density of states
in the Strutinsky method, the three-point odd-even mass difference
$\Delta_n(A)$ in Eq. (\ref{eq:30}) is a good indicator of the pairing gap,
which is approximately equal to the quasi-particle energy.
It is notable that, in contrast to the $A$-dependence $5.18/A^{1/3}$ of
$\Delta_n(A)$, the $A$-dependence of $w^{(n)}_{v=1}(A)=d_n(A)$ can
be expressed as $12/\sqrt{A}$, as shown in Fig. \ref{fig8}(a).
The curve $12/\sqrt{A}$ is known to represent the $A$-dependence of the pairing
energy of the semi-empirical mass formula \cite{Zeldes}, which is estimated
with the four-point odd-even mass difference.
The shifted quasi-particle energy $d_n(A)$, therefore, corresponds to
the pairing energy of the semi-empirical mass formula or the four-point
odd-even mass difference.
The classical mass formulas, having the pairing energy term $\delta_{pair}$
and the symmetry energy term $a_T T^2$, lead to the relation
$\Delta_n(A) \approx \delta_{pair}- a_T/4$. Combining this relation and
Eq. (\ref{eq:35}), we confirm the equivalence $d_n=\delta_{pair}$.
Equation (\ref{eq:35}) indicates that the so-called pairing energy
$d_n=\delta_{pair}$ contains a symmetry energy contribution.
We can now distinguish the two curves $5.18/A^{1/3}$ and $12/\sqrt{A}$:
The former represents the three-point odd-even mass difference $\Delta_n(A)$
(which is the quasi-particle energy or the pairing gap), and the latter
represents the four-point odd-even mass difference equal to $d_n(A)$,
including the symmetry energy contribution $a_{sym}^{(n)}/4$.
\subsection{Seniority $v=2$ states of odd-odd nuclei}
The remaining task is to determine whether the mass formula (\ref{eq:28})
is effective for odd-odd nuclei. The ground state of an odd-odd
nucleus is the seniority $v=2$ state, except in the case of some $N=Z$ nuclei.
(The exceptional state with $v=0$ and $T=1$ is the $1n1p$ pair state
$S_{K=0}^\dagger |\Phi_0(A_0+m\alpha) \rangle$ considered in Fig. \ref{fig5}.)
The seniority $v=2$ state is composed of a quasi-neutron and a quasi-proton,
\begin{equation}
|A=A_0+m\alpha+2l+n+p \rangle = a_n^\dagger a_p^\dagger
|0(lS)\otimes 0(A_0+m\alpha) \rangle . \label{eq:37}
\end{equation}
The energy $w_{v=2}(A)$ for this state is defined
by $w_{v=2}(A)=W(A)-w_{T=l}(A)$.
Let us evaluate its experimental value in a manner similar to Eq. (\ref{eq:30}):
\begin{eqnarray}
& {} & w_{v=2}(A=A_0+m\alpha+2l+n+p) \nonumber \\
& {} & \ \ \ = W(N,Z) - ( W(N-1,Z-1) + W(N+1,Z+1) ) /2. \ \ \label{eq:38}
\end{eqnarray}
The calculated values are plotted in Fig. \ref{fig9}.
It is seen that there is a difference between the odd-odd $N=Z$ nuclei
and the other odd-odd nuclei. The data indicate the relations
\begin{eqnarray}
w_{v=2}(N=Z) \approx d_n + d_p , \label{eq:39} \\
w_{v=2}(N \neq Z) \approx \Delta_n + \Delta_p . \label{eq:40}
\end{eqnarray}
\begin{figure}[t]
\begin{center}
\includegraphics[width=7cm,height=7cm]{fig9.eps}
\caption{Energies $w_{v=2}(A)$ for odd-odd nuclei.}
\label{fig9}
\end{center}
\end{figure}
The parameters $\Delta_n$ and $\Delta_p$ in Eq. (\ref{eq:36})
[$d_n$ and $d_p$ in Eq.~(\ref{eq:35})]
fitted for the odd-$A$ nuclei can reproduce the experimental energies
$w_{v=2}$ of odd-odd nuclei, though it is not clear why $w_{v=2}(N=Z)$
is different from $w_{v=2}(N \neq Z)$.
This point is possibly related to the condition that there is
no Cooper pair in odd-odd $N=Z$ nuclei, while the other odd-odd nuclei
have one or more Cooper pairs.
Sometimes, correction terms are added to mass formulas for odd-odd nuclei.
The correction for the odd-odd $N=Z$ nuclei is included
in Eq. (\ref{eq:39}) in contrast to Eq. (\ref{eq:40})
for odd-odd $N \neq Z$ nuclei. We ignore another correction,
which represents an additional $n-p$ interaction,
because the deviations from the fitted curves in Fig. \ref{fig9} are
of a similar or smaller magnitude than the deviations
in Figs. \ref{fig5}--\ref{fig9}.
\section{Concluding remarks}
We have shown the essential role of the $T=0$ $2n-2p$ correlations
in the nuclear mass by considering concrete nuclear structure based on
the $jj$ coupling shell model.
We find that explicitly taking account of the effects
of the $T=0$ $2n-2p$ correlations,
which have been overlooked in the past,
is important for understanding
the nuclear mass formula. We have rearranged the mass formula
by treating the $T=0$ $2n-2p$ correlations and the $T=1$ pair correlations
as the most important correlations in nuclei. Let us write it again:
\begin{equation}
B(A) = B_{VSC}(A) + \delta U_{pot}(A)
+ E_{T=0}(A) + w_T(A) + w_v(A). \nonumber
\end{equation}
We have discussed the fact that the last three terms $E_{T=0}(A)$, $w_T(A)$ and
$w_v(A)$ represent the three modules of the structured wave functions
sketched in Eqs. (\ref{eq:10}), (\ref{eq:22}) and (\ref{eq:33}).
The systematic formulation of the $T=0$ $2n-2p$ and $T=1$ pair correlations
on the same footing makes it clear that the energy $E_{T=0}(A)$ of
the multi-quartet structure should be added to the energy $w_T(A)$
of the multi-pair structure. The $T=0$ energy plane $E_{T=0}(A)$ supplies
the base level for the measurement of the $T$-dependent energy $w_T(A)$.
The interacting boson model for the $T=1$ Cooper pair on the $\alpha$-like
superfluid base provides a structural explanation for the origins
of the symmetry energy and Wigner energy.
The two standard curves $5.18/A^{1/3}$ and $12/\sqrt{A}$ for the pairing
energy are distinguished and identified as representing the quasi-particle
energy or the pairing gap (three-point odd-even mass difference) and
the shifted quasi-particle energy
(four-point odd-even mass difference), respectively.
The $E_{T=0}(A)$ term as the base level affects the binding energies
of all nuclei. Adding $E_{T=0}(A)$ to existing mass formulas
could improve the precision. We can estimate the precision
using the parameters in Eqs. (\ref{eq:26}), (\ref{eq:27}) and (\ref{eq:36})
and the experimental values of $E_{T=0}(A)$.
The average of the root-mean-square (rms) errors estimated
is 1.42 MeV for even-even nuclei, 1.37 MeV for odd-$A$ nuclei, and
1.11 MeV for odd-odd nuclei. These values are, of course, larger than
the rms errors for modern mass formulas. The average of the rms errors
for the FRDM, for instance, is 1.08 MeV for even-even nuclei,
1.13 MeV for odd-$A$ nuclei, and 1.12 MeV for odd-odd nuclei
in the region $17 \le Z,N \le 36$.
However, it should be noted that these FRDM values are larger than
the average of the rms errors for all nuclei, 0.67 MeV.
This suggests a flaw in the FRDM mass formula for $N \approx Z$ nuclei.
The advantage of our treatment is clear if we consider nuclei
near the $N=Z$ line.
For $T<4$ nuclei, the average of the rms errors becomes 0.63 MeV
for even-even nuclei and 1.07 MeV for odd-$A$ nuclei. The good
parallelism from the $T=0$ line to the $T=3$ line in Fig. \ref{fig4} reveals
this mechanism. By contrast, the FRDM mass formula does not
show such a reduction when the number of $T=|N-Z|/2$ is limited.
There seems to be room to take into account the energy $E_{T=0}(A)$
of the fundamental $T=0$ $2n-2p$ correlated structure
in the modern mass formulas.
|
{
"timestamp": "2005-05-21T02:49:56",
"yymm": "0503",
"arxiv_id": "nucl-th/0503006",
"language": "en",
"url": "https://arxiv.org/abs/nucl-th/0503006"
}
|
\section*{Introduction}
After J.J. Thomson [1] discovered the small corpuscle which soon
became known as the electron an enormous amount of theoretical work
has been done to explain the existence of the electron. Some of the
most distinguished physicists have participated in this effort. Lorentz [2],
Poincar\'{e} [3], Ehrenfest [4], Einstein [5], Pauli [6], and others showed
that it is fairly certain that the electron cannot be explained as a purely
electromagnetic particle. In particular it was not clear how the electrical
charge could be held together in its small volume because the internal
parts of the charge repel each other. Poincar\'{e} [7] did not leave it at
showing that such an electron could not be stable, but suggested a
solution for the problem by introducing what has become known as
the Poincar\'{e} stresses whose origin however remained unexplained.
These studies were concerned with the static properties of the
electron, its mass m(e$^\pm$) and its electric charge e.
In order to explain the electron with its existing mass and charge
it appears to be necessary to add to Maxwell's equations a
non-electromagnetic mass and a non-electromagnetic force which
could hold the electric charge together.
We shall see what this mass and force is.
The discovery of the spin of the electron by Uhlenbeck and
Goudsmit [8] increased the difficulties of the problem in so far as it now
had also to be explained how the angular momentum
$\hbar$/2 and the magnetic moment $ \mu_e$ come about.
The spin of a point-like electron seemed to be explained by Dirac's [9]
equation, however it turned out later [10] that Dirac type equations can be
constructed for any value of the spin. Afterwards Schr\"{o}dinger [11]
tried to explain the spin and the magnetic moment of the electron with
the so-called Zitterbewegung. Later on many other models of the electron
were proposed. On p.74 of his book ``The Enigmatic Electron" Mac Gregor
[12] lists more than thirty such models.
At the end none of these models has been completely successful
because the problem developed a seemingly insurmountable difficulty when
it was shown through electron-electron scattering experiments that the radius
of the electron must be smaller than $10^{-16}$\,cm, in other words that the
electron appears to be a point particle, at least by three orders of
magnitude smaller than the classical electron radius r$_e$ = e$^2$/mc$^2$ = 2.8179$\cdot10^{-13}$\,cm. This, of
course, makes it very difficult to explain how a particle can have
a finite angular momentum when the radius goes to zero, and how an
electric charge can be confined in an infinitesimally small volume. If the
elementary electrical
charge were contained in a volume with a radius of O($10^{-16}$)\,cm the
Coulomb self-energy would be orders of magnitude larger than the rest
mass of the electron, which is not realistic. The choice is between a
massless point charge and a finite size particle with a non-interacting mass
to which an elementary electrical charge is attached.
We propose in the following that the non-electromagnetic mass which
seems to be necessary in order to explain the mass of the electron
consists of neutrinos. This is actually a necessary consequence of our
standing wave model [13] of the masses of the mesons and baryons.
And we propose that the non-electromagnetic force required to
hold the electric charge and the neutrinos in the electron together is the
weak nuclear force which, as we
have suggested in [13], holds together the masses of the mesons
and baryons and also the mass of the muons. Since the range of the weak
nuclear force is on the order of $10^{-16}$\,cm the neutrinos can only be
arranged in a lattice with the weak force extending from each lattice point only
to the nearest neighbors. The size of the neutrino lattice in the electron does
not at all contradict the results of the scattering experiments, just as the
explanation of the mass of the muons with the standing wave model
does not contradict the apparent point particle characteristics of the muon,
because neutrinos are in a very good approximation non-interacting and
therefore are not noticed in scattering experiments with electrons.
\section{ The mass and charge of the electron}
The rest mass of the electron is m(e$^\pm$) = 0.510\,998\,92 $\pm$
4$\cdot10^{-8}$\,MeV/c$^2$ and the electrostatic charge
of the electron is e = 4.803\,204\,41$\cdot10^{-10}$\,esu, as stated
in the Review of Particle Physics [14]. Both are
known with great accuracy. The objective of a theory of the
electron must be the explanation of both values. We will first explain
the rest mass of the electron making use of what we
have learned from the standing wave model, in particular of what
we have learned about the explanation of the mass of the
$\mu^\pm$\,mesons in [13].
The muons are leptons, just as the electrons, that means that they interact
with other particles exclusively through the electric force. The muons
have a mass which is 206.768
times larger than the mass of the electron, but they have the same
elementary electric charge as the electron or positron and the same
spin. Scattering experiments tell that the $\mu^\pm$\,mesons are point
particles with a size $<$\,$10^{-16}$\,cm, just as the electron. In other
words, the muons have the same characteristics as the electrons and
positrons but for a mass which is about 200 times larger. Consequently
the muon is often referred to as a ``heavy" electron. If a
non-electromagnetic mass is required to explain
the mass of the electron then a non-electromagnetic mass 200 times
as large as in the electron is required to explain the mass of the muons.
These non-electromagnetic masses must be \emph{non-interacting},
otherwise scattering experiments could not find the size of either the
electron or the muon at 10$^{-16}$\,cm.
We have already explained the mass of the muons with the standing wave
model [13]. According to this model the muons consist of an elementary
electric charge and a lattice of neutrinos
which, as we know, do not interact with charge or mass. Neutrinos
are the only non-interacting matter we know of.
In the muon lattice are, according to [13],
(N\,-\,1)/4 = N$^\prime$/4 muon neutrinos $\nu_\mu$ (respectively anti-muon
neutrinos $\bar{\nu}_\mu$), N$^\prime$/4 electron neutrinos $\nu_e$
and the same number of anti-electron neutrinos $\bar{\nu}_e$, one
elementary electric charge and the energy of the lattice oscillations.
The letter N stands for the number of all neutrinos and antineutrinos in
the cubic lattice of the $\pi^\pm$ mesons [13,\,Eq.(15)]
\begin{equation} \mathrm{N} = 2.854\cdot10^9\,. \end{equation}
It is, according to [13], a necessary consequence of the decay of the
$\mu^-$ muon $\mu^- \rightarrow$ e$^- + \bar{\nu}_e + \nu_\mu$ that there
must be N$^\prime$/4 electron neutrinos $\nu_e$ in the emitted electron,
where N$^\prime$ = N - 1 $\cong$ N [13].
For the mass of the electron neutrinos and anti-electron neutrinos we found
in Eq.(34) of [13] that
\begin{equation} \mathrm{m}(\nu_e) = \mathrm{m}(\bar{\nu}_e) =
0.365 \,\mathrm{milli\,eV/c^2}\,. \end{equation}
\noindent
The sum of the energies in the rest masses of the N$^\prime$/4 neutrinos
or antineutrinos in the lattice of the electron or positron is then
\begin{equation} \sum{\,\mathrm{m(\nu_e)c^2}} =
\mathrm{N}^\prime/4\cdot\mathrm{m}(\nu_e)\mathrm{c}^2 =
0.260\,43\,\mathrm{MeV}
= 0.5096\,\mathrm{m(e^\pm)}\mathrm{c}^2\,. \end{equation}
To put this in other words, one half of the
rest mass of the electron comes from the rest masses of
electron neutrinos. The other half of the rest mass of the
electron must originate from the energy in the electric charge carried
by the electron. From pair production
$\gamma$ + M $\rightarrow$ e$^-$ + e$^+$ + M, (M being any
nucleus), and from conservation of neutrino numbers follows necessarily
that there must also be a neutrino lattice
composed of N$^\prime$/4 anti-electron neutrinos, which make up
the lattice of the positrons, which lattice has, because of Eq.(2), the
same rest mass as the neutrino lattice of the electron, as it must be for
the antiparticle of the electron.
Fourier analysis dictates that a continuum of high frequencies must be in
the electrons or positrons created by pair production in a timespan of
$10^{-23}$ seconds. We will now determine the energy E$_\nu$(e$^\pm$)
contained in the oscillations in the interior of the electron. Since
we want to explain the \emph{rest mass} of the electron we can only
consider the frequencies of non-progressive waves, either standing waves
or circular waves. The sum of the energies of the lattice
oscillations is, in the case of the $\pi^\pm$\,mesons, given by
\begin{equation} \mathrm{E}_\nu(\pi^\pm) =
\frac{\mathrm{h}\nu_0\mathrm{N}}{2\pi(\mathrm{e^{h\nu/kT}}\,\mathrm{-}\,1)}
\,\int\limits_{-\pi}^{\pi}\,\phi\,d\phi\,. \end{equation}
\noindent
This is Eq.(14) combined with Eq.(16) in [13] where they were used to
determine the oscillation energy
in the $\pi^0$ and $\pi^\pm$ mesons. This equation was introduced
by Born and v.\,Karman [15] in order to explain the internal
energy of cubic crystals. In Eq.(4) h is Planck's constant,
$\nu_0$ = c/2$\pi\emph{a}$ is the reference frequency with the lattice
constant \emph{a} = $10^{-16}$ cm, N is the number of all oscillations,
$\phi = 2\pi\emph{a}/\lambda$ and T is the temperature in the lattice,
for which we found in [13] the value T = 2.38\,$\cdot$\,$10^{14}$\,K.
If we apply Eq.(4) to
the oscillations in the electron which has N$^\prime$/4 electron neutrinos
$\nu_e$ we arrive at E$_\nu$(e$^\pm)$ = 1/4$\cdot$E$_\nu(\pi^\pm$),
which is mistaken because E$_\nu(\pi^\pm$) $\approx$ m($\pi^\pm$)c$^2$/2
and m($\pi^\pm$) $\approx$ 273\,m(e$^\pm$). Eq.(4) must
be modified in order to be suitable for the oscillations in the electron.
It turns out that we must use
\begin{equation} \mathrm{E}_\nu(\mathrm{e}^\pm) =
\frac{\mathrm{h}\nu_0\mathrm{N}\cdot\alpha_f}{2\pi(\mathrm{e^{h\nu/kT}}\,
\mathrm{-}\,1)}\,\int\limits_{-\pi}^{\pi}\,\phi\,d\phi\,,
\end{equation}
\noindent
where $\alpha_f$ is the fine structure constant. The appearance of
$\alpha_f$ in Eq.(5)
indicates that the nature of the oscillations in the electron is different
from the oscillations
in the $\pi^0$ or $\pi^\pm$ lattices. With $\alpha_f$ = e$^2/\hbar$c and
$\nu_0$ = c/2$\pi$\emph{a} we have
\begin{equation}h\nu_0\alpha_f = e^2/\emph{a}\, \end{equation}
that means that the oscillations in the electron are \emph{electric oscillations}.
There must be N$^\prime$/2 oscillations of the elements of the electric
charge in e$^\pm$, because we deal with non-progressive waves, the
superposition of two waves. As we will see later the spin requires
that the oscillations are circular. That means that 2$\times$N$^\prime$/4
$\cong$ N/2 oscillations are in Eq.(5). From Eqs.(4,5) then follows that
\begin{equation} \mathrm{E_\nu(e}^\pm) =
\alpha_f/2\cdot\mathrm{E}_\nu(\pi^\pm)\,. \end{equation}
\noindent
E$_\nu(\pi^\pm)$ is the oscillation energy in the $\pi^\pm$\,mesons which
can be calculated with Eq.(4). According to Eq.(27) of [13] it is
\begin{equation} \mathrm{E}_\nu(\pi^\pm) = 67.82 \,\mathrm{MeV} =
0.486\,\mathrm{m}(\pi^\pm)\mathrm{c}^2 \approx
\mathrm{m}(\pi^\pm)\mathrm{c}^2/2\,.
\end{equation}
With E$_\nu(\pi^\pm$) $\approx$ m($\pi^\pm$)c$^2$/2 = 139.57/2\,MeV
and $\alpha_f$ = 1/137.036 follows from Eq.(7) that
\begin{equation} \mathrm{E_\nu(e}^\pm) = \frac{\alpha_f}{2}\cdot
\frac{\mathrm{m}(\pi^\pm)\mathrm{c}^2}{2} = 0.254\,62\,\mathrm{MeV}
= 0.99657\,\mathrm{m(e^\pm)}\mathrm{c}^2/2\,.
\end{equation}
\noindent
We have determined the value of the oscillation energy in e$^\pm$ from
the product of the very accurately known fine structure constant and the
very accurately known rest mass of the $\pi^\pm$\,mesons. \emph{One half
of the energy in the rest mass of the electron comes from the electric oscillations
in the electron}. The other half of the energy in the rest mass of the electron is
in the rest masses of the neutrinos in the electron.
We can confirm Eq.(9)
using Eq.(5) or Eq.(13) with N/2 = 1.427$\cdot10^9$,
e = 4.803$\cdot10^{-10}$\,esu,
$\emph{a}$ = 1$\cdot10^{-16}$\,cm, f(T) = 1/1.305$\cdot10^{13}$, and
with the integral being $\pi^2$ we obtain E$_\nu$(e$^\pm$) =
0.968\,m($\mathrm{e}^\pm)$c$^2$/2.
This calculation involves more parameters than Eq.(9) and is consequently
less accurate than Eq.(9).
In a good approximation the oscillation energy of e$^\pm$ in Eq.(9) is
equal to the sum of the energies in the rest masses
of the electron neutrinos in the e$^\pm$ lattice in Eq.(3). Since
\begin{equation} \mathrm{m(e}^\pm)\mathrm{c}^2
= \mathrm{E}_\nu(\mathrm{e}^\pm) + \sum{\,\mathrm{m}(\nu_e)\mathrm{c}^2}
= \mathrm{E_\nu(e^\pm)} + \mathrm{N^\prime/4\cdot m(\nu_e)c^2}\,,
\end{equation}
\noindent
it follows from Eqs.(3) and (9) that
\begin{equation} \mathrm{m(e^\pm)c^2(theor)} = 0.5151\,\mathrm{MeV} =
1.0079\,\mathrm{m(e^\pm)c^2(exp)}\,. \end{equation}
The measured rest mass of the electron or positron agrees within the accuracy
of the parameters N and m($\nu_e)$ with the theoretically
predicted rest masses.
From Eq.(7) follows with E$_\nu(\pi^\pm)$ $\cong$ m($\pi^\pm$)c$^2$/2 that
\vspace{0.5cm}
\centerline{2E$_\nu$(e$^\pm)$ $\cong$ m(e$^\pm$)c$^2$
= $\alpha_f$E$_\nu(\pi^\pm)$ = $\alpha_f$m$(\pi^\pm)$c$^2$/2\,,}
\vspace{0.5cm}
\noindent
or that
\begin{equation} \mathrm{m(e^\pm)}\cdot2/\alpha_f =
274.072\,\mathrm{m(e^\pm)}
\cong \mathrm{m(\pi^\pm)}\,, \end{equation}
whereas the actual ratio of the mass of the $\pi^\pm$\,mesons to the mass
of the electron is
m($\pi^\pm$)/m(e$^\pm$) = 273.132 or 0.9965$\cdot$2/$\alpha_f$. We
have here recovered the ratio m($\pi^\pm$)/m(e$^\pm$) which we found
with the standing wave model of the $\pi^\pm$\,mesons, Eq.(65) of [13].
This seems to be a necessary condition for the validity of our model of
the electron.
We have thus shown that the \emph{rest mass of the electron can be
explained} by the sum of the rest masses of the electron neutrinos in a cubic
lattice with N$^\prime$/4 electron neutrinos $\nu_e$ and the mass in the
sum of the energy of N/2 electric oscillations in the lattice, Eq.(9). The one
oscillation added to the 2$\times$N$^\prime$/4 oscillations is the
oscillation at the center of the lattice, Fig.(1). From this model follows, since
it deals with a cubic neutrino lattice, that \emph{the electron is not a point
particle}, which is unlikely to begin with, because at a true point the self-energy
would be infinite. However, since neutrinos are non-interacting their presence
will not be detected in electron-electron scattering experiments.
\begin{figure}[h]
\vspace{0.5cm}
\hspace{2.2cm}
\includegraphics{elat.eps}
\vspace{-0.2cm}
\begin{quote}
Fig.1. Horizontal or vertical section through the central part of\\
\indent\hspace{1.1cm} the electron lattice.
\end{quote}
\end{figure}
The \emph{rest mass of the muon} has been explained similarly with an
oscillating lattice of muon and electron neutrinos [13].
We found that m($\mu^\pm)$/m(e$^\pm$) is\\
$\cong$ 3/2$\alpha_f$ = 205.55, nearly equal to the actual mass ratio
206.768, in agreement with what Nambu [16] found empirically.
The heavy weight of the muon is primarily a consequence
of the heavy weight of the N$^\prime$/4 muon neutrinos in the muon lattice.
The mass of the muon neutrino is related to the mass of the electron
neutrino through m($\nu_e)$ = $\alpha_f$m($\nu_\mu$), Eq.(39) of [13].
In order to confirm the \emph{validity} of our preceding explanation of the
mass of the electron we must show that the sum of the charges of the electric
oscillations in the interior of the electron is equal to the elementary electric
charge of the electron.
We recall that Fourier analysis requires that, after pair production,
there must be a continuum of frequencies in the electron and positron.
With h$\nu_0\alpha_f$ = e$^2$/\emph{a} from Eq.(6) follows
from Eq.(5) that the oscillation energy in e$^\pm$ is the sum of
2$\times$(N$^\prime$/4 + 1) $\cong$ N/2 electric oscillations
\begin{equation} \mathrm{E}_\nu(\mathrm{e}^\pm) =
\frac{\mathrm{N}}{2} \cdot \frac{\mathrm{e^2}}{\emph{a}}\cdot
\frac{f(T)}{2\pi}\,
\int\limits_{-\pi}^{\pi}\,\phi\,\mathrm{d}\phi\,, \end{equation}
with f(T) = 1/(e$^{h\nu/kT} \mathrm{-}$ 1) = 1/1.305$\cdot10^{13}$
from p.17 in [13]. Inserting the values for N, f(T) and \emph{a} we find
that E$_\nu$(e$^\pm$) = 0.968\,m(e$^\pm$)c$^2$/2. The discrepancy
between m(e$^\pm$)c$^2$/2 and E$_\nu$(e$^\pm$) so calculated
must originate from the uncertainty of the parameters N, f(T) and \emph{a}
in Eq.(13). We note that it follows from the factor e$^2$/\emph{a} in
Eq.(13) that the oscillation energy is the same for electrons and positrons,
as it must be.
We replace the integral divided by 2$\pi$ in Eq.(13), which has the value $\pi$/2,
by the sum $\Sigma\,\phi_k\Delta\,\phi$, where k is an integer number with the
maximal value k$_m$ = (N/4)$^{1/3}$. $\phi_k$ is equal to k$\pi$/k$_m$
and we have
\begin{displaymath} \Sigma\,\phi_k\,\Delta\phi = \sum_{k=1}^{k_m}\,\frac{k\pi}{k_m}\cdot\frac{1}{k_m}
= \frac{ k_m(k_m + 1)\pi}{2\,k_m^2} \cong \frac{\pi}{2}\,,
\end{displaymath}
\noindent as it must be. The energy in the individual electric oscillation with
index k is then
\begin{equation} \Delta\mathrm{E}_\nu(k) = \phi_k\,\Delta\phi = k\pi/k_m^2\,.
\end{equation}
Suppose that the energy of the electric oscillations is correctly
described by the self-energy of an electrical charge
\begin{equation}\mathrm{U} = 1/2\,\cdot\,\mathrm{e}^2/\mathrm{r}\,.
\end{equation}
The self-energy of the elementary electrical charge is normally used to
determine the mass of the electron from its charge, here we use Eq.(15)
the other way around, we determine the charge from the energy
in the oscillations.
The charge of the electron is contained in the electric oscillations.
That means that \emph{the electric charge is not concentrated in a
point} but is distributed over N/4 = O($10^9)$ charge elements
Q$_k$. \emph{The charge elements are distributed in a cubic lattice} and the
resulting electric field is cubic, not spherical. For distances large as compared
to the sidelength of the cube, (which is O($10^{-13}$)\,cm), say at the first
Bohr radius which is on the order of $10^{-8}$\,cm, the deviation of the cubic
field from the spherical field will be reduced by about $10^{-10}$.
The charge in all electric oscillations is
\begin{equation} \mathrm{Q} = \sum_{k}\,\mathrm{Q}_\mathrm{k}\,. \end{equation}
Setting the radius r in the formula for the self-energy equal to 2\,\emph{a} we find,
with Eqs.(13,14,15), that the charge in the individual electric oscillations is
\begin{equation} \mathrm{Q_k} = \pm\,\sqrt{2\pi\,N\,e^2f(T)/k_m^2}\,\cdot\,\sqrt{k}\,.
\end{equation}
\noindent
and with k$_m$ = 1/2\,(N/4)$^{1/3}$ = 447 and
\begin{displaymath} \sum_{k=1}^{k_m}\,\sqrt{k} = 6310.8\, \end{displaymath}
\noindent follows, after we have doubled the sum over $\sqrt{k}$, because for
each index k there is a second oscillation on the negative axis of $\phi$, that
\begin{equation} \mathrm{Q} = \Sigma\,\mathrm{Q_k} =
\pm\,5.027\cdot10^{-10}\,\,\mathrm{esu}\,,\end{equation}
\noindent whereas the elementary electrical charge is e =
$\pm$\,4.803\,$\cdot\,10^{-10}$\,esu. That means that our theoretical charge of the
electron is 1.047 times the elementary electrical charge. Within the uncertainty
of the parameters the theoretical charge of the electron agrees with
the experimental charge e. We have confirmed that it follows from our
explanation of the mass of the electron that the electron has, within a
5\% error, the correct electrical charge.
Each element of the charge distribution is surrounded in the horizontal
plane by four electron neutrinos as in Fig.(1), and in vertical direction by an
electron neutrino above and also below the element. The electron neutrinos
hold the charge elements in place. We must assume that the charge
elements are bound to the neutrinos by the weak nuclear force. The
weak nuclear force plays here a role similar to its role in holding, for example,
the $\pi^\pm$ or $\mu^\pm$ lattice together. It
is not possibe, in the absence of a definitive explanation
of the neutrinos, to give a theoretical explanation for the electro-weak
interaction between the electric oscillations and the neutrinos.
However, the presence of the
range \emph{a} of the weak nuclear force in e$^2$/\emph{a} is a sign that
the weak force is involved in the electric oscillations. The attraction of the
charge elements by the
neutrinos overcomes the Coulomb repulsion of the charge elements.
The weak nuclear force is the missing non-electromagnetic force or the
Poincar\'{e} stress which holds the elementary electric charge together.
The same considerations
apply for the positive electric charge of the positron, only that then the
electric oscillations are all of the positive sign and that they are
bound to anti-electron neutrinos.
Finally we learn that Eq.(13) precludes the possibility that the charge of
the electron sits only on its surface. The number N in Eq.(13) would then be
on the order of $10^6$, whereas N must be on the order of $10^9$ so that
E$_\nu$(e$^\pm$) can be m($\mathrm{e}^\pm)$c$^2$/2 as is necessary.
In other words, the charge of the electron must be distributed throughout the
interior of the electron, as we assumed.
Summing up: The rest mass of the electron and positron originates from
the sum of the rest masses of N$^\prime$/4 electron neutrinos or anti-electron
neutrinos in cubic lattices plus the mass in the energy of N$^\prime$/2 electric
oscillations in the neutrino lattices. That means that neither the electron nor the
positron are point particles. The electric oscillations are attached to the
neutrinos by the weak nuclear force. The sum of the charge elements
of the electric oscillations
accounts for the elementary charge of the electron, respectively positron.
\section{The spin and magnetic moment \\ of the electron}
The model of the electron we have proposed in the preceding
chapter has, in order to be valid, to pass a crucial test; the model has
to explain satisfactorily the spin and the magnetic moment of the
electron. When Uhlenbeck and Goudsmit [8] (U\&G) discovered
the existence of the spin of the electron they also proposed that the
electron has a magnetic moment with a value equal to Bohr's
magnetic moment $\mu_B$ = e$\hbar$/2m$(\mathrm{e}^\pm)$c. Bohr's
magnetic moment results from the motion of an electron on a circular
orbit around a proton. The magnetic moment of the electron postulated
by U\&G has been confirmed experimentally, but has been corrected by
about 0.11\% for the so-called anomalous magnetic moment.
If one tries to explain the magnetic moment of the electron
with an electric charge moving on a circular orbit around the
particle center, analogous to the magnetic moment of hydrogen, one ends
up with velocities larger than the velocity of light, which cannot be, as
already noted by U\&G. It remains to be explained how the magnetic
moment of the electron comes about.
We will have to explain the spin of the electron first. The spin, or
the intrinsic angular momentum of a particle is, of course, the sum of the
angular momentum vectors of all components of the particle. In the
electron these are the neutrinos and the electric oscillations. Each
neutrino has spin 1/2 and in order for the electron to have
s = 1/2 all, or all but one, of the spin vectors of the neutrinos in their
lattice must cancel.
If the neutrinos are in a simple cubic lattice as in Fig.(1) and the
center particle of the lattice is not a neutrino, as in Fig.(1), the
spin vectors of all neutrinos in the lattice cancel, $\Sigma\,j(n_i)$ = 0,
provided that the spin vectors of the
electron neutrinos of the lattice point in opposite direction at their
mirror points in the lattice. Otherwise the spin vectors of the neutrinos
would add up and make a very large angular momentum. We
follow here the procedure we used in [17] to explain the spin
of the muons. The spin vectors of all electron neutrinos in the electron
cancel just as the spin vectors of all muon and electron neutrinos
in the muons cancel because there is
a neutrino vacancy at the center of their lattices, (Fig.(1) of [17]).
We will now see whether the electric oscillations in the electron
contribute to its angular momentum.
As we said in context with Eq.(7) there must be two times
as many electric oscillations in the electron lattice than there are
neutrinos. The oscillation pairs can either be the two oscillations in a
standing wave or they can be two circular oscillations. Both the standing
waves and the circular oscillations are non-progressive and can be part of
the \emph{rest mass} of a particle. We will now assume that the electric
oscillations are circular. Circular oscillations have an
angular momentum $\vec{j} = m\,\vec{r}\times\vec{v}$. And, as in the case
of the spin vectors of the neutrinos, all or all but one of the O$(10^9)$
angular momentum vectors of the electric oscillations must cancel in order
for the electron to have spin 1/2. As in [13] we will describe the
superposition of the two circular oscillations by
\begin{equation} x(t) = exp[i\omega t] + exp[-\,i(\omega t +
\pi)]\,,\end{equation}
\begin{equation} y(t) = exp[i(\omega t + \pi/2)] + exp[-\,i(\omega t +
3\pi/2)]\,\,,
\end{equation}
\noindent
that means by the superposition of a circular oscillation with the
frequency $\omega$ and a second circular oscillation with the frequency
$\mathrm{-}\,\omega$. The latter oscillation is shifted in phase by
$\pi$. Negative frequencies are permitted solutions of the equations
of motion in a cubic lattice, Eqs.(7,13) of [13].
As is well-known oscillating electric charges should emit radiation.
However, this rule does already not hold in the hydrogen atom, so we
will assume that the rule does not hold in the electron either.
In circular oscillations the kinetic energy is always equal to the potential
energy and the sum of both is the total energy. From
\begin{equation} \mathrm{E}_{pot} + \mathrm{E}_{kin} = 2\,\mathrm{E}_{kin}
=\mathrm{E}_{tot} \end{equation}
follows with E$_{kin}$ = $\Theta\,\omega^2$/2 and E$_{tot}$ = $\hbar\omega$
that 2\,E$_{kin}$ = $\Theta\,\omega^2$ = $\hbar\omega$.
${\Theta}$ is the moment of inertia. When we superpose the two
circular oscillations with $\omega$ and $\mathrm{-}\,\omega$ of Eqs.(19,20)
we have
\begin{equation} 2\times2\,\mathrm{E}_{kin} = 2\,\Theta\,\omega^2 =
\hbar\omega\,,\end{equation}
from which follows that the angular momentum is
\begin{equation} j = \Theta\,\omega = \hbar/2\,. \end{equation}
That means that each of the O$(10^9$) pairs of superposed circular
oscillations has an angular momentum $\hbar$/2.
The circulation of the oscillation pairs in Eqs.(19,20) is opposite for
all $\omega$ of opposite sign. It follows from the equation for the
displacements u$_n$ of the lattice points
\begin{equation} u_n = Ae^{i(\omega\,t\, +\, n\phi)}\,,
\end{equation}
\noindent
(Eq.(5) in [13]) that the velocities of the lattice points are given by
\begin{equation} v_n = \dot{u}_n = i\,\omega_n\,u_n\,. \end{equation}
The sign of $\omega_n$ changes with the sign of $\phi$ because the
frequencies are given by Eq.(13) of [13], that means by
\begin{equation} \omega_n = \pm\,\omega_0\,[\,\phi_n + \phi_0\,]\,.
\end{equation}
Consequently the circulation of the electric oscillations is opposite to
the circulation at the
mirror points in the lattice and the angular momentum vectors cancel,
but for the angular momentum vector of the electric oscillation at
the\,\emph{ center of the lattice}. The center circular oscillation has, as all
other electric oscillations, the angular momentum $\hbar$/2 as Eq.(23)
says. The angular momentum of the entire electron lattice is therefore
\begin{equation} j(\mathrm{e}^\pm) = \sum\,j(n_i) + \sum\,j(el_i) = j(el_0)
=\hbar/2\,,
\end{equation}
as it must be for spin s = 1/2. The explanation of the spin of the electron given
here follows the explanation of the spin of the baryons in [13], as well
as the explanation of the absence of spin in the mesons. A valid explanation
of the spin must be applicable to all particles, in particular to the electron, the
prototype of a particle with spin.
We will now turn to the magnetic moment of the electron which is known
with extraordinary accuracy, $\mu(\mathrm{e}^\pm)$ =
1.001\,159\,652\,187\,$\mu_B$, according
to the Review of Particle Physics [14], with $\mu_B$ being the Bohr
magneton. The decimals after 1.00\,$\mu_B$
are caused by the anomalous magnetic moment which we will not consider.
As is well-known the magnetic dipole moment of a particle with spin is,
in Gaussian units, given by
\begin{equation} \vec{\mu} = g\,\frac{e\hbar}{2mc}\,\vec{s}\,, \end{equation}
where g is the dimensionless Land\'{e} factor, m the rest mass of the
particle and $\vec{s}$ the spin vector.
The g-factor has been introduced in order to bring the magnetic moment
of the electron into agreement with the experimental facts. As U\&G
postulated and as has been confirmed experimentally the
g-factor of the electron is 2. With the spin s = 1/2 of the electron the
magnetic dipole moment of the electron is then
\begin{equation} \mu(\mathrm{e}^\pm) =
\mathrm{e}\hbar/2\mathrm{m}(\mathrm{e}^\pm)c\,, \end{equation}
or one Bohr magneton in agreement with the experiments, neglecting the
anomalous moment. For a structureless
point particle Dirac [9] has explained why g = 2 for the electron. However
we consider here an electron with a finite size and which is at rest, which
means that the velocity of the center of mass is zero. When it is at
rest the electron has still its magnetic moment. Dirac's theory does
therefore not apply here.
The only part of Eq.(28) that can be changed in order to explain the
g-factor of an electron with structure is the ratio e/m which deals with the
spatial distribution of charge and mass. In the classical electron models
the mass originates from the charge.
However that is not necessarily always so. If part of the mass of the
electron is non-electrodynamic and the non-electrodynamic part of the mass
does not contribute to the magnetic moment of the electron, which to all
that we know is true for neutrinos, then the ratio e/m in Eq.(28) is not
e/m($\mathrm{e}^\pm$) in the case of the electron. The elementary charge
e certainly remains unchanged, but e/m depends on what fraction of the
mass is of electrodynamic origin and what fraction of m is
non-electrodynamic, just as the mass of a current loop does not contribute
to the magnetic moment of the loop. From the very accurately known values
of $\alpha_f$, m($\pi^\pm$)c$^2$ and m(e$^\pm$)c$^2$ and from Eq.(9)
for the energy in the electric oscillations in the electron E$_\nu$(e$^\pm$) =
$\alpha_f$/2\,$\cdot$\,m($\pi^\pm$)c$^2$/2 =
0.996570\,m(e$^\pm$)c$^2$/2 follows
that very nearly one half of the mass of the electron is of electric origin,
whereas the other half of m($\mathrm{e}^\pm)$ is made of neutrinos
which do not contribute to the magnetic moment. That means that in the
electron the mass in e/m is practically m($\mathrm{e}^\pm$)/2. The
magnetic moment of the electron is then
\begin{equation}
\vec{\mu}_e = g \frac{e\hbar}{2m(e^\pm)/2\cdot c}\vec{s}\,,
\end{equation}
and with s = 1/2 we have $\mu(\mathrm{e}^\pm)$ =
g\,e$\hbar$/2m($\mathrm{e}^\pm$)c. Because of
Eq.(29) the g-factor must be equal to one and is unnecessary. In other words,
if the electron is composed of the neutrino lattice and the electric oscillations
as we have suggested, then the electron has the correct magnetic moment
$\mu_e$ = e$\hbar/2\mathrm{m(e^\pm)c}$, if exactly 1/2 of the electron
mass consists of neutrinos.
The preceding explanation of the magnetic moment of the electron has
to pass a critical test, namely it has to be shown that the same
considerations lead to a correct explanation of the magnetic moment of
the muon $\mu_\mu$ = e$\hbar$/2m($\mu^\pm$)c, which is about 1/200th
of the magnetic moment of the electron but is known with nearly the same
accuracy as $\mu_e$. Both magnetic moments are related
through the equation
\begin{equation} \frac{\mu_\mu}{\mu_e} =
\frac{\mathrm{m(e}^\pm)}{\mathrm{m}(\mu^\pm)} = \frac{1}{206.768}\,,
\end{equation}
as follows from Eq.(28) applied to the electron and muon.
This equation agrees with the experimental results to the sixth decimal.
The muon has, as the electron, an anomalous magnetic moment of about
0.11\,\% $\mu_\mu$, which is too small to be considered here.
In the standing wave model [13] the muons consist of a lattice of
N$^\prime$/4 muon neutrinos $\nu_\mu$, respectively anti-muon
neutrinos $\bar{\nu}_\mu$, of N$^\prime$/4 electron neutrinos and
the same number of anti-electron neutrinos plus an elementary electric
charge. For the explanation of the magnetic moment of the muon we
follow the same reasoning we have used for the explanation of the
magnetic moment of the electron. We say that m($\mu^\pm)$ consists
of two parts, one part which causes the magnetic moment and another
part which does not contribute to the magnetic moment. The part
of m($\mu^\pm$) which causes the magnetic moment must contain
circular electric oscillations without which there would
be no magnetic moment. It becomes immediately clear from the small
mass of the electron neutrinos and from Eq.(5) for the energy of the
electric oscillations in the electron that $\Sigma$\,m($\nu_e$) and
E$_\nu$(e$^\pm$) are too small, as compared to the energy in the
rest masses of all neutrinos in the muons, to make up m($\mu^\pm)$/2.
However, the oscillations in the $\mu^\pm$\,mesons do not follow
Eq.(5) for the oscillation energy in the electron, but rather Eq.(4) for
the oscillation energy in the muons. Both differ by the factor $\alpha_f$
in Eq.(5). But even when the oscillation energy in the muons as given
by Eq.(4) is considered, the energy of the electric oscillations in the
muons would be only E$_\nu(\mu^\pm)$/4 = 16.955\,MeV, if the
electric oscillations are attached to N/4 electron neutrinos, as is the
case in the electron.
It appears to be necessary to consider the case that the electric
oscillations in the $\mu^\pm$\,mesons are attached to \emph{all}
neutrinos of the electron neutrino type in the $\mu^\pm$ lattice,
regardless whether they are
electron neutrinos or anti-electron neutrinos. That would mean that
the electric charge is distributed uniformly in the $\mu^\pm$ lattice.
There are, as has been shown in the paragraph below Eq.(31) of [13],
3/4$\cdot$N neutrinos of the electron neutrino type in the muons, of
which N/4 neutrinos originate from the charge e$^\pm$ carried by
$\mu^\pm$. If the electric oscillations are attached to 3/4$\cdot$N
electron neutrinos, regardless of their type, then the energy in all
electric oscilllations or the energy in the electric charge is, with Eq.(8)
and E$_\nu(\mu^\pm$) = E$_\nu(\pi^\pm$) = 67.82\,MeV from
Eq.(31) in [13], as well as with m($\mu^\pm$)c$^2$ = 105.6583\,MeV,
given by
\vspace{1cm}
\begin{eqnarray}
\lefteqn{3/4\cdot\mathrm{E}_\nu(\mu^\pm) =
3/4\cdot 67.82\,\mathrm{MeV} = 50.865\,\mathrm{MeV}}\nonumber\\&
= & 0.4814\,\mathrm{m}(\mu^\pm)\mathrm{c}^2 \cong
1/2\cdot \mathrm{m}(\mu^\pm)\mathrm{c}^2\,. \end{eqnarray}
In other words, the energy in the electric oscillations or the electric
charge makes up, in a good approximation, 1/2 of the mass of the muons.
The other half of the rest mass of the muons consists of the sum of the
rest masses of the neutrinos in the muon lattice plus the oscillation energy
of the muon neutrinos, neither of which contributes to the magnetic moment.
It is
\begin{equation} 1/4\cdot\mathrm{E}_\nu(\mu^\pm) +
\mathrm{N}/4\cdot\mathrm{m}(\nu_\mu)\mathrm{c}^2 +
3/4\cdot\mathrm{Nm}(\nu_e)\mathrm{c}^2 \\
= 53.347\,\mathrm{MeV} = 0.50490\,\mathrm{m}(\mu^\pm)\mathrm{c}^2\,.
\end{equation}
The theoretical total energy in the rest mass of the muons is then
E(m($\mu^\pm$)) = 0.9863\,m($\mu^\pm$)c$^2$(exp).
In simple terms, if E$_\nu(\pi^\pm)$ = E$_\nu(\mu^\pm$) =
1/2$\cdot$m($\pi^\pm$), not 0.486\,m($\pi^\pm$) as in Eq.(27)
of [13], then it follows from 3/4$\cdot$E$_\nu(\mu^\pm$) =
3/8$\cdot$m($\pi^\pm$) and from the neutral part of the muon mass in Eq.(33),
which is likewise $\approx$3/8$\cdot$m($\pi^\pm$), that the rest mass of the
muons is m($\mu^\pm$) $\cong$ 3/8$\cdot$m($\pi^\pm$) + 3/8$\cdot$m($\pi^\pm$)
= 3/4$\cdot$ m($\pi^\pm$), as it must be in a first approximation,
whereas the actual m($\mu^\pm$) is
1.00937$\cdot$3/4$\cdot$m($\pi^\pm$). That means that in a good
approximation the charged part of the rest mass of the muons is 1/2
of the mass of the muons.
If the charged part of the muon mass as expressed by Eq.(32) makes up
1/2 of the mass of the muons and if the other part of the muon mass does
not contribute to the magnetic moment, then the magnetic moment of the
muon is given by
\begin{equation} \vec{\mu}_\mu = \frac{\mathrm{e}\hbar}
{2\mathrm{m}(\mu^\pm)/2\cdot\mathrm{c}}\cdot\vec{s}\,. \end{equation}
With s = 1/2 we have $\mu_\mu$ = e$\hbar$/2m$(\mu^\pm)$c as it must
be, without the artificial g-factor.
\section*{Conclusions}
One hundred years of sophisticated theoretical work have made it abundantly
clear that the electron is not a purely electromagnetic particle. There must be
something else in the electron but electric charge. It is equally clear from the
most advanced scattering experiments that the ``something else" in the
electron must be non-interacting, otherwise it could not be that we find that
the radius of the electron must be smaller than $10^{-16}$\,cm. The only
non-interacting matter we know of with certainty are the neutrinos. So it seems
to be natural to ask whether neutrinos are not part of the electron. Actually
we have not introduced the neutrinos in an axiomatic manner but rather as a
consequence of our standing wave model of the stable mesons, baryons
and $\mu$-mesons. It follows necessarily from this model that after the
decay of the $\mu^-$\,meson there must be electron neutrinos in the emitted
electron, and that they make up one half of the mass of the electron. The other
half of the energy in the electron originates from the energy of electric
oscillations. The theoretical rest mass
of the electron agrees, within 1\% accuracy, with the experimental
value of m(e$^\pm$). We have learned that the charge of the electron
is not concentrated in a single point, but rather is distributed over O(10$^9$)
elements which are held together with the neutrinos by the weak nuclear force.
The sum of the charges in the electric oscillations is, within the accuracy of the
parameters, equal to the elementary electrical charge of the electron.
From the explanation of the mass and charge of the electron follows,
as it must be, the correct spin and magnetic moment of the electron, the
other two fundamental features of the electron. With a cubic lattice of anti-electron
neutrinos we also arrive with the same considerations as above at the correct
mass, charge, spin and magnetic moment of the positron.
\section*{Acknowledgements}
Contributions of Professor J. Zierep and of Dr. T. Koschmieder are gratefully
acknowledged.
\section*{References}
\noindent
[1] Thomson, J.J. Phil.Mag. {\bfseries44},293 (1897).
\smallskip
\noindent
[2] Lorentz, H.A. \emph{Enzykl.Math.Wiss.} Vol.{\bfseries5},188 (1903).
\noindent
[3] Poincar\'{e}, H. Compt.Rend. {\bfseries 140},1504 (1905). Translated
in:\\
\indent
Logunov, A.A. \emph{On The Articles by Henri Poincare\\
\indent
``On The Dynamics of the Electron"}, Dubna JINR (2001).
\smallskip
\noindent
[4] Ehrenfest, P. Ann.Phys. {\bfseries24},204 (1907).
\smallskip
\noindent
[5] Einstein, A. Sitzungsber.Preuss.Akad.Wiss. {\bfseries20},349 (1919).
\smallskip
\noindent
[6] Pauli, W. \emph{Relativit\"{a}tstheorie}, B.G. Teubner (1921).
Translated in:\\
\indent Theory of Relativity, Pergamon Press (1958).
\smallskip
\noindent
[7] Poincar\'{e}, H. Rend.Circ.Mat.Palermo {\bfseries21},129 (1906).
\smallskip
\noindent
[8] Uhlenbeck, G.E. and Goudsmit, S. Naturwiss. {\bfseries13},953 (1925).
\smallskip
\noindent
[9] Dirac, P.A.M. Proc.Roy.Soc.London A{\bfseries117},610 (1928).
\smallskip
\noindent
[10] Gottfried, K. and Weisskopf, V.F. \emph{Concepts of Particle
Physics},\\
\indent
\,\,Vol.1,\,p.38. Oxford University Press (1984).
\smallskip
\noindent
[11] Schr\"{o}dinger, E. Sitzungsber.Preuss.Akad.Wiss. {\bfseries24},418
(1930).
\smallskip
\noindent
[12] Mac Gregor, M.H. \emph{The Enigmatic Electron}, Kluwer (1992).
\smallskip
\noindent
[13] Koschmieder, E.L. http://arXiv.org/phys/0602037 (2006).
\smallskip
\noindent
[14] Eidelman, S. et al. Phys.Lett.B {\bfseries 592},1 (2004).
\smallskip
\noindent
[15] Born, M. and v.\,Karman, Th. Phys.Z. {\bfseries13},297 (1912).
\smallskip
\noindent
[16] Nambu, Y. Prog.Th.Phys. {\bfseries7},595 (1952).
\smallskip
\noindent
[17] Koschmieder, E.L. http://arXiv.org/physics/0308069 (2003),\\
\indent\,\, \emph{Muons: New Research}, Nova (2005).
\end{document}
|
{
"timestamp": "2006-09-26T18:26:57",
"yymm": "0503",
"arxiv_id": "physics/0503206",
"language": "en",
"url": "https://arxiv.org/abs/physics/0503206"
}
|
\section*{Introduction}
Symplectic geometry (like many fields of geometry before it) has received a tremendous infusion of ideas from work on a single physics-inspired example. The celebrated paper of Atiyah and Bott \cite{AB82} created a boom in symplectic geometry when they interpreted the moduli space of a Riemann surface as the symplectic reduction of the space of connections over that surface considered as a Hamiltonian space. They were able to read off information about its cohomology by considering the square of the moment map as a Morse function (that this Morse function is exactly the Yang-Mills action reveals part of the physics inspiration behind their work). Kirwan \cite{Kirwan84} built on these ideas to prove that for any compact Hamiltonian space the square of the moment map is an equivariantly perfect Morse function (or Kirwan-Morse function, a weaker but sufficient notion), and used this to give an algorithm for computing the Betti numbers of the symplectic quotient. Not long afterwards inspiration came a second time from the same example: Witten \cite{Witten92} used quantum field theory ideas and some inventive symplectic geometry to find surprising formulas for the intersection pairings of the cohomology of the moduli space of a Riemann surface.
Even for finite-dimensional Hamiltonian spaces Witten's techniques were not entirely rigorous, since he assumed regularities of the critical points of the squared moment map that do not typically hold. Jeffrey and Kirwan were able to reproduce his key results both for moduli space \cite{JK98a} and for general compact Hamiltonian spaces \cite{JK95} by replacing his main technique (which he called nonabelian localization) with an older technique of Duistermaat-Heckman \cite{DH83} called abelian localization. Specifically they were able to relate intersection pairings in the rational cohomology of a symplectic quotient to certain integrals of equivariant forms on the full Hamiltonian space. In the intervening decade symplectic geometry has been a booming field with much of the work centering on the cohomology of the symplectic quotient and its relationship to the topology of the original Hamiltonian space (a sampling includes \cite{Kalkman95,GK96,Vergne96,BV97,LMTW98,MS99,Paradan00,Kiem04,TW03,JKKW03,BTW04}).
A remarkable feature of the work that has been driven by this one example is that none of the work actually applies to the original example (this is not entirely true: as mentioned above Jeffrey and Kirwan \cite{JK98a} manage to prove Witten's formulas for moduli space, but the geometry of the space of connections which inspired these results is entirely circumvented). The Hamiltonian space of interest is the space of connections, a space which is not just noncompact but in fact infinite dimensional, and thus a far cry from the finite-dimensional compact Hamiltonian spaces on which we usually focus. The failure of the results to apply to noncompact Hamiltonian spaces is particularly striking since the two most basic examples of Hamiltonian manifolds, $T^*G$ for $G$ a Lie group and any symplectic representation of a Lie group, are both noncompact.
Naturally one would like to extend Jeffrey and Kirwan's approach to the noncompact setting. This may very well be possible, but one of the key benefits of reducing to the abelian case is the convexity results of Atiyah \cite{Atiyah82}, which apply only to compact spaces. More precisely, Witten's results apply to Hamiltonian spaces for which $0$ is a regular value of the moment map. In this situation a neighborhood of $\mu^{-1}(0)$ is always a Hamiltonian space with no fixed points for any subtorus, so any attempt to reduce questions of the topology of the reduced space to questions about the fixed points seems doomed. In particular, a number of authors have extended Duistermaat-Heckman localization to noncompact settings \cite{PW94,Paradan00,Libine05}. All these versions of Duistermaat-Heckman describe the induced measure on the Lie algebra at points away from $0,$ while in the case of a neighborhood of $\mu{-1}(0)$ when $0$ is regular the integrals in question give measures on the Lie algebra with support entirely at $0.$
Witten's original approach to nonabelian localization, however, makes no apparent reliance on compactness. In fact, if one were willing to join Witten in ignoring the analytic details and gave a sketchy introduction to equivariant cohomology in the Cartan model, Witten's argument could fit into a first year graduate course in differential geometry. The intrinsic simplicity of his argument suggests that even in the compact case it may be illuminating and productive to work out the analytic details, assuming they are tractable.
They are indeed tractable, and this is the approach we take in this paper. What makes them tractable, and in fact not very difficult, is the ability to avoid the central problem: That the critical points of the square of the moment map, to which the integrals in question are supposed to localize, are in general singular spaces to which the differential geometry of forms and integration do not readily apply. It seems likely that Witten's nonabelian localization has much more to tell us, but that to make further progress will require understanding these singular spaces. for example, Paradan \cite{Paradan00} argues that the contribution to the Basic Integral from $\mu^{-1}(0)$ is still a polynomial even when $0$ is not a regular value. This suggests that it still localizes to an integral of some sort of cohomology class over the (now singular) reduced space. Since the higher critical sets can be built easily from $\mu^{-1}(0)$ of a related Hamiltonian space, we could then hope that the same is true for all critical sets.
The paper is organized as follows. Section 1 gives the local characterization of an arbitrary Hamiltonian space due to Guillemin and Sternberg \cite{GS84a,GS84b}. It uses the local characterization to describe the critical points of the square of the moment map, and extend Kirwan's proof that the square of the moment map is an equivariantly perfect Morse-Kirwan function. Section 2 reviews the Cartan model for equivariant cohomology, expressing the Cartan and Kirwan maps explicitly in this language. Section 3 defines the Basic Integral and computes key estimates for it. Section 4 assumes that $0$ is a regular value of the moment map and proves the main theorem, that in this case the Basic Integral is the integral over the reduced space of the image under the Kirwan map of a certain form (which has polynomial dependence on $\epsilon$) plus additional contributions which are exponentially damped in $\epsilon.$ Thus cohomological integrals on the reduced space can be calculated by computing the Basic Integral over the full Hamiltonian space.
\section{The Local Structure Of Hamiltonian Spaces}\label{sc_local}
Let $M$ be a finite-dimensional smooth Hamiltonian space (not
necessarily compact): That is a smooth manifold with symplectic form $\omega,$ acted on symplectomorphically by the
compact Lie group $G$ with Lie algebra $\lieg,$ and with moment map
$\mu \colon M \to \lieg^*.$ If $p \in M$ and $\xi \in \lieg$ we will
write $V\!\xi$ for the vector field associated to the infinitesimal
action of $\lieg$ on $M$ and $V_p\xi$ for the value of this vector
field at the point $p.$ Then the moment map condition is
\begin{equation} \label{eq:moment}
\omega \intprod V\!\phi = d\mu \dotprod \phi
\end{equation}
for all $\phi \in \lieg,$ where we use $\intprod$ to represent the interior product between a vector field and a form (or a tangent vector and a form at a point) with the convention that $v \intprod \omega = (-1)^{\deg(\omega)}\omega \intprod v.$ Choose an invariant inner product
$\bracket{\, \cdot \, , \, \cdot \,}_\lieg$ on
$\lieg.$ This inner product determines an identification $\star
\colon \lieg \to \lieg^*$ whose inverse we will also call $\star,$ so that
$\mu^\star\colon M \to \lieg.$ Finally, choose an almost
complex structure for $M$ compatible with the group action, that is to
say an invariant metric $\bracket{\, \cdot \, , \, \cdot \,}_M$ and an operator $J$ on
the tangent space such that $J^2=-1$ and $\bracket{x,y}=Jx\intprod
\omega\intprod y.$
\subsection{Local Characterization}
Guillemin and Sternberg (\cite{GS84a,GS84b}) give a local characterization of
a Hamiltonian space which will be crucial for what follows. They
show that for any point $p \in M,$ the Hamiltonian
space $M$ is determined in an equivariant neighborhood of $p$ by
the value of $\mu$ at $p,$ the Lie subgroup $H$ fixing $p$ and its Lie
subalgebra $\lieh,$ and the symplectic action of $H$ on the tangent space
$T_pM.$ More specifically, let $p \in M,$ with isotropy group $H \subset G,$ whose Lie algebra
is $\lieh,$ and define
$\beta=\mu^\star_p$ and $K \subset G$
the stabilizer of $\beta,$ with $\liek$ its Lie algebra (so that
$\lieh \subset \liek$). Let $Y$ be the subspace of $T_pM$ of vectors which are omega-orthogonal and orthogonal to $V\liek,$ the space of directions $V\!\phi$ for $\phi \in \liek.$ This is an $H$ representation and the symplectic form restricts to a symplectic form $\omega_X$ on $X.$ On the space
\[G \times ( \liek \oplus X)\]
define the action of $G$ by the left action on the first component, and define the action of $H$ diagonally from the right action on the first component (applied to the inverse), the natural action on the second component, and the adjoint action on the third component. Define a closed invariant two-form $\omega$ at the point $(g,\nu + x)$ by
\begin{equation}\label{eq:standard_omega}
\omega=\bracket{d\nu - \frac{1}{2} [\beta+\nu,g^{-1}dg],g^{-1}dg}_G + \frac{1}{2} dx\intprod \omega_X \intprod dx.
\end{equation}
and moment maps for the two actions
\begin{equation}\label{eq:G_moment}
\mu_G \phi = \bracket{\Ad_g(\beta + \nu),\phi}_G
\end{equation}
\begin{equation}\label{eq:H_moment}
\mu_H \phi = \bracket{\nu,\phi}_G + \frac{1}{2}x \intprod \omega \intprod \phi x.
\end{equation}
The symplectic reduction by $\mu_H$ (i.e., the quotient of $\mu^{-1}(0)$ by the action of $H$ is $G$-Hamiltonian space isomorphic to
\[G \times_H (\liek/\lieh \oplus X)\]
where we will interpret $\liek/\lieh$ as the subspace of $\liek$ perpendicular to $\lieh.$
One easily checks that the point $(1,0) \in G \times ( \liek \oplus X)$ is in $\mu_H^{-1}(0),$ has isotropy group $H,$ a tangent space isomorphic to $T_pM$ as an $H$-space, and moment value $\mu_G=\beta^\star.$ Therefore by
\cite{GS84b}[Thm. 41.2] there is an isomorphism of Hamiltonian spaces from $G \times_H (\liek/\lieh \oplus X)$ to a neighborhood of $p$ in $M.$ In the future we will refer to the choice of such an isomorphism as ``choosing a standard neighborhood of $p.$''
\subsection{The Square of the Moment Map}
We are interested in the critical
sets of the nonnegative function $\bracket{\mu,\mu}_\lieg=|\mu|^2$ on $M.$
A point $ p\in M$ is a critical point for $|\mu|^2$ means that at $x,$
$d\bracket{\mu^\star,\mu^\star}_\lieg= 2\bracket{d\mu^\star,\mu^\star}_\lieg =d\mu
\mu^\star= \omega \intprod V\! \mu^\star =0.$ Since $\omega$ is
nondegenerate, to say that the one-form $V_p\mu^\star \intprod \omega$ is
zero at $p$ is to say that $V\!\mu^\star$ is zero at $p,$ and thus the
critical points of $|\mu|^2$ are exactly the zeros of the vector field
$V\!\mu^\star.$ Equivalently, critical points are the zeros of the one-form
\begin{equation}\label{eq_lambda}
\lambda\intprod v \defequals \bracket{V\!\mu^\star, v}_M
\end{equation}
for $v$ a tangent vector on $M.$
If $p \in M$ and $G \times_H (\liek/\lieh \oplus X)$ is a standard neighborhood around $p$ then $p$ is a critical point for
the square of the moment map if and only if the $H$-orbit of $(1,0)$
is critical in the standard neighborhood. This is equivalent to saying $V\!\beta=0,$ or
$\beta \in \lieh.$
\begin{proposition} \label{pr_local_critical} Let $Z$ be the set of $x
\in X$
such that $\beta x=0$ and $Q_x=0,$ where $\bracket{Q_x,\phi}= \frac{1}{2} x \intprod
\omega_X \intprod \phi x$ for all $\phi \in \lieh.$ The
connected component
of the critical set of $|\mu|^2$ in $G \times_H(\liek/\lieh \oplus X)$ containing the $H$-orbit
of $(1,0)$ is the $H$-orbit
of all points
$(g, z)$ where $g \in G $ and $z \in Z$.
This space is an algebraic variety.
\end{proposition}
\begin{pf}
Critical points of $|\mu|^2$ are points where $V\mu^\star=0.$ At a
point $(g, \nu + x)$ in $G \times (\liek/\lieh
\oplus X)$ we have
\[\mu \cdot \phi=\bracket{Ad_{a}(\beta + \nu),\phi} + \frac{1}{2} \phi x\intprod \omega_X \intprod x.\]
For a point $(g, \nu+x)$ in $\mu_H^{-1}(0)$ to descend to a point for which $V_G\phi=0$ means $V_G\phi$ is in $V_H \lieh.$ This
requires that $\Ad_{g^{-1}}\phi \in \lieh,$
$[\Ad_{g^{-1}}\phi,\nu]=0,$ and $\Ad_{g^{-1}}\phi\dotprod x=0.$
The first condition when $\phi=\mu^\star$ implies that $\nu=0.$ The second
implies nothing additional, and the third implies that
$(\beta+Q_x)x=0.$ Thus the critical points are in general those
for which $\nu=0$ and $(\beta+Q_x)x.$ The latter condition
implies that $\bracket{\beta+Q_x,Q_x}=0,$ which in turn implies that
$|\mu|^2 = |\beta+Q_x|^2 = |\beta|^2-|Q_x|^2.$ If there is a
path of critical points connecting this to $(1,0),$ the value of
$|\mu|^2$ would be constant, which implies that $Q_x=0$ and hence
$\beta x=0.$ Of course
all points of this form are critical and are obviously path
connected to $(1,0),$ so the connected component of the critical set
includes these points. We have only to show all other solutions are
separated from this set.
All other solutions have $\beta x \neq 0.$ Since $\beta$
commutes with $\lieh \subset \liek,$
write $X$ as a sum of orthogonal and $\omega$-orthogonal symplectic
submodules $X_\alpha,$ on each of which $|\beta x| =\beta_\alpha
|x|$ for some positive $\beta_\alpha.$ If $x$ satisfies $(\beta +
Q_x) x=0,$ but not $\beta x=0$ then breaking $x$ into its
components there is a nonzero $x_\alpha$ such that $-\beta
x_\alpha = Q_x x_\alpha \neq 0.$ Since $|\beta x_\alpha|=
\beta_\alpha |x_\alpha|$ at every solution $|Q_x| \geq
\min_\alpha \beta_\alpha$ holds. Thus $|Q_x| > \frac{2}{3} \min_\alpha
\beta_\alpha$ and $|Q_x| < \frac{1}{3} \min_\alpha
\beta_\alpha$ separate $Z$ from all other solutions.
\end{pf}
\begin{corollary} \label{cr:describe_critical}
The set of critical points of $|\mu|^2$ on the Hamiltonian space $M$ is a discrete union of closed connected components, each of which is locally an algebraic variety and on each one of which the value of $\mu$ lies in a single coadjoint orbit.
\end{corollary}
\begin{corollary}
If $\mu$ is proper (that is the inverse image of compact sets is compact) then $|\mu|^2$ is a minimally degenerate equivariantly perfect
Morse function in the sense of Kirwan \cite{Kirwan84}.
\end{corollary}
\begin{pf}
In order for $|\mu|^2$ to be minimally degenerate we need to show that the critical set is a discrete union of compact sets on each of which $|\mu|^2$ is constant, and that for each of the sets there is a locally closed submanifold $\Sigma$ containing the critical set as a minimum and at each point in the critical set the tangent space to $\Sigma$ is a maximal subspace of the full tangent space on which the Hessian of $|\mu|^2$ is positive semidefinite. The description of the critical sets is exactly the content of the previous corollary, together with the properness of $\mu.$ The existence of such a $\Sigma$ follows by the argument given by Kirwan unmodified, as does the equivariant perfection of this function.
\end{pf}
\section{Equivariant de Rham Cohomology}
An excellent reference on equivariant cohomology is \cite{GS99}, which gives a more complete and sophisticated treatment of everything in sections 2.1 and 2.2.
\subsection{Equivariant Forms}
Let $\mathcal{P}(\lieg)$ be the (graded) algebra of all complex-valued polynomial functions of $\lieg,$
$\mathcal{S}(\lieg)$ be the algebra of complex-valued Schwartz functions on $\lieg$ (that is, any combination of derivatives of the function times any power of $|\phi|$ approaches $0$ as $\phi\to \infty,$ with the supremums of these products as seminorms), $\mathcal{D}(\lieg)$ be the space of complex-valued tempered distributions, which is to say continuous linear functionals on $\mathcal{S}(\lieg),$ and $\mathcal{F}(\lieg)$ be the space of all continuous linear functionals on $\mathcal{P}(\lieg).$ Here and in the sequel we represent functions on
$\lieg$ as formulas in a dummy variable $\phi \in \lieg.$ Each of the function spaces ($\mathcal{P}(\lieg),$ $\mathcal{S}(\lieg)$) is an algebra and $\mathcal{P}(\lieg)$ acts by multiplication on $\mathcal{S}(\lieg),$ inducing various actions of the function spaces on the dual spaces ($\mathcal{D}(\lieg),$ $\mathcal{F}(\lieg)$) all represented by multiplication. Also there are natural embeddings $\mathcal{P} \subset \mathcal{D},$ $\mathcal{S} \subset \mathcal{D},$ and $\mathcal{S} \subset \mathcal{F},$ sending $f(\phi)$ to $f(\phi) \dint \phi,$ where $\dint\phi$ represents Haar measure on $\lieg.$ By analogy with this embedding we will represent the pairing between a function space and its dual by $\int_\lieg \,\cdot.$
If $M$ is a smooth
manifold and $\mathcal{X}$ represents one of $\mathcal{P}, \mathcal{S}, \mathcal{D}, \mathcal{F}$ we can define $\\Omega(M) \hat{\tensor}
\mathcal{X}(\lieg^*)$ to be smooth sections of
the bundle over $M$ which at each point $p \in M$
is the tensor product of $\Lambda(T_pM) \tensor \mathcal{X}(\lieg).$ Here smooth means that when any element of the given space dual to $\mathcal{X}$ is paired with the second factor, the result is a smooth ordinary form. When $\mathcal{X}$ is $\mathcal{P},$ this is an
algebra graded by the form degree plus twice the polynomial degree. If $G$ acts smoothly on $M$ then $G$ acts
naturally and consistently on these bundles (with the diagonal action of $G$ acting
naturally on forms and by the dual of the adjoint action on
functions on $\lieg$), so we may
speak of the the $G$-invariant elements of each space. These are
respectively the \emph{$\mathcal{X}$-equivariant forms on $M,$} though when $\mathcal{X}$ is $\mathcal{P}$ we drop the $\mathcal{P}$ and simply say \emph{equivariant forms on $M.$}
The exterior
derivative $d$ is defined on
all four bundles, so consider the \emph{equivariant derivative}
\begin{equation}\label{eq_equivariant_derivative}
D \alpha= d\alpha + i V\!\phi\intprod \alpha
\end{equation}
where $V\!\phi$ represents the linear map from $\lieg$ to vector
fields on $M.$ Note that $D$ is an equivariant map which
increases degree by one and
satisfies $D^2=0$ on invariant elements. Thus there are four equivariant cohomologies $H^{*,\mathcal{X}}_G(M),$ where in the case $\mathcal{X}=\mathcal{P}$ (the only case where the cohomology has an integer grading, the others have only a $\ZZ/2$ grading) we drop the $\mathcal{P}$ and write $H^{*}_G(M),$ the equivariant cohomology of $M.$ Equivariant differential forms give a
model for the cohomology with complex coefficients of the homotopy quotient $M_G,$ which is the
geometric significance of everything we do in this paper, but which is
mentioned for the last time here.
We say an $\mathcal{X}$-equivariant form has compact support if the closure of the set of points in $M$ where $\alpha$ is a nonzero function on $\lieg$ is compact. Equivariant $D$
preserves both these concepts and we call the cohomology generated by
compactly-supported $\mathcal{X}$-equivariant
forms $H^{*,\mathcal{X}}_{G,\compact}(M)$.
The various products among $\mathcal{P},$ $\mathcal{S},$ $\mathcal{D},$ and $\mathcal{F}$ extend to products on the various equivariant forms by wedging the form component. For example if $\alpha$ is an $\mathcal{S}$-equivariant form and $\beta$ is a $\mathcal{D}$-equivariant form then $\alpha\beta$ is an $\mathcal{F}$-equivariant form. Note that equivariant $D$ satisfies the Leibniz rule on all such products. Finally, an $\mathcal{F}$-equivariant form can be paired with $1$ (i.e. integrated) to get an ordinary form, which can be integrated over an invariant submanifold $N$ (assuming $M$ is oriented, and that either the original form was compactly supported or $N$ is compact) by taking on the component of appropriate degree. Because $V\!\phi \intprod$ lowers form degree,
\[\int_N \int_\lieg D\alpha \dint \phi= \int_N \int_\lieg d\alpha \dint \phi= \int_{\partial N} \int_\lieg \alpha \dint \phi\]
which is zero when $N$ has no boundary.
This fact can be viewed as an
equivariant version of Stokes theorem and when applied to $N=M$ descends for example to a well-defined pairing on cohomology
\[H^{*,\mathcal{S}}_{G}(M)\times H^{*,\mathcal{D}}_{G,\compact}(M) \to \CC\]
and likewise with the compact subscript on the other factor.
\subsection{The Cartan and Kirwan Maps}
If the group action is locally free, the homotopy quotient retracts to the ordinary quotient and thus the equivariant cohomology is isomorphic to the ordinary cohomology of the quotient. This isomorphism can be made completely explicit on the level of equivariant forms.
Let $P \to N$ be an orbifold principal $G$-bundle, which is to say locally $P$ can be identified with $G \times_H V$ where $H$ is a finite subgroup of $G$ and $V$ is an $H$-module, so that the $G$ orbit of each point in $G \times_H V$ is a fiber of the map $P \to N.$ Let $A$ be a connection for this bundle, i.e. an equivariant $\lieg$-valued one-form on $P$ such that $A\intprod V\!\phi=\phi$ for all $\phi \in \lieg.$ Let $P_A$ be the operator on $TP$ which sends a tangent vector $v$ to its projection onto the $A=0$ subspace, $P_Av=v - V\!A\intprod v.$ If $\alpha$ is a form on $P,$ define $P_A^*\alpha$ so that $v\intprod P_A^*\alpha = P_A^*(P_A(v) \intprod \alpha),$ i.e. $P_A^*\alpha$ is $\alpha$ projected onto the subspace of forms zero on all vertical vectors. This map extends naturally to equivariant forms. Define the \emph{Cartan map}
$\Cartan \colon \Omega(P) \hat{\tensor}
\mathcal{P}(\lieg) \to \Omega(P)$ by
\begin{equation}\label{weyl_map}
\Cartan(\alpha(\phi))=P_A^*(\alpha(iF_A))
\end{equation}
where $F_A$ refers to the $\lieg$-valued curvature two-form of the connection and its placement in parentheses denotes substituting its value for $\phi$ in the second tensor factor of $\alpha,$ thus producing a form to be wedged with the first tensor factor.
\begin{proposition}
The Cartan map descends to a grade-preserving isomorphism from the complex of equivariant forms to that of basic (i.e. horizontal and invariant) forms on $P$ inverting the natural imbedding. Composing with the natural isomorphism of the complex of basic forms on $P$ with ordinary forms on $N,$ we get a map which descends to an isomorphism
\[\Cartan \colon H^*_G(P)\to H^*(N).\]
\end{proposition}
\begin{pf}
If $\alpha$ is an equivariant form on $P,$ it is clear that $\Cartan(\alpha)$ is invariant, by the equivariance of $P_A$ and $F_A.$ It is also clear that $\Cartan(\alpha)$ is horizontal, since $F_A$ is horizontal and the range of $P_A^*$ is horizontal. Finally, it is clear that the Cartan map is an algebra homomorphism. So for the homomorphism of complexes we need only show that the Cartan map intertwines the equivariant and ordinary derivatives, which can be checked locally.
To do this consider a chart $V$ on which a finite subgroup $H$ of $G$ acts, and an equivariant isomorphism of $G \times_H V$ with a neighborhood in $P.$ $A$ defines an $H$-invariant one-form $\tau$ on $V$ with values in $\lieg,$ by $\tau_{v}\intprod \xi= A_{(1,v)}\intprod (0,\xi).$ A form on $G \times_H V$ is an $H$-invariant form on $G \times V.$ Since $G$ only acts on the first factor,
\[\left[\left(\Omega(G\times V)\right)^H \hat{\tensor} {\mathcal P}(\lieg^*)\right]^G \iso \left(\left[\Omega(G) \hat{\tensor} {\mathcal P}(\lieg^*) \right]^G\times \Omega(V)\right)^H\]
as complexes.
Since $D$ and $d$ satisfy the Leibniz rule, we can check the intertwining on generators of the complex. These are forms on $V,$ one-forms on $G$ $\bracket{\xi,g^{-1} dg }_G$ for $\xi \in \lieg,$ and functions $\bracket{\xi, g^{-1}\phi g}_G$ for $\xi \in \lieg.$ That $D$ and $d$ are intertwined on the first set of generators is obvious. For the second class
\begin{eqnarray*}
\Cartan(D(\bracket{\xi,g^{-1}dg })) &=& \Cartan(\bracket{\xi, g^{-1} dg g^{-1} dg } + i \bracket{\xi, g^{-1} \phi g})\\
&=& \bracket{\xi, \frac{1}{2}[\tau, \tau]} -\bracket{\xi, d\tau + \frac{1}{2}[\tau,\tau]}\\
&=& -\bracket{\xi,d\tau}\\
&=& -d(\bracket{\xi,\tau})\\
&=& d(\Cartan(\bracket{\xi,g^{-1}dg })).
\end{eqnarray*}
For the third class
\begin{eqnarray*}
\Cartan(D(\bracket{\xi,g^{-1} \phi g }))&=& \Cartan(\bracket{\xi, [g^{-1}dg ,g^{-1} \phi g] } \\
&=& -i\bracket{\xi, [\tau, d\tau +\frac{1}{2} [\tau, \tau]]}\\
&=& i \bracket{\xi, [d\tau,\tau]}\\
&=& i d(\bracket{\xi,d\tau + \frac{1}{2}[\tau,\tau]})\\
&=& d(\Cartan(\bracket{\xi,g^{-1} \phi g })).
\end{eqnarray*}
Finally, to see that its inverse is the natural embedding of basic forms into equivariant forms, since it is the identity on basic forms, we need only check that every closed equivariant form is cohomologous to a basic form. This requires defining certain operators on the complex of equivariant forms.
We write $\frac{\partial}{\partial \phi}$ for the formal derivative with respect to $\phi,$ which we view as a function on $\lieg$ with values in equivariant forms. Thus the operator
\[\Phi=A\cdot \frac{\partial }{\partial \phi}\]
denotes (viewing the connection $A$ as a form tensored with a Lie algebra element) applying this operator on the equivariant form to the second tensor factor and wedging the first tensor factor with the result. By a similar logic $VA \intprod $ applies $V$ to the second tensor factor to get a tangent vector, takes the interior product with the form on which the operator acts to get a new form, and wedges the first factor with the result. Now a straightforward calculation shows
\[ D \Phi + \Phi D =
dA \cdot \frac{\partial}{\partial \phi} + i( \phi \cdot \frac{\partial}{\partial \phi} + V\!A \intprod \,)\]
where the two new operators in the above expression are defined similarly. The first operator in the parentheses ($\phi \cdot \frac{\partial}{\partial \phi}$) multiplies any homogenous polynomial by its degree, and thus gives a grading of the space of equivariant forms into eigenvalues. Similarly the second operator in the parentheses ($V\!A \intprod\,$) grades the space into eigenspaces with nonnegative integral eigenvalues, representing the ``number of form degrees in vertical directions.'' Since the two commute, they give a grading by their sum, call it the total degree, such that the total degree zero piece consists of basic forms on $P.$ Notice that the term not in parentheses ($dA \cdot \frac{\partial}{\partial \phi}$) strictly lowers total degree. Thus if $\alpha$ is a closed equivariant form whose maximum total degree piece has degree $p>0,$ then $\alpha + \frac{i}{p}D(\Phi\alpha)$ has strictly lower degree, and thus by induction $\alpha$ is cohomologous to a total degree zero form.
\end{pf}
Now suppose that $M$ is a Hamiltonian space with a proper moment map, and that $0$ is a regular value of $M,$ i.e. that $d\mu$ is onto for all points with $\mu=0.$
\begin{proposition}\label{pr:orbifold}
If $d\mu$ is onto for each point of $\mu^{-1}(0),$ then $G$ acts on $\mu^{-1}(0)$ with finite stabilizers. In this case $\mu^{-1}(0)$ is a smooth manifold and an orbifold principal bundle over the quotient $M_{\text{red}}=\mu^{-1}(0)/G,$ which has an orbifold symplectic structure $\omega_0.$ \end{proposition}
\begin{pf}
If $d\mu$ is onto at some point, then by the moment map condition $V\!\phi$ is nonzero for all $\phi \in \lieg,$ so that the isotropy group must be finite.
If the isotropy group is finite at some $z$ with $\mu_z=0,$ then a standard neighborhood looks like
\[G \times_H (\lieg \oplus X)\]
where $H$ is a finite subgroup and $X$ is a symplectic vector space on which $H$ acts. The subspace on which $\mu=0$ is $G \times_H X.$ The image of this space in the quotient by $G$ is isomorphic to $X/H,$ the quotient of a vector space by a finite-dimensional group action. If another standard neighborhood $G \times_K X'$ contains $z,$ we argue the diffeomorphism of standard neighborhoods lifts to a local diffeomorphism of $X$ and $X'.$ This guarantees that a covering collection of standard neighborhoods form an orbifold atlas for $M_{\text{red}}=\mu^{-1}(0)/G.$
To see this, we can assume by equivariance that the standard neighborhood $G \times_K X'$ is chosen so that $z$ is the image of a point $(1,x).$ Then $H$ is the subgroup of $K$ which fixes $x,$ so that a neighborhood of $x$ is a representation $V$ of $H,$ and does not intersect with its image under any other elements of $K.$ Then $G \times_K X'$ is diffeomorphic locally to $G \times_H V,$ and this induces a diffeomorphism between $V$ and $X.$
A symplectic structure on an orbifold is a choice of $H$-invariant symplectic form on $V$ for each chart $(V,H),$ which is preserved by the overlap maps. Clearly $\omega_X$ is an invariant form on each vector space $X,$ and it is immediate that it is preserved by the overlap map.
\end{pf}
The imbedding of $\mu^{-1}(0)$ into $M$ gives a map of equivariant cohomology which when composed with the Cartan map gives the \emph{Kirwan map} $\Kirwan \colon H^*_G(M) \to H^*(M_{\text{red}}).$ The fact that the square of the moment map is equivariantly perfect means that this map is surjective.
\section{Equivariant Integration and Localization}
For this section let $M$ be a Hamiltonian space with a proper moment map, and $\epsilon$ be a positive real parameter.
\subsection{Localization}
The moment map condition guarantees that the equivariant form
\[\omega + i \mu \dotprod \phi\]
is closed, and thus represents an element of $H^*_G(M).$ Here the exponentiation is interpreted as its power series. Suppose now that $\alpha$ is an equivariant form on $M$, so that
\[\alpha \exp(\omega + i \mu \dotprod \phi - \frac{\epsilon}{2} |\phi|^2)\]
is an $\mathcal{S}$-equivariant form which is closed and/or compactly-supported if $\alpha$ is.
On the other hand consider an invariant ordinary one-form $\lambda$ on $M$ (which is therefore also an equivariant one-form).
\begin{lemma}\label{lm:Dlambda}
For each nonnegative real $t$
\[\int_0^t \exp(sD\lambda) \dint s \lambda\]
gives a $\mathcal{D}$-equivariant form satisfying $D\left(\int_0^t \exp(sD\lambda) \dint s \lambda\right)= \exp(tD\lambda)-1.$ Thus $\exp(tD\lambda)$
is a closed $\mathcal{D}$-equivariant form which is $\mathcal{D}$-cohomologous to $1.$
Further, on a submanifold of $M$ on which $\lambda\intprod V\!\phi$ is never the zero functional on $\phi,$ the limit of this integral as $t$ approaches infinity exists in the $\mathcal{D}$-topology and satisfies $D\left(\int_0^\infty \exp(sD\lambda) \dint s \lambda\right)= -1.$
\end{lemma}
\begin{pf}
We interpret the exponential and the integral in terms of power series, and at a point in $M$ write $\lambda \intprod V\!\phi$ as $\bracket{\xi,\phi}$ for some $\xi\in \lieg,$ so that the integral is a sum of terms of the form
\[\bracket{\text{FORM}} \int_0^t s^k \exp(i s \bracket{\xi, \phi}) \dint s \]
which as a functional on some test function $f(\phi) \in \mathcal{S}(\lieg)$ is
\[\bracket{\text{FORM}}\int_0^t s^k \int_\lieg f(\phi) \exp(is \bracket{\xi, \phi}) \dint \phi \dint s = \int_0^t s^k \widehat{f}(s\xi) \dint s \]
where $\widehat{f}$ is the Fourier transform of $f$ (ignoring arbitrary constants) and thus is well-defined. So $\int_0^t \exp(sD\lambda) \dint s \lambda$ is a $\mathcal{D}$-equivariant form whose equivariant derivative is
\[\int_0^t \exp(sD\lambda) D\lambda \dint s = 1-\exp(tD\lambda).\]
If $\lambda \intprod V\!\phi$ is never zero then $\xi\in \lieg$ as defined in the previous paragraph is never zero, so we get
\[\int_0^\infty s^k \widehat{f}(s\xi) \dint s\]
which converges since $\widehat{f}$ is Schwartz.
\end{pf}
\subsection{The Basic Integral}
Since $\mu$ is proper by Corollary \ref{cr:describe_critical} identify the critical values of $|\mu|^2$ as
\[0 \leq r_1 < r_2 < \cdots\]
(the sequence may be finite or infinite) and as long as $r\in \RR^+$ is regular, i.e. satisfies $r \neq r_i$ $\forall i \in \NN$ then
\[M_r \defequals \{p \in M \,|\, |\mu_p|^2 \leq r\}\]
is a compact manifold with compact boundary.
Recall that the symplectic form gives a natural orientation to $M$ and hence $M_r$ and thus integration over $M_r$ when $r$ is a regular value of $|\mu|^2$ is well-defined.
Let $\lambda$ be the invariant one-form on $M$ which for any tangent vector $v$ gives
\begin{equation}\label{eq:lambda_def}
\lambda \intprod v = \bracket{V\!\mu^\star, v}.
\end{equation}
The $\lambda\intprod V\!\phi$ is zero exactly when $V\!\mu^\star$ is zero, which in turn happens exactly at the critical points of $|\mu|^2.$
For a equivariant form $\alpha,$ for any nonnegative real number $t$ and for any regular value $r$ of $|\mu|^2$ define the \emph{Basic Integral}
\begin{equation}\label{eq:regularized_int}
\BI(\alpha, r, t) \defequals \frac{1}{K}\int_\lieg \int_{M_r} \alpha \exp[\omega + i \mu \dotprod \phi -\frac{\epsilon}{2} |\phi|^2 + t D\lambda] \dint \phi
\end{equation}
where $\lambda$ defined in Equation (\ref{eq:lambda_def}) and
\begin{equation}\label{eq:int_factor}
K= \vol(G) (2\pi)^{\dim(G)}.
\end{equation}
The following estimates are crucial to the calculations that follow.
\begin{lemma}\label{lm:bound_higher}
Suppose $\alpha$ is an equivariant form and $r$ and $s$ are regular values of $|\mu|^2$ with $s<r.$ Then
\[|\BI(\alpha,r,0)-\BI(\alpha,s,0)| <
\text{POLYNOMIAL}(\epsilon^{\pm 1/2})\exp(-\frac{s}{2\epsilon})\]
where the coefficients of the polynomial depend on $r.$ In other words the contribution to the Basic Integral at $t=0$ of points with large values of $|\mu|$ is exponentially damped.
\end{lemma}
\begin{pf}
\begin{eqnarray*}
& &\left|\frac{1}{K}\int_\lieg \int_{M_r-M_s} \alpha(\phi) \exp(\omega + i \mu \dotprod \phi - \frac{\epsilon}{2} |\phi|^2)\dint \phi\right| \\
&=& \frac{1}{K} \left| \int_{M_r-M_s} \exp(\omega) \int_\lieg \alpha(\phi) \exp(i \mu \dotprod \phi - \frac{\epsilon}{2} |\phi|^2) \dint \phi \right|\\
&=& \frac{1}{K} \left| \int_{M_r-M_s} \exp(\omega) \exp(-\frac{1}{2\epsilon} |\mu|^2) \int_\lieg \alpha(\phi+i \mu^\star/\epsilon) \exp(-\frac{\epsilon}{2} |\phi|^2) \dint \phi
\right|\\
&=& \frac{1}{K} \left| \int_{M_r-M_s} \exp(\omega) \exp(-\frac{1}{2\epsilon} |\mu|^2)
\text{POLYNOMIAL}(\epsilon^{\pm1/2})
\right|\\
&\leq& \exp(-\frac{s}{2\epsilon})\left| \text{POLYNOMIAL}(\epsilon^{\pm1/2})\right|.
\end{eqnarray*}
Here the coefficients of the polynomial can be bounded by certain integrals over $M_r.$
\end{pf}
\begin{lemma}\label{lm:large_t}
Suppose that $\alpha$ is a \emph{closed} equivariant form and that $r$ is a regular value of $|\mu|^2.$ Then
\[ \lim_{t\to \infty} \BI(\alpha, r, t)\]
exists and differs from $\BI(\alpha, r, 0)$ by
\[ \text{POLYNOMIAL}(\epsilon^{\pm 1/2})\exp(-\frac{C}{2\epsilon})\]
where the coefficients of the polynomial and $C$ depend on $r.$
\end{lemma}
\begin{pf} Suppose $t_1<t_2 \in \RR.$
\begin{eqnarray*}
&&|\BI(\alpha,r, t_2)-\BI(\alpha, r, t_1)|\\
&=&\frac{1}{K}\left| \int _\lieg \int_{M_r} \alpha(\phi)
\exp(\omega + i \mu \dotprod \phi -\frac{\epsilon}{2} |\phi|^2)
(\exp(t_2D\lambda)-\exp( t_1 D\lambda) ) \dint \phi \right| \\
&=& \frac{1}{K}\left| \int_\lieg \int_{M_r} \alpha(\phi)
\exp(\omega + i \mu \dotprod \phi -\frac{\epsilon}{2} |\phi|^2)
D\left(\int_{t_1}^{t_2} \exp(sD\lambda) \lambda \dint s\right) \dint \phi\right| \\
&=&\frac{1}{K}\left| \int_\lieg \int_{\partial {M_r}} \alpha(\phi)
\exp(\omega + i \mu \dotprod \phi -\frac{\epsilon}{2} |\phi|^2)
\int_{t_1}^{t_2} \exp(sD\lambda) \lambda \dint s \dint \phi \right| \\
&=& \frac{1}{K}\Big| \int_{\partial {M_r}} \int_{t_1}^{t_2} \exp(\omega +sd\lambda)\lambda
\int_\lieg \alpha(\phi)
\exp(i \bracket{\mu^\star + sV^\star V\!\mu^\star, \phi} -\frac{\epsilon}{2} |\phi|^2)
\dint \phi \dint s\Big|.
\end{eqnarray*}
Completing the square yields
\begin{eqnarray*}
&=& \frac{1}{K}\Big| \int_{\partial {M_r}} \int_{t_1}^{t_2} \exp(\omega +sd\lambda)\lambda
\int_\lieg \alpha(\phi + \frac{1}{2\epsilon}(\mu^\star + sV^\star V\!\mu^\star)
\exp(-\frac{\epsilon}{2} |\phi|^2)\dint \phi \\
&&\qquad \cdot \exp( -\frac{1}{2\epsilon} |\mu|^2 -\frac{s}{2\epsilon} |V\!\mu^\star|^2 -\frac{s^2}{2\epsilon} |V^\star V\!\mu^\star|^2) \dint s
\Big| \\
&=& \frac{1}{K}\Big| \int_{\partial {M_r}} \int_{t_1}^{t_2} \exp(\omega +sd\lambda)\lambda
\text{POLYNOMIAL}(\epsilon^{\pm1/2},s) \\
&&\qquad \cdot \exp( -\frac{1}{2\epsilon} |\mu|^2 -\frac{s}{2\epsilon} |V\!\mu^\star|^2 -\frac{s^2}{2\epsilon} |V^\star V\!\mu^\star|^2) \dint s
\Big| \\
&\leq & \frac{1}{K}\Big| \int_{\partial {M_r}} \exp( -\frac{1}{2\epsilon} |\mu|^2)
\text{FORM} \int_{t_1}^{t_2}
\text{POLYNOMIAL}(\epsilon^{\pm1/2},s) \\
&&\qquad \cdot \exp( -\frac{s^2}{2\epsilon} |V^\star V\!\mu^\star|^2) \dint s
\Big|.
\end{eqnarray*}
For a fixed $\epsilon,$ since the $s$ integral is a polynomial times a Gaussian, this quantity is bounded by $\exp(-t_1^2 \min(|V^\star V\!\mu^\star|^2)/(2\epsilon)),$ where $ |V^\star V\!\mu^\star|^2$ is bounded below since $r$ is a regular value. The limit of the difference can be written as a telescoping sum of such differences, which decrease hypergeometrically and hence the sum converges. On the other hand choosing $t_1=0$ we see that there is a polynomial times $\epsilon^{\pm 1/2}$ which times $\exp(-C/(2\epsilon))$ bounds the difference regardless of $\epsilon$ or $t_2.$
\end{pf}
\subsection{The Basic Integral as a Sum of Contributions} The large $t$ limit of the Basic Integral is a sum of contributions from the critical points of $|\mu|^2,$ as is illustrated in the following.
\begin{lemma} \label{lm:r_independence}
If $r$ and $s$ are regular values of $|\mu|^2$ with no critical values between them and $\alpha$ is a closed equivariant form then
\[\lim_{t\to \infty} \BI(\alpha, r, t)=\lim_{t\to \infty} \BI(\alpha, s, t).\]
\end{lemma}
\begin{pf}
This follows directly from Lemma \ref{lm:Dlambda}.
\end{pf}
\begin{corollary} \label{cr:contribution}
For each $i$ choose $r_i'$ and $r_i''$ such that $r_{i-1}<r_i' < r_i < r_i'' < r_{i+1}.$ Define $r_1'=0$ and if $r_i$ is the maximum critical value choose any $r_i''>r_i.$ Then given a closed equivariant form $\alpha$ the quantity
\[C_i(\alpha)=\lim_{t\to \infty} \BI(\alpha, r_i'',t) - \BI(\alpha, r_i'',t)\]
exists and is independent of the choice of $r_i'$ and $r_i''.$ Further, for any regular value $r$ of $|\mu|^2$
\[\lim_{t\to \infty} \BI(\alpha, r, t)= \sum_{r_i<r} C_i.\]
In other words the large $t$ limit of the Basic Integral up to $r$ is the sum of the contributions from each critical set below $r.$ The contribution $C_i(\alpha)$ when $r_i>0$ is bounded by
\[\text{POLYNOMIAL}(\epsilon^{\pm 1/2} \exp(-\frac{r_i-\delta}{2\epsilon})\]
where $\delta$ can be made as small as we like.
\end{corollary}
\begin{lemma} \label{lm:lambda_invariance}
Let $\alpha$ be any closed equivariant form, let $r_i$ be a critical value of $|\mu|^2,$ let $N$ be a compact manifold with boundary containing a neighborhood of the critical set corresponding to $r_i$ and no other critical points of $|\mu|^2,$ and let $\lambda'$ be the result of an isotopy of $\lambda$ such that the points of $M$ at which $\lambda \colon V\!\phi$ is the zero functional on $\lieg$ remain fixed through the isotopy. Then
\[
C_i(\alpha) = \lim_{t\to \infty} \int_\lieg \int_N \alpha \exp(\omega + i \mu \phi + t D\lambda' -\frac{\epsilon}{2}|\phi|^2) \dint \phi.
\]
\end{lemma}
\begin{pf} By Lemma \ref{lm:Dlambda} the limit above with $\lambda$ replacing $\lambda'$ is equal to $C_i(\alpha).$ Define $\lambda''$ to agree with $\lambda'$ in a neighborhood of the critical set but to agree with $\lambda $ near the boundary of $N.$ Then
\begin{eqnarray*}
&&\frac{1}{K}\int_\lieg \int_{N} \alpha \exp(\omega + i \mu \dotprod \phi -\frac{\epsilon}{2} |\phi|^2) \\
&& \qquad \cdot \left(\exp(tD\lambda) - \exp(tD\lambda'')\right)\dint \phi\\
&=&
\frac{1}{K}\int_\lieg \int_{N} D\Big(
\alpha \exp(\omega + i \mu \dotprod \phi -\frac{\epsilon}{2} |\phi|^2) \\
&&\qquad \cdot \left( \int_0^t \exp(sD\lambda) \dint s \lambda - \int_0^t \exp(sD\lambda'') \dint s \lambda'' \right)\Big)\dint \phi\\
&=&
\frac{1}{K}\int_\lieg \int_{\partial N}
\alpha \exp(\omega + i \mu \dotprod \phi -\frac{\epsilon}{2} |\phi|^2)\\
&&\qquad \cdot \left( \int_0^t \exp(sD\lambda) \dint s \lambda - \int_0^t \exp(sD\lambda'') \dint s \lambda'' \right)\dint \phi\\
&=&
0
\end{eqnarray*}
so that $\lambda$ and $\lambda''$ give the same contribution. On the other hand by replacing $N$ by a smaller neighborhood (again by Lemma \ref{lm:Dlambda}), we can assure that $\lambda'$ and $\lambda''$ agree on $N$ and thus give the same contribution.
\end{pf}
\begin{proposition} \label{pr:bound_higher}
Suppose that $\alpha$ is a closed equivariant form and $r $ is a regular value of $|\mu|^2.$ Then the large $t$ limit of the Basic Integral (\ref{eq:regularized_int}) is equal to its contribution $C_0(\alpha)$ of the critical set with $\mu=0$ (as defined in Corollary \ref{cr:contribution}) plus a contribution bounded by $\exp(-c/\epsilon)$ for some $c.$
\end{proposition}
\begin{pf}
This follows immediately from Lemma \ref{lm:bound_higher}.
\end{pf}
\section{When Zero is a Regular Value of the Moment Map}
The proof of the following result in the case of trivial isotropy group appears in \cite{GS84b}, the full statement appears in \cite{Jeffrey99}. While the statement and proof are widely known to experts, to the author's knowledge no proof appears in the literature, so for the sake of completeness it is included here.
\begin{proposition}\label{pr:normal_form}
If $0$ is a regular value of $\mu$ (i.e. if $d\mu$ is onto for each point of $\mu^{-1}(0)$) recall by Proposition \ref{pr:orbifold} the map $\pi\colon \mu^{-1}(0) \to M_{\text{red}}=\mu^{-1}(0)/G$ is a principal orbifold bundle and $M_{\text{red}}$ has an orbifold symplectic structure $\omega_0.$ Given a connection $A$ on this bundle, there is an isomorphism of Hamiltonian spaces between a neighborhood of $\mu^{-1}(0)$ in $M$ and the Hamiltonian space $\mu^{-1}(0) \times \lieg,$ with symplectic form and moment map at $(p,\nu) \in \mu^{-1}(0) \times \lieg$ given by
\begin{equation}
\widetilde{\omega}=\pi^* \omega_0 + d\bracket{\nu,A}
\end{equation}
\begin{equation}
\widetilde{\mu}=\nu^\star.
\end{equation}
\end{proposition}
\begin{pf} One readily checks that $\widetilde{\omega}$ defines a closed form which is nondegenerate at $\mu^{-1}(0),$ and therefore in a neighborhood. Also $\widetilde{\omega}$ is manifestly $G$-invariant (with the diagonal action of $G$ on $\mu^{-1}(0) \times \lieg$) and satisfies the moment map condition with $\widetilde{\mu}.$ By Guilleman and Sternberg's local characterization \cite{GS84b}[Thm.41.2], it suffices to give an equivariant symplectic isomorphism between the zeros of the moment map in each case, and then extend it to an equivariant identification of the normal bundles which preserves $d\mu.$
The equivariant symplectic isomorphism is of course the natural imbedding of ${\widetilde{\mu}}^{-1}(0)= \mu^{-1}(0) \times \{0\}$ into $M.$ Its equivariance is by naturality and it preserves $\omega$ by inspection. Because $d\mu$ is onto at every point it gives a trivialization of the normal bundle, identifying it with $\mu^{-1}(0) \times \lieg.$ This identification clearly is equivariant and takes $d\widetilde{\mu}$ to $d\mu.$
\end{pf}
\begin{theorem} \label{th:zero_contribution}
Suppose $\alpha$ is a closed equiviariant form, and $0$ is a regular value for the moment map. Then the contribution $C_0(\alpha)$ to the Basic Integral (\ref{eq:regularized_int}) from $\mu^{-1}(0)$ is
\[\int_\lieg \dint \phi \int_{M_{\text{red}}} \Kirwan(\alpha) exp(\omega_0 + \frac{\epsilon}{2} c_2)\]
where $\Kirwan$ is the Kirwan map, $M_{\text{red}}$ is the orbifold quotient $\mu^{-1}(0)/G$ and $c_2$ is the second Chern class of the bundle $\mu^{-1}(0) \to M_{\text{red}}.$ In particular it has polynomial dependence on $\epsilon.$
\end{theorem}
\begin{pf}
The contribution to the basic integral of $\mu^{-1}(0)$ is
\[\lim_{t\to \infty}\frac{1}{K} \int_N \int_\lieg \alpha(\phi)\exp(\omega + i \mu \cdot \phi - \frac{\epsilon}{2}|\phi|^2 + t D\lambda) \dint \phi\]
where $N$ is a neighborhood of $\mu^{-1}(0)$ containing no other critical points in its closure. By Proposition \ref{pr:normal_form} we can take $N$ isomorphic to a neighborhood of $\mu^{-1}(0)$ in $\mu^{-1}(0) \times \lieg.$ The integral is unchanged if we replace $\alpha$ by something cohomologous, so using an equivariant homotopy we can replace $\alpha$ with a form that agrees with $\iota^*(\alpha)\times 1$ in a neighborhood of $\mu^{-1}(0)$ in $\mu^{-1}(0) \times \lieg$ ($\iota$ being the inclusion of $\mu^{-1}(0)$). By making $N$ sufficiently small this form agrees with $\iota^*(\alpha)\times 1$ (which we will abbreviate $\iota^*(\alpha)$) everywhere. Thus
\[=\lim_{t\to \infty}\frac{1}{K} \int_{N \subset \mu^{-1}(0) \times \lieg} \int_\lieg \iota^*(\alpha)(\phi) \exp(\pi^* \omega_0 + d\bracket{\nu,A} + i \bracket{\nu,\phi} - \frac{\epsilon}{2}|\phi|^2 + t D\lambda) \dint \phi.\]
By Lemma \ref{lm:lambda_invariance}, we may isotope $\lambda$ provided the zeros of $\lambda\intprod V\!\phi$ do not change. Since $\bracket{V\!\mu^\star, \,\cdot \,}_M$ and $\bracket{\nu, A}_G$ are both positive on the vector $V\!\mu^\star,$ interpolating between them linearly does not change the zeros. Thus replacing $\lambda$ with $\bracket{\nu,A}$ does not change the limit, giving
\begin{eqnarray*}
&=&
\lim_{t\to \infty}\frac{1}{K} \int_{N \subset \mu^{-1}(0) \times \lieg} \int_\lieg \iota^*(\alpha)(\phi) \\
&& \qquad \cdot \exp(\pi^* \omega_0 + d\bracket{\nu,A} + td\bracket{\nu,A} + i \bracket{\nu,\phi} + it \bracket{\nu,\phi} - \frac{\epsilon}{2}|\phi|^2) \dint \phi.
\end{eqnarray*}
Notice that $\nu$ occurs throughout with the factor $(1+t)$ (because $\iota^*(\alpha)$ does not depend $\nu$) and thus we can rescale to eliminate $t$ except for the dependence of the region of integration. In the large $t$ limit this becomes the integral over all $\lieg$
\begin{eqnarray*}
&=&
\frac{1}{K} \int_{\mu^{-1}(0) \times \lieg} \int_\lieg \iota^*(\alpha(\phi)) \\
&&\qquad \cdot \exp(\pi^* \omega_0 + d\bracket{\nu,A} + i \bracket{\nu,\phi} - \frac{\epsilon}{2}|\phi|^2) \dint \phi.
\end{eqnarray*}
Completing the square
\begin{eqnarray*}
&=&\frac{1}{K} \int_{\mu^{-1}(0) \times \lieg} \left(\int_\lieg \iota^*(\alpha)(\phi+\frac{i}{\epsilon} \nu)
\exp(-\frac{\epsilon}{2}|\phi|^2) \dint \phi\right)\\
&&\qquad \cdot \exp(\pi^* \omega_0 + d\bracket{\nu,A} - \frac{1}{2\epsilon}|\nu|^2)\\
&=&
\frac{1}{K} \int_{\mu^{-1}(0) \times \lieg} \left(\int_\lieg \iota^*(\alpha)(\phi+ \frac{i}{\epsilon} \nu) \exp(-\frac{\epsilon}{2}|\phi|^2) \dint \phi\right)\\
&&\qquad \cdot \exp(\pi^* \omega_0 + \bracket{d\nu, A} + \bracket{\nu,dA} - \frac{1}{2\epsilon}|\nu|^2).
\end{eqnarray*}
Note the only occurrence of $d\nu$ is in $\exp(\bracket{d\nu,A}).$ Consider a basis of tangent vector at some point in $\mu^{-1}(0) \times \lieg$ which consists of an orthonormal basis of $\lieg,$ the image of this orthonormal basis under $V,$ and a basis of $A$-horizontal vectors in $\mu^{-1}(0).$ The top dimensional piece of this multiform is a sum of terms with $\bracket{d\nu,A}$ raised to various powers, but the only terms which are nonzero when applied to this basis are those where $\bracket{d\nu,A}$ is raised to $\dim(G),$ and on those terms the value is unchanged if $P_A^*$ is applied to all other forms in the product. Thus
\begin{eqnarray*}
&=&\frac{1}{K} \int_{\mu^{-1}(0) \times \lieg} \left(\int_\lieg P_A^*\circ \iota^*(\alpha)(\phi+ \frac{i}{\epsilon} \nu)
\exp(-\frac{\epsilon}{2}|\phi|^2) \dint \phi\right)\\
&&\qquad \cdot \exp(\pi^* \omega_0 + \bracket{\nu,P_A^*(dA)} - \frac{1}{2\epsilon}|\nu|^2)\bracket{d\nu,A}^{\dim(G)}\\
&=&
\frac{1}{K} \int_{\mu^{-1}(0) \times \lieg} \left(\int_\lieg P_A^*\circ \iota^*(\alpha)(\phi+ \frac{i}{\epsilon} \nu)
\exp(-\frac{\epsilon}{2}|\phi|^2) \dint \phi\right)\\
&&\qquad \cdot \exp(\pi^* \omega_0 + \bracket{\nu,F_A} - \frac{1}{2\epsilon}|\nu|^2)\bracket{d\nu,A}^{\dim(G)}\\
&=&
\frac{1}{K} \int_{\mu^{-1}(0) \times \lieg} \left(\int_\lieg P_A^*\circ \iota^*(\alpha)(\phi+ \frac{i}{\epsilon} \nu +i F_A)
\exp(-\frac{\epsilon}{2}|\phi|^2- \frac{1}{2\epsilon}|\nu|^2) \dint \phi\right)\\
&&\qquad \cdot \exp(\pi^* \omega_0 + \frac{\epsilon}{2} |F_A|^2 ) \bracket{d\nu,A}^{\dim(G)}\\
&=&\frac{\vol(G)}{K} \int_{\mu^{-1}(0)/G} \left(\int_\lieg \int_\lieg P_A^*\circ \iota^*(\alpha)(\phi+ \frac{i}{\epsilon} \nu +i F_A)
\exp(-\frac{\epsilon}{2}|\phi|^2 - \frac{1}{2\epsilon}|\nu|^2) \dint \phi \dint \nu\right)\\
&&\qquad \cdot \exp( \omega_0 + \frac{\epsilon}{2} |F_A|^2 )
\end{eqnarray*}
where we have completed the square on $\nu$ and integrated the result over the vertical fibers, noting that the integral is constant in these directions and that the measure $ \bracket{d\nu,A}^{\dim(G)}$ is equal to Haar measure on the vertical fiber times Lebesgue measure $\dint \nu$ on $\nu .$ Now changing the $\nu$ and $\phi$ variables to a single complex variable $z=\sqrt{\epsilon} \phi + i \nu/\sqrt{\epsilon}$ and noting that the integral of any complex polynomial against a complex Gaussian measure gives its constant term yields
\begin{eqnarray*}
&=&\frac{\vol(G)}{K} \int_{\mu^{-1}(0)/G} \left(\int_{\lieg + i \lieg} P_A^*\circ \iota^*(\alpha)(\frac{1}{\sqrt{\epsilon}}z +i F_A) \exp(-|z|^2) \dint z\right)\exp( \omega_0 + \frac{\epsilon}{2} |F_A|^2 ) \\
&=&\frac{\vol(G)(2 \pi)^{\dim(G)})}{K} \int_{\mu^{-1}(0)/G} P_A^*\circ \iota^*(\alpha)(i F_A) \exp( \omega_0 + \frac{\epsilon}{2} |F_A|^2 ) \\
&=&\frac{\vol(G)(2 \pi)^{\dim(G)})}{K} \int_{\mu^{-1}(0)/G} \Kirwan(\alpha) \exp( \omega_0 + \frac{\epsilon}{2} |F_A|^2 )
\end{eqnarray*}
\end{pf}
\begin{corollary}\label{cr:reduction_of_integral}
If $\alpha$ is a closed equivariant form, $r$ is a regular value of $|\mu|^2,$ $\mu$ is proper and $0$ is a regular value of $\mu$ then the Basic Integral $\BI(\alpha, r, 0)$ can be written uniquely as a sum of a polynomial in $\epsilon$ plus a term bounded by $\exp(-c/\epsilon)$ for some $c>0,$ the polynomial piece representing the contribution from $\mu^{-1}(0)$ as in Theorem \ref{th:zero_contribution}.
\end{corollary}
\begin{pf}
We know that $\BI(\alpha, r, 0)$ differs from the large $t$ limit $\lim_{t\to \infty} \BI(\alpha, r, t)$ by a quantity bounded by $\exp(-c/\epsilon)$ for some $c>0$ by Lemma \ref{lm:large_t}. On the other hand the large $t$ limit is a sum of contributions from $r=0$ and higher critical values by Corollary \ref{cr:contribution}. The former is a polynomial in $\epsilon$ by Theorem \ref{th:zero_contribution}, the latter is bounded by $\exp(-c/\epsilon)$ for some $c>0$ by Proposition \ref{pr:bound_higher}. Since a function can only be written in one way as a polynomial plus a term bounded by $\exp(-c/\epsilon),$ the result follows.
\end{pf}
In general we have no reason to believe that the integral over all of $M,$ that is the large $r$ limit of $\BI(\alpha, r, t)$ exists for a fixed $t$ or the large $t$ limit. however, if it exists and converges sufficiently rapidly, the same results as above apply. For example
\begin{proposition}
Suppose that $M$ is a Hamiltonian space with proper moment map and $0$ is a regular value of $\mu.$ Suppose also the symplectic volume of $M_r$ as a function of $r$ is such that $|\partial \Vol(M_r)/\partial r | < \exp(c \sqrt{r})$ for some $c>0.$ Suppose also that for some almost complex structure the supremum over all of $M_r$ of the norm of $\alpha$ (the norm as an ordinary form at each point times the norm as a symmetric tensor in $\lieg^*$) is also bounded by $\exp(c \sqrt{r}).$ Then
\[\lim_{r\to \infty} \int_{M_r} \int_\lieg \alpha \exp(\omega + i \mu \phi -\frac{\epsilon}{2}|\phi|^2)\dint \phi\]
exists and is of the form a polynomial in $\epsilon$ plus a term exponentially damped in $\epsilon,$ the polynomial
\end{proposition}
\begin{pf} Fix a regular value $r_0$ of $|\mu|^2.$
\begin{eqnarray*}
&&\left| \lim_{r\to \infty}
\BI(\alpha, r, 0)- \BI(\alpha, r_0, 0)\right|\\
&=& \frac{1}{K} \int_{M-M_{r_0}} \int_{\lieg} \alpha(\phi) \exp(\omega + i \mu \phi -\frac{\epsilon}{2}|\phi|^2)\dint \phi\\
&=& \frac{1}{K} \left|\int_{M-M_{r_0}} \exp(\omega - \frac{1}{2\epsilon}|\mu|^2)\int_{\lieg} \alpha(\phi +_ i \mu^\star/\epsilon) \exp( -\frac{\epsilon}{2}|\phi|^2)\dint \phi\right|\\
&\leq& \frac{1}{K} \left|\int_{r_0}^\infty \exp(-\frac{r}{2\epsilon} + c \sqrt{r}) \text{POLY}(\sqrt{r}, \epsilon^{\pm 1/2}) \partial \Vol(M_r)/\partial r \dint r\right|\\
&\leq& \frac{1}{K} \left|\int_{r_0}^\infty \exp(-\frac{r}{2\epsilon} + 2c \sqrt{r}) \text{POLY}(\sqrt{r}, \epsilon^{\pm 1/2}) \dint r\right|\\
&\leq& \exp(-\frac{k}{\epsilon}) \text{POLY}( \epsilon^{\pm 1/2})
\end{eqnarray*}
for some positive constant $k.$ Applying this to Corollary \ref{cr:reduction_of_integral} gives the result.
\end{pf}
\bibliographystyle{alpha}
\def$'${$'$}
|
{
"timestamp": "2005-03-18T18:16:22",
"yymm": "0503",
"arxiv_id": "math/0503385",
"language": "en",
"url": "https://arxiv.org/abs/math/0503385"
}
|
\section{Introduction}
\label{intro}
Pattern dynamics in non-equilibrium systems have been studied over decades, in fluid, chemical reaction-diffusion systems, liquid crystal, and so forth.
The extensive studies in the field have elucidated a rich variety of pattern dynamics, together with the advances in theoretical analysis \cite{mikhailov 3} $\sim$ \cite{pearson}.
One field in the pattern dynamics that is not so well explored in comparison with the above examples, is a discharge system.
When a strong voltage is applied between the electrodes, electric discharge appears through the gas filled in the chamber.
The discharge often forms a complex spatiotemporal pattern, as has been studied theoretically and experimentally (\cite{2 layers model 1}$\sim$\cite{Germany group 3}).
In such discharge system with a variable resister, there exists some global constraint among each local discharge processes.
Viewed as pattern dynamics, this means existence of global coupling among local dynamical processes.
On the other hand, global coupling among nonlinear units often shows a non-trivial collective motion, as has been extensively studied as collective motion in globally coupled dynamical systems.
Now in the discharge system, there exists interplay between collective motion in globally coupled systems and pattern dynamics by local nonlinear dynamics.
As a study of nonlinear system, it is interesting to search for some novel dynamic state, due to this interplay.
Indeed, there is a beautiful experiment by Nasuno, suggesting such novel, non-trivial dynamics\cite{Nasuno's experiment}\cite{Nasuno's experiment2}.
He set up an experiment consisting of two parallel plates between which electric charge flows.
By controlling the voltage and current, he found formation of spots, organization of 'molecular-like' structures of spots, complex motions of them, including the dynamics what he called teleportation.
In the present paper we introduce a phenomenological model on 2-dimensional (2D) gas discharge system, as a coupled electric circuit subjecting a global constraint, to describe the salient features observed in Nasuno's experiment, and to make further predictions.
As a theoretical model, we adopt a two-layer model that describes well a glow discharge system \cite{2 layers model 1}$\sim$ \cite{gas pressure effect 2}.
This two-layer model consists of a non-linear resistive part and a linear resistive part.
By extending this two-layer model, we construct a coupled dynamical system consisting of elementary circuit systems, each of which represents the process at the interior between square electrodes.
By taking spatial continuum limit, this model is reduced to a two-dimensional reaction diffusion (RD) system with global constraint.
Through extensive simulations of the model, we will show several phases of the pattern dynamics, and reproduce some salient features in Nasuno's experiment.
On the one hand, our model is an abstract and simplified model, and it may not completely correspond to the discharge system by Nasuno.
On the other hand, the model shows several novel interesting patten dynamics as a reaction-diffusion system with global coupling, which itself deserves investigation.
The outline of the paper is as follows.
In Sec.\ref{Experiment}, the experiment by Nasuno is briefly described.
In Sec.\ref{Model}, we present some assumptions in order to reproduce the experiment and then construct a coupled circuit system as a model on the gas discharge experiment.
In Sec.\ref{Phenomena}, results of extensive simulations are presented, with classification of distinctive pattern dynamics as in the phase diagram.
Formation of spots, and a cluster of spots are numerically found, as well as the dynamics of creation and annihilation of spots, while the dynamical systems mechanism of the process is discussed.
In Sec.\ref{S and D}, summary, discussion and future perspectives are presented.
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=0.53]{fig1.jpg}
\caption{Schematic figure of 2 layers model for glow discharge.
We consider that the glow discharge structure consists of two parts, the non-linear resistive region and the linear one.}
\label{glow discharge}
\end{center}
\end{figure}
\section{Experiment}
\label{Experiment}
Here we briefly describe a remarkable experiment by Nasuno \cite{Nasuno's experiment},\cite{Nasuno's experiment2}.
In the experiment, two square electrodes are immersed in low pressure air $P$ and connected to a dc power source $\epsilon$ and a variable resistor $R_{ex}$. The power source $\epsilon$ can supply either a constant voltage of up to 2 kV or a constant current of up to 150 mA via a series resistor $Rex$ 35.2 k$\Omega$.
He adopted the parameter region of the gas discharge experiment so that the product of pressure $P$ and distance $d$ between electrodes satisfies $P \times d \sim$0.04 (Torr$\cdot$cm), by using $Air$.
The parameter region is located in the vicinity of the minimum of paschen curve, according to Fig.7.3 of chapter7 in \cite{discharge physics text}.
It is known that, under the experimental condition, current-voltage characteristics has monotonous curve \cite{discharge physics text}$\sim$\cite{negative slope problems 2}, and that no pattern dynamics is usually observed.
He fixed the total current $I$ instead of voltage, which is different from the conventional discharge experiments.
However, when the current $I$ is fixed as a constant, $V$ is determined uniquely by a monotonous current-voltage characteristic so that $V$ is also constant.
Indeed, according to the experiment, current-voltage characteristic $I-V$ is monotonous, so that $I$ and $V$ remain fixed\cite{Nasuno's experiment2}.
However, local current and charge density between electrodes can change in space and time, to produce nontrivial dynamics.
\paragraph{}
By controlling $I$ and $P$, complex pattern dynamics in glow discharge region was observed.
By increasing current $I$, the pattern dynamics of discharge changes as follows.
\begin{itemize}
\item Just after onset of glow discharge, one isolated spot with a typical size (that is, a humped bell-like light intensity distribution) appears.
The spot wanders irregularly on the square plane.
\item With the increase of $I$, the number of isolated and wandering spots increases.
\item As $I$ is increased further, some spots form a molecule-like localized structure, which is called \em cluster\rm.
This cluster also wanders through the plane.
\item With the further increase of $I$, the number of spots in the cluster increases leading to various clusters with a variety of configurations of spots.
Switching among different clusters occurs intermittently.
\item At much higher $I$, the excited domain (the area with high light intensity) forms a string or a closed loop.
\item A remarkable phenomenon which he termed 'Teleportation' was reported, where annihilation of a spot and immediate recreation of a different spot at a distant place was observed.
For example, when 3 spots existed, one of them suddenly was disappeared, and then a new spot immediately appeared at a distant position, located in the neighbor of a remaining spot, so that the 3 spot-state is recovered \cite{Nasuno's experiment}.
\end{itemize}
The phase diagram for the above behaviors is displayed in terms of $I$ and $P$ in the paper \cite{Nasuno's experiment}.
\section{Model}
\label{Model}
\subsection{Assumption}
Here we introduce an electric circuit model corresponding to the above experiment.
The model is essentially based on a circuit model introduced by Purwins et al.\cite{2 layers model 3}$\sim$\cite{gas pressure effect 2}, derived from an analogy between an electric circuit and a reduced equation from plasma physics.
The model is at a macroscopic phenomenological level, where discharge process between 2D electrodes is described as \em charge transfer process by a coupled circuit system \rm.
For each element, we adopt the so-called ``2 layer model'' which consists of non-linear resistive part and linear resistive one, as schematically shown in Fig.\ref{glow discharge}.
The model has succeeded in explaining several discharge experiments\cite{2 layers model 1}$\sim$\cite{gas pressure effect 2}.
Here we revise their model so that it meets with the experimental condition by Nasuno.
To set up the coupled circuit model, the following assumptions are made following Purwins et al.
\begin{itemize}
\item The experimental condition is set at the region of glow discharge.
\item We describe the phenomena in terms of current density $i(x,y)$ and space charge density $q(x,y)$ because we focus only on the \em charge transfer process \rm between two electrodes.
\item \em Gas pressure effect \rm between electrodes known experimentally is taken into account.
According to \cite{gas pressure effect 1}\cite{gas pressure effect 2}, this effect is effectively described as diffusion of current density that flows between electrodes.
Based on the experimental results, we assume a monotonous relation between gas pressure and the diffusion of current density.
\end{itemize}
In considering Nasuno's experiment, we make further assumptions:
\begin{itemize}
\item \em Global \rm current-voltage characteristic shows a \em monotonous \rm cubic curve in a logarithmic scale \cite{footnote 2}, as will be given later(Fig.\ref{nullcline}(b)).
\item The total current that flows into electrodes is kept as \em constant \rm.
Therefore, the voltage drop between electrodes also remains constant.
\item In electrodes (represented by 2-dimensional planes), local voltage-current density characteristic is assumed to be cubic in a logarithmic scale with a \em negative slope. \rm
\end{itemize}
\subsection{Specific Model}
The first set of assumptions underlie the coupled circuit model by Purwins et al.\cite{2 layers model 3}$\sim$\cite{gas pressure effect 2}, while we arrange it to meet with the latter set of assumptions to adapt to Nasuno's experiment.
\paragraph{}
The discharge model which we consider here consists of two square electrodes (area $S$) and the interior, which are under ``External circuit'' that consists of the dc power source $\epsilon$ and the variable resistor $R_{ex}$, as shown in Fig.\ref{circuit model}.
The total current $I$ flowing into the electrodes is controlled to be constant\cite{Nasuno's experiment}.
Thus $V$ is uniquely determined to be a constant.
Further, each elementary circuit $k$ is connected with the neighboring ones by linear resistor $R$ and is also contacted with ``External circuit''.
As shown in Fig.\ref{circuit model}, this elementary circuit consists of a linear resistor, capacitance, linear coil and a nonlinear resistor given by a specific $i$-$v$ characteristic.
The linear resistor $r$, the region U and the intermediate region correspond to linear resistive layer, the non-linear resistive layer, and the interface in 2-layer model, respectively.
Each part is characterized by a linear resistor $r$, capacitance C, the constant self-inductance $l$ of the coil, while the voltage drop $v_{k}$ at the non-linear layer follows the $i$-$v$ characteristic given by a log-scaled cubic function of $i_{k}$.
The total current $I$ is distributed to each element with a current $J_k$, while its current at B and U is given by $j_{k}$ and $i_{k} + \dot{q_{k}}$ respectively, as shown in Fig.\ref{circuit model}.
Now, the main discharge system is described by a set of $N$ ordinary differential equations as $N$ coupled circuits, as shown in Fig \ref{circuit model}.
In the model it is assumed that there are some charge flow into the interface (i.e. at "B" in the figure) from the non-linear resistive layer.
This flow of charge density is represented \em by \rm $\dot{s_{k}}$.
The voltage drop at the interface is represented by $\frac{s_{k}}{C}$
Since the total space charge $Q$ should be constant in time between electrodes, there is a global constraint on $\sum_k s_k$ as will be discussed again in
Sec.\ref{Flowing charge density}.
Considering the law of the conservation of charge (the first Kirchoff's rule) at the point A, we get
\begin{eqnarray}
\label{model-1}
J_{k}-j_{k}+[-\frac{1}{RC}\{(q_{k}+s_{k})-(q_{k-1}+s_{k-1}) \\
\nonumber +(q_{k}+s_{k})-(q_{k+1}+s_{k+1})\}]=0,
\end{eqnarray}
while the law of the conservation of charge at the point B gives
\begin{equation}
\label{model-2}
j_{k}-(i_{k}+\frac{dq_{k}}{dt})=\frac{ds_{k}}{dt}.
\end{equation}
At the region U, the voltage balance (the second Kirchoff's rule) leads to the following equation
\begin{equation}
\label{model-3}
l\frac{di_{k}}{dt}+v_{k}-\frac{q_{k}}{C}=0.
\end{equation}
In addition to (\ref{model-1})$\sim$(\ref{model-3}), we need to consider the constraint on the constant total current $I$ and constant voltage drop $V$.
\begin{gather}
\label{model-I}
I=\sum_{k=1}^{N}J_{k}=const. \\
\label{model-V}
V=rJ_{k}+\frac{q_{k}}{C}+\frac{s_{k}}{C}=const.
\end{gather}
Here, to solve the above equations, we assume a constraint for the description of $s_{k}$ as a function of $\{i_{m},q_{m}\},(m=1,2,...,N)$.
Accordingly we must define $s_{k}$ satisfying
(\ref{model-1})$\sim$(\ref{model-V}).
After some calculations, with $v_{k}=v(i_{k})$ and $q^{\prime}_{k}=q_{k}+s_{k}$, $s_{k}$ are determined so as to satisfy
\begin{gather}
\frac{dq^{\prime}_{k}}{dt}=\frac{V}{r}-\frac{q^{\prime}_{k}}{rC}-i_{k}+\frac{1}{RC}(q^{\prime}_{k-1}+q^{\prime}_{k+1}-2q^{\prime}_{k}) \tag*{}
\\
\frac{di_{k}}{dt}=\frac{q^{\prime}_{k}}{lC}-\frac{s_{k}}{lC}-\frac{v(i_{k})}{l} \tag*{}
\\
I=\sum_{k=1}^{N}i_{k}=const. \tag*{} \\
Q^{\prime}=\sum_{k=1}^{N}q^{\prime}_{k}=Cv(\frac{I}{N})N=const.\tag*{} \\
V=r\frac{I}{N}+\frac{1}{C}\frac{Q^{\prime}}{N}=const.\tag*{}
\end{gather}
We define $s_{k}$ as
\[s_{k}\equiv\frac{i_{k}}{I}(Q^{\prime}-C\sum_{k=1}^{N}v(i_{k})).\]
(See appendix for details.)
By taking spatial continuum limit, we obtain the following 2-component RD equation, with
$D_{q^{\prime}}= \frac{1}{RC}, Q^{\prime}=\int q^{\prime} dS$, as
\begin{subequations}
\begin{eqnarray}
\label{model-4}
\frac{dq^{\prime}}{dt}=\frac{V}{r}-\frac{q^{\prime}}{rC}-i+D_{q^{\prime}}\triangle q^{\prime} \\
\label{model-5}
\frac{di}{dt}=\frac{q^{\prime}}{lC}-\frac{s}{lC}-\frac{v(i)}{l}+D_{i}\triangle i \\
\label{model-6}
s\equiv\frac{i}{I}\{Q^{\prime}-C\int v(i) dS\} \\
\label{model-7}
I=\int i dS =const. \hspace{.2in} \\
\label{model-8}
Q^{\prime}=\int q^{\prime} dS =Cv(\frac{I}{S})S=const. \\
\label{model-9}
V=r\frac{I}{S}+v(\frac{I}{S})=const.
\end{eqnarray}
\end{subequations}
By using dimensionless variables
\paragraph{}
$\tilde{i}=\frac{i}{i_{c}}$, $\tilde{v}=\frac{v}{v_{c}}$,
$\tilde{q}=\frac{q^{\prime}}{Cv_{c}}$, $\tilde{t}=\frac{t}{\frac{l}{r}}$
\hspace{.1in}and redefining
$\tilde{i}\rightarrow i$, $\tilde{v}\rightarrow v$,
$\tilde{q}\rightarrow q$, $\tilde{t}\rightarrow t$,\hspace{.2in}
the RD equation (\ref{model-4})$\sim$(\ref{model-9}) is written as
\begin{subequations}
\begin{eqnarray}
\label{mujigen-1}
\tau\frac{dq}{dt}=V-q-ai+\triangle q \\
\label{mujigen-2}
\frac{di}{dt}=\frac{1}{a}\{q-s-v(i)\}+D\triangle i \\
\label{mujigen-3}
s\equiv\frac{i}{I}\{Q-\int v(i) dS\} \\
\label{mujigen-4}
I=\int i dS =const. \hspace{.2in} \\
\label{mujigen-5}
Q=\int q dS =v(\frac{I}{S})S=const. \\
\label{mujigen-6}
V=a\frac{I}{S}+v(\frac{I}{S})=const.,
\end{eqnarray}
\end{subequations}
\paragraph{}
where
$\tau=\frac{rC}{\frac{l}{r}},a=r\frac{i_{c}}{v_{c}},D^{\prime}=\frac{D_{i}}{D_{q}},D=\frac{D^{\prime}}{\tau},\\
\hspace{.32in}\xi^{2}=rCD_{q}$ (characteristic length),
$S=\frac{1}{\xi^{2}}$.
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=0.55]{fig2.jpg}
\caption{Schematic figure of our 2-dimensional discharge model.
Our model system consists of $N$ coupled circuits.
The element $k$ is connected with neighbor ones by linear resistor $R$ and with ``External circuit''.
Although displayed in a one-dimensional representation, what we actually simulated in the paper is a two-dimensional case. }
\label{circuit model}
\end{center}
\end{figure}
\subsection{Flowing charge density}
\label{Flowing charge density}
Here we make some remarks on the flowing charge $s_{k}$ at the point B in Fig \ref{circuit model}.
The point B corresponds to the interface of the 2-layer model, that is, the position between the ``Negative glow" and Faraday dark space, which exists for the steady glow discharge (Fig \ref{glow discharge}).
In this region, electrons flowing from Cathode layer (non-linear resistive layer) combine with positive ions flowing from ``Positive column", or anode (i.e., the linear resistive layer).
There, some complex processes occur through diffusion, excitation, ionization, and recombination of molecules.
Hence there is a longitudinal flow of charge, and the flowing charge $s_{k}$ at B is distributed.
For a homogeneous steady glow discharge, there is no charge in this region.
Furthermore, $s_{k}$ at B is zero, as the initial state before the discharge.
Hence, if a spatially homogeneous state were stable, the flow term $s_{k}$ would remain to be 0.
However, as will be shown, for almost all the parameter regions, such homogeneous state is unstable, where charge density $q$ is distributed inhomogeneously in the non-linear resistive layer.
In this case, flowing charge density $s$ is also distributed in the interface.
Hence we need to consider this term.
After taking a spatial continuum limit, the total flowing charge is given by the spatial integration,
\begin{equation}
h\equiv \int s dS (= Q-\int v(i) dS ),
\end{equation}
where $h$ means the total charge flowing from the non-linear resistive layer into the interface, which is a macroscopic dynamical variable characterizing the global charge transfer process between the non-linear layer and the interface.
\paragraph{}
In Nasuno's experiment \cite{Nasuno's experiment},\cite{Nasuno's experiment2}, due to the short distance between electrodes, the Positive column does not appear.
Hence the anode and the anode surface correspond to the linear part and the interface, respectively.
Since the anode has very small resistance in the experiment, $s$ is assumed to be distributed over the anode surface.
\subsection{Parameter setting}
Hereafter we study the behaviors of the present model given by the equations (\ref{mujigen-1})$\sim$(\ref{mujigen-6}), by controlling the parameters $I$ and $D$, which are the total current into the system, and a parameter for the gas pressure effect, respectively.
Hence the control parameters $I$ and $D$ correspond to those in the experiment, i.e., the total current and the pressure $P$, respectively.
As initial conditions of $i(x,y)$ and $q(x,y)$, we mostly choose ``uniform state $i_{u}, q_{u}$" with small fluctuations $\eta(x,y)$ as a uniform random number over [-0.01,0.01].
Accordingly total current $I$ (= $i_{u}S$), total charge $Q (=\int q dS)$ and voltage $V$ are determined initially (See appendix).
The phenomena to be discussed is always observed from these initial conditions, \em i.e. \rm the pattern dynamics is attracted to a global attractor.
Unless otherwise mentioned, the other parameters are fixed as
\paragraph{}
$\tau=20, \xi^{2}=0.004 $ (characteristic length),$ a=400\frac{i_{c}}{v_{c}}=0.0108$,$ S=\frac{1}{\xi^{2}}=250$.
\paragraph{}
Numerical simulations are carried out by the mesh size $128\times128$.
Unless otherwise mentioned, we choose a periodic boundary condition.
These parameters are chosen so that a single element satisfies the
behavior for a simple discharge (without spatial pattern).
Before describing the pattern dynamics of the model, we first show how a
single element behaves.
Fig \ref{nullcline} is the nullcline of the single element dynamics (Fig.\ref{nullcline}(a)), and global $I-V$ characteristic (Fig.\ref{nullcline}(b)).
We mainly focus on the case that the steady state ($i_{u},q_{u}$) is among the critical point (1,1) in Fig.\ref{nullcline}(a) \cite{icvc}.
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=0.7]{fig3.jpg}
\caption{Nullcline of the single element dynamics (a), and global $I-V$ characteristic (b).
We mainly focus on the case that the steady state ($i_{u},q_{u}$) is among the critical point (1,1), where it is stable if $i_{u} < 1$, whereas destabilized if $i_{u}> 1$ and then the element shows limit cycle oscillation.
$I-V$ characteristic is monotonous so that a steady state is realized.
Q is also a function of $i_{u}$.
$v(i)$ is cubic and log-scaled function of $i$.
Note the relation $I=i_{u}S$.}
\label{nullcline}
\end{center}
\end{figure}
\section{Phenomena}
\label{Phenomena}
\subsection{Global phase diagram}
According to our interpretation of the present model (\ref{mujigen-1})$\sim$(\ref{mujigen-6}), the discharge occurs at local site if $i(x,y) \geq i_{u}$.
Hence we show the pattern dynamics by displaying only the pixels that satisfy $i(x,y)>i_{u}$, an active region with discharge, by gray.
Here we study the pattern dynamics by changing $i_{u} (=\frac{I}{S})$ and $D$, by focusing on the parameter region with $i_{u} \sim 1$, where the discharge phenomena occur.
As a 2-dimensional phase diagram with regard to $i_{u}$ and $D$, the pattern dynamics we have observed is classified into five phases, as displayed in Fig.\ref{global phase diagram}.
We first give a brief description of each phase, and will discuss the characteristic of each phase in detail later.
For small $i_{u}(\sim 1.0)$, there are three phases, depending on the value of $D$, that are "distributed spot phase (DS)", "localized spots phase (LS)", and moving spot phase (MS), respectively, as shown in Fig \ref{global phase diagram}.
In these phases, spots are formed.
Discharge occurs locally, only within these spots.
Indeed, when $i(x,y)$ is plotted in space and time, it shows a stepwise increase at the border of each spot, and shows a high plateau within each spot.
The pattern dynamics here will be discussed from the configuration of spot patterns and their motion, with which the three phases are classified.
For all these three phases, the number of spots increases with $i_u$.
For a larger value of $i_{u}$, there appears a periodic pattern in space, which we call Periodic Wave (PW) phase.
For a further large value of $i_u$, there is a uniform glow (UG), without spatial inhomogeneity.
The behavior of each phase is summarized as follows;
\begin{itemize}
\item DS phase --- Spots are arranged with some distance on the average, which decreases with the increase of $i_u$.
For small $i_u$, a few spots are isolated with some distance, while hexagonal pattern of spots appears as $i_{u}$ is increased.
After transient time, these spots are fixed, and do not move.
\item LS phase ---
Spots form several clusters similar to molecular structures, as reported in the experiment by Nasuno.
The clusters show a dynamic, complex behavior with the formation, collapse, and regeneration.
\item MS phase --- A spot can move by itself in the space, which shows a soliton-like behavior without collapse.
\item PW phase --- There appears a periodic wave pattern in space and time.
\item UG phase --- There is a uniform glow, without spatial inhomogeneity.
\end{itemize}
\begin{figure}
\begin{center}
\includegraphics[scale=0.45]{fig4.jpg}
\caption{Phase diagram as function of $i_{u} (=\frac{I}{S})$ and $D$.
Vertical axis is log-scaled.
Global property of the model is roughly classified into five phases under $i_{u}> 0.9 $.}
\label{global phase diagram}
\end{center}
\end{figure}
\paragraph{Collective variable -- $\int{s}dS$ ($\equiv h$) --}
As a global measure characterizing the pattern dynamics, the integration of $s$, $h \equiv \int{s}dS$ is often useful.
In the subsequent subsections, the pattern dynamics will be characterized by the motion of $h$, which is a collective variable.
Now we discuss the pattern dynamics, by referring to the change of this collective variable $h$.
(Note that $h \ll Q$ in the present parameter setting.)
\subsection{DS phase}
\paragraph{Isolated steady spots}
Starting with initial condition $i(x,y)=i_{u}+\eta(x,y)$, competition for current among elements occurs, resulting in instability of the homogeneous state.
This leads to the formation of spots with localized high current.
These spots are of the same size.
Initially, each spot moves very slowly, and is arranged so that each is located with an equal distance, until a regular, stationary spot pattern is formed.
\paragraph{Hexagonal pattern}
As $i_{u}$ is increased, the number of spots is increased, so that the distance between spots is decreased.
As $i_u$ is increased from $1.5$ to $2.0$, a regular hexagonal pattern of spots is formed, as shown in Fig.\ref{alpha-phase}.
The formation of this regular lattice of spots is understood through weak repulsion between neighboring spots.
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=0.35]{fig5.jpg}
\caption{Steady patterns observed in the DS phase.
Spots are of the same size under fixed $D$.
Initially, each spot moves very slowly, and is arranged so that it is located with the equal distance.
As $i_{u}$ is increased, the number of spots is increased.
As the number is increased with the increase of $i_u$, the distance between spots is decreased.
Left--Isolated steady spots.
$i_{u}=$0.87,0.93,1.0 in the order from the top to bottom with, $D$=0.01.
Right--Hexagonal spots pattern.
$i_{u}=$1.2,1.5,2.0 in the order from the top to bottom with $D$=0.01.
The mesh size here is 256$\times$256 }
\label{alpha-phase}
\end{center}
\end{figure}
\paragraph{Behaviors at the boundary between DS and other phases}
As $i_{u}$ is increased toward the boundary to the PW phase, the spots start to breathe, and their sizes (and shape) change, while the distance between spots still remains almost equal as in Fig.\ref{intermittency}(a-1).
The collective variable $h$ changes intermittently, corresponding to irregular pattern dynamics (Fig.\ref{intermittency}(a-2)).
Around the boundary between the DS and LS phases, the characteristic wavelength disappears, and the configuration of spots is irregular, while spots start to show a bursting behavior, as well as splitting, as in (Fig \ref{intermittency}(b-1)).
The change of $h$ as well as the pattern dynamics is irregular(Fig \ref{intermittency}(b-2)).
\paragraph{}
In the experiment by Nasuno, the behavior corresponding to this phase is not observed.
We expect that the range of the change of pressure in the experiment may not be sufficient to cover low values needed and to detect this phase.
\begin{figure}
\begin{center}
\includegraphics[scale=0.47]{fig6.jpg}
\caption{Behaviors at the boundary between the DS and PW phases(a) and at the boundary between the DS and LS phases (b).
The left figures (1) give snapshots of pattern dynamics, while the time series of the integration of flowing charge $h(=\int{s}dS)$ corresponding to them are plotted in (2)(right).
In (a), the spots start to breathe, so that their sizes change, while the distance still remains almost equal.
In (b), the distance is no longer equal, and an irregular pattern is formed.\hspace{.2in}(a) $i_{u}=2.4,D=0.01$,\hspace{.1in} (b) $i_{u}=2.0,D=0.012$, mesh size 256$\times$256 }
\label{intermittency}
\end{center}
\end{figure}
\subsection{LS phase}
\label{betaphase}
For small $i_{u}$($\sim$ 0.9), a single spot appears.
As $i_{u}$ is increased, the number of spots increases, through successive split of the original spot.
Spots are not completely stable in time, and some of them collapse after breathing.
As replication and extinction of spots are repeated, a cluster of spots is formed.
For small $i_u$ , the number of spots changes between 1 and 2, while the range of the number of spots that the system can take increases with $i_u$.
Some of the generation processes are displayed in Fig \ref{split}.
\paragraph{Phase diagram within the LS phase}
By measuring the dependence of the maximal number of spots on $i_u$ and $D$, the phase diagram within the LS phase is depicted as shown in Fig.\ref{beta phase diagram}, where the maximal number of spots is plotted in the $D-i_{u}$ parameter space.
Hereafter denote \em the case ``$M_{k}$'' \rm when $k$ spots exist at maximum.
On the other hand, if $n$ spots exist at a moment, we call \em the state ``$sp_{n}$''\rm .
Indeed, the configuration of the phases in the diagram of Fig.\ref{beta phase diagram} obtained from our model agree rather well with that obtained experimentally by Nasuno\cite{Nasuno's experiment}.
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=0.55]{fig7.jpg}
\caption{Phase diagram of the LS phase.
The behaviors in the LS phase are classified in terms of the maximal number of spots, and plotted as a phase diagram in the $D-i_{u}$ parameter space.
Note that, in this phase diagram, ``$M_{n}$'' shows the case in which the maximal number of spots is $n$.}
\label{beta phase diagram}
\end{center}
\end{figure}
\paragraph{Localized cluster of spots; molecule-like structure}
In the following, we discuss dynamics of a localized cluster of spots.
After spots are generated by splitting, they separate up to some distance.
Thus they form a chain-like structure, a localized structure like a molecule.
As the distance between spots is increased and a spot is separated from a cluster very slowly, it starts breathing.
Following the amplification of this oscillation, the spot collapses.
After this collapse, split of some other spot(s) follows, and the number of spots returns to the maximal under the given value of $i_{u}$.
In this way, a cluster of spots is sustained.
When the maximal number of spots $\geq$ 6, there appears a variety of configurations for the cluster, as shown in Fig \ref{localized structures}.
Depending on which spot splits to which direction, there appears a different configuration.
Over a long time span, one can observe a variety of spot configurations.
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=0.55]{fig8.jpg}
\caption{Snapshots of typical clusters,$sp_{2}\sim sp_{8}$.
They are the snapshots of the clusters with the maximal number of spots under the given $i_{u}$ and $D$.
$D$=$0.014.\hspace{.03in}sp_{2}i_{u}$=$0.93$,$sp_{3}i_{u}$=$0.934$,$sp_{4}$(a)$i_{u}$=$0.98,\hspace{.02in}
\newline sp_{4}$(b)$i_{u}$=$1.01,\hspace{.02in}sp_{5}$(a)$i_{u}$=$1.1$
$sp_{5}$(b)$i_{u}$=$1.06, sp_{6}$(a)
\newline $i_{u}$=$1.1,\hspace{.02in}sp_{6}$(b)$i_{u}$=$1.1$,$sp_{6}$(c)$i_{u}$=$1.1,\hspace{.02in}sp_{6}$(d)$i_{u}$=$1.1,\hspace{.02in}
\newline sp_{8}$(a)$i_{u}$=$1.15$,$sp_{6}$(b)$i_{u}$=$1.15$.}
\label{localized structures}
\end{center}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=0.7]{fig9.jpg}
\caption{Generation processes of the clusters $sp_{3} \sim sp_{8}$ through successive split of the spots.
$sp_{3}$ : $sp_{1} \rightarrow sp_{2} \Rightarrow sp_{2}\rightarrow
sp_{3} \Rightarrow sp_{3}$ cluster ($i_{u}=0.939,D=0.014$).
$sp_{4}$ : $sp_{1} \rightarrow sp_{2} \Rightarrow sp_{2} \rightarrow
sp_{4} \Rightarrow sp_{4}$ cluster ($i_{u}=0.98,D=0.014$).
$sp_{5}$ : $sp_{2} \rightarrow sp_{4} \Rightarrow sp_{4} \rightarrow
sp_{5} \Rightarrow sp_{5}$ cluster ($i_{u}=1.06,D=0.013$).
$sp_{6}$ : $sp_{2} \rightarrow sp_{4} \Rightarrow sp_{4} \rightarrow
sp_{6} \Rightarrow sp_{6}$ cluster ($i_{u}=1.08,D=0.013$).
$sp_{7}$ : $sp_{2} \rightarrow sp_{4} \Rightarrow sp_{4} \rightarrow
sp_{7} \Rightarrow sp_{7}$ cluster ($i_{u}=1.1,D=0.013$).
$sp_{8}$ : $sp_{2} \rightarrow sp_{4} \Rightarrow sp_{4} \rightarrow
sp_{8} \Rightarrow sp_{8}$ cluster ($i_{u}=1.1,D=0.013$).}
\label{split}
\end{center}
\end{figure}
In a long time scale, a cluster wanders slowly on a 2-dimensional space, since the configuration after reconstruction of a cluster is slightly different from the original, even if they are similar.
After the split of a spot, there remains some asymmetry between the two spots, which causes a wandering motion of a cluster.
We show three examples of such wandering cluster, by displaying successive snapshot patterns in Fig. \ref{sp2-snap-series}, where the maximum number of spots is 2 ($M_{2}$) for Fig \ref{sp2-snap-series}(a), 3 ($M_{3}$) for Fig \ref{sp2-snap-series}(b) and 4 ($M_{4}$) for Fig, \ref{sp2-snap-series} (c), respectively.
The motion of cluster is not unidirectional, but its direction changes irregularly.
This irregular motion arises since the two spots after split are not completely identical.
\paragraph{Dynamics of the collective variable $h$ corresponding to the change in the spot number}
When the number of spots changes, the transfer of charge is altered drastically. Hence, it is relevant to study the number change in relationship with the collective variable $h$, i.e., the integration of the flowing charge.
In Fig \ref{Time series of h for sp2 to sp4}(a),(b), and (c), we show the time series $h$ for $M_{2}$, $M_{3}$ and $M_{4}$ cases.
We plot the time series of $h$ (in (a-1),(b-1),(c-1)), as well as the orbit in the phase space, by embedding the time series of $h$ into three dimensional phase space ($h(t),h(t+1),h(t+2)$).
For all the cases, $h$ increases with the spot number $n$.
When the spot number stays at some value, $h$ remains almost constant.
Note that $h$ value is mainly determined by the spot number of the moment $sp_n$, and is almost independent of the maximal number $M_k$ for a given condition.
For example, $h$ for $sp_{2}$ under $M_{2}$ almost equals to that of $sp_{2}$ under $M_{4}$.
If the system stays at a state with given $sp_{n}$ ($n \leq N$), it is clearly seen as each plateau in the time series of $h$, or a region with residence of an orbit around a fixed point in the phase space.
With the breathing of spots, $h$ also starts to oscillate, and in the phase space picture, the orbit spirals out of a fixed point, leading to a switch to a state with a different number of spots.
Consider the case M3 or M4.
There the decrease in the number of spots occurs as $(sp_{4}\rightarrow ) sp_{3}\rightarrow sp_{2} \rightarrow sp_{1}$, successively.
During this decrease, the system stays at each state with an intermediate spot number, for some time interval, while the return process $sp_{1}\rightarrow sp_{2} \rightarrow sp_{3}$ (or $sp_{4}$), is relatively rapid, without staying long at each intermediate state.
(The state $sp_1$ is unstable, and the orbit exits immediately).
These decrease and increase in the spot number are repeated.
We also note that the switching process observed here is true for a state with a higher number of spots, as have been numerically confirmed up to $sp_{8}$.
In Fig.\ref{M6-M8}, we display an example for $M_{6}\sim M_{8}$ case.
In the three-dimensional representation of the phase space from the time series of $h$, each state of a given spot number seems to be regarded as a saddle which has one-dimensional stable manifold, and two-dimensional unstable manifold, with an unstable focus.
The attraction and repulsion of each orbit around a saddle appears whenever the orbit passes through a given $sp_{n}$ state.
Each state of a given spot number is approached from a certain direction, and the orbit spirals out from it.
This process is analogous to that observed in Shilnikov chaos\cite{Shilnikov chaos}.
In the Shilnikov chaos, however, the spiral motion is stable, giving a stable 2-dimensional manifold for a focus, while a third direction gives an unstable manifold.
In contrast, in the present case, the former is unstable, and the third direction gives a stable manifold.
It is expected that the instability here leads to irregular wandering of spots.
However, when $h$ increases, the orbits path through several saddle points with three real eigenvalues in this representation of the phase space, as long as $sp_n$ is less than its maximal spot number (as can be seen in Fig. \ref{Time series of h for sp2 to sp4}).
Note that the orbits approaching the saddle and leaving it take different paths, depending on either if $h$ is increased or decreased, as shown in Fig.\ref{saddle}. In other words, the three-dimensional phase-space representation is insufficient, and the high-dimensionality in the original dynamics leads to the dependence on the history of the change in $h$.
In this sense, the itinerant motion over different spot numbers will be better described as chaotic itinerancy\cite{GCM,chaotic scenario}, in the sense that several low dimensional ordered states are visited through high-dimensional chaotic motion.
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=0.6]{fig10.jpg}
\caption{A sequence of snapshots for $sp_{2},sp_{3},sp_{4}$ state.
We see that the positions of $sp_{2},sp_{3},sp_{4}$ shift in time due to the asymmetry for splitting.
In a long time scale, the clusters appear to wander.
(a) $sp_{2}$:$i_{u}$=$0.93$($M_{2}$case), (b) $sp_{3}$:$i_{u}$=$0.939$($M_{3}$case), (c) $sp_{4}$:$i_{u}$=$0.98$($M_{4}$case), $D$=$0.014$}
\label{sp2-snap-series}
\end{center}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=0.42]{fig11.jpg}
\caption{Dynamics of $h$. (1) Its time series and (2) orbits embedded into a 3-dimensional phase.
(a) The case $M_{2}$ ($i_{u}$=0.93), (b) The case $M_{3}$ ($i_{u}$=0.939), (c) The case $M_{4}$ ($i_{u}$=0.98).
Each plateau value of $h$ corresponds to each state $sp_{1}\sim sp_{4}$ with respect to different $i_{u}$ under fixed $D$.
$D=0.014$ }
\label{Time series of h for sp2 to sp4}
\end{center}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=0.4]{fig12.jpg}
\caption{Motion around a saddle in high dimensional phase space.
The nature of the saddle point $sp_{m}$ is different depending on if $h$ increases or decreases. The approach to and deviation from a saddle changes according to the in(de)crease of $h$.
$i_{u}=0.939,D=0.014$ ($M_{3}$ case, m=2) }
\label{saddle}
\end{center}
\end{figure}
\paragraph{Transition rules between several clusters}
Here we study transition-rules between clusters in some detail.
As the increase in the spot number occurs through the splitting of a spot, there exists a certain transition rule from a state with a lower number of spots to that with a higher number.
Here we display the paths of generating $sp_{n}(n=2,...,8)$ through the splitting process (see Fig.\ref{keito}).
As shown, each cluster with a larger number of spots is systematically generated from a \em specific \rm state with a smaller number.
Note that there are several states with different configuration of spots for a given number of them, when it is larger than or equal to 6 \cite{footnote 3}.
This transition rule of clusters is enriched with the increase of $i_{u}$.
In Fig.\ref{process}, Schematic figure for the change of the transition rule with $i_{u}$ is displayed, corresponding to the pattern in Fig \ref{keito}, where the solid upward and broken downward arrows show multiplication of spots by splitting and their collapse by breathing, respectively, while the thickness of the arrows shows the frequency of such processes observed.
The transition rules are summarized as follows.
\begin{itemize}
\item When $i_{u}$ is small, only the transition $sp_{1}\rightleftharpoons sp_{2}$ occurs.
\item As $i_{u}$ is increased, the states $sp_{3}$ and $sp_{4}^{(1)}$ (which is one type of the 4-spot state) appear, with the transitions $sp_{1}\sim sp_{3}$, $sp_{1}\sim sp_{4}^{(1)}$.
\item With the further increase of $i_{u}$, another 4-spot cluster with asymmetric configuration appears, denoted by $sp_{4}^{(2)}$.
Now there are transitions $sp_{1}\sim sp_{3}$, $sp_{1}\sim sp^{(1)}_{4}$, $sp_{1}\sim sp^{(2)}_{4}$.
\item As $i_{u}$ is increased further, the states $sp_{5}\sim sp_{8}$ appear, with a variety of transitions.
These transitions are divided into two groups as in Fig.\ref{keito}, that is, those among $sp_{1}\sim sp^{(1)}_{4}$ and among $sp_{1}\sim sp^{(2)}_{4}$.
\end{itemize}
Note that with the increase in the spot number, both the configurations and transitions are diversified, while $h$ shows complex dynamics corresponding to the diversification (Fig.\ref{M6-M8}).
\begin{itemize}
\item When the maximum number of spots is larger than or equal to 4, there exist clusters with the same spot number but different configurations.
Those states give almost the same value of $h$, and they are degenerated in the representation of $h$.
\item When the maximum number of spots is larger than or equal to 6, the collapse process is diversified, and indeed is more diverse than that displayed in Fig \ref{process}.
\item The higher the symmetry in the configuration of the clusters is, the higher is the frequency of the appearance of such configuration.
Indeed, this frequent appearance of configuration with higher symmetry is also observed in the experiment (\cite{Nasuno's experiment}\cite{Nasuno's experiment2}).
\item The transitions between the degenerate states, i.e., different configuration states with the same number of spots, cannot occur directly.
Only after the increase or decrease in the spot number, they can mutually change, as in Fig. \ref{process}.
\end{itemize}
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=0.42]{fig13.jpg}
\caption{Time series of $h$ for $i_{u}$=$1.1,D$=$0.013.$
$h$ shows complex dynamics, corresponding to the diversification of the cluster configurations with the increase of the spot number.}
\label{M6-M8}
\end{center}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=0.4]{fig14.jpg}
\caption{Paths to create state with a higher number of spots, $sp_{n}$ (n=2,...,8) by splitting.
Note that only the states generated by splitting are displayed, and some other complicated ones, which are generated as a result of extinction of spots by breathing, are not displayed here.}
\label{keito}
\end{center}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=0.38]{fig15.jpg}
\caption{Schematic figure for transitions among quasi-stationary states.
The solid upward and broken downward arrows show multiplication processes of spots by splitting and extinction process of them by breathing, respectively.
The thickness of the arrows indicates the frequency of the processes.
The typical cluster configurations in the present figure are shown in Fig \ref{keito}.}
\label{process}
\end{center}
\end{figure}
\paragraph{Correspondence with the experiment}
\label{Correspondence with the experiment}
The pattern dynamics we observed here agree with that observed experimentally \cite{Nasuno's experiment}\cite{Nasuno's experiment2} as is summarized as follows.
\begin{itemize}
\item The global phase diagram on the spot number with regards to $i_u$ and $D$ (pressure) agree rather well.
\item The cluster of spots forms a molecule-like structure, whose structures are identical.
These structures are similar to that observed on the experiment.
(In the experiment, the molecular state is so far reported up to the spot number 6 mainly).
In our case, the number of spots shows intermittent decrease, while by comparing the state that is dominant in time, the agreement is clear.
\item When the maximal spot number is larger than or equal to 6, a variety of forms of clusters appear, both in our model and in experiment.
\item An interpretation of teleportation in the term of Nasuno: the reported phenomena in which one of the two spots disappears, and right after it, a new spot appears around the remaining spot.
Since it is observed as if one of the spots moved to a distant place within a very short time scale, he called this phenomenon as 'teleportation'.
According to our result, we can give a possible interpretation to this phenomenon.
In the $M_2$ regime, one of the spots disappears intermittently as we have mentioned.
Following this disappearance of one spot, the remaining spot divides into two.
Note that the time interval between the disappearance of one spot and the division of the remaining spot is very short.
Then, within the resolution of experimental measurement, this process is observed as if one spot teleportated to the location close to another spot.
\end{itemize}
\subsection{Moving spot phase}
At this phase a single spot moves in the space.
For $0.93\leq i_{u}\leq 1.0$ in Fig \ref{global phase diagram}, a single \em moving spot \rm appears, while multiple moving spots are observed with the increase of $i_{u}$.
\paragraph{Soliton-like spot}
Here a single spot (abbreviated by \em ``$ms_{1}$'' \rm) moves without split or collapse.
The locus of a spot is displayed in Fig \ref{Dependence under periodic condition}, with a periodic boundary condition, where a solid line shows the center of mass of $ms_{1}$.
The center of mass is defined by the mean position among sites with $i(x,y)\geq i_{u}$.
Of course, if the spot is completely symmetric in shape, it cannot move.
The motion of a spot is due to asymmetry in the shape of $ms_{1}$, and indeed the spot is deviated from a circle slightly.
The nature of motion changes with the increase of $i_u$.
For small $i_u$ value, that is just above the onset of the appearance of $ms_{1}$, the motion is linear with a constant speed.
(See Fig \ref{Dependence under periodic condition},$i_{u}=0.925$).
With the increase of $i_{u}$, the motion starts to have a curvature, and the locus is bended.
This curvature is increased with $i_u$, and the locus shows a circle, as in Fig \ref{Dependence under periodic condition}, $i_{u}=0.934,0.94$.
With the increase of curvature, the speed also increases.
For this motion of spot, the choice of boundary condition may be more crucial. For example, to compare with an experiment, the periodic boundary condition may not be relevant.
Hence, we have also made some simulations with Neumann boundary condition, as shown in Fig \ref{The orbits of center of mass on Neumann boundary condition}. With reflection at the boundary, the motion becomes more complex.
\begin{figure}
\begin{center}
\includegraphics[scale=0.45]{fig16.jpg}
\caption{Dependence of the orbits of a single moving spot on the value of $i_{u}$, under periodic boundary condition where a solid line shows the center of mass of the spot.
We show the direction of movement of ms1 with an arrow.
$D=$0.027, mesh size 256$\times$256.}
\label{Dependence under periodic condition}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[scale=0.5]{fig17.jpg}
\caption{The loci of a single moving spot under the Neumann boundary condition.
Arrows show the time course of the spot center.
Because the spot reflects at the boundary, the motion is complex as shown for $i_{u}=0.94$.
On the other hand, it often shows periodic orbits as given for the loci for $i_{u}=0.93,0.935,0.945.$
For $i_{u}=0.95,0.97$, it shows almost circular motion, but the center of the circular orbits shifts gradually.
These behaviors depend on the ratio of the radius of curvature of the spot locus to the electrode size $S^{\frac{1}{2}}$. $D=$0.027 }
\label{The orbits of center of mass on Neumann boundary condition}
\end{center}
\end{figure}
\paragraph{Multiple moving spots}
As $i_{u}$ is increased further, a single moving spot splits, leading to moving multiple spots(Fig \ref{ms2-snap}).
Here, the moving spot is first distorted to extend its tail as spiral.
When the tail is small, the spot rotates with it, while as it is larger, the spot is divided into two, which moves to the opposite direction (\em generation of $ms_{2}$\rm).
After the motion of these two spots, collapse of one spot occurs as in the LS phase, and the system comes back to a single spot state.
These processes are repeated to lead to a complex motion of spots.
\begin{figure}
\begin{center}
\includegraphics[scale=0.4]{fig18.jpg}
\caption{Successive snapshots of multiple moving spots.
We show the direction of movement of the moving spots with arrows.
(1)A single moving spot just before splitting.
Here the size starts to expand.
(2)It splits into two spots, but one of them is too small and vanishes.
The other distorted spot survives and soon begins splitting again.
(3)Just before splitting of the distorted spot.
(4)Just after splitting of the distorted spot, leading to the state $ms_{2}$.
Each spot is separated from each other in the opposite direction.
(5)The state $ms_{2}$ is maintained, but one of them starts breathing and will vanish before long.
$i_{u}=1.05,D=0.03$ }
\label{ms2-snap}
\end{center}
\end{figure}
As shown in Fig.\ref{ms2-snap}, the spot shape starts to be distorted.
This distortion is much clearer, as $i_u$ is further increased, where the system approaches the boundary with the PW phase.
There the spot size is larger, and forms a distorted "banana-like" shape (Fig.\ref{banana-snap}).
This "banana-like" spot moves straight without collapse.
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=0.5]{fig19.jpg}
\caption{Banana-like spot.
A spot is spread in the direction perpendicular to the direction of its motion, and finally the banana-like shape is formed.
The banana-like spot moves straight without collapse.
$i_{u}=1.2,D=0.03$ }
\label{banana-snap}
\end{center}
\end{figure}
\paragraph{Periodic wave phase}
In general, as $i_{u}$ is increased, each site tends to synchronize with each other, since the effective (attractive) coupling among oscillator elements is stronger.
Accordingly, in this regime of a large current, spots no longer exist, and they are replaced by a pattern periodic both in space and time (Fig \ref{delta-phase}(a)).
\paragraph{Traveling wave}
One typical pattern observed here is plane wave(Fig \ref{delta-phase}(b)).
As shown, an extended string of discharged region propagates with a constant speed here.
A string of concentrated electric discharge is formed over the plate, and it travels with a constant speed.
\begin{figure}
\begin{center}
\includegraphics[scale=0.42]{fig20.jpg}
\caption{ (a) Periodic pattern.
The parameter values are $i_{u}=5.0$, $D=0.015$, while the similar patterns appear over a wide range of parameters.
(b) Plane wave. $i_{u}=5.0$, $D=0.03$.
This plane wave moves in one direction as shown by the arrow.}
\label{delta-phase}
\end{center}
\end{figure}
\paragraph{Stripe pattern}
Here we briefly describe pattern observed at a different set of parameter values of $\tau$ and $\xi$ than that adopted so far.
For some parameter values of $\tau$ and $\xi$, we have found a steady pattern, which is different from the DS phase.
As an example, we discuss briefly the pattern observed for $\tau=35$ and $\xi$=0.008.
As $i_u$ is increased, the connected spots form a string and form a stripe pattern as shown in Fig \ref{Stripe pattern}.
The length of string is increased with $i_{u}$, and finally a stripe pattern covers the whole space as in Fig.\ref{Stripe pattern}.
Through the time evolution, a steady pattern is formed, without temporal change.
Such pattern formation is also common to that observed in Benard convection with a large aspect ratio.
\begin{figure}[htbp]
\begin{center}
\includegraphics[scale=0.4]{fig21.jpg}
\caption{Steady stripe patten.
Spots are extended and connected to form a \em string \rm ($i_{u}=2.2,4.2$).
As $i_{u}\nearrow$, many strings are formed, and as a result they are connected with the neighboring strings as shown (\em stripe\rm).
Plotted for $i_{u}=6.2,8.2$.}
\label{Stripe pattern}
\end{center}
\end{figure}
\section{Summary and Discussion}
\label{S and D}
\subsection{Summary}
To sum up we have introduced a coupled circuit model $(\ref{mujigen-1})\sim (\ref{mujigen-6})$, corresponding to the discharge experiment by Nasuno.
Mathematically, the model belongs to a class of reaction-diffusion system with global coupling due to the global constraint on the conservation of charge and current.
When the total current is large, the system shows a homogeneous glow, while this homogeneous state is destabilized with the decrease in the current, and a variety of pattern dynamics of discharge is observed, as are classified into four phases, depending on the value of total current and the diffusion constant that corresponds to the pressure value in the discharge experiment.
For the regime with a lower current, the regions with concentrated discharge form spots.
They are classified as
\begin{itemize}
\item
{\bf Distributed Spot phase}:
Spots are arranged with some distance, to form a regular array of spots.
The phase appears for small $D$.
\item
{\bf Localized Spot phase}:
A few number of local spots exist, which form a cluster.
With the increase in the current the spot number increases.
In the cluster spots are arranged like a molecule, while there are several configurations of spots when their number is larger than or equal to 4.
Also, we have observed intermittent collapse of spots to decrease their number, and immediate recovery by the splitting of spots.
These processes are repeated as a cycle, and with this cycle, the cluster of spots wanders throughout the space.
\item
{\bf Moving Spot phase}:
For a larger value of $D$, a shape of a spot is asymmetric, and each spot starts to move by itself.
With the increase of the total current the motion changes from linear to circular, while complex motion is observed when the boundary condition is not periodic.
\end{itemize}
Besides these spot phases, we have observed another phase at a high current region:
\begin{itemize}
\item {\bf Periodic Wave phase}, where a string of discharged region propagates in space.
\end{itemize}
These pattern dynamics are characterized by introducing the collective variable $h=\int s dS$, the integration of the flowing charge $s_{k}$, which expresses the global charge transfer between the non-linear resistive region and the interface in the direction of the gap.
Indeed, $h$ turns out to characterize the number of spots, and the birth-and-death dynamics of spots are represented by the temporal change of $h$.
\subsection{Comparison with the experiment by Nasuno}
The behaviors observed in the LS phase reproduce the phenomena observed in experiments.
They include:
\begin{itemize}
\item
Increase of spot number with the current.
\item
Molecule-like structures of spots and their variety in shapes.
\item
Disappearance and recreation of spots, as is termed 'Teleportation' by Nasuno.
\item
Wandering motion of spots.
\end{itemize}
\paragraph{\em loop \rm}
In the experiment, at a high current region, a loop structure that moves in space is observed.
Although a connected string observed in the PW phase is similar to the loops observed in the experiment, one difference here is that in our case the strings are connected through the whole space.
Since we have chosen the periodic boundary condition, the edges of the traveling string in Fig \ref{delta-phase} are connected, to form a loop, but such traveling string observed is extended to the whole space in our model.
So far, we are not yet confident if the strings in the PW phase correspond to the loops in the experiment.
As the loops are not small in contrast to localized spot structures, the influence of the boundary may be important.
A suitable choice of boundary condition should be necessary.
Instead of periodic or Neumann boundary condition, the Dirichlet boundary may be more relevant to make more accurate correspondence with the experiment.
This problem is left for the future.
\paragraph{Related studies}
\paragraph{}
There are some recent studies concerning Nasuno's experiment on gas discharge.
Static pattern of particle structure is discussed by modifying
Gray-Scott equation in \cite{kobayashi}, while its relationship with
discharge system is not so straightfoward.
In \cite{obstructed discharge}, pattern dynamics caused by local current heating of the gas are studied by using the left-branch of Paschen
curve, termed as obstructed discharge. In our study, we consider surface charge on the electrode in the obstructed discharge in Subsection \ref{Flowing charge density}, and study instability of the homogeneous steady state at the left-branch of Paschen curve.
\subsection{Discussion}
\paragraph{Novel phenomena in reaction-diffusion system}
Since our model belongs to a class of reaction-diffusion systems, many of
the pattern dynamics observed here are common with those studied
therein; formation of spots, array of spots, and strings.
Still, the molecule structure of spot cluster, cyclic process of
collapse and split of spots, wandering motion of spots are rather novel
and characteristic to the present model.
For these phenomena, inclusion of global coupling into local
reaction-diffusion equations is essential.
Systems both with local and global couplings have also been studied in
surface catalytic reactions \cite{mikhailov 1}, coupled
maps\cite{Glocal}, and so forth.
Search for a novel class of pattern dynamics by global coupling will be
interesting in the future.
\paragraph{Itinerancy over clusters with different configurations of spots}
Through split and collapse of spots, their number changes, and also
transitions between quasi-stationary states with different clusters take place.
This transition among quasi-stationary states is found to be governed by
a specific rule with regard to the increase or decrease in the spot number.
With the aid of the phase-space representation by the macroscopic
variable $h$, this process of itinerancy over different quasi-stationary
states with different spot configurations is understood as follows:
Each quasi-stationary state corresponds to a state with a different
configuration or a different number of spots, $sp_n$.
This state is also represented by a ``saddle'' point in the high
dimensional dynamical systems of the collective coordinates $h$.
There is a stable manifold to this saddle connecting from a state with a
different number of spots, while the orbit spirals out of the saddle,
corresponding to the breathing motion of a spot.
Although this low-dimensional dynamical description of quasi-stationary
state seems to be rather effective, the transition, indeed, occurs in a
high-dimensional dynamical system.
Such switching among effectively low-dimensional states within
high-dimensional dynamical system is studied as chaotic
itinerancy\cite{GCM,chaotic scenario,milnor}.
The present model gives a novel example of chaotic itinerancy, whose
mechanism has to be elucidated in future.
When the number of spots is larger than or equal to 6, a variety of transition processes appear due to drastic increase in possible configurations of spots.
As the spot number increases beyond 6, the combinatorial explosion of the cofiguration of spots sets in, so that stable manifolds connecting many saddle points come to be entangled.
This leads to a variety of transitions over quasi-stable spot states, resulting in chaotic itinerancy.
In \cite{magic7_1,magic7_2} it is shown that combinatorial explosion which appears beyond the degrees of freedom $5\sim 9$ leads to complex dynamics behavior with chaotic itinerancy. Complex dynamics for the spot number beyond 6 may be discussed from this viewpoint.
\paragraph{Acknowledge}
We would like to thank Masaki Sano, Masashii Tachikawa, Akinori Awazu, Koichi Fujimoto, Shin'ichi Sasa, and Teruhisa Komatsu, for discussions and suggestions.
The present paper is dedicated to the memory of the late Dr. Satoru Nasuno.
|
{
"timestamp": "2009-04-30T18:29:19",
"yymm": "0503",
"arxiv_id": "nlin/0503034",
"language": "en",
"url": "https://arxiv.org/abs/nlin/0503034"
}
|
\section{Introduction}
\label{sect1}
\input{cakeintro}
\setcounter{equation}{0}
\section{Cake Baking as a Diffusion Process}
\label{sect2}
\subsection{Cake Baking from a Quantitative Point of View}
\label{sect2_1}
\input{cake2_1}
\subsection{The G\'enoise}
\label{sect2_2}
\input{cake2_2}
\subsection{Theory from a Naive Perspective}
\label{sect2_3}
\input{cake2_3}
\setcounter{equation}{0}
\section{Experiment}
\label{sect3}
\subsection{Procedure}
\label{sect3_1}
\input{cake3_1}
\subsection{Revised Model}
\label{sect3_3}
\input{cake3_2}
\input{cake3_3}
\setcounter{equation}{0}
\section{Estimating the Baking Time of a Cake}
\label{sect4}
\input{cake4}
\setcounter{equation}{0}
\section{An Irrelevant but Intriguing Digression}
\label{sect5}
\input{cake5}
\setcounter{equation}{0}
\section{Conclusions}
\label{sect6}
\input{cakeconcl}
\newpage
|
{
"timestamp": "2005-11-14T17:06:49",
"yymm": "0503",
"arxiv_id": "physics/0503210",
"language": "en",
"url": "https://arxiv.org/abs/physics/0503210"
}
|
\section{Introduction}
Elastic hadron electromagnetic form factors (FFs) are fundamental quantities for the understanding of nucleon structure. They contain information on the nucleon ground state, and constitute a further severe test for the models of nucleon structure, which already reproduce the static properties of the nucleon, such as masses and magnetic moments. Moreover, the dependence of FFs on the momentum transfer squared, $q^2=-Q^2$, should reflect the transition from the non perturbative regime, where effective degrees of freedom describe the nucleon structure, to the asymptotic region, where QCD applies.
The magnetic proton FF, which is the dominant term in the elastic $ep$ cross section, has been measured at $Q^2$ values up to 31 GeV$^2$ in the space-like (SL) region \cite{Ar86}, and from $p \bar p$ or $e^+e^-$ annihilation up to 18 GeV$^2$ in the time-like (TL) region \cite{An03}. Large progress has been recently done in the determination of the electric and magnetic proton form factors, based on the idea, firstly suggested in Ref. \cite{Re68}, to measure the polarization of the recoil proton in $\vec e p$ elastic scattering, when the electron is longitudinally polarized.
Experiments, based on this method, have been performed at JLab up to $Q^2=5.6$ GeV$^2$ \cite{Jo00,Ga02}. A similar method, applied to the reaction $d(e,e'n)$p in quasi-elastic kinematics, has allowed the measurement of the neutron electric FF up to $Q^2$=1 GeV$^2$ using a polarized deuteron target \cite{Day} and up to $ Q^2$=1.47 GeV$^2$, measuring the polarization of the outgoing neutron \cite{Madey}. The polarization method has been also successfully applied at low $Q^2$, for a precise determination of the neutron FFs, at Mainz, and shows that $G_{En}$ is definitely different from zero (\cite{Gl04} and refs therein).
These results have been obtained thanks to the availability of high intensity, highly polarized electron beams and polarized targets, and to the optimization of hadron polarimeters in the GeV range. An extension of the measurement of the polarization transfer in $\vec e +p\to e+\vec p$ up to 9 GeV$^2$ is in preparation \cite{04108}.
More data are expected in future, in SL region, after the upgrade of Jlab, and in TL region, at Frascati and at the future FAIR facility at Darmstadt \cite{GSI}.
In the TL region \cite{An03}, due to the poor statistics, the determination of FFs requires to integrate the differential cross section over a wide angular
range. One typically assumes that the $G_E$ contribution plays a minor role in the cross section at large $q^2$ and the
experimental results are usually given
in terms of $|G_M|$, under the hypothesis that $G_E=0$ or $|G_E|=|G_M|$. The first hypothesis is an arbitrary one. The second hypothesis is strictly
valid at threshold only, i.e., for $\tau=q^2/(4m^2)=1$, but there is no
theoretical argument which justifies its validity at any other momentum
transfer, where $q^2\neq 4m^2$ ($m$ is the nucleon mass).
The measurement of the differential
cross section for the process $p+\overline{p}\to \ell^+ +\ell^-$ at a fixed value of the total energy $s$, and for two different angles $\theta$, allowing the separation of the two FFs, $|G_M|^2$ and $|G_E|^2$, is equivalent to the well known Rosenbluth separation for the elastic $ep$-scattering. However, in TL region, this procedure is simpler, as it requires to change only one kinematical variable, $\cos\theta$, whereas, in SL region
it is necessary to change simultaneously two kinematical variables: the energy of the initial electron and the electron scattering angle, fixing the momentum transfer squared, $Q^2$. Due to the limited statistics, the Rosenbluth separation of the $|G_E|^2$ and $|G_M|^2$ contributions has not yet been realized in TL region. Early attempts showed that the large error bars prevent to discriminate between the two hypothesis on $|G_E|$ and $|G_M|$ quoted above \cite{Bi83,Ba94}.
The $|G_M|$ values depend, in principle, on the kinematics where the
measurement was performed and the angular
range of integration. However, it turns out that these two assumptions for $G_E$
lead to comparable values for $|G_M|$.
In the SL region the situation is different. The cross section for the elastic
scattering
of electrons on protons is sufficiently large to allow the measurements of the
angular
distribution and/or of polarization observables. Data on $G_M$ are available up to the highest
measured value, $Q^2\simeq$ 31 GeV$^2$ \cite{Ar86} and this FF is often approximated according to a dipole behavior:
\begin{equation}
G_M(Q^2)/\mu_p=G_d,~\mbox{with}~
G_d=\left [1+{Q^2}/{ m_d^2 }\right ]^{-2},~m_d^2=0.71~\mbox{GeV}^2,
\label{eq:dipole}
\end{equation}
where $\mu_p$ is the magnetic moment of the proton.
It should be noted that the independent determination of both FFs, $G_M$ and $G_E$, from the unpolarized $e^- +p$-cross section, has been done up to $Q^2=$ 8.7 GeV$^2$ \cite{And94}, and the further extraction of $G_M$ assumes $G_E=G_M/\mu_p$.
The behavior of $G_{Ep}$, deduced from polarization experiments, in which, more precisely, the ratio $G_{Ep}/G_{Mp}$ is directly related to the longitudinal and transversal component of the scattered proton polarization, differs from $G_M/\mu_p$,
with a deviation up to 70\% at $Q^2$=5.6 GeV$^2$ \cite{Ga02}. This is the maximum momentum at which new, precise data are available.
The recent experimental data have inspired many new theoretical
developments, and shown the necessity of a global representation of FFs in the full region of momentum transfer squared.
FFs are analytical functions of $q^2$,
being real functions in the SL region (due to the hermiticity of the
electromagnetic Hamiltonian) and complex functions in the
TL region. The discussion of the constraints and consequences of a description in the full kinematical domain was firstly done in Ref. \cite{Bi93} and more recently in Refs. \cite{ETG01,Br03,Ia03,Wa04}.
The extension of the nucleon models developed for the SL region to the TL region is straightforward for VMD inspired models, which may give a good description of all FFs in the whole kinematical region, after a fitting procedure involving a certain number of parameters \cite{Du03,Bij04}.
The purpose of this paper is to update and compare some of the available models on the world data set in both TL and SL regions, and to predict time-like polarization observables, in framework of these models. The paper is organized as follows. In section II the expressions for the relevant polarization observables, in the process $\bar p+p\to \ell^+ +\ell^- $, $\ell=e$ or $\mu$, are given as a function of the electromagnetic FFs, in Section III we update some of the fits of nucleon FFs on the available data, and discuss their extension to the TL region. In section IV we give the predictions of the considered models in TL region.
\section{Observables in TL region}
We develop a simple and transparent formalism for the study of polarization phenomena for
$p+\overline{p}\to \ell^+ +\ell^-$, in framework of one-photon mechanism.
The calculations of the cross section and of the polarization observables for the process $\bar p+p\to \ell^+ +\ell^- $, $\ell=e$ or $\mu$, in the annihilation channel are more conveniently performed in the center of mass system (CMS), Fig. \ref{fig:cms}. The momenta of the particles are indicated in the figure and $|\vec k_1|=|\vec k_2| =|\vec k|$ and $|\vec p_1|=|\vec p_2| =|\vec p|$. Let us choose the $z$ axis along the direction of the incoming antiproton, the $y$ axis normal to the scattering plane, and the $x$ axis to form a left-handed coordinate system. The components of the unity vectors are therefore $\hat{\vec p}=(0,0,1)$ and $\hat{\vec k}=(\sin\theta,0,\cos\theta)$ with $\hat{\vec p}\cdot\hat{\vec k}=\cos\theta$, where $\theta$ is the electron production angle in CMS. The relevant kinematical variable is the antiproton energy, $E$, which is related to the four momentum transfer, $q^2=s=(k_1+k_2)^2=4E^2$, as, in CMS, $\vec k_1+\vec k_2=0$. In the laboratory (Lab) system, one finds $q^2=2m^2+2mE_L$, where $E_L$ is the Lab antiproton energy. The observables are calculated in the approximation of zero electron mass.
The starting point of the analysis of the reaction $p+\overline{p}\to e^+ +e-$ is the standard expression of the matrix element in framework of one-photon exchange mechanism:
\begin{equation}
{\cal M}=\displaystyle\frac {e^2}{q^2}\overline{u}(-k_2)\gamma_{\mu}u(k_1) \overline{u}(p_2)\left [F_{1N}(q^2)\gamma_{\mu}-
\displaystyle\frac{\sigma_{\mu\nu}q_{\nu}}{2m}F_{2N}(q^2)\right] u(-p_1),
\label{eq:mat}
\end{equation}
where $p_1$, $p_2$, $k_1$ and $k_2$ are the four-momenta of initial antiproton and proton and the final electron and positron respectively, $q^2>4m^2$, $q=k_1+k_2=p_1+p_2$. $F_{1N}$ and $F_{2N}$ are the Dirac and Pauli nucleon electromagnetic FFs, which are complex functions of the variable $q^2$ - in the TL region of momentum transfer.
\begin{center}
\begin{figure}[ht]
\mbox{\epsfxsize=8cm\leavevmode\epsffile{cms.eps}}
\caption{The kinematics of the process $p+\overline{p}\to e^- + e^+$ in the reaction CMS.}
\label{fig:cms}
\end{figure}
\end{center}
In framework of one-photon exchange, the matrix element is written as the product of the leptonic and hadronic currents:
\begin{equation}
{\cal M}=\displaystyle\frac{e^2}{q^2} \ell_{\mu}{\cal J}_{\mu}=
\displaystyle\frac{e^2}{q^2} (\ell_0{\cal J}_0- \vec\ell\cdot\vec{\cal J})
=-\displaystyle\frac{e^2}{q^2} \vec\ell\cdot\vec{\cal J},
\label{eq:eq1}
\end{equation}
where $\ell_0{\cal J}_0=0$, due to the conservation of the leptonic and hadronic currents\footnote{The conservation of the current implies that $\ell\cdot q=0$, i.e., $\ell_0 q_0-\vec\ell\cdot\vec q =0$, but $\vec q=\vec k_1+\vec k_2=0 $ in CMS. Therefore, $\ell_0 q_0=0$ for any energy $q_0$, i.e., $\ell _0=0.$}. The expression for the leptonic current is:
\begin{equation}
\vec\ell=\sqrt{q^2}\phi^{\dagger}_2(\vec\sigma-\hat{\vec k}\vec\sigma\cdot\hat{\vec k})\phi_1,
\label{eq:eq2}
\end{equation}
where $\phi_1(\phi_2)$ is the two-component spinor of the electron (positron), $\hat{\vec k}$ is the unit vector along the final electron three-momentum, and for the hadronic current:
\begin{equation}
\vec{\cal J}=\sqrt{q^2}\chi^{\dagger}_2\left [ G_M(q^2)(\vec\sigma-
\hat{\vec p}\vec\sigma\cdot\hat{\vec p})+\displaystyle\frac{1}{\sqrt\tau}G_E(q^2)\hat{\vec p}\vec\sigma\cdot\hat{\vec p} \right ]\chi_1,
\label{eq:eq3}
\end{equation}
where $\chi_1$ and $\chi_2$ are the two-component spinors of the antiproton and the proton, $\hat{\vec p} $ is the unit vector along the three momentum of the antiproton in CMS.
\begin{center}
\begin{figure}
\mbox{\epsfxsize=9.cm\leavevmode\epsffile{borns.ps}}
\caption{One-photon mechanism for $p+\overline{p}\to e^- + e^+$ (with notation of four particle four-momenta).}
\label{fig:borns}
\end{figure}
\end{center}
From this expression one can see the physical meaning of the particular relation between the nucleon electromagnetic FFs at threshold:
$$
G_{E}(q^2)=G_{M}(q^2),~q^2= 4 m^2.
$$
The structure $\hat{\vec p}\vec\sigma\cdot\hat{\vec p}$ describes the $\overline{p}
+p$ annihilation from $D$-wave, i.e., with angular momentum $\ell$=2. At threshold, where $\tau\to 1$, the finite radius of the strong interaction allows only the S-state, and $G_{M}(q^2)-\displaystyle\frac{1}{\sqrt\tau}G_{E}(q^2)=0$.
From Eqs. (\ref{eq:eq1}), (\ref{eq:eq2}), and (\ref{eq:eq3}) one can find the formulas for the unpolarized cross section, the angular asymmetry and all the polarization observables.
\subsection{The cross section}
To calculate the cross section when all particles are unpolarized, one has to sum over the polarization of the final particles and to average over the polarization of initial particles:
$$
\left (\displaystyle\frac{d\sigma}{d\Omega}\right )_0=\displaystyle\frac{| \overline{\cal M}|^2}{64\pi^2 q^2}
\displaystyle\frac{k}{p},~ k=\displaystyle\frac{\sqrt{(q^2)}}{2},~p=\sqrt{\displaystyle\frac{(q^2)}{4}-m^2},
$$
\begin{equation}
| \overline{\cal M}|^2=\displaystyle\frac{1}{4}\displaystyle\frac{e^4}{q^4}
\ell_{ab} {\cal J}_{ab},~\ell_{ab}=\ell_a\ell_b^*,~
{\cal J}_{ab}={\cal J}_a{\cal J}_b^*.
\label{eq:eq12}
\end{equation}
Using the expressions (\ref{eq:eq2}) and (\ref{eq:eq3}), the formula for the cross section in CMS is:
\begin{equation}
\left (\displaystyle\frac{d\sigma}{d\Omega}\right )_0={\cal N}\left [(1+\cos^2\theta)|G_M|^2+\displaystyle\frac{1}{\tau}\sin^2\theta|G_E|^2\right ],
\label{eq:eq7}
\end{equation}
where
${\cal N}=\displaystyle\frac{\alpha^2}{4\sqrt{q^2(q^2-4m^2)}}$, $\alpha=e^2/(4\pi)\simeq 1/137 $, is a kinematical factor. This formula was firstly obtained in Ref. \cite{Zi62}.
The angular dependence of the cross section, Eq. (\ref{eq:eq7}), results
directly from the assumption of one-photon exchange, where the photon has spin 1 and the electromagnetic hadron interaction satisfies the
$P-$invariance.
Therefore, the measurement of the differential
cross section at three angles (or more) would also allow to test the presence of $2\gamma$ exchange \cite{Re03}.
The electric and the magnetic FFs are weighted by different angular terms, in the cross section, Eq. (\ref{eq:eq7}). One can define an angular asymmetry, ${\cal R}$, with respect to the differential cross section measured at $\theta=\pi/2$, $\sigma_0$ \cite{ETG01}:
\begin{equation}
\left (\displaystyle\frac{d\sigma}{d\Omega}\right )_0=
\sigma_0\left [ 1+{\cal R} \cos^2\theta \right ],
\label{eq:asym}
\end{equation}
where ${\cal R}$ can be expressed as a function of FFs:
\begin{equation}
{\cal R}=\displaystyle\frac{\tau|G_M|^2-|G_E|^2}{\tau|G_M|^2+|G_E|^2}.
\end{equation}
This observable should be very sensitive to the different underlying
assumptions on FFs, therefore, a precise measurement of this quantity, which does not require polarized particles, would be very interesting.
The $q^2$ dependence of the total cross section can be presented as follows:
\begin{equation}
\sigma(q^2)={\cal N}\displaystyle\frac{8}{3}\pi \left [2|G_M|^2+ \displaystyle\frac{1}{\tau}|G_E|^2\right ].
\label{eq:eq13}
\end{equation}
Polarization phenomena will be especially important in $p+\overline{p}\to \ell^+ +\ell^-$. The dependence of the cross section on the polarizations $\vec P_1$ and $\vec P_2$ of the colliding antiproton and proton can be written as follows:
\begin{eqnarray}
\left (\displaystyle\frac{d\sigma}{d\Omega}\right )_0(\vec P_1,\vec P_2)
&=& \left (\displaystyle\frac{d\sigma}{d\Omega}\right )_0
[1+A_y(P_{1y}+ P_{2y})+A_{xx} P_{1x}P_{2x}+A_{yy} P_{1y}P_{2y}+A_{zz} P_{1z}P_{2z}\nonumber \\
&&
+A_{xz} (P_{1x}P_{2z}+P_{1z}P_{2x})],
\label{eq:eq13a}
\end{eqnarray}
where the coefficients $A_i$ and $A_{ij}$ $(i,j=x,y,z)$, analyzing powers and correlation coefficients, depend on the nucleon FFs. Their explicit form is given in the following sections. The dependence (\ref{eq:eq13a}) results from the P-invariance of hadron electrodynamics.
\subsection{Single spin polarization observables}
In case of polarized antiproton beam with polarization $\vec P_1$, the contribution to the cross section can be calculated as:
\begin{equation}
\left (\displaystyle\frac{d\sigma}{d\Omega}\right )_0 \vec A_1=-\ell_{ab}\displaystyle\frac{1}{4} Tr {\cal J}_a\vec\sigma {\cal J}_b^*.
\label{eq:eq14}
\end{equation}
Here the terms related to $|G_E|^2$ and $|G_M|^2$ vanish. For the interference terms, the only non zero analyzing power is related to $P_y$:
\begin{equation}
\left (\displaystyle\frac{d\sigma}{d\Omega}\right )_0 A_{1,y}=\displaystyle\frac{\cal N}{\sqrt{\tau}}\sin2\theta Im(G_MG_E^*).
\label{eq:eq15}
\end{equation}
When the target is polarized, one writes:
$$
\left (\displaystyle\frac{d\sigma}{d\Omega}\right )_0\vec A_2=\ell_{ab}\displaystyle\frac{1}{4} Tr {\cal J}_a{\cal J}_b^*
\vec\sigma. $$
Again the terms related to $|G_E|^2$ and $|G_M|^2$ vanish. Moreover, one can find $\vec A_2=\vec A_1=\vec A$.
Eq. (\ref{eq:eq15}) has been proved also in Ref. \cite{Zi62}. One can see that this analyzing power, being T-odd, does not vanish in $p+\overline{p}\to \ell^+ +\ell^-$, even in one-photon approximation, due to the fact FFs are complex in time-like region. This is a principal difference with elastic $ep$ scattering. Let us note also that the assumption $G_E=G_M$ implies $A_y=0$, independently from any model taken for the calculation of FFs.
\subsection{Double spin polarization observables}
The contribution to the cross section, when both colliding particles are polarized is calculated through the following expression:
$$
\left (\displaystyle\frac{d\sigma}{d\Omega}\right )_0
A_{ab}=-\displaystyle\frac{1}{4} \ell_{mn} Tr {\cal J}_m\sigma_a{\cal J}_n^{\dagger}\sigma_b,
$$
where $a$ and $b=x,y,z$ refer to the $a(b)$ component of the projectile (target) polarization. Among the nine possible terms, $A_{xy}=A_{yx}=A_{zy}=A_{yz}=0$, and the nonzero components are:
\begin{eqnarray}
\left (\displaystyle\frac{d\sigma}{d\Omega}\right )_0A_{xx}&=&
\sin^2\theta\left (|G_M|^2 +\displaystyle\frac{1}{\tau}|G_E|^2\right ){\cal N},\nonumber \\
\left (\displaystyle\frac{d\sigma}{d\Omega}\right )_0A_{yy}&=&
-\sin^2\theta\left (|G_M|^2 -\displaystyle\frac{1}{\tau}|G_E|^2\right ){\cal N},\nonumber\\
\left (\displaystyle\frac{d\sigma}{d\Omega}\right )_0A_{zz}&=&
\left [(1+\cos^2\theta)|G_M|^2-
\displaystyle\frac{1}{\tau}\sin^2\theta |G_E|^2\right ]{\cal N},\nonumber\\
\left (\displaystyle\frac{d\sigma}{d\Omega}\right )_0A_{xz}&=&\left (\displaystyle\frac{d\sigma}{d\Omega}\right )_0A_{zx}=
\displaystyle\frac{1}{\sqrt{\tau}}\sin 2\theta Re G_E G_M^* {\cal N}. \label{eq:pol}
\end{eqnarray}
One can see that the double spin observables depend on the moduli squared of FFs, besides $A_{xz}$. Therefore, in order to determine the relative phase of FFs, in TL region, the interesting observables are $A_y$, and $A_{xz}$ which contain, respectively, the imaginary and the real part of the product $G_EG_M^*$.
\section{Results and discussion}
\subsection{The data}
The nucleon FFs world data were collected and listed in Table I for proton FFs and Table II for neutron FFs in SL region.
In Fig. \ref{fig:fig1} the nucleon FFs world data in SL region are shown: the ratio $\mu_p G_{Ep}/G_{Mp} $ (Fig. \ref{fig:fig1}a) , the magnetic proton FF normalized to the dipole FF and divided by $\mu_p$ (Fig. \ref{fig:fig1}b), the electric neutron FF (Fig. \ref{fig:fig1}c), and the magnetic neutron FF normalized to the dipole FF and divided by $\mu_n$ (Fig. \ref{fig:fig1}d).
For the electric proton FF, the discrepancy among the data measured with the Rosenbluth methods (stars) and the polarization method (solid squares) appears clearly in Fig. \ref{fig:fig1}a. This problem has widely been discussed in the literature, (for a recent discussion see, for instance, Ref. \cite{Pu05}) and rises fundamental issues. If the trend indicated by polarization measurements is confirmed at higher $Q^2$ \cite{04108}, not only the electric and magnetic charge distribution in the nucleus are different and deviate, classically, from an exponential charge distribution, but also the electric FF has a zero and becomes eventually negative. This scenario will change our view on the nucleon structure and will favor VMD inspired models like \cite{Du03,Bij04}, which can reproduce such behavior.
We included data issued from both kind of measurements in the fit, although if a consensus seems to appear that FFs extracted from polarization measurements are more reliable, as less affected by all kinds of radiative corrections. Our purpose here is not to get the best $\chi^2$, but to get a global description of the overall data. The precision and the number of points is very different for the different FFs, therefore one can obtain a good $\chi^2$ for a model that reproduces well, for example, the electric and magnetic FFs in the SL region and fails in giving the trend of $G_{En}$ in TL region.
We included in the fit the data on proton magnetic FFs which were published after 1973, and we did not include the data on the neutron electric FFs from Refs. \cite{Ha73,St66,Br95,Hu65,Ak64} as data of much better precision were, later, available, in the same $Q^2$ range.
The data in the TL region are drawn in
In Fig. \ref{fig:fig2}a, b for the proton and in Fig. \ref{fig:fig2}c, d for the neutron, respectively and summarized in Table III. As no separation has been done for electric and magnetic FFs, the data are extracted under the hypothesis that $|G_{EN}|=|G_{MN}|$.
Concerning the neutron, the first and still unique measurement was done at Frascati, by the collaboration FENICE \cite{An98}.
\subsection{The models}
Among the existing models of nucleon FFs, we consider some parametrizations, which have an analytical expression that can be continued in TL region: predictions of pQCD, in a form generally used as simple fit to experimental data, a model based on vector meson dominance (VMD) \cite{Ia73}, and a third model based on an extension of VMD, with additional terms in order to satisfy the asymptotic predictions of QCD \cite{Lomon}, in the form called GKex(02L). We also considered the Hohler parametrization \cite{Ho76} and the Bosted empirical fit \cite{Bo95}.
In order to help the reader, we report in the Appendix the explicit forms of the parametrizations previously published, with the parameters corresponding to the present fit, compared to the published ones.
The pQCD prediction, based on counting rules, follows the dipole behavior (\ref{eq:dipole}) in SL region, and can be extended in TL region as \cite{Le80}:
\begin{equation}
|G_M|=\frac{A(N)}{q^4\ln^2(q^2/\Lambda^2)},
\label{eq:eqtp}
\end{equation}
where $\Lambda=0.3$ GeV is the QCD scale parameter and $A$ is a free parameter.
This simple parametrization is taken to be the same for proton and neutron. The best fit ( Fig. \ref{fig:fig2}, dashed line) is obtained with a parameter $A(p)$= 56.3 GeV$^4$ for proton and
$A(n)$= 77.15 GeV$^4$ for neutron, which reflects the fact that in TL region, neutron FFs are larger than for proton. One should note that errors are also larger in TL region.
A possible explanation of the fact that FFs are systematically larger in TL region than in SL region (which is true also in the proton case) is the presence of a resonance in the $N\overline{N}$ system, just below the $N\overline{N}$ threshold \cite{Ga96}.
More pQCD inspired parametrizations exist for the form factor ratio $F_2/F_1$, which include logarithmic corrections, and have been recently discussed in Ref. \cite{Br03}. However, some of these analytical forms have problems related to the asymptotic behavior. This will be discussed in a future paper.
The analytical continuation to TL region of the other models is based on the following relations:
\begin{equation}
Q^2=-q^2=q^2e^{-i\pi}~\Longrightarrow~\left\{\begin{array}{c}
\ln(Q^2)=ln(q^2)-i\pi\\
\sqrt{Q^2}=e^{\frac{-i\pi}{2}}\sqrt{q^2}\\
\end{array} \right.
\end{equation}
Most of the models predict a different behavior for the electric and the magnetic FFs in TL region, whereas, as already mentioned, no individual determination of electric and magnetic FFs has been done yet. We chose to fit the data assuming that they correspond to the magnetic FFs for proton and neutron, Fig. \ref{fig:fig2}a and \ref{fig:fig2}c, respectively. Therefore, the curves for the electric FFs, in Figs. \ref{fig:fig2}b and \ref{fig:fig2}d have to be considered predictions from the models. Including or not the data on neutron FFs, in TL region, influence very little the fitting procedure.
The parametrization from Ref. \cite{Ia73} is shown as a dotted line, in
Figs. \ref{fig:fig1} and \ref{fig:fig2}. This model is based on a view of the nucleon as composed by an inner core with a small radius (described by a dipole term) surrounded by a meson cloud.
While it reproduces very well the proton data in SL region (and particularly the polarization measurements), it fails in reproducing the large $Q^2$ behaviour of the magnetic neutron FF in SL region. The present fit constrained on the TL data and on the recent SL data does not improve the situation. In framework of this model a good global fit in SL region has been obtained with a modification including a phase in the common dipole term. However, the TL region is less well reproduced \cite{Bij04}. Therefore, the curves drawn in all the figures correspond to the original parameters, which give, in our opinion, a better representation of the whole set of data.
The result from an update fit based on the parametrization GKex(02L) \cite{Lomon} is shown in Figs. \ref{fig:fig1} and \ref{fig:fig2} (solid line). It is possible to find a good overall parametrization, with parameters not far from those found in the original paper for the SL region only. The agreement is very good, for both proton and neutron FFs.
The Hohler parametrization \cite{Ho76}, contains also pole terms with adjustable parameters. The $\rho$-exchange contribution, however, is fully determined, with constants fixed on $\pi N$ data. The model contains 17 parameters, already, so we did not try to readjust or refit the $\rho$-contribution.
As noted in the original paper, such model is not suited to the extrapolation to TL region, because poles appear in the physical region. Constraining the parameters, in order to avoid these instabilities, worsens the description in the SL region.
Therefore, we give only a fit on all FFs, in SL region, Fig. \ref{fig:fig1} (dash-dotted line), corresponding to $\chi^2/ndf\simeq 1.7$. The formulas as well as the original and updated parameters are also given in Appendix. Parametrization \cite{Du03} can be considered a successful generalization, in TL region, based on unitarity and analyticity. It requires the modelization of ten resonances, five isoscalar and five isovector.
The Bosted parametrization \cite{Bo95} is an empirical fit to nucleon FFs, in the SL region, based on simple formulas which are useful for fast estimations. It does not seem possible to find a unique function, which describes satisfactorily both the magnetic nucleon FFs and the electric proton FF, so the parameters are specific to each FFs. In the extension to TL region, as for the Hohler parametrization, one can not avoid poles and instabilities, and attempts to obtain a description in SL and TL regions remained unsuccessful. Therefore, we give the fit for the SL region, only, as dashed line in Fig. \ref{fig:fig1}, and report in the Appendix the useful formulas and the updated parameters. As one can see from the table, they do not differ more than 20\% from the published ones and the fit corresponds to $\chi^2/ndf\simeq 2$.
\section{Predictions in TL region}
We give the predictions for the cross section asymmetry and the polarization observables, for those models, described above, which give a good overall description of the available FFs data in SL and TL regions. The calculation is based on Eqs. (\ref{eq:asym}), (\ref{eq:eq15}) and (\ref{eq:pol}), for a fixed value of the angle $\theta=\pi/4$.
As shown in Fig. \ref{fig:fig3}, all these observables are, generally, quite large. The model \cite{Ia73} predicts the largest (absolute) value at $q^2\simeq$ 15 GeV $^2$ for all observables, except $A_{xz}$, which has two pronounced extrema.
All observables manifest a different behavior, according to the different models. The sign, also, can be opposite for VMD inspired models and pQCD. The model \cite{Lomon} is somehow intermediate between the two representations, as it contains the asymptotic predictions of QCD (at the expenses of a large number of parameters).
The fact that single spin observables in annihilation reactions are discriminative towards models, especially at threshold, was already pointed out in Ref. \cite{Dub96}, for the process $e^++e^-\to p+\overline{p}$ on the basis of two versions of a unitary VDM model. The present results, (Fig. \ref{fig:fig3}), for the inverse reaction $p+\overline{p}\to e^++e^-$ confirm this trend and show that experimental data will be extremely useful, particularly in the kinematical region around $q^2\simeq$ 15 GeV $^2$.
\section{Summary}
The measurement of polarization observables and the possibility to access individual nucleon FFs in TL and SL regions at larger $Q^2$ and/or with higher precision is foreseen in next future.
A general analysis of the experimental data on nucleon electromagnetic FFs, extracted from elastic scattering and annihilation reactions, has been performed in the available kinematical region.
Expressions of the experimental observables in the reaction $p+\overline{p}\to e^++e^-$ have been derived in terms of the electromagnetic FFs, as a function of the momentum transfer squared.
Some of the models on nucleon FFs have been reviewed, extended in TL region and used to give predictions on experimental observables which should be useful to plan future experiments.
Many questions are still open. Recent data in the SL region show that the ratio $G_{Ep}/G_{Mp}$ deviates from the expected dipole behavior. In the TL region, the values of $|G_M|$, obtained under the assumption that $G_E=G_M$, are larger than the corresponding SL values. This has been considered as a proof of the non applicability of the Phr\`agmen-Lindel\"of theorem, (up to $s$=18 GeV$^2$, at least) or as an evidence that the asymptotic regime is not reached \cite{Bi93}. The presence of a large relative phase of magnetic and electric proton FFs
in
the TL region, if experimentally proved at relatively large momentum transfer,
can be considered a strong
indication that these FFs have a different behavior. In particular, it will allow a test of the Phr\`agmen-Lindel\"of theorem \cite{Bi93}.
Large progress in view of a global interpretation of the nucleon FFs is expected from future experiments with antiproton beams: it will be possible, at the future FAIR facility at GSI, to separate the electric and magnetic FFs in a wide region of $s$ and to extend the measurement of FFs up to the largest available energy, corresponding to $s\simeq 30$ GeV$^2$.
The angular distribution of the produced leptons will allow the separation of the electric and magnetic FFs. The measurement of the asymmetry ${\cal R}$ (from the angular dependence of the
differential cross section for $p+\overline{p}\leftrightarrow \ell^+ +\ell^-$) is sensitive to the relative value of $|G_M|$ and $|G_E|$.
In particular, the $\theta$-dependence of the single spin and double spin polarization observables is very sensitive to existing models of the nucleon FFs, which reproduce equally well the data in SL region.
Similar information can be obtained from the final polarization in $\ell^++\ell^- \to \vec p+\overline{p}$ \cite{Dub96}, but in this case one has to deal with the problem of hadron polarimetry, in conditions of very small cross sections.
Only the study of the processes $p+\overline{p}\to \pi^0+ \ell^+ +\ell^-$ and $p+\overline{p}\to \pi^++\pi^-+\ell^+ +\ell^-$, \cite{Re65,Dub95} will allow to measure proton FFs in the unphysical region (for $s\le 4m^2$, where the vector meson contribution plays an important role) and to determine the relative phase of pion and nucleon FFs.
\section{Appendix}
The Sachs FFs are expressed in terms of the Pauli and Dirac FFs as:
$$ G^N_{E}=F_1^N(Q^2)+\tau F_2^N(Q^2),~G^N_{M}=F_1^N(Q^2)+F_2^N(Q^2).$$
One can introduce the isoscalar and isovector FFs $F_i^{s}$ and $F_i^{v}$, $i=1,2$ as:
$2F^p_i=F_i^{s}+F_i^{v}$, $2F^n_i=F_i^{s}-F_i^{v}$.
Then, the isoscalar and isovector currents can be parametrized in terms of meson propagators,
effective FFs, and/or terms which insure specific properties, according to the different models.
\subsection{Model from Iachello, Jackson and Land\'e \protect\cite{Ia73}
and Iachello and Wan \protect\cite{Wa04} }
FFs are parametrized following the work \cite{Ia73} , with a modification that consists in adding a phase in the dipole term, $g(Q^2)$, for the extension in TL region.
\begin{eqnarray*}
F_1^s(Q^2)&=&
\displaystyle\frac{g(Q^2)}{2}
\left[(1-\beta_\omega-\beta_\phi)+\beta_\omega\displaystyle\frac{\mu_\omega^2}{\mu_\omega^2+Q^2}+\beta_\phi
\displaystyle\frac{\mu_\phi^2}{\mu_\phi^2+Q^2}\right],\\
F_1^v(Q^2)&=&\displaystyle\frac{g(Q^2)}{2}
\left[(1-\beta_\rho)+\beta_\rho
\displaystyle\frac{\mu_\rho^2+8\Gamma_\rho\mu_\pi/\pi}
{(\mu_\rho^2+Q^2)+(4\mu_\pi^2+Q^2)\Gamma_\rho\alpha(Q^2)/\mu_\pi}\right],\\
F_2^s(Q^2)&=&
\displaystyle\frac{g(Q^2)}{2}
\left[(\mu_p+\mu_n-1-\alpha_\phi)
\displaystyle\frac{\mu_\omega^2}
{\mu_\omega^2+Q^2}+\alpha_\phi\displaystyle\frac{\mu_\phi^2}{\mu_\phi^2+Q^2}\right],\\
F_2^v(Q^2)&=&\displaystyle\frac{g(Q^2)}{2}
\left[(\mu_p-\mu_n-1)
\displaystyle\frac{\mu_\rho^2+8\Gamma_\rho\mu_\pi/\pi}{(\mu_\rho^2+Q^2)+(4\mu_\pi^2+Q^2)
\Gamma_\rho\alpha(Q^2)/\mu_\pi}\right],
\end{eqnarray*}
with $g(Q^2)=\displaystyle\frac{1}{(1+\gamma e^{i\theta}Q^2)^2}$
and $\alpha(Q^2)=\displaystyle\frac{2}{\pi}
\sqrt{\displaystyle\frac{Q^2+4\mu_\pi^2}{Q^2}}
ln\left[\displaystyle\frac{\sqrt{(Q^2+4\mu_\pi^2)}+\sqrt{Q^2}}{2\mu_\pi}\right]$, with the standard values of the masses $m=0.939$~GeV, $\mu_\rho=0.77$~GeV, $\mu_\omega=0.78$~GeV, $\mu_\phi=1.02$~GeV, $\mu_\pi=0.139$~GeV and the $\rho$ width $\Gamma_\rho=0.112$~GeV.
The values of the six parameters are given in Table \ref{table:IJL}.
\subsection{Model from Lomon \protect\cite{Lomon}}
\begin{eqnarray*}
F_1^{v}(Q^2)&=&
\displaystyle\frac{N}{2}
\left [
\displaystyle\frac{1.0317+0.0875(1+Q^2/0.3176)^{-2}}{(1+Q^2/0.5496)}+
\frac{g_{\rho'}}{f_{\rho'}}\displaystyle\frac{m_{\rho'}^2}{m_{\rho'}^2+Q^2}
\right ]
F_1^\rho(Q^2)+\\
&&\left(1-1.1192\displaystyle\frac{N}{2}-\displaystyle\frac{g_{\rho'}}{f_{\rho'}}\right)F_1^D(Q^2),\\
F_2^{v}(Q^2)&=&\displaystyle\frac{N}{2}
\left [\displaystyle\frac{5.7824+0.3907(1+Q^2/0.1422)^{-1}}{(1+Q^2/0.5362)}+
\kappa_{\rho'}\displaystyle\frac{g_{\rho'}}{f_{\rho'}}\displaystyle\frac{m_{\rho'}^2}{m_{\rho'}^2+Q^2}
\right ]
F_2^\rho(Q^2)+\\
&&
\left(\kappa_\nu-6.1731\displaystyle\frac{N}{2}-\kappa_{\rho'}\displaystyle\frac{g_{\rho'}}{f_{\rho'}}\right)F_2^D(Q^2),\\
F_1^{s}(Q^2)&=& \left (\displaystyle\frac{g_\omega}{f_\omega}\displaystyle\frac{m_{\omega}^2}{m_{\omega}^2+Q^2}+
\displaystyle\frac{g_{\omega '}}{f_{\omega '}}\displaystyle\frac{m_{\omega '}^2}{m_{\omega '}^2+Q^2}\right )
F_1^\omega(Q^2)+\\
&&\displaystyle\frac{g_\phi}{f_\phi}\displaystyle\frac{m_{\phi}^2}{m_{\phi}^2+Q^2}F_1^\phi(Q^2)+
\left(1-\displaystyle\frac{g_\omega}{f_\omega} -\displaystyle\frac{g_{\omega '}}{f_{\omega '}} \right)F_1^D(Q^2),\\
F_2^{s}(Q^2)&=& \left (\kappa_\omega\displaystyle\frac{g_\omega}{f_\omega}\displaystyle\frac{m_{\omega}^2}{m_{\omega}^2+Q^2}
+\kappa_{\omega '}\displaystyle\frac{g_{\omega '}}{f_{\omega '}}
\displaystyle\frac{m_{\omega '}^2}{m_{\omega '}^2+Q^2}\right )
F_2^\omega(Q^2)+\kappa_\phi\displaystyle\frac{g_\phi}{f_\phi}\displaystyle\frac{m_{\phi}^2}{m_{\phi}^2+Q^2}F_2^\phi(Q^2)+\\
&&
\left(\kappa_s-
\kappa_\omega\displaystyle\frac{g_\omega}{f_\omega}-
\kappa_{\omega '}\displaystyle\frac{g_{\omega '}}{f_{\omega '}}-
\kappa_\phi\displaystyle\frac{g_\phi}{f_\phi}\right)F_2^D(Q^2),\\
\end{eqnarray*}
with
\begin{eqnarray*}
F_1^{\alpha,D}(Q^2)&=&\displaystyle\frac{\Lambda_{1,D}^2}{\Lambda_{1,D}^2+\widetilde Q^2}\displaystyle\frac{\Lambda_{2}^2}{\Lambda_{2}^2+\widetilde Q^2}, ~\alpha=\rho,~ \omega~ and~ \Lambda_{1,D}\equiv \Lambda_1~ \mbox{for} F_i^\alpha, ~\Lambda_{1,D}\equiv\Lambda_D ~for~F_i^D \\
F_2^{\alpha,D}(Q^2)&=&\displaystyle\frac{\Lambda_{1,D}^2}{\Lambda_{1,D}^2+\widetilde Q^2}\left(\displaystyle\frac{\Lambda_{2}^2}{\Lambda_{2}^2+\widetilde Q^2}\right)^2,~
F_1^\phi(Q^2)=F_1^\alpha\left(\displaystyle\frac{Q^2}{\Lambda_1^2+Q^2}\right)^{1.5},~\\
F_2^\phi(Q^2)&=&F_2^\alpha\left(\displaystyle\frac{\Lambda_1^2}{\mu_\phi^2}\displaystyle\frac{Q^2+\mu_\phi^2}{\Lambda_1^2+Q^2}\right)^{1.5} ,~
\widetilde Q^2=Q^2\displaystyle\frac{ln[(\Lambda_D^2+Q^2)/\Lambda_{QCD}^2]}{ln(\Lambda_D^2/\Lambda_{QCD}^2)}.
\end{eqnarray*}
The set of parameters is reported in Table \ref{table:lomon}.
\subsection{Model from Hohler \protect\cite{Ho76}}
This model is also based on a VMD parametrization:
\begin {eqnarray*}
F_1^\rho (Q^2)& = & 0.5 \left [0.955+\displaystyle\frac{0.09}{\left(1+{Q^2}/{0.355} \right)^2} \right] \displaystyle\frac{1}{1+{Q^2}/{0.536}},
\\
F_2^\rho (Q^2)& = & 0.5 \left [5.335+\displaystyle\frac{0.962}{\left(1+{Q^2}/{0.268} \right)^2}\right] \displaystyle\frac{1}{1+{Q^2}/{1.603}},
\\
F_{i}^{(s)}(Q^2) & = & \sum_j\displaystyle\frac{a_j^{(i,s)}}{b_j^{(s)}+Q^2},
\\
F_{i}^{(v)}(Q^2) & = & F_{i}^\rho(Q^2)+ \sum_j\displaystyle\frac{a_j^{(i,v)}}{b_j^{(v)}+Q^2}.
\end{eqnarray*}
The parameters are given in Table \ref{table:Hohler}.
\subsection{Model from Bosted \protect\cite{Bo95}}
The analytical expressions are inverse of polynomes as functions of $Q$, whereas $G_{En}$ is described by a different function, as suggested by Galster \cite{Ga71}:
\begin{equation}
F^j=\displaystyle\frac{1}{1+\sum_i a^j_i Q^{i}},
\end{equation}
\begin{equation}
G_E^n=\displaystyle\frac{\alpha\mu_n\tau G_D(Q^2)}{1+\beta\tau },
\end{equation}
with $a^j_i$, $\alpha$ and $\beta$ free parameters. In the present notation $j=1,2,3$ corresponds to $G_{Ep}$ and $G_{Mn}$ and $G_{Mp}$, respectively.
The inverse polynomes are of fourth order for $G_{Ep}$ and $G_{Mn}$ and of fifth order for $G_{Mp}$.
The parameters are given in Table \ref{table:Bosted}.
\begin{figure}[pht]
\begin{center}
\includegraphics[width=16cm]{sl.eps}
\caption{\label{fig:fig1} Nucleon Form Factors in Space-Like region: (a) proton electric FF, scaled by $\mu_p G_{Mp}$ (b) proton magnetic FF scaled by $\mu_p G_D$ , (c) neutron electric FF, (d) neutron magnetic FF, scaled by $\mu_n G_D$. The predictions of the models are drawn: from Ref. \cite{Ia73} (dotted line), from Ref. \cite{Lomon} (solid line), model from Ref. Ref. \cite{Ho76} (dash-dotted line), from \cite{Bo95} (dashed line). }
\end{center}
\end{figure}
\begin{figure}[pht]
\begin{center}
\includegraphics[width=16cm]{tl.eps}
\caption{\label{fig:fig2} Form Factors in Time-Like region and predictions of the models: pQCD-inspired (dashed line), from Ref. \cite{Ia73} (dotted line), from Ref. \cite{Lomon} (solid line).}
\end{center}
\end{figure}
\begin{figure}[pht]
\begin{center}
\includegraphics[width=16cm]{obs.eps}
\caption{\label{fig:fig3} Angular asymmetry and polarization observables, according to Eqs. (\protect\ref{eq:eq15}) and (\protect\ref{eq:pol}), for a fixed value of $\theta=45^0$. Notations as in Fig. \protect\ref{fig:fig2}.}
\end{center}
\end{figure}
|
{
"timestamp": "2005-03-01T15:25:09",
"yymm": "0503",
"arxiv_id": "nucl-th/0503001",
"language": "en",
"url": "https://arxiv.org/abs/nucl-th/0503001"
}
|
\section{Introduction} \label{S:intro}
Let $\mathcal{M}$ be a von Neumann algebra, $\mathcal{P}(\mathcal{M})$ its projections, and $\sim$ the relation of Murray-von Neumann equivalence on $\mathcal{P}(\mathcal{M})$. The description of the quotient $\mbox{$(\p(\M)/\sim)$}$ is known as the \textit{dimension theory} for $\mathcal{M}$. In this paper we prove basic results about three aspects of dimension theory: topology, parameterization, and order.
The second section of the paper contains background which is relevant for all three topics. Section 3 deals with topology; Sections 4 and 5 with parameterization; Sections 6 and 7 with order structure. Except for one or two references, these three groupings are independent from each other. In the remainder of this introduction we explain the problems which motivate our investigations.
\smallskip
\textsc{Topology.} The first goal requires little explanation.
\begin{problem}
Study the topology that $\mbox{$(\p(\M)/\sim)$}$ inherits from the strong (equivalently, the weak) topology on $\mathcal{M}$.
\end{problem}
\noindent Some of the results are used in the author's recent work on unitary orbits (\cite{S}).
\smallskip
\textsc{Parameterization.} It is easy to check that $\mbox{$(\p(\M)/\sim)$}$ also inherits a well-defined partial order from $\mathcal{P}(\mathcal{M})$. Classical work of Murray and von Neumann (\cite{MvN1}) and Dixmier (\cite{Di1949,Di1952}) shows that $\mbox{$(\p(\M)/\sim)$}$ can be naturally parameterized by a subset of the extended positive cone of the center, at least when $\mathcal{M}$ is $\sigma$-finite. This parameterization map, called a \textit{dimension function}, can be extended to all of $\mathcal{M}_+$, and the extension is called an \textit{extended center-valued trace}. The existence of a dimension function on a non-$\sigma$-finite von Neumann algebra is also classical, though less-known. It was originally studied in connection with spatial isomorphisms by Griffin (\cite{G1953,G1955}) and Pallu de la Barri\`{e}re (\cite{P}), and eventually given a representation-free foundation by Tomiyama (\cite{To}).
There is a noticeable gap between the last two objects.
\begin{problem}
Is there a version of the extended center-valued trace which extends the dimension function on a non-$\sigma$-finite von Neumann algebra?
\end{problem}
\noindent One might expect (and dread) technical constructions involving cardinals and limits. We show how to avoid most of this by simply marrying Tomiyama's dimension function to the equivalence relation of Kadison and Pedersen (\cite{KP}). In fact, the main point to settle does not involve cardinals.
\smallskip
\textsc{Order.} The range of Tomiyama's map consists of certain cardinal-valued order-continuous functions on the spectrum of the center. Tomi-yama assumed pointwise order and arithmetic on the range, then gave some examples to show that his map lacks basic continuity properties. In fact the pointwise operations (on infinite sets of functions) do not behave well, and it seems to us that these are essentially the wrong operations to be considering. Our viewpoint here is more algebraic. This repairs certain degeneracies and allows us to resolve affirmatively the basic
\begin{problem} \label{P:complat}
Is $\mbox{$(\p(\M)/\sim)$}$ always a complete lattice?
\end{problem}
We recall that a \textit{lattice} (resp. \textit{complete lattice}) is a partially-ordered set in which one may take meets and joins of finitely (resp. arbitrarily) many elements. $\mathcal{P}(\mathcal{M})$ is a complete lattice, but it does not induce lattice operations on $\mbox{$(\p(\M)/\sim)$}$: for example, $[p] \wedge [q]$ is not well-defined as $[p \wedge q]$. Nonetheless the comparison theorem for projections readily implies that $\mbox{$(\p(\M)/\sim)$}$ is a lattice. And in a finite von Neumann algebra, the dimension function identifies $\mbox{$(\p(\M)/\sim)$}$ with a complete sublattice of $\mathcal{Z}(\mathcal{M})_1^+$. Problem \ref{P:complat} asks about the existence of meets and joins of arbitrarily large sets of equivalence classes coming from arbitrarily large von Neumann algebras. Its answer has a somewhat surprising reformulation in terms of representations.
\section{Background} \label{S:back}
Let $\mathcal{M}$ be a von Neumann algebra of arbitrary type and cardinality. We write $\mathcal{Z}(\mathcal{M})$ for its center, and we occasionally symbolize the strong and weak topologies by $s$ and $w$. The central support of an operator is $c(\cdot)$.
We use the standard terminology and results from \cite[Section V.1]{T} for projections, including $p^\perp$ for $(1-p)$. Besides $p \sim q$, we write $p \preccurlyeq q$ for subequivalence, and $p \prec q$ for $p \preccurlyeq q$ but not $p \sim q$.
Notice that for pairwise orthogonal sets $\{p_\alpha\}, \{q_\alpha\} \subset \mathcal{P}(\mathcal{M})$,
\begin{equation} \label{E:addeq}
p_\alpha \sim q_\alpha, \: \forall \alpha \Rightarrow \left(\sum p_\alpha\right) \sim \left(\sum q_\alpha\right),
\end{equation}
\begin{equation} \label{E:addeq2}
p_\alpha \preccurlyeq q_\alpha, \: \forall \alpha \Rightarrow \left(\sum p_\alpha\right) \preccurlyeq \left(\sum q_\alpha\right).
\end{equation}
Among the many adjectives which may be applied to a single projection, we specify one which may cause confusion. A nonzero projection $p$ is \textit{properly infinite} if $zp$ is infinite or zero for any central projection $z$. (An alternative definition: $p$ is properly infinite if it can be decomposed into a countably infinite sum of projections, each of which is equivalent to $p$.) Any adjective can be applied to an algebra when the adjective describes the identity projection of the algebra.
According to \eqref{E:addeq}, we can sum unambiguously any set in $\mbox{$(\p(\M)/\sim)$}$ for which there are mutually orthogonal representatives, simply by taking the equivalence class of the sum of representatives. This determines a partial order on $(\mathcal{P}(\mathcal{M})/\sim)$: $[p] \leq [q]$ if there exists a projection $r$ with $[p] + [r] = [q]$. One may also induce the same order directly, since the quotient operation respects the order in $\mathcal{P}(\mathcal{M})$. By this we mean
$$[p_1] \leq [p_2] \iff \exists q_1, q_2 \text{ with } q_1 \sim p_1, \: q_2 \sim p_2, \: q_1 \leq q_2.$$
So $[p_1] \leq [p_2]$ means nothing other than $p_1 \preccurlyeq p_2$.
Actually the comparison theorem for projections (\cite[Theorem V.1.8]{T}) implies that $(\mathcal{P}(\mathcal{M})/\sim)$ is a lattice. For $p,q \in \mathcal{P}(\mathcal{M})$, let $z$ be a central projection with $zp \preccurlyeq zq$, $z^\perp p \succcurlyeq z^\perp q$. Then
\begin{equation} \label{E:lattice}
[p] \wedge [q] = [zp + z^\perp q], \qquad [p] \vee [q] = [z^\perp p + zq].
\end{equation}
Next we recall basic properties of the extended center-valued trace. This material is due to Dixmier (\cite{Di1949,Di1952}), but for the reader's convenience (presumably), we give citations from Takesaki's book \cite{T}.
\begin{definition} (\cite[Definition V.2.33]{T}) Let $\mathcal{M}$ be an arbitrary von Neumann algebra, and let $\Omega(\mathcal{Z}(\M))$ be the spectrum of the abelian $C^*$-algebra $\mathcal{Z}(\mathcal{M})$. By $\widehat{\mathcal{Z}(\mathcal{M})}_+$ we mean the partially-ordered monoid of $[0,+\infty]$-valued continuous functions on $\Omega(\mathcal{Z}(\M))$. $\mathcal{Z}(\mathcal{M})_+$ is contained in $\widehat{\mathcal{Z}(\mathcal{M})}_+$ and acts on it by multiplication.
An \textbf{extended center-valued trace} on $\mathcal{M}$ is an additive map $T: \mathcal{M}_+ \to \widehat{\mathcal{Z}(\mathcal{M})}_+$ which commutes with the action of $\mathcal{Z}(\mathcal{M})_+$ and satisfies $T(x^*x) = T(xx^*)$ for $x \in \mathcal{M}_+$.
$T$ is \textit{faithful} if $T(x^*x)=0 \Rightarrow x=0, \: \forall x \in \mathcal{M}_+$. $T$ is \textit{normal} if
\begin{equation} \label{E:normal}
T(\sup x_\alpha) = \sup T(x_\alpha)
\end{equation}
for any bounded increasing net $\{x_\alpha\} \subset \mathcal{M}_+$. $T$ is \textit{semifinite} if $\{x \in \mathcal{M} \mid T(x^*x) \in \mathcal{Z}(\mathcal{M})_+\}$ is $\sigma$-weakly dense in $\mathcal{M}$.
\end{definition}
Here we wish to draw attention to a point which will be amplified in Sections \ref{S:cont} and \ref{S:comp}. What is the meaning of the expression $\sup T(x_\alpha)$ in \eqref{E:normal}? The pointwise supremum of an increasing family of $[0,+\infty]$-valued continuous functions on $\Omega(\mathcal{Z}(\M))$ may not be continuous, and some kind of algebraic supremum is required instead. Dixmier showed that such a supremum exists, using the fact that $\Omega(\mathcal{Z}(\M))$ is stonean (\cite{Di1951}). He also mentions specifically that other methods, including a purely formal one, could reach the same goal (\cite[p.25]{Di1952}). We suppose that our technique in Section \ref{S:comp} is similar to the formal approach that he had in mind.
Semifinite von Neumann algebras - those with no summand of type III - are characterized by the existence of a faithful normal semifinite extended center-valued trace (\cite[Theorem V.2.34]{T}). Such a map $T$ is unique up to multiplication by an element of $\widehat{\mathcal{Z}(\mathcal{M})}_+$ which takes finite values on an open dense subset of $\Omega(\mathcal{Z}(\M))$, so all are equally useful in calculations. A projection $p$ is finite if and only if $T(p)$ takes finite values on an open dense subset of $\Omega(\mathcal{Z}(\M))$ (\cite[Proposition V.2.35]{T}). From all this $p \preccurlyeq q \Rightarrow T(p) \leq T(q)$, and the converse holds if $p$ is finite.
If $\mathcal{M}$ is finite, there is a \textit{unique} faithful extended center-valued trace $T$ with $T(1_\mathcal{M}) = 1_\mathcal{M}$ (\cite[Theorem V.2.6]{T}). Such a map is automatically normal, and the linear extension which is defined on all of $\mathcal{M}$ is called simply a \textit{center-valued trace}.
\begin{convention} \label{C}
Whenever we talk of an ``extended center-valued trace" $T$ on $\mathcal{M}_+$ in the sequel, it is assumed that
\begin{itemize}
\item $T$ is normal and faithful;
\item on the finite summand of $\mathcal{M}$, $T$ agrees with the center-valued trace;
\item on the semifinite summand of $\mathcal{M}$, $T$ is semifinite;
\item on the infinite type I summand of $\mathcal{M}$, $T$ maps an abelian projection to its central support.
\end{itemize}
Therefore $T(p) = (+\infty) c(p)$ for a projection supported on the type III summand.
\end{convention}
A word about operator topologies on $\mathcal{M}$: the strong, $\sigma$-strong, weak, and $\sigma$-weak topologies can all be defined spatially. The $\sigma$-strong and $\sigma$-weak topologies are independent of the choice of (faithful normal) representation, and this is not true for the strong and the weak. But on \textit{bounded} sets, we have the agreements strong=$\sigma$-strong and weak=$\sigma$-weak; we therefore permit ourselves the small linguistic abuse of referring to the strong (or weak) topology on a bounded subset of $\mathcal{M}$.
For $\mathcal{M}$ finite, the normality of the center-valued trace is equivalent to $\sigma$-weak-$\sigma$-weak continuity. It will be more useful for us that this map is also $\sigma$-strong-$\sigma$-strong continuous, and therefore strong-strong continuous on bounded sets. (See \cite[Theorem 13]{G1953}, \cite[I.4.Th\'{e}or\`{e}me 2 and p.250]{Di1969}, or \cite{R} in connection with this. In fact the strong-strong or weak-weak continuity on all of $\mathcal{M}$ does depend on the representation (\cite[Theorem 8]{G1953}).)
\bigskip
Here are some examples of $\mbox{$(\p(\M)/\sim)$}$.
\begin{enumerate}
\item When $\mathcal{M}$ is a type $\text{I}_n$ factor, $(\mathcal{P}(\mathcal{M})/\sim)$ is isomorphic to the initial segment of cardinals $\leq n$, via the map that sends a projection to its rank.
\item When $\mathcal{M}$ is a type $\text{II}_1$ factor, $(\mathcal{P}(\mathcal{M})/\sim) \simeq [0,1]$.
\item When $\mathcal{M}$ is a $\sigma$-finite type $\text{II}_\infty$ factor, $(\mathcal{P}(\mathcal{M})/\sim) \simeq [0, +\infty]$.
\item When $\mathcal{M}$ is a $\sigma$-finite type $\text{III}$ factor, $(\mathcal{P}(\mathcal{M})/\sim) \simeq \{0, +\infty\}$.
\end{enumerate}
The isomorphisms in (2) and (3) are implemented by a (bounded or unbounded) trace. When $\mathcal{M}$ is a non-factor with separable predual, $(\mathcal{P}(\mathcal{M})/\sim)$ is naturally viewed as a direct integral of the lattices above. When $\mathcal{M}$ is finite, $(\mathcal{P}(\mathcal{M})/\sim)$ is isomorphic to a sublattice of $\mathcal{Z}(\mathcal{M})_1^+$ via the center-valued trace (see \cite[Theorem 8.4.4]{KR2}).
Continuous (type II) and degenerate (type III) dimension theory were part of the original appeal for Murray and von Neumann: what happens at large cardinality? Since $\mbox{$(\p(\M)/\sim)$}$ is totally ordered if and only if $\mathcal{M}$ is a factor, this is the scenario closest to set theory. Do type II and III factors contain ``quantum cardinal arithmetic" which diverges from the usual cardinal arithmetic of a type I factor?
\smallskip
The questions above are answered neatly by the parameterization of $\mbox{$(\p(\M)/\sim)$}$ as developed by Griffin (\cite{G1953,G1955}), Pallu de la Barri\`{e}re (\cite{P}), and especially as formulated by Tomiyama (\cite{To}). The main point is a structure theorem allowing us to break a properly infinite von Neumann algebra into direct summands, each of which has a well-defined size. This is in direct analogy to the structure theorem for type I von Neumann algebras, but we use $\sigma$-finiteness instead of abelianness as the ``unit of measurement".
\begin{definition} \label{D:homog} $\text{(\cite[Definition 1]{To})}$
Let $\kappa$ be a cardinal. We say that a nonzero projection $p$ in a von Neumann algebra $\mathcal{M}$ is \textbf{$\kappa$-homogeneous} if $p$ is the sum of $\kappa$ mutually equivalent projections, each of which is the sum of centrally orthogonal $\sigma$-finite projections. We also define
$$\kappa_\mathcal{M} = \sup \{\kappa \mid \text{$\mathcal{M}$ contains a $\kappa$-homogeneous projection}\}.$$
\end{definition}
\begin{remark} The terminology here is conflicting. We follow Tomiyama, but elsewhere ``$\kappa$-homogeneous projection" means a central projection which is the sum of $\kappa$ equivalent abelian projections (e.g. \cite[p.299]{T}).
\end{remark}
A projection can be $\kappa$-homogeneous for at most one $\kappa \geq \aleph_0$; also for $\kappa \geq \aleph_0$, two $\kappa$-homogeneous projections with identical central support are necessarily equivalent (\cite{G1955,To}). $\kappa_\mathcal{M}$ is not larger than the dimension of a Hilbert space on which $\mathcal{M}$ is faithfully represented.
The fundamental result for us is a m\'{e}lange of two theorems of Griffin, one covering the semifinite case (slightly adapted to our setting, and also proved by Pallu de la Barri\`{e}re) and one covering the purely infinite. It was rewritten in the non-spatial setting by Tomiyama.
\begin{theorem} \label{T:griffin} $($\cite[Theorem 3]{G1953}, \cite[Theorem 1]{G1955}, also \cite[I.5]{P} and \cite[Theorem 1]{To}$)$
Let $\mathcal{M}$ be a properly infinite von Neumann algebra. Then uniquely
$$1_\mathcal{M} = \sum_{\aleph_0 \leq \kappa \leq \kappa_\mathcal{M}} z_\kappa,$$
where each $z_\kappa$ is either zero or a $\kappa$-homogeneous central projection.
\end{theorem}
Let $T$ be an extended center-valued trace on a von Neumann algebra $\mathcal{M}$ (following Convention \ref{C}). Given any $p \in \mathcal{P}(\mathcal{M})$, let $z^f$ be the largest central projection such that $z^f p$ is finite. By applying Theorem \ref{T:griffin} to $(1-z^f) p\mathcal{M} p$, there are unique central projections $(z_\kappa)_{\aleph_0 \leq \kappa \leq \kappa_\mathcal{M}}$ such that $\sum z_\kappa p = (1-z^f)p$ and any nonzero $z_\kappa p $ is $\kappa$-homogeneous. Make the formal assignment
\begin{equation} \label{E:gdf}
p = \left( z^f p + \sum_{\aleph_0 \leq \kappa \leq \kappa_\mathcal{M}} z_\kappa p \right) \mapsto \left( T(z^f p) + \sum_{\aleph_0 \leq \kappa \leq \kappa_\mathcal{M}} \kappa z_\kappa \right).
\end{equation}
From our earlier comments this assignment is a complete invariant for the equivalence class of $p$.
Under the isomorphism $\mathcal{Z}(\mathcal{M}) \simeq C(\Omega(\mathcal{Z}(\M)))$, projections correspond to clopen subsets of $\Omega(\mathcal{Z}(\M))$, so elements on the right-hand side of \eqref{E:gdf} can be interpreted as partially-defined functions on $\Omega(\mathcal{Z}(\M))$. The range is in $([0,+\infty) \cup \{\kappa \mid \aleph_0 \leq \kappa \leq \kappa_\mathcal{M}\})$, and the functions are (order) continuous on their domains, which are easily shown to be open and dense. Tomiyama showed (\cite[Lemma 5]{To}) that such functions extend uniquely to continuous functions on all of $\Omega(\mathcal{Z}(\M))$.
\begin{definition} \label{T:defgdf} (\cite{To})
The assignment described above, from $\mathcal{P}(\mathcal{M})$ to the continuous $([0,+\infty) \cup \{\kappa \mid \aleph_0 \leq \kappa \leq \kappa_\mathcal{M}\})$-valued functions on $\Omega(\mathcal{Z}(\M))$, is a \textbf{(generalized) dimension function} of $\mathcal{M}$.
\end{definition}
\begin{theorem} \label{T:gdf} $($\cite{To}$)$
Let $D$ be a dimension function of $\mathcal{M}$. Then $D$ is additive on pairs of orthogonal projections, provided that one incorporates the positive reals into cardinal arithmetic in the obvious way. We have
$$p \preccurlyeq q \iff D(p) \leq D(q), \qquad \forall p,q \in \mathcal{P}(\mathcal{M}),$$
where we use the pointwise ordering of functions on the right-hand side.
\end{theorem}
It follows that $D$ factors as
$$\mathcal{P}(\mathcal{M}) \twoheadrightarrow \mbox{$(\p(\M)/\sim)$} \overset{\sim}{\to} D(\mathcal{P}(\mathcal{M})).$$
Here the second map is an embedding in a function space, preserving order, sums (when they exist), and the multiplicative $\mathcal{P}(\mathcal{Z}(\mathcal{M}))$-action.
\begin{corollary} \label{T:factor} ${}$
\begin{enumerate}
\item In a factor of type $\text{II}_\infty$, the totally ordered set $(\mathcal{P}(\mathcal{M})/\sim)$ is isomorphic to
$$[0, +\infty) \cup \{\kappa \mid \aleph_0 \leq \kappa \leq \kappa_\mathcal{M}\}.$$
\item In a factor of type III, the totally ordered set $(\mathcal{P}(\mathcal{M})/\sim)$ is isomorphic to
$$\{0\} \cup \{\kappa \mid \aleph_0 \leq \kappa \leq \kappa_\mathcal{M}\}.$$
\end{enumerate}
\end{corollary}
So any interest in ``quantum cardinal arithmetic" wanes here: infinite quantum cardinals are (isomorphically) just cardinals. For the reader interested in axiomatic treatments of $\mbox{$(\p(\M)/\sim)$}$ and more general algebraic structures obtained as quotients of lattices, see \cite{L,M,F}.
\section{The topology of $(\mathcal{P}(\mathcal{M})/\sim)$} \label{S:top}
If we want $\mbox{$(\p(\M)/\sim)$}$ to inherit a topology from $\mathcal{P}(\mathcal{M})$, there really are not so many interesting choices. The quotient of the norm topology is the discrete topology, since $\|p - q\| < 1$ implies that $p$ and $q$ are unitarily equivalent (\cite[5.2.6-10]{W-O}). And all of the ``operator" topologies (notably, the strong and the weak) are equivalent when restricted to $\mathcal{P}(\mathcal{M})$ (\cite[Ex. 5.7.4]{KR1}). We point out, however, that $(\mathcal{P}(\mathcal{M}), \text{strong})$ is complete, while $(\mathcal{P}(\mathcal{M}), \text{weak})$ may not be; completeness is not a topological property.
We will denote the resulting quotient strong/weak operator topology on $(\mathcal{P}(\mathcal{M})/\sim)$ by ``$QOT$". In the rest of this section, all closures and convergences in $\mbox{$(\p(\M)/\sim)$}$ are to be understood in this topology.
We need a few lemmas.
\begin{lemma} \label{T:liminf}
Let $\{x_\alpha\}$ be a net in a semifinite von Neumann algebra $\mathcal{M}$ equipped with an extended center-valued trace $T$. If $x_\alpha^* x_\alpha = y_1$ is fixed, while $x_\alpha x_\alpha^* \overset{w}{\to} y_2$, then $T(y_1) \geq T(y_2)$ in $\widehat{\mathcal{Z}(\mathcal{M})}_+$.
\end{lemma}
\begin{proof}
Fix any $\varphi \in \mathcal{Z}(\mathcal{M})_*^+$. Then $\varphi \circ T$ is a semifinite normal weight, so weakly lower-semicontinuous (\cite{H}). We have
\begin{align*}
\varphi \circ T(y_2) &= \varphi \circ T(w-\lim x_\alpha x_\alpha^*) \leq \liminf \varphi \circ T(x_\alpha x_\alpha^*) \\ &= \liminf \varphi \circ T(x_\alpha^* x_\alpha) = \varphi \circ T(y_1).
\end{align*}
Since $\varphi$ is arbitrary, the conclusion follows.
\end{proof}
\begin{lemma} \label{T:proj}
Let $p,q,r \in \mathcal{P}(\mathcal{M})$, with $\mathcal{M}$ and $p$ properly infinite.
\begin{enumerate}
\item If $p \sim q_j$ for a countable set $\{q_j\}$, then $p \sim \vee q_j$.
\item If $zq \prec zr$ for all nonzero central projections $z$, then $q^\perp \sim 1_\mathcal{M}$.
\end{enumerate}
\end{lemma}
\begin{proof} $\quad$
(1) It is clear that $p \preccurlyeq \vee q_j$. Write $p = \sum p_j$, where each $p_j \sim p$. Let $v_j$ be a partial isometry between $p_j$ and $q_j$. The operator $\sum v_j/2^j$ has right support $\vee q_j$ and left support $\leq p$, so also $p \succcurlyeq \vee q_j$.
(2) We compare $q$ and $q^\perp$. If there were a nonzero central projection $z$ with $zq \succcurlyeq zq^\perp$, then $zq$ would be properly infinite (else a nonzero central projection would be the sum of two finite projections). Write $zq = zq_1 + zq_2$, where $zq_1 \sim zq_2 \sim zq$. By \eqref{E:addeq2},
$$zq \preccurlyeq z = (zq + zq^\perp) \preccurlyeq (zq_1 + zq_2) = zq,$$
so that $zq \sim z$. Now $zr \succ zq \sim z$, which is impossible.
Thus $q \preccurlyeq q^\perp$. By the same argument, $q^\perp$ is properly infinite and $q^\perp \sim 1_\mathcal{M}$.
\end{proof}
\begin{theorem} \label{T:sinfin}
If $\mathcal{M}$ is a finite von Neumann algebra, the center-valued trace induces a homeomorphism from $(\mbox{$(\p(\M)/\sim)$}, QOT)$ to a subspace of $(\mathcal{Z}(\mathcal{M})_1^+, \text{\textnormal{strong}})$. Consequently
$$\overline{\{[p]\}} = \{[p]\}, \qquad p \in \mathcal{P}(\mathcal{M}).$$
\end{theorem}
\begin{proof} Let $T$ be the center-valued trace. If $[p_\alpha] \to [p]$, then there exist $q_\alpha \sim p_\alpha$ with $q_\alpha \overset{s}{\to} p$. By the strong-strong continuity of $T$ on bounded sets, we have $T(p_\alpha) = T(q_\alpha) \overset{s}{\to} T(p)$.
On the other hand, suppose $p_\alpha, p$ are projections such that $T(p_\alpha) \overset{s}{\to} T(p)$. Let $q_\alpha \leq p$ be projections with $T(q_\alpha) = T(p_\alpha) \wedge T(p)$, where the meet is taken in $\mathcal{Z}(\mathcal{M})_1^+$. Let $r_\alpha \perp q_\alpha$ be projections with $T(r_\alpha) = T(p_\alpha - p) \vee 0$. It follows that $r_\alpha$ is centrally orthogonal to $(p-q_\alpha)$, and by comparing center-valued traces $(q_\alpha + r_\alpha) \sim p_\alpha$.
When $\mathcal{M}$ is $\sigma$-finite, the strong topology on bounded sets is generated by the norm $x \mapsto \tau(x^*x)^{1/2}$, for $\tau$ any faithful tracial state (\cite[Proposition III.V.3]{T}). A general finite algebra is a direct sum of $\sigma$-finite ones (\cite[Corollary V.2.9]{T}), so it suffices to show convergence for the seminorms coming from a family of traces, each of which is faithful on a $\sigma$-finite summand.
We now take such a trace $\tau$ and compute
\begin{align*}
\tau([(q_\alpha + r_\alpha) - p]^2) &= \tau([r_\alpha - (p - q_\alpha)]^2) = \tau (r_\alpha + (p-q_\alpha)) \\ &= \tau (T(r_\alpha) + T(p-q_\alpha)) = \tau(|T(r_\alpha) - T(p-q_\alpha)|) \\ &= \tau(|T((q_\alpha + r_\alpha) - p)|) = \tau(|T(p_\alpha - p)|) \\ &\leq \tau(|T(p_\alpha - p)|^2)^{1/2} \to 0. \qedhere
\end{align*}
\end{proof}
Regarding Theorem \ref{T:sinfin}, we remind the reader that typically we do \textit{not} have an equivalence between the strong and weak topologies on $\mathcal{Z}(\mathcal{M})_1^+$.
\begin{theorem} \label{T:sininf}
Let $p$ be a projection in a properly infinite von Neumann algebra $\mathcal{M}$. If $p$ is finite,
\begin{equation} \label{E:finclos}
\overline{\{[p]\}} = \{[q] \mid [q] \leq [p]\}.
\end{equation}
If $p$ is properly infinite and $c(p)=1_\mathcal{M}$,
\begin{equation} \label{E:infclos}
\overline{\{[p]\}} = (\mathcal{P}(\mathcal{M})/\sim).
\end{equation}
Equations \eqref{E:finclos} and \eqref{E:infclos} may be synthesized into
\begin{equation} \label{E:seg}
\overline{\{[p]\}} = \{[q] \mid T(q) \leq T(p)\}, \qquad \forall p \in \mathcal{P}(\mathcal{M}),
\end{equation}
for any extended center-valued trace $T$.
\end{theorem}
\begin{proof} First consider a finite projection $p$. We may assume that $c(p) = 1$ and so $\mathcal{M}$ is semifinite; let $T$ be an extended center-valued trace. If $p_\alpha \sim p$ and $p_\alpha \overset{w}{\to} q$, then by Lemma \ref{T:liminf}, $T(q) \leq T(p)$. We have assumed $p$ finite, so $q$ is as well and $p \succcurlyeq q$. For the other containment, choose any $q$ with $[q] \leq [p]$. Write $p = p_0 + p_1$, with $p_0 \sim q$. Since $q$ is finite, $q^\perp \sim 1$ is properly infinite, and we may write $q^\perp = \sum_{k=1}^\infty q_k$, with $q_k \sim q^\perp \sim 1$. Let $p_1 \sim r_k \leq q_k$. Then $p = (p_0 + p_1) \sim (q + r_k) \overset{s}{\to} q$. This proves \eqref{E:finclos}.
Now consider arbitrary $q$ and properly infinite $p$ with $c(p)=1$. Find the largest central projection $z$ with $zp \preccurlyeq zq$. Consider the nonempty net $\{zp_\alpha \mid zp \sim zp_\alpha \leq zq\}$, with order inherited from $\mathcal{P}(\mathcal{M})$. It is upward directed by Lemma \ref{T:proj}(1), applied to two projections. Its supremum is $zq$.
By Lemma \ref{T:proj}(2) $z^\perp q^\perp \sim z^\perp$, which is properly infinite, so we may write $z^\perp q^\perp$ as the countable sum $\sum q_j$, with each $q_j \sim z^\perp q^\perp \sim z^\perp$. Write $z^\perp p = z^\perp p_0 + z^\perp p_1$, where $z^\perp p_0 \sim z^\perp q$. Also let $z^\perp p_1 \sim r_j \leq q_j$. Then $z^\perp p = (z^\perp p_0 + z^\perp p_1) \sim (z^\perp q + r_k) \overset{w}{\to} z^\perp q$.
Combining the results for $zp$ and $z^\perp p$ and considering the product net, we see that $q$ is a strong limit of projections equivalent to $p$. This proves \eqref{E:infclos}.
Equation \eqref{E:seg} follows from \eqref{E:finclos} and \eqref{E:infclos} by breaking off the largest central summand where $p$ is properly infinite with full central support.
\end{proof}
\begin{corollary} \label{T:Eclos}
Let $\mathcal{M}$ be a factor and $E \subseteq \mbox{$(\p(\M)/\sim)$}$. We consider an extended center-valued trace $T$ on $\mathcal{M}$ to be a $[0,+\infty]$-valued function.
If $\mathcal{M}$ is finite,
$$\overline{E} = \{[q] \mid T(q) \in \overline{\{T(p) \mid [p] \in E\}}\}.$$
If $\mathcal{M}$ is properly infinite,
$$\overline{E} = \{[q] \mid T(q) \leq \sup_{[p] \in E} T(p)\}.$$
\end{corollary}
Corollary \ref{T:Eclos} follows readily from the preceding arguments, and its easy proof is left to the interested reader.
\begin{corollary}
$QOT$ is a $T_1$ topology exactly when $\mathcal{M}$ is finite.
\end{corollary}
\begin{proof}
This is a direct consequence of Theorems \ref{T:sinfin} and \ref{T:sininf}.
A topology is $T_1$ if for any two distinct points $x,y$, there is a closed set which contains $x$ and not $y$. If $\mathcal{M}$ is not finite, let $x$ be the equivalence class of a properly infinite projection, and let $y$ be $[0]$. Since $y$ belongs to the closure of $x$, no such separating closed set exists. (In general, a topology is $T_1$ iff singletons are closed.)
\end{proof}
It turns out to be more useful for our applications elsewhere (\cite{S}) to know when $QOT$ is $T_0$. A topology is $T_0$ if for any two distinct points, there exists a closed set which contains exactly one of them.
\begin{proposition} \label{T:T0}
For a von Neumann algebra $\mathcal{M}$, the following conditions are equivalent.
\begin{enumerate}
\item $QOT$ is a $T_0$ topology on $(\mathcal{P}(\mathcal{M})/\sim)$.
\item For any $p,q \in \mathcal{P}(\mathcal{M})$, $[p] \in \overline{\{[q]\}} \Rightarrow p \preccurlyeq q$.
\item $\kappa_\mathcal{M} \leq \aleph_0$.
\item $\mathcal{M}$ is a (possibly uncountable) direct sum of $\sigma$-finite von Neumann algebras.
\item $\mathcal{M}$ does not contain $\mathcal{B}(\mathfrak{H}_1)$, where $\mathfrak{H}_1$ is a Hilbert space of dimension $\aleph_1$.
\end{enumerate}
\end{proposition}
\begin{proof} The equivalence of conditions (3)-(5) follows from the definitions and Theorem \ref{T:griffin}. We therefore focus on the equivalence of (1)-(3).
(1) $\to$ (3): If (3) fails, let $q$ be an $\aleph_1$-homogeneous projection, and let $p$ be an $\aleph_0$-homogeneous projection with $c(p)=c(q)$. Then $[p] \in \overline{\{[q]\}}$ and $[q] \in \overline{\{[p]\}}$, but $[p] \neq [q]$. Clearly there is no closed set separating the two.
(3) $\to$ (2): When $\kappa_\mathcal{M} \leq \aleph_0$, $T |_{\mathcal{P}(\mathcal{M})}$ can be identified with $D$. By Theorems \ref{T:sinfin} and \ref{T:sininf} we have
$$[p] \in \overline{\{[q]\}} \Rightarrow T(p) \leq T(q) \Rightarrow D(p) \leq D(q) \Rightarrow p \preccurlyeq q.$$
(2) $\to$ (1): Suppose (2) holds. Given $[p],[q] \in \mbox{$(\p(\M)/\sim)$}$, they can be separated by a closed set if $[p] \notin \overline{\{[q]\}}$ or $[q] \notin \overline{\{[p]\}}$. If neither of these is true, then
$$[p] \in \overline{\{[q]\}}, \: [q] \in \overline{\{[p]\}} \: \Rightarrow p \preccurlyeq q, \: q \preccurlyeq p \: \Rightarrow [p]=[q]. \qedhere$$
\end{proof}
\section{From dimension function to trace in full generality} \label{S:trace}
Let $T$ be an extended center-valued trace on a von Neumann algebra $\mathcal{M}$, with $D$ the induced dimension function. We will create a map which extends $D$ to the entire positive cone and so is a trace which distinguishes among infinite cardinalities. (In case $\kappa_\mathcal{M} \leq \aleph_0$, this process simply recovers $T$.) The main tool is
\begin{definition} (\cite{KP})
For two elements $h,k \in \mathcal{M}_+$, we write $h \approx k$ if and only if there exists a family $\{x_\alpha\} \subset \mathcal{M}$ such that $h = \sum x_\alpha^* x_\alpha$ and $k = \sum x_\alpha x_\alpha^*$.
We write $h \lessapprox k$ to mean that there exists $k' \leq k$ with $h \approx k'$.
For $h \in \mathcal{M}_+$, we say that $h$ is \textit{finite} if $h \approx k \leq h \Rightarrow k=h$.
\end{definition}
The following facts are shown in \cite{KP}.
\begin{itemize}
\item The relation $\approx$ is an equivalence relation. It is homogeneous ($h \approx k \Rightarrow \lambda h \approx \lambda k, \: \lambda \in \mathbb{R}_+$) and completely additive in the sense that
$$h_\alpha \approx k_\alpha, \: \forall \alpha \quad \Rightarrow \quad \sum h_\alpha \approx \sum k_\alpha$$
(when the two sums exist in $\mathcal{M}$).
\item The relation $\lessapprox$ gives a partial order on equivalence classes. In particular,
\begin{equation} \label{E:pord}
h \lessapprox k, \: k \lessapprox h \: \Rightarrow \: h \approx k, \qquad h,k \in \mathcal{M}_+.
\end{equation}
\item For projections, $p \approx q \iff p \sim q$.
\item For $h,k \in \mathcal{M}_+$, $h \lessapprox k \Rightarrow T(h) \leq T(k)$, and the converse holds if $h$ is finite.
\end{itemize}
We will also say that nonzero $h \in \mathcal{M}_+$ is \textit{properly infinite} if $zh$ is finite and nonzero for no central projection $z$. For projections, the usage here of ``finite" and ``properly infinite" coincides with the usual meaning; in fact proper infiniteness of (nonzero) $h$ in either case is characterized by $T(h)$ being $\{0,+\infty\}$-valued.
\begin{lemma} \label{T:mult} ${}$
\begin{enumerate}
\item Let $\lambda \in ((0,1) \cup (1,\infty))$, and let $p$ be a projection. Then
$$p \text{ is properly infinite } \iff p \approx \lambda p.$$
\item Let $h,k \in \mathcal{M}_+$ have equal central support, with $k$ properly infinite and $h$ a countable sum of finite elements. Then $h \lessapprox k$.
\item Let $h,k \in \mathcal{M}_+$ be properly infinite with equal central support, and suppose that each is a countably infinite sum of finite elements. Then $h \approx k$.
\end{enumerate}
\end{lemma}
\begin{proof} (1) If $p \approx \lambda p$, then $T(p)$ must be $\{0,+\infty\}$-valued. For the opposite implication, we first check rational multiples. Let $m,n \in \mathbb{N}$. By proper infiniteness, we may write
$$p = \sum_{i=1}^m p_i = \sum_{j=1}^n p'_n, \qquad p_i \sim p \sim p'_j, \: \forall i,j.$$
Then
\begin{align*}
p &= \sum_{i=1}^m p_i \approx \sum_{i=1}^m p = mp = \left( \frac{m}{n} \right) np = \left( \frac{m}{n} \right) \left(\sum_{j=1}^n p \right) \\ &\approx \left( \frac{m}{n} \right) \left( \sum_{j=1}^n p'_n \right) = \left( \frac{m}{n} \right) p.
\end{align*}
Find two positive rationals $\lambda_1, \lambda_2$ with $\lambda_1 \leq \lambda \leq \lambda_2$:
$$p \approx \lambda_1 p \leq \lambda p \leq \lambda_2 p \approx p \; \Rightarrow \; p \approx \lambda p,$$
using \eqref{E:pord}.
(2) Write $h = \sum_{j=1}^\infty h_j$, where each $h_j$ is finite. Since $T(h_1) \leq T(k)$, there is an operator $k_1$ with $h_1 \approx k_1 \leq k$. We continue in this way: since $T(h_n) \leq T(k - \sum_{j=1}^{n-1} k_j)$, find $k_n$ with $h_n \approx k_n \leq (h - \sum_{j=1}^{n-1} k_j)$.
Now each $(\sum_{j=1}^n h_j) \approx (\sum_{j=1}^n k_j)$, and these terms are finite and increasing to $h$ and some $k'$, respectively. It follows from \cite[Lemma 3.3]{KP} that $h \approx k' \leq k$.
(3) Both $h \lessapprox k$ and $h \gtrapprox k$ follow from the previous part; apply \eqref{E:pord}.
\end{proof}
\begin{proposition} \label{T:appproj}
Let $h \in \mathcal{M}_+$ be properly infinite. Then there exists $p \in \mathcal{P}(\mathcal{M})$ such that $h \approx p$.
\end{proposition}
\begin{proof} It does no harm to assume that $h$ has full central support, and therefore $\mathcal{M}$ is properly infinite. Write the identity as $1_\mathcal{M} = \sum_{n=-\infty}^\infty p_n, \: 1_\mathcal{M} \sim p_n$, and let $r_0 \leq p_0$ be an $\aleph_0$-homogeneous projection with full central support.
Now make the decomposition
$$h = \sum_{n=1}^\infty (2^{-n} \|h\|) q_n,$$
where $q_n$ is the spectral projection for $h$ corresponding to
$$\bigcup_{j=1}^{2^{n-1}} \left( (2j-1)2^{-n}\|h\|, (2j) 2^{-n} \|h\| \right].$$
For each $n \geq 1$, let $z^f_n$ be the largest central projection such that $z^f_n q_n$ is finite. Using Lemma \ref{T:mult}(1) and then conjugating by a partial isometry from $(1 - z^f_n)$ to $(1-z^f_n)p_n$, find a projection $r_n$ with
$$(1-z_n^f)(2^{-n} \|h\|) q_n \approx (1-z^f_n) q_n \sim r_n \leq p_n.$$
Conjugating by a partial isometry from $z^f_n$ to $z^f_n p_{-n}$, let $r_{-n}$ be any operator (necessarily finite, but not necessarily a projection) with
$$z^f_n (2^{-n} \|h\|) q_n \approx r_{-n} \in p_{-n} \mathcal{M} p_{-n}.$$
By construction we have $h \approx \sum_{n=1}^\infty (r_n + r_{-n})$.
Set $z_0 = \wedge z^f_n$. We will complete the proof by showing that $z_0 h$ and $z_0^\perp h$ are both (Kadison-Pedersen) equivalent to projections.
First,
$$z_0 h \approx z_0 \left( \sum_{n=1}^\infty (r_n + r_{-n}) \right) = z_0 \left( \sum_{n=1}^\infty r_{-n} \right).$$
The left-hand side has central support $z_0$, and is either zero or properly infinite because $h$ is properly infinite. The right-hand side is a countable sum of finite elements. By Lemma \ref{T:mult}(3),
$$z_0 h \approx z_0 r_0.$$
Second,
$$z_0^\perp \left( \sum_{n=1}^\infty r_n \right) \sim z_0^\perp \left( r_0 + \sum_{n=1}^\infty r_n \right),$$
since the central supports are equal and the left-hand side is a properly infinite projection. (For example, this follows by evaluating the dimension function on both sides and noting that adding $\aleph_0$ does not change an infinite cardinal.) On the other hand, Lemma \ref{T:mult}(2) implies
$$z_0^\perp \left( \sum_{n=1}^\infty r_{-n} \right) \lessapprox z_0^\perp r_0.$$
We put these together:
\begin{align*}
z_0^\perp h &\approx z_0^\perp \left( \sum_{n=1}^\infty (r_n + r_{-n}) \right) \lessapprox z_0^\perp \left( r_0 + \sum_{n=1}^\infty r_n \right) \\ &\sim z_0^\perp \left(\sum_{n=1}^\infty r_n \right) \lessapprox z_0^\perp \left( \sum_{n=1}^\infty (r_n + r_{-n}) \right) \approx z_0^\perp h.
\end{align*}
Then all terms above are (Kadison-Pedersen) equivalent, and the middle two are projections.
\end{proof}
\begin{corollary} \label{T:absorb}
Under the same hypotheses as in Lemma \ref{T:mult}(2), $k \approx \lambda k$ for any $\lambda \in (0, \infty)$, and $(h + k) \approx k$.
\end{corollary}
\begin{proof}
By Proposition \ref{T:appproj} and Lemma \ref{T:mult}(1), there is a properly infinite projection $p$ with $k \approx p \approx \lambda p \approx \lambda k$. By Lemma \ref{T:mult}(2),
$$(h + k) \lessapprox 2k \approx k \lessapprox (h+k) \: \Rightarrow \: (h+k) \approx k.$$
\end{proof}
We are now ready to define our map.
\begin{definition}
With $T$ (and $D$) given, we construct a \textbf{fully extended center-valued trace} $\widehat{T}$ on $\mathcal{M}$ as follows.
For any $h \in \mathcal{M}_+$, let $z^f$ be the largest central projection so that $z^f h$ is finite. Let $p$ be a projection with $p \approx (1 -z^f) h$. Such a $p$ exists by Proposition \ref{T:appproj}, and all choices belong to the same Murray-von Neumann equivalence class.
We define
\begin{equation}
\widehat{T}(h) = T(z^f h) + D((1- z^f) p),
\end{equation}
which we view as a continuous $([0,+\infty) \cup \{\kappa \mid \aleph_0 \leq \kappa \leq \kappa_\mathcal{M}\})$-valued function on $\Omega(\mathcal{Z}(\M))$.
\end{definition}
\begin{theorem} \label{T:Tfull}
The map $\widehat{T}$ extends $D$, is additive, commutes with the multiplicative action of $\mathcal{Z}(\mathcal{M})_+$, and satisfies
\begin{equation} \label{E:pres}
h \lessapprox k \iff \widehat{T}(h) \leq \widehat{T}(k), \qquad h,k \in \mathcal{M}_+.
\end{equation}
(We are allowing cardinal arithmetic to incorporate the positive reals in the obvious way.)
\end{theorem}
\begin{proof}
Clearly $\widehat{T}$ extends $D$. By the properties of $D$ and $T$ we have $h \approx k \iff \widehat{T}(h) = \widehat{T}(k)$.
In saying that $\widehat{T}$ is additive, we mean that
\begin{equation} \label{E:add}
\widehat{T}(h + k) = \widehat{T}(h) + \widehat{T}(k), \qquad h,k \in \mathcal{M}_+.
\end{equation}
For $h,k$ finite, \eqref{E:add} follows from additivity of $T$. For $h,k$ properly infinite, the projection representing $h + k$ may be constructed as the sum of orthogonal representing projections for $h$ and $k$; \eqref{E:add} then follows from the additivity of $D$. Finally, let $h$ and $k$ have the same central support, with $h$ finite and $k$ properly infinite. In this case $\widehat{T}(h)$ is bounded above by $\aleph_0$, while $\widehat{T}(k) \geq \aleph_0$ where it is nonzero. So $\widehat{T}(h) + \widehat{T}(k) = \widehat{T}(k)$. Since $(h + k) \approx k$ by Corollary \ref{T:absorb}, $\widehat{T}(h + k) = \widehat{T}(k)$ as well.
In saying that $\widehat{T}$ commutes with the action of $\mathcal{Z}(\mathcal{M})_+$, we mean
\begin{equation} \label{E:modmap}
y \widehat{T}(h) = \widehat{T}(yh), \qquad y \in \mathcal{Z}(\mathcal{M})_+, \: h \in \mathcal{M}_+.
\end{equation}
Clearly \eqref{E:modmap} holds for finite elements, since the analogous formula is true for $T$. It therefore suffices to prove \eqref{E:modmap} under the assumption that $h$ and $y$ have full central support, with $h$ properly infinite. In this case $y \widehat{T}(h) = \widehat{T}(h)$, so we are left to show that $yh \approx h$. If $y \geq \lambda c(y)$ for some $\lambda > 0$, then by Corollary \ref{T:absorb}
$$h \approx \lambda h \leq yh \leq \|y\| h \approx h \: \Rightarrow \: h \approx yh.$$
The general conclusion follows by writing $y$ as a central sum of operators which are invertible on their supports.
As for \eqref{E:pres}, the forward implication is a consequence of additivity. For the reverse implication, we look at central summands: where $h$ is finite, this is a property of $T$; where $h$ and $k$ are both infinite, this is a property of $D$.
\end{proof}
From Theorem \ref{T:Tfull}, we see that $\widehat{T}$ factors as
$$\mathcal{M}_+ \twoheadrightarrow (\mathcal{M}_+/\approx) \overset{\sim}{\to} \widehat{T}(\mathcal{M}_+).$$
Here the second map is an embedding in a function space, preserving order, sums, and the multiplicative $\mathcal{Z}(\mathcal{M})_+$-action.
More generally, we may say that an arbitrary completely additive map on $\mathcal{M}_+$ which respects the $\mathbb{R}_+$-action is \textit{tracial} if and only if it factors through the quotient $\mathcal{M}_+ \twoheadrightarrow (\mathcal{M}_+/\approx)$. Numerical (completely additive) traces result when the range is $[0,+\infty]$; they are ``one-dimensional representations" of $(\mathcal{M}_+/\approx)$.
\begin{remark} \label{R:trace} Kadison and Pedersen observed that all extended center-valued traces on semifinite algebras can be generated in the following manner (\cite[Theorem 3.8]{KP}). Fix a finite projection $p$ with full central support, and assume that $p$ is the identity on the finite summand and abelian on the infinite type I summand (to match Convention \ref{C}). Then for finite $h \in \mathcal{M}_+$, $T(h)$ is the unique element of the extended center with $h \approx T(h)p$. Already this requires a small extension of $\approx$ to unbounded sums.
With a further extension involving cardinals, $\widehat{T}$ can also be defined in this way. For general $\mathcal{M}$, let $p$ be the identity on the finite summand, abelian on the infinite type I summand, finite on the type II summand, and $\aleph_0$-homogeneous on the type III summand; of course $p$ should have full central support. For $h \in \mathcal{M}_+$, one can define $\widehat{T}(h)$ as the unique formal sum (as in \eqref{E:gdf}) such that $h \approx \widehat{T}(h)p$ and $\widehat{T}(h)$ takes no finite nonzero values on the type III summand. Probably this is more interesting to mention than to carry out, so we omit the details.
\end{remark}
\section{Continuity} \label{S:cont}
In the remaininder of the paper we assume that $T$, $D$, and $\widehat{T}$ are given on $\mathcal{M}$.
The order-preserving embeddings of $\mbox{$(\p(\M)/\sim)$}$ and $(\mathcal{M}_+/\approx)$ in a function space (albeit cardinal-valued) make pointwise operations available. From Theorems \ref{T:gdf} and \ref{T:Tfull} we know that for finite sets, addition in the quotient structures agrees with addition of functions. One may likewise add up infinite sets of functions, but there is no guarantee that the sum will be continuous. Tomiyama gave an example (\cite[Example 2]{To}) to show that for a pairwise orthogonal set $\{p_\alpha\}$, one cannot expect an identity between $\sum D(p_\alpha)$ and $D(\sum p_\alpha)$, so that $D$ is not completely additive.
This is really an artifact of the function representation. $\mbox{$(\p(\M)/\sim)$}$ carries a natural (partially-defined) sum operation, given by
$$\sum [p_\alpha] \triangleq \left[ \sum q_\alpha \right]$$
whenever there exists a set of pairwise orthogonal projections $\{q_\alpha\}$ with $q_\alpha \sim p_\alpha$. A similar definition is possible for sums in $\widehat{T}(\mathcal{M}_+)$, where we simply require that the representatives sum to an element of $\mathcal{M}_+$. Note that there is no ambiguity in these definitions, by \eqref{E:addeq} and the definition of $\approx$, and as an immediate consequence, the maps $\mathcal{P}(\mathcal{M}) \twoheadrightarrow \mbox{$(\p(\M)/\sim)$}$ and $\mathcal{M}_+ \twoheadrightarrow (\mathcal{M}_+/\approx)$ are completely additive. It is of course possible to transport these sum operations to $D(\mathcal{P}(\mathcal{M}))$ and $\widehat{T}(\mathcal{M}_+)$.
\smallskip
Pointwise lattice operations on pairs in $D(\mathcal{P}(\mathcal{M}))$ match \eqref{E:lattice} and so agree with the operations in $\mbox{$(\p(\M)/\sim)$}$, but meets and joins of infinite sets of continuous functions need not be continuous. For bounded real-valued functions on a stonean space, a regularization corrects this problem (\cite[Section III.1]{T}), but the situation for cardinal-valued functions is less clear.
Normality for $D$ and $\widehat{T}$ means an appropriate analogue of \eqref{E:normal}. So how do we interpret an expression like $\sup D(p_\alpha)$, where $\{p_\alpha\}$ is an increasing net in $\mathcal{P}(\mathcal{M})$? As we just mentioned, the pointwise supremum need not lie in $D(\mathcal{P}(\mathcal{M}))$. In the next section we show that the supremum always does make sense in $\mbox{$(\p(\M)/\sim)$}$, but unfortunately normality is to much to ask. Tomiyama gave a simple example (\cite[Example 1]{To}) of an uncountable increasing family of projections $\{p_\alpha\}$ for which the pointwise supremum of $D(p_\alpha)$ does lie in $D(\mathcal{P}(\mathcal{M}))$, and yet $\sup D(p_\alpha) \neq D(\sup p_\alpha)$.
This represents a phenomenon which really occurs in the quotient map $\mathcal{P}(\mathcal{M}) \twoheadrightarrow \mbox{$(\p(\M)/\sim)$}$, as we already saw in the proof of \eqref{E:infclos}. For $p$ a properly infinite projection, the elements of $[p]$, under the operator ordering, form an increasing net which converges strongly to $c(p)$. One obtains a counterexample to normality whenever $c(p) \notin [p]$, and such counterexamples exist when $\kappa_\mathcal{M} > \aleph_0$. On the other hand, if $\kappa_\mathcal{M} \leq \aleph_0$, the quotient maps are given by the extended center-valued trace, which we know to be normal. We conclude
\begin{proposition} \label{T:normal}
Another equivalent condition in Proposition \ref{T:T0} is
\begin{enumerate}
\item[(6)] The quotient maps $\mathcal{P}(\mathcal{M}) \twoheadrightarrow \mbox{$(\p(\M)/\sim)$}$ and $\mathcal{M}_+ \twoheadrightarrow (\mathcal{M}_+/\approx~)$ are normal.
\end{enumerate}
\end{proposition}
In contrast, a pointwise criterion for normality of $D$ and $\widehat{T}$ holds if and only if $\kappa_\mathcal{M} \leq \aleph_0$ and the center of $\mathcal{M}$ is finite-dimensional. We do not bother to prove this explicitly, but we mention an example. Let $\mathcal{M} = \ell^\infty$, and take $p_n$ to be the sum of the first $n$ elements of the standard basis. Since $\sup D(p_n)$ does not agree with $D(\sup p_n)$ at any point of $(\beta \mathbb{N} \setminus \mathbb{N}) \subset \beta \mathbb{N} \simeq \Omega(\mathcal{Z}(\M))$, pointwise normality fails. And here $D$ is the identity!
\smallskip
Our conclusion from all this is that the pointwise lattice and addition operations on functions in the range of $D$ and $\widehat{T}$ should be shelved in favor of the induced quotient structures on $\mbox{$(\p(\M)/\sim)$}$ and $(\mathcal{M}_+/\approx)$. With this interpretation the assertion ``$D$ and $\widehat{T}$ are normal" is also equivalent to the conditions in Proposition \ref{T:T0}.
\section{$\mbox{$(\p(\M)/\sim)$}$ is a complete lattice} \label{S:comp}
Having just been warned about the degeneracies of the pointwise ordering, we omit the last step of Tomiyama's construction for $D$ and stick with a more algebraic language. We follow the right-hand side of \eqref{E:gdf}, further dividing $T(z^f p)$ into the pieces where it lies between consecutive finite cardinals. This allows us to write the typical element of $\widehat{T}(\mathcal{M}_+)$ as
\begin{equation} \label{E:simpler}
\sum_{\kappa \leq \kappa_\mathcal{M}} g_\kappa z_\kappa.
\end{equation}
The meaning of this expression is as follows. If $\kappa$ is an infinite cardinal, then $g_\kappa = \kappa$. If $\kappa$ is a nonnegative integer, $g_\kappa$ is an element of $\mathcal{Z}(\mathcal{M})_+$ satisfying $(\kappa-1)z_\kappa \leq g_\kappa \leq \kappa z_\kappa$ and $c(g_\kappa - (\kappa-1)z_\kappa) = z_\kappa$. The central projections $z_\kappa$ sum to 1, and the decomposition is unique.
The partial order, pairwise sum operation, and pairwise lattice operations are all easily implemented for expressions of the form \eqref{E:simpler}. Conversely, such an expression belongs to $\widehat{T}(\mathcal{M}_+)$ if it is $\{0, +\infty\}$-valued on the type III summand and less than some finite multiple of $\widehat{T}(1_\mathcal{M})$. To belong to $D(\mathcal{P}(\mathcal{M})) \subseteq \widehat{T}(\mathcal{M}_+)$, an expression must be $\leq D(1_\mathcal{M})$ and appropriately valued on both the type I and type III summands.
\begin{theorem} \label{T:complat}
$(\mathcal{M}_+/\approx)$ and $\mbox{$(\p(\M)/\sim)$}$ are complete lattices.
\end{theorem}
\begin{proof}
We show how to perform lattice operations on formal sums of the form \eqref{E:simpler}. Our constructions will preserve all conditions mentioned in the paragraph before Theorem \ref{T:complat}, so they are well-defined operations in $(\mathcal{M}_+/\approx)$ and $\mbox{$(\p(\M)/\sim)$}$.
Let us find the supremum of an arbitrary set $\{f^\alpha\}$, where
$$f^\alpha = \sum g_\kappa^\alpha z_\kappa^\alpha.$$
For each cardinal $\kappa \leq \kappa_\mathcal{M}$, set
$$y_{\leq \kappa} = \bigwedge_\alpha \left(\sum_{\lambda \leq \kappa} z^\alpha_\lambda \right);$$
$y_{\leq \kappa}$ is ``where all $f^\alpha$ are $\leq \kappa$". Note that $y_{\leq \kappa}$ is increasing in $\kappa$ and $y_{\leq \kappa_\mathcal{M}} = 1$. Next define, for each cardinal $\kappa \leq \kappa_\mathcal{M}$,
$$z_\kappa = y_{\leq \kappa} - \bigvee_{\lambda < \kappa} y_{\leq \lambda}.$$
The $z_\kappa$ are pairwise disjoint: if $\kappa_1 < \kappa_2$, then $$z_{\kappa_1} \leq y_{\leq \kappa_1} \perp z_{\kappa_2}.$$
Notice also that $\sum z_\kappa = 1$. For if there were $z \in \mathcal{P}(\mathcal{Z}(\mathcal{M}))$ with $z \perp (\sum z_\kappa)$, then let $\lambda$ be the least cardinal with $z y_{\leq \lambda} \neq 0$; by definition $z z_\lambda \neq 0$ as well, which contradicts the assumption.
We claim that
\begin{equation} \label{E:sup}
\sup_\alpha f^\alpha = \sum g_\kappa z_\kappa \triangleq f,
\end{equation}
where $g_\kappa = \kappa$ when $\kappa$ is infinite, and otherwise $g_\kappa = \sup_\alpha (g_\kappa^\alpha z_\kappa)$, which exists as the supremum of a bounded set in $\mathcal{Z}(\mathcal{M})_+$.
Next we show that $f \geq f^\alpha$ for any $\alpha$. Fixing a cardinal $\lambda \leq \kappa_\mathcal{M}$,
\begin{equation} \label{E:cut}
z_\lambda f^\alpha = z_\lambda \left( \sum g_\kappa^\alpha z_\kappa^\alpha \right) = (z_\lambda y_{\leq \lambda}) \left( \sum g_\kappa^\alpha z_\kappa^\alpha \right) \leq z_\lambda \left( \sum_{\kappa \leq \lambda} g_\kappa^\alpha z_\kappa^\alpha \right).
\end{equation}
When $\lambda$ is infinite, we continue \eqref{E:cut} as
$$\leq \lambda z_\lambda = z_\lambda f.$$
When $\lambda$ is finite, we continue \eqref{E:cut} as
$$\leq z_\lambda (\lambda - 1) \left( \sum_{\kappa < \lambda} z_\kappa^\alpha \right) + z_\lambda g_\lambda^\alpha z_\lambda^\alpha \leq z_\lambda g_\lambda = z_\lambda f.$$
Since $z_\lambda f^\alpha \leq z_\lambda f$ for all $\lambda$, $f \geq f^\alpha$.
Finally we check that if $h = \sum h_\kappa x_\kappa$ satisfies $h \geq f^\alpha$, $\forall \alpha$, then necessarily $h \geq f$. Fixing a cardinal $\lambda \leq \kappa_\mathcal{M}$,
\begin{align*}
h_\lambda x_\lambda = x_\lambda h \geq x_\lambda f^\alpha, \: \forall \alpha \quad &\Rightarrow \quad x_\lambda \leq \sum_{\kappa \leq \lambda} z^\alpha_\kappa, \: \forall \alpha \\ &\Rightarrow \quad x_\lambda \leq \bigwedge_\alpha \left(\sum_{\kappa \leq \lambda} z^\alpha_\kappa \right) = y_{\leq \lambda}.
\end{align*}
This last inequality implies
\begin{equation} \label{E:cut2}
x_\lambda f = x_\lambda y_{\leq \lambda} f \leq x_\lambda \left( \sum_{\kappa \leq \lambda} g_\kappa z_\kappa \right).
\end{equation}
When $\lambda$ is infinite, we continue \eqref{E:cut2} as
$$\leq \lambda x_\lambda = x_\lambda h.$$
When $\lambda$ is finite, we continue \eqref{E:cut2} as
$$\leq x_\lambda (\lambda - 1) \left( \sum_{\kappa < \lambda} z_\kappa \right) + x_\lambda g_\lambda z_\lambda$$
and the inequality $h \geq f^\alpha$, $\forall \alpha$, allows us to compute further
$$ = x_\lambda (\lambda - 1) \left( \sum_{\kappa < \lambda} z_\kappa \right) + x_\lambda \left( \sup_\alpha g_\lambda^\alpha z_\lambda \right) \leq x_\lambda h.$$
Since $x_\lambda f \leq x_\lambda h$ for all $\lambda$, $f \leq h$.
This completes the proof that $f = \sup f^\alpha$.
As for the infimum of the $f^\alpha$, we first point out that we cannot write anything like
$$\bigwedge f^\alpha = 1 - \left(\bigvee (1-f^\alpha)\right),$$
which is a useful duality in $\mathcal{P}(\mathcal{M})$. There is no complementation in the lattices $\mbox{$(\p(\M)/\sim)$}$ and $(\mathcal{M}_+/\approx)$, at least when $\mathcal{M}$ is not finite. Instead we define
$$y_{\leq \kappa} = \bigvee_\alpha \left(\sum_{\lambda \leq \kappa} z_\lambda^\alpha \right)$$
and complete the rest of the proof similarly to the proof for the supremum. (The substitute for \eqref{E:cut} should begin with ``$z_\lambda^\alpha f = \dots$"; for \eqref{E:cut2} should begin with ``$z_\lambda h = \dots$".)
\end{proof}
\section{Application to representation theory} \label{S:rep}
In this section we reinterpret Theorem \ref{T:complat} in terms of the (normal Hilbert space) representations of $\mathcal{M}$. Unless noted otherwise, we use ``isomorphism" in the sense of normed $\mathcal{M}$-modules, i.e.
$$\{\pi_1, \mathfrak{H}_1\} \simeq \{\pi_2, \mathfrak{H}_2\} \iff$$
$$\exists \text{ unitary }U: \mathfrak{H}_1 \to \mathfrak{H}_2: \qquad U\pi_1(x)U^* = \pi_2(x), \qquad \forall x \in \mathcal{M}.$$
It follows from the basic theory (see \cite[Sections 2.1-2]{JS} or \cite[Section 2]{S2003}) that any representation is (isomorphically) contained in a direct sum of copies of the standard form $\{\text{id}, L^2(\mathcal{M})\}$. We view $\oplus_I L^2(\mathcal{M})$ as a row vector and think of the $\mathcal{M}$-action as multiplication on the left. The commutant is right multiplication by $\mathcal{B}(\ell^2_I) \overline{\otimes} \mathcal{M}$, and the closed submodules are of the form $(\oplus_I L^2(\mathcal{M})) q$, where $q \in \mathcal{P}(\mathcal{B}(\ell^2_I) \overline{\otimes} \mathcal{M})$. Two submodules are isomorphic if and only if the corresponding projections are equivalent.
This means that the isomorphism class of a representation corresponds to an equivalence class of projections in some amplification of $\mathcal{M}$. Adding representations corresponds to adding equivalence classes. As we have mentioned, the partial order can be defined in terms of the sum, so provided we make some kind of size restriction, we get an isomorphism of ordered monoids. For example, if $\mathcal{M}$ is $\sigma$-finite, we obtain an identification between $(\mathcal{P}(\mathcal{B}(\ell^2) \overline{\otimes} \mathcal{M})/\sim)$ and isomorphism classes of countably generated Hilbert $\mathcal{M}$-modules. This all works for $L^p$ modules (\cite{JuS}), too, and is closely related to the $\mathcal{K}_0$ functor (\cite{Han,W-O}).
(Most of the ideas of the preceding two paragraphs were discussed by Breuer (\cite{B1968,B1969}), without making reference to standard forms. He focused on the monoid generated by equivalence classes of finite projections, because the associated Grothendieck group, called the \textit{index group} of $\mathcal{M}$, is the natural carrier for the Fredholm theory of $\mathcal{M}$. Olsen (\cite{O}) later combined Breuer's work with Tomiyama's dimension function to give a very general version of index theory in von Neumann algebras.)
\begin{corollary}
Let $\{\pi_\alpha, \mathfrak{H}_\alpha\}$ be a set of representations of a fixed von Neumann algebra $\mathcal{M}$. Then there is a maximal representation of $\mathcal{M}$ which is (isomorphically) contained in all of these, and there is a minimal representation which (isomorphically) contains all of these. Both are unique up to $\mathcal{M}$-module isomorphism.
\end{corollary}
\begin{proof}
Choose a large enough set $I$ so that for all $\alpha$, $\{\pi_\alpha, \mathfrak{H}_\alpha\}$ is a subrepresentation of $\{\text{id}, \oplus_I L^2(\mathcal{M})\}$. The corollary follows from the preceding discussion and the fact that $(\mathcal{P}(\mathcal{B}(\ell^2_I) \overline{\otimes} \mathcal{M})/\sim)$ is a complete lattice.
\end{proof}
In the early years of the subject, von Neumann algebras were generally given on Hilbert spaces, and the notion of $\mathcal{M}$-module isomorphism was therefore not in use. Instead, one classified represented algebras up to the slightly weaker notion of \textit{spatial isomorphism}, which allows for an arbitrary isomorphism between the algebras. (An $\mathcal{M}$-module isomorphism between representations $\{\pi_1, \mathfrak{H}_1\}$ and $\{\pi_2, \mathfrak{H}_2\}$ is a spatial isomorphism between von Neumann algebras $\{\pi_1(\mathcal{M}), \mathfrak{H}_1\}$ and $\{\pi_2(\mathcal{M}), \mathfrak{H}_2\}$ which induces the algebra isomorphism $\pi_2 \circ \pi_1^{-1}$.) The question ``When is an algebraic isomorphism of represented von Neumann algebras spatial?", which is a noncommutative version of the fundamental problem of unitary equivalence for normal operators, is treated in detail in \cite{K1957}. Also see \cite{Dig} for a projection-based approach to the existence of spatial isomorphisms.
Having said that, equivalence classes of representations/represented algebras were first studied by Murray and von Neumann (\cite[Chapter III]{MvN4}), using the coupling constant for finite factors. The generalizations to coupling functions and arbitrary algebras were the motivations for the Griffin and Pallu de la Barri\`{e}re results featured in Section \ref{S:trace}. The space-free approach was notably developed by the Japanese school of the 1950's.
Modulo spatial isomorphism, the set of equivalence classes of representations of a fixed von Neumann algebra may not even be partially ordered. We mention the relevant example. Let $\mathcal{M}$ be a type $\text{II}_\infty$ factor with dimension function $D$ and fundamental group $\Gamma \notin \{\{1\},(0,\infty)\}$. (The existence of such an $\mathcal{M}$ remained in doubt until a breakthrough of Connes in 1980 (\cite{C}). The fundamental group of a $\text{II}_\infty$ factor can be defined as
$$\{\lambda \in (0,\infty) \mid \exists \alpha \in \text{Aut}(\mathcal{M}), \; D \circ \alpha = \lambda D\},$$
with the group operation being multiplication.) Kadison (\cite{K1955}) showed that for nonzero finite projections $p,q$, $L^2(\mathcal{M})p$ is spatially isomorphic to $L^2(\mathcal{M})q$ if and only if $\frac{D(p)}{D(q)} \in \Gamma$. Since $\Gamma \neq (0,\infty)$, we may find nonzero finite projections $p,p'$ with $\frac{D(p)}{D(p')} \notin \Gamma$. And $\Gamma \neq \{1\}$, so we may find spatial isomorphisms $L^2(\mathcal{M})q_1 \simeq L^2(\mathcal{M})p' \simeq L^2(\mathcal{M})q_2$ with $q_1 \lneqq p \lneqq q_2$. Therefore the spatial equivalence class of $L^2(\mathcal{M})p$ both dominates and is dominated by that of $L^2(\mathcal{M})p'$, yet the two are not equal.
At least for factors, this kind of pairing - $\text{II}_\infty$ algebra, $\text{II}_1$ commutant - is the only case where the two notions of equivalence differ. Not coincidentally, the only choice required for $T$, $D$, and $\widehat{T}$ which cannot be standardized is the normalization on the finite elements in a $\text{II}_\infty$ summand. (On a $\text{II}_\infty$ summand, one possible definition for ``normalization" is the inverse image of the identity, which is nothing but the equivalence class of the projection $p$ discussed in Remark \ref{R:trace}.)
|
{
"timestamp": "2005-03-31T20:16:20",
"yymm": "0503",
"arxiv_id": "math/0503747",
"language": "en",
"url": "https://arxiv.org/abs/math/0503747"
}
|
\section{Introduction and statement of main results}
We consider volume-preserving or symplectic diffeomorphisms on a
compact connected Riemannian manifold $M$. Let $\mbox{{\rm Diff}$_\mu^r(M)$}$ be the the set
of all $C^r$ diffeomorphisms preserving a smooth volume $\mu$ on
$M$. If $r$ is not an integer, $r= k + \alpha$ for some positive
integer $k$ and $0 < \alpha < 1$, it is understood that the functions
in $\mbox{{\rm Diff}$_\mu^r(M)$}$ are $C^k$ functions with $\alpha$-H\"older $k$-th
derivatives.
An invariant set $\Lambda \subset M$ is said to be {\em hyperbolic}\/
if there is a continuous splitting of $T_xM = E^s_x \oplus E^u_x$ for
every $x \in \Lambda$ and constants $C>0$, $\lambda >1$ such that
\begin{eqnarray}
df_x(E^s_x) &=& E^s_{f(x)} \; \mbox{ and } \; df_x(E^u_x) = E^u_{f(x)}
\nonumber \\ |df^n_x v^s_x| &\leq& C \lambda^{-n} |v^s|,\; \mbox{ for all } v^s
\in E^s_x, \; n \in \mathbb{N} \nonumber \\ |df^{-n}_x v^u_x| &\leq& C \lambda^{-n}
|v^u|,\; \mbox{ for all } v^u \in E^u_x, \; n \in \mathbb{N} \nonumber \end{eqnarray}
If the whole manifold $M$ is hyperbolic for some $f \in \mbox{{\rm Diff}$_\mu^r(M)$}$, then
$f$ is said to be {\em Anosov}. Not all manifolds can support
Anosov diffeomorphisms.
Typical examples of hyperbolic invariant sets are Cantor sets as in
Smale's horseshoe map. The following simple proposition explains why
this is the case.
\begin{prop}
Let $f \in \mbox{{\rm Diff}$_\mu^r(M)$}$, $r \geq 1$, be a volume-preserving
diffeomorphism on a compact manifold $M$. Let $\Lambda \subset M$ be a
closed hyperbolic invariant set. If the interior of $\Lambda$ is
non-empty, then $f$ is Anosov on $M$ and $\Lambda =M$.
\label{prop}
\end{prop}
This proposition and its simple proof, given in the next section, will
motivate our main result of this paper. The proof uses the fact that
the recurrent points are dense on the manifold. This is a consequence
of the volume-preserving property. Without the volume-preserving or
the dense recurrent points condition, the proposition is not true, we
refer to Fisher \cite{Fisher04} for a counter-example. Fisher also
give a proof of the above proposition. On the other hand, it is an
open problem whether there are any Anosov diffeomporphisms with
wandering domains.
A natural question one asks is whether there is any hyperbolic
invariant set with a positive measure for a volume-preserving
non-Anosov diffeomorphism. The answer is yes for $C^1$
diffeomorphisms, as Bowen's example of fat horseshoe shows
\cite{Bowen75}, see also Robinson \& Young \cite{RY80}. However, if
the map is assumed to be $C^{1 + \alpha}$ for some $\alpha >0$, then
the answer is no. This is the main result of this paper.
\begin{thm}
Let $f \in \mbox{{\rm Diff}$_\mu^r(M)$}$, $r>1$, be a volume preserving diffeomorphism on a
compact manifold $M$. Let $\Lambda \subset M$ be a closed hyperbolic
invariant set. If $\mu(\Lambda) >0$, then $f$ is Anosov on $M$ and
$\Lambda = M$.
\label{thm}
\end{thm}
It is not surprising that the map is required to be $C^{1+
\alpha}$. As various examples show, measure-theoretical properties
are often not respected by $C^1$ maps. Additional smoothness, even
though very little, guarantees certain regularities in measure.
Our proof uses a special type of measure density points different from
the Lebesgue density points. The density basis for our density points
are dynamically defined. It is similar to the juliennes defined by
Pugh \& Shub \cite{PS00} \cite{PS03}. But our case is much simpler.
Our method also provides a direct proof of the ergodicity of $C^{1 +
\alpha}$ volume-preserving Anosov diffeomorphisms, without using the
Hopf arguments or the Birkhoff ergodic theorem. However, we do use the
absolute continuity of stable and unstable foliations. This is given
in the last section of the paper.
Another result of this paper Lemma \ref{lemf} is of interest in its
own right. We showed that for a $C^{1+ \alpha}$ hyperbolic or
partially hyperbolic volume-preserving diffeomorphism, if a set is
invariant, then almost every point of the stable manifold (or unstable
manifold) of almost every point is in the invariant set. This is also
true for non-unifomly hyperbolic invariant set in Pesin theory. This
result can find its applications in various other problems.
\vspace{1ex} We are grateful to M. Viana for pointing out that
Theorem \ref{thm} was also proved, with a different method, by
Bochi and Viana \cite{BV03}. We are also grateful to F. Ledrappier
for showing us another possible proof of Theorem \ref{thm}.
\section{Proof of the proposition}
In this section, we give a simple proof of the Proposition \ref{prop}.
Let $U$ be the interior of the hyperbolic invariant set $\Lambda$. By
the assumption of the proposition, $U \neq \emptyset$. Clearly, $U$ is
invariant. We want to prove that the closure of $U$, $\bar{U}$ is the
whole manifold. We know that $\bar{U}$ is closed, it suffices to show
that $\bar{U}$ is also open.
For any $x \in \bar{U}$, there exists a sequence of points $x_n \in
U$, $n \in \mathbb{N}$, such that $x_n \rightarrow x$ as $n \rightarrow \infty$. As $f$ is volume
preserving, by Poincar\'e recurrence theorem, almost every point is
both forward and backward recurrent. Moreover, the set of periodic
points is dense in $U$, since $U$ is hyperbolic. We may choose
$\{x_n\}_{n \in \mathbb{N}}$ to be periodic points. Since $U$ is invariant and
each $x_n$ is an interior point in $U$, then $W^s(x_n)$ and $W^u(x_n)$
are in $U$ for all $n \in \mathbb{N}$. For each fixed $\delta >0$ small, let
$W^s_\delta(x)$ and $W^u_\delta(x)$ be, respectively, the local stable
manifold and unstable manifold of $x$. As $x_n \rightarrow x$, as $n \rightarrow \infty$,
we have that $W^s_\delta(x_n) \rightarrow W^s_\delta(x)$ and $W^u_\delta(x_n)
\rightarrow W^u_\delta(x)$ as $n \rightarrow \infty$. This implies that each point on
$W^s_\delta(x)$ or on $W^u_\delta(x)$ is also in the closure of
$U$. Let $y \in W^s_\delta(x)$ and $z \in W^u_\delta(x)$, the same
argument shows that $W^u_\delta(y)$ and $W^s_\delta(z)$ are both in
the closure of $U$. Consequently,
$$W^u_\delta(y) \cap W^s_\delta(z) \in \bar{U},$$ i.e., $\bar{U}$ has
the product structure. This implies that $x$ is in the interior of
$\bar{U}$. Consequently, the set $\bar{U}$ is open. Since $\bar{U}$ is
also closed and $M$ is connected, we have $\bar{U} = M$. i.e., $f$ is
hyperbolic on $M$.
This proves the proposition.
\section{Proof of the Theorem}
The proof of Theorem \ref{thm} uses a similar idea to the proof of
Proposition \ref{prop}, but the details are much more
complicated. Here the interior points are replaced by density
points. One may regard the density points as measure theoretical
interior points for a set with positive measure.
We need some preliminary results from standard smooth ergodic theory.
It is well-known that the stable and unstable foliations for a $C^1$
Anosov diffeomorphism may not be absolutely continuous. However, for
$C^{1 +\alpha}$ diffeomorphisms, these foliations are absolutely
continuous (Anosov \cite{Anosov67}). Moreover, the stable and unstable
foliations over a hyperbolic (even non-uniformly, cf Pesin
\cite{Pesin77}) invariant set are also absolutely continuous for $C^{1
+ \alpha}$ diffeomorphisms. In fact, the absolute continuity of the
foliations is proved by showing that the holonomy maps of these
foliations are absolutely continuous.
We also need some results on density basis and density points of a
measurable set. Let $A \subset \mathbb{R}^n$ be a measurable set with the
standard Lebesgue measure $m$. A point $x \in \mathbb{R}^n$ is said to be a
Lebesgue density point if $$\lim_{\epsilon \rightarrow 0} \frac{m(B(x, \epsilon) \cap
A)}{m(B(x, \epsilon))} =1,$$
where $B(x, \epsilon)$ is the $\epsilon$-ball
centered at $x$. Lebesgue density theorem states that almost every
point of $A$ is a density point for $A$.
To prove our theorem, we need a different definition of density point.
The Lebesgue density point is defined by a basis of $\epsilon$-balls. We
replace it by a dynamically defined basis. Let $\Lambda \subset M$ be
a hyperbolic invariant set, we first define a basis on the unstable
manifold for each point $\Lambda$.
For a fixed small real number $\delta >0$, let $W^u_\delta(x)$ be the
local unstable manifold of a point $x \in \Lambda$. Let $\mu_u$ and
$\mu_s$ respectively be the induced measures of the smooth volume form
$\mu$ on the unstable leaves and stable leaves. Let $n_u$ and $n_s$
respectively be the dimensions be the unstable and stable leaves. For
any positive integer $k$, let $B^u_k(x)$ be a subset of $W^u(x)$
defined by
$$B^u_k(x) = f^{-k}(W^u_\delta(f^k(x))). $$
Clearly, the cubes
$B^u_k(x)$, $k \in \mathbb{N}$ shrinks to the point $x$ as $k \rightarrow \infty$. We call
the collection of the sets $\{B^u_k(x) \; | \; k \in \mathbb{N}, \; x \in
\Lambda \}$ the unstable density basis.
Similarly, we can define the stable density basis $\{B^s_k(x) \; | \;
k \in \mathbb{N}, \; x \in \Lambda \}$, by defining
$$B^s_k(x) = f^{k}(W^u_\delta(f^{-k}(x))). $$ The density basis we
defined has infinite eccentricity.
A point $x \in \Lambda$ is said to be a dynamical density point, or
simply density point, on the unstable foliation if
$$ \lim_{k \rightarrow \infty} \frac{\mu_u(B^u_k(x) \cap
\Lambda)}{\mu_u(B^u_k(x))} =1.$$ Similarly, we can define the
dynamical density points on the stable foliation.
\begin{prop}
The set of points on $\Lambda$ that are both density points on the
stable foliations and unstable foliations has the full measure in
$\Lambda$.
\label{propd}
\end{prop}
Our definitions of density points can be regarded as simplified
versions of the juliennes density points defined by Pugh \& Shub
\cite{PS00} \cite{PS03}. The Pugh-Shub density points are defined by
the julienne density basis and they are much more complicated than
what we have here. The above proposition follows from the proof for
the Pugh-Shub's juliennes density point. The proof itself is
similar to the proof of the Lebesgue Density Theorem. The key
properties for the density basis are scaling and engulfing defined as
follows.
(a) Scaling: for any fixed $k \geq 0$, $m(B^u_n(x)) /m( B^u_{n+k}(x))$
is unformly bounded as $n \rightarrow \infty$.
(b) Engulfing: there is a unifom $L$ such that $$B^u_{n+L}(x) \cap
B^u_{n+L}(y) \neq \emptyset \; \Rightarrow \; B^u_{n+L}(x) \cup
B^u_{n+L}(y) \subset B^u_n(x). $$
These two properties can be easily verified.
We remark that our density points are defined on the stable and
unstable foliations, we freely used the fact that the stable and
unstable foliations are absolutely continuous.
Let $A$ be a subset of $\Lambda$ such that for any $x \in A$, $x$ is a
density point of $\Lambda$ on both stable foliation and unstable
foliation; and $x$ is a recurrent point, both forward and backward. By
Poincar\'e recurrence theorem, Proposition \ref{propd} and the
absolute continuity of the foliations, the set $A$ has the full
measure in $\Lambda$.
The following is the main lemma in proving our theorem.
\begin{lem}
Assume that $f$ is a $C^{1+\alpha}$ volume-preserving
diffeomorphism, for some positive number $\alpha >0$. Fix $x \in A$
and a positive number $\delta >0$. Then for any $\epsilon >0$, there
exists a positive integer $k_0$, depending on $x$ and $\epsilon$, such
that for $k \geq k_0$, $$\mu_s(W^s_\delta(f^{-k}(x)) \cap A) \geq
(1-\epsilon) \mu_s(W^s_\delta(f^{-k}(x)))$$
and
$$\mu_u(W^u_\delta(f^k(x)) \cap A) \geq (1-\epsilon)
\mu_u(W^u_\delta(f^k(x)))$$
i.e., for sufficiently large $k$, the
set $A$ has a very high density in $W^s_\delta(f^{-k}(x))$ and
$W^u_\delta(f^k(x))$.
\end{lem}
\noindent {\it Proof of the lemma}: We first prove the lemma for the
unstable foliation. Let $n_u$ be the dimension of the leaves of the
foliation, local unstable manifold $W^u_\delta(x)$ can be identified
with a cube in $E^u_x = \mathbb{R}^{n_u}$ by the exponential map from $E^u_x$
to $W^u(x)$. Since the leaves of the unstable foliation is smooth,
the conditional measure $\mu_u$ are smoothly equivalent to the
standard Lebesgue measure $m$ on $\mathbb{R}^{n_u}$. i.e., for any point $x
\in \Lambda$, there is a smooth function $g_u(y)$ defined for $y \in
W^u_\delta(x)$ on the local unstable manifold, uniformly bounded away
from zero and infinity, such that
$$\mu_u(E) = \int_{E} g_u dm,$$
where $E$ is a measurable set in
$W^u_\delta(x)$ and $m$ is the standard Lebesgue measure in
$\mathbb{R}^{n_u}$.
For any positive integer $k$, we want to estimate the measure of the
set $W^u_\delta(f^k(x)) \cap A$. Let $B^k_0 =
f^{-k}(W^u_\delta(f^k(x)))$, obviously $B^k_0 \subset W^u_\delta(x)$.
In fact, $B^k_0$ is the set $B^u_k(x)$ in our definition of density
basis. We iterate $B^k_0$ under $f$ and obtain a sequence of sets
$B^k_i = f^i(B^k_0)$, for $i=1, 2, \ldots, k$. The last set in the
sequence is $B^k_k = W^u_\delta(f^k(x))$.
Let $\eta_k = 1- \mu_u(B^k_0 \cap A) / \mu_u(B^k_0)$, then $0 \le \eta_k
\leq 1$. As $x$ is a density point on the unstable foliation,
$$\lim_{k \rightarrow \infty} \frac{\mu_u(B^k_0 \cap A)}{\mu_u(B^k_0)} =1,$$
The
number $\eta_k$ is small for large $k$, and $\lim_{k \rightarrow \infty} \eta_k
=0$. Since $f$ is $C^{1 + \alpha}$, there exists a constant $C_1 > 0$
such that $||df_y - df_z|| \leq C_1 |y-z|^\alpha$. Here we abuse the
notation a little by writing $|y-z|$ as the distance between $y$ and
$z$.
Let $\rho^k_i$ be the maximum distance from $f^i(x)$ to the boundary
of $B^k_i$, i.e., $$\rho^k_i = \max_{y \in B^k_i} \{ d(f^i(x), y) \}.
$$
For any $y \in B^k_0$, $||df_y - df_x|| \leq C_1(\rho^k_0)^\alpha$.
In general, for any $y \in B^k_i$, $||df_y - df_{f^{i}(x)}|| \leq C_1
(\rho^k_i)^\alpha$.
Let $J_u(y)=|\det(df_y|_{E^u_y})|$ be the Jacobian of the map $f$ at $y$
restricted on the unstable manifold of $y$. Then $|J_u(x)-J_u(y)| \leq
C_2 |x-y|^\alpha$, for some positive constant $C_2 >0$.
Let $D_0= B^k_0 \backslash A$ and $D_i = B^k_i \backslash A$, for $i
=1, 2, \ldots, k$. These are the complements of $A$ in $B^k_i$. By the
definition of $\eta_k$, $\mu_u(D_0) = \eta_k \mu_u(B^k_0)$. We need to
estimate the measure of $D_1$. For any set $E \subset B^k_0$,
$$\mu_u(f(E)) = \int_{f(E)} g_u dm =\int_{E} J_u (g_u\cdot f) dm =
\int_{E} J_u (g_u\cdot f) g_u^{-1} d\mu_u$$
Since the functions $g_u$ and $g_u^{-1}$ are smooth on any
unstable manifold, the integrand
in the above integral is $C^{1+\alpha}$, there is a constant $C_3 >0$
such that $$|J_u(y) (g_u\cdot f)(y) g_u^{-1}(y) - J_u(x) (g_u\cdot f)(x)
g_u^{-1}(x)| \leq C_3 |x - y|^\alpha , $$ for all $x\in \Lambda$, $y \in
W^u_\delta(x)$. Therefore,
$$ |\mu_u(f(E)) - (J_u(x) (g_u\cdot f)(x)
g_u^{-1}(x)) \mu_u(E)| \leq C_3 (\rho^k_0)^\alpha \mu_u(E).$$
Consequently,
$$\mu_u(D_1) \leq (J_u(x) (g_u\cdot f)(x)
g_u^{-1}(x)) \mu_u(D_0) + C_3 (\rho^k_0)^\alpha \mu_u(D_0)$$
and $$\mu_u(B^k_1\backslash D_1) \geq (J_u(x) (g_u\cdot f)(x)
g_u^{-1}(x)) \mu_u(B^k_0 \backslash D_0) - C_3 (\rho^k_0)^\alpha
\mu_u(B^k_0\backslash D_0)$$
and therefore
$$\mu_u(D_1) \leq \eta_k \frac{(1 + C_3
(\rho^k_0)^\alpha)}{(1 - C_3
(\rho^k_0)^\alpha)} \mu_u(B^k_1).$$ By induction on $i$, we have
$$\mu_u(D_k) \leq \eta_k \frac{(1 + C_3
(\rho^k_0)^\alpha)}{(1 - C_3
(\rho^k_0)^\alpha)}\frac{(1 + C_3
(\rho^k_1)^\alpha)}{(1 - C_3
(\rho^k_1)^\alpha)} \cdots \frac{(1 + C_3
(\rho^k_{k-1})^\alpha)}{(1 - C_3
(\rho^k_{k-1})^\alpha)}\mu_u(B^k_k). $$
The map $df: T_{\Lambda}M \rightarrow T_{\Lambda}M$ uniformly expands vectors
on the unstable splitting. That uniform expansion extends to local
unstable manifolds $W^u_\delta(x)$, $x \in \Lambda$ if $\delta$ is
chosen small enough. There exist positive real numbers $C_4 > C >0$
and $\lambda> \lambda_1 >1$ (here $C$ and $\lambda$ are the same as
those in the definition of the hyperbolic invariant set) such that
$\rho^k_{k-1} \leq C_4 \lambda_1^{-1} \delta$ and $\rho^k_{i} \leq C_4
\lambda_1^{-(k-i)} \delta$, for $i=0, 1, \ldots, k-1$. This implies
that \begin{eqnarray} \mu_u(D_k) & \leq & \eta_k \frac{(1 + C_3
(C_4 \lambda_1^{-k}\delta)^\alpha)}{(1 - C_3
(C_4 \lambda_1^{-k}\delta)^\alpha)}\frac{(1 + C_3
(C_4 \lambda_1^{-k+1}\delta)^\alpha)}{(1 - C_3
(C_4 \lambda_1^{-k+1}\delta)^\alpha)} \nonumber \\
&& \cdots \frac{(1 + C_3
(C_4 \lambda_1^{-1}\delta)^\alpha)}{(1 - C_3
(C_4 \lambda_1^{-1}\delta)^\alpha)}\mu_u(B^k_k) \nonumber \\
&<& \eta_k \mu_u(B^k_k) \prod_{i=1}^\infty (1+
C_3(C_4\lambda_1^{-i}\delta)^\alpha )/ \prod_{i=1}^\infty
(1- C_3(C_4\lambda_1^{-i}\delta)^\alpha ) \nonumber\end{eqnarray}
Since $\lambda > 1$, then $\lambda^\alpha >1$, the infinite products
converge. We have
$$\mu_u(D_k) < C_5 \eta_k \mu_u(B^k_k).$$ for some constant $C_5 >0$.
Choose a positive integer $k_0$ such that for $k \geq k_0$, $\eta_k <
\epsilon /C_5$, then we have
$$\mu_u(W^u_\delta(f^k(x))
\cap A) \geq (1-\epsilon) \mu_u(W^u_\delta(f^k(x))),$$
for all $k \geq k_0$.
This proves the statement of the lemma on the unstable foliation. The
part on the stable foliation can be proved in the same way by
considering $f^{-1}$.
This proves the lemma. \vs{1ex}
We return to the proof of the theorem. Let $E$ be the closure of $A$
in $M$. We claim that if $y \in E$, then $W^s_\delta(y) \subset E$ and
$W^u_\delta(y) \subset E$.
Suppose that this is not true. i.e., there is a point $z \in W^u(y)$
and a small $\epsilon_1$-ball around $z$, $B(z, \epsilon_1) \subset M$ such that
$B(z, \epsilon_1) \cap A = \emptyset$. Consequently, there are constants
$\epsilon_2>0$, depending on $\epsilon_1$, and $C_6 >0$, independent of $\epsilon_1$,
such that if $x \in \Lambda$, $|x-y| \leq \epsilon_2$, then
$$\mu_u(W^u_\delta(f^k(x)) \cap A) < (1- C_6 \epsilon^{n_u}_1)
\mu_u(W^u_\delta(f^k(x))),$$ for all $k \in \mathbb{N}$.
On the other hand, since $y \in E$, there is a sequence of points $x_i
\in A$, $i \in \mathbb{N}$ such that $x_i \rightarrow y$ and $i \rightarrow \infty$. For the above
$\epsilon_1$, there is positive integer $i_0$ such that if $i \geq i_0$,
$d(x_i, y) \leq \epsilon_1/3$. For any fixed $\epsilon >0$, by the lemma above
and the recurrence of $x_i$, there is a positive integer $k >0$ such
that $d(x_i, f^k(x_i)) \leq \epsilon_1 /3$ and
$$\mu_u(W^u_\delta(f^k(x_i)) \cap A) \geq (1-\epsilon)
\mu_u(W^u_\delta(f^k(x_i))).$$ Since $d(y, f^k(x_i)) \leq
\frac{2\epsilon_2}{3} < \epsilon_2$, choosing $\epsilon = C_6 \epsilon^{n_u}_1$ leads to a
contradiction. This contradiction show that if $y \in E$, then
$W^u_\delta(y) \subset E$. Similarly by considering $f^{-1}$, we have
$W^s_\delta(y) \subset E$.
To conclude our proof, for any $y \in E$, $W^u_\delta(y) \subset E$
and $$V = \bigcup_{z \in W^u_\delta(y)} W^s_\delta(z) \subset E.$$
Since $V$ is hyperbolic, $y$ is in the interior of $V$. This implies
that $E$ is an open set. But $V$ is also closed and non-empty. The
connectness of $M$ implies that $E=M$. This implies that $f$ is
Anosov.
Finally, the reason that $\Lambda=M$ in the first place is that $f$ is
ergodic, any invariant set with positive measure must have full
measure and its closure must be the whole manifold.
This proves the theorem.
\section{Ergodicity of volume preserving Anosov diffeomorphisms}
In this section, we give a direct proof of the ergodicity of $C^{1 +
\alpha}$ volume-preserving Anosov diffeomorphisms, without using the
usual Hopf arguments or the Birkhoff ergodic theorem.
We first prove the following lemma.
\begin{lem}
Let $f \in \mbox{{\rm Diff}$_\mu^r(M)$}$, $r>1$, be a volume preserving diffeomorphism on a
compact manifold $M$. Let $\Lambda \subset M$ be a
hyperbolic invariant set (not neccessarily closed). If $\mu(\Lambda)
>0$, then for a.e. $x \in \Lambda$, $W^s(x) \subset \Lambda$ modulus a
$\mu_s$-measure zero set in $W^s(x)$ and $W^u(x) \subset \Lambda$ modulus a
$\mu_u$-measure zero set in $W^u(x)$, where $\mu_s$ and $\mu_u$ are
respectively the induced measures of $\mu$ on the stable and unstable
foliations.
\label{lemf}
\end{lem}
\noindent {\it Proof}: For any fixed positive integer $k$
and positive number $\eta>0$, let $\Lambda_{(\eta, k)} \subset \Lambda$
be the set such that $$\frac{\mu_u(B^u_i(x) \cap
\Lambda)}{\mu_u(B^u_i(x))} > (1 -\eta), \; \mbox{ for all } i \geq k$$
Since almost every point of $\Lambda$ is a density point of $\Lambda$,
for any $\eta>0$, $$\lim_{k\rightarrow \infty} \mu(\Lambda_{(\eta, k)}) =
\mu(\Lambda).$$
By Poincar\'e recurrence theorem, for a.e. $x \in \Lambda_{(\eta,
k)}$, there exists a sequence of integers $n_i \rightarrow \infty$ such that
$f^{-n_i}(x) \in \Lambda_{(\eta, k)}$. By the distortion estimates
from the last section, $$\mu_u(W^u_\delta(x) \cap \Lambda) \geq
(1-C_5\eta) \mu_u(W^u_\delta(x)),$$ where $C_5$ and $\delta$ are fixed
constants from the last section. Since the above estimate is independent
of $k$, it must hold for almost all $x \in \cup_{k=1}^\infty \Lambda_{(\eta,
k)}$. Since $\mu(\cup_{k=1}^\infty \Lambda_{(\eta, k)}) = \mu(\Lambda)$
for any fixed $\eta >0$, this implies that for a.e. $x \in \Lambda$,
$$\mu_u(W^u_\delta(x) \cap \Lambda) \geq
(1-C_5\eta) \mu_u(W^u_\delta(x)).$$
This is true for all $\eta >0$, therefore, for a.e. $x \in \Lambda$,
$$\mu_u(W^u_\delta(x) \cap \Lambda) = \mu_u(W^u_\delta(x)).$$
This proves the lemma for the unstable foliations. The results on the
stable foliation can be proved in the same way by considering
$f^{-1}$.
This proves the lemma.
\vs{1ex}
Now we can prove the following ergodicity theorem.
\begin{thm}
Let $f \in \mbox{{\rm Diff}$_\mu^r(M)$}$, $r>1$, be an Anosov volume preserving diffeomorphism
on a compact manifold $M$. Then $f$ is ergodic.
\end{thm}
\noindent {\it Proof}: Let $\Lambda \subset M$ be an invariant set and
$\mu(\Lambda) >0$. Let $\Lambda_s \subset \Lambda$ be the set such
that for each $x \in \Lambda_s$, a.e. ($\mu_s$) $y$ in $W^s_\delta(x)$
is in $\Lambda$. The above lemma shows that $\mu(\Lambda_s) = \mu
(\Lambda)$. Also by the above lemma, for a.e. $z \in \Lambda$,
a.e. ($\mu_u$) $x$ in $W^u_\delta(z)$ is in $\Lambda_s$. By the
absolute continuity of the stable and unstable foliations, the set
$$\bigcup_{y \in W^u_\delta(z)} W^s_\delta(y)$$ has the full measure in
the $\delta$ neighborhood of $z$ and therefore $\Lambda$ has the full
measure in a $\delta$ neighborhood of $z$ for a.e. $z \in
\Lambda$. This implies that $\Lambda$ has the full measure in
$M$. Since $\Lambda$ is an arbitrary positive measure set, this
implies ergodicity.
This proves the theorem.
Finally we remark that Lemma \ref{lemf} is also true for partially
hyperbolic invariant and non-uniformly hyperbolic invariant sets (sets with
non-zero Liapunov exponents). The proof is exactly the same.
|
{
"timestamp": "2005-08-26T22:13:29",
"yymm": "0503",
"arxiv_id": "math/0503437",
"language": "en",
"url": "https://arxiv.org/abs/math/0503437"
}
|
\chapter{Analysis Methods for pions, kaons, proton and anti-proton}
\chapter{Analysis Methods}
\label{chp:analysis}
\section{Trigger}
The detector used for these studies was the Solenoidal Tracker at
RHIC (STAR). The main tracking device is the Time Projection
Chamber (TPC) which provides momentum information and particle
identification for charged particles up to $p_{T}\sim1.1$ GeV/c by
measuring their ionization energy loss ({\it dE/dx})~\cite{tpc}.
Detailed descriptions of the TPC and d+Au run conditions have been
presented in Ref.~\cite{stardau,tpc}. A prototype time-of-flight detector
(TOFr) based on multi-gap resistive plate chambers
(MRPC)~\cite{startof} was installed in STAR for the d+Au and p+p
runs. It extends particle identification up to $p_{T}\sim3$ GeV/c
for $p$ and $\bar{p}$.
TOFr covers $\pi/30$ in azimuth and $-1\!<\!\eta\!<\!0$ in
pseudorapidity at a radius of $\sim220$ cm. It contains 28 MRPC
modules which were partially instrumented during the 2003 run.
Since the acceptance of TOFr is small, a special trigger selected
events with a valid pVPD coincidence and at least one TOFr hit. A
total of 1.89 million and 1.08 million events were used for the
analysis from TOFr triggered d+Au and non-singly diffractive (NSD)
p+p collisions, representing an integrated luminosity of about 40
$\mathrm{{\mu}b}^{-1}$ and 30 $\mathrm{nb}^{-1}$, respectively.
Minimum-bias d+Au and p+p collisions that did not require pVPD and
TOFr hits were also used to study the trigger bias and enhancement,
and the TOFr efficiency and acceptance. The d+Au minimum-bias
trigger required an equivalent energy deposition of about 15 GeV in
the Zero Degree Calorimeter in the Au beam
direction~\cite{stardau}. The trigger efficiency was determined to
be $95\pm3\%$. Minimum-bias p+p events were triggered by the
coincidence of two beam-beam counters (BBC) covering $3.3<
|\eta|<5.0$~\cite{starhighpt}. The NSD cross section was measured to
be $30.0\pm3.5$ mb by a van der Meer scan and PYTHIA~\cite{pythia}
simulation of the BBC acceptance~\cite{starhighpt}.
\subsection{Centrality tagging}
Centrality tagging of d+Au collisions was based on the charged
particle multiplicity in $-3.8<\eta<-2.8$, measured by the Forward
Time Projection Chamber in the Au beam
direction~\cite{stardau,ftpc}. The TOFr triggered d+Au events were
divided into three centralities: most central $20\%$, $20-40\%$
and $40-\sim100\%$ of the hadronic cross section. The average
number of binary collisions $\langle N_{bin}\rangle$ for each
centrality class and for the combined minimum-bias event sample is
derived from Glauber model calculations and listed in
Table~\ref{centrality}.
Table~\ref{centrality} also lists the uncorrected FTPC east
reference multiplicity ranges for centrality definitions.
\begin{table}[h]
\centering
\begin{tabular}{|c|c|c|c|}
\hline
Centrality Bin & Uncorr. FTPCRefMult Range & Uncorr. $N_{charge}$ & $N_{bin}$ \\ \hline
M.B. & & 10.2 & $7.5\pm0.4$ \\ \hline
0\%-20\% & FTPCRefMult $\geq$ 17 & 17.58 & $15.0\pm1.1$ \\ \hline
20\%-40\% & 10 $\leq$ FTPCRefMult $<$ 17 & 12.55 & $10.2\pm1.0$ \\ \hline
40\%-100\% & 0 $\leq$ FTPCRefMult $<$ 10 & 6.17 & $4.0\pm0.3$ \\ \hline
\end{tabular}
\caption{Centrality definitions for different uncorrected FTPC
east reference multiplicity ranges. Uncorrected $N_{charge}$
stands for the average value of uncorrected reference multiplicity
in certain centrality bin. The fourth column represents the number
of binary collisons $\langle N_{bin}\rangle$ calculated from
Glauber model.} \label{centrality}
\end{table}
\subsection{Trigger bias study}
Since we set up a special trigger which selected events with a
valid pVPD coincidence and at least one TOFr hit, the study of
$p_{T}$ dependence of trigger bias is necessary.
Figure~\ref{PtRatioRealTOFAcceptance} shows there is negligible
trigger bias on $p_{T}$ dependence at $p_{T}>$ 0.3 GeV/c from
simulation. In this figure, pVPD means that pVPD is required to
fire in minimum-bias collisions. TOF means that TOFr is required
to fire in minimum-bias collisions, and pVPD $\&$ TOF means that
pVPD and TOFr are required to fire in minimum-bias collisions.
From this figure, if we required pVPD and TOFr to fire, we can see
the ratio is flat with $p_{T}$ when $p_{T}$ is larger than 0.3
GeV/c by comparison through the $p_{T}$ distribution in
minimum-bias collisions. That means the trigger bias for $p_T$
distribution is negligible at $p_{T}>$ 0.3 GeV/c.
\begin{figure}[h]
\centering
\includegraphics[height=18pc,width=24pc]{PtRatioRealTOFAcceptanceMod.eps}
\caption{The $p_{T}$ dependence plot of the trigger bias.}
\label{PtRatioRealTOFAcceptance}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[height=18pc,width=24pc]{NchbiasNew.eps}
\caption{The enhancement factor and $\langle N_{ch}\rangle$ bias
in minimum-bias and centrality selected d+Au collisions. }
\label{NchbiasNew}
\end{figure}
Minimum-bias d+Au and p+p collisions are used to study the trigger
bias and enhancement. Figure~\ref{NchbiasNew} shows the trigger
bias and enhancement in d+Au minimum-bias collisions and three
centrality bins. In this figure, TOFr means that TOFr is required
to fire in minimum-bias events. pVPD means that TOFr and pVPD are
required to fire in minimum-bias events. Minbias means the
minimum-bias triggered events. For enhancement study, TOFr/pVPD is
the ratio of the number of events in which TOFr is required to
fire over the number of events in which TOFr and pVPD are required
to fire, and Minbias/pVPD is the ratio of the number of
minimum-bias triggered events over the number of events in which
TOFr and pVPD are required to fire. The enhancement factor for
TOFr is (Minbias/pVPD)/(TOFr/pVPD). For example, in minimum-bias
collision, Minbias/pVPD is equal to 28.7, while TOFr/pVPD is 2.87,
so in minimum-bias collisions, the enhancement of TOFr trigger is
10. For $\langle N_{ch}\rangle$ bias study, TOFr/pVPD is the ratio
of $\langle N_{ch}\rangle$ in the events where TOFr is required to
fire over the $\langle N_{ch}\rangle$ in the events where TOFr and
pVPD are required to fire. Since in our triggered events, TOFr and
pVPD are required to fire, TOFr/pVPD is our $\langle
N_{ch}\rangle$ bias factor. The curves in this figure show the
charged particle multiplicity at mid-rapidity in TOFr events and
in TOFr and pVPD events individually. Table~\ref{triggerbiastable}
lists the enhancement factor and trigger bias in minimum-bias,
centrality selected d+Au collisions and minimum-bias p+p
collisions.
\begin{table}[h]
\begin{tabular}{|c|c|c|c|}
\hline
Centrality Bin & TOFr triggered events & enhancement factor & $\langle N_{ch}\rangle$ bias \\ \hline
0\%-100\% & 1.80 M & 10.0 & 1.02 \\ \hline
0\%-20\% & 0.523 M & 5.75 & 1.04 \\ \hline
20\%-40\% & 0.500 M & 8.03 & 1.03 \\ \hline
40\%-100\% & 0.479 M & 15.8 & 0.965 \\ \hline
p+p & 0.995 M& 37.4 & 1.19 \\ \hline
\end{tabular}
\caption{Trigger bias study. The $\langle N_{ch}\rangle$ bias and
enhancement factor in minimum-bias, centrality selected d+Au
collisions and minimum-bias p+p collisions.}
\label{triggerbiastable}
\end{table}
\section{Track selection and calibration}
The TPC and TOFr are two independent systems. In the analysis,
hits from particles traversing the TPC were reconstructed as
tracks with well defined geometry, momentum, and {\it dE/dx}
~\cite{tpc}.
The particle trajectory was then extended outward to the TOFr
detector plane. The pad with the largest signal within one pad
distance to the projected point was associated with the track for
further time-of-flight and velocity ($\beta$) calculations.
\subsection{Calibration}
\subsubsection{pVPD calibration}
For TOFr, we use pVPD as our start-timing detector. In d+Au and
p+p collisions, at least one east pVPD and one west pVPD were
required to fire. In d+Au collisions, to calibrate east pVPD, we
required 3 east pVPD to fire; to calibrate west pVPD, we required
3 west pVPD to fire. In p+p collisions, to calibrate east pVPD, we
required 2 east pVPD to fire; to calibrate west pVPD, we required
2 west pVPD to fire. Let's take the east pVPD calibration in d+Au
collisions as an example. The label for 3 pVPD are pVPD1, pVPD2,
pVPD3, the adc and tdc value for pVPD1 are $a1$, $t1$, and the
slewing correction function is $f1$; the adc and tdc value for
pVPD2 are $a2$, $t2$, and the slewing correction function is $f2$;
the adc and tdc value for pVPD3 are $a3$, $t3$, and the slewing
correction function is $f3$. We use $t1-((t2-f2)+(t3-f3))/2$ vs
$a1$ to get the slewing correction for pVPD1; use
$t2-((t3-f3)+(t1-f1))/2$ vs $a2$ to get the slewing correction for
pVPD2; use $t3-((t1-f1)+(t2-f2))/2$ vs $a3$ to get the slewing
correction for pVPD3. At the beginning, $f1=f2=f3=0$, we got 3
curves of $t1-((t2-f2)+(t3-f3))/2$ vs $a1$,
$t2-((t3-f3)+(t1-f1))/2$ vs $a2$ and $t3-((t1-f1)+(t2-f2))/2$ vs
$a3$. The 3 curves corresponded to the 3 slewing functions $f(a1),
f(a2), f(a3)$; For the second step, $f1=f(a1), f2=f(a2),
f3=f(a3)$, also plot $t1-((t2-f2)+(t3-f3))/2$ vs $a1$,
$t2-((t3-f3)+(t1-f1))/2$ vs $a2$ and $t3-((t1-f1)+(t2-f2))/2$ vs
$a3$. And we got the new three slewing curves $f'(a1), f'(a2),
f'(a3)$. For the third step, $f1=f'(a1), f2=f'(a2), f3=f'(a3)$,
also plot $t1-((t2-f2)+(t3-f3))/2$ vs $a1$,
$t2-((t3-f3)+(t1-f1))/2$ vs $a2$ and $t3-((t1-f1)+(t2-f2))/2$ vs
$a3$. And we got another new three slewing curves $f''(a1),
f''(a2), f''(a3)$. And so on and so forth till the resolution of
$t1-f1-((t2-f2)+(t3-f3))/2, t2-f2-((t3-f3)+(t1-f1))/2$ and
$t3-f3-((t1-f1)+(t2-f2))/2$ converged. The looping method is to
subtract the correlation of different pVPD tubes in the same
direction. The function for the slewing correction we use is
$y=par[0]+par[1]/\sqrt{x}+par[2]/x+par[3]\times{x}$. In
Figure~\ref{pvpdslewingplotforthesis}, the left plot shows the
pVPD2 slewing plot and the right plot shows that the timing is
independent on the ADC value after the slewing correction.
\begin{figure}[h]
\begin{minipage}[t]{80mm}
\includegraphics[height=13pc,width=18pc]{pvpdslewingplot.eps}
\end{minipage}
\hspace{\fill}
\begin{minipage}[t]{80mm}
\includegraphics[height=13pc,width=18pc]{pvpdslewingplot1.eps}
\end{minipage}
\caption{pVPD slewing correction.}
\label{pvpdslewingplotforthesis}
\end{figure}
After the slewing correction, we got the corrected timing of east
pVPD and west pVPD. For each side, the timing difference should be
shifted to zero. That's to say the mean value in the distribution
of $t1-f1-(t2-f2)$ and $t1-f1-(t3-f3)$ were shifted to zero. Also
we need to correct for the effect caused by the different numbers
of fired pVPD in different events. What we did was shifting the
mean value of the distribution of
($\sum{te})/Ne-(\sum{tw})/Nw-2.\times{Vz/c}$ to zero, where the
$\sum{te}$, $\sum{tw}$ means the sum of the corrected timing of
east fired pVPD and west fired pVPD respectively, $Ne, Nw$ means
the number of east fired pVPD and west fired pVPD, $Vz$ is the $z$
value of primary vertex of the event, and $c$ is the light
velocity.
\subsubsection{TOFr calibration}
After the slewing correction for pVPD, we use this variable as our
start timing:
\begin{equation}
T_{start}=\frac{{\sum_{i=1}^{Ne}{te}}+{\sum_{i=1}^{Nw}{tw}}-(Ne-Nw)\times{Vz}/c}{Ne+Nw}
\end{equation}
\begin{figure}[h]
\centering
\includegraphics[height=18pc,width=24pc]{dedxplottmp.eps}
\caption{dE/dx vs $p$ plot from d+Au collisions. The line
represents that $dE/dx=0.028\times{10^{-4}}$ GeV/cm in this
momentum range $0.3<p<0.6$ GeV/c.} \label{dAudedxplot}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[height=18pc,width=24pc]{slewingplot.eps}
\caption{The slewing correction.} \label{slewingplot}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[height=18pc,width=24pc]{ZFit_forthesis.eps}
\caption{The z position correction.} \label{ZFit_forthesis}
\end{figure}
\begin{figure}[h]
\begin{minipage}[t]{80mm}
\includegraphics[height=13pc,width=18pc]{dAutiming.eps}
\end{minipage}
\hspace{\fill}
\begin{minipage}[t]{80mm}
\includegraphics[height=13pc,width=18pc]{pptiming.eps}
\end{minipage}
\caption{The overall timing resolution after the calibration.}
\label{timeresolution}
\end{figure}
The difference between TOFr timing $T_{tofr}$ and start timing
$T_{start}$ is our time of flight $tof=T_{tofr}-T_{start}$. To
calibrate the $tof$, the pure pion sample was chosen by selecting
the particle energy loss $dE/dx$ in TPC at
$dE/dx<0.028\times{10^{-4}}$ GeV/cm in the momentum range
$0.3<p<0.6$ GeV/c. Figure~\ref{dAudedxplot} shows dE/dx vs $p$
plot from d+Au collisions. Firstly the so called $T_{0}$
correction was done due to the different cable lengths for
different read-out channels, which was done by shifting the mean
value of the distribution of $tof-T_{\pi}$ to zero channel by
channel, where $T_{\pi}$ is the calculation timing assuming the
particle was pion particle. Secondly, the slewing correction due
to correlation between timing and signal amplitude of the
electronics was done by getting the curve of $tof'-T_{\pi}$ vs
$adc$ for each channel, where the $tof'$ was the time of flight
after the $T_{0}$ correction and $adc$ was the ADC value of TOFr.
The slewing curve is like the plot shown in
Figure~\ref{slewingplot}. The function of the slewing correction
is
$y=par[0]+par[1]/\sqrt{x}+par[2]/x+par[3]/\sqrt{x}/x+par[4]/x/x$.
The z position correction was also done since the different hit
positions on the read-out strip will generate different
transmission timing. This was done by getting the function of
$tof''-T_{\pi}$ versus $Z_{local}$, where the $tof''$ is the time
of flight after the $T_{0}$ and slewing correction, and
$Z_{local}$ is the the hit local z position of the TOFr. The
function for the z position correction is
$y=\sum_{i=0}^{7}{(par[i]\times{x^{i}})}$. The z position
correction for all the channels is shown in
Figure~\ref{ZFit_forthesis}. After the z position was done, the
calibration for TOFr was finished. The overall resolution of TOFr
was 120 ps and 160 ps in d+Au and p+p collisions respectively,
where the effective timing resolution of the pVPDs was 85 ps and
140 ps, respectively. Figure~\ref{timeresolution} shows the
overall resolution of TOFr in d+Au and p+p collisions.
\section{Raw yield}
\begin{figure}[h]
\centering
\includegraphics[height=18pc,width=24pc]{tofr_beta_p_prplot0910.eps}
\caption{$1/\beta$ vs. momentum for $\pi^{\pm}$, $K^{\pm}$, and
$p(\bar{p})$ from 200 GeV d+Au collisions. Separations between
pions and kaons, kaons and protons are achieved up to
$p_{T}\simeq1.6$ and $3.0$ GeV/c, respectively. The insert shows
$m^{2}=p^{2}(1/\beta^{2}-1)$ for $1.2<p_{T}<1.4$ GeV/c. Clear
separation of $\pi$, $K$ and $p$ is seen.} \label{beta}
\end{figure}
From the timing information $t$ from TOFr after the calibration
and the pathlength $L$ from TPC, the velocity $\beta$ of the
particle can be easily got by $\beta=L/t/c$. Figure~\ref{beta}
shows $1/\beta$ from TOFr measurement as a function of momentum
($p$) calculated from TPC tracking in TOFr triggered d+Au
collisions. The raw yields of $\pi^{\pm}$, $K^{\pm}$, $p$ and
$\bar{p}$ are obtained from Gaussian fits to the distributions in
$m^{2}=p^{2}(1/\beta^{2}-1)$ in each $p_{T}$ bin.
\subsection{$\pi$ raw yield extraction}
For $\pi^{\pm}$, the rapidity range is $-0.5<y_{\pi}<0.$. After
$|N_{\sigma\pi}|<2$ was required, the mass squared
$m^{2}=p^{2}(1/\beta^{2}-1)$ distributions in different $p_{T}$
bin in d+Au minimum-bias collisions are shown is
Figure~\ref{pionplusrawyieldplot} and
Figure~\ref{pionminusrawyieldplot}. At $p_{T}<0.8$ GeV/c, the
single Gaussian function was used to fit the distribution of
$m^{2}$ to get the raw yield. At the same time, the counting
result by counting the track number at the range $-0.1<m^{2}<0.1$
$(GeV/c^{2})^2$ was also used to compare with the raw yield from
the fitting method. The difference between them was found in one
sigma range. The raw yield we quote is from the fitting method. At
$p_{T}>0.8$ GeV/c, the double Gaussian function was used to
extract the raw yield. The raw signals in each $P_{T}$ bin are
shown in Table~\ref{pionplustable} and Table~\ref{pionminustable}.
Also shown in the tables are those in centrality selected d+Au
collisions and minimum-bias p+p collisions.
\subsection{$K$ raw yield extraction}
For $K^{\pm}$, the rapidity range is $-0.5<y_{K}<0$. After
$|N_{\sigma K}|<2$ was required, the mass squared
$m^{2}=p^{2}(1/\beta^{2}-1)$ distributions in different $p_{T}$
bin in d+Au minimum-bias collisions are shown is
Figure~\ref{kaonplusrawyieldplot} and
Figure~\ref{kaonminusrawyieldplot}. At $p_{T}<0.8$ GeV/c, the
single Gaussian function was used to fit the distribution of
$m^{2}$ to get the raw yield. At the same time, the counting
result by counting the track number at the range $0.16<m^{2}<0.36$
$(GeV/c^{2})^2$ was also used to compare with the raw yield from
the fitting method. The difference between them was found in one
sigma range. The raw yield we quote is from the fitting method. At
$p_{T}>0.8$ GeV/c, the double Gaussian function was used to
extract the raw yield. The raw signals in each $P_{T}$ bin are
shown in Table~\ref{kaonplustable} and Table~\ref{kaonminustable}.
Also shown in the tables are those in centrality selected d+Au
collisions and minimum-bias p+p collisions.
\subsection{$p$ and $\bar{p}$ raw yield extraction}
\begin{figure}[h]
\centering
\includegraphics[height=18pc,width=24pc]{pbardca1.0.eps}
\caption{the ratio of $\bar{p}$ at $dca<1.0$ cm over $\bar{p}$ at
$dca<3.0$ cm.} \label{pbardcaratio}
\end{figure}
For $\bar{p}$, the rapidity range is $-0.5<y_{\bar{p}}<0$. After
$|N_{\sigma p}|<2$ was required, the mass squared
$m^{2}=p^{2}(1/\beta^{2}-1)$ distributions in different $p_{T}$
bin in d+Au minimum-bias collisions are shown is
Figure~\ref{pbarrawyieldplot}. At $p_{T}<1.6$ GeV/c, the single
Gaussian function was used to fit the distribution of $m^{2}$ to
get the raw yield. At the same time, the counting result by
counting the track number at the range $0.64<m^{2}<1.44$
$(GeV/c^{2})^2$ was also used to compare with the raw yield from
the fitting method. The difference between them was found in one
sigma range. The raw yield we quote is from the fitting method. At
$p_{T}>1.6$ GeV/c, the double Gaussian function was used to
extract the raw yield. The raw signals in each $P_{T}$ bin are
shown in Table~\ref{pbartable}. For the $p$, the raw yield
extraction method is the same as $\bar{p}$ except that at
$p_{T}<1.6$
GeV/c, we use the method
$Np=Np_{dca<1.cm}\times{(N\bar{p}_{dca<3.cm}/N\bar{p}_{dca<1.cm})
}$ to reject the background, where $Np$ and $N\bar{p}$ are the
number of the $p$ and $\bar{p}$ tracks individually, and
$N\bar{p}_{dca<1.cm}/N\bar{p}_{dca<3.cm}$ is the ratio of
$\bar{p}$ tracks at $dca<1.0$ cm over those at $dca<3.0$ cm. In
Figure~\ref{protonrawyieldplot}, the first 10 $p_{T}$ bins are for
$dca<1.0$ cm, the last 4 $p_{T}$ bins are for $dca<3.0$ cm.
Figure~\ref{pbardcaratio} shows the ratio of $\bar{p}$ tracks at
$dca<1.0$ cm over those at $dca<3.0$ cm. After this correction of
$Np=Np_{dca<1.cm}\times{(N\bar{p}_{dca<3.cm}/N\bar{p}_{dca<1.cm})
}$, the $p$ raw signals in each $P_{T}$ bin are shown in
Table~\ref{protontable}.
\section{Efficiency and acceptance correction}
\begin{figure}[h]
\begin{minipage}[t]{80mm}
\includegraphics[height=13pc,width=18pc]{pluseff.eps}
\end{minipage}
\hspace{\fill}
\begin{minipage}[t]{80mm}
\includegraphics[height=13pc,width=18pc]{minuseff.eps}
\end{minipage}
\caption{TPC reconstruction efficiency of $\pi^{\pm}$, $K^{\pm}$,
$p$ and $\bar{p}$ as a function of $p_{T}$. The left plot for
charged plus particle and the right for charged minus particle. }
\label{tpceff}
\end{figure}
\begin{figure}[h]
\begin{minipage}[t]{80mm}
\includegraphics[height=13pc,width=18pc]{plusMatchingEff.eps}
\end{minipage}
\hspace{\fill}
\begin{minipage}[t]{80mm}
\includegraphics[height=13pc,width=18pc]{minusMatchingEff.eps}
\end{minipage}
\caption{Matching efficiency from TOFr to TPC of $\pi^{\pm}$,
$K^{\pm}$, $p$ and $\bar{p}$ as a function of $p_{T}$, including
detector response. The left plot for charged plus particle and the
right for charged minus particle.} \label{matcheff}
\end{figure}
Acceptance and efficiency were studied by Monte Carlo simulations
and by matching TPC track and TOFr hits in real data.
TPC tracking efficiency was studied by Monte Carlo simulations.
The simulated $\pi^{\pm}$, $K^{\pm}$, $p$ and $\bar{p}$ are
generated using a flat $p_T$ and a flat $y$ distribution and pass
through GSTAR~\cite{long:01} (the framework software package to
run the STAR detector simulation using
GEANT~\cite{geant:01,geant:02}) and TRS (the TPC Response
Simulator~\cite{long:01}). The simulated $\pi^{\pm}$, $K^{\pm}$,
$p$ and $\bar{p}$ are then combined with a real raw event and we
call this combined event a simulated event. This simulated event
is then passed through the standard STAR reconstruction chain and
we call this event after reconstruction a reconstructed event. The
reconstructed information of those particles in the reconstructed
event is then associated with the Monte-Carlo information in the
simulated event. And then we get the total number of simulated
$\pi^{\pm}$, $K^{\pm}$, $p$ and $\bar{p}$ from simulated events in
a certain transverse momentum bin. Also we can get the total
number of associated tracks in the reconstructed events in this
transverse momentum bin~\cite{Haibin:03}. In the end, take the
ratio of the number of associated $\pi^{\pm}$, $K^{\pm}$, $p$ and
$\bar{p}$ over the number of simulated $\pi^{\pm}$, $K^{\pm}$, $p$
and $\bar{p}$ and this ratio is the TPC reconstruction efficiency
for a certain transverse momentum bin in the mid-rapidity range.
Figure~\ref{tpceff} shows the TPC reconstruction efficiency of
$\pi^{\pm}$, $K^{\pm}$, $p$ and $\bar{p}$ as a function of
$p_{T}$.
\begin{figure}[h]
\centering
\includegraphics[height=18pc,width=24pc]{tofr_detecting_efficiency.eps}
\caption{The TOFr response efficiency as a function of $p_{T}$.}
\label{detectorresponse}
\end{figure}
The Matching Efficiency from TPC to TOFr were studied in
real data, and the formula are
\begin{equation}
Eff_{Match}=\frac{TofrMatchedTracks/dAuTOFrEvents}
{(MinBiasTracks/MinBiasEvents)_{pVPD}\times{factor1}\times{factor2}}
\end{equation}
where the $TofrMatchedTracks/dAuTOFrEvents$ is the number of TOFr
matched tracks per dAuTOFr trigger event,
$(MinBiasTracks/MinBiasEvents)_{pVPD}$ is the number of
minimum-bias tracks per minimum-bias event by requiring the pVPD
to fire, $factor1$ is the enhancement factor of dAuTOFr trigger,
and $factor2$ is the other factors such as the TOFr trip factor.
The $Eff_{Match}$ includes the detector response efficiency.
Figure~\ref{matcheff} shows the matching efficiency of different
particle species including the detector response versus $p_{T}$.
The detector response efficiency, including the material
absorption and scattering effect between TPC and TOFr, as a
function of $p_{T}$ is shown in Figure~\ref{detectorresponse},
which is around 90\% at $p_{T}>$ 0.3 GeV/c. After the material
absorption and scattering effect correction, the detector response
efficiency is around 95\%.
\section{Background correction}
\begin{figure}[h]
\centering
\includegraphics[height=18pc,width=18pc]{pionbackgroundMod.eps}
\caption{$\pi$ background contribution as a function of $p_{T}$.
The circled symbols represent the total $\pi$ background
contribution including feed-down and $\mu$ misidentification. The
squared and triangled symbols represent the week-decay and $\mu$
misidentification contributions individually.}
\label{pionbackground}
\end{figure}
Weak-decay feeddown (e.g. $K_{s}^{0}\rightarrow\pi^{+}\pi^{-}$) to
pions is $\sim12\%$ at low $p_{T}$ and $\sim5\%$ at high $p_{T}$,
and was corrected for using PYTHIA~\cite{pythia} and
HIJING~\cite{hijing} simulations, as shown in
Figure~\ref{pionbackground}. For $\pi$ spectra, the $\mu$
misidentification was also corrected for, which is also shown in
Figure~\ref{pionbackground}.
\begin{figure}[h]
\centering
\includegraphics[height=18pc,width=24pc]{protonScatter_dca1.0.eps}
\caption{The $p$ scattering effect contribution when we cut
$dca<1.0$ cm.} \label{protonScatter}
\end{figure}
Inclusive $p$ and $\bar{p}$ production is presented without
hyperon feeddown correction. $p$ and $\bar{p}$ from hyperon decays
have the same detection efficiency as primary $p$ and
$\bar{p}$~\cite{antiproton} and contribute about 20\% to the
inclusive $p$ and $\bar{p}$ yield, as estimated from the
simulation. However, for $p$, there is still some scattering
contribution which comes from the beam pipe interaction after the
cut of $dca<1.0$ cm. Figure~\ref{protonScatter} shows the
contribution of scattering effect for proton when we cut $dca<1.0$
cm. The correction is done at $p_{T}<$ 1.1 GeV/c and negligible at
higher $p_{T}$.
\section{Energy loss correction}
The energy loss effect due to the interaction with the detector
material was also corrected for. This was studied by simulation.
Figure~\ref{eloss} shows the momentum and transverse momentum
correction for energy loss effect. At $p_{T}>$0.35 GeV/c, for
$\pi$, the energy loss effect is negligible while for kaon and
proton, the energy loss correction is non-negligible at lower
$p_{T}$ and negligible at higher $p_{T}$. The correction was done
by shifting the position of $p_{T}$ in the $p_{T}$ spectra.
\begin{figure}[h]
\begin{minipage}[t]{80mm}
\includegraphics[height=13pc,width=18pc]{ptotthetadiff.eps}
\end{minipage}
\hspace{\fill}
\begin{minipage}[t]{80mm}
\includegraphics[height=13pc,width=18pc]{ptdiff.eps}
\end{minipage}
\caption{(left) p energy loss correction of different particle
species as a function of p. $p_{rec}$ is the reconstructed
momentum before the energy loss correction, $p_{MC}$ is the
momentum after energy loss correction from simulation, $\theta$ is
the angle between the reconstructed momentum and beam line.
(right) $p_{T}$ energy loss correction of different particle
species as a function of $p_{T}$. $p_{T}(rec)$ is the
reconstructed transverse momentum before the energy loss
correction, $p_{T}(MC)$ is the transverse momentum after energy
loss correction from simulation. } \label{eloss}
\end{figure}
\section{Normalization}
The efficiency including vertex efficiency and trigger efficiency
is 91\% in d+Au minimum-bias collisions and 85\% in p+p and
40-100\% d+Au collisions. In 0\%-20\% and 20\%-40\% d+Au
collisions, the efficiency is 100\%. Since the statistic of p+p
minimum-bias events in run 3 is not good enough for us to get very
precise enhancement factor and $N_{ch}$ bias factor. We compare
the $\pi$ spectra in the first 5 $p_{T}$ bin with those from the
paper~\cite{olga} and get the additional normalization factor for
p+p collisions.
\begin{figure}[h]
\hspace{-3pc}
\includegraphics[height=40pc,width=40pc]{pionplusrawyield_forthesis.eps}
\caption{$\pi^{+}$ raw yields versus mass squared distribution.
The histograms are our data. The curves are Gaussian fits.}
\label{pionplusrawyieldplot}
\end{figure}
\begin{figure}[h]
\hspace{-3pc}
\includegraphics[height=40pc,width=40pc]{pionminusrawyield_forthesis.eps}
\caption{$\pi^{-}$ raw yields versus mass squared distribution.
The histograms are our data. The curves are Gaussian fits.}
\label{pionminusrawyieldplot}
\end{figure}
\begin{figure}[h]
\hspace{-3pc}
\includegraphics[height=40pc,width=40pc]{kaonplusrawyield_forthesis.eps}
\caption{$K^{+}$ raw yields versus mass squared distribution. The
histograms are our data. The curves are Gaussian fits.}
\label{kaonplusrawyieldplot}
\end{figure}
\begin{figure}[h]
\hspace{-3pc}
\includegraphics[height=40pc,width=40pc]{kaonminusrawyield_forthesis.eps}
\caption{$K^{-}$ raw yields versus mass squared distribution. The
histograms are our data. The curves are Gaussian fits.}
\label{kaonminusrawyieldplot}
\end{figure}
\begin{figure}[h]
\hspace{-3pc}
\includegraphics[height=40pc,width=40pc]{protonrawyield_forthesis.eps}
\caption{$p$ raw yields versus mass squared distribution. The
histograms are our data. The curves are Gaussian fits.}
\label{protonrawyieldplot}
\end{figure}
\begin{figure}[h]
\hspace{-3pc}
\includegraphics[height=40pc,width=40pc]{pbarrawyield_forthesis.eps}
\caption{$\bar{p}$ raw yields versus mass squared distribution.
The histograms are our data. The curves are Gaussian fits.}
\label{pbarrawyieldplot}
\end{figure}
\begin{table}[h]
\begin{scriptsize}
\centering
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$p_{T}$ (GeV/c) & d+Au Trigger & 0\%-20\% & 20\%-40\% & 40\%-100\% & p+p \\ \hline
0.3-0.4 & $2.929e+04\pm171.4$ & $9219\pm96.21$ & $8735\pm93.6$ & $8604\pm92.84$ & $1.806e+04\pm134.4$ \\ \hline
0.4-0.5 & $2.185e+04\pm147.8$ & $6894\pm83.03$ & $6657\pm81.59$ & $6325\pm79.53$ & $1.274e+04\pm114.3$ \\ \hline
0.5-0.6 & $1.592e+04\pm126.2$ & $5162\pm71.85$ & $4901\pm70.3$ & $4534\pm67.34$ & $9180\pm95.81$ \\ \hline
0.6-0.7 & $1.166e+04\pm108$ & $3832\pm62.19$ & $3556\pm59.64$ & $3311\pm57.54$ & $6531\pm80.82$ \\ \hline
0.7-0.8 & $8556\pm92.5$ & $2909\pm53.93$ & $2628\pm51.26$ & $2368\pm48.67$ & $4447\pm66.74$ \\ \hline
0.8-0.9 & $6198\pm78.86$ & $2099\pm45.85$ & $1936\pm44.33$ & $1693\pm41.17$ & $2973\pm54.57$ \\ \hline
0.9-1 & $4520\pm67.25$ & $1487\pm38.57$ & $1361\pm36.9$ & $1276\pm35.74$ & $2132\pm46.31$ \\ \hline
1-1.1 & $3312\pm57.61$ & $1147\pm33.9$ & $1033\pm32.17$ & $845.9\pm29.15$ & $1386\pm37.71$ \\ \hline
1.1-1.2 & $2406\pm49.35$ & $788.6\pm28.19$ & $752.5\pm27.58$ & $652.7\pm25.7$ & $959.2\pm31.93$ \\ \hline
1.2-1.4 & $3227\pm58.17$ & $1132\pm34.28$ & $934.4\pm30.98$ & $831.5\pm29.82$ & $1183\pm40.11$ \\ \hline
1.4-1.6 & $1756\pm45.2$ & $573.7\pm26.2$ & $543.7\pm24$ & $412.8\pm21.83$ & $625.5\pm30.16$ \\ \hline
1.6-1.8 & $1046\pm39$ & $337.9\pm20.42$ & $309.8\pm18.62$ & $234\pm16.58$ & $364.5\pm32.64$ \\ \hline
\end{tabular}
\caption{$\pi^{+}$ raw signal table in minimum-bias, centrality
selected d+Au collisions and minimum-bias p+p collisions.}
\label{pionplustable}
\end{scriptsize}
\end{table}
\begin{table}[h]
\begin{scriptsize}
\centering
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$p_{T}$ (GeV/c) & d+Au Trigger & 0\%-20\% & 20\%-40\% & 40\%-100\% & p+p \\ \hline
0.3-0.4 & $2.861e+04\pm169.4$ & $8922\pm94.66$ & $8507\pm92.38$ & $8519\pm92.4$ & $1.715e+04\pm131$ \\ \hline
0.4-0.5 & $2.139e+04\pm146.3$ & $6805\pm82.49$ & $6458\pm80.35$ & $6306\pm79.41$ & $1.28e+04\pm113.1$ \\ \hline
0.5-0.6 & $1.611e+04\pm126.9$ & $5327\pm72.98$ & $4873\pm69.8$ & $4605\pm67.86$ & $9189\pm95.86$ \\ \hline
0.6-0.7 & $1.166e+04\pm108$ & $3831\pm61.9$ & $3550\pm59.5$ & $3355\pm57.92$ & $6362\pm79.69$ \\ \hline
0.7-0.8 & $8447\pm91.91$ & $2837\pm53.64$ & $2540\pm50.4$ & $2387\pm48.86$ & $4154\pm64.5$ \\ \hline
0.8-0.9 & $5950\pm77.17$ & $2076\pm45.61$ & $1780\pm42.2$ & $1646\pm40.68$ & $2899\pm53.87$ \\ \hline
0.9-1 & $4284\pm65.46$ & $1446\pm38.03$ & $1317\pm36.31$ & $1171\pm34.29$ & $1924\pm44.01$ \\ \hline
1-1.1 & $3296\pm57.47$ & $1123\pm33.55$ & $1014\pm31.9$ & $897.5\pm30.02$ & $1372\pm37.77$ \\ \hline
1.1-1.2 & $2464\pm49.88$ & $812.4\pm28.58$ & $762.4\pm27.69$ & $650.8\pm25.72$ & $1005\pm33.16$ \\ \hline
1.2-1.4 & $3136\pm57.28$ & $1027\pm32.54$ & $972.9\pm31.71$ & $828.8\pm29.65$ & $1243\pm39.78$ \\ \hline
1.4-1.6 & $1716\pm45.79$ & $612.5\pm25.87$ & $539.3\pm25.67$ & $422.7\pm24.22$ & $603.8\pm30.49$ \\ \hline
1.6-1.8 & $1033\pm39.74$ & $375.3\pm21.21$ & $306.1\pm19.59$ & $239.9\pm48.55$ & $337.1\pm28.82$ \\ \hline
\end{tabular}
\caption{$\pi^{-}$ raw signal table in minimum-bias, centrality
selected d+Au collisions and minimum-bias p+p collisions.}
\label{pionminustable}
\end{scriptsize}
\end{table}
\begin{table}[h]
\begin{scriptsize}
\centering
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$p_{T}$ (GeV/c) & d+Au Trigger & 0\%-20\% & 20\%-40\% & 40\%-100\% & p+p \\ \hline
0.4-0.5 & $1410\pm37.54$ & $417.2\pm20.44$ & $420.9\pm20.52$ & $354.9\pm18.84$ & $753.2\pm27.44$ \\ \hline
0.5-0.6 & $1588\pm39.85$ & $486.3\pm22.06$ & $461\pm21.48$ & $435\pm20.87$ & $729.3\pm27$ \\ \hline
0.6-0.7 & $1499\pm38.71$ & $465.9\pm21.59$ & $445.5\pm21.17$ & $395\pm19.87$ & $710.1\pm26.65$ \\ \hline
0.7-0.8 & $1346\pm36.69$ & $423.9\pm20.59$ & $419.8\pm20.62$ & $335.7\pm18.32$ & $579\pm24.06$ \\ \hline
0.8-0.9 & $1105\pm33.59$ & $369.7\pm19.3$ & $317.9\pm18.43$ & $282.5\pm17.03$ & $496.2\pm22.44$ \\ \hline
0.9-1 & $969.1\pm31.19$ & $283.9\pm16.86$ & $305.1\pm17.52$ & $258.7\pm16.23$ & $381.2\pm19.87$ \\ \hline
1-1.1 & $799.3\pm28.41$ & $278.6\pm16.79$ & $224\pm15.04$ & $192.3\pm14.04$ & $301.5\pm18.39$ \\ \hline
1.1-1.2 & $656.7\pm26.24$ & $199.1\pm14.33$ & $186.2\pm14$ & $155.7\pm12.94$ & $267.8\pm18.71$ \\ \hline
1.2-1.4 & $1013\pm34.43$ & $335\pm19.51$ & $283.1\pm17.71$ & $234.1\pm17.68$ & $421.1\pm29.14$ \\ \hline
1.4-1.6 & $605.9\pm30.14$ & $191.1\pm17.77$ & $174.1\pm14.77$ & $148.8\pm15.48$ & $238.8\pm13.97$ \\ \hline
1.6-1.8 & $382.9\pm28.9$ & $---$ & $---$ & $---$ & $---$ \\ \hline
\end{tabular}
\caption{$K^{+}$ raw signal table in minimum-bias, centrality
selected d+Au collisions and minimum-bias p+p collisions.}
\label{kaonplustable}
\end{scriptsize}
\end{table}
\begin{table}[h]
\begin{scriptsize}
\centering
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$p_{T}$ (GeV/c) & d+Au Trigger & 0\%-20\% & 20\%-40\% & 40\%-100\% & p+p \\ \hline
0.4-0.5 & $1341\pm36.62$ & $411.6\pm20.29$ & $367.6\pm19.19$ & $378\pm19.44$ & $682.6\pm26.13$ \\ \hline
0.5-0.6 & $1498\pm38.7$ & $460\pm21.45$ & $411.5\pm20.32$ & $420.5\pm20.51$ & $740.7\pm27.21$ \\ \hline
0.6-0.7 & $1410\pm37.55$ & $436.2\pm20.89$ & $398.4\pm19.97$ & $361.9\pm19.02$ & $616.8\pm24.83$ \\ \hline
0.7-0.8 & $1207\pm34.74$ & $366\pm19.14$ & $349.7\pm18.7$ & $350\pm18.75$ & $557.4\pm23.61$ \\ \hline
0.8-0.9 & $1057\pm32.66$ & $317.4\pm18.12$ & $332.4\pm18.37$ & $268.2\pm16.66$ & $432.9\pm20.93$ \\ \hline
0.9-1 & $863.7\pm29.42$ & $256.9\pm16.09$ & $267.2\pm16.43$ & $223.8\pm15.03$ & $368.9\pm19.59$ \\ \hline
1-1.1 & $635.2\pm25.35$ & $198.2\pm14.35$ & $183.1\pm13.61$ & $187.9\pm13.88$ & $320.4\pm19.42$ \\ \hline
1.1-1.2 & $543\pm23.92$ & $166.4\pm13.14$ & $143\pm12.22$ & $154.1\pm12.89$ & $248.5\pm18.84$ \\ \hline
1.2-1.4 & $895\pm32.45$ & $302.1\pm18.3$ & $258\pm17.06$ & $206.7\pm16.17$ & $377.1\pm26.67$ \\ \hline
1.4-1.6 & $645.5\pm31.61$ & $202.4\pm16.78$ & $141\pm15.58$ & $166.6\pm17.87$ & $237.6\pm20.52$ \\ \hline
1.6-1.8 & $351.3\pm29.79$ & $---$ & $---$ & $---$ & $---$ \\ \hline
\end{tabular}
\caption{$K^{-}$ raw signal table in minimum-bias, centrality
selected d+Au collisions and minimum-bias p+p collisions.}
\label{kaonminustable}
\end{scriptsize}
\end{table}
\begin{table}[h]
\begin{scriptsize}
\centering
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$p_{T}$ (GeV/c) & d+Au Trigger & 0\%-20\% & 20\%-40\% & 40\%-100\% & p+p \\ \hline
0.4-0.5 & $1377\pm107.5$ & $412.6\pm40.73$ & $403.5\pm40.11$ & $422.7\pm41.42$ & $657.5\pm64.94$ \\ \hline
0.5-0.6 & $1527\pm89.71$ & $492.2\pm36.58$ & $428.6\pm33.09$ & $424.8\pm32.89$ & $752.6\pm59.91$ \\ \hline
0.6-0.7 & $1456\pm80.28$ & $437.9\pm31.47$ & $420\pm30.56$ & $401.3\pm29.6$ & $704.9\pm54.1$ \\ \hline
0.7-0.8 & $1336\pm73.68$ & $410.6\pm29.43$ & $387.9\pm28.28$ & $385\pm28.14$ & $670\pm55.36$ \\ \hline
0.8-0.9 & $1278\pm72.45$ & $387.6\pm28.57$ & $371.4\pm27.72$ & $312\pm24.57$ & $498.8\pm45.17$ \\ \hline
0.9-1 & $1124\pm68.87$ & $349.1\pm27.39$ & $322.1\pm25.87$ & $288.1\pm23.94$ & $448.3\pm44.89$ \\ \hline
1-1.1 & $954.5\pm62.39$ & $288.2\pm24.38$ & $285.3\pm24.23$ & $265.9\pm23.11$ & $365\pm40.02$ \\ \hline
1.1-1.2 & $832.1\pm58.34$ & $257.2\pm23.06$ & $245.4\pm22.25$ & $213.3\pm20.25$ & $273.8\pm34.08$ \\ \hline
1.2-1.4 & $1268\pm72.57$ & $441.9\pm31.19$ & $362.6\pm27.1$ & $320.2\pm24.77$ & $393.8\pm41.64$ \\ \hline
1.4-1.6 & $806.8\pm57.58$ & $306.9\pm26.38$ & $221.5\pm20.77$ & $190.8\pm18.71$ & $210.3\pm31.59$ \\ \hline
1.6-1.8 & $540.8\pm23.27$ & $170.7\pm13.06$ & $146.2\pm12.14$ & $116.3\pm10.81$ & $126\pm11.31$ \\ \hline
1.8-2 & $314.2\pm17.8$ & $119.2\pm10.97$ & $81.7\pm9.764$ & $68.35\pm9.093$ & $93.98\pm10.15$ \\ \hline
2-2.5 & $388.1\pm21.21$ & $148.4\pm12.48$ & $135.7\pm12.01$ & $89.74\pm10.02$ & $109\pm12.33$ \\ \hline
2.5-3 & $109.1\pm12.92$ & $36.33\pm6.809$ & $34.3\pm8.488$ & $30.64\pm7.487$ & $24.22\pm5.422$ \\ \hline
3-4 & $82.18\pm12.30$ & $---$ & $---$ & $---$ & $---$ \\ \hline
\end{tabular}
\caption{$p$ raw signal table in minimum-bias, centrality selected
d+Au collisions and minimum-bias p+p collisions.}
\label{protontable}
\end{scriptsize}
\end{table}
\begin{table}[h]
\begin{scriptsize}
\centering
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$p_{T}$ (GeV/c) & d+Au Trigger & 0\%-20\% & 20\%-40\% & 40\%-100\% & p+p \\ \hline
0.4-0.5 & $692.6\pm26.33$ & $215.1\pm14.67$ & $183\pm13.53$ & $202.8\pm14.26$ & $421.7\pm20.56$ \\ \hline
0.5-0.6 & $1009\pm31.76$ & $310.8\pm17.63$ & $304\pm17.43$ & $268.5\pm16.39$ & $526.6\pm22.95$ \\ \hline
0.6-0.7 & $1098\pm33.17$ & $317.9\pm17.84$ & $305.6\pm17.51$ & $327\pm18.11$ & $561.5\pm23.71$ \\ \hline
0.7-0.8 & $1062\pm32.59$ & $340.2\pm18.44$ & $307\pm17.53$ & $285.5\pm16.9$ & $435.1\pm20.86$ \\ \hline
0.8-0.9 & $992.2\pm31.5$ & $315.1\pm17.81$ & $284.4\pm16.96$ & $244.4\pm15.63$ & $376.4\pm19.4$ \\ \hline
0.9-1 & $827.5\pm28.76$ & $288.9\pm17$ & $225\pm15.01$ & $202.6\pm14.24$ & $310.4\pm17.62$ \\ \hline
1-1.1 & $724\pm26.91$ & $240.2\pm15.5$ & $181.5\pm13.48$ & $192.2\pm13.87$ & $246.4\pm15.7$ \\ \hline
1.1-1.2 & $608.5\pm24.67$ & $161.3\pm12.7$ & $184.3\pm13.61$ & $149.8\pm12.25$ & $192\pm13.87$ \\ \hline
1.2-1.4 & $914.9\pm30.24$ & $301.5\pm17.36$ & $269.7\pm16.42$ & $214.1\pm14.63$ & $269.6\pm16.42$ \\ \hline
1.4-1.6 & $575.8\pm24$ & $204.9\pm14.32$ & $160.9\pm12.71$ & $120.5\pm10.98$ & $138.6\pm12.01$ \\ \hline
1.6-1.8 & $407.2\pm20.18$ & $127.3\pm11.29$ & $108.1\pm10.43$ & $89.9\pm9.497$ & $100.6\pm10.63$ \\ \hline
1.8-2 & $257.3\pm16.26$ & $73.85\pm8.992$ & $92.22\pm9.69$ & $46.81\pm7.802$ & $71.23\pm8.92$ \\ \hline
2-2.5 & $305.6\pm18.45$ & $114\pm11.01$ & $83.43\pm9.464$ & $77.84\pm9.16$ & $64.02\pm10.44$ \\ \hline
2.5-3 & $111\pm12.79$ & $28.91\pm6.26$ & $29.29\pm8.869$ & $20.55\pm7.198$ & $25.71\pm6.856$ \\ \hline
3-4 & $67.05\pm11.87$ & $---$ & $---$ & $---$ & $---$ \\ \hline
\end{tabular}
\caption{$\bar{p}$ raw signal table in minimum-bias, centrality
selected d+Au collisions and minimum-bias p+p collisions.}
\label{pbartable}
\end{scriptsize}
\end{table}
\chapter{{\hspace{3.5cm}STAR Collaboration}}
\begin{figure}[htb]
\hspace{-6pc}
\includegraphics[width=45pc]{sci-july03-01.eps}
\end{figure}
\begin{figure}[htb]
\hspace{-6pc}
\includegraphics[width=45pc]{sci-july03-02.eps}
\end{figure}
\chapter{Conclusion and Outlook}
\label{chp:conclusion}
\section{Conclusion}
In summary, we have reported the identified particle spectra of
pions, kaons, protons and anti-protons at mid-rapidity from 200
GeV minimum-bias, centrality selected d+Au collisions and NSD p+p
collisions. The time-of-flight detector, based on novel multi-gap
resistive plate chamber technology, was used for particle
identification. This is the first time that MRPC detector was
installed to take data as a time-of-flight detector in the
collider experiment. The calibration method was set up in the STAR
experiment for the first time and has been applied to the data
taken later successfully. The intrinsic timing resolution of the
MRPC was 85 ps after the calibration. In 2003 run, the pion/kaon
can be separated up to transverse momentum 1.6 GeV/c while proton
can be identified up to 3.0 GeV/c.
The spectra of $\pi^{\pm}$, $K^{\pm}$, $p$ and $\bar{p}$ in d+Au
and p+p collisions provide an important reference for those in
Au+Au collisions. The initial state in d+Au collisions is similar
to that in Au+Au collisions, and, it's believed that the
quark-gluon plasma doesn't exist in d+Au collisions. These results
from d+Au collisions are very important for us to judge whether
the quark-gluon plasma exists in Au+Au collisions or not and to
understand the property of the dense matter created in Au+Au
collisions. We observe that the spectra of $\pi^{\pm}$, $K^{\pm}$,
$p$ and $\bar{p}$ are considerably harder in d+Au than those in
p+p collisions. In $\sqrt{s_{_{NN}}} = 200$ GeV d+Au collisions,
the $R_{dAu}$ of protons rise faster than $R_{dAu}$ of pions and
kaons. The $R_{dAu}$ of proton is larger than 1 at intermediate
$p_T$ while the proton production follows binary scaling at the
same $p_T$ range in 200 GeV Au+Au collisions. These results
further prove that the suppression observed in Au+Au collisions at
intermediate and high $p_T$ is due to final state interactions in
a dense and dissipative medium produced during the collision and
not due to the initial state wave function of the Au nucleus.
Additionally, the particle-species dependence of the Cronin effect
is found to be significantly smaller than that from lower energy
p+A collisions. In $\sqrt{s_{_{NN}}} = 200$ GeV d+Au collisions,
the ratio of the nuclear modification factor $R_{dAu}$ between
$(p+\bar{p})$ and charged hadrons ($h$) in the $p_{T}$ range $1.2<
p_{T}<3.0$ GeV/c was measured to be
$1.19\pm0.05$(stat)$\pm0.03$(syst) in minimum-bias collisions.
Both the $R_{dAu}$ values and $(p+\bar{p})/h$ ratios show little
centrality dependence, in contrast to previous measurements in
Au+Au collisions at $\sqrt{s_{NN}}$ = 130 and 200 GeV. The ratios
of protons over charged hadrons in d+Au and p+p collisions are
found to be about a factor of 2 lower than that from Au+Au
collisions, indicating that the relative baryon enhancement
observed in heavy ion collisions at RHIC is due to the final state
effects in Au+Au collisions.
The identified particle spectra in d+Au and p+p collisions not
only provide the reference for those in Au+Au collisions, but also
provide a chance to see the mechanism of the Cronin effect itself
clearly. Usually the Cronin effect has been explained to be the
initial state effect only since 1970s~\cite{accardi}. However, we
compare our pion and proton spectra in minimum-bias and
centrality-selected d+Au collisions with the recombination
model~\cite{hwayang}. The recombination model can reproduce both
the pion spectra and proton spectra. This recombination model is
built on the hadronization process, which is a final-state effect,
while the initial multiple parton scattering model~\cite{accardi}
can't reproduce the difference of the Cronin effect between pions
and protons. From these comparisons, we conclude that the Cronin
effect in $\sqrt{s_{_{NN}}} = 200$ GeV d+Au collisions is not the
initial state effect only, and that final state effect plays an
important role.
The integral yield $dN/dy$ and $\langle p_T \rangle$ in p+p and
d+Au collisions were estimated from the power law fit and thermal
model fit. The integral yield $R_{dAu}$ of $\pi^{\pm}$, $K^{\pm}$,
$p$ and $\bar{p}$ are observed to be smaller than 1 while those of
$p$ and $\bar{p}$ are close to 1. The $\pi^{-}/\pi^{+}$,
$K^{-}/K^{+}$ and $\bar{p}/p$ ratios as a function of $p_{T}$ are
observed to be flat with $p_T$ within the errors in d+Au and p+p
minimum-bias collisions and show little centrality dependence in
d+Au collisions. The integral yield ratios of $K^{-}/\pi^{-}$ and
$\bar{p}/\pi^{-}$ as a function of $dN/d\eta$ were also presented
in p+p and d+Au collisions.
\section{Outlook}
For the outlook, I will discuss whether the Cronin effect is mass
dependent or baryon/meson dependent at 200 GeV. What other physics
topic have we done from MRPC-TOFr in d+Au and p+p collisions in
2003 run? If we have the full time-of-flight (Full-TOF) coverage,
what can we do? Also I will discuss a little bit about the low
energy 63 GeV Au+Au run.
\subsection{Cronin effect at 200 GeV: Mass dependent or baryon/meson dependent?}
We know that recombination model can reproduce the spectra of
pions and protons in d+Au collisions. Also the $R_{CP}$ of
identified particles in Au+Au collisions suggest that the degree
of suppression depends on particle species(baryon/meson) at
intermediate $p_T$. Does the Cronin effect in 200 GeV d+Au
collisions depend on the particle species (baryon/meson) or depend
on the particle mass? From our data, it shows the Cronin effect
for proton is bigger than those for pion and kaon. And the Cronin
effect of pion shows little difference from that of kaon at
$p_T<1.5$ GeV/c. In order to see the Cronin effect is baryon/meson
dependent or mass dependent, we can compare the Cronin effect of
proton with those of $K^*$ and $\phi$ since the mass of $K^*$ and
$\phi$ are close to that of proton while $K^*$ and $\phi$ are
mesons and proton is a baryon. The preliminary results show that
the Cronin effect of $K^*$ and $\phi$~\cite{kstarphi} are similar
to that of pion and different from that of proton. However, the
final results from $K^*$ and $\phi$ are needed to confirm this
issue.
\subsection{Electron PID from MRPC-TOFr}
The production and spectra of hadrons with heavy flavor are
sensitive to initial conditions and the later stage dynamical
evolution in high energy nuclear collisions, and may be less
affected by the non-perturbative complication in theoretical
calculations~\cite{charm1}. Charm production has been proposed as
a sensitive measurement of parton distribution function in nucleon
and the nuclear shadowing effect by systematically studying p+p,
and p+A collisions~\cite{lin96}. The relatively reduced energy
loss of heavy quark traversing a quark-gluon plasma will help us
distinguish the medium in which the jet loses its
energy~\cite{dokshitzer01}. A possible enhancement of charmonia
($J/\Psi$) production can be present at RHIC energies~\cite{jpsi}
due to the coalescence of the copiously produced charm
quarks~\cite{opencharm}.
\begin{figure}[h]
\centering
\includegraphics[height=20pc,width=24pc]{electronpid.eps}
\caption{$dE/dx$ in TPC versus $p$ without(the upper panel) or
with (the lower pannel) the TOFr velocity cut $|1/\beta-
1|\le0.03$. The insert shows $dE/dx$ distribution for 1 $\le p
\le$ 1.5 GeV/c. } \label{electron}
\end{figure}
The recent STAR results on the absolute open charm cross section
measurements from direct charmed hadron $D^0$
reconstruction~\cite{Haibin:03} in d+Au collisions and electrons
from charm semileptonic decay in both p+p and d+Au collisions at
200 GeV were presented~\cite{opencharm}. Based on the capability
of hadron identification~\cite{startof1} from the MRPC-TOFr tray
in 2003, electrons could be identified at low momentum
($p_{T}\le3$ GeV/c) by the combination of velocity ($\beta$) from
TOFr~\cite{startof} and the particle ionization energy loss
($dE/dx$) from TPC~\cite{tpc}. Figure~\ref{electron} shows that
the electrons are clearly identified as a separate band in the
$dE/dx$ versus momentum ($p$) with a selection on $\beta$ at
$|1/\beta-1|\le0.03$ in d+Au collisions. At higher $p_{T}$ (2--4
GeV/c), negative electrons were also identified directly by TPC
since hadrons have lower $dE/dx$ due to the relativisitic rise of
electron $dE/dx$. Based on the clear electron identification, the
open-charm-decayed electron spectra was derived~\cite{opencharm}.
Combined with $D^0$ measurement from TPC, the total charm cross
section was obtained~\cite{opencharm} in d+Au collisions.
\subsection{Full-TOF Physics}
Based on the hadron PID and electron PID of MRPC-TOFr in 2003, we
can imagine how many physics we can do if we have full
time-of-flight coverage based on MRPC technology. The
proposal~\cite{tofproposal} for large area time-of-flight system
for STAR has been proposed. Since the pion/kaon can be separated
up to transverse momentum 1.6 GeV/c and proton can be identified
up to 3.0 GeV/c from time-of-flight system. The resonance spectra
measured from hadronic decay will be extended to much higher
$p_T$. The direct open charm spectra from its hadronic decay
channel will reach higher $p_T$ with much more precise
measurement. Since the electron can be clearly identified up to
transverse momentum 3$\sim$4 GeV/c by the combination of velocity
($\beta$) from TOFr ~\cite{startof} and the particle ionization
energy loss ($dE/dx$) from TPC ~\cite{tpc}, the electron spectra
from charm-semi-leptonic decay will be measured precisely. As we
know that the measurement of the di-leptonic decays of vector
mesons are very difficult since the branch ratios are too small
and it's really hard to subtract the background. But with TOF
upgrading together with the SVT and micro-vertex detector
upgrading, the di-leptonic decays of vector mesons will be
measured much more easily, which will bring the direct information
of QGP since the electron is a lepton and the cross section of
interaction between electrons and hadrons is little. Thus we can
see directly the property of quark-gluon plasma such as the
temperature and the chiral symmetry restoration. This will be the
most interesting and meaningful thing for the QGP
search~\cite{xzb}. Besides, there are many other physics
topics~\cite{tofproposal} such as identified particle correlation
and fluctuation, particle composition of jet fragmentation, and
anti-nuclei etc.
\subsection{63 GeV Au+Au collisions at RHIC}
The bulk properties such as elliptic flow $v_2$ and particle
production show smooth trend from AGS, SPS to RHIC energy. One
energy point $\sqrt{s_{_{NN}}} = 63$ GeV, which is between SPS and
full RHIC energy, was selected since high quality charged-particle
and $\pi^0$ inclusive spectra have been measured in p+p collisions
at 63 GeV at Intersecting Storage Rings (ISR) and will serve as
the reference spectra for computing the nuclear modification
factor for Au+Au collisions measured at the same energy. The $v_2$
and particle production at 63 GeV will be studied at RHIC.
Besides, the nuclear modification factor as a function of $p_T$
will also be studied in this collision system in which the hard
scattering component has been significantly reduced. The results
from 63 GeV Au+Au collisions will be helpful for us to understand
the property of dense medium created in 200 GeV Au+Au collisions.
\chapter{Discussion}
\label{chp:discussion}
\section{Cronin effect}
The identified particle spectra in d+Au and p+p collisions not
only provide the reference for those in Au+Au collisions at 200
GeV, but also provide a chance to see the mechanism of the Cronin
effect itself clearly. Cronin effect was observed 30 years
ago~\cite{cronin}. It is the enhancement of particle production at
high $p_{T}$. The enhancement was explained by initial multiple
parton scattering. Also the recent experimental results of Cronin
effect on inclusive charged hadron are consistent with the
predictions based on initial multiple parton
scattering~\cite{accardi}. It suggests the suppression at
intermediate $p_{T}$ in Au+Au collisions is due to final state
effects. However, the initial multiple parton scattering with the
independent fragmentation function will result in the same Cronin
effect for $p(\bar{p})$ and for pions, while experimentally the
Cronin effect for $p(\bar{p})$ is larger than that for $\pi$.
That's to say the initial multiple scattering with the independent
fragmentation function can't account for the Cronin effect
observed. Maybe in the initial multiple parton scattering, the
broadening for gluon and for quark/antiquark are not the
same~\cite{xinniancommu}. Or maybe the fragmentation processes in
p+A collisions are not the same as those in p+p
collisions~\cite{qiucommu}. Whether the Cronin effect is initial
state effect or final state effect will be discussed below.
\subsection{Model comparison: initial state effect?}
The initial multiple parton scattering model predicts that the
Cronin effect on deuteron beam outgoing side is larger than that
on Au beam outgoing side since the deuteron traverses a much
larger nucleus~\cite{xinnian:dAu}. Figure~\ref{etaasymmetry}
(left) shows the predictions for the Cronin effect at different
rapidity range. The different curves correspond to the prediction
results from different shadowings. The $y=1$ is on the deuteron
beam outgoing side. The $y=-1$ is on the Au beam outgoing side.
The $y=0$ is at mid-rapidity. We can see that the $R_{dAu}$ on
deuteron beam side ($y=1$) increases faster than that on Au beam
side ($y=-1$). If we take the ratio of $R_{dAu}$ on Au beam side
over $R_{dAu}$ on deuteron beam side, it will result in a minimum
value at $p_{T}\sim3.5$ GeV/c, as shown in the curves on the right
plot of Figure~\ref{etaasymmetry}. The solid symbol on the right
plot of Figure~\ref{etaasymmetry} represents the data
points~\cite{johan}, which is the ratio of $R_{dAu}$ on Au beam
side at $-1<\eta<-0.5$ over $R_{dAu}$ on deuteron beam side at
$0.5<\eta<1$. We observe the $\eta$ asymmetry from experiment
reaches a maximum value firstly and then decreases. This is
different from the predictions. That means, the model based on
initial multiple parton scattering only, can't reproduce the
experimental results. Recently, Qiu and Vitev have come up with
the idea of coherent multiple scattering and applied it to the
RHIC experiments~\cite{coherent}. In this picture, the hard probe
may interact coherently with many low x parton inside different
nucleons inside the nucleus. As a result, this process will lead
to the suppression of the total cross section. This coherent
effect will play an important role in p+A collisions at forward
rapidity. In the deuteron outgoing beam direction, the coherent
effect is non-negligible since the Au nucleus is big while on the
Au side, the coherent effect is not big since the deuteron is of a
small size. This will result in bigger suppression on the deuteron
side than on the Au side. It may qualitatively reproduce the data.
This coherent multiple scattering is a final-state effect.
\begin{figure}[h]
\begin{minipage}[t]{80mm}
\includegraphics[height=18pc,width=14pc]{r200.eps}
\end{minipage}
\hspace{-2cm}
\begin{minipage}[t]{80mm}
\includegraphics[height=17pc,width=24pc]{wongRatio_hipt_one_panel_v4-1.eps}
\end{minipage}
\caption{(left) The Cronin effect at different rapidity as a
function of $p_{T}$. The different curve in each panel shows the
different shadowing. This figure is from ~\cite{xinnian:dAu}.
(right) The $\eta$ asymmetry of the Cronin effect: the ratio of
Cronin effect in Au beam outgoing direction over the Cronin effect
in deuteron beam outgoing direction. This figure is from
~\cite{johan}.} \label{etaasymmetry}
\end{figure}
As we all known, in Au+Au collisions, the suppression at
intermediate $p_{T}$ can be reproduced by the initial multiple
scattering and jet quenching
qualitatively~\cite{starhighpt,jetquench}. However, the model
based on the initial multiple scattering, jet quenching and
independent fragmentation will result in the same suppression for
baryons and mesons at intermediate $p_{T}$ in Au+Au collisions.
Experimentally $R_{cp}$ for baryons are larger than $R_{cp}$ for
mesons at intermediate $p_{T}$. This difference can be reproduced
by coalescence or recombination models~\cite{hwa,fries,ko}.
Recently the recombination model~\cite{hwayang} has been applied
to d+Au system to see whether it can reproduce the Cronin effect
or not.
With the help of Prof. C.B. Yang~\cite{yang}, I also compare our
pion and proton spectra in d+Au collisions with the recombination
model~\cite{hwayang}. In the following the recombination
model~\cite{hwayang} will be discussed and the comparison between
the data and the model will be presented in detail.
\subsection{Model comparison: recombination}
The inclusive distribution for the production of pions can be
written in the recombination model~\cite{hwayang}, when mass
effects are negligible, in the invariant form
\begin{eqnarray} p{dN_{\pi} \over dp} = \int {dp_1 \over
p_1}{dp_2 \over p_2}F_{q\bar{q}} (p_1, p_2) R_{\pi}(p_1, p_2, p) ,
\label{1}
\end{eqnarray} where $F_{q\bar{q}} (p_1, p_2)$ is the joint
distribution of a $q$ and $\bar q$ at $p_1$ and $p_2$, and
$R_{\pi}(p_1, p_2, p)$ is the recombination function for forming a
pion at $p$: $R_{\pi}(p_1, p_2, p) = (p_1p_2/p)\delta (p_1+p_2-
p)$. $F_{q\bar{q}}$ depends on the colliding hadron/nuclei. In
general, $F_{q\bar{q}}$ has four contributing components
represented schematically by
\begin{eqnarray} F_{q\bar{q}} = {\cal TT} + {\cal TS} + ({\cal SS})
_1 + ({\cal SS})_2
\end{eqnarray} where $\cal{ T}$ denotes thermal distribution and
$\cal{S}$ shower distribution. $({\cal SS})_1$ signifies two
shower partons in the same hard-parton jet, while $({\cal SS})_2$
stands for two shower partons from two nearby jets~\cite{hwayang}.
For $p+A$ collisions it may not be appropriate to refer to any
partons as thermal in the sense of a hot plasma as in heavy-ion
collisions. Here in d+Au collisions, the symbol $\cal{ T}$
represents the soft parton distribution at low $k_T$. At low $p_T$
the observed pion distribution is exponential; we identify it with
the contribution of the ${\cal TT}$ term~\cite{hwayang}.
\begin{eqnarray}
{dN^{{\cal TT}}_{\pi} \over pdp} = {C^2 \over 6} exp (-p/T)
\end{eqnarray}
where $T$ is the inverse slope. We shall determine $C$ and $T$ by
fitting the d+Au data at low $p_T$. The pion spectra for different
centralities can be calculated from thermal-thermal ($pion_{tt}$),
thermal-shower ($pion_{ts}$) and shower-shower ($pion_{ss}$)
contributions by using parameters $C$ and $N_{bin}$:
$dN/p_Tdp_T=C\times C\times pion_{tt}+2.5 \times C \times N_{bin}
\times pion_{ts}+2.5 \times N_{bin} \times pion_{ss}$, where C is
determined by fitting the d+Au data at $0.4<p_T<1.0$ GeV/c, and
$N_{bin}$ is the number of binary collisions. The data points of
$pion_{tt}, pion_{ts}$ and $pion_{ss}$ are from Prof. C.B.
Yang~\cite{yang}. The $C$ values for minimum-bias, 0-20\%, 20-40\%
and 40-$\sim$100\% d+Au collisions are 8.85, 13.08, 10.96 and 6.84
individually. The $T$ value of 0.21 GeV is used in the low $p_{T}$
fit. Figure~\ref{pirecombination} shows the $\pi^{+}$ spectra in
d+Au collisions as well as those from recombination model. This
figure shows that the recombination model can reproduce the
spectra of pion in minimum-bias and centrality selected d+Au
collisions.
\begin{figure}[h]
\begin{minipage}[t]{80mm}
\includegraphics[height=18pc,width=18pc]{pipluscent_recombine_new.eps}
\end{minipage}
\hspace{0mm}
\begin{minipage}[t]{80mm}
\includegraphics[height=18pc,width=18pc]{piplusall_recombine.eps}
\end{minipage}
\caption{(left) The invariant yield for $\pi^{+}$ at 0\%-20\% d+Au
collisions as a function of $p_{T}$. The open circles are our data
points. The curves are the calculation results from recombination
model. Sum represents the total contribution from recombination
model. Thermal-thermal represents the soft contribution. The
thermal-shower represents the contribution from the interplay
between soft and hard components. The shower-shower represents the
hard contribution. (right) The invariant yields for $\pi^{+}$ in
minimum-bias and centrality selected d+Au collisions as a function
of $p_{T}$. The symbols represent our data points. The curves on
the top of the symbols are the corresponding calculation results
from recombination model. } \label{pirecombination}
\end{figure}
\begin{figure}[h]
\begin{minipage}[t]{80mm}
\includegraphics[height=18pc,width=18pc]{ppluscent_recombine_new.eps}
\end{minipage}
\hspace{0mm}
\begin{minipage}[t]{80mm}
\includegraphics[height=18pc,width=18pc]{pplusall_recombine.eps}
\end{minipage}
\caption{(left) The invariant yield for $p$ at 0\%-20\% d+Au
collisions as a function of $p_{T}$. The open circles are our data
points. The curves are the calculation results from recombination
model. Sum represents the total contribution from recombination
model. TTT represents the soft contribution. The TTS+TSS
represents the contribution from the interplay between soft and
hard components. The SSS represents the hard contribution. (right)
The invariant yields for $p$ in minimum-bias and centrality
selected d+Au collisions as a function of $p_{T}$. The symbols
represent our data points. The curves on the top of the symbols
are the corresponding calculation results from recombination
model. } \label{precombination}
\end{figure}
The invariant inclusive distribution for proton formation at
midrapidity in the recombination model ~\cite{hwayang}
\begin{eqnarray}
p^0{dN_p \over dp} = \int {dp_1 \over p_1}{dp_2 \over p_2} F
(p_1, p_2, p_3) R_p(p_1, p_2, p_3, p)
\end{eqnarray}
where all momentum variables $p_i$ and $p$ are transverse momenta,
and $p^0$ denotes the energy of the proton. $F (p_1, p_2, p_3)$ is
the joint distribution of $u, u,$ and $d$ quarks at $p_1, p_2$ and
$p_3$, respectively. $R_p(p_1, p_2, p_3, p)$ is the recombination
function for a proton with momentum $p$. We write schematically
\begin{eqnarray}
F = {\cal TTT} + {\cal TTS} + {\cal TSS} + {\cal SSS}
\end{eqnarray}
where all the shower partons $\cal{S}$ are from one hard parton
jet. Shower partons from different jets are ignored here for RHIC
energies. In d+Au collisions, $\cal{ T}$ denotes the soft partons
that are not associated with the shower components of a hard
parton. The ${\cal SSS}$ term is regarded as the fragmentation of
a hard parton into a proton. The ${\cal TTT}$ term comes entirely
from the soft partons, while ${\cal TTS}$ and ${\cal TSS}$
accounts for the interplay between the soft and shower partons.
The soft contribution to the proton spectrum arising from ${\cal
TTT}$ recombination is
\begin{eqnarray}
{dN^{\rm th}_{proton} \over pdp} = {C^3 \over 6} {p^2 \over
p^0} e^{-p/T} { B (\alpha + 2, \gamma +2) B (\alpha + 2,\alpha +
\gamma +4) \over B (\alpha + 1, \gamma +1) B (\alpha + 1,\alpha +
\gamma +2) }
\end{eqnarray}.
Where $C$ and $T$ are determined by fitting the proton spectra at
low $p_{T}$, the $\alpha$ is equal to 1.75, $\gamma$ is equal to
1.05, $B(x,y)$ is the beta function~\cite{hwayang}. For the
invariant yield of proton, there are 4 different contributions:
soft-soft-soft ($proton_{ttt}$), soft-soft-shower
($proton_{tts}$), soft-shower-shower ($proton_{tss}$), and
shower-shower-shower ($proton_{sss}$). The total contributions are
$dN/p_Tdp_T=C \times C \times C \times proton_{ttt}+C \times C
\times N_{bin} \times proton_{tts}+C \times N_{bin} \times
proton_{tss}+N_{bin} \times proton_{sss}$, where $C$ is determined
by fitting the d+Au data at $0.5<p_T<1.5$ GeV/c, and $N_{bin}$ is
the number of binary collisions. The data points of $proton_{ttt},
proton _{tts}, proton_{tss}$ and $proton_{sss}$ are from Prof.
C.B. Yang~\cite{yang}. The $C$ values for minimum-bias, 0-20\%,
20-40\% and 40-$\sim$100\% d+Au collisions are 9.67, 12.34, 10.92
and 7.91 individually. The $T$ value of 0.21 GeV is used in the
low $p_{T}$ fit. Figure~\ref{precombination} shows the proton
spectra in d+Au collisions as well as those from recombination
model. This figure shows that the recombination model can
reproduce the spectra of proton in minimum-bias and centrality
selected d+Au collisions. From the comparison between our data and
the calculation results from the recombination model, we know that
the recombination model actually can reproduce both the proton and
pion spectra in d+Au collisions, while as we have mentioned above,
the initial multiple parton scattering model~\cite{accardi} with
independent fragmentation can't reproduce the difference of Cronin
effect between proton and pion. Besides, the initial multiple
parton scattering model with independent fragmentation can't
reproduce the $\eta$ asymmetry of the Cronin effect. In the
recombination model~\cite{hwayang}, the number of such soft
partons on the Au outgoing side is larger than that on the
deuteron outgoing side. This will result in the Cronin effect on
the Au side larger than that on the deuteron side~\cite{hwayang}.
Qualitatively the recombination model can reproduce the $\eta$
asymmetry of the Cronin effect. As we know, the recombination
model is a final-state effect model. These all seem to indicate
that the Cronin effect is not initial-state effect only. The
final-state effect plays an important role too. To directly
confirm the Cronin effect is initial or final state effect, it's
necessary for us to compare the Cronin effect of Drell-Yan process
with those of pion, kaon and proton. I will come to this later.
\subsection{Integral yield $R_{dAu}$: shadowing effect?}
\begin{figure}[h]
\centering
\includegraphics[height=24pc,width=24pc]{integral_rdau.eps}
\caption{Integral yield $R_{dAu}$ as a function of $dN/d\eta$ in
minimum-bias and centrality selected d+Au collisions at
mid-rapidity. Statistic errors and systematic uncertainties have
been added in quadrature. The shadowing represents the
normalization uncertainty.} \label{integralrdau}
\end{figure}
The initial multiple elastic scattering only changes the $p_{T}$
distribution while the total cross section should not change. Thus
we can look at the integral yield $dN/dy$ $R_{dAu}$, which are
measured through comparison to the integral yield $dN/dy$ in p+p
collisions, scaled by the number of binary collisions $N_{bin}$.
Figure~\ref{integralrdau} shows that integral yield $R_{dAu}$ of
pion, kaon and proton as a function of $dN/d\eta$ in minimum-bias
and centrality selected d+Au collisions at mid-rapidity. The
integral yield $R_{dAu}$ of pion and kaon are less than 1 while
that of proton is close to 1. This may be the indication of
shadowing effect at 200 GeV. The integral yield $R_{dAu}$ for
proton is larger than that for kaon and a little bit more larger
than that for pion. This may be the indication that the shadowing
effect is mass dependent at 200 GeV.
\subsection{Initial or final state effect: Drell-Yan process}
In order to see the Cronin effect is initial or final state
effect, we may look into the Drell-Yan process since there is
little final state effect in Drell-Yan process. If there is no
enhancement at high $p_{T}$ for Drell-Yan process, the enhancement
for $\pi, K, p$ is due to final-state effect.
Figure~\ref{drellyan} shows the integral yield Cronin ratio as a
function of atomic weight at p-A fixed target
experiment~\cite{drell}. The proton incident energy is 800 GeV. We
can see there is no enhancement for Drell-Yan process. However,
this is the total cross section while what we want to compare is
Cronin ratio as a function of $p_{T}$. It will be better if we
have the $p_{T}$ dependence of Cronin ratio of Drell-Yan process.
However, at the same $p_T$ range with the same proton incident
energy, the Cronin ratio of Drell-Yan is not available in p+A
collisions. It's hard to compare the Cronin ratio of Drell-Yan
process with those of $\pi, K, p$.
\begin{figure}[h]
\centering
\includegraphics[height=24pc,width=24pc]{fig16.eps}
\caption{Integral yield Cronin ratio as a function of atomic
weight in p+A fixed target experiment for Drell-Yan process, etc.
This figure is from~\cite{drell}.} \label{drellyan}
\end{figure}
\section{Baryon excess in Au+Au collisions}
Now let's come to another important physics from d+Au collisions.
We know that the $(p+\bar{p})/h$ ratio from minimum-bias Au+Au
collisions~\cite{phenixpid} at a similar energy is about a factor
of 2 higher than that in d+Au and p+p collisions for
$p_{T}{}^{>}_{\sim}2.0$ GeV/c. This enhancement is most likely
due to final-state effects in Au+Au collisions. There are many
models trying to explain this baryon excess in Au+Au
collisions~\cite{jetquench,junction,derekhydro,pisahydro,fries,ko}.
In the following baryon production mechanism will be discussed.
\subsection{$\bar{p}/p$ ratio vs $p_T$}
\begin{figure}[h]
\centering
\includegraphics[height=18pc,width=24pc]{pbarp_pQCD.eps}
\caption{$\bar{p}/p$ ratio as a function of $p_{T}$ in d+Au and
p+p minimum-bias collisions. The open squared symbols are for p+p
collisions and the solid circled symbols for d+Au collisions. The
triangled symbols represent the result from Au+Au minimum-bias
collision~\cite{ex0307022}. The curve is the pQCD calculation
results from~\cite{junction} in p+p collisions. Errors are
statistical.} \label{pbarpdiscussion}
\end{figure}
In 200 GeV Au+Au collisions, $\bar{p}/p$ ratio was observed to be
flat with $p_{T}$ till intermediate $p_{T}$
range~\cite{ex0307022}, as shown in Figure~\ref{pbarpdiscussion}.
The baryon junction model~\cite{junction} tried to explain it by
using junction anti-junction production with jet quenching, on the
basis of pQCD calculation~\cite{junction} where the $\bar{p}/p$
ratio decreases with $p_{T}$ in p+p collisions. The curve from
pQCD calculation~\cite{junction} is also shown in
Figure~\ref{pbarpdiscussion}. However, $\bar{p}/p$ ratios in d+Au
and p+p collisions in our data show to be flat with $p_{T}$ within
errors. Anyway, the precise measurement with more statistics in
p+p and d+Au collisions is needed to address this issue.
\subsection{Baryon production at RHIC: multi-gluon dynamics?}
\begin{figure}[h]
\centering
\includegraphics[height=18pc,width=18pc]{pnch_ee.eps}
\caption{Minimum-bias ratios of ($p+\bar{p}$) over charged hadrons
at $-0.5\!<\!\eta\!<\!0.0$ from $\sqrt{s_{_{NN}}} =200$ GeV p+p
(open diamonds) and d+Au (filled triangles) collisions. Also shown
are the $(p+\bar{p})/h$ ratios in $e^{+}e^{-}$ collisions at
ARGUS~\cite{argus}. The solid line represents the $(p+\bar{p})/h$
ratio from three gluon hadronization while the dashed line for the
ratio from quark and antiquark fragmentation~\cite{argus}. Errors
are statistical.} \label{pbarpnchcomparison}
\end{figure}
Let's compare the $(p+\bar{p})/h$ ratio in p+p, d+Au and Au+Au
collisions at RHIC energy 200 GeV with the ratio in $e^{+}e^{-}$
collisions at ARGUS~\cite{argus}. Using the ARGUS detector at the
$e^{+}e^{-}$ storage ring DORIS II, the inclusive production of
pion, kaon and proton in multihadron events at 9.98 GeV and in
direct decays of the $\Upsilon(1S)$ meson were
investigated~\cite{argus}. Multihadron final states in
$e^{+}e^{-}$ annihilation are produced via quark and antiquark
fragmentation, and those from direct $\Upsilon(1S)$ decays
originate from the hadronization of three gluons~\cite{argus}.
Figure~\ref{pbarpnchcomparison} shows the $(p+\bar{p})/h$ ratio in
200 GeV p+p collisions together with the ratio in $e^{+}e^{-}$
collisions at ARGUS~\cite{argus}. The plot shows that the
$(p+\bar{p})/h$ ratio from three gluon hadronization is a factor
of 3 higher than that from quark and antiquark fragmentation at
ARGUS. Our data from 200 GeV p+p collisions is close to
$(p+\bar{p})/h$ ratio from 3 gluon hadronization. This may be the
indication that in the heavy ion collisions at RHIC energy,
multi-gluon hadronization plays an important role for the particle
production.
\chapter{Physics}
\label{chp:physics}
\section{Deconfinement and phase diagram}
The theory which describes the interaction of the color charges of
quarks and gluons is called Quantum Chromodynamics (QCD). In
phenomenological quark models, mesons can be described as
quark-antiquark bound states, while baryons can be considered as
three quark bound states. Up to now, it's found that all the
hadron states which can be observed in isolation is colorless
singlet states. Experimentally, no single quark, which is
described by a color-triplet state, has ever been isolated. The
absence of the observation of a single quark in isolation suggests
that the interaction between quarks and gluons must be strong on
large distance scale. In the other extreme, much insight into the
nature of the interaction between quarks and gluons on short
distance scales was provides by deep inelastic scattering
experiments. In these experiments, the incident electron interacts
with a quark within a hadron and is accompanied by the momentum
transfer from the electron to the quark. The measurement of the
electron momentum before and after the interaction allows a probe
of the momentum distribution of the parton inside the nucleon. It
was found that with very large momentum transfer, the quarks
inside the hadron behave as if they were almost free~\cite{QCD}.
The strong coupling between quarks and gluons at large distances
and asymptotic freedom are the two remarkable features of QCD.
When the energy density is high enough either due to the high
temperature or high baryon density, the quark or gluon may be
deconfined from a hadron. The thermalized quark gluon system is
what we called quark-gluon plasma. Lattice QCD calculations,
considering two light quark flavors, predict a phase transition
from a confined phase, hadronic matter, to a deconfined phase, or
quark-gluon plasma (QGP), at a temperature of approximately
\begin{figure}[h]
\centering
\includegraphics[height=28pc,width=28pc]{phase_diagram.eps}
\caption{Phase diagram of hadronic and partonic matter. Figure is
taken from~\cite{pbm:01}.}
\end{figure}
160 MeV~\cite{harris:98}. Figure 1.1 shows the phase diagram of
the hadronic and partonic matter. A phase transition from the
confined hadronic matter to the deconfined QGP matter is expected
to happen at either high temperature or large baryon chemical
potential $\mu_B$. Recent Lattice QCD calculations show that the
QGP is far from ideal below 3 $T_{c}$. The nonideal nature of this
strongly coupled QGP is also seen from the deviation of the
pressure, $P(T)$, and energy density $\epsilon(T)$ from the Stefan
Boltzmann limit as shown in Figure from~\cite{Fodor}.
\begin{figure}[h]
\centering
\includegraphics[height=16pc,width=28pc]{FodorKatz_P_e_mu0.eps}
\caption{A recent Lattice QCD calculation \protect{\cite{Fodor}}
of the pressure, $P(T)/T^4$, and a measure of the deviation
from the ideal Stefan-Boltzmann limit $(\epsilon(T)-3 P(T))/T^4$.}
\label{Fodorplot}
\end{figure}
Experiments on relativistic heavy ion collisions are designed to
search for and study the deconfined QGP matter.
\section{Relativistic Heavy Ion Collisions}
The experimental programs in relativistic heavy ions started in
1986 using the Alternating Gradient Synchrotron (AGS) at
Brookhaven National Lab (BNL) and the Super Proton Synchrotron
(SPS) at European laboratory for particle physics (CERN). At BNL,
ion beams of silicon and gold, accelerated to momenta of 14 and 11
GeV/c per nucleon, respectively, have been utilized in 10
fixed-target experiments. There have been 15 heavy ion experiments
at CERN utilizing beams of oxygen at 60 and 200 GeV/c per nucleon,
sulphur at 200 GeV/c per nucleon and Pb at 160 GeV/c
per nucleon~\cite{harris:98}.\\
The Relativistic Heavy Ion Collider (RHIC) at BNL is designed for
head-on Au+Au collisions at $\sqrt{s_{NN}}$ = 200 GeV. The first
RHIC run was performed in 2000 with Au+Au collisions at
$\sqrt{s_{NN}}$ = 130 GeV/c in four experiments, STAR, PHENIX,
PHOBOS and BRAHMS. The second RHIC run was in 2001 and 2002 with
Au+Au and p+p collisions at $\sqrt{s_{NN}}$ = 200 GeV. The third
RHIC run was in 2002 and 2003 with
d+Au and p+p collisions at $\sqrt{s_{NN}}$ = 200 GeV.\\
The above mentioned relativistic heavy ion collision experiments
are designed for the search and study of the possible deconfined
high energy density matter, quark-gluon plasma. In head-on
relativistic heavy ion collisions, two nuclei can be represented
as two thin disks approaching each other at high speed because of
the Lorentz contraction effect in the moving direction. During the
initial stage of the collisions, the energy density is higher than
the critical energy density from the Lattice QCD calculation, so
the quarks and gluons will be de-confined from nucleons and form
the quarks and gluons system. The large cross section of
interaction may lead to the thermalization of the quarks and
gluons system. That's what we called the formation of quark-gluon
plasma (QGP). In this stage, the high transverse momentum
($p_{T}$) jets and $c\bar{c}$ pair will be produced due to the
large momentum transfer. After that, the QGP will expand and cool
down and enter into the mixed-phase expansion. The chemical freeze
out point will be formed after the inelastic interactions stop.
That means that the particle yields and ratios will not change.
After the chemical freeze out, the elastic interactions between
hadrons will change the $p_{T}$ distribution of particles. The
particles will freeze out finally from the system after the
elastic interactions stop. That's what we called the kinetic
freeze out point. In the following the important results from RHIC
will be addressed.
\section{The experimental results at RHIC}
\subsection{Flow}
In non-central Au+Au collisions, the spatial space asymmetry will
be transferred into the momentum space asymmetry by the azimuthal
asymmetry of pressure gradients.
\begin{figure}[h]
\centering
\includegraphics[height=16pc,width=22pc]{Ks_Lam_fig1_color_2.eps}
\caption{The minimum-bias (0--80\% of the collision cross section)
$v_{2}(p_T)$ for $K_{S}^{0}$, $\Lambda+\overline{\Lambda}$ and
$h^{\pm}$. The error bars shown include statistical and
point-to-point systematic uncertainties from the background. The
additional non-flow systematic uncertainties are approximately
-20\%. Hydrodynamical calculations of $v_2$ for pions, kaons,
protons and lambdas are also plotted~\cite{hydroPasi01}. Figure is
taken from~\cite{starv2raa}.} \label{KsLamv2}
\end{figure}
The azimuthal particle distributions in
momentum space can be expanded in a form of Fourier series
\begin{equation}
E\frac{d^3N}{d^3p}=\frac{1}{2\pi}\frac{d^2N}{p_Tdp_Tdy}(1+
\sum^{\infty}_{n=1}2v_n\cos[n(\phi-\Psi_r)])
\end{equation}
where $\Psi_r$ denotes the reaction plane angle. The Fourier
expansion coefficient $v_n$ stands for the $n$th harmonic of the
event azimuthal anisotropies. $v_1$ is so called direct flow and
$v_2$ is the elliptic flow. The elliptic flow is generated mainly
during the highest density phase of the evolution before the
initial geometry asymmetry of the plasma disappears.
Hydrodynamical calculations~\cite{derekhydro} show most of $v_2$
is produced before 3 fm/c at RHIC.\\
Figure~\ref{KsLamv2} shows that The $v_2$ of $K_{S}^{0}$,
$\Lambda+\overline{\Lambda}$ and charged hadrons ($h^{\pm}$) as a
function of $p_T$ for 0--80\% of the collision cross
section~\cite{starv2raa}. Also shown are the $v_2$ of pions,
kaons, protons and lambdas from hydrodynamical
model~\cite{hydroPasi01}. The $v_2$ from hydrodynamical model
shows strong mass dependence, which fits the $K_{S}^{0}$ $v_2$ up
to $p_{T}\sim1$ GeV/c and fits the $\Lambda+\overline{\Lambda}$
$v_2$ up to $p_{T}\sim2.5$ GeV/c. Even though the $v_2$ from
hydrodynamical model shows consistency with data at low $p_T$,
however, the $v_2$ from experimental results show saturation at
intermediate $p_{T}$ while hydrodynamical predictions show rising
trend at the same $p_T$ range.
\subsection{High $p_{T}$ suppression and di-hadron azimuthal correlation}
The $v_2$ from hydrodynamical models show consistency with data at
lower $p_T$ and fail to reproduce data at higher $p_{T}$. At high
$p_{T}$, the suppression for charged hadron production was
observed in Au+Au collisions at RHIC energy. The comparison of the
spectra in Au+Au collisions through those in p+p collisions,
scaled by the number of binary nucleon nucleon collisions is the
nuclear modification factor $R_{AA}$.
\begin{equation}
R_{AA}(p_T)=\frac{d^2N^{AA}/dp_Td\eta}{T_{AA}d^2\sigma^{NN}/dp_Td\eta}
\end{equation}
where $T_{AA}=\langle N_{\text{bin}} \rangle
/\sigma^{NN}_{\text{inel}}$ accounts for the collision geometry,
averaged over the event centrality class. $\langle N_{\text{bin}}
\rangle$, the equivalent number of binary $NN$ collisions, is
calculated using a Glauber model. The $R_{AA}$ is an experimental
variable. The high $p_T$ hadron suppression in central Au+Au
collisions can also be investigated by comparing the hadron
spectra in central and peripheral Au+Au collisions. That's what we
called $R_{CP}$. $R_{CP}$ is defined as
\begin{equation}
R_{CP}=\frac{\langle N_{\text{bin}}^{\text{peripheral}} \rangle
d^2N^{\text{central}}/dp_Td\eta}{\langle
N_{\text{bin}}^{\text{central}} \rangle
d^2N^{\text{peripheral}}/dp_Td\eta}.
\end{equation}\\
\begin{figure}[h]
\centering
\includegraphics[height=20pc,width=26pc]{highpt_200.eps}
\caption{$R_{AA}(p_T)$ of inclusive charged hadron for various
centrality bins. Figure is taken from ~\cite{starhighpt}.}
\label{raa200}
\end{figure}
Figure~\ref{raa200} shows $R_{AA}(p_T)$ of inclusive charged
hadron for various centrality bins in Au+Au collisions at
$\sqrt{s_{NN}}$=200 GeV. $R_{AA}(p_T)$ increases monotonically for
$p_T<$ 2 GeV/c at all centralities and saturates near unity for
$p_T>$ 2 GeV/c in the most peripheral bins. In contrast,
$R_{AA}(p_T)$ for the central bins reaches a maximum and then
decreases strongly above $p_T$ = 2 GeV/c, showing the suppression
of the charged hadron yield relative the $NN$
reference~\cite{starhighpt}.\\
Suppression of high $p_{T}$ hadron production in central Au+Au
collisions relative to p+p collisions
~\cite{starhighpt,phenixhighpt} has been interpreted as energy
loss of the energetic partons traversing the produced hot and
dense medium~\cite{jetquench}, that's so called jet quenching. If
a dense partonic matter is formed during the initial stage of a
heavy-ion collision with a large volume and a long life time
(relative to the confinement scale $1/\Lambda_{\rm QCD}$), the
produced large $E_T$ parton will interact with this dense medium
and will lose its energy via induced radiation. The energy loss
depends on the parton density of the medium. Therefore, the study
of parton energy loss can shed light on the properties of the
dense matter in the early stage of heavy-ion
collisions~\cite{jetquench}. At sufficiently high beam energy,
gluon saturation is also expected to result in a relative
suppression of hadron yield at high $p_{T}$ in A+A
collisions~\cite{cgc}. Also shown in the Figure~\ref{raa200} are
the results from perturbative QCD (pQCD) calculations. The
Full-pQCD calculations include the partonic energy loss, the
Cronin enhancement(due to initial multiple scattering) and the
nuclear shadowing effect. The suppression is predicted to be $p_T$
independent when $p_T$ is larger than 6 GeV/c, which is consistent
with our data. However, the discrepancy at 2-6 GeV/c was observed
between the prediction and the experimental data. This discrepancy
may be due to different mechanism for particle production at
intermediate $p_T$. The particle production at intermediate $p_T$
will be discussed later in this chapter.\\
\begin{figure}[tbh]
\begin{minipage}{0.49\textwidth}
\includegraphics[width=0.95\textwidth,angle=-90]{auau_reactionplane.eps}
\end{minipage}\hfill\hspace{2.5cm}
\begin{minipage}{0.49\textwidth}\vspace{-0.8cm}
\includegraphics[width=0.80\textwidth]{QMproc_highlight_4_Kai.eps}
\end{minipage}
\caption{(a) Azimuthal distribution of particles with respect to a
trigger particle for p+p collisions (solid line), and mid-central
Au+Au collisions within the reaction plane (squares) and
out-of-plane (circles) at 200 GeV~\cite{Aihong}. (b) Mean
transverse momentum for particles around the away-side region as a
function of number of charged particles~\cite{Fuqiang}. The solid
line shows the mean transverse momentum of inclusive hadrons.}
\label{jetplot}
\end{figure}
A more differential probe of parton energy loss is the measurement
of high $p_T$ di-hadron azimuthal correlation relative to the
reaction plane orientation. The trigger hadron is in the range
$4<p_{T}<6$ GeV/c and the associated particle is at $2<p_{T}<4$
GeV/c. Figure~\ref{jetplot} (left) shows the high $p_T$ di-hadron
correlation when the trigger particle is selected in the azimuthal
quadrants centered either in the reaction plane (in plane) or
orthogonal to it(out of plane). The near side di-hadron azimuthal
correlations in both cases were observed to be the same as that in
p+p collisions, while the suppression of back to back correlation
shows strong dependence on the relative angle between the
triggered high $p_T$ hadron and the reaction plane. This
systematic dependence is consistent with the picture of parton
energy loss: the path length for a dijet oriented out of plane is
longer than that for a dijet oriented in plane, leading to a
stronger suppression of parton energy loss in the out of plane.
The dependence of parton energy loss on the path length is
predicted to be substantially larger than
linear~\cite{jetquench}.\\
The energy lost by away side partons traversing the collision
matter must in the form of the excess of softer emerging particles
due to the transverse momentum conservation. An analysis of
azimuthal correlations between soft and hard particles has been
performed for both 200 GeV p+p and Au+Au collisions~\cite{Fuqiang}
at STAR as a first of attempt to trace the degree of the
degradation on the away side. With triggered hadron still in the
range $4<p_{T}^{trig}<$ 6 GeV/c, but the associated hadrons now
sought over $0.15<p_{T}<4$ GeV/c, combinatorial coincidences
dominate this correlation and they must be subtracted carefully by
mixed-event technique and also the elliptic flow effect was also
subtracted by hand~\cite{Fuqiang}. The results demonstrate that,
in comparison with the p+p and peripheral Au+Au collisions, the
momentum-balancing hadrons opposite to the high $p_T$ triggered
particle in central Au+Au are greater in number, much more widely
dispersed in azimuthal angle, and significantly softer in
momentum. Figure~\ref{jetplot} (right) shows the $\langle p_{T}
\rangle$ of the momentum-balancing hadrons opposite to the high
$p_T$ trigger as a function of centrality. The $\langle
p_{T}\rangle$ were observed to decrease from peripheral to central
Au+Au collisions. Also shown in the Figure~\ref{jetplot} (right)
is the $\langle p_{T}\rangle$ of the inclusive hadrons as a
function of centrality. This study will be extended to higher
$p_T$ trigger particle. The results may suggest that the
moderately hard parton traversing a significant path length
through the collision matter makes substantial progress toward
equilibrium with the bulk. The rapid attainment of thermalization
via multitude of softer parton-parton interactions in the earliest
collision stages would then not be so
surprising~\cite{starwhitepaper}.
\subsection{Particle composition in Au+Au at intermediate $p_{T}$}
As we have mentioned above, for $R_{AA}$, the pQCD model including
the parton energy loss, Cronin enhancement and nuclear shadowing
can qualitatively fit the trend of data at $2<p_{T}<6$ GeV/c,
however, the quantitative discrepancy between the model and the
data is also obvious. In the intermediate $p_{T}$, the mechanism
for particle production may be different from that at
high $p_{T}$.\\
Figure~\ref{phenixAuAuspectra} (left) shows the $\pi$, K, p
spectra in 0\%-5\% and 60\%-92\% Au+Au 200 GeV collisions
from~\cite{ex0307022}. It shows that the shapes of the spectra
show clear mass dependence. And in central collisions, the $\pi$,
K, p yields are close to each other at $p_{T}>2$ GeV/c while it's
not the case in peripheral collisions.
Figure~\ref{phenixAuAuspectra} (right) shows proton/pion (top) and
anti-proton/pion (bottom) ratios for central 0--10\%, mid-central
20--30\% and peripheral 60--92\% in Au+Au collisions at 200
GeV~\cite{ex0307022}. It shows that the $p(\bar{p})/\pi$ ratios
increase fast from peripheral to central collisions. In the 0-10\%
centrality bin, the proton yield is even larger than pion yield at
intermediate $p_T$. Figure~\ref{KsLamRcp} shows the ratio $R_{CP}$
for identified mesons and baryons at mid-rapidity calculated using
centrality intervals, 0--5\% vs. 40--60\% of the collision cross
section from STAR measurement~\cite{starhighlight}. It seems that
for meson, the $R_{CP}$ follows a common trend and for baryon, the
$R_{CP}$ also follows a common trend, which is different from that
for mesons. The $R_{CP}$ for baryons is observed to be larger than
that for mesons.\\
\begin{figure}[h]
\begin{minipage}[t]{80mm}
\includegraphics[height=16pc,width=16pc]{pt_spectra_all.eps}
\end{minipage}
\hspace{\fill}
\begin{minipage}[t]{80mm}
\includegraphics[height=16pc,width=16pc]{ppi_ratio_all.eps}
\end{minipage}
\caption{(left) The $\pi$, K, p spectra in 0\%-5\% and 60\%-92\%
Au+Au 200 GeV collisions from~\cite{ex0307022}. (right)
Proton/pion (top) and anti-proton/pion (bottom) ratios for central
0--10\%, mid-central 20--30\% and peripheral 60--92\% in Au+Au
collisions at 200 GeV. Open (filled) points are for charged
(neutral) pions. The data at $\sqrt{s} = 53 $~GeV p+p
collisions~\cite{ISR} are also shown. The solid line is the
$(\bar{p} + p)/(\pi^{+} + \pi^{-})$ ratio measured in gluon
jets~\cite{DELPHI}. This figure is from~\cite{ex0307022}. }
\label{phenixAuAuspectra}
\end{figure}
These experimental results suggest that the degree of suppression
depends on particle species(baryon/meson) at intermediate $p_T$.
The spectra of baryons (protons and lambdas) are less suppressed
than those of mesons (pions, kaons) ~\cite{starv2raa,phenixpid} in
the $p_{T}$ range $2<p_{T}<5$ GeV/c. The baryon content in the
hadrons at intermediate $p_{T}$ depends strongly on the impact
parameter (centrality) of the Au+Au collisions with about 40\% of
the hadrons being baryons in the minimum-bias collisions and 20\%
in very peripheral collisions~\cite{starv2raa,phenixpid}.
Hydrodynamics~\cite{derekhydro,pisahydro}, parton coalescence at
hadronization~\cite{hwa,fries,ko} and gluon
junctions~\cite{junction} have been suggested as explanations for
the observed particle-species dependence.
\begin{figure}[tbph]
\centering\mbox{
\includegraphics[width=0.9\textwidth]{plot_rcp_24jan04_1.eps}}
\caption{ The ratio $R_{CP}$ for identified mesons and baryons at
mid-rapidity calculated using centrality intervals, 0--5\% vs.
40--60\% of the collision cross section. The bands represent the
uncertainties in the model calculations of $\mathrm{N_{bin}}$. We
also show the charged hadron $R_{CP}$ measured by STAR for
$\sqrt{s_{_{NN}}}=200$~GeV~\cite{starhighpt}. This figure is
from~\cite{starhighlight}.} \label{KsLamRcp}
\end{figure}
In these models, recombination/coalescence models successfully
reproduce $R_{AA}$ of baryons and mesons at intermediate $p_T$, as
well as showing consistency with the $v2$ measurement in the same
$p_T$ range.
\subsubsection{Recombination model}
The concept of quark recombination was introduced to describe
hadron production at forward rapidity in p+p
collisions~\cite{ref50:whitepaper}. At forward rapidity, this
mechanism allows a fast quark resulting from a hard parton
scattering to recombine with a slow anti-quark, which could be one
in the original sea in the incident hadron, or one incited by a
gluon~\cite{ref50:whitepaper}. If a QGP is formed in the
relativistic heavy ion collisions, then one might expect
coalescence of the abundant thermal partons to provide another
important hadron production mechanism, active over a wide range of
rapidity and transverse momentum~\cite{ref51:whitepaper}. In
particular, at moderate $p_T$ values(above the realm of
hydrodynamics applicability), the hadron production from
recombination of lower $p_T$ partons from thermal
bath~\cite{hwa,fries,ko} has been predicted to be competitive with
the production from fragmentation of higher $p_T$ scattered
partons. It has been suggested~\cite{ref53:whitepaper} that the
need for substantial recombination to explain the observed hadron
yield and flow may be taken as a signature of QGP
formation.\\
In order to explain the features of RHIC collisions, the
recombination models~\cite{ref51:whitepaper,hwa,fries,ko} make the
central assumption that coalescence proceeds via constituent
quarks, whose number in a hadron determines its production rate.
The constituent quarks are presumed to follow a thermal
(exponential) momentum spectrum and to carry a collective
transverse velocity distributions. This picture leads to clear
predicted effects on baryon and meson production rates, with the
former depending on the spectrum of thermal constituent quarks and
antiquarks at roughly one-third the baryon $p_T$, and the latter
determined by the spectrum at roughly one-half the meson $p_T$.
Indeed, the recombination model was recently was re-introduced at
RHIC context, precisely to explain the abnormal abundance of
baryon vs meson observed at intermediate
$p_T$~\cite{hwa,fries,ko}. If the observed saturated elliptic flow
values of hadrons in this momentum range result from coalescence
of collectively flowing constituent quarks, then one expect a
similarly simple baryon vs meson relationship~\cite{hwa,fries,ko}:
the baryon (meson) flow would be 3 (2) times the quark flow at
roughly one-third (one-half) the baryon (meson)
$p_T$~\cite{starwhitepaper}.
\subsection{Summary}
In summary, the several important results from RHIC have been
introduced. The elliptic flow $v2$ can be reproduced by
hydrodynamics at low $p_T$. At intermediate $p_T$, $v2$ from data
show saturation and deviate from hydrodynamical model predictions.
At the same time, $v2$ from data show baryon or meson species
dependence. High $p_T$ suppression can be reproduced by pQCD model
and gluon saturation model. The gluon saturation model is also
called color glass condensate model (CGC). The production rate
dependence on baryon or meson species has been observed at
intermediate $p_{T}$, which can be reproduced by the recombination
model.
\section{Cronin effect}
\subsection{Why we need d+Au run at RHIC}
In order to see the intermediate and high $p_T$ suppression is due
to the final-state effect or initial state effect, the
measurements from d+Au collisions will provide the essential
proof. Since the initial state in d+Au collisions is similar to
that in Au+Au collisions, and, it's believed that the quark-gluon
plasma doesn't exist in d+Au collisions, the results from d+Au
collisions will be very important for us to judge whether the
quark-gluon plasma exists in Au+Au collisions or not and to
understand the property of the dense matter created in Au+Au
collisions. Besides, if the identified particle spectra in d+Au
and p+p collisions are measured, they will not only provide the
reference for those in Au+Au collisions at 200 GeV, but also
provide a chance to see the mechanism of the Cronin effect itself
clearly at 200 GeV. Cronin effect was observed 30 years ago
experimentally and the study of this effect was only limited to
lower energy fixed target experiments. Before we go to the d+Au
collisions, let's look back on the p+A collisions at lower energy
fixed target experiment.
\subsection{Lower energy}
The hadron $p_{T}$ spectra have been observed to depend on the
target atomic weight ($A$) and the produced particle species in
lower energy p+A collisions~\cite{cronin}. This is known as the
``Cronin Effect'', a generic term for the experimentally observed
broadening of the transverse momentum distributions at
intermediate $p_{T}$ in p+A collisions as compared to those in p+p
collisions~\cite{cronin,petersson83,accardi}. The effect can be
characterized as a dependence of the yield on the target atomic
weight as $A^{\alpha}$. At energies of $\sqrt{s} \simeq$ 30 GeV,
$\alpha$ depends on $p_{T}$ and is greater than unity at high
$p_{T}$~\cite{cronin}, indicating an enhancement of the production
cross section. As shown in Figure~\ref{poweralphaplot}, the
$\alpha$ is larger than 1 in the intermediate $p_{T}$ and shows
strong particle-species dependence. The $\alpha$ for proton and
antiproton are larger than those for kaon and pion. And $\alpha$
for kaon is larger than that for pion. This effect has been
interpreted as partonic scatterings at the initial
impact~\cite{petersson83,accardi}.
\begin{figure}[h]
\begin{minipage}[t]{80mm}
\includegraphics[height=18pc,width=18pc]{poweralpha.eps}
\end{minipage}
\hspace{\fill}
\begin{minipage}[t]{80mm}
\includegraphics[height=18pc,width=18pc]{R_woverbe_energy.eps}
\end{minipage}
\caption{(left) The power alpha of A dependence from 300 GeV
incident proton-fixed target experiment. This figure is
from~\cite{cronin}. (right) The Cronin ratio $R_{W/B_{e}}$ at
$p_{T}=4.61$ GeV/c versus energy. This plot is
from~\cite{cronin}.} \label{poweralphaplot}
\end{figure}
Besides, the lower energy data suggest the power $\alpha$
decreases with energy, as shown in Figure~\ref{poweralphaplot}.
However, the energy dependence study of Cronin effect is limited
to fixed target experiment at lower energy. What's the
extrapolation of Cronin effect at higher energy such as RHIC
energy 200 GeV. At higher energies, multiple parton collisions are
possible even in p+p collisions~\cite{e735kno}. This combined with
the hardening of the spectra with increasing beam energy would
reduce the Cronin effect~\cite{accardi}. There are several models
which give different predictions of Cronin effect at 200 GeV.
\subsection{Predictions: RHIC energy}
One of the models is the initial multiple parton scattering model.
In this model, the transverse momentum of the parton inside the
proton will be broadened when the proton traverses the Au nucleus
due to the multiple scattering between the proton and the nucleons
inside the Au nucleus. In these models, the Cronin ratio will
increases to a maximum value between 1 and 2 at 2.5$<p_T<$4.5
GeV/c and then decreases with $p_T$ increasing~\cite{accardi}. The
Cronin effect is predicted to be larger in central d+Au collisions
than in d+Au peripheral collisions~\cite{Vitev03}. Another model
is the gluon saturation model. At sufficiently high beam energy,
gluon saturation is expected to result in a relative suppression
of hadron yield at high $p_{T}$ in both p+A and A+A collisions and
in a substantial decrease and finally in the disappearance of the
Cronin effect~\cite{cgc}.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{dAu_Fig3.eps}
\caption{ $R_{AB}$ for minimum bias and central d+Au collisions,
and central Au+Au collisions~\cite{starhighpt}. The minimum bias
d+Au data are displaced 100 MeV/c to the right for clarity. The
bands show the normalization uncertainties, which are highly
correlated point-to-point and between the two d+Au distributions.
This Figure is from~\cite{stardau}.} \label{FigThree}
\end{figure}
Figure~\ref{FigThree} shows the $R_{dAu}$ of charged hadron vs
$p_{T}$ from STAR. We can see that the Cronin ratio increases to a
maximum value around 1.5 at $3<p_{T}<4$ GeV/c and then decreases
again~\cite{stardau}. This is consistent with the initial multiple
parton scattering model~\cite{accardi}. These results on inclusive
hadron production from d+Au collisions indicate that hadron
suppression at intermediate and high $p_{T}$ in Au+Au collisions
is due to final state interactions in a dense and dissipative
medium produced during the collision and not due to the
initial state wave function of the Au nucleus~\cite{stardau,otherdau}.\\
Now we know that the hadron suppression at intermediate $p_{T}$ in
Au+Au collisions is due to final-state
effects~\cite{stardau,otherdau}. What's the effect on particle
composition at the same $p_{T}$ range in Au+Au collisions? Another
question is whether there is any Cronin effect dependence on
particle-species in d+Au collisions or not. In order to further
understand the mechanisms responsible for the particle dependence
of $p_{T}$ spectra in heavy ion collisions, and to separate the
effects of initial and final partonic rescatterings, we measured
the $p_{T}$ distributions of $\pi^{\pm}$, $K^{\pm}$, $p$ and
$\bar{p}$ from 200 GeV d+Au and p+p collisions. In this thesis, we
discuss the dependence of particle production on $p_{T}$,
collision energy, and target atomic weight. And we compare the
Cronin effect of $\pi^{\pm}$, $K^{\pm}$, $p$ and $\bar{p}$ with
models to address the mechanism for Cronin effect in d+Au
collisions at $\sqrt{s_{_{NN}}} = 200$ GeV.
\chapter{{\hspace{3.5cm}How to make MRPC}}
This appendix is based on the procedure of the MRPC production in
USTC. I will introduce the material preparations and then the
chamber installation.
\section{Preparations}
\subsection{Glass} (1) Check the glass very carefully by eye. The
glass with scrapes is not accepted. (2) Measure the size of the
glass with the digital vernier caliper. The errors of the length
and width are required to be within 0.1 mm. Measure the thickness
in several different places. The precision of the thickness is
required to be 0.01 mm for each glass. (3) Use the micrometer to
measure the flatness, which is required to be less than 0.01 mm.
Use the mirror and observe the stripes of interference. (4) Grind
the edge and the corner of the glass, and clean it. The size of
outer glass is $78(width)\times 206(length) \times 1.1(height)$
$mm^3$ and the size of inner glass is $61\times 200\times 0.54$
$mm^3$.
\subsection{Graphite Layer} (1) Stick the layer in the middle of the outer
glass. Squeeze the air out. (2) Stick a small copper tape, which
is for high voltage (HV) applying, on to the graphite layer, which
is in the middle of the long side, and 0$\sim$0.5 mm away from the
edge of the glass. The size of the graphite layer is $74\times202$
$mm^2$. The size of the copper tape (the rectangle with the round
angle) is $6\times10$ $mm^2$.
\subsection{Mylar layer} Cut the mylar layer, and see if there is any tiny holes or
scrapes. If yes, don't use it. The size of mylar is
$84\times212\times0.35$ $mm^3$.
\subsection{Honeycomb board} Measure the size and flatness. The
error of the length and width is required to be within 0.2 mm, the
error of the thickness is required to be within 0.05 mm. The
flatness is required to be within 0.1 mm. The size of honeycomb
board is $84\times 208\times 4$ $mm^3$.
\subsection{The printed circuit board (PCB)} (1) Check the surface of the metal
which is used as read-out strips carefully, and see the position
of the HV-holes is right or not. Check the size of the metal
holes, whose diameters are required to be larger than 0.9 mm. (2)
Use double side tape to stick the PCB board with the Mylar. The
size of the double side tape is the same as the PCB board. The
length of the mylar is 1 mm longer than that of the PCB board. (3)
Use sealing ion to open a $\phi$ 3 mm hole, the center of the hole
is in the middle of the HV holes. The size of PCB is $94\times
210\times 1.5$ $mm^3$. The size of metal holes are $\phi$ 1 mm.
\subsection{Lucite cylinder}
Use digital vernier caliper to measure the length of the Lucite
cylinder. Clean it and stick a double side tape on one side. The
size is $\phi$ 3 mm and $3.87<length<3.93$ mm.
\subsection{Other stuff}
Besides, we also need pins, fish line and little plastic cannula.
The size of the pin is 2-2.1 cm long. The fish line is $\phi$ 0.22
mm. The plastic cannula is $\phi$ 1 mm. One kind of the cannula is
7 mm long, and the other is 5.6 mm long. Table~\ref{mrpcmaterial}
lists the main materials for 1 MRPC.
\section{Installation}
\subsection{The outer glass and mylar and PCB}
(1) stick the outer glass on to the center of the mylar. (2) Use
the sealing ion to connect the HV conductive line with the copper
tape. Apply the HV to measure the noise rate and dark current. (3)
Between the mylar and outer glass edge, on each side, use silica
gel to seal. Attention: keep the surface clean and smooth.
Attention: If one side is done, wait till the silica gel becomes
solid. (4) Stick pins. Seal the pins which are used for the fish
line coiling, into the metal holes of the PCB board. (5) Use the
inner glass to fix on the position of Lucite cylinders, and keep
them away from the pins for fish line. Then stick the 8$\sim$10
Lucite cylinders onto the outer glass.
\subsection{Inner glass and fish-line coiling} (1) Pre-install.
Don't use fish-line. Pay attention to adjust the position of the
pins. (2) This is now the real installation and fish line coiling.
Clean the outer glass and inner glass carefully, coil a loop of
fish line, add a piece of glass, then coil another loop of fish
line, add another piece of glass, and so on and so forth.
Attention: clean the fish line before it coils, and blow the
surface of glass to protect it from the dirt with nitrogen jet.
(3) Another time for pre-installation. Pay attention to the
position of the upper and lower electrodes and adjust the position
of pins. (4) Paste 3140 RTV coating onto the surface of Lucite
cylinders. (5) Connect the two electrodes. Make sure all the pins
connect right into the metal holes. Then lay the whole flat, and
put on a block which is 4 kilogram weight. (6) After 2 hours,
stick the honeycomb. (7) Measure the thickness of the whole. Make
sure the precision is within 0.05 mm. (8) Connect the
conductive-line for the read-out strips, and then put the whole
into a bag. Attention: the conductive-line should not be broken.
\begin{table*}
\begin{scriptsize}
\centering
\begin{tabular}{|c|c|c|c|c|} \hline
material& type & character & number & source \\ \hline
outer glass& window glass & 1.1 mm thick, VR: $8.7\times 10^{12}$
ohm.cm & 2 & Shanghai \\ \hline inner glass&window glass & 0.54 mm
thick, VR: $8.5\times 10^{12}$ ohm.cm & 5 & USA \\ \hline graphite
layer& T9149& 0.13 mm thick, SR : 2M ohm/square & 2 & Japan \\
\hline Mylar& M0 & 0.35 mm thick& 2& Dupont Corp.
\\ \hline
honeycomb & & 4 mm thick & 2& Shanghai \\ \hline PCB & gold & 1.58
mm thick& 2 & Shenzhen \\ \hline copper tape & & 0.08 mm thick & 2
& 3M Comp. \\ \hline LC & Lucite& $\phi$ 3 mm,3.9 mm long & 8-10 &
processing \\ \hline pins (single) &metal pin&21.5 mm long& 14 &
\\ \hline pins (pair) &metal pin&21.5 mm long& 12 &
\\ \hline cannula & F-plastic & $\phi$ 1.4 mm & 38 & \\ \hline
fish line & top line & $\phi$ 0.22 mm & &Switzerland \\
\hline DST & 9690& 0.13 mm thick && 3M Comp. \\ \hline silica gel
& CAF4 & high-voltage insulation & &Switzerland \\ \hline
\end{tabular}
\caption{The material for 1 MRPC model. VR is the volume
resistivity and SR is the surface resistivity. LC is the Lucite
cylinder. DST is the double side tape.} \label{mrpcmaterial}
\end{scriptsize}
\end{table*}
\chapter{{\hspace{3.5cm}List of Publications}}
\hspace{0.7cm}1. \emph{Pion, kaon, proton and anti-proton
transverse momentum distributions from p+p and d+Au collisions at
$\sqrt{s_{NN}}$ = 200 GeV,} STAR Collaboration, e-Print Archives
(nu-ex/0309012), submitted.
2. \emph{Open Charm Yields in 200 GeV p+p and d+Au Collisions at
RHIC,} Lijuan Ruan (for the STAR Collaboration), Journal of
Physics G, 30 (2004) S1197-S1200, contributed to 17th
International Conference on Ultra Relativistic Nucleus-Nucleus
Collisions (Quark Matter 2004).
3. \emph{A Monte Carlo Simulation of Multi-gap Resistive Plate
Chamber and comparision with Experimental Results,} RUAN Li-Juan,
SHAO Ming, CHEN Hong-Fang, {\it et al.}, HEP and NP, Vol. 27, No.
8 (2003) 712-715.
4. \emph{ Monte Carlo Study of the Property of Multi-gap
Resistive Plate Chambers,} Shao Ming, Ruan Lijuan, Chen Hongfang,
{\it et al.}, HEP and NP, Vol. 27, No. 1 (2003) 67-71, (in
Chinese).
5. \emph{Study on Light Collection and its Uniformity of Long
Lead Tungstate crystal by Monte Carlo Method,} Ruan Lijuan, Shao
Ming, Xu Tong, {\it et al.}, Chinese Journal of Computational
Physics, Vol. 19, No. 5 (2002) 453-458, (in Chinese).
6. \emph{Beam test results of two kinds of multi-gap resistive
plate chambers,} M. Shao, L. J. Ruan, H. F. Chen, J. Wu, , C. Li,
Z. Z. Xu, X. L. Wang, S.L. Huang, Z. M. Wang and Z. P. Zhang,
Nucl. Instri. and Meth. A 492 (2002) 344-350.
7. \emph{The Study of the Resistive Property of the Electrode
Material of MRPC,} Ruan Lijuan, Wang Xiaolian, Li Cheng , {\it et
al.}, to be published in Journal of University of Science and
Technology of China (in Chinese).
8. \emph{The Calibration Method of TOFr in the STAR Experiment,}
RUAN Lijuan, WU Jian, DONG Xin , {\it et al.}, to be published in
HEP and NP (in Chinese).
9. \emph{Spectra of $\pi$ K p $K^{*}$ $\phi$ from Au+Au
Collisions at 62.4 GeV,} Lijuan Ruan (for the STAR Collaboration),
to be published in Journal of Physics G, Contributed to 8th
International Conference on Strangeness in Quark Matter (SQM
2004).
10. \emph{Pseudorapidity Asymmetry and Centrality Dependence of
Charged Hadron Spectra in d+Au Collisions at $\sqrt{s_{NN}}$ = 200
GeV,} STAR Collaboration: e-Print Archives (nu-ex/0408016),
submitted.
11. \emph{Transverse momentum correlations and minijet dissipation
in Au-Au collisions at $\sqrt{s_{NN}}$ = 130 GeV,} STAR
Collaboration, e-Print Archives (nu-ex/0408012), submitted.
12. \emph{Azimuthal anisotropy and correlations at large
transverse momenta in p+p and Au+Au collisions at $\sqrt{s_{NN}}$=
200 GeV,} STAR Collaboration, e-Print Archives (nu-ex/0407007),
submitted.
13. \emph{Open charm yields in d+Au collisions at $\sqrt{s_{NN}}$
= 200 GeV,} STAR Collaboration, e-Print Archives (nu-ex/0407006),
submitted.
14. \emph{Measurements of transverse energy distributions in Au+Au
collisions at $\sqrt{s_{NN}}$ = 200 GeV,} STAR Collaboration,
e-Print Archives (nu-ex/0407003), submitted.
15. \emph{Transverse-momentum dependent modification of dynamic
texture in central Au+Au collisions at $\sqrt{s_{NN}}$ = 200 GeV,}
STAR Collaboration, e-Print Archives (nu-ex/0407001), submitted.
16. \emph{Hadronization geometry and charge-dependent number
autocorrelations on axial momentum space in Au-Au collisions at
$\sqrt{s_{NN}}$ = 130 GeV,} STAR Collaboration, e-Print
Archives(nu-ex/0406035), submitted.
17. \emph{Phi meson production in Au+Au and p+p collisions at
sqrt(s)=200 GeV,} STAR Collaboration, e-Print Archives
(nu-ex/0406003), submitted.
18. \emph{Centrality and pseudorapidity dependence of charged
hadron production at intermediate pT in Au+Au collisions at
$\sqrt{s_{NN}}$ = 130 GeV,} STAR Collaboration, e-Print Archives
(nu-ex/0404020), to be published in Physical Review C.
19. \emph{Production of e$+$e$-$ Pairs Accompanied by Nuclear
Dissociation in Ultra-Peripheral Heavy Ion Collision,} STAR
Collaboration, e-Print Archives (nu-ex/0404012), to be published
in Physical Review C.
20. \emph{Photon and neutral pion production in Au+Au collisions
at $\sqrt{s_{NN}}$ = 130 GeV,} STAR Collaboration, e-Print
Archives (nu-ex/0401008), to be published in Physical Review C.
21. \emph{Azimuthally sensitive HBT in Au+Au collisions at
$\sqrt{s_{NN}}$ = 200 GeV,} STAR Collaboration, Phys. Rev. Lett.
93, 012301 (2004).
22. \emph{Production of Charged Pions and Hadrons in Au+Au
Collisions at $\sqrt{s_{NN}}$=130 GeV,} STAR Collaboration,
e-Print Archives (nu-ex/0311017), submitted.
23. \emph{Azimuthal anisotropy at RHIC: the first and fourth
harmonics,} STAR Collaboration, Phys. Rev. Lett. 92, 062301
(2004).
24. \emph{Cross Sections and Transverse Single-Spin Asymmetries
in Forward Neutral Pion Production from Proton Collisions at
sqrt(s) = 200 GeV,}
STAR Collaboration, Phys. Rev. Lett. 92, 171801 (2004).
25. \emph{Identified particle distributions in pp and Au+Au
collisions at sqrt{snn}=200 GeV,} STAR Collaboration, Phys. Rev.
Lett. 92, 112301 (2004).
26. \emph{Event-by-Event (pt) fluctuations in Au-Au collisions at
$\sqrt{s_{NN}}$ = 130 GeV,} STAR Collaboration, e-Print Archives
(nu-ex/0308033), submitted.
27. \emph{Multi-strange baryon production in Au-Au collisions at
$\sqrt{s_{NN}}$ = 130 GeV,} STAR Collaboration, Phys. Rev. Lett.
92, 182301 (2004).
28. \emph{Pion-Kaon Correlations in Central Au+Au Collisions at
$\sqrt{s_{NN}}$ = 130 GeV,} STAR Collaboration, Phys. Rev. Lett.
91, 262302 (2003).
29. \emph{rho-0 Production and Possible Modification in Au+Au and
p+p Collisions at $\sqrt{s_{NN}}$ = 200 GeV,} STAR Collaboration,
Phys. Rev. Lett. 92, 092301 (2004).
30. \emph{Net charge fluctuations in Au+Au collisions at
$\sqrt{s_{NN}}$ = 130 GeV,} STAR Collaboration, Phys. Rev. C 68,
044905 (2003).
31. \emph{Rapidity and Centrality Dependence of Proton and
Anti-proton Production from Au+Au Collisions at $\sqrt{s_{NN}}$ =
130 GeV,} STAR Collaboration, e-Print Archives (nu-ex/0306029),
submitted.
32. \emph{Three-Pion Hanbury Brown-Twiss Correlations in
Relativistic Heavy-Ion Collisions from the STAR Experiment,} STAR
Collaboration, Phys. Rev. Lett. 91, 262301 (2003).
33. \emph{Evidence from d+Au measurements for final-state
suppression of high pT hadrons in Au+Au collisions at RHIC,} STAR
Collaboration, Phys. Rev. Lett. 91, 072304 (2003).
34. \emph{Particle-type dependence of azimuthal anisotropy and
nuclear modification of particle production in Au+Au collisions at
$\sqrt{s_{NN}}$ = 200 GeV,} STAR Collaboration, Phys. Rev. Lett.
92, 052302 (2004).
35. \emph{Transverse momentum and collision energy dependence of
high pT hadron suppression in Au+Au collisions at
ultrarelativistic energies,} STAR Collaboration, Phys. Rev. Lett.
91, 172302 (2003).
\chapter{Results}
\label{chp:results}
\section{$\pi, K, p$ and $\bar{p}$ spectra in d+Au and p+p collisions at mid-rapidity}
\begin{figure}[h]
\centering
\includegraphics[height=18pc,width=24pc]{spectra_dAu_TOFr1215.eps}
\caption{The invariant yields of pions (filled circles), kaons
(open squares), protons (filled triangles) and their
anti-particles as a function of $p_{T}$ from d+Au and NSD p+p
events at 200 GeV. The rapidity range was $-0.5<y<0.0$ with the
direction of the outgoing Au ions as negative rapidity. Errors
are statistical.} \label{spectra}
\end{figure}
The invariant yields $\frac{1}{2\pi p_T}\frac{d^2N}{dydp_T}$ of
$\pi^{\pm}$, $K^{\pm}$, $p$ and $\bar{p}$ from both NSD p+p and
minimum-bias d+Au events at mid-rapidity $-0.5<y<0$ are shown in
Figure\ref{spectra}, where $N$ is the corrected signal number per
minimum-bias event in each $p_{T}$ bin.
$N=\frac{N_{raw}\times{factor3}\times{factor4}\times{factor5}}{N_{total}\times{factor1}\times{factor2}}$,
where $N_{raw}$ is the raw signal number in each $p_{T}$ bin,
$N_{total}$ is the total TOFr triggered events, $factor1$ is the
enhancement factor of TOFr trigger, $factor2$ is the TPC
efficiency times TOFr matching efficiency, $factor3$ is the
background correction factor, $factor4$ is the $\langle N_{ch}
\rangle$ bias factor, and $factor5$ is the vertex efficiency times
trigger efficiency and normalization factor.
\subsection{Systematic uncertainty}
For the invariant yield of $\pi^{\pm}$, $K^{\pm}$, $p$ and
$\bar{p}$, the average bin-to-bin systematic uncertainty was
estimated to be of the order of 8\%. The systematic uncertainty is
dominated by the uncertainty in the detector response in Monte
Carlo simulations ($\pm7\%$). Additional factors contributing to
the total systematic uncertainty include the background correction
($\pm3\%$), the small $\eta$ acceptance of the TOFr ($\pm2\%$),
TOFr response ($\pm2\%$), the correction for energy loss in the
detector (${}^{<}_{\sim}10\pm10\%$ at $p_{T}<0.6$ GeV/c for the
$p$ and $\bar{p}$, much smaller for other species and negligible
at higher $p_{T}$), absorption of $\bar{p}$ in the material
($\pm3\%$), and the momentum resolution correction
($\simeq5\pm2\%$). The normalization uncertainties in d+Au
minimum-bias and p+p NSD collisions are $10\%$ and $14\%$,
respectively~\cite{starhighpt,stardau}. The charged pion yields
are consistent with $\pi^0$ yields measured by the PHENIX
collaboration in the overlapping $p_{T}$
range~\cite{phenixhighpt,otherdau}. The invariant yields of
$\pi^{\pm}$, $K^{\pm}$, $p$ and $\bar{p}$ in minimum-bias,
centrality selected d+Au and minimum-bias p+p collisions, are
listed in the tables in Appendix A with statistical errors and
systematic uncertainties.
\section{Cronin effect}
\begin{figure}[h]
\centering
\includegraphics[height=18pc,width=24pc]{pidratio_dAu_TOFr1216.eps}
\caption{The identified particle $R_{dAu}$ for minimum-bias and
top 20\% d+Au collisions. The filled triangles are for
$p+\bar{p}$, the filled circles are for $\pi^{+}+\pi^{-}$ and the
open squares are for $K^{+}+K^{-}$. Dashed lines are $R_{dAu}$ of
inclusive charged hadrons from~\cite{stardau}. The open triangles
and open circles are $R_{CP}$ of $p+\bar{p}$ and $\pi^{0}$ in
Au+Au collisions measured by PHENIX~\cite{phenixpid}. Errors are
statistical. The gray band represents the normalization
uncertainty of 16\%.} \label{Rdau}
\end{figure}
Nuclear effects on hadron production in d+Au collisions are
measured through comparison to the p+p spectrum, scaled by the
number of underlying nucleon-nucleon inelastic collisions using
the ratio
\[R_{dAu}=\frac{d^{2}N/(2{\pi}p_{T}dp_{T}dy)}{T_{dAu}d^{2}\sigma^{pp}_{inel}/(2{\pi}p_{T}dp_{T}dy)} ,\]
where $T_{dAu}={\langle N_{bin}\rangle}/\sigma^{pp}_{inel}$
describes the nuclear geometry, and
$d^{2}\sigma^{pp}_{inel}/(2{\pi}p_{T}dp_{T}dy)$ for p+p inelastic
collisions is derived from the measured p+p NSD cross section. The
difference between NSD and inelastic differential cross sections
at mid-rapidity, as estimated from PYTHIA~\cite{pythia}, is $5\%$
at low $p_{T}$ and negligible at $p_{T}>1.0$ GeV/c.
Figure.~\ref{Rdau} shows $R_{dAu}$ of $\pi^{+}+\pi^{-}$,
$K^{+}+K^{-}$ and $p+\bar{p}$ for minimum-bias and central d+Au
collisions. The systematic uncertainties on $R_{dAu}$ are of the
order of 16\%, dominated by the uncertainty in normalization. The
$R_{dAu}$ of the same particle species are similar between
minimum-bias and top 20\% d+Au collisions. In both cases, the
$R_{dAu}$ of protons rise faster than $R_{dAu}$ of pions and
kaons. We observe that the spectra of $\pi^{\pm}$, $K^{\pm}$, $p$
and $\bar{p}$ are considerably harder in d+Au than those in p+p
collisions. The $R_{dAu}$ of the identified particles has
characteristics of the Cronin effect~\cite{cronin,accardi} in
particle production with $R_{dAu}$ less than unity at low $p_{T}$
and above unity at $p_{T}{}^{>}_{\sim} 1.0$ GeV/c.
\section{$p+\bar{p}/h$ ratio in d+Au and p+p collisions at
middle pseudo-rapidity}
\begin{figure}[h]
\centering
\includegraphics[height=18pc,width=24pc]{baryon_nch_cronin_TOFr0910.eps}
\caption{Minimum-bias ratios of ($p+\bar{p}$) over charged hadrons
at $-0.5\!<\!\eta\!<\!0.0$ from $\sqrt{s_{_{NN}}} =200$ GeV p+p
(open diamonds), d+Au (filled triangles) and $\sqrt{s_{_{NN}}}
=130$ GeV Au+Au~\cite{phenixpid} (asterisks) collisions. Results
of $\mathrm{p+\bar{p}}$ collisions at $\sqrt{s_{_{NN}}} = 1.8$
TeV~\cite{e735} are shown as open stars. Dashed lines are results
of $p/h^{+}$ ratios from $\sqrt{s_{_{NN}}} = 23.8$ GeV p+p
(short-dashed lines) and p+W (dot-dashed)
collisions~\cite{cronin}. Errors are statistical. }
\label{bnchratio}
\end{figure}
Figure~\ref{bnchratio} depicts $(p+\bar{p})/h$, the ratio of
$p+\bar{p}$ over inclusive charged hadrons as a function of
$p_{T}$ in d+Au and p+p minimum-bias collisions at
$\sqrt{s_{_{NN}}} = 200$ GeV, and $p/h^{+}$ ratios in p+p and p+W
minimum-bias collisions at $\sqrt{s_{_{NN}}} = 23.8$
GeV~\cite{cronin}. Although the relative yields of particles and
anti-particles are very different at $\sqrt{s}<40$ GeV due to the
valence quark effects from target and projectile, the Cronin
effects are similar. The systematic uncertainties on these ratios
were estimated to be of the order of 10\% for
$p_{T}{}^{<}_{\sim}1.0$ GeV/c, decreasing to 3\% at higher
$p_{T}$. At RHIC energies, the anti-particle to particle ratios
approach unity ($\bar{p}/p=0.81\pm0.02\pm0.04$ in d+Au
minimum-bias collisions) and their nuclear modification factors
are similar. The difference between $R_{dAu}$ at $\sqrt{s_{_{NN}}}
= 200$ GeV for $p+\bar{p}$ and $h$ can be obtained from the
$(p+\bar{p})/h$ ratios in d+Au and p+p collisions.
Table~\ref{pbarpnchratio} shows $R_{dAu}^{p+\bar{p}}/R_{dAu}^{h}$
determined by averaging over the bins within $1.2<p_{T}<3.0$
GeV/c. At lower energy, the $\alpha$ parameter in the power law
dependence on target atomic weight $A^{\alpha}$ of identified
particle production falls with $\sqrt{s}$~\cite{cronin}. From the
ratios of $R_{dAu}$ between $p+\bar{p}$ and $h$, we may further
derive the $\alpha_{p}-\alpha_{\pi}$ for $1.2< p_{T}< 3.0$ GeV/c
to be $0.041\pm0.010$(stat)$\pm0.006$(syst) under the assumptions
that $\alpha_{K}\simeq\alpha_{\pi}$ and that $(p+\bar{p})/{\pi}$
and $K/{\pi}$ are between 0.1 and 0.4 in p+p collisions. This
result is significantly smaller than the value $0.095\pm0.004$ in
the same $p_{T}$ range found at lower energies~\cite{cronin}.\\
\begin{table}[h]
\caption{\label{pbarpnchratio}$\langle N_{bin}\rangle$ from a
Glauber model calculation, $(p+\bar{p})/h$ averaged over the bins
within $1.2<p_{T}<2.0$ GeV/c (left column) and within
$2.0<p_{T}<3.0$ GeV/c (right column) and the $R_{dAu}$ ratios
between $p+\bar{p}$ and $h$ averaged over $1.2<p_{T}<3.0$ GeV/c
for minimum-bias, centrality selected d+Au collisions and
minimum-bias p+p collisions. A p+p inelastic cross section of
$\sigma_{inel}=42$ mb was used in the calculation. For $R_{dAu}$
ratios, only statistical errors are shown and the systematic
uncertainties are 0.03 for all centrality bins. } {\centering
{\begin{tabular}{c|c|c|c|c} \hline \hline centrality &
$\langle N_{bin}\rangle$ & \multicolumn{2}{c|} {$(p+\bar{p})/h$} &
${R_{dAu}^{p+\bar{p}}}/{R_{dAu}^h}$\\ \hline min. bias & $7.5\pm0.4$
&$0.21\pm0.01$ &$0.24\pm0.01$ & $1.19\pm0.05$\\ 0--20\% &
$15.0\pm1.1$ &$0.21\pm0.01$ &$0.24\pm0.02$ & $1.18\pm0.06$\\ 20--40\%
& $10.2\pm1.0$ &$0.20\pm0.01$ &$0.24\pm0.02$ & $1.16\pm0.06$\\
40--$\sim$100\% & $4.0^{+0.8}_{-0.3}$ &$0.20\pm0.01$ &$0.23\pm0.02$ &
$1.13\pm0.06$\\ \hline p+p & $1.0$ &$0.17\pm0.01$ &$0.21\pm0.02$ &
--- \\ \hline \hline
\end{tabular}
}
\par}
\label{Tab:D}
\end{table}
\begin{figure}[h]
\centering
\includegraphics[height=18pc,width=24pc]{baryon_nch_cronin_TOFr_centrality_1023_color_new.eps}
\caption{Minimum-bias ratios of ($p+\bar{p}$) over charged hadrons
at $-0.5\!<\!\eta\!<\!0.0$ from $\sqrt{s_{_{NN}}} =200$ GeV
minimum-bias and centrality selected d+Au collisions. Errors are
statistical. } \label{bnchratiocentrality}
\end{figure}
Also shown is $(p+\bar{p})/h$ ratio from the Au+Au minimum-bias
collisions at $\sqrt{s_{_{NN}}} = 130$ GeV~\cite{phenixpid}. The
$(p+\bar{p})/h$ ratio from minimum-bias Au+Au
collisions~\cite{phenixpid} at a similar energy is about a factor
of 2 higher than that in d+Au and p+p collisions for
$p_{T}{}^{>}_{\sim}2.0$ GeV/c. This enhancement is most likely
due to final-state effects in Au+Au
collisions~\cite{jetquench,junction,derekhydro,pisahydro,fries,ko}.
The ratios show little centrality dependence in d+Au collisions,
as shown in Table~\ref{Tab:D} and
Figure~\ref{bnchratiocentrality}. For $p_{T}<2.0$ GeV/c, the ratio
in $\mathrm{p+\bar{p}}$ collisions at $\sqrt{s_{_{NN}}} = 1.8$
TeV~\cite{e735} is very similar to those in d+Au and p+p
collisions at $\sqrt{s_{_{NN}}} = 200$ GeV.
\section{$K/\pi$, $p/\pi$ and anti-particle to particle ratios}
\begin{figure}[h]
\begin{minipage}[t]{50mm}
\includegraphics[height=11pc,width=13pc]{piratio.eps}
\end{minipage}
\begin{minipage}[t]{50mm}
\includegraphics[height=11pc,width=13pc]{kratio.eps}
\end{minipage}
\begin{minipage}[t]{50mm}
\includegraphics[height=11pc,width=13pc]{pratio.eps}
\end{minipage}
\caption{$\pi^{-}$/$\pi^{+}$, $K^{-}$/$K^{+}$ and $\bar{p}/p$
ratios as a function of $p_{T}$ in d+Au and p+p minimum-bias
collisions. The open symbols are for p+p collisions and the solid
symbols for d+Au collisions. Errors are statistical.}
\label{antiparticleratio}
\end{figure}
\begin{table}[h]
\begin{scriptsize} \centering
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Centrality Bin & $\pi^{-}$/$\pi^{+}$ & $X^{2}/ndf$ & $K^{-}$/$K^{+}$ & $X^{2}/ndf$ & $\bar{p}/p$ & $X^{2}/ndf$ \\ \hline
d+Au M.B. & $1.01\pm0.01$ & $0.88$ & $0.94\pm0.02$ & $1.78$ & $0.81\pm0.02$ & $0.85$\\ \hline
0\%-20\% & $1.01\pm0.01$ & $0.80$ & $0.93\pm0.03$ & $1.43$ & $0.80\pm0.03$ & $0.70$ \\ \hline
20\%-40\% & $1.00\pm0.01$ & $0.98$ & $0.91\pm0.03$ & $1.19$ & $0.79\pm0.03$ & $1.14$ \\ \hline
40\%-100\% & $1.02\pm0.01$ & $0.81$ & $1.02\pm0.03$ & $0.45$ & $0.78\pm0.03$ & $0.70$ \\ \hline
p+p & $1.00\pm0.01$ & $1.24$ & $0.98\pm0.02$ & $0.71$ & $0.79\pm0.03$ & $0.73$ \\ \hline
\end{tabular}
\caption{$\pi^{-}$/$\pi^{+}$, $K^{-}$/$K^{+}$ and $\bar{p}/p$
ratios in p+p and d+Au minimum-bias collisions. Also shows in the
table are the ratios in centrality selected d+Au collisions.
Errors are statistical. } \label{antiparticletoparticleratio}
\end{scriptsize}
\end{table}
Figure~\ref{antiparticleratio} shows the $\pi^{-}/\pi^{+}$,
$K^{-}/K^{+}$ and $\bar{p}/p$ ratios as a function of $p_{T}$ in
d+Au and p+p minimum-bias collisions. It shows the anti-particle
to particle ratios are flat with $p_{T}$. The zero order
polynominal function was used to fit the data and get the
anti-particle to particle ratios. The results are list in
Table~\ref{antiparticletoparticleratio}. In centrality selected
d+Au collisions, the anti-particle to particle ratios are also
flat with $p_{T}$ and show little centrality dependence. The
results are also shown in the Table
~\ref{antiparticletoparticleratio}.
\begin{figure}[h]
\begin{minipage}[t]{80mm}
\includegraphics[height=13pc,width=14pc]{kppipratio.eps}
\end{minipage}
\begin{minipage}[t]{80mm}
\includegraphics[height=13pc,width=14pc]{kmpimratio.eps}
\end{minipage}
\caption{$K/\pi$ ratios as a function of $p_{T}$ in d+Au and p+p
minimum-bias collisions. The open symbols are for p+p collisions
and the solid symbols for d+Au collisions. Errors are
statistical.} \label{kpiratio}
\end{figure}
\begin{figure}[h]
\begin{minipage}[t]{80mm}
\includegraphics[height=13pc,width=14pc]{pppipratio.eps}
\end{minipage}
\begin{minipage}[t]{80mm}
\includegraphics[height=13pc,width=14pc]{pmpimratio.eps}
\end{minipage}
\caption{$p(\bar{p})/\pi$ ratios as a function of $p_{T}$ in d+Au
and p+p minimum-bias collisions. The open symbols are for p+p
collisions and the solid symbols for d+Au collisions. Errors are
statistical.} \label{ppiratio}
\end{figure}
The $K/\pi$ and $p/\pi$ ratios are shown in Figure~\ref{kpiratio}
and Figure~\ref{ppiratio} individually. From the plots, the
$K/\pi$ ratios increase with $p_{T}$ in both d+Au and p+p
collisions and the increasing trend is the same within our errors.
The $p/\pi$ ratios increase with $p_{T}$ in both d+Au and p+p
collisions and the increasing in d+Au collisions is faster than
that in p+p collisions. The trends of the $K/\pi$ and $p/\pi$ as a
function of $p_{T}$ show little centrality dependence in d+Au
collisions.
\section{$dN/dy$, $\langle p_T \rangle$, and model fits}
\begin{figure}[h]
\begin{minipage}[t]{80mm}
\includegraphics[height=17pc,width=18pc]{piplusyield.eps}
\end{minipage}
\hspace{\fill}
\begin{minipage}[t]{80mm}
\includegraphics[height=17pc,width=18pc]{piminusyield.eps}
\end{minipage}
\caption{The re-scaled $\pi^{+}$ and $\pi^{-}$ spectra in
minimum-bias, centrality selected d+Au collisions and also in p+p
collisions. The errors are statistical.} \label{pionspectra}
\end{figure}
\begin{figure}[h]
\begin{minipage}[t]{80mm}
\includegraphics[height=17pc,width=18pc]{kplusyield.eps}
\end{minipage}
\hspace{\fill}
\begin{minipage}[t]{80mm}
\includegraphics[height=17pc,width=18pc]{kminusyield.eps}
\end{minipage}
\caption{The re-scaled $K^{+}$ and $K^{-}$ spectra in
minimum-bias, centrality selected d+Au collisions and also in p+p
collisions. The errors are statistical.} \label{kaonspectra}
\end{figure}
The spectra in minimum-bias and centrality selected d+Au
collisions and also in p+p collisions are shown in
Figure~\ref{pionspectra},
Figure~\ref{kaonspectra} and Figure~\ref{protonspectra}. The spectra
show little centrality dependence for each particle in d+Au
collisions but harder than those in p+p collisions. The power law
function was used to fit the spectra and get the $dN/dy$ and
$\langle p_T \rangle$. The power law fit function is:
\begin{equation}
\frac{1}{2\pi p_T}\frac{d^2N}{dydp_T}=a(1+\frac{p_T}{\langle p_T
\rangle \frac{n-3}{2}})^{-n}
\end{equation}
Where the parameter $a$ is a constant value proportional to the
mid-rapidity yield $dN/dy$, the parameter $n$ is the order of the
power law and $\langle p_T \rangle$ is the mean value of the
transverse momentum which is extracted from the fit.
Figure~\ref{powerlawfit} shows power law fit to the spectra of
minimum-bias d+Au and p+p collisions.
Figure~\ref{3centralitypowerlawfit} shows power law fit to the
spectra of 3 centrality selected d+Au collisions. The power law
fit results are listed in Table~\ref{dndypowerlawfit} and
Table~\ref{meanptpowerlawfit} individually. The thermal
model~\cite{thermal} was also used to fit the spectra. The final
$dN/dy$ and $\langle p_T \rangle$ are shown in
Table~\ref{finaldndy} and Table~\ref{finalmeanpt} respectively,
which were obtained by averaging the results from the power law
fit and thermal fit. Half of the differences in them are taken as
the systematic errors due to the extrapolation to low $p_{T}$
region. The errors in this table include the systematic
uncertainties and statistical errors.
\begin{table}[h]
\begin{scriptsize}
\centering
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Centrality Bin & $\pi^{-}$ & $\pi^{+}$ & $K^{-}$ & $K^{+}$ & $\bar{p}$ & $p$ \\ \hline
d+Au M.B. & $0.403\pm0.004$ & $0.405\pm0.004$ & $0.609\pm0.009$ & $0.629\pm0.009$ & $0.714\pm0.008$ & $0.677\pm0.010$\\ \hline
0\%-20\% & $0.421\pm0.004$ & $0.421\pm0.004$ & $0.626\pm0.018$ & $0.658\pm0.016$ & $0.727\pm0.013$ & $0.705\pm0.014$ \\ \hline
20\%-40\% & $0.408\pm0.004$ & $0.411\pm0.004$ & $0.604\pm0.015$ & $0.625\pm0.015$ & $0.725\pm0.013$ & $0.691\pm0.015$ \\ \hline
40\%-100\% & $0.387\pm0.005$ & $0.391\pm0.004$ & $0.589\pm0.016$ & $0.616\pm0.016$ & $0.667\pm0.013$ & $0.646\pm0.014$ \\ \hline
p+p & $0.357\pm0.004$ & $0.361\pm0.004$ & $0.571\pm0.013$ & $0.571\pm0.013$ & $0.567\pm0.010$ & $0.569\pm0.012$ \\ \hline
\end{tabular}
\caption{$\langle p_T \rangle$ of $\pi^{-}$, $\pi^{+}$, $K^{-}$,
$K^{+}$, $\bar{p}$ and $p$ from power law fit in minimum-bias,
centrality selected d+Au collisions and also in p+p collisions.
The errors are from the power law fit. The unit of $p_{T}$ is
GeV/c.} \label{meanptpowerlawfit}
\end{scriptsize}
\end{table}
\begin{table}[h]
\begin{scriptsize}
\centering
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Centrality Bin & $\pi^{-}$ & $\pi^{+}$ & $K^{-}$ & $K^{+}$ & $\bar{p}$ & $p$ \\ \hline
d+Au M.B. & $5.078\pm0.080$ & $5.032\pm0.080$ & $0.685\pm0.013$ & $0.703\pm0.012$ & $0.466\pm0.009$ & $0.594\pm0.019$\\ \hline
0\%-20\% & $10.657\pm0.190$ & $10.521\pm0.187$ & $1.448\pm0.085$ & $1.453\pm0.035$ & $0.972\pm0.026$ & $1.222\pm0.045$ \\ \hline
20\%-40\% & $7.631\pm0.148$ & $7.515\pm0.139$ & $0.988\pm0.028$ & $1.051\pm0.027$ & $0.651\pm0.018$ & $0.842\pm0.033$ \\ \hline
40\%-100\% & $3.153\pm0.069$ & $3.024\pm0.060$ & $0.399\pm0.012$ & $0.379\pm0.011$ & $0.261\pm0.008$ & $0.338\pm0.014$ \\ \hline
p+p & $1.524\pm0.027$ & $1.504\pm0.027$ & $0.166\pm0.004$ & $0.173\pm0.009$ & $0.113\pm0.003$ & $0.137\pm0.007$ \\ \hline
\end{tabular}
\caption{$dN/dy$ of $\pi^{-}$, $\pi^{+}$, $K^{-}$, $K^{+}$,
$\bar{p}$ and $p$ from power law fit in minimum-bias, centrality
selected d+Au collisions and also in p+p collisions. The errors
are from the power law fit.} \label{dndypowerlawfit}
\end{scriptsize}
\end{table}
\begin{table}[h]
\begin{scriptsize}
\centering
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Centrality Bin & $\pi^{-}$ & $\pi^{+}$ & $K^{-}$ & $K^{+}$ & $\bar{p}$ & $p$ \\ \hline
d+Au M.B. & $0.420\pm0.019$ & $0.422\pm0.019$ & $0.613\pm0.025$ & $0.625\pm0.025$ & $0.761\pm0.056$ & $0.739\pm0.069$\\ \hline
0\%-20\% & $0.435\pm0.017$ & $0.436\pm0.017$ & $0.627\pm0.025$ & $0.646\pm0.028$ & $0.774\pm0.056$ & $0.761\pm0.063$ \\ \hline
20\%-40\% & $0.425\pm0.019$ & $0.427\pm0.018$ & $0.610\pm0.025$ & $0.622\pm0.025$ & $0.766\pm0.052$ & $0.744\pm0.061$ \\ \hline
40\%-100\% & $0.405\pm0.020$ & $0.408\pm0.019$ & $0.591\pm0.024$ & $0.608\pm0.026$ & $0.715\pm0.056$ & $0.703\pm0.063$ \\ \hline
p+p & $0.377\pm0.021$ & $0.379\pm0.020$ & $0.565\pm0.023$ & $0.565\pm0.023$ & $0.627\pm0.065$ & $0.634\pm0.070$ \\ \hline
\end{tabular}
\caption{The final $\langle p_T \rangle$ of $\pi^{-}$, $\pi^{+}$,
$K^{-}$, $K^{+}$, $\bar{p}$ and $p$ in minimum-bias, centrality
selected d+Au collisions and also in p+p collisions. The unit of
$p_{T}$ is GeV/c.} \label{finalmeanpt}
\end{scriptsize}
\end{table}
\begin{table}[h]
\begin{scriptsize}
\centering
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
Centrality Bin & $\pi^{-}$ & $\pi^{+}$ & $K^{-}$ & $K^{+}$ & $\bar{p}$ & $p$ \\ \hline
d+Au M.B. & $4.731\pm0.359$ & $4.668\pm0.356$ & $0.662\pm0.040$ & $0.684\pm0.039$ & $0.425\pm0.044$ & $0.531\pm0.067$\\ \hline
0\%-20\% & $10.063\pm0.628$ & $9.932\pm0.621$ & $1.383\pm0.095$ & $1.418\pm0.079$ & $0.896\pm0.084$ & $1.114\pm0.117$ \\ \hline
20\%-40\% & $7.137\pm0.514$ & $7.074\pm0.464$ & $0.952\pm0.059$ & $1.020\pm0.059$ & $0.603\pm0.054$ & $0.765\pm0.083$ \\ \hline
40\%-100\% & $2.925\pm0.236$ & $2.829\pm0.203$ & $0.387\pm0.023$ & $0.371\pm0.020$ & $0.238\pm0.025$ & $0.304\pm0.036$ \\ \hline
p+p & $1.411\pm0.116$ & $1.400\pm0.108$ & $0.163\pm0.009$ & $0.168\pm0.010$ & $0.099\pm0.015$ & $0.120\pm0.018$ \\ \hline
\end{tabular}
\caption{The final $dN/dy$ of $\pi^{-}$, $\pi^{+}$, $K^{-}$,
$K^{+}$, $\bar{p}$ and $p$ in minimum-bias, centrality selected
d+Au collisions and also in p+p collisions.} \label{finaldndy}
\end{scriptsize}
\end{table}
\begin{figure}[h]
\begin{minipage}[t]{80mm}
\includegraphics[height=17pc,width=18pc]{protonyield.eps}
\end{minipage}
\hspace{\fill}
\begin{minipage}[t]{80mm}
\includegraphics[height=17pc,width=18pc]{pbaryield.eps}
\end{minipage}
\caption{The re-scaled $p$ and $\bar{p}$ spectra in minimum-bias,
centrality selected d+Au collisions and also in p+p collisions.
The errors are statistical.} \label{protonspectra}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[height=18pc,width=24pc]{daupppowerlawfit.eps}
\caption{The spectra of $\pi^{-}$, $\pi^{+}$, $K^{-}$, $K^{+}$,
$\bar{p}$ and $p$ in d+Au and p+p minimum-bias collisions. The
curves are from power law fit.} \label{powerlawfit}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[height=24pc,width=24pc]{3centralitypowerlawfit.eps}
\caption{The spectra of $\pi^{-}$, $\pi^{+}$, $K^{-}$, $K^{+}$,
$\bar{p}$ and $p$ in three centrality selected d+Au collisions.
The curves are from power law fit.} \label{3centralitypowerlawfit}
\end{figure}
\section{System comparison}
\begin{figure}[h]
\centering
\includegraphics[height=24pc,width=24pc]{meanpt_new.eps}
\caption{$\langle p_T \rangle$ as a function of $dN/d\eta$. The
squared, circled and triangled symbols are from~\cite{olga} in p+p
and Au+Au collisions. The cross, star and diamond are our data
points in p+p and d+Au collisions. Statistic errors and systematic
uncertainties have been added in quadrature.} \label{meanptNch}
\end{figure}
Figure~\ref{meanptNch} shows the $\langle p_T \rangle$ of
$\pi^{-}$, $K^{-}$ and $\bar{p}$ as a function of charged particle
multiplicity at mid-rapidity. From p+p to d+Au collisions, the
$\langle p_T \rangle$ increase with charged particle multiplicity
smoothly. We observed the $\langle p_T \rangle$ in 0\%-20\% d+Au
collisions are larger than those in peripheral Au+Au collisions.
\begin{figure}[h]
\centering
\includegraphics[height=24pc,width=24pc]{kaonprotontopionratio_new.eps}
\caption{$K^{-}/\pi^{-}$ and $\bar{p}/\pi^{-}$ as a function of
$dN/d\eta$. The circled and triangled symbols are
from~\cite{olga} in p+p and Au+Au collisions. The star and diamond
are our data points in p+p and d+Au collisions. Statistic errors
and systematic uncertainties have been added in quadrature.}
\label{kaonpbarpionratio}
\end{figure}
The $K^{-}/\pi^{-}$ and $\bar{p}/\pi^{-}$ as a function of charged
particle multiplicity at mid-rapidity are shown in
Figure~\ref{kaonpbarpionratio}. The $K^{-}/\pi^{-}$ and
$\bar{p}/\pi^{-}$ ratios were derived by taking the ratios of the
dN/dy of $K^{-}$ or $\bar{p}$ over the dN/dy of $\pi^{-}$ in
table~\ref{finaldndy}. These ratios increase with charged particle
multiplicity from p+p, d+Au to Au+Au collisions smoothly. The
kinetic freeze out temperature $T_{kin}$ and flow velocity
$\langle \beta \rangle$ from thermal fit as a function of charged
particle multiplicity are shown in Figure~\ref{freezeoutT}. We can
see the $T_{kin}$ is flat from p+p to d+Au and then decreases from
d+Au to Au+Au collisions and the $\langle \beta \rangle$ increases
from p+p, d+Au to Au+Au collisions.
\begin{figure}[h]
\begin{minipage}[t]{80mm}
\includegraphics[height=17pc,width=18pc]{temperature.eps}
\end{minipage}
\hspace{\fill}
\begin{minipage}[t]{80mm}
\includegraphics[height=17pc,width=18pc]{beta_new.eps}
\end{minipage}
\caption{The kinetic freeze out temperature $T_{kin}$ (left) and
flow velocity $\langle \beta \rangle$ (right) from thermal fit as
a function of charged particle multiplicity. The circled symbols
are from ~\cite{olga} in p+p and Au+Au collisions. The star are
our data points in p+p and d+Au collisions. Errors are
systematic.} \label{freezeoutT}
\end{figure}
\chapter{The STAR Experiment} \label{chp:star} \section{The RHIC
Accelerator} The Relativistic Heavy Ion Collider (RHIC) at
Brookhaven National Lab (BNL) is the first hadron accelerator and
collider consisting of two independent ring. It is designed to
operate at high collision luminosity over a wide range of beam
energies and particle species ranging from polarized proton to
heavy ion~\cite{rhic:01,rhic:02}, where the top energy of the
colliding center-of-mass energy per nucleon-nucleon pair is
$\sqrt{s_{NN}}$ = 200 GeV. The RHIC facility consists of two
super-conducting magnets, each with a circumference of 3.8 km,
which focus and guide the beams. \\
Figure 2.1 shows the BNL accelerator complex including the
accelerators used to bring the gold ions up to RHIC injection
energy. In the first, gold ions are accelerated to 15 MeV/nucleon
in the Tandem Van de Graaff facility. Then the beam is transferred
to the Booster Synchrotron and accelerated to 95 MeV/nucleon
through the Tandem-to-Booster line. Then the gold ions are
transferred to the Alternating Gradient Synchrotron (AGS) and
accelerated to 10.8 GeV/nucleon. Finally they are injected to RHIC
and accelerated to the collision energy 100 GeV/nucleon.\\
\begin{figure} \centering
\includegraphics[height=35pc,width=32pc]{rhic.eps} \caption{A
diagram of the Brookhaven National Laboratory collider complex
including the accelerators that bring the nuclear ions up to RHIC
injection energy (10.8 GeV/nucleon for $^{197}$Au). Figure is
taken from~\cite{sorenson:01,Haibin:03}.}
\end{figure}
RHIC's 3.8 km ring has six intersection points where its two rings
of accelerating magnets cross, allowing the particle beams to
collide. The collisions produce the fleeting signals that, when
captured by one of RHIC's experimental detectors, provide
physicists with information about the most fundamental workings of
nature. If RHIC's ring is thought of as a clock face, the four
current experiments are at 6 o'clock (STAR), 8 o'clock (PHENIX),
10 o'clock (PHOBOS) and 2 o'clock (BRAHMS). There are two
additional intersection points at 12 and 4 o'clock where future
experiments may be placed~\cite{rhic:01}.
\section{The STAR Detector}
\begin{figure}[h] \centering
\includegraphics[height=18pc,width=28pc]{star_1.eps}
\caption{Perspective view of the STAR detector, with a cutaway for
viewing inner detector
systems. Figure is taken
from ~\cite{detector:01}.}
\label{starfigure1}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[height=22pc,width=28pc]{star_2.eps}
\caption{Cutaway side view
of the STAR detector as
configured in 2001. Figure
is taken
from~\cite{detector:01}.}
\label{starfigure2}
\end{figure} The Solenoidal
Tracker at RHIC (STAR) is one of the two large detector systems
constructed at the Relativistic Heavy Ion Collider (RHIC) at
Brookhaven National Laboratory. STAR was constructed to
investigate the behavior of strongly interacting matter at high
energy density and to search for signatures of quark-gluon plasma
(QGP) formation. Key features of the nuclear environment at
RHIC are a large number of produced particles (up to
approximately one thousand per unit pseudo-rapidity) and high
momentum particles from hard parton-parton scattering. STAR can
measure many observables simultaneously to study signatures of
a possible QGP phase transition and to understand the space-time
evolution of the collision process in
ultra-relativistic heavy ion collisions. The goal is to
obtain a fundamental understanding of the microscopic
structure of these hadronic interactions at high energy
densities. In order to accomplish this, STAR was designed
primarily for measurements of hadron production over a large solid
angle, featuring detector systems for high precision tracking,
momentum analysis, and particle identification at the center of
mass (c.m.) rapidity. The large acceptance of STAR makes it
particularly well suited for event-by-event characterizations of
heavy ion collisions and for the detection of hadron jets~\cite{detector:01}.\\
The layout of the STAR experiment~\cite{STAR CDR} is shown in
Figure~\ref{starfigure1}. A cutaway side view of the STAR detector
as configured for the RHIC 2001 run is displayed in
Figure~\ref{starfigure2}. A room temperature solenoidal
magnet~\cite{brown} with a maximum magnetic field of 0.5 T
provides a uniform magnetic field for charged particle momentum
analysis. Charged particle tracking close to the interaction
region is accomplished by a Silicon Vertex Tracker~\cite{bellwied}
(SVT). The Silicon Drift Detectors~\cite{baudot} (SDD) installed
after 2001 is also for the inner tracking. The silicon detectors
cover a pseudo-rapidity range $\mid {\eta }\mid \leq 1$ with
complete azimuthal symmetry ($\Delta \phi = 2\pi$). Silicon
tracking close to the interaction allows precision localization of
the primary interaction vertex and identification of secondary
vertices from weak decays of, for example, $\Lambda$, $\Xi$, and
$\Omega$. A large volume Time Projection
Chamber~\cite{wieman,tpc} (TPC) for charged particle tracking and
particle identification is located at a radial distance from 50 to
200 cm from the beam axis. The TPC is 4 meters long and it covers
a pseudo-rapidity range $\mid {\eta}\mid \leq 1.8$ for tracking
with complete azimuthal symmetry ($\Delta \phi = 2\pi$). Both the
SVT and TPC contribute to particle identification using ionization
energy loss, with an anticipated combined energy loss resolution
(dE/dx) of 7 \% ($\sigma$). The momentum resolution of the SVT
and TPC reach a value of $\delta $p/p = 0.02 for a majority of the
tracks in the TPC. The $\delta $p/p resolution improves as the
number of hit points along the track increases and
as the particle's momentum decreases, as expected~\cite{detector:01}. \\
To extend the tracking to the forward region, a radial-drift TPC
(FTPC)~\cite{eckardt} is installed covering $2.5<\mid{\eta }\mid <
4$, also with complete azimuthal coverage and symmetry. To extend
the particle identification in STAR to larger momenta over a small
solid angle for identified single-particle spectra at
mid-rapidity, a ring imaging Cherenkov detector
~\cite{ALICE_HMPID} covering $\mid\eta\mid < 0.3$ and $\Delta \phi
= 0.11\pi$, and a time-of-flight patch (TOFp)~\cite{pVPD} covering
$-1<\eta <0$ and $\Delta\phi = 0.04\pi $ (as shown in
Figure~\ref{starfigure2}) was installed at STAR in
2001~\cite{detector:01}. In 2003, a time-of-flight tray (TOFr)
based on multi-gap resistive plate chamber (MRPC)
technology~\cite{startof} was installed in STAR detector, covering
$-1<\eta <0$ and $\Delta\phi = \pi/30 $. For the time-of-flight
system, the Pseudo-Vertex Position Detectors (pVPD) was installed
as the start-timing detector, which was 5.4 m away from TPC center
and covers $4.4<|\eta|<4.9$ with the azimuthal coverage
19\%~\cite{pVPD} in 2003.\\
The fast detectors that provide input to the trigger system are a
central trigger barrel (CTB) at $|\eta|<1$ and two zero-degree
calorimeters (ZDC) located in the forward directions at $\theta<2$
mrad. The CTB surrounds the outer cylinder of the TPC, and
triggers on the flux of charged particles in the mid-rapidity
region. The ZDCs are used for determining the energy in neutral
particles remaining in the forward directions~\cite{detector:01}.
A minimum bias trigger was obtained by selecting events with a
pulse height larger than that of one neutron in each of the
forward ZDCs, which corresponds to 95 percent of the geometrical
cross section~\cite{detector:01}.
\subsection{The Time Projection
Chamber} The STAR detector~\cite{STAR CDR} uses the TPC as
its primary tracking device. The TPC records the tracks of
particles, measures their momenta, and identifies the particles
by measuring their ionization energy loss ($dE/dx$). Particles
are identified over a momentum range from 100 MeV/c to greater
than 1 GeV/c and momenta are measured over a range of 100
MeV/c to 30 GeV/c~\cite{tpc}.\\
The STAR TPC is shown schematically in Figure~\ref{tpcman}. It is
a volume of gas in a well defined uniform electric field of
$\approx$ 135 V/cm. The working gas of TPC is P10 gas (10\%
methane, 90\% argon) regulated at 2 mbar above atmospheric
pressure\cite{gas}. This gas has long been used in TPCs. Its
primary attribute is a fast drift velocity which peaks at a low
electric field. Operating on the peak of the velocity curve makes
the drift velocity stable and insensitive to small variations
in temperature and pressure~\cite{tpc}. The paths of primary
ionizing particles passing through the gas volume are
reconstructed with high precision from the released secondary
electrons which drift to the readout end caps at the ends of
the chamber. The drift velocity of electrons is 5.45 cm/$\mu$s.
The uniform electric field which is required to drift the
electrons is defined by a thin conductive Central Membrane (CM) at
the center of the TPC, concentric field cage cylinders and the
read out end caps~\cite{tpc}. The readout system is based on Multi
Wire Proportional Chambers (MWPC) with readout pads. The drifting
electrons avalanche in the high fields at the 20 $\mu$m anode
wires providing an amplification of 1000 to 3000. The induced
charge from an avalanche is shared over several adjacent pads,
so the original track position can be reconstructed to a small
fraction of a pad width. There are a total of 136,608 pads in the
readout system~\cite{tpc}, which give $x$-$y$ coordinate
information. The $z$ position
information is provided by 512 time buckets.\\
\begin{figure}[htb]
\includegraphics[width=14cm]{tpcman.eps}
\caption{The STAR TPC surrounds a beam-beam interaction region at
RHIC. The collisions take place near the center of the TPC.}
\label{tpcman}
\end{figure}
At the Data Acquisition (DAQ) stage, raw events containing
millions of ADC values and TDC values were recorded. Raw data were
then reconstructed into hits, tracks, vertices, and the collision
vertex through the reconstruction chain of TPC~\cite{starsoftware}
by Kalman method. The collision vertex are called the primary
vertex. The tracks are called the global tracks. If the
3-dimensional distance of closest approach (DCA/dca) of the global
track to the primary vertex is less than 3 cm, this track will be
chosen for a re-fit by forcing a new track helix ending at the
primary vertex. These newly reconstructed helices are called
primary tracks~\cite{Haibin:03}. As expected, the vertex
resolution decreases as the square root of the number of tracks
used in the calculation. The vertex resolution is 350 $\mu$m
when there are more than 1,000 tracks~\cite{tpc}.
Figure~\ref{eventshow} shows the beam's eye view of a central
Au+Au collision event in the STAR TPC.
\begin{figure}[h]
\centering
\includegraphics[height=14pc,width=18pc]{event.eps} \caption{Beam's
eye view of a central Au+Au collision event in the STAR Time
Projection Chamber. This event was drawn by the STAR online
display. Figure is taken from~\cite{detector:01}.}
\label{eventshow} \end{figure}
\subsubsection{Particle Identification (PID) of TPC by
dE/dx}Energy lost in the TPC gas is a valuable tool for
identifying particle species. It works especially well for low
momentum particles but as the particle energy rises, the energy
loss becomes less mass-dependent and it is hard to separate
particles with velocities $v>0.7$c~\cite{tpc}. For a particle with
charge $z$ (in units of $e$) and speed $\beta=v/c$ passing through
a medium with density $\rho$, the mean energy loss it suffers can
be described by the Bethe-Bloch formula
\begin{equation} \langle \frac{dE}{dx} \rangle = 2\pi
N_0r_e^2m_ec^2\rho\frac{Zz^2}{A\beta^2}
[\text{ln}\frac{2m_e\gamma^2v^2E_M}{I^2}-2\beta^2] \end{equation}
where $N_0$ is Avogadro's number, $m_e$ is the electron mass,
$r_e$ ($=e^2/m_e$) is the classical electron radius, $c$ is the
speed of light, $Z$ is the atomic number of the absorber, $A$ is
the atomic weight of the absorber, $\gamma=1/\sqrt{1-\beta^2}$,
$I$ is the mean excitation energy, and $E_M$
($=2m_ec^2\beta^2/(1-\beta^2)$) is the maximum transferable energy
in a single collision~\cite{tang:01,Haibin:03}. From the above
equation, we can see that different charged particles (electron,
muon, pion, kaon, proton or deuteron) with the same momentum $p$
passing through the TPC gas can result in different energy loss.
Figure~\ref{fdedx} shows the energy loss for particles in the TPC
as a function of the particle momentum, which includes both
primary and secondary particles. We can see that charged pions
and kaons can be identified up to about transverse momentum 0.75
GeV/c and protons and anti-protons can be identified to 1.1 GeV/c.
\begin{figure}[h]
\centering
\includegraphics[width=22pc]{dedxPlotAllBands.eps}
\caption{The energy loss distribution for primary and
secondary particles in the STAR TPC as a function of the $p_T$
of the primary particle. This figure is taken from~\cite{tpc}.}
\label{fdedx}
\end{figure}
In order to quantitatively describe the particle identification,
we define the variable $N_{\sigma\pi}$ (in the case of charged
pion identification) as
\begin{equation}
N_{\sigma\pi}=[\frac{dE}{dx}_{meas.}-\langle\frac{dE}{dx}\rangle_\pi]/
[\frac{0.55}{\sqrt{N}}\frac{dE}{dx}_{meas.}] \end{equation} in
which $N$ is the number of hits for a track in the TPC,
$\frac{dE}{dx}_{meas.}$ is the measured energy loss of a track and
$\langle\frac{dE}{dx}\rangle_\pi$ is the mean energy loss for
charged pions. In order to identify charged kaons, protons and
anti-protons, we can have similar definition of $N_{\sigma K}$ and
$N_{\sigma p}$. Thus we can cut on the variables $N_{\sigma\pi}$,
$N_{\sigma K}$ and $N_{\sigma p}$ to select different particle
species~\cite{Haibin:03}.\\
A specific part of the particle
identification is the topological identification of neutral
particles, such as the $K_S^0$ and $\Lambda$. These neutral
particles can be reconstructed by identifying the secondary
vertex, commonly called V0 vertex, of their charged daughter decay
modes, $K_S^0\rightarrow\pi^+\pi^-$ and $\Lambda\rightarrow p
\pi^-$~\cite{Haibin:03}.
\subsection{The time-of-flight tray based on MRPC technology}
\begin{figure}[h] \centering
\includegraphics[height=24pc,width=32pc]{traymodule-instar.eps}
\caption{Tray structure. Figure is taken from
~\cite{tofproposal}.} \label{tofrtray}
\end{figure}
In 2003, the time-of-flight tray (TOFr) based on multi-gap
resistive plate chamber (MRPC) technology~\cite{startof} was
installed in STAR detector. It extends particle identification up
to $p_{T}\sim3$ GeV/c for $p$ and $\bar{p}$. This tray was
installed on the Au beam outgoing direction. MRPC technology was
first developed by the CERN ALICE group~\cite{williams} to provide
a cost-effective solution for large-area time-of-flight coverage.
For full time-of-flight coverage at STAR, there will be 120 trays,
with 60 on east side and 60 on west side. For each tray, there
will be 33 MRPCs. For each MRPC, there are 6 read-out channels.
Figure~\ref{tofrtray} shows the tray which indicates the position
of each MRPC module. The MRPCs are tilted differently so that each
MRPC is most projective to the average primary vertex location at
Z=0. In 2003 d+Au and p+p run, only 28 MRPCs were installed in the
tray and 12 out of 28 were instrumented with the electronics,
representing 0.3\% of TPC coverage. If we number the 33 MRPCs in
the tray from 1 to 33, with 1 close to TPC center and 33 far from
TPC center, the numbers of 12 modules instrumented with the
electronics in 2003 are 3,4,5,7,9,10,11,12,13,14,26 and 32.
\subsubsection{The introduction of MRPC}
\begin{figure}[h]
\begin{minipage}[t]{1.0\linewidth}
\includegraphics[height=16pc,width=32pc]{Augusttmp1.eps}
\end{minipage}
\hspace{\fill}
\begin{minipage}[t]{1.0\linewidth}
\includegraphics[height=11.5pc,width=32pc]{Augusttmp2.eps}
\end{minipage}
\caption{Two side views of MRPC. The upper (lower) is for long
(short) side view. The two plots are not at the same scale. Figure
is taken from ~\cite{tofproposal}.} \label{mrpcstru}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[height=12pc,width=24pc]{readout.eps}
\caption{The shape of the 6 read-out strips for each MRPC.}
\label{readout}
\end{figure}
Resistive Plate Chambers (RPCs) were developed in
1980s~\cite{mysimu:01}, and were originally operated in streamer
mode. This operation mode allows us to get high detection
efficiency ($>$95\%) and time resolution (~1 ns), with low fluxes
of incident particles. At higher fluxes ($>$200 $Hz/cm^2$), RPCs
begin to lose their efficiency. A way to overcome this problem is
to operate RPCs in avalanche mode. The Multi-gap Resistive Plate
Chamber (MRPC) was developed less than 10 years
ago~\cite{mysimu:02}. It consists of a stack of resistive plates,
spaced one from the other with equal sized spacers creating a
series of gas gaps. Electrodes are connected to the outer surfaces
of the stack of resistive plates while all the internal plates are
left electrically floating. Initially the voltage on these
internal plates is given by electrostatics, but they are kept at
the correct voltage due to the flow of electrons and ions created
in the avalanches. Figure~\ref{mrpcstru} shows the structure of
MRPC detector. For each MRPC, there are 6 read-out strips.
Figure~\ref{readout} shows the shape of the read-out strip.
The detailed production process can be found at Appendix B.\\
MRPC, as a new kind of detector for time of fight system, operated
in avalanche mode with a non flammable gas mixture of 90\% F134A,
5\% isobutane, 5\% SF6, can fulfill all these requirements: high
efficiency ($>$95\%), excellent intrinsic time resolution ($<$100
ps)~\cite{mysimu:13,startof,mysimu:15,mysimu:16,mysimu:17}, high
rate capability (~500 $Hz/cm^2$), high modularity and simplicity
for construction, good uniformity of response, high
granularity/low occupancy, and large acceptance.
\subsubsection{Simulation: the work principle of this chamber}
A detailed description of the model used in the simulation was
reported in these
papers~\cite{mysimu:03,mysimu:04,mysimu:05,mysimu:06}, here just
the main items will be repeated. The program starts from
considering an ionizing particle which crosses the gas gaps and
generates a certain number of clusters of ion-electron pairs. The
electrons contained in the clusters drift towards the anode and,
if the electric field is sufficiently high, give rise to the
avalanche processes. \\
The primary cluster numbers and the avalanche growth are assumed
to follow, respectively, simple Poisson statistics and the usual
exponential law. Avalanche gain fluctuations have been taken into
account using a Polya distribution~\cite{mysimu:07}. After the
simulation of the drifting avalanches, the program computes, by
means of Ramo~\cite{mysimu:08} theorem, the charge $q_{ind}$
induced on the external pick-up electrodes (strips or pads) by the
avalanche motion. Under certain approximations, this is given by
the formula
\begin{equation} q_{ind} =
\frac{q_{e}}{\eta{d}}\triangle{V}_{w}{\sum_{j=1}^{n_{cl}}{n_{j}M(e^{\eta{(d-x_{j})}}-1)}}
\end{equation} where $q_e$ is the
electron charge, $\eta$ 1st effective Townsend coefficient
$\eta=\alpha-\beta$, $\alpha$ is the Townsend coefficient, $\beta$
is the attachment coefficient, $x_j$ the $j_{th}$ cluster initial
distances from the anode, $d$ the gap width, $n_j$ the number of
initial electrons in the considered $j_{th}$ cluster, $M$ the
avalanche gain fluctuations factor, and $\triangle{V}_{w}/d=E_{w}$
is the normalized weighting field. In addition to $q_{ind}$, the
current $i_{ind}(t)$ induced on the same electrodes by the
drifting charge $q_d(t)$ may be computed as
\begin{equation} i_{ind}(t) =
\triangle{V}_{w}\frac{v_{d}}{d}q_d(t)Me^{\eta{v_{d}t}}
\end{equation}, where $v_d$ is the electron drift velocity. The computation of
$i_{ind}$ allows us to reproduce the whole information coming out
from MRPC, such as time
distribution.\\
\textbf{Charge Spectrum Simulation}: The almost Gaussian charge
distribution obtained with the MRPC is a key ingredient to its
performance. If the avalanches grew following Townsend's formula
the charge distribution would be exponential in shape. Thus the
space charge effects must be considered in the simulation. \\
The input parameters for the simulation program are: the Townsend
coefficient $\alpha$, the attachment coefficient $\beta$, the
average distance between clusters $\lambda$ and the probability
distribution of the number of electrons per cluster. These pieces
of information can be obtained, for a given gas mixture and given
conditions (pressure and temperature) and electric field, by the
programs HEED~\cite{mysimu:09} and
MAGBOLTZ~\cite{mysimu:10,mysimu:11}. In addition, a maximum number
of electrons in an avalanche (cutoff value) is specified. \\
In a given gap, we generate a number of clusters with distances
exponentially distributed with average distance $\lambda$. For
each cluster, we then generate a certain number of electrons,
according to the distribution obtained by the program HEED. Each
electron from the primary cluster will give rise to a number of
electrons, generated according to an exponential probability
law.\\
\begin{figure}[h]
\centering
\includegraphics[height=18pc,width=24pc]{tmp_ano.eps}
\caption{Simulated 1st effective Townsend coefficient curve and
normalized charge distribution for a 6 and 10 gap MRPC.}
\label{chargedistribution}
\end{figure}
For each cluster, the avalanche growth is stopped when the total
charge reaches a certain cutoff value, as originally suggested in
ref.~\cite{mysimu:12} to take into account space charge effects in
the avalanche development. This cutoff value has been set to be
$1.6\times10^7$ electrons.\\
In Figure~\ref{chargedistribution} we show the results of
simulations, Figure~\ref{chargedistribution}(a) is the simulated
curve of the 1st effective Townsend coefficient $\eta$ versus the
electric field, which is generated by Magboltz. The curve shows
that the correlation between $\eta$ and the electric field is
almost linear when MRPC is operated at high electric field for the
gas mixture. Figure~\ref{chargedistribution}(b) is the charge
spectrum for a 6 gap chamber and (c) (d) for a 10 gap chamber
compared to experimental data~\cite{mysimu:13,startof}, and the
number under each plot shows the electric field $E$ in the gas gap
for MRPC. In both cases the gap size is 220 $\mu{m}$. The gas
mixture was 90\% F134A, 5\% isobutane and 5\% SF6 in normal
conditions of pressure and temperature. The value of $\lambda$
used was 0.1 $mm$, derived from HEED program. \\
The charge distribution has an almost Gaussian form, especially
for the 10 gap MRPC. The left side of the distribution (very few
events at values near zero) is due to the fact that the MRPC
operates at high gain $\eta \times{d}\sim30$. This means that
avalanches starting in the middle of the gap width, which only
avalanche over half the distance, give a detectable signal. The
charge distribution is the superposition of several probability
distributions which, according to the central limit theorem, will
tend to a Gaussian form. The right side of the charge distribution
(the fact that the tails are not very long) indicates that indeed
the space charge effects stop the development of the avalanche.\\
\begin{figure}[h]
\centering
\includegraphics[height=18pc,width=24pc]{6gap-time-velocity.eps}
\caption{Simulated results of a 6 gap MRPC.}
\label{timedistribution}
\end{figure}
\textbf{Time Distribution Simulation:} We then proceed to simulate
the time distribution of these same chambers. The electron drift
velocity can be obtained from HEED. When the total induced charge
signal is over threshold, the time is recorded. In this paper, the
threshold is 13 fc for the 6 gap MRPC and 26 fc for the 10 gap
MRPC. Fig.2 is the simulated results for a 6 gap chamber.
Figure~\ref{timedistribution} (a) is the simulated curve of the
electron drift velocity versus the electric field, which is
generated by Magboltz. Figure~\ref{timedistribution} (b) is the
time distribution of a 6 gap MRPC. The intrinsic time resolution
is only 19 ps or so. If we consider other contributions, such as
front-end electronics 30 ps, TDC resolution 25 ps, fanout start
signal 10 ps, beam size (1cm) 15 ps, we can get the MRPC
resolution is $\sqrt{20^2+30^2+25^2+10^2+15^2}=47$ ps. This value
is similar to the experimental result~\cite{mysimu:13,startof}.
For a 10 gap MRPC, the intrinsic time resolution is about 15
ps.\\
From the simulation, we can get the bottom line of MRPC time
resolution $\sim20$ ps. And we need to keep control of all these
contributions to ensure best time resolution.
\subsubsection{MRPC for this tray installed in 2003}
In 2003, for the MRPCs in the TOFr, the inner glass thickness is
0.54 mm, the outer glass is 1.1 mm. The gas gap is 0.22 mm. Both
the volume resistivity ($10^{12-13} ohm.cm$) of the glass plates
and the surface resistivity(2M ohm per square) of carbon layer at
room temperature are presented in~\cite{resistivity}. It is found
the volume resistivity of the plate decreases with the temperature
increasing. And the radiation will decrease the volume resistivity
of the plate~\cite{resistivity}. In order not to pollute the
working gas of TPC, SF6 is not used as part of the working gas of
TOFr. The working gas of MRPC-TOFr at STAR is 95\% freon and 5\%
iso-butane at normal atmospheric pressure. The high voltage
applied to the electrodes is 14.0 kV.
\chapter{{\hspace{3.5cm}Tables of the Invariant Yields}}
\begin{table*}
\begin{scriptsize}
\centering
\begin{tabular} {|c|c|c|c|} \hline
$p_{T}$ & $p_{T}$ width & M.B. & 0\%-20\% \\ \hline
3.50e-01& 1.00e-01 & $3.14e+00\pm3.65e-02\pm2.51e-01$ &
$6.70e+00\pm9.69e-02\pm5.36e-01$ \\ \hline 4.50e-01& 1.00e-01 &
$1.82e+00\pm2.45e-02\pm1.45e-01$ &
$3.88e+00\pm6.51e-02\pm3.10e-01$ \\ \hline 5.50e-01& 1.00e-01 &
$1.08e+00\pm1.73e-02\pm8.67e-02$ &
$2.38e+00\pm4.68e-02\pm1.90e-01$ \\ \hline 6.50e-01& 1.00e-01 &
$6.18e-01\pm1.22e-02\pm4.94e-02$ &
$1.38e+00\pm3.28e-02\pm1.10e-01$ \\ \hline 7.50e-01& 1.00e-01 &
$4.13e-01\pm9.15e-03\pm3.31e-02$ &
$9.53e-01\pm2.56e-02\pm7.63e-02$ \\ \hline 8.50e-01& 1.00e-01 &
$2.66e-01\pm6.98e-03\pm2.13e-02$ &
$6.11e-01\pm1.93e-02\pm4.88e-02$ \\ \hline 9.50e-01& 1.00e-01 &
$1.70e-01\pm5.36e-03\pm1.36e-02$ &
$3.79e-01\pm1.44e-02\pm3.03e-02$ \\ \hline 1.05e+00& 1.00e-01 &
$1.14e-01\pm4.07e-03\pm9.09e-03$ &
$2.68e-01\pm1.15e-02\pm2.14e-02$ \\ \hline 1.15e+00& 1.00e-01 &
$7.70e-02\pm3.27e-03\pm6.16e-03$ &
$1.71e-01\pm8.82e-03\pm1.37e-02$ \\ \hline 1.30e+00& 2.00e-01 &
$4.43e-02\pm1.66e-03\pm3.55e-03$ &
$1.06e-01\pm4.71e-03\pm8.45e-03$ \\ \hline 1.50e+00& 2.00e-01 &
$2.10e-02\pm1.07e-03\pm1.68e-03$ &
$4.65e-02\pm2.96e-03\pm3.72e-03$ \\ \hline 1.70e+00& 2.00e-01 &
$1.05e-02\pm7.19e-04\pm8.40e-04$ &
$2.29e-02\pm1.92e-03\pm1.83e-03$ \\ \hline
\end{tabular}
\caption{$\pi^{+}$ spectra in minimum-bias and 0-20\% d+Au
collisions. The unit of $p_{T}$ and $p_{T}$ width is $GeV/c$. }
\label{pionplusspectratable1}
\end{scriptsize}
\end{table*}
\begin{table*}
\begin{scriptsize}
\centering
\begin{tabular}{|c|c|c|c|} \hline
$p_{T}$ & $p_{T}$ width & 20\%-40\% & 40\%-100\% \\ \hline
3.50e-01& 1.00e-01 & $4.73e+00\pm6.94e-02\pm3.78e-01$ &
$1.93e+00\pm2.85e-02\pm1.55e-01$ \\ \hline 4.50e-01& 1.00e-01 &
$2.80e+00\pm4.73e-02\pm2.24e-01$ &
$1.10e+00\pm1.89e-02\pm8.81e-02$ \\ \hline 5.50e-01& 1.00e-01 &
$1.68e+00\pm3.36e-02\pm1.34e-01$ &
$6.43e-01\pm1.31e-02\pm5.14e-02$ \\ \hline 6.50e-01& 1.00e-01 &
$9.50e-01\pm2.30e-02\pm7.60e-02$ &
$3.68e-01\pm9.03e-03\pm2.94e-02$ \\ \hline 7.50e-01& 1.00e-01 &
$6.41e-01\pm1.76e-02\pm5.13e-02$ &
$2.40e-01\pm6.80e-03\pm1.92e-02$ \\ \hline 8.50e-01& 1.00e-01 &
$4.20e-01\pm1.36e-02\pm3.36e-02$ &
$1.52e-01\pm5.09e-03\pm1.22e-02$ \\ \hline 9.50e-01& 1.00e-01 &
$2.58e-01\pm1.00e-02\pm2.07e-02$ &
$1.01e-01\pm3.97e-03\pm8.05e-03$ \\ \hline 1.05e+00& 1.00e-01 &
$1.80e-01\pm7.92e-03\pm1.44e-02$ &
$6.10e-02\pm2.84e-03\pm4.88e-03$ \\ \hline 1.15e+00& 1.00e-01 &
$1.21e-01\pm6.35e-03\pm9.72e-03$ &
$4.38e-02\pm2.37e-03\pm3.50e-03$ \\ \hline 1.30e+00& 2.00e-01 &
$6.49e-02\pm3.02e-03\pm5.19e-03$ &
$2.40e-02\pm1.16e-03\pm1.92e-03$ \\ \hline 1.50e+00& 2.00e-01 &
$3.28e-02\pm2.05e-03\pm2.63e-03$ &
$1.03e-02\pm7.12e-04\pm8.27e-04$ \\ \hline 1.70e+00& 2.00e-01 &
$1.57e-02\pm1.31e-03\pm1.26e-03$ &
$4.92e-03\pm4.49e-04\pm3.94e-04$ \\ \hline
\end{tabular}
\caption{$\pi^{+}$ spectra in 20-40\% and 40-100\% d+Au
collisions. The unit of $p_{T}$ and $p_{T}$ width is $GeV/c$.}
\label{pionplusspectratable2}
\end{scriptsize}
\end{table*}
\begin{table}[h]
\begin{scriptsize}
\centering
\begin{tabular}{|c|c|c|} \hline
$p_{T}$ & $p_{T}$ width & p+p \\ \hline
3.50e-01& 1.00e-01 & $9.71e-01\pm1.21e-02\pm7.76e-02$ \\ \hline
4.50e-01& 1.00e-01 & $5.32e-01\pm7.84e-03\pm4.26e-02$ \\ \hline
5.50e-01& 1.00e-01 & $3.14e-01\pm5.46e-03\pm2.51e-02$ \\ \hline
6.50e-01& 1.00e-01 & $1.74e-01\pm3.72e-03\pm1.40e-02$ \\ \hline
7.50e-01& 1.00e-01 & $1.08e-01\pm2.65e-03\pm8.64e-03$ \\ \hline
8.50e-01& 1.00e-01 & $6.42e-02\pm1.89e-03\pm5.14e-03$ \\ \hline
9.50e-01& 1.00e-01 & $4.03e-02\pm1.42e-03\pm3.22e-03$ \\ \hline
1.05e+00& 1.00e-01 & $2.40e-02\pm9.94e-04\pm1.92e-03$ \\ \hline
1.15e+00& 1.00e-01 & $1.55e-02\pm7.71e-04\pm1.24e-03$ \\ \hline
1.30e+00& 2.00e-01 & $8.19e-03\pm3.86e-04\pm6.55e-04$ \\ \hline
1.50e+00& 2.00e-01 & $3.77e-03\pm2.46e-04\pm3.02e-04$ \\ \hline
1.70e+00& 2.00e-01 & $1.84e-03\pm1.96e-04\pm1.47e-04$ \\ \hline
\end{tabular}
\caption{$\pi^{+}$ spectra in p+p collisions. The unit of $p_{T}$
and $p_{T}$ width is $GeV/c$.} \label{pionplusspectratable3}
\end{scriptsize}
\end{table}
\begin{table}[h]
\begin{scriptsize}
\centering
\begin{tabular}{|c|c|c|c|} \hline
$p_{T}$ & $p_{T}$ width & M.B. & 0\%-20\% \\ \hline
3.50e-01& 1.00e-01 & $3.20e+00\pm3.73e-02\pm2.56e-01$ &
$6.77e+00\pm9.88e-02\pm5.42e-01$ \\ \hline 4.50e-01& 1.00e-01 &
$1.82e+00\pm2.46e-02\pm1.45e-01$ &
$3.92e+00\pm6.60e-02\pm3.13e-01$ \\ \hline 5.50e-01& 1.00e-01 &
$1.08e+00\pm1.72e-02\pm8.60e-02$ &
$2.42e+00\pm4.71e-02\pm1.94e-01$ \\ \hline 6.50e-01& 1.00e-01 &
$6.70e-01\pm1.30e-02\pm5.36e-02$ &
$1.49e+00\pm3.51e-02\pm1.20e-01$ \\ \hline 7.50e-01& 1.00e-01 &
$4.05e-01\pm9.06e-03\pm3.24e-02$ &
$9.22e-01\pm2.50e-02\pm7.38e-02$ \\ \hline 8.50e-01& 1.00e-01 &
$2.59e-01\pm6.89e-03\pm2.07e-02$ &
$6.13e-01\pm1.96e-02\pm4.91e-02$ \\ \hline 9.50e-01& 1.00e-01 &
$1.68e-01\pm5.38e-03\pm1.34e-02$ &
$3.85e-01\pm1.48e-02\pm3.08e-02$ \\ \hline 1.05e+00& 1.00e-01 &
$1.16e-01\pm4.23e-03\pm9.29e-03$ &
$2.70e-01\pm1.18e-02\pm2.16e-02$ \\ \hline 1.15e+00& 1.00e-01 &
$7.63e-02\pm3.28e-03\pm6.11e-03$ &
$1.71e-01\pm8.83e-03\pm1.37e-02$ \\ \hline 1.30e+00& 2.00e-01 &
$4.36e-02\pm1.66e-03\pm3.49e-03$ &
$9.67e-02\pm4.45e-03\pm7.73e-03$ \\ \hline 1.50e+00& 2.00e-01 &
$2.04e-02\pm1.07e-03\pm1.63e-03$ &
$4.94e-02\pm3.06e-03\pm3.96e-03$ \\ \hline 1.70e+00& 2.00e-01 &
$1.03e-02\pm7.23e-04\pm8.26e-04$ &
$2.54e-02\pm2.07e-03\pm2.03e-03$ \\ \hline
\end{tabular}
\caption{$\pi^{-}$ spectra in minimum-bias and 0-20\% d+Au
collisions. The unit of $p_{T}$ and $p_{T}$ width is $GeV/c$. }
\label{pionminusspectratable1}
\end{scriptsize}
\end{table}
\begin{table}[h]
\begin{scriptsize}
\centering
\begin{tabular}{|c|c|c|c|} \hline
$p_{T}$ & $p_{T}$ width & 20\%-40\% & 40\%-100\% \\ \hline
3.50e-01& 1.00e-01 & $4.81e+00\pm7.11e-02\pm3.85e-01$ &
$2.00e+00\pm2.95e-02\pm1.60e-01$ \\ \hline 4.50e-01& 1.00e-01 &
$2.76e+00\pm4.73e-02\pm2.21e-01$ &
$1.12e+00\pm1.93e-02\pm8.95e-02$ \\ \hline 5.50e-01& 1.00e-01 &
$1.65e+00\pm3.28e-02\pm1.32e-01$ &
$6.43e-01\pm1.31e-02\pm5.14e-02$ \\ \hline 6.50e-01& 1.00e-01 &
$1.03e+00\pm2.47e-02\pm8.23e-02$ &
$4.04e-01\pm9.83e-03\pm3.23e-02$ \\ \hline 7.50e-01& 1.00e-01 &
$6.15e-01\pm1.71e-02\pm4.92e-02$ &
$2.40e-01\pm6.80e-03\pm1.92e-02$ \\ \hline 8.50e-01& 1.00e-01 &
$3.92e-01\pm1.30e-02\pm3.13e-02$ &
$1.51e-01\pm5.10e-03\pm1.21e-02$ \\ \hline 9.50e-01& 1.00e-01 &
$2.61e-01\pm1.03e-02\pm2.09e-02$ &
$9.62e-02\pm3.91e-03\pm7.70e-03$ \\ \hline 1.05e+00& 1.00e-01 &
$1.81e-01\pm8.11e-03\pm1.45e-02$ &
$6.65e-02\pm3.07e-03\pm5.32e-03$ \\ \hline 1.15e+00& 1.00e-01 &
$1.20e-01\pm6.26e-03\pm9.58e-03$ &
$4.23e-02\pm2.31e-03\pm3.38e-03$ \\ \hline 1.30e+00& 2.00e-01 &
$6.82e-02\pm3.18e-03\pm5.46e-03$ &
$2.42e-02\pm1.18e-03\pm1.93e-03$ \\ \hline 1.50e+00& 2.00e-01 &
$3.24e-02\pm2.13e-03\pm2.59e-03$ &
$1.05e-02\pm7.69e-04\pm8.42e-04$ \\ \hline 1.70e+00& 2.00e-01 &
$1.55e-02\pm1.34e-03\pm1.24e-03$ &
$5.02e-03\pm1.06e-03\pm4.01e-04$ \\ \hline
\end{tabular}
\caption{$\pi^{-}$ spectra in 20-40\% and 40-100\% d+Au
collisions. The unit of $p_{T}$ and $p_{T}$ width is $GeV/c$.}
\label{pionminusspectratable2}
\end{scriptsize}
\end{table}
\begin{table}[h]
\begin{scriptsize}
\centering
\begin{tabular}{|c|c|c|} \hline
$p_{T}$ & $p_{T}$ width & p+p \\ \hline
3.50e-01& 1.00e-01 & $9.71e-01\pm1.22e-02\pm7.76e-02$ \\ \hline
4.50e-01& 1.00e-01 & $5.47e-01\pm8.02e-03\pm4.37e-02$ \\ \hline
5.50e-01& 1.00e-01 & $3.09e-01\pm5.38e-03\pm2.47e-02$ \\ \hline
6.50e-01& 1.00e-01 & $1.84e-01\pm3.89e-03\pm1.47e-02$ \\ \hline
7.50e-01& 1.00e-01 & $1.00e-01\pm2.50e-03\pm8.01e-03$ \\ \hline
8.50e-01& 1.00e-01 & $6.36e-02\pm1.89e-03\pm5.09e-03$ \\ \hline
9.50e-01& 1.00e-01 & $3.80e-02\pm1.38e-03\pm3.04e-03$ \\ \hline
1.05e+00& 1.00e-01 & $2.44e-02\pm1.03e-03\pm1.95e-03$ \\ \hline
1.15e+00& 1.00e-01 & $1.57e-02\pm7.87e-04\pm1.25e-03$ \\ \hline
1.30e+00& 2.00e-01 & $8.70e-03\pm4.02e-04\pm6.96e-04$ \\ \hline
1.50e+00& 2.00e-01 & $3.62e-03\pm2.45e-04\pm2.90e-04$ \\ \hline
1.70e+00& 2.00e-01 & $1.69e-03\pm1.76e-04\pm1.35e-04$ \\ \hline
\end{tabular}
\caption{$\pi^{-}$ spectra in p+p collisions. The unit of $p_{T}$
and $p_{T}$ width is $GeV/c$.} \label{pionminusspectratable3}
\end{scriptsize}
\end{table}
\begin{table}[h]
\begin{scriptsize}
\centering
\begin{tabular}{|c|c|c|c|} \hline
$p_{T}$ & $p_{T}$ width & M.B. & 0\%-20\% \\ \hline
4.57e-01& 1.00e-01 & $2.41e-01\pm9.47e-03\pm1.93e-02$ &
$4.83e-01\pm2.75e-02\pm3.86e-02$ \\ \hline 5.56e-01& 1.00e-01 &
$1.87e-01\pm7.53e-03\pm1.49e-02$ &
$3.87e-01\pm2.14e-02\pm3.10e-02$ \\ \hline 6.55e-01& 1.00e-01 &
$1.33e-01\pm5.14e-03\pm1.07e-02$ &
$2.81e-01\pm1.53e-02\pm2.25e-02$ \\ \hline 7.54e-01& 1.00e-01 &
$1.01e-01\pm3.18e-03\pm8.07e-03$ &
$2.14e-01\pm1.09e-02\pm1.72e-02$ \\ \hline 8.54e-01& 1.00e-01 &
$6.97e-02\pm2.48e-03\pm5.57e-03$ &
$1.58e-01\pm8.74e-03\pm1.27e-02$ \\ \hline 9.54e-01& 1.00e-01 &
$5.01e-02\pm1.95e-03\pm4.01e-03$ &
$9.95e-02\pm6.30e-03\pm7.96e-03$ \\ \hline 1.05e+00& 1.00e-01 &
$3.77e-02\pm1.65e-03\pm3.02e-03$ &
$8.92e-02\pm5.83e-03\pm7.14e-03$ \\ \hline 1.15e+00& 1.00e-01 &
$2.73e-02\pm1.31e-03\pm2.18e-03$ &
$5.61e-02\pm4.30e-03\pm4.49e-03$ \\ \hline 1.30e+00& 2.00e-01 &
$1.78e-02\pm7.25e-04\pm1.42e-03$ &
$4.00e-02\pm2.49e-03\pm3.20e-03$ \\ \hline 1.50e+00& 2.00e-01 &
$9.12e-03\pm5.25e-04\pm7.30e-04$ &
$1.95e-02\pm1.89e-03\pm1.56e-03$ \\ \hline 1.70e+00& 2.00e-01 &
$4.68e-03\pm3.96e-04\pm3.75e-04$ & $---$ \\ \hline
\end{tabular}
\caption{$K^{+}$ spectra in minimum-bias and 0-20\% d+Au
collisions. The unit of $p_{T}$ and $p_{T}$ width is $GeV/c$. }
\label{kaonplusspectratable1}
\end{scriptsize}
\end{table}
\begin{table}[h]
\begin{scriptsize}
\centering
\begin{tabular}{|c|c|c|c|} \hline
$p_{T}$ & $p_{T}$ width & 20\%-40\% & 40\%-100\% \\ \hline
4.57e-01& 1.00e-01 & $3.63e-01\pm2.06e-02\pm2.91e-02$ &
$1.27e-01\pm7.70e-03\pm1.02e-02$ \\ \hline 5.56e-01& 1.00e-01 &
$2.74e-01\pm1.54e-02\pm2.19e-02$ &
$1.07e-01\pm6.16e-03\pm8.60e-03$ \\ \hline 6.55e-01& 1.00e-01 &
$2.01e-01\pm1.11e-02\pm1.60e-02$ &
$7.37e-02\pm4.26e-03\pm5.89e-03$ \\ \hline 7.54e-01& 1.00e-01 &
$1.58e-01\pm8.18e-03\pm1.27e-02$ &
$5.25e-02\pm2.99e-03\pm4.20e-03$ \\ \hline 8.54e-01& 1.00e-01 &
$1.01e-01\pm6.16e-03\pm8.06e-03$ &
$3.73e-02\pm2.35e-03\pm2.98e-03$ \\ \hline 9.54e-01& 1.00e-01 &
$7.97e-02\pm4.89e-03\pm6.37e-03$ &
$2.80e-02\pm1.86e-03\pm2.24e-03$ \\ \hline 1.05e+00& 1.00e-01 &
$5.34e-02\pm3.83e-03\pm4.27e-03$ &
$1.91e-02\pm1.47e-03\pm1.52e-03$ \\ \hline 1.15e+00& 1.00e-01 &
$3.91e-02\pm3.12e-03\pm3.13e-03$ &
$1.36e-02\pm1.18e-03\pm1.09e-03$ \\ \hline 1.30e+00& 2.00e-01 &
$2.51e-02\pm1.67e-03\pm2.01e-03$ &
$8.65e-03\pm6.83e-04\pm6.92e-04$ \\ \hline 1.50e+00& 2.00e-01 &
$1.32e-02\pm1.19e-03\pm1.05e-03$ &
$4.69e-03\pm5.07e-04\pm3.75e-04$ \\ \hline
\end{tabular}
\caption{$K^{+}$ spectra in 20-40\% and 40-100\% d+Au collisions.
The unit of $p_{T}$ and $p_{T}$ width is $GeV/c$.}
\label{kaonplusspectratable2}
\end{scriptsize}
\end{table}
\begin{table}[h]
\begin{scriptsize}
\centering
\begin{tabular}{|c|c|c|} \hline
$p_{T}$ & $p_{T}$ width & p+p \\ \hline
4.57e-01& 1.00e-01 & $6.47e-02\pm3.01e-03\pm5.18e-03$ \\ \hline
5.56e-01& 1.00e-01 & $4.32e-02\pm2.10e-03\pm3.45e-03$ \\ \hline
6.55e-01& 1.00e-01 & $3.18e-02\pm1.51e-03\pm2.54e-03$ \\ \hline
7.54e-01& 1.00e-01 & $2.18e-02\pm9.70e-04\pm1.74e-03$ \\ \hline
8.54e-01& 1.00e-01 & $1.57e-02\pm7.69e-04\pm1.25e-03$ \\ \hline
9.54e-01& 1.00e-01 & $9.92e-03\pm5.60e-04\pm7.93e-04$ \\ \hline
1.05e+00& 1.00e-01 & $7.17e-03\pm4.74e-04\pm5.74e-04$ \\ \hline
1.15e+00& 1.00e-01 & $5.60e-03\pm4.18e-04\pm4.48e-04$ \\ \hline
1.30e+00& 2.00e-01 & $3.73e-03\pm2.72e-04\pm2.98e-04$ \\ \hline
1.50e+00& 2.00e-01 & $1.81e-03\pm1.18e-04\pm1.45e-04$ \\ \hline
\end{tabular}
\caption{$K^{+}$ spectra in p+p collisions. The unit of $p_{T}$
and $p_{T}$ width is $GeV/c$.} \label{kaonplusspectratable3}
\end{scriptsize}
\end{table}
\begin{table}[h]
\begin{scriptsize}
\centering
\begin{tabular}{|c|c|c|c|} \hline
$p_{T}$ & $p_{T}$ width & M.B. & 0\%-20\% \\ \hline
4.57e-01& 1.00e-01 & $2.34e-01\pm9.47e-03\pm1.87e-02$ &
$4.87e-01\pm2.81e-02\pm3.90e-02$ \\ \hline 5.56e-01& 1.00e-01 &
$1.92e-01\pm7.98e-03\pm1.54e-02$ &
$4.01e-01\pm2.28e-02\pm3.21e-02$ \\ \hline 6.55e-01& 1.00e-01 &
$1.39e-01\pm5.81e-03\pm1.12e-02$ &
$2.91e-01\pm1.69e-02\pm2.33e-02$ \\ \hline 7.54e-01& 1.00e-01 &
$9.21e-02\pm3.10e-03\pm7.37e-03$ &
$1.90e-01\pm1.04e-02\pm1.52e-02$ \\ \hline 8.54e-01& 1.00e-01 &
$7.11e-02\pm2.61e-03\pm5.69e-03$ &
$1.45e-01\pm8.76e-03\pm1.16e-02$ \\ \hline 9.54e-01& 1.00e-01 &
$4.70e-02\pm1.94e-03\pm3.76e-03$ &
$9.49e-02\pm6.34e-03\pm7.59e-03$ \\ \hline 1.05e+00& 1.00e-01 &
$3.11e-02\pm1.49e-03\pm2.48e-03$ &
$6.56e-02\pm5.07e-03\pm5.25e-03$ \\ \hline 1.15e+00& 1.00e-01 &
$2.28e-02\pm1.21e-03\pm1.82e-03$ &
$4.74e-02\pm4.00e-03\pm3.79e-03$ \\ \hline 1.30e+00& 2.00e-01 &
$1.61e-02\pm7.10e-04\pm1.29e-03$ &
$3.70e-02\pm2.41e-03\pm2.96e-03$ \\ \hline 1.50e+00& 2.00e-01 &
$9.47e-03\pm5.54e-04\pm7.58e-04$ &
$2.01e-02\pm1.78e-03\pm1.61e-03$ \\ \hline 1.70e+00& 2.00e-01 &
$4.44e-03\pm4.19e-04\pm3.55e-04$ & $---$ \\ \hline
\end{tabular}
\caption{$K^{-}$ spectra in minimum-bias and 0-20\% d+Au
collisions. The unit of $p_{T}$ and $p_{T}$ width is $GeV/c$. }
\label{kaonminusspectratable1}
\end{scriptsize}
\end{table}
\begin{table}[h]
\begin{scriptsize}
\centering
\begin{tabular}{|c|c|c|c|} \hline
$p_{T}$ & $p_{T}$ width & 20\%-40\% & 40\%-100\% \\ \hline
4.57e-01& 1.00e-01 & $3.24e-01\pm1.95e-02\pm2.59e-02$ &
$1.39e-01\pm8.25e-03\pm1.11e-02$ \\ \hline 5.56e-01& 1.00e-01 &
$2.66e-01\pm1.58e-02\pm2.13e-02$ &
$1.13e-01\pm6.63e-03\pm9.03e-03$ \\ \hline 6.55e-01& 1.00e-01 &
$1.99e-01\pm1.18e-02\pm1.59e-02$ &
$7.50e-02\pm4.61e-03\pm6.00e-03$ \\ \hline 7.54e-01& 1.00e-01 &
$1.36e-01\pm7.60e-03\pm1.08e-02$ &
$5.62e-02\pm3.16e-03\pm4.50e-03$ \\ \hline 8.54e-01& 1.00e-01 &
$1.13e-01\pm6.64e-03\pm9.04e-03$ &
$3.78e-02\pm2.47e-03\pm3.03e-03$ \\ \hline 9.54e-01& 1.00e-01 &
$7.35e-02\pm4.83e-03\pm5.88e-03$ &
$2.55e-02\pm1.82e-03\pm2.04e-03$ \\ \hline 1.05e+00& 1.00e-01 &
$4.51e-02\pm3.57e-03\pm3.61e-03$ &
$1.93e-02\pm1.51e-03\pm1.54e-03$ \\ \hline 1.15e+00& 1.00e-01 &
$3.03e-02\pm2.74e-03\pm2.43e-03$ &
$1.36e-02\pm1.20e-03\pm1.09e-03$ \\ \hline 1.30e+00& 2.00e-01 &
$2.35e-02\pm1.66e-03\pm1.88e-03$ &
$7.84e-03\pm6.41e-04\pm6.27e-04$ \\ \hline 1.50e+00& 2.00e-01 &
$1.05e-02\pm1.20e-03\pm8.36e-04$ &
$5.10e-03\pm5.72e-04\pm4.08e-04$ \\ \hline
\end{tabular}
\caption{$K^{-}$ spectra in 20-40\% and 40-100\% d+Au collisions.
The unit of $p_{T}$ and $p_{T}$ width is $GeV/c$.}
\label{kaonminusspectratable2}
\end{scriptsize}
\end{table}
\begin{table}[h]
\begin{scriptsize}
\centering
\begin{tabular}{|c|c|c|} \hline
$p_{T}$ & $p_{T}$ width & p+p \\ \hline
4.57e-01& 1.00e-01 & $5.99e-02\pm2.91e-03\pm4.79e-03$ \\ \hline
5.56e-01& 1.00e-01 & $4.79e-02\pm2.35e-03\pm3.83e-03$ \\ \hline
6.55e-01& 1.00e-01 & $3.06e-02\pm1.58e-03\pm2.45e-03$ \\ \hline
7.54e-01& 1.00e-01 & $2.15e-02\pm9.81e-04\pm1.72e-03$ \\ \hline
8.54e-01& 1.00e-01 & $1.46e-02\pm7.67e-04\pm1.17e-03$ \\ \hline
9.54e-01& 1.00e-01 & $1.01e-02\pm5.86e-04\pm8.09e-04$ \\ \hline
1.05e+00& 1.00e-01 & $7.87e-03\pm5.22e-04\pm6.30e-04$ \\ \hline
1.15e+00& 1.00e-01 & $5.25e-03\pm4.27e-04\pm4.20e-04$ \\ \hline
1.30e+00& 2.00e-01 & $3.42e-03\pm2.57e-04\pm2.74e-04$ \\ \hline
1.50e+00& 2.00e-01 & $1.75e-03\pm1.61e-04\pm1.40e-04$ \\ \hline
\end{tabular}
\caption{$K^{-}$ spectra in p+p collisions. The unit of $p_{T}$
and $p_{T}$ width is $GeV/c$.} \label{kaonminusspectratable3}
\end{scriptsize}
\end{table}
\begin{table}[h]
\begin{scriptsize}
\centering
\begin{tabular}{|c|c|c|c|}
\hline
$p_{T}$ & $p_{T}$ width & M.B. & 0\%-20\% \\ \hline
4.68e-01& 1.00e-01 & $1.88e-01\pm1.87e-02\pm2.44e-02$ &
$3.96e-01\pm4.51e-02\pm5.15e-02$ \\ \hline 5.63e-01& 1.00e-01 &
$1.35e-01\pm1.11e-02\pm1.75e-02$ &
$2.80e-01\pm2.67e-02\pm3.64e-02$ \\ \hline 6.61e-01& 1.00e-01 &
$1.07e-01\pm8.89e-03\pm1.39e-02$ &
$2.10e-01\pm2.01e-02\pm2.73e-02$ \\ \hline 7.59e-01& 1.00e-01 &
$8.02e-02\pm6.78e-03\pm1.04e-02$ &
$1.74e-01\pm1.67e-02\pm2.27e-02$ \\ \hline 8.58e-01& 1.00e-01 &
$5.82e-02\pm5.20e-03\pm7.57e-03$ &
$1.25e-01\pm1.27e-02\pm1.63e-02$ \\ \hline 9.57e-01& 1.00e-01 &
$4.45e-02\pm4.85e-03\pm5.78e-03$ &
$1.05e-01\pm1.25e-02\pm1.37e-02$ \\ \hline 1.06e+00& 1.00e-01 &
$3.63e-02\pm4.11e-03\pm2.90e-03$ &
$8.16e-02\pm1.02e-02\pm6.53e-03$ \\ \hline 1.16e+00& 1.00e-01 &
$2.82e-02\pm2.11e-03\pm2.26e-03$ &
$5.08e-02\pm5.11e-03\pm4.06e-03$ \\ \hline 1.31e+00& 2.00e-01 &
$1.86e-02\pm1.14e-03\pm1.49e-03$ &
$4.14e-02\pm3.21e-03\pm3.31e-03$ \\ \hline 1.50e+00& 2.00e-01 &
$1.02e-02\pm7.85e-04\pm8.14e-04$ &
$2.46e-02\pm2.34e-03\pm1.97e-03$ \\ \hline 1.70e+00& 2.00e-01 &
$5.64e-03\pm3.24e-04\pm4.51e-04$ &
$1.21e-02\pm1.03e-03\pm9.66e-04$ \\ \hline 1.90e+00& 2.00e-01 &
$3.14e-03\pm2.33e-04\pm2.51e-04$ &
$8.08e-03\pm8.38e-04\pm6.46e-04$ \\ \hline 2.25e+00& 5.00e-01 &
$1.39e-03\pm9.47e-05\pm1.12e-04$ &
$3.37e-03\pm3.21e-04\pm2.70e-04$ \\ \hline 2.75e+00& 5.00e-01 &
$2.75e-04\pm3.88e-05\pm2.20e-05$ &
$6.19e-04\pm1.26e-04\pm4.95e-05$ \\ \hline 3.50e+00& 1.00e+00 &
$8.13e-05\pm1.37e-05\pm6.50e-06$ & $---$ \\ \hline
\end{tabular}
\caption{$p$ spectra in minimum-bias and 0-20\% d+Au collisions.
The unit of $p_{T}$ and $p_{T}$ width is $GeV/c$. }
\label{protonspectratable1}
\end{scriptsize}
\end{table}
\begin{table}[h]
\begin{scriptsize}
\centering
\begin{tabular}{|c|c|c|c|}
\hline
$p_{T}$ & $p_{T}$ width & 20\%-40\% & 40\%-100\% \\ \hline
4.68e-01& 1.00e-01 & $2.78e-01\pm3.24e-02\pm3.61e-02$ &
$1.21e-01\pm1.40e-02\pm1.57e-02$ \\ \hline 5.63e-01& 1.00e-01 &
$1.90e-01\pm1.84e-02\pm2.48e-02$ &
$7.84e-02\pm7.61e-03\pm1.02e-02$ \\ \hline 6.61e-01& 1.00e-01 &
$1.56e-01\pm1.49e-02\pm2.03e-02$ &
$6.18e-02\pm5.96e-03\pm8.03e-03$ \\ \hline 7.59e-01& 1.00e-01 &
$1.18e-01\pm1.14e-02\pm1.53e-02$ &
$4.84e-02\pm4.71e-03\pm6.29e-03$ \\ \hline 8.58e-01& 1.00e-01 &
$8.54e-02\pm8.68e-03\pm1.11e-02$ &
$2.98e-02\pm3.12e-03\pm3.87e-03$ \\ \hline 9.57e-01& 1.00e-01 &
$6.45e-02\pm7.77e-03\pm8.38e-03$ &
$2.39e-02\pm2.93e-03\pm3.11e-03$ \\ \hline 1.06e+00& 1.00e-01 &
$5.48e-02\pm6.88e-03\pm4.38e-03$ &
$2.12e-02\pm2.69e-03\pm1.69e-03$ \\ \hline 1.16e+00& 1.00e-01 &
$4.21e-02\pm3.97e-03\pm3.37e-03$ &
$1.52e-02\pm1.49e-03\pm1.22e-03$ \\ \hline 1.31e+00& 2.00e-01 &
$2.68e-02\pm2.09e-03\pm2.15e-03$ &
$9.80e-03\pm7.91e-04\pm7.84e-04$ \\ \hline 1.50e+00& 2.00e-01 &
$1.41e-02\pm1.38e-03\pm1.13e-03$ &
$5.05e-03\pm5.17e-04\pm4.04e-04$ \\ \hline 1.70e+00& 2.00e-01 &
$7.69e-03\pm7.02e-04\pm6.15e-04$ &
$2.54e-03\pm2.55e-04\pm2.03e-04$ \\ \hline 1.90e+00& 2.00e-01 &
$4.12e-03\pm5.31e-04\pm3.30e-04$ &
$1.43e-03\pm2.03e-04\pm1.15e-04$ \\ \hline 2.25e+00& 5.00e-01 &
$2.30e-03\pm2.28e-04\pm1.84e-04$ &
$6.31e-04\pm7.57e-05\pm5.05e-05$ \\ \hline 2.75e+00& 5.00e-01 &
$4.36e-04\pm1.13e-04\pm3.49e-05$ &
$1.62e-04\pm4.14e-05\pm1.29e-05$ \\ \hline
\end{tabular}
\caption{$p$ spectra in 20-40\% and 40-100\% d+Au collisions. The
unit of $p_{T}$ and $p_{T}$ width is $GeV/c$.}
\label{protonspectratable2}
\end{scriptsize}
\end{table}
\begin{table}[h]
\begin{scriptsize}
\centering
\begin{tabular}{|c|c|c|}
\hline
$p_{T}$ & $p_{T}$ width & p+p \\ \hline
4.68e-01& 1.00e-01 & $4.51e-02\pm5.24e-03\pm5.86e-03$ \\ \hline
5.63e-01& 1.00e-01 & $3.33e-02\pm3.29e-03\pm4.33e-03$ \\ \hline
6.61e-01& 1.00e-01 & $2.60e-02\pm2.57e-03\pm3.38e-03$ \\ \hline
7.59e-01& 1.00e-01 & $2.03e-02\pm2.12e-03\pm2.64e-03$ \\ \hline
8.58e-01& 1.00e-01 & $1.14e-02\pm1.30e-03\pm1.49e-03$ \\ \hline
9.57e-01& 1.00e-01 & $8.93e-03\pm1.20e-03\pm1.16e-03$ \\ \hline
1.06e+00& 1.00e-01 & $6.98e-03\pm1.00e-03\pm5.59e-04$ \\ \hline
1.16e+00& 1.00e-01 & $4.68e-03\pm5.95e-04\pm3.75e-04$ \\ \hline
1.31e+00& 2.00e-01 & $2.90e-03\pm3.14e-04\pm2.32e-04$ \\ \hline
1.50e+00& 2.00e-01 & $1.34e-03\pm2.04e-04\pm1.07e-04$ \\ \hline
1.70e+00& 2.00e-01 & $6.61e-04\pm6.44e-05\pm5.29e-05$ \\ \hline
1.90e+00& 2.00e-01 & $4.73e-04\pm5.58e-05\pm3.78e-05$ \\ \hline
2.25e+00& 5.00e-01 & $1.84e-04\pm2.23e-05\pm1.47e-05$ \\ \hline
2.75e+00& 5.00e-01 & $3.06e-05\pm7.26e-06\pm2.45e-06$ \\ \hline
\end{tabular}
\caption{$p$ spectra in p+p collisions. The unit of $p_{T}$ and
$p_{T}$ width is $GeV/c$.} \label{protonspectratable3}
\end{scriptsize}
\end{table}
\begin{table}[h]
\begin{scriptsize}
\centering
\begin{tabular}{|c|c|c|c|}
\hline
$p_{T}$ & $p_{T}$ width & M.B. & 0\%-20\% \\ \hline
4.68e-01& 1.00e-01 & $1.22e-01\pm7.62e-03\pm1.59e-02$ &
$2.57e-01\pm2.17e-02\pm3.34e-02$ \\ \hline 5.63e-01& 1.00e-01 &
$1.08e-01\pm5.87e-03\pm1.40e-02$ &
$2.26e-01\pm1.62e-02\pm2.94e-02$ \\ \hline 6.61e-01& 1.00e-01 &
$8.86e-02\pm4.69e-03\pm1.15e-02$ &
$1.73e-01\pm1.23e-02\pm2.25e-02$ \\ \hline 7.59e-01& 1.00e-01 &
$6.40e-02\pm3.47e-03\pm8.32e-03$ &
$1.39e-01\pm9.76e-03\pm1.81e-02$ \\ \hline 8.58e-01& 1.00e-01 &
$5.51e-02\pm3.04e-03\pm7.16e-03$ &
$1.19e-01\pm8.58e-03\pm1.54e-02$ \\ \hline 9.57e-01& 1.00e-01 &
$3.80e-02\pm2.30e-03\pm4.94e-03$ &
$8.99e-02\pm6.92e-03\pm1.17e-02$ \\ \hline 1.06e+00& 1.00e-01 &
$3.27e-02\pm1.41e-03\pm2.62e-03$ &
$7.36e-02\pm5.01e-03\pm5.89e-03$ \\ \hline 1.16e+00& 1.00e-01 &
$2.36e-02\pm1.12e-03\pm1.89e-03$ &
$4.24e-02\pm3.50e-03\pm3.39e-03$ \\ \hline 1.31e+00& 2.00e-01 &
$1.55e-02\pm6.16e-04\pm1.24e-03$ &
$3.48e-02\pm2.14e-03\pm2.78e-03$ \\ \hline 1.50e+00& 2.00e-01 &
$8.00e-03\pm4.04e-04\pm6.40e-04$ &
$1.93e-02\pm1.46e-03\pm1.54e-03$ \\ \hline 1.70e+00& 2.00e-01 &
$4.78e-03\pm2.97e-04\pm3.82e-04$ &
$1.01e-02\pm9.74e-04\pm8.11e-04$ \\ \hline 1.90e+00& 2.00e-01 &
$2.56e-03\pm2.04e-04\pm2.05e-04$ &
$4.99e-03\pm6.54e-04\pm3.99e-04$ \\ \hline 2.25e+00& 5.00e-01 &
$1.06e-03\pm7.97e-05\pm8.49e-05$ &
$2.69e-03\pm2.85e-04\pm2.15e-04$ \\ \hline 2.75e+00& 5.00e-01 &
$3.32e-04\pm4.61e-05\pm2.66e-05$ &
$5.86e-04\pm1.35e-04\pm4.69e-05$ \\ \hline 3.50e+00& 1.00e+00 &
$7.89e-05\pm1.53e-05\pm6.31e-06$ & $---$ \\ \hline
\end{tabular}
\caption{$\bar{p}$ spectra in minimum-bias and 0-20\% d+Au
collisions. The unit of $p_{T}$ and $p_{T}$ width is $GeV/c$.}
\label{pbarspectratable1}
\end{scriptsize}
\end{table}
\begin{table}[h]
\begin{scriptsize}
\centering
\begin{tabular}{|c|c|c|c|}
\hline
$p_{T}$ & $p_{T}$ width & 20\%-40\% & 40\%-100\% \\ \hline
4.68e-01& 1.00e-01 & $1.63e-01\pm1.45e-02\pm2.12e-02$ &
$7.50e-02\pm6.44e-03\pm9.75e-03$ \\ \hline 5.63e-01& 1.00e-01 &
$1.65e-01\pm1.20e-02\pm2.14e-02$ &
$6.04e-02\pm4.55e-03\pm7.85e-03$ \\ \hline 6.61e-01& 1.00e-01 &
$1.24e-01\pm8.94e-03\pm1.62e-02$ &
$5.51e-02\pm3.89e-03\pm7.16e-03$ \\ \hline 7.59e-01& 1.00e-01 &
$9.35e-02\pm6.78e-03\pm1.22e-02$ &
$3.61e-02\pm2.68e-03\pm4.69e-03$ \\ \hline 8.58e-01& 1.00e-01 &
$7.98e-02\pm5.96e-03\pm1.04e-02$ &
$2.85e-02\pm2.22e-03\pm3.70e-03$ \\ \hline 9.57e-01& 1.00e-01 &
$5.22e-02\pm4.33e-03\pm6.78e-03$ &
$1.95e-02\pm1.68e-03\pm2.54e-03$ \\ \hline 1.06e+00& 1.00e-01 &
$4.14e-02\pm3.20e-03\pm3.31e-03$ &
$1.82e-02\pm1.37e-03\pm1.45e-03$ \\ \hline 1.16e+00& 1.00e-01 &
$3.61e-02\pm2.82e-03\pm2.89e-03$ &
$1.22e-02\pm1.04e-03\pm9.73e-04$ \\ \hline 1.31e+00& 2.00e-01 &
$2.32e-02\pm1.50e-03\pm1.85e-03$ &
$7.64e-03\pm5.47e-04\pm6.11e-04$ \\ \hline 1.50e+00& 2.00e-01 &
$1.13e-02\pm9.51e-04\pm9.04e-04$ &
$3.51e-03\pm3.35e-04\pm2.81e-04$ \\ \hline 1.70e+00& 2.00e-01 &
$6.41e-03\pm6.63e-04\pm5.13e-04$ &
$2.21e-03\pm2.48e-04\pm1.77e-04$ \\ \hline 1.90e+00& 2.00e-01 &
$4.64e-03\pm5.37e-04\pm3.71e-04$ &
$9.80e-04\pm1.70e-04\pm7.84e-05$ \\ \hline 2.25e+00& 5.00e-01 &
$1.46e-03\pm1.78e-04\pm1.17e-04$ &
$5.66e-04\pm7.10e-05\pm4.53e-05$ \\ \hline 2.75e+00& 5.00e-01 &
$4.43e-04\pm1.38e-04\pm3.54e-05$ &
$1.29e-04\pm4.62e-05\pm1.03e-05$ \\ \hline
\end{tabular}
\caption{$\bar{p}$ spectra in 20-40\% and 40-100\% d+Au
collisions. The unit of $p_{T}$ and $p_{T}$ width is $GeV/c$.}
\label{pbarspectratable2}
\end{scriptsize}
\end{table}
\begin{table}[h]
\begin{scriptsize}
\centering
\begin{tabular}{|c|c|c|}
\hline
$p_{T}$ & $p_{T}$ width & p+p \\ \hline
4.68e-01& 1.00e-01 & $3.74e-02\pm2.60e-03\pm4.86e-03$ \\ \hline
5.63e-01& 1.00e-01 & $2.84e-02\pm1.77e-03\pm3.70e-03$ \\ \hline
6.61e-01& 1.00e-01 & $2.27e-02\pm1.38e-03\pm2.96e-03$ \\ \hline
7.59e-01& 1.00e-01 & $1.32e-02\pm8.65e-04\pm1.71e-03$ \\ \hline
8.58e-01& 1.00e-01 & $1.05e-02\pm7.20e-04\pm1.37e-03$ \\ \hline
9.57e-01& 1.00e-01 & $7.17e-03\pm5.40e-04\pm9.32e-04$ \\ \hline
1.06e+00& 1.00e-01 & $5.60e-03\pm3.77e-04\pm4.48e-04$ \\ \hline
1.16e+00& 1.00e-01 & $3.74e-03\pm2.86e-04\pm2.99e-04$ \\ \hline
1.31e+00& 2.00e-01 & $2.31e-03\pm1.49e-04\pm1.84e-04$ \\ \hline
1.50e+00& 2.00e-01 & $9.69e-04\pm8.84e-05\pm7.75e-05$ \\ \hline
1.70e+00& 2.00e-01 & $5.95e-04\pm6.65e-05\pm4.76e-05$ \\ \hline
1.90e+00& 2.00e-01 & $3.57e-04\pm4.79e-05\pm2.86e-05$ \\ \hline
2.25e+00& 5.00e-01 & $1.12e-04\pm1.90e-05\pm8.93e-06$ \\ \hline
2.75e+00& 5.00e-01 & $3.87e-05\pm1.07e-05\pm3.10e-06$ \\ \hline
\end{tabular}
\caption{$\bar{p}$ spectra in p+p collisions. The unit of $p_{T}$
and $p_{T}$ width is $GeV/c$.} \label{pbarspectratable3}
\end{scriptsize}
\end{table}
|
{
"timestamp": "2005-03-23T14:47:43",
"yymm": "0503",
"arxiv_id": "nucl-ex/0503018",
"language": "en",
"url": "https://arxiv.org/abs/nucl-ex/0503018"
}
|
\section{Introduction }
During the last two centuries in peaceful rich countries, people lived on
average longer and longer, while during the last few decades the number of
chidren born per women during her lifetime has sunken below the replacement
rate of slightly above 2. Also in many poorer countries the number of births
has fallen and the life expectancy increased. Thus the fear of overpopulation
of our planet Earth has to be modified by fear of old-age poverty: In the
year 2030 only those goods and services can be consumed by retired people
which have
been produced by working-age people. A million dollars of old-age savings
can be halved by a ten-percent inflation rate over seven years, if not
enough young people help me to live. This Econosociobiophysics problem is one
of demography, not of money.
We present in an appendix details of the assumptions for our extrapolations into
the future. In the next section we deal with conditions as are typical for
Western Europe, to be followed by a section on the different problems of
Algeria. More literature on ageing models, including one applied to our
demography \cite{cebrat}, is given in \cite{vancouver}.
\section{Western Europe}
Around 1970, the contraceptive pill reduced in the then two German states
the average number of babies born by a women during her lifetime below the
replacement level of two, to about 1.4. Spain and Italy followed later but
levelled at a lower plateau, while in France the number is higher, about
1.7. Life expectancy rises further though slower than during the first half of
the 20th century. Thus if people retire at an age of about 62 years, and if
around 2030 the strongest age cohort in Germany are the 70-year olds, problems
lie ahead. Only in recent years were they discussed in general newspapers.
As in science in general, we need open publications of extrapolation methods
and results. Only if many different simulations are compared can we see to what
extent they agree and thus may be relied upon.
The top curve in Fig.1 shows what happens if nothing is done: The average
retirement age is 62 years, and immigration and emigration cancel each other.
Then \cite{bomsdorf,stauffer,martins} the number of old people to be supported
by working-age people will increase drastically, while the total population
will decrease. We added here the number of children (up to age 20) to the
pensioneers since both groups are not fully ''working'' in the usual sense.
For the middle curve we assumed a net immigration of 0.38 percent per year,
starting now, and an increase of the retirement age by about half of the
increase of the life expectancy. Thus for every year which medical progress
gives us, about six month are given like a tax to the labour market, while the
other six months are
leasure time after retirement. Now the ratio and the population are more stable.
If we do not count in the latter simulation the children (bottom curve), then
the ratio of
pensioneers only to working age people is lower \cite{martins}. However, the
reduction of the expenses for children is mainly an effect of the past, not of
the future.
\begin{figure}
\begin{center}
\includegraphics[angle=-90,scale=0.5]{granadar2.eps}
\end{center}
\caption{Ratio of number of pensioneers to number of working age people (+)
and ratio of number of pensioneers plus number of children to working age
people (x). (Algeria)
}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[angle=-90,scale=0.5]{granadar3.eps}
\end{center}
\caption{Ratio of number of babies died before reaching the age one year
to number of total birth(+) (data of the National Office of the Statistics
ONS Algeria)
the fit line shows fractions increase of about 20 percent from 1980 until
1901.
}
\end{figure}
\section{Algeria}
During the first half of the previous century, the fertility was very
large in North Africa compared to Europe. It reached the value 8.1 during
the seventies in Algeria because of a low
average age of marriage in this country. Thirty years later, the average
number of births per women (during her lifetime) becomes close to 2, whereas in
France the fertility needed two centuries to pass from 6 in the middle of the
17th century to 2 in the 1930's. Algerian people are thus young.
Figure 2 shows that the
number of children (up to age $20$) added to the pensioneers
(the retirement age in Algeria is $60$ years) obliged workers to
support about two times their number until the year $2000$. Sixty years
afterwards the population will be older but the fractions remain constant (no
fear of increasing). We assumed in Fig.2 the Gompertz slope $b$ (see appendix)
to increase with time from $0.07$ in 1901 to $0.082$ in 1971 and to remain
constant thereafter. Only fertility data from 1950 on is available in Algeria.
The fertility is constant with a mean value of $7.3$ from $1950$ to $1980$ and
then decreases abruptly til 2004 to reach a value $2.04$; it is assumed to
stay constant at this value thereafter. The sixty years period necessary to
reach the steady state, corresponds to the age of retirement. In figure 3, we
show that the ratio of the number of babies dying in their first year to the
total number of births decreased by about $20$ percent from 1901 to 1980. Thus,
we made a correction on the fertility data (in fig.2) by reducing them by the
number of children dying before they reach maturity.
We noticed also that the
greatest emigration rate of Algerian people was between 1950 and 1970 but
remains weak compared to the rate of births and does not influence the
population evolution. In our simulation
we then neglected the emigration in such calculations. However, this simulation
did not account the rate of unemployeds
which was very small during the period of socialism but reaches now 17 percent
of population. However, the main prediction of Fig. 2 is an increase of the
social load for old age by 400 percent starting from 2020, while that for
children and old age combined will stabilize at the level around the year 2000.
\section{Summary}
With rising life expectencies and falling births, the demographic problems of
rich countries can be alleviated by controlled immigration and a moderate
increase of the average retirement age. That policy requires that first the
unemployment is reduced appreciably. For Algeria, on the other hand, emigration
could not affect sensitively the evolution of pensioneers, but their rate should
be multiplied by a factor four after 15 years from now on which would create
a real economic problem were it not offset by a reduction of the number of
children.
\bigskip
LZ thanks the DAAD for supporting a one-year part of his thesis work in
Cologne.
We thank W.J. Paul for suggesting to add the children to the
pensioneers.
\section{Appendix}
According to the Azbel lectures at this seminar, in all different countries and
centuries, the probability of humans to survive up to a fixed age is a
universal function of the life expectancy; we do not have to apply this
universality to yeast cells for the purpose of human demography. Thus we use
Germany as typical Western European country, without taking into account the
effects of World War II.
The mortality function $\mu = - d \ln S(a)/da$, where $S(a)$ is the number of
survivors from birth to age $a$, is assumed to follow a Gompertz law for adults:
$m \propto b\exp[(a-X)b]$ since the deviations at young age occur at such low
mortalities that they are not relevant if we want to be accurate within a few
percent. The deviations at old age \cite{robine} are not yet reliably
established and may also be negligible as long as the fraction of centenarians
among pensioneers is very small.
The Gompertz slope $b$ was assumed to increase linearly with time from 0.07 in
1821 to 0.093 in 1971 and to stay constant thereafter, in contrast to Bomsdorf
\cite{bomsdorf} and Azbel \cite{azbel} but in agreement with Yashin et al
\cite{yashin}; see also Wilmoth et al \cite{wilmoth}. Instead, the
characteristic age $X$ was constant at 103 years until 1971 and then increased
each year by 0.15 years to give a rising life expectancy.
Also these deviations from universality are not yet established reliably.
(Therefore we ignored the effect for Algeria, keeping $X=103$ constant there.)
Babies are born by mothers of age 21 to 40 with age-independent probability.
The average number of children born per women over her lifetime and reaching
adult age is assumed to be $2.2 - 0.4\tanh[(t-1971)/3]$ recently.
Immigrants are assumed to be 6 to 40 years old with equal probability, and their
number per year equals a fraction $c = 0.38 \%$ of the population, adjusted to
give a constant total population.
After the year 2010, the retirement age is increased by 60 percent of the
increase of life expectancy at birth to 73 in 2100 at a life expectancy then
of 99 years; for the problem year 2030 these ages are 64 and 84 years.
The program is available from stauffer@thp.uni-koeln.de as rente16.f.
|
{
"timestamp": "2005-03-11T14:36:05",
"yymm": "0503",
"arxiv_id": "q-bio/0503015",
"language": "en",
"url": "https://arxiv.org/abs/q-bio/0503015"
}
|
\section{Introduction}\label{sec:intro}
Least squares fitting is a well-known and powerful method for combining
information from a set of related experimental measurements to estimate the
underlying theoretical parameters (see, for instance, Reference~\cite{pdg}).
We discuss a specific
implementation of this method for use in high-energy physics experiments,
where the free parameters, denoted by the vector $\mathbf{m}$, are extracted
from event yields for signal processes. Typically, these yields are subject
to corrections for background, crossfeed, and efficiency. Because the sizes
of these corrections depend on the values of the free parameters, we make all
yield adjustments directly in the fit. Often, the uncertainties on these
corrections are ignored during the fit and are propagated to the free
parameters afterwards.
However, if these uncertainties modify the relative weights of the
measurements, then the above two-step procedure would bias both the fitted
central values and the estimated uncertainties.
Therefore, we build the $\chi^2$ variable from a
full description of the uncertainties, statistical and systematic, as well
as their correlations, on both the yields and their corrections.
Thus, the input measurements --- event yields, signal efficiencies,
parameters quantifying the background processes, and background efficiencies
--- and their uncertainties are all treated in a uniform fashion.
In the $\chi^2$ minimization, we account for the $\mathbf{m}$
dependence of the yield corrections.
\section{Formalism}\label{sec:formalism}
Below, we denote matrices by upper case bold letters and one-dimensional
vectors by lower case bold letters.
Let $\mathbf{n}$ represent a set of $N$ event yield measurements,
each for a different signal process.
Each measurement may receive crossfeed contributions from other
signal processes as well as backgrounds from non-signal sources.
The background processes are described by $\mathbf{b}$, a vector of $B$
estimated
production yields, which can be functions of experimentally measured
quantities, such as branching fractions, cross sections, and luminosities.
In principle, the free parameters $\mathbf{m}$ can also appear in $\mathbf{b}$,
although no additional degrees of freedom are introduced by $\mathbf{b}$.
The rates at which these background processes contaminate the signal yields
are given by the $N\times B$ background efficiency matrix, $\mathbf{F}$.
Thus, the vector $\mathbf{s}\equiv \mathbf{n} - \mathbf{Fb}$ represents
the background-subtracted yields.
We use an $N\times N$ signal efficiency matrix, $\mathbf{E}$, to
describe simultaneously detection efficiencies (diagonal elements) and
crossfeed probabilities (off-diagonal elements). The elements $E_{ij}$ are
defined to be the probabilities that an event of signal process $j$
is reconstructed and counted in yield $i$. The corrected yields, denoted by
$\mathbf{c}$, are obtained by acting on $\mathbf{s}$ with the inverse of
$\mathbf{E}$:
\begin{equation}\label{eq:correctedYields}
\mathbf{c} = \mathbf{E}^{-1} \mathbf{s} =
\mathbf{E}^{-1}( \mathbf{n} - \mathbf{Fb} ).
\end{equation}
Thus, $\mathbf{c}$ encapsulates all the experimental measurements.
The variance matrix of $\mathbf{c}$, denoted by $\mathbf{V_c}$, receives
contributions,
both statistical and systematic, from each element of $\mathbf{n}$,
$\mathbf{b}$, $\mathbf{E}$, and $\mathbf{F}$.
In the least squares fit, we define
$\chi^2 \equiv \left(\mathbf{c}-\mathbf{\widetilde c}\right)^T \mathbf{V}_{\mathbf{c}}^{-1} \left(\mathbf{c}-\mathbf{\widetilde c}\right)$,
where $\mathbf{\widetilde c}$ is the vector of predicted yields, which are
also functions of $\mathbf{m}$.
Because both $\mathbf{\widetilde c}$ and $\mathbf{c}$ (through $\mathbf{b}$)
depend on $\mathbf{m}$, minimizing this $\chi^2$ amounts to a nonlinear
version of the total least squares method~\cite{tls}. We solve this problem
by extending the conventional least squares fit to include contributions from
both $\mathbf{\widetilde c}$ and $\mathbf{c}$ in
$\partial\chi^2/\partial\mathbf{m}$. Given a set of seed values,
$\mathbf{m}_0$, the optimized estimate,
$\mathbf{\widehat m}$, and its variance matrix, $\mathbf{V_m}$, are
\begin{eqnarray}
\label{eq:fittedParameters}
\mathbf{\widehat m} &=& \mathbf{m}_0 +
\left(\mathbf{D}\mathbf{V}_{\mathbf{c}}^{-1}\mathbf{D}^T\right)^{-1}
\mathbf{D}\mathbf{V}_{\mathbf{c}}^{-1}
\left[\mathbf{c}(\mathbf{m}_0)-\mathbf{\widetilde c}(\mathbf{m}_0)\right]\\
\label{eq:fittedError}
\mathbf{V_m} &=& \frac{1}{2}\frac{\partial^2\chi^2}{\partial\mathbf{m}\,
\partial\mathbf{m}^T} =
\left(\mathbf{D}\mathbf{V}_{\mathbf{c}}^{-1}\mathbf{D}^T\right)^{-1},
\end{eqnarray}
where the $M\times N$ derivative matrix $\mathbf{D}$ is defined to be
\begin{equation}
\mathbf{D}\equiv
\frac{\partial\mathbf{\widetilde c}}{\partial\mathbf{m}} -
\frac{\partial\mathbf{c}}{\partial\mathbf{m}} =
\frac{\partial\mathbf{\widetilde c}}{\partial\mathbf{m}} +
\frac{\partial\mathbf{b}}{\partial\mathbf{m}}
\mathbf{F}^T\left(\mathbf{E}^{-1}\right)^T.
\end{equation}
In general, $\mathbf{\widetilde c}$ and $\mathbf{c}$ are nonlinear
functions of $\mathbf{m}$, so the linearized solution
$\mathbf{\widehat m}$ is approximate, and the above procedure is
iterated until the $\chi^2$ converges. Between iterations, all the
fit inputs that depend on $\mathbf{m}$ are reevaluated with the updated
values of $\mathbf{\widehat m}$.
Nonlinearities also occur when $\mathbf{V_c}$ contains multiplicative
or Poisson uncertainties that depend on the measurement values.
With the least squares method, these nonlinearities result in biased
estimators unless these variable uncertainties are
evaluated using the predicted yields $\mathbf{\widetilde c}$ instead of the
measured $\mathbf{c}$. Therefore, all three ingredients in the $\chi^2$ ---
$\mathbf{c}$, $\mathbf{\widetilde c}$, and $\mathbf{V_c}$ --- are functions of
$\mathbf{m}$. However, we do not include the derivatives
$\partial\mathbf{V_c}/\partial\mathbf{m}$ in $\mathbf{D}$ because
doing so would generate biases in $\mathbf{\widehat m}$.
For a simple demonstration of the aforementioned biases, we
consider two measured yields, $c_1$ and $c_2$, which are both estimators of
a true yield $\bar c$. We assume that the uncertainties on $c_1$ and $c_2$
are uncorrelated, multiplicative, and of the same fractional size, $\lambda$.
We construct an improved estimator, $\widehat c$, by minimizing
$\chi^2 = (c_1-c)^2/\sigma_{c_1}^2 + (c_2-c)^2/\sigma_{c_2}^2$ with respect to
$c$. If, following the prescription given above, we neglect the
$\partial\sigma_{c_i}^2/\partial c$ terms in
$\partial\chi^2/\partial c$ and assign (iteratively) the uncertainties
$\sigma_{c_1}=\sigma_{c_2}=\lambda\widehat c$,
then $c_1$ and $c_2$ are equally weighted, and $\widehat c$ is an unbiased
estimate of $\bar c$:
\begin{eqnarray}
\widehat c_{\rm unbiased} &=& \frac{c_1+c_2}{2} \\
\chi^2_{\rm unbiased} &=&
\frac{2}{\lambda^2}\left(\frac{c_1-c_2}{c_1+c_2}\right)^2 .
\end{eqnarray}
On the other hand, including the $\partial\sigma_{c_i}^2/\partial c$ terms in
$\partial\chi^2/\partial c$ results in an upward bias:
\begin{eqnarray}
\widehat c_{\rm biased1} &=& \frac{c_1^2+c_2^2}{c_1+c_2} =
\widehat c_{\rm unbiased}\left(1+\frac{\lambda^2\chi^2_{\rm unbiased}}{2}\right) \\
\chi^2_{\rm biased1} &=& \frac{(c_1-c_2)^2}{\lambda^2(c_1^2+c_2^2)} .
\end{eqnarray}
Finally, if we assign uncertainties based on the measured yields, not the
predicted yields, such that $\sigma_{c_1}=\lambda c_1$,
$\sigma_{c_2}=\lambda c_2$, and $\partial\sigma_{c_i}^2/\partial c=0$,
then the resulting estimate is biased low:
\begin{eqnarray}
\widehat c_{\rm biased2} &=& \frac{c_1 c_2 (c_1+c_2)}{c_1^2+c_2^2} =
\widehat c_{\rm unbiased}(1-\lambda^2\chi^2_{\rm biased1}) \\
\chi^2_{\rm biased2} &=& \chi^2_{\rm biased1}.
\end{eqnarray}
Thus, even though $\chi^2_{\rm biased1}$ and $\chi^2_{\rm biased2}$ are
smaller than $\chi^2_{\rm unbiased}$, the corresponding estimators possess
undesired properties.
\section{\boldmath Input Variance Matrix}
\label{sec:inputVarianceMatrix}
The uncertainties on the $N$ elements of $\mathbf{n}$ and the $B$ elements
of $\mathbf{b}$ are characterized by the $N\times N$ matrix
$\mathbf{V_n}$ and the $B\times B$ matrix $\mathbf{V_b}$, respectively.
Usually, the elements of $\mathbf{E}$ and $\mathbf{F}$ share many common
correlated systematic uncertainties, so we construct a joint
variance matrix from the submatrices $\mathbf{V_E}$, $\mathbf{V_F}$, and
$\mathbf{C_{EF}}$, where
$\mathbf{V_E}$ ($N^2\times N^2$) and $\mathbf{V_F}$ ($NB\times NB$) are
the variance matrices for the elements of $\mathbf{E}$ and $\mathbf{F}$,
respectively, and $\mathbf{C_{EF}}$ ($N^2\times NB$)
contains the correlations between $\mathbf{E}$ and $\mathbf{F}$.
Below, we label each element of $\mathbf{E}$
or $\mathbf{F}$ by two indices ($E_{ij}$ or $F_{ij}$), and the two dimensions
of $\mathbf{E}$ or $\mathbf{F}$ are mapped onto one dimension of
$\mathbf{V_E}$ or $\mathbf{V_F}$.
We form $\mathbf{V_c}$ by propagating the statistical and systematic
uncertainties on $\mathbf{n}$, $\mathbf{b}$, $\mathbf{E}$, and $\mathbf{F}$
to $\mathbf{c}$ via
\begin{equation}
\label{eq:errorPropagation1}
\mathbf{V_c} =
\frac{\partial\mathbf{c}}{\partial\mathbf{n}}^T \mathbf{V_n}
\frac{\partial\mathbf{c}}{\partial\mathbf{n}} +
\frac{\partial\mathbf{c}}{\partial\mathbf{b}}^T \mathbf{V_b}
\frac{\partial\mathbf{c}}{\partial\mathbf{b}} +
\left(\begin{array}{cc}
(\partial\mathbf{c}/\partial\mathbf{E})^T &
(\partial\mathbf{c}/\partial\mathbf{F})^T
\end{array}\right)
\left(\begin{array}{cc}
\mathbf{V_E} & \mathbf{C_{EF}} \\
\mathbf{C}_{\mathbf{EF}}^T & \mathbf{V_F}
\end{array}\right)
\left(\begin{array}{c}
\partial\mathbf{c}/\partial\mathbf{E} \\
\partial\mathbf{c}/\partial\mathbf{F}
\end{array}\right).
\end{equation}
Where appropriate, we substitute $\mathbf{\widetilde c}$ for $\mathbf{c}$, as
discussed in Section~\ref{sec:formalism}.
The first term of Equation~\ref{eq:errorPropagation1} is simply
$\mathbf{E}^{-1}\mathbf{V_n} (\mathbf{E}^{-1})^T$, and the
second term is
$\mathbf{E}^{-1}\mathbf{F}\mathbf{V_b}\mathbf{F}^T (\mathbf{E}^{-1})^T$.
For the third term, we evaluate the partial derivatives and find
\begin{eqnarray}
\frac{\partial\mathbf{c}}{\partial\mathbf{E}} &=&
\mathbf{s}^T
\left(\frac{\partial\mathbf{E}^{-1}}{\partial\mathbf{E}}\right)^T =
-\mathbf{s}^T \left(\mathbf{E}^{-1}\right)^T
\left(\frac{\partial\mathbf{E}}{\partial\mathbf{E}}\right)^T
\left(\mathbf{E}^{-1}\right)^T =
-\mathbf{A}\left(\mathbf{E}^{-1}\right)^T \\
\frac{\partial\mathbf{c}}{\partial\mathbf{F}} &=&
-\mathbf{B}\left(\mathbf{E}^{-1}\right)^T,
\end{eqnarray}
where
$\mathbf{A}\equiv\mathbf{c}^T (\partial\mathbf{E}/\partial\mathbf{E})^T$ and
$\mathbf{B}\equiv\mathbf{b}^T (\partial\mathbf{F}/\partial\mathbf{F})^T$, with
elements given in terms of the Kronecker delta ($\delta_{ij}$):
$\partial E_{kl}/\partial E_{ij}=\partial F_{kl}/\partial F_{ij}=\delta_{ik}\delta_{jl}$.
The matrices $\mathbf{A}$ and $\mathbf{B}$ have rows labeled by two indices,
which refer to the elements of $\mathbf{E}$ and $\mathbf{F}$, respectively,
and columns labeled by one index, which refers to the elements of $\mathbf{c}$.
In other words, the $ij$-th row of $\mathbf{A}$ is given by
$\mathbf{c}^T (\partial\mathbf{E}/\partial E_{ij})^T$, where
$(\partial\mathbf{E}/\partial E_{ij})_{kl} = \partial E_{kl}/\partial E_{ij}$.
Therefore, the elements of $\mathbf{A}$ and $\mathbf{B}$ are
$A_{ij,k} = \delta_{ik} \widetilde c_j$ and
$B_{ij,k} = \delta_{ik} b_j$.
For $N=B=2$, these matrices are
\begin{equation}
\label{eq:Adefinition}
\mathbf{A} =
\left(\begin{array}{cc}
\widetilde c_1 & 0 \\
\widetilde c_2 & 0 \\
0 & \widetilde c_1 \\
0 & \widetilde c_2 \\
\end{array}\right)
\hspace{1cm}{\rm and}\hspace{1cm}
\mathbf{B} =
\left(\begin{array}{cc}
b_1 & 0 \\
b_2 & 0 \\
0 & b_1 \\
0 & b_2 \\
\end{array}\right).
\end{equation}
This treatment of error propagation in matrix inversion agrees with
that derived in Reference~\cite{Lefebvre:1999yu}.
The above relations allow us to reexpress $\mathbf{V_c}$ as
\begin{equation}
\label{eq:errorPropagation2}
\mathbf{V_c} = \mathbf{E}^{-1}\mathbf{V_{\Delta n}}
\left(\mathbf{E}^{-1}\right)^T,
\end{equation}
where $\mathbf{V_{\Delta n}}\equiv \mathbf{V_n} + \mathbf{F}\mathbf{V_b}\mathbf{F}^T + \mathbf{A}^T \mathbf{V_E} \mathbf{A} + \mathbf{B}^T \mathbf{V_F} \mathbf{B} + \mathbf{A}^T \mathbf{C_{EF}} \mathbf{B} + \mathbf{B}^T \mathbf{C}_{\mathbf{EF}}^T \mathbf{A}$.
As a result, we have
$\chi^2 = \mathbf{\Delta n}^T \mathbf{V}_{\mathbf{\Delta n}}^{-1}\mathbf{\Delta n}$,
where $\mathbf{\Delta n}\equiv\mathbf{n}-\mathbf{E\widetilde c}-\mathbf{Fb}$.
Thus, the $\chi^2$ minimization can be formulated equivalently in terms of
$\mathbf{n}$ instead of $\mathbf{c}$:
$\mathbf{V_m} = \left(\mathbf{D'}\mathbf{V}_{\mathbf{\Delta n}}^{-1}\mathbf{D'}^T\right)^{-1}$ and
$\mathbf{\widehat m} = \mathbf{m}_0 + \mathbf{V_m} \mathbf{D'}\mathbf{V}_{\mathbf{\Delta n}}^{-1} \mathbf{\Delta n}$,
where $\mathbf{D'}\equiv \mathbf{D}\mathbf{E}^T = (\partial\mathbf{\widetilde c}/\partial\mathbf{m})\mathbf{E}^T +(\partial\mathbf{b}/\partial\mathbf{m})\mathbf{F}^T$.
Systematic uncertainties on the efficiencies are often
multiplicative and belong to one of three categories: those that depend
only on the reconstructed mode (row-wise),
those that depend only on the generated mode (column-wise),
and those that are uncorrelated among elements of
$\mathbf{E}$ and $\mathbf{F}$.
For row-wise efficiency uncertainties, all the elements in any given
row of $\mathbf{E}$ and $\mathbf{F}$ have the same fractional uncertainty,
which we denote by
$\lambda_i \equiv \sigma_{E_{ij}}/E_{ij} = \sigma_{F_{ij}}/F_{ij}$.
The correlation coefficients between elements of different rows are
$\lambda_{ij}^2 / (\lambda_i\lambda_j)$,
where $\lambda_{ij}$ characterizes the uncertainties common to
$c_i$ and $c_j$. For instance, if $\lambda_{\rm track}$ is the fractional
uncertainty associated with the charged particle tracking efficiency, then
$\lambda_i = t_i\lambda_{\rm track}$ and
$\lambda_{ij}^2 = t_i t_j \lambda_{\rm track}^2$, where
$t_i$ and $t_j$ are the track multiplicities in modes $i$ and $j$,
respectively. Note that $\lambda_{ii} = \lambda_i$. Similarly, for
column-wise uncertainties, we define the fractional uncertainties
$\mu_j\equiv \sigma_{E_{ij}}/E_{ij} = \sigma_{F_{ij}}/F_{ij}$ and correlation
coefficients $\mu_{ij}^2/(\mu_i\mu_j)$. We denote the uncorrelated
fractional uncertainty on any element of $\mathbf{E}$ or $\mathbf{F}$
by $\nu_{ik, jl}$. Table~\ref{tab:vefElements} gives expressions
for the elements of $\mathbf{V_E}$, $\mathbf{V_F}$, and $\mathbf{C_{EF}}$,
as well as their contributions to $\mathbf{V_c}$ for row-wise, column-wise,
and uncorrelated uncertainties.
\begin{table}[ht]
\begin{center}
\caption{Expressions for the elements of $\mathbf{V_E}$, $\mathbf{V_F}$,
and $\mathbf{C_{EF}}$, as well as their contributions to $\mathbf{V_c}$.
Repeated external indices are not summed over.}
\label{tab:vefElements}
\begin{tabular}{cccc}
\hline\hline
Quantity & Row-wise & Column-wise & Uncorrelated \\
\hline
$(\mathbf{V_E})_{ik, jl}$ &
$\lambda_{ij}^2 E_{ik}E_{jl}$ &
$\mu_{kl}^2 E_{ik}E_{jl}$ &
$\nu_{ik,jl}^2 E_{ik}E_{jl}\delta_{ij}\delta_{kl}$ \\
$(\mathbf{V_F})_{ik, jl}$ &
$\lambda_{ij}^2 F_{ik}F_{jl}$ &
$\mu_{kl}^2 F_{ik}F_{jl}$ &
$\nu_{ik,jl}^2 F_{ik}F_{jl}\delta_{ij}\delta_{kl}$ \\
$(\mathbf{C_{EF}})_{ik, jl}$ &
$\lambda_{ij}^2 E_{ik}F_{jl}$ &
$\mu_{kl}^2 E_{ik}F_{jl}$ &
0 \\
\hline
$(\mathbf{A}^T \mathbf{V_E} \mathbf{A})_{ij}$ &
$\lambda_{ij}^2 \widetilde s_i \widetilde s_j$ &
$\mu_{kl}^2 E_{ik} \widetilde c_k E_{jl} \widetilde c_l$ &
$\delta_{ij} \sigma^2_{E_{jk}} \widetilde c_k^2$ \\
$(\mathbf{B}^T \mathbf{V_F} \mathbf{B})_{ij}$ &
$\lambda_{ij}^2 F_{ik}b_k F_{jl}b_l$ &
$\mu_{kl}^2 F_{ik}b_k F_{jl}b_l$ &
$\delta_{ij} \sigma_{F_{jk}}^2 b_k^2$ \\
$(\mathbf{A}^T \mathbf{C_{EF}} \mathbf{B})_{ij}$&
$\lambda_{ij}^2 \widetilde s_i F_{jk}b_k$ &
$\mu_{kl}^2 E_{ik} \widetilde c_k F_{jl}b_l$ &
0 \\
\hline\hline
\end{tabular}
\end{center}
\end{table}
\section{\boldmath Example: Hadronic $D$ Meson Branching Fractions}
The least squares method described in the previous sections has been employed
by the CLEO-c collaboration~\cite{cleoc-dhad} to measure absolute branching
fractions for hadronic $D$ meson decays. Using $D\bar D$ pairs produced
through the $\psi(3770)$ resonance, the branching fraction for mode $i$,
denoted by ${\cal B}_i$, is measured by comparing the number of events
where a single $D\to i$ decay is reconstructed (called single tag,
denoted by $x_i$) with the number of events where both $D$ and $\bar D$ are
reconstructed via $D\to i$ and $\bar D\to j$ (called double tag, denoted by
$y_{ij}$). These yield measurements form the vector $\mathbf{n}$.
The free parameters $\mathbf{m}$ are the ${\cal B}_i$
and the numbers of $D^0\bar D^0$ and $D^+D^-$ pairs produced, denoted by
${\cal N}^{00}$ and ${\cal N}^{+-}$, respectively, and denoted generically
by ${\cal N}$.
Yields for charge conjugate modes are measured separately, so the predicted
corrected yields $\mathbf{\widetilde c}$ are ${\cal N}{\cal B}_i$ for single
tags and ${\cal N}{\cal B}_i{\cal B}_j$ for double tags.
Thus, ${\cal B}_i$ and ${\cal N}$ can be
extracted from various products and ratios of $x_i$, $x_j$, and $y_{ij}$:
${\cal B}_i \sim y_{ij}/x_j$, ${\cal N}\sim x_i x_j/y_{ij}$, up to
corrections for efficiency, crossfeed, and background.
The matrix $\mathbf{V_n}$ describes the statistical uncertainties and
correlations among the $x_i$ and $y_{ij}$. The $y_{ij}$ are uncorrelated,
but because any given event can contain both single tag and double tag
candidates, the $x_i$ are correlated among themselves as well as with the
$y_{ij}$. If the selection criteria for single and double tags are the same,
then the events (signal and background) used to estimate $y_{ij}$ are a proper
subset of those for $x_i$ and $x_j$. Thus, any single tag yield is a sum of
exclusive single tags ($x_i^{\rm excl}$) and double tags:
$x_{\{i,j\}} = x_{\{i,j\}}^{\rm excl} + y_{ij}$.
Propagating the uncertainties on the independent variables,
$x_i^{\rm excl}$, $x_j^{\rm excl}$, and $y_{ij}$, gives the following elements
for $\mathbf{V_n}$:
\begin{eqnarray}
\label{eq:singleVar}
\langle \Delta x_i \Delta x_j \rangle &=&
\delta_{ij} \sigma_{x_i} \sigma_{x_j} +
(1-\delta_{ij}) \sigma^2_{y_{ij}} \\
\label{eq:doubleVar}
\langle \Delta y_{ij} \Delta y_{kl} \rangle
&=& \delta_{ik}\delta_{jl}\sigma_{y_{ij}}\sigma_{y_{kl}} \\
\label{eq:singleDoubleCovar}
\langle \Delta x_i \Delta y_{jk} \rangle &=&
(\delta_{ij} + \delta_{ik}) \sigma^2_{y_{jk}},
\end{eqnarray}
where $\Delta x_i\equiv x_i - \langle x_i\rangle$,
$\Delta y_{ij} \equiv y_{ij} - \langle y_{ij}\rangle$, and
$\sigma_{x_{\{i,j\}}}^2=\sigma_{x_{\{i,j\}}^{\rm excl}}^2 + \sigma_{y_{ij}}^2$.
Thus, for any two single tag yields and the corresponding double tag
yield, the three off-diagonal elements of $\mathbf{V_n}$ are all
given by the uncertainty on the number of overlapping events.
In addition to these statistical uncertainties,
$\mathbf{V_n}$ can also receive contributions from additive systematic
uncertainties.
Some of the sources of background we consider are non-signal $D$ decays,
$e^+e^-\to q\bar q$ events, and $e^+e^-\to\tau^+\tau^-$ events.
If there are two non-signal $D$ backgrounds with branching fractions
${\cal C}_1$ and ${\cal C}_2$, then the vector $\mathbf{b}$ is given by
\begin{equation}\label{eq:backrounds}
\mathbf{b} = \left(\begin{array}{c}
{\cal N}{\cal C}_1 \\
{\cal N}{\cal C}_2 \\
{\cal L} X_{q\bar q} \\
{\cal L} X_{\tau^+\tau^-}
\end{array}\right),
\end{equation}
where $X_{q\bar q}$ and $X_{\tau^+\tau^-}$ are the cross sections
for $q\bar q$ and $\tau^+\tau^-$ production, respectively, and ${\cal L}$ is
the integrated luminosity of the data sample.
Because of the non-signal $D$ decays, the free parameter ${\cal N}$ appears in
$\mathbf{b}$ but does not contribute any additional terms to the variance
matrix $\mathbf{V_b}$, which takes the following block diagonal form:
\begin{equation}
\mathbf{V_b} = \left(\begin{array}{cccc}
{\cal N}^2\sigma_{{\cal C}_1}^2 & 0 & 0 & 0 \\
0 & {\cal N}^2\sigma_{{\cal C}_2}^2 & 0 & 0 \\
0 & 0 & {\cal L}^2\sigma_{X_{q\bar q}}^2 + X_{q\bar q}^2\sigma_{\cal L}^2 &
X_{q\bar q} X_{\tau^+\tau^-} \sigma_{\cal L}^2 \\
0 & 0 & X_{q\bar q} X_{\tau^+\tau^-} \sigma_{\cal L}^2 &
{\cal L}^2\sigma_{X_{\tau^+\tau^-}}^2 + X_{\tau^+\tau^-}^2
\sigma_{\cal L}^2
\end{array}\right).
\end{equation}
Also, the matrix
$\partial\mathbf{b}/\partial\mathbf{m}$ is nontrivial and is incorporated into
the $\chi^2$ minimization.
In the joint variance matrix for $\mathbf{E}$ and $\mathbf{F}$, uncertainties
of all three types discussed in Section~\ref{sec:inputVarianceMatrix} are
present. Row-wise effects arise from systematic uncertainties on simulated
reconstruction efficiencies for charged tracks, $\pi^0\to\gamma\gamma$
decays, $K^0_S\to\pi^+\pi^-$ decays, and
particle identification (PID) for charged pions and kaons.
Column-wise uncertainties reflect the poorly known resonant substructure in
multi-body final states. Uncorrelated contributions come from statistical
uncertainties due to the finite Monte Carlo (MC) simulated samples used to
determine $\mathbf{E}$ and $\mathbf{F}$.
Thus, for example, if mode $i$ is $D^0\to K^-\pi^+\pi^0$ and mode $j$ is
$D^+\to K^0_S\pi^+$, then the row-wise uncertainties are given by
\begin{eqnarray}
\lambda_i^2 & = & ( 2\lambda_{\rm track} )^2 + \lambda_{\pi^0}^2 +
\lambda_{\pi^\pm {\rm PID}}^2 + \lambda_{K^\pm {\rm PID}}^2 \\
\lambda_j^2 & = & ( 3\lambda_{\rm track} )^2 +
\lambda_{\pi^\pm {\rm PID}}^2 \\
\lambda_{ij}^2 & = & 6\lambda_{\rm track}^2 + \lambda_{\pi^\pm {\rm PID}}^2.
\end{eqnarray}
Because these row-wise and column-wise uncertainties are completely correlated
among the yields to which they pertain, they degrade the precision of
${\cal B}_i$ but not ${\cal N}$. Furthermore, they have no effect
on the central values of $\mathbf{\widehat m}$ because
the relative weight of each yield is unaltered by these
uncertainties. However, they can introduce large systematic correlations
among the fit parameters, even between statistically independent branching
fractions of different charge.
\subsection{Toy Monte Carlo Study}\label{sec:toyMC}
We test the method presented above using a toy MC simulation with Gaussian
smearing of the fit inputs. We generate data for five decay modes,
$D^0\to K^-\pi^+$, $D^0\to K^-\pi^+\pi^0$, $D^0\to K^-\pi^+\pi^-\pi^+$,
$D^+\to K^-\pi^+\pi^+$, and $D^+\to K^0_S\pi^+$ (charge conjugate particles
are implied), for which there are ten single tag and thirteen double tag
yields. The fit determines seven free parameters: ${\cal N}^{00}$,
${\cal N}^{+-}$, and five charge-averaged branching fractions. The input
branching fractions are taken to be the world-average values given in
Reference~\cite{pdg}, and we use ${\cal N}^{00}=2.0\times 10^5$ and
${\cal N}^{+-}=1.5\times 10^5$. The efficiencies are mode-dependent:
30\%--70\% for single tags and 10\%--50\% for double tags, with fractional
statistical uncertainties of 0.5\%--1.0\%. The yield uncertainties are
specified to be close to the Poisson limit, and backgrounds correspond roughly
to those expected in 60 ${\rm pb}^{-1}$ of $e^+e^-$ collisions at the
$\psi(3770)$. Also, we apply correlated systematic efficiency uncertainties of
1\% for tracking, 2\% for $\pi^0$ reconstruction, 2\% for $K^0_S$
reconstruction, and 1\% for charged pion and kaon PID.
The fit reproduces the input parameters well.
Figure~\ref{fig:toyMCPulls} shows the pull distributions for the seven
fit parameters and the fit confidence level for 10000 toy MC trials.
All the pull distributions are unbiased and have widths consistent with unity.
Also, the confidence level is flat. Table~\ref{tab:correlations} gives the
correlation coefficients among the fit parameters. Branching fractions tend
to be positively correlated with each other and negatively correlated with
${\cal N}^{00}$ and ${\cal N}^{+-}$. In particular, the $D^0$ branching
fractions are correlated with those for $D^+$.
In the absence of correlated efficiency uncertainties, the $D^0$ and $D^+$
free parameters would essentially be independent.
\begin{figure}
\includegraphics*[width=0.5\linewidth]{3950205-001.eps}
\caption{Toy MC fit pull distributions for ${\cal N}^{00}$ (a),
${\cal B}(D^0\to K^-\pi^+)$ (b), ${\cal B}(D^0\to K^-\pi^+\pi^0)$ (c),
${\cal B}(D^0\to K^-\pi^+\pi^-\pi^+)$ (d), ${\cal N}^{+-}$ (e),
${\cal B}(D^+\to K^-\pi^+\pi^+)$ (f), and ${\cal B}(D^+\to K^0_S\pi^+)$ (g),
overlaid with Gaussian curves with zero mean and unit width. The fit
confidence level distribution (h) is overlaid with a line with zero slope.}
\label{fig:toyMCPulls}
\end{figure}
\begin{table}[htb]
\caption{Correlation coefficients, including systematic uncertainties,
for the free parameters determined by the fit to toy MC samples.}
\label{tab:correlations}
\begin{center}
\begin{tabular}{l|ccccccc}
\hline\hline
& ~~${\cal N}^{00}$~~ & ~~$K^-\pi^+$~~ & ~~$K^-\pi^+\pi^0$~~ &
~~$K^-\pi^+\pi^-\pi^+$~~ &
~~${\cal N}^{+-}$~~ & ~~$K^-\pi^+\pi^+$~~ & ~~$K^0_S\pi^+$~~ \\
\hline
${\cal N}^{00}$ & 1 & $-0.63$ & $-0.52$ & $-0.38$
& $-0.01$ & $-0.01$ & $-0.01$ \\
$K^-\pi^+$ & & 1 & 0.79 & 0.87 & $-0.01$ & 0.40 & 0.29 \\
$K^-\pi^+\pi^0$ & & & 1 & 0.77 & $-0.01$ & 0.37 & 0.27 \\
$K^-\pi^+\pi^-\pi^+$ & & & & 1 & $-0.01$ & 0.53 & 0.39 \\
${\cal N}^{+-}$ & & & & & 1 & $-0.82$ & $-0.77$ \\
$K^-\pi^+\pi^+$ & & & & & & 1 & 0.87 \\
$K^0_S\pi^+$ & & & & & & & 1\\
\hline\hline
\end{tabular}
\end{center}
\end{table}
Slight asymmetries can be observed in the pull distributions, especially
in those for ${\cal N}^{00}$ and ${\cal N}^{+-}$. These asymmetries are
caused by the nonlinear nature of the multiplicative efficiency uncertainties
and of the functions $\mathbf{\widetilde c}(\mathbf{m})$. Because the fit
parameters are effectively estimated from ratios of the input yields, Gaussian
fluctuations in the denominators produce non-Gaussian fluctuations in the
ratios, which are most visible in ${\cal N}^{00}$ and ${\cal N}^{+-}$, where
the uncertainties in the denominators are dominant.
Similarly, multiplicative uncertainties, which affect only the
branching fractions, scale with the fitted values and, therefore, give rise to
asymmetric ${\cal B}$ pulls. In both cases, larger fractional uncertainties
would heighten the asymmetries.
If we form the matrix $\mathbf{A}$ in Equation~\ref{eq:Adefinition} using the
measured yields $\mathbf{c}$ rather than the predicted yields
$\mathbf{\widetilde c}$, then the variance matrix $\mathbf{V_c}$ need not be
reevaluated after each fit iteration. However, in this case, the pull
distributions become significantly biased, as shown in
Figure~\ref{fig:toyMCPullsBiased}. Thus, obtaining
unbiased fit results and the correct uncertainties requires proper handling
of the efficiency variance matrices $\mathbf{V_E}$ and $\mathbf{V_F}$.
\begin{figure}
\includegraphics*[width=0.5\linewidth]{3950805-002.eps}
\caption{Toy MC fit pull distributions, with $\mathbf{V_c}$ calculated using
$\mathbf{c}$ instead of $\mathbf{\widetilde c}$, for ${\cal N}^{00}$ (a),
${\cal B}(D^0\to K^-\pi^+)$ (b), ${\cal B}(D^0\to K^-\pi^+\pi^0)$ (c),
${\cal B}(D^0\to K^-\pi^+\pi^-\pi^+)$ (d), ${\cal N}^{+-}$ (e),
${\cal B}(D^+\to K^-\pi^+\pi^+)$ (f), and ${\cal B}(D^+\to K^0_S\pi^+)$ (g),
overlaid with Gaussian curves with zero mean and unit width. The fit
confidence level distribution (h) is overlaid with a line with zero slope.}
\label{fig:toyMCPullsBiased}
\end{figure}
\section{Summary}
We have developed a least squares fit that simultaneously incorporates
statistical and systematic uncertainties, as well as their correlations,
on all the input
experimental measurements. Biases from nonlinearities are reduced by
introducing fit parameter dependence in the input variance matrix.
This fitting method is used to measure absolute branching fractions of
hadronic $D$ meson decays, and toy Monte Carlo studies validate the performance
of the fitter. By including all known sources of measurement uncertainty in
the $\chi^2$, we obtain unbiased fit parameters with correct estimated
uncertainties.
\begin{acknowledgments}
We wish to thank Roy Briere, David Cassel, Lawrence Gibbons, Wolfgang Rolke,
Anders Ryd, and Ian Shipsey for many helpful discussions.
This work was supported in part by the National Science Foundation under
Grant No. PHY-0202078.
\end{acknowledgments}
|
{
"timestamp": "2005-12-20T00:30:07",
"yymm": "0503",
"arxiv_id": "physics/0503050",
"language": "en",
"url": "https://arxiv.org/abs/physics/0503050"
}
|
\section{Introduction}
\label{sec:intro}
Various new challenging problems in shape matching have been
appearing from different scientific areas including Bioinformatics
and Image Analysis.
In a class of problems in Shape Analysis,
one assumes that the points in two or more configurations are
labelled and these configurations are to be matched after filtering
out some transformation. Usually the transformation is a rigid
transformation or similarity transformation. Several new
problems are appearing where the points of configuration are either
not labelled or the labelling is ambiguous, and in which
some points do not appear in each of the configurations.
An example of ambiguous labelling arises in understanding
the secondary structure of proteins, where we are
given not only the 3-dimensional molecular configuration but also the
type of molecules (amino acids) at each point. A
generic problem is to match such two configurations, where the matching has
to be invariant under some transformation group. Descriptions of such
problems can be found in the review article by
Mardia, Taylor and Westhead (2003).
We now describe two datasets related to protein structure. One is
of 2-dimensional gel data where each point is a protein itself and the
transformation group is affine. In this case we have a partial
matching identified already by experts, that we can use to assess
our procedures.
In the second example we have a 3-dimensional configuration of
two active sites of two proteins which has also additional chemical
information. Here the underlying transformation to be filtered out is
rigid motion. In this protein structure problem, one of the main aims
is to take a query active site and find matches to a given database,
in some ranking order. The matches will give some idea of functions of
the unknown proteins, leading to the design of new enzymes for example.
There are other related examples from Image Analysis such as matching buildings
when one has multiple 2-dimensional views of 3-dimensional objects
(see, for example, Cross and Hancock, 1998). The problem here requires
filtering out the projective transformations before matching. Other
examples involve matching outlines or
surfaces (see, for example, Chui and Rangarajan, 2000, and Pedersen, 2002). Here
there is no labelling of points involved, and we are dealing with a continuous
contour or surface rather than a finite number of points. Such problems
are not addressed in this paper.
In Section 2 we build a hierarchical Bayesian model for the point
configurations and derive inferential procedure for its parameters.
In particular, modelling hidden point locations as a Poisson process
leads to a considerable simplification. We discuss in particular
the problem when only a linear or affine transformation has to be
filtered out. In Section 3 we discuss prior specifications, and provide an
implementation of the resulting
methodology by means of Markov chain Monte Carlo (MCMC) samplers.
Under a broad parametric family of loss functions, an optimal Bayesian point
estimate of the matching matrix can be constructed,
which turns out to depend on a single parameter of the family.
We also discuss a modification to the likelihood in our model
to make use of partial label (`colour') information at the points.
Finally here there is a note on the possibilities
for an alternative computational
approach using the EM algorithm.
Section 4 describes application of our methods to the two examples from
Bioinformatics mentioned above: matching Protein gels in 2 dimensions and
aligning active sites of Proteins in 3 dimensions. The paper concludes
with a Discussion of some open problems and future directions, and comparisons with
other methods.
The principal innovations in our approach are (a) the fully model-based
approach to alignment, (b) the model formulation allowing integrating
out of the hidden point locations, (c) the prior specification for the
rotation matrix, and (d) the MCMC algorithm.
\section{Hierarchical modelling of alignment and matching problems}
\label{sec:models}
We will build a hierarchical model for the observed point configurations,
and derive inferential procedures for its parameters, including the
unknown matching between the configurations, according to the
Bayesian paradigm.
\subsection{Point process model, with geometrical transformation and random thinning}
Suppose we are given two point configurations in $d$-dimensional
space $\mathcal{R}^d$: $\{x_j, j=1,2,\ldots,m\}$ and $\{y_k, k=1,2,\ldots,n\}$.
The points are labelled for identification, but arbitrarily.
Both point sets are regarded as noisy observations on subsets of
a set of true locations
$\{\mu_i\}$, where we do not know the mappings from $j$ and $k$ to $i$.
There may be a geometrical transformation between the $x$-space
and the $y$-space, which may also be unknown.
The objective is to make model-based inference about these
mappings, and in particular make probability statements
about matching -- which pairs $(j,k)$ correspond to the same
true location?
The geometrical transformation between the $x$-space
and the $y$-space will be denoted $\mathcal{A}$; thus $y$ in $y$-space
corresponds to $x=\mathcal{A} y$ in $x$-space. The notation does not
imply that the transformation $\mathcal{A}$ is necessarily linear.
It may be a rotation or more general linear transformation,
a translation, both of these,
or some non-rigid motion. We regard the true locations
$\{\mu_i\}$ as being in $x$-space.
The mappings between the indexing of the $\{\mu_i\}$ and that of
the data $\{x_j\}$ and $\{y_k\}$ are captured by indexing arrays
$\{\xi_j\}$ and $\{\eta_k\}$; specifically we assume that
\bel{likx}
x_j=\mu_{\xi_j}+\varepsilon_{1j}
\end{equation}
for $j=1,2,\ldots,m$, where $\{\varepsilon_{1j}\}$ have
probability density $f_1$, and
\bel{liky}
\mathcal{A} y_k=\mu_{\eta_k}+\varepsilon_{2k}
\end{equation}
for $k=1,2,\ldots,n$, where $\{\varepsilon_{2k}\}$ have
density $f_2$. Multiple matches are excluded, thus
each hidden point $\mu_i$ is observed at most once in
each of the $x$ and $y$ configurations; equivalently, the
$\xi_j$ are distinct, as are the $\eta_k$.
All $\{\varepsilon_{1j}\}$ and $\{\varepsilon_{2k}\}$
are independent of each other, and independent of the $\{\mu_i\}$.
\subsection{Formulation of Poisson process prior}
\label{sec:pp}
Suppose that the set of true locations $\{\mu_i\}$
forms a homogeneous Poisson process with rate $\lambda$
over a region $V\subset\mathcal{R}^d$ of volume $v$, and that there
are $N$ points realised in this region. Some of these
give rise to both $x$ and $y$ points, some to points of
one kind and not the other, and some are not observed at all.
We suppose these four possibilities occur independently
for each realised point, with probabilities parameterised so
that with probabilities $(1-p_{\rm x}-p_{\rm y}-\rhop_{\rm x}p_{\rm y},p_{\rm x},
p_{\rm y},\rhop_{\rm x}p_{\rm y})$ we observe neither, $x$ alone,
$y$ alone, or both $x$ and $y$, respectively.
The parameter $\rho$ is a certain
measure of the tendency {\it a priori} for points to be matched:
the random thinnings leading to the observed $x$ and $y$ configurations
can be dependent, but remain independent from point to point.
Given $N$, $m$ and $n$, there are $L$ matched pairs of points in our sample
if and only if the
numbers of these four kinds of occurrence among the $N$ points
are $(N-m-n+L,m-L,n-L,L)$. Under the assumptions above these four counts will
be independent Poisson distributed variables, with means
$(\lambda v (1-p_{\rm x}-p_{\rm y}-\rhop_{\rm x}p_{\rm y}),
\lambda v p_{\rm x},\lambda v p_{\rm y},\lambda v \rhop_{\rm x}p_{\rm y})$.
The prior probability
distribution of $L$ conditional on $m$ and $n$ is therefore
proportional to
$$
\frac{e^{-\lambda v p_{\rm x}}(\lambda v p_{\rm x})^{m-L}}{(m-L)!}\times
\frac{e^{-\lambda v p_{\rm y}}(\lambda v p_{\rm y})^{n-L}}{(n-L)!}\times
\frac{e^{-\lambda v \rhop_{\rm x}p_{\rm y}}(\lambda v \rhop_{\rm x}p_{\rm y})^L}{L!}
$$
so that
\bel{lprior}
p(L) \propto \frac{(\rho/\lambda v)^L}{(m-L)!(n-L)!L!}
\end{equation}
for $L=0,1,\ldots,\min\{m,n\}$. The normalising constant here
is the reciprocal of
$H(m,n,\rho/(\lambda v))$, where $H$ can be written
in terms of the confluent hypergeometric function
$$
H(m,n,d) = \frac{d^m}{m!(n-m)!} \: \mbox{}_1\!F_1(-m,n-m+1,-1/d),
$$
assuming without loss of generality that $n>m$;
see Abramowitz and Stegun (1970, p. 504).
Here and later,
we use the generic $p(\cdot)$ notation for distributions
and conditional distributions in our hierarchical model.
The matching of the configurations is represented by the
{\it matching matrix} $M$, where $M_{jk}$ indicates whether $x_j$ and $y_k$
are derived from the same $\mu_i$ point, or not, that is,
$$
M_{jk} =\cases {1 & if $\xi_j=\eta_k$ \cr
0 & otherwise \cr}.
$$
Note that $\sum_{j,k} M_{jk}=L$, and that,
since multiple matches are ruled out, there is at most
one 1 in each row and in each column of $M$: $\sum_j M_{jk}\leq 1 \forall k$,
$\sum_k M_{jk}\leq 1 \forall j$.
We assume for the moment that conditional
on $L$, $M$ is {\it a priori} uniform: there are $L! {m \choose L} {n \choose L}$
different $M$ matrices consistent with a given value of $L$, and these
are taken as equally likely. Thus
$$
p(M) = p(L)p(M|L) \propto \frac{(\rho/\lambda v)^L}{(m-L)!(n-L)!L!}
\left\{L! {m \choose L} {n \choose L}\right\}^{-1}
\propto (\rho/\lambda v)^L,
$$
(where here and later `$\propto$' means proportional to, as functions of
the variable(s) to the left of the conditioning $|$, in this case, $M$).
Thus
\bel{pm}
p(M) = \frac{(\rho/\lambda v)^L}
{\sum_{\ell=0}^{\min\{m,n\}} \ell! {m \choose \ell} {n \choose \ell}(\rho/\lambda v)^\ell}.
\end{equation}
Note that, because of the choice of parameterisation for the probabilities
that hidden points are observed, this expression does not involve
$p_{\rm x}$ and $p_{\rm y}$.
\begin{figure}[htbp]
\centering
\resizebox{3.5in}{!}{\rotatebox{0}{\includegraphics{align.eps}}}
\caption{Directed acyclic graph representing our model,
showing all data and parameters treated as variable. \label{fig:align}}
\end{figure}
\subsection{Likelihood of data}
We now have to specify the likelihood of the observed configurations of points,
given $M$. For simplicity, we will henceforth assume that $\mathcal{A}$
is an affine transformation: $\mathcal{A} y=Ay+\tau$. From (\ref{likx}) and
(\ref{liky}), the densities of $x_j$ and $y_k$, conditional
on $A$, $\tau$, $\{\mu_i\}$, $\{\xi_j\}$ and $\{\eta_k\}$ are
$f_1(x_j-\mu_{\xi_j})$ and $|A|f_2(Ay_k+\tau-\mu_{\eta_k})$, respectively,
$|A|$ denoting the absolute value of the determinant of $A$.
The locations $\{\mu_i\}$ of the $m-L$ points that
generate an $x$ observation but not a $y$ observation are independently
uniformly distributed over the region $V$, so that the likelihood contribution
of these $m-L$ observations, namely $\{x_j:M_{jk}=0\forall k\}$, is
$$
\prod_{j: M_{jk}=0\forall k} v^{-1} \int_V f_1(x_j-\mu) d\mu
$$
Similarly, the contributions from the unmatched $y$ observations, and from the
matched pairs are
$$
\prod_{k: M_{jk}=0\forall j} v^{-1} \int_V |A|f_2(Ay_k+\tau-\mu) d\mu
\quad\mbox{and}\quad
\prod_{j,k: M_{jk}=1} v^{-1} \int_V f_1(x_j-\mu) |A|f_2(Ay_k+\tau-\mu) d\mu
$$
respectively. These integrals all exhibit `edge effects'
from the boundary of the region $V$, which can be neglected if
$V$ is large relative to the supports of $f_1$ and $f_2$. In this
case these three expressions approximate to
$$
v^{-(m-L)}, (|A|/v)^{n-L}, \quad\mbox{and}\quad
(|A|/v)^L \prod_{j,k: M_{jk}=1}\int_{\mathcal{R}^d} f_1(x_j-\mu) f_2(Ay_k+\tau-\mu) d\mu
$$
respectively. The last expression can be written
$$
(|A|/v)^L \prod_{j,k: M_{jk}=1} g(x_j-Ay_k-\tau)
$$
where $g(z)=\int f_1(z+u)f_2(u)du$ (the density of $\varepsilon_{1j}-\varepsilon_{2k}$).
Combining these terms, the complete likelihood is
\bel{lik}
p(x,y|M,\mathcal{A}) = v^{-(m+n)} |A|^n \prod_{j,k: M_{jk}=1} g(x_j-Ay_k-\tau).
\end{equation}
Multiplying (\ref{pm}) and (\ref{lik}), we then have
$$
p(M,x,y|\mathcal{A}) \propto |A|^n \prod_{j,k: M_{jk}=1} \{(\rho/\lambda)g(x_j-Ay_k-\tau)\}.
$$
Note that the constant of proportionality involves $m$, $n$, $\lambda$, $\rho$,
and $v$, but not $A$, $\tau$, any parameters in $f_1$ or $f_2$, or $M$ of course.
If we further specialise by making assumptions of spherical normality
for $f_1$ and $f_2$:
$$
x_j \sim N_d(\mu_{\xi_j},\sigma_{\rm x}^2I)
\qquad\mbox{and}\qquad
Ay_k+\tau \sim N_d(\mu_{\eta_k},\sigma_{\rm y}^2I),
$$
with $\sigma_{\rm x}=\sigma_{\rm y}=\sigma$, say, then
$$
g(z)=\frac{1}{(\sigma\surd 2)^d} \phi(z/\sigma\surd 2)
$$
where $\phi$ is the standard normal density in $\mathcal{R}^d$, and
our final joint model is
\bel{post}
p(M,A,\tau,\sigma,x,y) \propto
|A|^{n} p(A) p(\tau) p(\sigma)\prod_{j,k:M_{jk}=1} \left(
\frac{\rho\phi(\{x_j-Ay_k-\tau\}/\sigma\surd 2)}{\lambda(\sigma\surd 2)^d}\right).
\end{equation}
Note that not only $p_{\rm x}$ and $p_{\rm y}$ but also $v$ does not appear in this
expression, principally from our choice of parameterisation,
and that only the ratio $\rho/\lambda$ is identifiable. The directed acyclic
graph representing this joint probability model, including the
variables ($\mu$, $\xi$ and $\eta$) that we have integrated out, is displayed in
Figure \ref{fig:align}.
\section{Prior distributions and computational implementation}
\label{sec:mcmc}
We will henceforth treat $\rho$ and $\lambda$ as fixed,
and consider inference for the remaining unknowns
$M$, $\tau$, $\sigma^2$ and sometimes $A$, given
the data $\{x_j\}$ and $\{y_k\}$. Markov chain Monte Carlo
methods must be used for the computation; several introductions
and overviews of MCMC are available, for example, the primer
in Green (2001). In Section \ref{sec:em}, we discuss the relevance
and applicability of an EM algorithm for making inference with
an approximation of our model.
We suppose
that prior information about $\tau$, $\sigma^2$ and $A$ will be
at best weak, and so we concentrate on generic prior
formulations that facilitate the posterior analysis.
Prior assumptions are therefore discussed in parallel
with MCMC implementation. Note that our formulation has some
affinity with mixture models, the matching matrix $M$ playing a similar
role to the allocation variables often used in computing
with mixtures; see, for example, Richardson and Green (1997).
As in that paper, the fully Bayesian analysis here aims at
simultaneous joint inference about both the discrete and
continuously varying unknowns, in contrast to frequentist
approaches.
Our model has another similarity with a mixture
formulation, in that as $M$ varies, the number of
hidden points needed to generate all the observed
data also varies, and thus there seems to be a
`variable-dimension' aspect to the model.
However, here our approach of integrating out the
hidden point locations eliminates the variable-dimension
parameter, so that reversible jump MCMC is not needed.
\subsection{Priors and MCMC updating for a rotation matrix}
\label{sec:rotmat}
We are interested in alignment and matching problems in which
either $A$ is given, and treated as fixed, or in which it is
one of the objects of inference. In the latter case, we consider in this
paper only the case of rotation matrices in two and three
dimensions. We therefore focus on the full
conditional distribution for $A$, which from (\ref{post}) is
$$
p(A|M,\tau,\sigma,x,y) \propto
|A|^{n} p(A) \prod_{j,k:M_{jk}=1}
\phi(\{x_j-Ay_k-\tau\}/\sigma\surd 2).
$$
Viewing this as a density for $A$, we are still free to choose the
dominating measure for $p(A)$, which is arbitrary: this full conditional
density is then with respect to the same measure.
Let us restrict attention to {\it rotations}: orthogonal matrices $A$,
(those with $A^{-1}$ = $A^T$) with positive determinant,
so that $|A|=1$.
Expanding the expression above, we then find
$$
p(A|M,\tau,\sigma,x,y) \propto
p(A) \exp\left(\sum_{j,k:M_{jk}=1}
-0.5(||x_j-Ay_k-\tau||/\sigma\surd 2)^2
\right)
$$
$$
\propto
p(A) \exp \left(\mbox{tr}\left\{ (1/2\sigma^2)\sum_{j,k:M_{jk}=1} y_k(x_j-\tau)^TA\right\} \right).
$$
Note a remarkable opportunity for (conditional) conjugacy -- if $p(A)$
has the form $p(A)\propto \exp(\mbox{tr}(F_0^TA))$ for some matrix $F_0$, then
the posterior has the same form with $F_0$ replaced by
$$
F=F_0+(1/2\sigma^2)\sum_{j,k:M_{jk}=1} (x_j-\tau)y_k^T.
$$
This form of $p(A)$ is known as the matrix Fisher distribution (Downs, 1972; Mardia and Jupp, 2000, p. 289).
To the best of our knowledge, this unique role of the matrix Fisher distribution
(or in the two-dimensional case, the von Mises distribution) as the
prior distribution for a rotation conjugate to spherical
Gaussian error distributions has not previously been noted.
(Although Mardia and El-Atoum (1976) have identified the von Mises--Fisher
distribution as the conjugate prior for the mean direction).
This may have relevance in models for other situations,
including the simpler case where there is no uncertainty in the
matching. The conjugacy is presumably related to the
interpretation of the matrix Fisher distribution
as a conditional multivariate Gaussian (see Mardia and Jupp,
2000, p.289).
\subsubsection*{Two-dimensional case}
Now consider the two-dimensional case, $d=2$. An arbitrary
rotation matrix $A$ can be written
$$
A=\left(
\begin{array}{rr}
\cos \theta & -\sin \theta \\
\sin \theta& \cos \theta \\
\end{array}
\right)
$$
and the natural dominating measure for $\theta$ is
Lebesgue on $(0,2\pi)$. Then
a uniformly distributed choice of $A$ corresponds to $p(A)\propto 1$.
More generally, the von Mises distribution for $\theta$
$$
p(\theta) \propto \exp(\kappa\cos(\theta-\nu))=\exp(\kappa\cos\nu\cos\theta+\kappa\sin\nu\sin\theta)
$$
can indeed be expressed as $p(A)\propto \exp(\mbox{tr}(F_0^TA))$, where
a (non-unique) choice for $F_0$ is
$$
F_0=\kappa/2\left(
\begin{array}{rr}
\cos \nu & -\sin \nu \\
\sin \nu& \cos \nu \\
\end{array}
\right).
$$
Thus the full conditional distribution for $\theta$ is of the same
von Mises form, with
$\kappa\cos\nu$ updated to $(\kappa\cos\nu+S_{11}+S_{22})$, and
$\kappa\sin\nu$ to $(\kappa\sin\nu-S_{12}+S_{21})$,
where $S$ is the $2\times 2$ matrix $(1/2\sigma^2)\sum_{j,k:M_{jk}=1} (x_j-\tau)y_k^T$.
It is therefore trivial to implement a Gibbs sampler
move to allow inference about $A$, assuming a von Mises
prior distribution on the rotation angle $\theta$
(including the uniform case, $\kappa=0$). We can use the
Best/Fisher algorithm, an efficient rejection method
(see Mardia and Jupp, 2000, p.43),
to sample from the full conditional for $\theta$.
\subsubsection*{Three-dimensional case}
In the three-dimensional case, we can represent $A$ as the
product of elementary rotations
\bel{geneul}
A=A_{12}(\theta_{12})A_{13}(\theta_{13})A_{23}(\theta_{23})
\end{equation}
as in Raffenetti and Ruedenberg (1970), and Khatri and Mardia (1977).
Here, for $i<j$, $A_{ij}(\theta_{ij})$ is the matrix with
$m_{ii}=m_{jj}=\cos\theta_{ij}$, $-m_{ij}=m_{ji}=\sin\theta_{ij}$,
$m_{rr}=1$ for $r\neq i,j$ and other entries 0.
We can then update each of the generalised Euler angles
$\theta_{ij}$ in turn, conditioning on the other two angles and the
other variables ($M,\tau,\sigma,x,y$) entering the expression for
$F$.
The joint full conditional density of the Euler angles is
$$
\propto \exp[\mbox{tr}\{F^TA\}] \cos\theta_{13}
$$
for $\theta_{12},\theta_{23}\in(-\pi,\pi)$ and $\theta_{13}\in(-\pi/2,\pi/2)$.
The cosine term arises since the natural dominating measure,
corresponding to uniform distribution of rotation,
has volume element $\cos\theta_{13} \:\d\theta_{12}\:\d\theta_{13}\:\d\theta_{23}$
in these coordinates.
Substituting the representation (\ref{geneul}), and simplifying, we
find that the trace can be written variously as
$\mbox{tr}\{F^TA\}=a_{12}\cos\theta_{12}+b_{12}\sin\theta_{12}+c_{12}
=a_{13}\cos\theta_{13}+b_{13}\sin\theta_{13}+c_{13}
=a_{23}\cos\theta_{23}+b_{23}\sin\theta_{23}+c_{23}$
where
\begin{eqnarray*}
a_{12}& = &(F_{22}-\sin\theta_{13}F_{13})\cos\theta_{23}
+(-F_{23}-\sin\theta_{13}F_{12})\sin\theta_{23}
+\cos\theta_{13}F_{11}
\\
b_{12}& = &(-\sin\theta_{13}F_{23}-F_{12})\cos\theta_{23}
+(F_{13}-\sin\theta_{13}F_{22})\sin\theta_{23}
+\cos\theta_{13}F_{21}
\\
a_{13}& = &\sin\theta_{12}F_{21}+\cos\theta_{12}F_{11}+\sin\theta_{23}F_{{32}}+\cos\theta_{23}F_{33}
\\
b_{13}& = &
(-\sin\theta_{23}F_{12}-\cos\theta_{23}F_{13})\cos\theta_{12}
+(-\sin\theta_{23}F_{22}-\cos\theta_{23}F_{23})\sin\theta_{12}+F_{31}
\\
a_{23}& = &(F_{22}-\sin\theta_{13}F_{13})\cos\theta_{12}+(-\sin\theta_{13}F_{23}-F_{12})\sin\theta_{12}
+\cos\theta_{13}F_{33}
\\
b_{23}& = &(-F_{23}-\sin\theta_{13}F_{12})\cos\theta_{12}+(F_{13}-\sin\theta_{13}F_{22})\sin\theta_{12}
+\cos\theta_{13}F_{32}
\end{eqnarray*}
and the $c_{ij}$ can be ignored, combined into the normalising constants.
Thus the full conditionals for $\theta_{12}$ and $\theta_{23}$ are von Mises distributions,
and so these two variables can be updated by Gibbs sampling.
That of $\theta_{13}$ is
proportional to
$$
\exp[a_{13}\cos\theta_{13}+b_{13}\sin\theta_{13}]\cos\theta_{13}
$$
and we use a random walk Metropolis update for this variable,
with a perturbation uniformly distributed on $[-0.1,0.1]$.
The latter distribution has been studied in Mardia and
Gadsden (1977) but with no discussion on how to
simulate from it.
\subsection{Priors and updating for other parameters}
\label{sec:prior}
We make the standard normal/inverse gamma assumptions:
$$
\tau \sim N_d(\mu_\tau,\sigma_\tau^2I)
\qquad\mbox{and}\qquad
\sigma^{-2} \sim \Gamma(\alpha,\beta).
$$
Under the assumptions of (\ref{post}), there
is conjugacy for $\tau$ and $\sigma$, and we have explicit
full conditionals:
$$
\tau|M,A,\sigma,x,y \sim N_d\left(
\frac{\mu_\tau/\sigma_\tau^2+\sum_{j,k:M_{jk}=1} (x_j-Ay_k)/2\sigma^2}
{1/\sigma_\tau^2+L/2\sigma^2}
,
\frac{1}{1/\sigma_\tau^2+L/2\sigma^2}I
\right)
$$
$$
\sigma^{-2}|M, A,\tau,x,y \sim \Gamma\left(\alpha+(d/2)L,
\beta+(1/4)\sum_{j,k:M_{jk}=1} ||x_j-Ay_k-\tau||^2\right),
$$
and so it is trivial to implement Gibbs sampler updates for these parameters.
\subsection{Updating $M$}
\label{sec:updatem}
The matching matrix $M$ is updated in detailed balance using Metropolis-Hastings
moves that only propose changes to a few entries: the number of matches
$L=\sum_{j,k}M_{jk}$ can only increase or decrease by 1 at a time, or
stay the same. The possible changes are
\begin{enumerate}
\item[(a)] adding a match: changing one entry $M_{jk}$ from 0 to 1
\item[(b)] deleting a match: changing one entry $M_{jk}$ from 1 to 0
\item[(c)] switching a match: simultaneously changing one entry from 0 to 1, and another {\it in the same row or column} from 1 to 0.
\end{enumerate}
The proposal proceeds as follows: first a uniform random choice is made
from all the $m+n$ data points $x_1,x_2,\ldots,x_m,y_1,y_2,\ldots,y_n$.
Suppose without loss of generality, by the symmetry of the set-up, that
an $x$ is chosen, say $x_j$. There are two possibilities: either $x_j$ is
currently matched ($\exists k$ such that $M_{jk}=1$) or not (there is no
such $k$).
If $x_j$ is matched to $y_k$, with probability $p^\star$ we
propose {\it deleting} the match, and with probability $1-p^\star$ we
propose {\it switching}
it from $y_k$ to $y_{k'}$, where $k'$ is drawn uniformly at random from
the currently unmatched $y$ points. On the other hand, if $x_j$
is not currently matched, we propose {\it adding} a match between
$x_j$ and a $y_{k}$, where again $k$ is drawn uniformly at random from
the currently unmatched $y$ points.
The acceptance probabilities for these three possibilities are easily
derived from the expression (\ref{post}) for the joint distribution, since in each case the proposed new matching matrix $M'$ is only slightly perturbed from $M$,
so that the ratio $p(M',\tau,\sigma|x,y)/p(M,\tau,\sigma|x,y)$ has only
a few factors. Taking into account also the proposal probabilities,
whose ratio is $(1/n_{\rm u})\div p^\star$,
where $n_{\rm u}=\#\{k\in 1,2,\ldots,n: M_{jk}=0\forall j\}$ is the number of
unmatched $y$ points in $M$, we find
that the acceptance probability for
adding a match $(j,k)$ is
\bel{mhadd}
\min\left\{1,
\frac
{\rho\phi(\{x_j-Ay_k-\tau\}/\sigma\surd{2})p^\star n_{\rm u}}
{\lambda(\sigma\surd{2})^d}
\right\}.
\end{equation}
Similarly, the acceptance probability for switching the match of $x_j$ from
$y_k$ to $y_{k'}$ is
\bel{mhswitch}
\min\left\{1,
\frac{\phi(\{x_j-Ay_{k'}-\tau\}/\sigma\surd{2})}
{\phi(\{x_j-Ay_k-\tau\}/\sigma\surd{2})}\right\}
\end{equation}
and for
deleting the match $(j,k)$ it is
$$
\min\left\{1,
\frac{\lambda(\sigma\surd{2})^d}
{\rho\phi(\{x_j-Ay_k-\tau\}/\sigma\surd{2})p^\star n_{\rm u}'}\right\},
$$
where $n_{\rm u}'=\#\{k\in 1,2,\ldots,n: M_{jk}'=0\forall j\}=n_{\rm u}+1$.
Along with just one of each of the other updates,
we typically make several moves updating $M$ per sweep,
since the changes effected are so modest.
\subsection{Loss functions}
The output from the MCMC sampler derived above, once equilibrated,
is a sample from the posterior distribution determined by (\ref{post}).
As always with sample-based computation, this
provides an extremely flexible basis for reporting
aspects of the full joint posterior that are of interest.
The matching matrix $M$ will often be of particular inferential
interest, and for some purposes a point estimate is desirable; in this
section we discuss how to obtain a Bayesian point estimate of the
matching matrix $M$.
The most easily understood estimator of $M$ would
be its posterior mode, the {\it maximum a posteriori} (MAP)
estimator. However, there are difficulties here. First, the notion is
itself ambiguous -- the unknown `parameter' in our model
consists of the matching matrix $M$, and some real parameters.
`MAP' might refer to the $M$ component of the overall maximum,
or the mode of the marginal posterior for $M$ alone. Secondly,
the posterior is multi-modal, and different modes may have
different `widths', appropriately measured. So there is no
intrinsic attraction to the MAP estimate. We should return
to basic principles.
By standard theory, this requires specification of
a loss function, $L(M,\widehat{M})$, giving the cost
incurred in declaring the matching matrix to be
$\widehat{M}$ when it is in fact $M$. The optimal estimate
given data $(x,y)$ is the matching matrix $\widehat{M}$ that
minimises the posterior expected loss
$$
E[L(M,\widehat{M})|x,y],
$$
the expectation over $M$ being taken with respect to the posterior
determined by (\ref{post}). In this language, the MAP estimator
is optimal for the `zero--one' loss function under which a fixed
total cost is paid if there is a single error in any value
$M_{jk}$; this is logically unappealing, and a further
argument against using MAP.
We consider instead loss functions $L(M,\widehat{M})$ that penalise different
kinds of error and do so cumulatively.
The simplest of these are additive over pairs $(j,k)$. Suppose that the loss when
$M_{jk}=a$ and $\widehat{M}_{jk}=b$, for $a,b=0,1$
is $\ell_{ab}$; for example, $\ell_{01}$ is the loss associated
with declaring a match between $x_j$ and $y_k$ when there is really none,
that is, a `false positive'. Then it is readily shown that
$$
E[L(M,\widehat{M})|x,y] = -(\ell_{10}+\ell_{01}-\ell_{11}-\ell_{00})
\sum_{j,k:\widehat{M}_{jk}=1} (p_{jk}-K)
$$
where
$$
K=(\ell_{01}-\ell_{00})/
(\ell_{10}+\ell_{01}-\ell_{11}-\ell_{00}),
$$
and
$p_{jk}=p(M_{jk}=1|x,y)$ is the posterior probability that $(j,k)$
is a match, which is estimated from an MCMC run by the
empirical frequency of this match.
Thus, provided that $\ell_{10}+\ell_{01}-\ell_{11}-\ell_{00}>0$ and
$\ell_{01}-\ell_{00}>0$, as is natural, the optimal estimate is that maximising
the sum of marginal posterior probabilities of the declared matches
$\sum_{j,k:\widehat{M}_{jk}=1} p_{jk}$, penalised by a multiple $K$ times the number of matches.
The optimal match therefore depends on the four loss function parameters
only through the cost ratio $K$. If false positive and false negative matches
are equally undesirable, one can simply choose $K=0.5$.
Computation of the optimal match $\widehat{M}$ would be trivial but for the constraint that
there can be at most one positive entry in each row and column of the array.
For modest-sized problems, the optimal match can be found by informal
heuristic methods. These may not even be necessary, especially if $K$ is not too small.
In particular, it is immediate that if the set of all $(j,k)$ pairs
for which $p_{jk}>K$
includes no duplicated $j$ or $k$ values, the optimal $\widehat{M}$ consists of precisely
these pairs.
We could also consider loss functions that penalise mismatches differently from
the sum of the losses of the individual errors. For example, declaring
$(j,k)$ to be a match when it should be $(j,k')$ might deserve a relative
loss greater or lesser
than $(\ell_{10}+\ell_{01}-\ell_{11}-\ell_{00})$, depending
on context. Such loss functions
could be handled in a broadly similar way, but this is left for future work.
\subsection{Using partial labelling information}
\label{modlik}
When the points in each configuration are `coloured', with the
interpretation that like-coloured points are more likely to be matched
than unlike-coloured ones, it is appropriate to use a modified likelihood
that allows us to exploit such information. Let the colours for the $x$ and $y$
points be $\{r^{\rm x}_j,j=1,2,\ldots,m\}$ and
$\{r^{\rm y}_k,k=1,2,\ldots,n\}$ respectively. The hidden point
model is augmented to generate the point colours, as follows.
Independently for each hidden point, with probability
$(1-p_{\rm x}-p_{\rm y}-\rhop_{\rm x}p_{\rm y})$ we observe neither
$x$ nor $y$ point, as before. With probabilities
$p_{\rm x}\pi^{\rm x}_r$ and $p_{\rm y}\pi^{\rm y}_r$,
respectively, we observe only an $x$ or $y$ point,
with colour $r$ from an appropriate finite set.
With probability
$$
\rhop_{\rm x}p_{\rm y}\pi^{\rm x}_r\pi^{\rm y}_s
\exp\{\gamma I[r=s]+ \delta I[r \neq s]\},
$$
we observe an $x$ point coloured $r$ and a $y$ point
coloured $s$.
Our original likelihood is equivalent to the case $\gamma=\delta=0$,
where colours are independent and so carry no information about matching.
If $\gamma$ and $\delta$ increase, then matches are more probable,
{\it a posteriori}, and if $\gamma>\delta$, matches between like-coloured
points are more likely than those between unlike-coloured ones.
The case $\delta\to-\infty$ allows the prohibition
of matches between unlike-coloured points, a feature that might be adapted
to other contexts such as the matching of shapes with
given landmarks.
In implementation of this modified likelihood, the
MCMC acceptance ratios in Section \ref{sec:updatem}
have to be modified accordingly. For example,
if $r^{\rm x}_j=r^{\rm y}_k$ and $r^{\rm x}_j\neq r^{\rm y}_{k'}$,
then (\ref{mhadd}) has to be multiplied by
$\exp(-\gamma)$ and (\ref{mhswitch}) by $\exp(\delta-\gamma)$.
Other, more complicated, colouring distributions where the log probability
can be expressed linearly in entries of $M$ can be handled
similarly.
\subsection{Alternative approach using the EM algorithm}
\label{sec:em}
The interplay between matching (allocation) and parameter uncertainty
has something in common with mixture estimation. This might suggest considering
maximisation of the posterior by using the EM algorithm,
which could of course in principle be applied either to maximum likelihood estimation
based on (\ref{lik}) or to MAP estimation based on (\ref{post}).
For the EM formulation, the `missing data' are the matches.
In an
exponential family, the EM algorithm alternates between
between finding expectations of missing values
given data, at current parameter values, and
maximising the log-posterior, with missing values
replaced by these expectations.
The `expectations of missing values' are just probabilities of
matching. These are only tractable if we were to drop the
assumption that a point can only be matched with at most one other
point -- that is, that $\sum_j M_{jk}\leq 1 \forall k$, $\sum_k M_{jk} \leq 1 \forall j$.
Making this approximation, the E-step is trivial:
the expectation of $I[M_{jk}=1]$ is
$p_{jk}=w_{jk}/(1+w_{jk})$ where $w_{jk}$ is the $(j,k)$ factor in the joint model, i.e.
$$
w_{jk}=\{(\rho/\lambda)g_\sigma(x_j-Ay_k-\tau)\}
$$
The M-step then requires maximising (for given $p_{jk}$)
$$
\log\left[|A|^{n} p(A) p(\tau) p(\sigma)\right]
+\sum_{j,k} p_{jk} \log \{w_{jk}(A,\tau,\sigma)\}
$$
over $A$, $\tau$, $\sigma$ -- note that here $w_{jk}$ is a function
of all three. Although for some individual parameters this seems to
be explicit, in the general case we need numerical optimisation.
In summary, EM allows us to study only certain aspects of an approximate
version of our model, and is not trivial numerically -- so we
do not pursue this approach. Obtaining the complete posterior by MCMC sampling
gives much greater freedom in inference.
\section{Applications}
\subsection{Matching protein gels}
The objective in this example is to match two electrophoretic gels
automatically, given the locations of the centres of 35
proteins on each of the two gels. The data are presented
in the supplementary information on the web.
The correspondence between pairs of proteins, one
protein from each gel, is unknown, so our aim is to match the two gels
based on these sets of unlabelled points. We suppose it
is known that the transformation between the gels is affine.
In this case, experts have already identified 10
points; see Horgan et al (1992).
Based on these 10 matches, the linear part of the
transformation is estimated {\it a priori} to be
\bel{gelA}
A=\left(
\begin{array}{rr}
0.9731 & 0.0394 \\
-0.0231 & 0.9040 \\
\end{array}
\right).
\end{equation}
(Dryden and Mardia, 1998, pp. 20--21, 292--296).
\begin{table}
\caption{The 20 marginally most probable matches in the analysis of the gel data.
\label{gelmatches}}
\footnotesize
\vspace*{5mm}\centering\leavevmode
\begin{tabular}{crrl}
rank & $j$ & $k$ & $p_{jk}$ \\
\hline
1 & 15 & 21 & 1 \\
2 & 19 & 19 & 1 \\
3 & 8 & 8 & 1 \\
4 & 3 & 3 & 1 \\
5 & 2 & 2 & 1 \\
6 & 31 & 30 & 0.9989 \\
7 & 6 & 6 & 0.9987 \\
8 & 4 & 4 & 0.9966 \\
9 & 5 & 5 & 0.9946 \\
10 & 10 & 10 & 0.9927 \\
11 & 24 & 23 & 0.9855 \\
12 & 7 & 7 & 0.9824 \\
13 & 32 & 31 & 0.9776 \\
14 & 1 & 1 & 0.9763 \\
15 & 9 & 9 & 0.9677 \\
16 & 26 & 32 & 0.7910 \\
17 & 12 & 13 & 0.7552 \\
18 & 21 & 33 & 0.3998 \\
19 & 26 & 27 & 0.1931 \\
20 & 35 & 35 & 0.0025 \\
\hline
\end{tabular}
\end{table}
\myfig{gelconf}{The 17 most probable matches in the gel data, the optimal
match for any $K\in(0.3998,0.7552)$;
+ symbols signify $x$ points, o symbols the $y$ points, linearly transformed
by premultiplication by the fixed affine transformation $A$ given in (\ref{gelA}).
The solid line for each of the 17 matches
joins the matched points, and represents the inferred translation $\tau$
plus noise.}{5}
Here, we have only
to make inference on the translation $\tau$ and the unknown matching
between certain of the proteins.
The model (\ref{post}) will therefore be taken to apply, with $d=2$
and with $A$ held fixed at (\ref{gelA}).
The MCMC sampler described in Section \ref{sec:mcmc} was run for
100 000 sweeps, after a burn-in period of 20 000 sweeps, considered on the
basis of an informal visual assessment of time series traces to be adequate
for convergence.
Prior and hyperprior settings were: $\alpha=1$, $\beta=16$, $\mu_\tau=(0,0)^T$,
$\sigma_\tau=20.0$ and $\lambda/\rho=0.0001$. The sampler parameter $p^\star$ was set to 0.5.
Such a run took about 2 seconds on a 800MHz PC. Acceptance rates for the moves updating
$M$ were between 0.6\% and 2.1\%.
The posterior expectation and variance of $\tau$ were estimated to
be $(-35.950,66.685)^T$
(to be compared with $(-36.08,66.64)^T$ obtained by Dryden and Mardia (1998))
and
$$
\left(
\begin{array}{rr}
0.5776 & -0.0227 \\
-0.0227 & 0.6345 \\
\end{array}
\right).
$$
The posterior mean and variance of $\sigma$ are 2.050 and 0.1192.
The 20 most probable matches between $x$ and $y$ points are listed in
Table \ref{gelmatches};
note that there is no duplication in their indices until the 19th
match: $j=26$ also appears in the 16th match (recall that there is a
simple rule for identifying the optimal $\widehat{M}$
if there are no duplicates among the matches with $p_{jk}$ above the threshold $K$).
We can conclude that for all values of
$K=(\ell_{01}-\ell_{00})/
(\ell_{10}+\ell_{01}-\ell_{11}-\ell_{00})$
from 1 down to 0.1112, the optimal Bayesian matching is given
by an appropriate subset of Table \ref{gelmatches}, reading down from the top.
For example if this cost ratio is 0.8 we take the first 15
rows of the table, while if the ratio is 0.6 or 0.4 we include the 16th and 17th rows
as well. The 17 most probable matches are displayed graphically
in Figure \ref{fig:gelconf}.
It will be noted that all of the expert-identified matches, points 1 to
10 in each set, are declared to be matches with high probability in the
Bayesian analysis. We also repeated the analysis with these 10 pairs
held fixed. The next 9 most probable matches, together with these 10,
are identical to those in the first 19 lines of Table \ref{gelmatches}, and
the posterior probabilities differ by less than 0.037 in all 19 cases.
\subsection{Aligning proteins in three dimensions}
\label{sec:3deg}
We now apply the matching method to a problem in three dimensional
structural biology, previously considered by Gold et al (2002).
The problem consists of finding the matches for two Active sites 1 and 2 corresponding to two Proteins A and B respectively. The corresponding coordinates $x$ and $y$ of these sites
are presented in the supplementary information; these coordinates are the centres of gravity
of the amino acids of the two sites. Here $m= 40$ and $n=63$. The biological details of the two proteins are as follows. Protein 1 is the human protein `17--beta hydroxysteroid dehydrogenase' and is involved in the synthesis of oestrogens. This protein binds the ligands (molecules comparatively smaller than proteins) oestradiol and NADP. Protein 2 is the mouse protein `carbonyl reductase' and is involved in metabolism of carbonyl compounds. This protein binds the ligands 2--Propanol and NADP. The common element between these two sets of ligands is NADP. From chemical properties of the sites, the relevant matching should be invariant under rigid transformation.
\label{sec:3dalign}
\myfig{nicolaconf42}{The optimal alignment (36 matches)
when $K=0.5$ for the protein alignment analysis data, without using
colouring information; + symbols signify $x$ points,
o symbols the $y$ points, rotated according to the
inferred $\widehat{A}$ matrix given by (\ref{meana1}). The entire joint configuration
has been rotated to its first two principal axes.
Solid lines represent the 36 marginally most probable matches, and
indicate the inferred translation $\tau$ plus noise.}{5}
\myfig{nicplall}{Time series traces and histograms of the MCMC run
of Section \ref{sec:3deg}, based on a thinned sub-sample of 2000 after burn-in.}{5}
There is information about the identities of the amino acids
in the two configurations: we defer use of this to Section \ref{sec:amino}.
The MCMC sampler described in Section \ref{sec:mcmc} was run for
1 000 000 sweeps, after a burn-in period of 200 000 sweeps, considered on the
basis of an informal visual assessment of time series traces to be adequate
for convergence.
Prior and hyperprior settings were: $\alpha=1$, $\beta=36$, $\mu_\tau=(0,0,0)^T$,
$\sigma_\tau=50.0$, $\lambda/\rho=0.003$ and the matrix $F_0$ defining the prior
for $A$ set to the zero matrix. The sampler parameter $p^\star$ was set to 0.5, and
we made updates to $M$ 10 times in each sweep.
Such a run took about 71 seconds on a 800MHz PC. Acceptance rates for the moves updating
$M$ were between 0.41\% and 5.6\%.
The posterior expectation and variance of $\tau$ were estimated to
be $(31.60,8.89,17.44)^T$ and
$$
\left(
\begin{array}{rrr}
0.227 & 0.120 & -0.044 \\
0.120 & 0.307 & 0.176 \\
-0.044 & 0.176 & 0.428 \\
\end{array}
\right)
$$
The posterior mean and variance of $\sigma$ are 1.051 and 0.00996.
In representing the centre of the posterior
distribution for the rotation matrix $A$, we
we need to use a definition of mean appropriate to
the geometry. We form the mean elementwise
from a thinned sample of 2000
values of $A$ from the post-burn-in MCMC run.
This mean matrix $\overline{A}$ is of course not a rotation matrix,
but post-multiplication by the positive definite symmetric
square root of $\overline{A}^T\overline{A}$
yields a rotation matrix that
is known as its polar part (see Mardia and Jupp, p. 286, 290).
This is an appropriate measure of location
of the posterior, and takes the value
\bel{meana1}
\widehat{A}=\left(
\begin{array}{rrr}
0.4339 & -0.8444 & 0.3140 \\
-0.7118 & -0.5350 & -0.4550 \\
0.5522 & -0.0261 & -0.8333 \\
\end{array}
\right)
\end{equation}
in this case.
The 40 most probable matches between $x$ and $y$ points are listed in
supplementary information;
there is no duplication in their indices until the 39th
match: $k=12$ also appears in the 38th match.
We can conclude that for all values of $K$
greater than 0.2895 (the marginal posterior probability associated
with the 39th match), the optimal Bayesian matching is given
by an appropriate leading subset of the matches.
For example if this cost ratio is 0.5 we take the first 36
matches; these are displayed graphically
in Figure \ref{fig:nicolaconf42}; in this 3-dimensional example, the axes signify
the first two principle coordinates of the combined cloud of data.
As would be anticipated, simultaneous inference for the rotation $A$
and the matching matrix $M$ (as well as $\tau$ and $\sigma$)
is a considerably greater challenge
for MCMC than is the problem of the previous section, where
the rotation matrix is held fixed. It is clear that there is a
possibility of severe multi-modality in the posterior, with the
conditional posterior for $M$ and $\tau$ given $A$ depending
strongly on $A$. This challenge is quantified empirically by a
heavy-tailed distribution of times to convergence, and by `meta-stability'
in the time series plots of various monitoring statistics
against simulation time. We found the log-posterior
to be a useful summary statistic for quality of fit, and
pilot runs provided experience to choose a threshold value,
exceedance of which we hypothesised diagnosed convergence to
the main mode of the posterior.
To investigate
multimodality and convergence time, we conducted a study in which
the MCMC run described was repeated -- with the same parameters --
from 100 different initial configurations, obtained by independent
random rotations as initial settings for $A$. After short runs of
50 000 sweeps, we tested whether the threshold log-posterior value had
been exceeded, and if not the run was abandoned. 83 out of the 100
runs passed this test, and these were allowed to run on for a further
450 000 sweeps. Every one of these 83 long runs provided exactly the
same set of 36 most probable matches, and we therefore felt justified to
conclude that they had not been trapped in a subsidiary mode of
the posterior, and that it was safe to draw inference from the
results. This conclusion is specific to the data set and
parameter settings used, and it would be straightforward to contrive
artificial data where multiple modes were more equal in probability
content. In such cases more sophisticated MCMC samplers would be needed.
\subsection{Prior settings and sensitivity}
\label{sec:sensitivity}
Our analysis depends of course on the settings of the
hyperparameters $\lambda/\rho$ (see Section \ref{sec:pp}), $F_0$ (Section \ref{sec:rotmat}),
and $\mu_\tau$, $\sigma_\tau$, $\alpha$, $\beta$ (Section \ref{sec:prior}).
These allow the provision of real prior information from the experimental context,
if it is available.
For a default analysis in the absence of such information, we would
set $F_0$ to the zero matrix (a uniform prior on $A$),
$\mu_\tau$ to be the zero vector, and $\sigma_\tau$ of the order of twice the
distance between the centres of gravity of the two configurations.
We fix $\alpha=1$, giving an exponential prior distribution
for $\sigma^{-2}$.
Here we briefly discuss settings of, and sensitivity to, the remaining two
parameters, the scalars $\lambda/\rho$ and $\beta$.
Sensitivity to $\lambda/\rho$ is
pronounced, as might be anticipated. This parameter ratio
has a very direct role in determining whether an $(x_j,y_k)$ pair
are noisy observations of the same hidden $\mu_i$
point or not, after transformation, since it controls the density of
hidden points. In practice, we should not expect to
be able to draw inference about matching without
real prior knowledge about this ratio or an equivalent measure of
the prior tendency of points to be matched.
The prior for the number of matches $L$ is parameterised
by $\lambda/\rho$: see (\ref{lprior}).
This distribution is non-standard, but
very well-behaved. It is clear from inspection that
setting $\lambda/\rho$ equal to
$(m-\overline{L})(n-\overline{L})/\overline{L} v$ yields a mode of
$L$ that is within 1 of $\overline{L}$, and numerical
calculation in the context of the example
in Section \ref{sec:3deg}, verifies that for all possible
`prior guesses' $\overline{L}$ for $L$, the prior expectation and median
are also both equal to $\overline{L}$ to the nearest integer.
Thus prior information about $L$ is directly informative
about the parameter ratio $\lambda/\rho$.
As long as $v$ is known, or at least a representative value
provided, and the analyst is able to make a prior
guess $\overline{L}$ at the number of matches,
this suggests a reasonable way to specify $\lambda/\rho$.
The posterior distribution for $L$ tracks the prior
rather closely, confirming that the raw data carry little information
about the number of matches.
The hyperparameter $\beta$ is an inverse scale parameter for
the precision of the noise terms $\varepsilon$; thus as $\beta$ increases,
we expect that $\sigma^2=\mbox{var}(\varepsilon)$ increases too. The runs we have presented
used $\beta=36$; reducing this by a factor of 2 makes
minimal difference to the posterior inference for
either $\sigma^2$ or $M$. However, increasing $\beta$ by a factor of
2 leads to a 3-fold increase in $\sigma$ and a sharp reduction in the
number of matches -- the posterior expectation of $L$ goes down from
around 34 to 26. The latter observation is perhaps counter-intuitive,
until one realises that when $\sigma$ is larger, it becomes relatively
less likely that points that are nearly coincident (after transformation)
are in fact matched.
Finally, it would be desirable to assess the sensitivity to the Poisson assumption
for the hidden point model, but this would be extremely onerous to
do directly, since alternatives would require a substantially modified formulation
and implementation. There is scientific reason to doubt
the Poisson assumption; for
example, the minimum spacing between the centres of gravity of the amino acids
in proteins is approximately 3.8 Angstroms. However,
experiments reported in Mardia, Nyirongo and Westhead (2005)
do at least suggest strongly that the ability of our method to
detect matches is little affected by real hard-core effects.
\subsection{Using information about types of amino acid}
\label{sec:amino}
The protein alignment data includes identifiers of the type of amino acid
at each point (see supplementary information).
There are 20 different types, which can be categorised
into 4 groups: hydrophobic, charged, polar and glycine;
we use the group identifiers as colours
in defining a modified likelihood as in Section \ref{modlik}.
The parameter values taken were
$\gamma=1.0$ and $\delta=-0.5$, providing a strong preference for
like-coloured matching ($\exp(\gamma-\delta)\approx 4.48$).
The analysis was repeated with this modified model, leaving all other
details unchanged.
The 40 most marginally probable matches are listed in supplementary information,
along with displayes of the optimal alignment.
The 36 most probable matches, which together form the optimal
matching whtn $K=0.4$,
are identical to those found in the previous section; however, there are modest variations in the posterior
probabilities attached to individual matches.
The posterior expectation and variance of $\tau$ were now estimated to
be $(31.94,8.94,17.61)^T$ (slightly shifted from that obtained in
the analysis of the previous section) and
$$
\left(
\begin{array}{rrr}
1.284 & -0.763 & -0.118 \\
-0.763 & 3.534 & -0.015 \\
-0.118 & -0.015 & 1.320 \\
\end{array}
\right)
$$
The posterior mean and variance of $\sigma$ are 1.3122 and 0.1984.
The increased estimate of $\sigma$ is perhaps anticipated.
The centre of the posterior distribution of $A$
is in this case:
\bel{meana2}
\widehat{A}=\left(
\begin{array}{rrr}
0.4240 & -0.8512 & 0.3092 \\
-0.7235 & -0.5237 & -0.4497 \\
0.5447 & -0.0331 & -0.8379 \\
\end{array}
\right).
\end{equation}
\myfig{comb43}{The optimal matching (36 matches), when $K=0.4$, in the protein alignment analysis data,
using colouring information, with $\gamma=1.0$ and $\delta=-0.5$;
matches are signified by line segments joining the sequence number
of the point in the $x$ configuration to that of the
matched point in the $y$ configuration.
The solid lines indicate the 27 matches identified by Gold et al (2002);
our method discovers all of these, together with the 9
further matches shown with broken lines.
The height of the vertical bars indicate the marginal
probabilities of each match.
The + symbols denote points that are present in either configuration
but are not matched.}{6.5}
In the approach to the analysis of these data taken by Gold et al (2002),
the matching between the configurations
was performed in two stages, and is not driven by an
explicit probability model. First, inter-point distances
$d(\cdot,\cdot)$ were calculated within each configuration.
These distances are invariant under the rigid body motions
considered here. A maximal set of pairs of indices
$\{(j_1,k_1),(j_2,k_2),\ldots\}$, with no ties
among the $j$s or $k$s, is found such that
$|d(x_{j_r},x_{j_s})-d(y_{k_r},y_{k_s})|$ is less than
some threshold, for all $s\neq r$. This is done using graph
theoretical algorithms of Bron and Kerbosch (1973) and
Carraghan and Pardos (1990), applied to a product graph
whose vertices are labelled with $(j,k)$ pairs.
This first stage of the matching alogrithm was formulated
by Kuhl et al (1984).
In the second stage, the matches are scored
using the amino acid information, assigning a score of 1
for identity of the amino acids, and 0.5 when the amino acids
are different but fall in the same group. The initial list
of matches from stage one is then permuted so as to
maximise the total score.
Once the matches are found the
rigid body transformation is estimated by Procrustes
analysis; for example, see Dryden and Mardia (1998, pp 176-178).
It is interesting to compare the rotation
matrix resulting from this method, namely
$$
A=\left(\begin{array}{rrr}
0.441 & -0.841 & 0.312 \\
-0.678 & -0.541 & -0.498 \\
0.588 & 0.008 & -0.809 \\
\end{array}
\right)
$$
with that obtained by our method. The trace of the orthogonal
matrix taking $A$ to $\widehat{A}$ is approximately
$1+2\cos 0.07$, so the two differ by
a rotation of only 0.07 radians.
Figure \ref{fig:comb43}
provides a comparison between the matchings achieved by the two
approaches. Of the 27 matches identified by Gold et al,
14 are among the most probable 20 that we find, and all 27 are
among the first 35.
A referee has raised with us the role of sequence ordering along
the protein in inference about alignment and matching.
The example in this section concerns ligand binding site
matching, in which biologically relevant matches do not necessarily preserve
sequential ordering, in contrast to the more familiar
situation of aligning protein backbones;
see for example Eidhammer et al (2004, pp. 333--334).
Examples are trypsin-subtilisin with
similar active sites and unrelated folds, and many adenine binding
sites in different folds. Somewhat remarkably, although sequence ordering
is not used in our analysis, the resulting matches do perfectly
respect this ordering. This is visualised in Figure \ref{fig:comb43},
which also reveals that some but not all of the matches revealed
by our analysis additional to those of Gold et al (2002) extend
already matched segments.
In this particular data set, the sites must come from very closely
related folds and would probably also be alignable by sequence-preserving
methods aligning full structures. Intriguingly, in this example at least,
knowledge of the sequence ordering would provide no additional
information beyond that extracted from the point coordinates and
amino-acid groups by our approach.
\section{Discussion}
The main conclusion of this paper is that a probability
model based approach is successful in allowing
simultaneous inference about
partial matching between two point configurations, and
a geometrical transformation between the coordinate systems
in which the configurations are measured. This seems
an advance over previous more ad-hoc methods.
We have only used the translation and rigid motion groups in illustrating our
methodology. However, the formulation allows inference about various other group
transformations such as affine transformation, and so on.
The fairly straightforward MCMC implementation presented here has
proved adequate for the models and data sets considered, although
allowing rotations did increase the needed run lengths considerably.
We anticipate that, at least for
models allowing rotations, dealing with larger data sets
will be much more challenging, since small
rotational perturbations generate large displacements at sites far
from the axis of rotation; moves that simultaneously perturb
allocations and geometrical and error distribution parameters
will be necessary for good performance. We also anticipate
more severe difficulties from multi-modality that were exposed
in Section \ref{sec:3deg}.
An important task left for future work is a formulation that
allows smooth nonparametric transformations between coordinate
systems, setting warping into a model-based framework;
this would be important in dealing more comprehensively with
gel matching problems.
We have only used pairwise
comparisons but there is scope for taking multiple combinations such as
triads. The transformations considered above
are parametric but some non-parametric alternatives such as non-linear
deformations may be useful in some cases, e.g. to deal with dynamic aspects
of the atoms in a protein. We have
considered only two configurations but a natural extension
would be to take three or more point configurations
simultaneously, and make joint inference about patterns of matching between
the configurations and the various geometrical transformations involved.
More straightforward extensions would be
to allow for non-Gaussian noise, other types of prior and so on.
Kent et al (2004) have treated the unlabelled case by using a
different model. While matching two configurations, one of them
is taken as the population and the second as a random sample
from this population after an unknown transformation.
This approach is different from the symmetrical model for the
two configurations proposed here. Further the emphasis in
Kent et al (2004) is on maximum likelihood inference using the EM-algorithm.
Recent independent work by Dryden, Hirst and Melville (2005),
addresses a similar problem of matching unlabelled point sets.
Their approach has some substantial differences, for example
there is assymmetry
in comparing two configurations, one being treated as a perturbation
of the other. The geometrical transformation parameters are given
uniform priors and maximised out, using standard ideas from
shape analysis, rather than integrated out as in our fully
Bayesian approach. Neither the loss function basis for
estimating matches, nor the treatment of partial labelling,
appear.
There is other statistical work on alignment and matching in
proteins by Wu et al (1998) and Schmidler (2004),
which in contrast does use sequence information.
Further work is needed to clarify the
relationships between all these methods and their comparative
performance.
Finally, in the context of using methods such as ours in
database search, often the reason for assessing protein alignment,
there are issues related to multiple comparisons. These are not discussed here,
but the answers will depend on the size of the database as well as the number of points in
the query site.
\section*{Acknowledgements}
We are grateful to Nicola Gold and Dave Westhead for their
many helpful discussions, and in particular for the data in
Example 2, and to Vysaul Nyirongo and Charles Taylor for
various helpful comments.
\section*{References}
\begin{list}{}{\setlength{\itemindent}{-\leftmargin}}
\item Abramowitz, M. and Stegun, I. A. (1970).
{Handbook of Mathematical Functions}. Dover, New York.
\item Bron, C. and Kerbosch, J. (1973).
Algorithm 457; finding all cliques of an undirected graph.
{\it Communication of the ACM}, {\bf 16}, 575--577.
\item Carraghan, R. and Pardalos, P. M. (1990).
Exact algorithm for the minimal clique problem.
{\it Operations Research Letters}, {\bf 9}, 375.
\item Chui, H. and Rangarajan, A. (2000). A new algorithm for non-rigid point matching. {\em IEEE Conference on Computer Vision and Pattern Recognition.} {\bf 2}, 44--51.
\item Cross, A. D. J. and Hancock, E. R. (1998). Graph matching with dual-step
{EM} algorithm. {\em IEEE transactions on pattern analysis and machine
intelligence.} {\bf 20}, 1236--1253.
\item Downs, T. D. (1972). Orientation statistics.
{\it Biometrika}, {\bf 59}, 665--676.
\item Dryden, I. L., Hirst, J. D. and Melville, J. L. (2005).
Statistical analysis of unlabelled point sets: comparing
molecules in chemoinformatics.
Under revision for {\it Biometrics}.
\item Dryden, I. L. and Mardia, K. V. (1998). {\em Statistical shape analysis}.
Wiley, Chichester.
\item Eidhammer, T., Jonassen, T. and Taylor, W. R. (2004).
{\it Protein Bioinformatics}.
Wiley, Chichester.
\item Gold, N. D., Pickering, S. J., and Westhead, D. R. (2002).
Protein functional site matching using graph theory techniques.
In {\it Proceedings of
the International Conference on Bioinformatics}, Bangkok, Thailand,
page 79.
\item Green, P. J. (2001).
A Primer on Markov chain Monte Carlo, pp. 1--62 of
{\it Complex Stochastic Systems}, Barndorff-Nielsen, O. E., Cox, D. R. and Kl\"{u}ppelberg, C. (eds.), Chapman and Hall, London.
\item Horgan, G. W., Creasey, A. and Fenton, B. (1992).
Superimposing two
dimensional gels to study genetic variation in malaria parasites.
{\it Electrophoresis}, {\bf 13},871--875.
\item Kent, J. T., Mardia, K. V. and Taylor, C. C. (2004).
Matching problems for unlabelled configurations.
In {\it Bioinformatics, Images and Wavelets}, edited by
Aykroyd, R.G., Barber, S. and Mardia, K.V.
Proceedings of LASR 2004, Leeds University Press, Leeds.
\item Khatri, C. G. and Mardia, K. V. (1977). The von Mises--Fisher distribution
in orientation statistics.
{\it Journal of the Royal Statistical Society}, B, {\bf 39}, 95--106.
\item Kuhl, F. S., Crippen, G. M. and Friesen, D. K. (1984).
A combinatorial algorithm for calculating ligand binding.
{\it Journal of Computational Chemistry}, {\bf 5}, 24--34.
\item Mardia, K. V. and El-Atoum, S. A. M. (1976).
Bayesian inference for the von Mises--Fisher distribution.
{\it Biometrika}, {\bf 63}, 203--205.
\item Mardia, K. V. and Gadsden, R. J. (1977).
A circle of best fit for spherical data and areas of
vulcanism.
{\it Applied Statistics},{ \bf 26}, 238--245.
\item Mardia, K. V. and Jupp, P. E. (2000).
{\it Directional Statistics}, Wiley, Chichester.
\item Mardia K. V., Taylor, C. C, and Westhead, D. R. (2003). Structural bioinformatics revisited. In {\it LASR2003}, pp11--18. Leeds University Press.
\item Mardia, K. V., Nyirongo, V., and Westhead, D.R. (2005).
EM algorithm, Bayesian and distance approaches to matching active sites
{\it Mathematical and Statistical Annual Meeting in Bioinformatics}, Rothamsted, March 2005, Abstracts pp13-14.
\item Pedersen, L. (2002). {\em Analysis of two-dimensional electrophoresis gel images.} Ph.D thesis, IMM Technical University of Denmark.
\item Raffenetti, R. C. and Ruedenberg, K. (1970).
Parameterization of an orthogonal matrix
in terms of generalized Eulerian angles.
{\it International Journal of Quantum Chemistry}, {\bf IIIS}, 625--634.
\item Richardson, S. and Green, P. J. (1997).
On Bayesian analysis of mixtures with an unknown number of components
(with discussion).
{\it Journal of the Royal Statistical Society}, B, {\bf 59}, 731--792.
\item Schmidler, S. C. (2004).
{\it Bayesian shape matching and structural alignment}.
Presentation at the 6th World Congress of the Bernoulli Society,
Barcelona, July 2004.
\item Wu, T. D., Schmidler, S. C., Hastie, T. and Brutlag, G. (1998).
Regression analysis of multiple protein structures.
{\it Journal of Computational Biology}, {\bf 5}, pp 585--595.
\end{list}
\end{document}
|
{
"timestamp": "2005-07-01T16:37:22",
"yymm": "0503",
"arxiv_id": "math/0503712",
"language": "en",
"url": "https://arxiv.org/abs/math/0503712"
}
|
\section{Introduction}
Random matrices play an important role in physics and mathematics \cite{Mehta, courseynard, BI, DGZ, Guhr, Moerbeke:2000, DeiftBook}.
It has been observed more and more in the recent years how deeply random matrices are related to integrability ($\tau$-functions),
and algebraic geometry.
Here, we consider the computation of large n asymptotics for orhogonal polynomials as an example of a problem where the concepts of integrability,
isomonodromy and algebraic geometry appear and combine.
The method presented here below, is not, to that date, rigorous mathematicaly. It is based on the asumption that an integral with a large number of variables
can be approximated by a saddle-point method. This asumption was never proven rigorously, it is mostly based on ``physical intuition''.
However, the results given by that method have been rigorously proven by another method, namely the Riemann--Hilbert method \cite{BlIt, BlIt1, dkmvz, dkmvz2}.
The method presented below was presented in many works \cite{eynchain, eynchaint, BEHAMS, eynbetapol, eynhabilit}.
\section{Definitions}
Here we consider the 1-Hermitean matrix model with polynomial potential:
\begin{eqnarray}\label{defZ}
Z_N&:=& \int_{H_N} dM\, {\mathbf e}^{-N{\rm tr}\, V(M)}\cr
&=& \int_{{\mathbb{R}}^N} dx_1\dots dx_N\,\, \left(\Delta(x_1,\dots,x_N)\right)^2 \,\, \prod_{i=1}^N {\mathbf e}^{-NV(x_i)}
\end{eqnarray}
where $\Delta(x_1,\dots,x_N):=\prod_{i>j} (x_i-x_j)$, and the $x_i$'s are the eigenvalues of the matrix $M$,
and $V(x)$ is a polynomial called the potential:
\begin{equation}\label{defV}
V(x) = \sum_{k=0}^{\deg V} g_k x^k
\end{equation}
\begin{remark}
All the calculations which are presented below, can be extended to a more general setting, with no big fundamental changes:
- one can consider $V'(x)$ any rational fraction \cite{BEHsemiclas} instead of polynomial, in particular one can add logarithmic terms to the potential $V(x)$.
- one can consider arbitrary paths (or homology class of paths) of integrations $\Gamma^N$ insteaf of ${\mathbb{R}}^N$,
in particular finite segments \cite{marcopath} ...
- one can study non hermitean matrix models \cite{eynbetapol}, where the Vandermonde $\Delta^2$ is replaced by $\Delta^\beta$ where $\beta=1,2,4$.
- one can consider multi-matrix models, in particular 2-matrix model \cite{BEHAMS, eynchain, eynchaint}.
\end{remark}
\section{Orthogonal polynomials}
Consider the family of monic polynomials $p_n(x)=x^n + O(x^{n-1})$, defined by the orthogonality relation:
\begin{equation}
\int_{{\mathbb{R}}} p_n(x) p_m(x) {\mathbf e}^{-NV(x)} dx = h_n \delta_{nm}
\end{equation}
It is well known that the partition function is given by \cite{Mehta}:
\begin{equation}
Z_N = N! \prod_{n=0}^{N-1} h_n
\end{equation}
Such an orthogonal family always exists if the integration path is ${\mathbb{R}}$ or a subset of ${\mathbb{R}}$, and if the potential is a real polynomial.
In the more general setting, the orthogonal polynomials ``nearly always'' exist (for arbitrary potentials, the set of paths for which they don't exist is enumerable).
\medskip
We define the kernel:
\begin{equation}
K(x,y):=\sum_{n=0}^{N-1} {p_n(x) p_n(y)\over h_n}
\end{equation}
One has the following usefull theorems:
\begin{theorem} Dyson's theorem \cite{thDyson}:
any correlation function of eigenvalues, can be written in terms of the kernel $K$:
\begin{equation}
\rho(\l_1,\dots, \l_k) = \det(K(\l_i,\l_j))
\end{equation}
\end{theorem}
Thus, if one knows the orthogonal polynomials, then one knows all the correlation functions.
\begin{theorem} Christoffel-Darboux theorem \cite{Mehta, Szego}:
The kernel $K(x,y)$ can be written:
\begin{equation}
K(x,y) = \gamma_N\,{p_N(x)p_{N-1}(y)-p_N(y)p_{N-1}(x)\over x-y}
\end{equation}
\end{theorem}
Thus, if one knows the polynomials $p_N$ and $p_{N-1}$, then one knows all the correlation functions.
\medskip
Our goal now, is to find large $N$ ''strong'' asymptotics for $p_N$ and $p_{N-1}$, in order to have the large $N$ behaviours of any correlation functions.
\medskip
{\bf Notation:}
we define the wave functions:
\begin{equation}
\psi_n(x) := {1\over \sqrt{h_n}}\, p_n(x)\, {\mathbf e}^{-{N\over 2}V(x)}
\end{equation}
they are orthonormal:
\begin{equation}
\int \psi_n(x)\psi_m(x) = \delta_{nm}
\end{equation}
\section{Differential equations and integrability}
It can be proven that $(\psi_n,\psi_{n-1})$ obey a differential equation of the form \cite{bonan, BlIt, Mehta, TW2, BEHtauiso}:
\begin{equation}
-{1\over N} \,{\partial \over \partial x} \pmatrix{\psi_n(x) \cr \psi_{n-1}(x)} = {{\cal D}}_n(x)\,\pmatrix{\psi_n(x) \cr \psi_{n-1}(x)}
\end{equation}
where ${\cal D}_n(x)$ is a $2\times 2$ matrix, whose coefficients are polynomial in $x$, of degree at most $\deg V'$.
(In case $V'$ is a rational function, then ${\cal D}$ is a rational function with the same poles).
\medskip
$(\psi_n,\psi_{n-1})$ also obeys differential equations with respect to the parameters of the model \cite{BlIt, BEHtauiso}, i.e. the coupling constants, i.e. the
$g_k$'s defined in \ref{defV}:
\begin{equation}
{1\over N} \,{\partial \over \partial g_k} \pmatrix{\psi_n(x) \cr \psi_{n-1}(x)} ={\cal U}_{n,k}(x)\,\pmatrix{\psi_n(x) \cr \psi_{n-1}(x)}
\end{equation}
where ${\cal U}_{n,k}(x)$ is a $2\times 2$ matrix, whose coefficients are polynomial in $x$, of degree at most $k$.
\medskip
It is also possible to find some discrete recursion relation in $n$ (see \cite{BEHtauiso}).
\medskip
The compatibility of these differential systems, i.e. ${\partial\over \partial x}{\partial\over \partial g_k}={\partial\over \partial g_k}{\partial\over \partial x}$,
${\partial\over \partial g_j}{\partial\over \partial g_k}={\partial\over \partial g_k}{\partial\over \partial g_j}$, as well as compatibility with the discrete recursion, imply {\bf integrability},
and allows to define a $\tau$-function \cite{MiwaJimbo, BEHtauiso}.
\bigskip
We define the spectral curve as the locus of eigenvalues of ${\cal D}_n(x)$:
\begin{equation}
E_n(x,y):=\det(y{\bf 1} - {\cal D}_n(x))
\end{equation}
\begin{remark}\rm\small
In the 1-hermitean-matrix model, ${\cal D}_n$ is a $2\times 2$ matrix, and thus $\deg_y E_n(x,y)=2$, i.e. the curve $E_n(x,y)=0$ is an {\bf hyperelliptical curve}.
In other matrix models, one gets algebraic curves which are not hyperelliptical.
\end{remark}
\begin{remark}\rm\small
What we will se below, is that the curve $E_N(x,y)$ has a large $N$ limit $E(x,y)$, which is also an hyperelliptical curve.
In general, the matrix ${\cal D}_N(x)$ has no large $N$ limit.
\end{remark}
\section{Riemann-Hilbert problems and isomonodromies}
The $2\times2$ system ${\cal D}_N$ has $2$ independent solutions:
\begin{equation}
-{1\over N} \,{\partial \over \partial x} \pmatrix{\psi_n(x) \cr \psi_{n-1}(x)} = {{\cal D}}_n(x)\,\pmatrix{\psi_n(x) \cr \psi_{n-1}(x)}
\quad , \quad
-{1\over N} \,{\partial \over \partial x} \pmatrix{\phi_n(x) \cr \phi_{n-1}(x)} = {{\cal D}}_n(x)\,\pmatrix{\phi_n(x) \cr \phi_{n-1}(x)}
\end{equation}
where the wronskian is non-vanishing: $\det\pmatrix{\psi_n(x) & \phi_n(x) \cr \psi_{n-1}(x) & \phi_{n-1}(x)}\neq 0$.
We define the matrix of fundamental solutions:
\begin{equation}
\Psi_n(x):=\pmatrix{\psi_n(x) & \phi_n(x) \cr \psi_{n-1}(x) & \phi_{n-1}(x)}
\end{equation}
it obeys the same differential equation:
\begin{equation}
-{1\over N} \,{\partial \over \partial x} \Psi_n(x) = {{\cal D}}_n(x)\,\Psi_n(x)
\end{equation}
\medskip
Here, the second solution can be constructed explicitely:
\begin{equation}
\phi_n(x) = {\mathbf e}^{+{N\over 2}V(x)}\,\int {dx'\over x-x'}\,\psi_n(x') {\mathbf e}^{-{N\over 2}V(x')}
\end{equation}
Notice that $\phi_n(x)$ is discontinuous along the integration path of $x'$ (i.e. the real axis in the most simple case), the discontinuity is simply $2i\pi \psi_n(x)$.
In terms of fundamental solutions, one has the jump relation:
\begin{equation}\label{JumpRH}
\Psi_n(x+i0) = \Psi_n(x-i0)\,\pmatrix{1 & 2i\pi \cr 0 & 1}
\end{equation}
Finding an invertible piecewise analytical matrix, with given large $x$ behaviours, with given jumps on the borders between analytical domains, is called a {\bf Riemann--Hilbert problem} \cite{BlIt, BlIt1, BEHRH}.
It is known that the Riemann--Hilbert problem has a unique solution, and that if two R-H problems differ by $\epsilon$ (i.e. the difference between jumps and behaviours at $\infty$ is bounded
by $\epsilon$), then the two solutions differ by at most $\epsilon$ (roughly speeking, harmonic functions have their extremum on the boundaries).
Thus, this approach can be used \cite{BlIt, dkmvz, dkmvz2} in order to find large $N$ asymptotics of orthogonal polynomials:
The authors of \cite{BlIt} considered a guess for the asymptotics, which satisfies another R-H problem, which differs from this one by $O(1/N)$.
\bigskip
Notice that the jump matrix in \ref{JumpRH} is independent of $x$, of $n$ and of the potential, it is a constant.
The jump matrix is also called a monodromy, and the fact that the monodromy is a constant, is called {\bf isomonodromy} property \cite{MiwaJimbo}.
Consider an invertible, piecewise analytical matrix $\Psi_n(x)$, with appropriate behaviours\footnote{The behaviours at $\infty$ are far beyond the scope of this short lecture.
They are easily obtained by computing $\phi_n(x)$ by saddle point method at large $x$.} at $\infty$,
which satisfies \ref{JumpRH}, then, it is clear that the matrix
$-{1\over N} \Psi_n'(x) (\Psi_n(x))^{-1}$, has no discontinuity, and given its behaviour at $\infty$, it must be a polynomial.
Thus, we can prove that $\Psi_n(x)$ must satisfy a differential system ${\cal D}_n(x)$ with polynomial coefficients.
Similarly, the fact that the monodromy is independent of $g_k$ and $n$ implies the deformation equations, as well as the discrete recursion relations.
Thus, the isomonodromy property, implies the existence of compatible differential systems, and integrability \cite{BI, FIK, stringIts, MiwaJimbo, TW2, BEHtauiso}.
\section{WKB--like asymptotics and spectral curve}
\label{secasympWKBformal}
Let us look for a formal solution of the form:
\begin{equation}\label{asympWKBformal}
\Psi_N(x) = A_N(x) \, {\mathbf e}^{-N T(x)} B_N
\end{equation}
where $T(x)={\rm diag}(T_1(x),T_2(x))$ is a diagonal matrix, and $B_N$ is independent of $x$.
The differential system ${\cal D}_N(x)$ is such that:
\begin{eqnarray}
{\cal D}_N(x) &=& -{1\over N}\Psi_N'\Psi_N^{-1} = A_N(x) T'(x) A_N^{-1}(x) - {1\over N} A'_N(x) A_N^{-1}(x) \cr
& =& A_N(x) T'(x) A_N^{-1}(x) + O({1\over N})
\end{eqnarray}
this means, that, under the asumption that $A_N(x)$ has a large $N$ limit $A(x)$, $T'_1(x)$ and $T'_2(x)$ are the large $N$ limits of the eigenvalues of ${\cal D}_N(x)$.
With such an hypothesis, one gets for the orthogonal polynomials:
\begin{equation}
\psi_N(x) \sim A_{11}{\mathbf e}^{-NT_1(x)} B_{1,1} + A_{12}{\mathbf e}^{-NT_2(x)} B_{2,1}
\end{equation}
We are now going to show how to derive such a formula.
\section{Orthogonal polynomials as matrix integrals}
\subsection{Heine's formula}
\begin{theorem} Heine's theorem \cite{Szego}.
The orthogonal polynomials $p-n(x)$ are given by:
\begin{eqnarray}
p_n(\xi) &=& {\int dx_1\dots dx_N\,\, \prod_{i=1}^N (\xi-x_i)\,\, (\Delta(x_1,\dots,x_N))^2\,\, \prod_{i=1}^N {\mathbf e}^{-NV(x_i)}\over
\int dx_1\dots dx_N\,\, (\Delta(x_1,\dots,x_N))^2\,\, \prod_{i=1}^N {\mathbf e}^{-NV(x_i)}} \cr
&=& \left<\det(\xi{\bf 1} - M)\right>
\end{eqnarray}
\end{theorem}
i.e. the orthogonal polynomial is the average of the characteristic polynomial of the random matrix.
Thus, we can define the orthogonal polynomials as matrix integrals, similar to the partition function $Z$ define in \ref{defZ}.
\subsection{Another matrix model}
Define the potential:
\begin{equation}
V_h(x):=V(x)-h\ln{(\xi-x)}
\end{equation}
and the partition function:
\begin{equation}
Z_n(h,T):={\mathbf e}^{-{n^2\over T^2}F_n(h,T)}
:=\int dx_1\dots dx_n\,\, (\Delta(x_1,\dots,x_n))^2\,\, \prod_{i=1}^n {\mathbf e}^{-{n\over T}V_h(x_i)}
\end{equation}
i.e. $Z_N(0,1)=Z$ is our initial partition function.
Heine's formula reads:
\begin{equation}
p_n(\xi) = {Z_n({1\over N},{n\over N})\over Z_n(0,{n\over N})}
= {\mathbf e}^{-N^2(F_n({1\over N},{n\over N})-F_n(0,{n\over N}))}
\end{equation}
The idea, is to perform a Taylor expansion in $h$ close to $0$ and $T$ close to $1$.
\subsubsection{Taylor expansion}
We are interested in $n=N$ and $n=N-1$, thus $T={n\over N}=1+{n-N\over N}=1+O(1/N)$ and $h=0$ or $h=1/N$, i.e. $h=O(1/N)$:
\begin{equation}
T=1+O(1/N)
\quad , \quad
h=O(1/N)
\end{equation}
Roughly speaking:
\begin{eqnarray}\label{asymp1}
p_n(\xi)
&\sim& {\mathbf e}^{-N^2\left( h {\partial F\over \partial h}+(T-1)h{\partial^2 F\over \partial h\partial T}+{h^2\over 2}{\partial^2 F\over \partial h^2}+O(1/N^3)\right)} \cr
&\sim& {\mathbf e}^{-N {\partial F\over \partial h}}\,{\mathbf e}^{-(n-N){\partial^2 F\over \partial h\partial T}}\,{\mathbf e}^{-{1\over 2}{\partial^2 F\over \partial h^2}}\,\,(1+O(1/N))
\end{eqnarray}
where all the derivatives are computed at $T=1$ and $h=0$.
\subsubsection{Topological expansion}
Imagine that $F_n$ has a $1/n^2$ expansion of the form:
\begin{equation}
F = F^{(0)} + {1\over n^2} F^{(1)} + O({1\over n^3})
\end{equation}
where all $F^{(0)}$ and $F^{(1)}$ are analytical functions of $T$ and $h$, than one needs only $F^{(0)}$ in order to compute the asymptotics \ref{asymp1}.
\smallskip
Actualy, that hypothesis is not always true. It is wrong in the so called ''mutlicut'' case.
But it can be adapted in that case, we will come back to it in section \ref{sectmulticutasymp}.
For the moment, let us conduct the calculation only with $F^{(0)}$.
\section{Computation of derivatives of $F^{(0)}$}
We have defined:
\begin{equation}
Z_n(h,T)={\mathbf e}^{-{n^2\over T^2}F_n(h,T)}
= \int dM_{n\times n} {\mathbf e}^{-{n\over T}{\rm tr}\, V(M)}\, {\mathbf e}^{h{n\over T}\ln{(\xi-M)}}
\end{equation}
this implies that:
\begin{equation}
-{n^2\over T^2} {\partial F_n\over \partial h} = \left<{n\over T}{\rm tr}\, \ln{(\xi-M)}\right>_{V_h}
\end{equation}
i.e.
\begin{eqnarray}
{\partial F_n\over \partial h}
&=& -{T\over n}\left<{\rm tr}\, \ln{(\xi-M)}\right>_{V_h} \cr
\end{eqnarray}
It is a primitive of $-{T\over n}\left<{\rm tr}\, \ln{(x-M)}\right>_{V_h}$, which behaves as $-{T\over n}\ln{x}+O(1/x)$ at large $x$.
Therefore, we define the resolvent $W(x)$:
\begin{equation}\label{defWVh}
W(x):={T\over n}\left<{\rm tr}\, {1\over x-M}\right>_{V_h}
\end{equation}
Notice that it depends on $\xi$ through the potential $V_h$, i.e. through the average $<.>$.
And we define the effective potential:
\begin{equation}
{V_{\rm eff}}(x)= V_h(x)-2T\ln{x}-2\int_{\infty}^x (W(x')-{T\over x'}) dx'
\end{equation}
which is a primitive of $V'_h(x)-2W(x)$.
Thus , we have:
\begin{equation}
{\partial F_n\over \partial h} = {1\over 2}\left({V_{\rm eff}}(\xi)-V_h(\xi)\right)
\end{equation}
We also introduce:
\begin{equation}\label{defOm}
\Omega(x):={\partial W(x)\over \partial T}
\quad , \quad
\ln{\Lambda(x)}:=\ln{x}+\int_{\infty}^x(\Omega(x')-{1\over x'})dx' = -{1\over 2}{\partial \over \partial T}V_{\rm eff}(x)
\end{equation}
\begin{equation}\label{defH}
H(x,\xi):={\partial W(x)\over \partial h}
\quad , \quad
\ln{H(\xi)}:=\int_{\infty}^\xi H(x',\xi)dx'
\end{equation}
i.e.
\begin{equation}
{\partial^2 F_n\over \partial h^2} = -\ln{H(\xi)}
\quad , \quad
{\partial^2 F_n\over \partial h\partial T} = -\ln{\Lambda(\xi)}
\end{equation}
With these notations, the asymptotics are:
\begin{equation}
\psi_n(\xi)
\sim
\sqrt{H(\xi)}\,\,\left(\Lambda(\xi)\right)^{n-N}\,\,{\mathbf e}^{-{N\over 2}{V_{\rm eff}}(\xi)}\,\,(1+O(1/N))
\end{equation}
Now, we are going to compute $W$, $\Lambda$, $H$, etc, in terms of geometric properties of an hyperelliptical curve.
\begin{remark}\rm\small
This is so far only a sketch of the derivation, valid only in the 1-cut case.
In general, $F_n$ has no $1/n^2$ expansion, and that case will be addressed in section \ref{sectmulticutasymp}.
\end{remark}
\begin{remark}\rm\small
These asymptoics are of the form of \ref{asympWKBformal} in section.\ref{secasympWKBformal}, and thus,
${1\over 2}V'(x)-W(x)$ is the limit of the eigenvalues of ${\cal D}_N(x)$.
\end{remark}
\section{Saddle point method}
There exists many ways of computing the resolvent and its derivatives with respect to $h$, $T$, or other parameters.
The loop equation method is a very good method, but there is not enough time to present it here.
There are several saddle-point methods, which all coincide to leading order.
We are going to present one of them, very intuitive, but not very rigorous on a mathematical ground, and not very appropriate for next to leading computations.
However, it gives the correct answer to leading order.
\bigskip
Write:
\begin{equation}
Z_n(h,T)={\mathbf e}^{-{n^2\over T^2}F_n(h,T)}=\int dx_1\dots dx_n {\mathbf e}^{-{n^2\over T^2}{\cal S}(x_1,\dots,x_n)}
\end{equation}
where the action is:
\begin{equation}
{\cal S}(x_1,\dots,x_n):={T\over n}\sum_{i=1}^n V_h(x_i) -2{T^2\over n^2}\sum_{i>j} \ln{(x_i-x_j)}
\end{equation}
The saddle point method consists in finding configurations $x_i=\overline{x}_i$ where ${\cal S}$ is extremal,
i.e.
\begin{equation}
\forall i=1,\dots n, \qquad \left.{\partial {\cal S}\over \partial x_i}\right|_{x_j=\overline{x}_j}=0
\end{equation}
i.e., we have the {\bf saddle point equation}:
\begin{equation}
\forall i=1,\dots n, \qquad
V'_h(\overline{x}_i) = 2{T\over n}\sum_{j\neq i} {1\over \overline{x}_i-\overline{x}_j}
\end{equation}
The saddle point approximation\footnote{The validity of the saddle point approximation is not proven rigorously for large number of variables.
But here, we have many evidences that we can trust the results it gives. The asymptotics we are going to find have been proven rigorously by other methods.
Basicaly, it is expected to work because the number of variables $n$ is small compared to the large parameter $n^2$ in the action.}
consists in writting:
\begin{equation}
Z_n(h,T) \sim {1\over \sqrt{\det\left(\partial {\cal S}\over \partial x_i\partial x_j\right)}}\,\,{\mathbf e}^{-{n^2\over T^2}{\cal S}(\overline{x}_1,\dots,\overline{x}_n)}\,\,(1+O(1/n))
\end{equation}
where $(\overline{x}_1,\dots,\overline{x}_n)$ is the solution of the saddlepoint equation which minimizes $\Re {\cal S}$.
\begin{remark}\rm\small
The saddle point equation may have more than one minimal solution $(\overline{x})$.
- in particular if $\xi\in {\mathbb{R}}$, there are two solutions, complex conjugate of each other.
- in the multicut case, there are many saddlepoints with near-minimal action.
In all cases, one needs to sum over all the saddle points.
Let us call $\{\overline{x}\}_I$, the collection of saddle points. We have:
\begin{equation}
Z_n \sim \sum_I {C_I\over \sqrt{{\cal S}''(\{\overline{x}\}_I)}}\,\,{\mathbf e}^{-{n^2\over T^2}{\cal S}(\{\overline{x}\}_I)}\,\,(1+O(1/n))
\end{equation}
Each saddle point $\{\overline{x}\}_I$ corresponds to a particular minimal $n$-dimensional integration path in ${\mathbb{C}}^n$,noted $\Gamma_I$,
and the coefficients $C_I\in {\mathbb{Z}}$ are such that:
\begin{equation}
{\mathbb{R}}^n = \sum_I C_I \Gamma_I
\end{equation}
\end{remark}
\section{Solution of the saddlepoint equation}
We recall the saddle point equation:
\begin{equation}\label{sadlepointxbar}
\forall i=1,\dots n, \qquad
V'_h(\overline{x}_i) = 2{T\over n}\sum_{j\neq i} {1\over \overline{x}_i-\overline{x}_j}
\end{equation}
We introduce the function:
\begin{equation}\label{defom}
\omega(x):={T\over n}\sum_{j=1}^n {1\over x-\overline{x}_j}
\end{equation}
in the large $N$ limit, $\omega(x)$ is expected to tend toward the resolvent, at least in the case there is only one minimal saddle point.
Indeed, the $\overline{x}_i$'s are the position of the eigenvalues minimizing the action, i.e. the most probable positions of eigenvalues of $M$, and thus
\ref{defom} should be close to ${T\over n}{\rm tr}\, {1\over x-M}$.
\subsection{Algebraic method}
Compute $\omega^2(x)+{T\over n}\omega'(x)$, you find:
\begin{eqnarray}
\omega^2(x)+{T\over n}\omega'(x)
&=& {T^2\over n^2} \sum_{i=1}^n \sum_{j=1}^n {1\over (x-\overline{x}_i)(x-\overline{x}_j)} - {T^2\over n^2} \sum_{i=1}^n {1\over (x-\overline{x}_i)^2} \cr
&=&{T^2\over n^2} \sum_{i\neq j}^n {1\over (x-\overline{x}_i)(x-\overline{x}_j)} \cr
&=&{T^2\over n^2} \sum_{i\neq j}^n \left({1\over x-\overline{x}_i}-{1\over x-\overline{x}_j}\right)\,{1\over \overline{x}_i-\overline{x}_j} \cr
&=&{2T^2\over n^2} \sum_{i=1}^n {1\over x-\overline{x}_i}\,\sum_{j\neq i}^n {1\over \overline{x}_i-\overline{x}_j} \cr
&=&{T\over n} \sum_{i=1}^n {V'_h(\overline{x}_i)\over x-\overline{x}_i} \cr
&=&{T\over n} \sum_{i=1}^n {V'_h(x)-(V'_h(x)-V'_h(\overline{x}_i))\over x-\overline{x}_i} \cr
&=& V'_h(x)\omega(x)-{T\over n} \sum_{i=1}^n {V'_h(x)-V'_h(\overline{x}_i)\over x-\overline{x}_i} \cr
&=&(V'(x)-{h\over x-\xi})\omega(x)-{T\over n} \sum_{i=1}^n {V'(x)-V'(\overline{x}_i)\over x-\overline{x}_i} + h{\omega(\xi)\over x-\xi}\cr
\end{eqnarray}
i.e. we get the equation:
\begin{equation}
\omega^2(x)+{T\over n}\omega'(x)
= V'(x)\omega(x)- P(x) - h{\omega(x)-\omega(\xi)\over x-\xi}
\end{equation}
where $P(x):={T\over n} \sum_{i=1}^n {V'(x)-V'(\overline{x}_i)\over x-\overline{x}_i}$ is a polynomial in $x$ of degree at most $\deg V-2$.
In the large $N$ limit, if we assume\footnote{It is possible to do the calculation without droping the $1/N$ term. One gets a Ricati equation, which is equivalent to a Schroedinger equation.
If one is interested in a large N limit for the resolvent, the asymptotic analysis of that Schroedinger equation (Stokes phenomenon) gives, to leading order, the same thing as when one drops the $1/N$ term. If one whishes to go beyond leading order, many subtleties occur.} that we can drop the $1/N W'(x)$ term, we get an algebraic equation, which is in this case
an hyperelliptical curve.
In particular at $h=0$ and $T=1$:
\begin{equation}
\omega(x) = {1\over 2}\left(V'(x)-\sqrt{V'^2(x)-4P(x)}\right)
\end{equation}
The properties of this algebraic equation have been studied by many authors, and the $T$ and $h$ derivatives, as well as other derivatives were computed in various works.
Here, we briefly sketch the method.
See \cite{kriechever, KazMar, eynmultimat} for more details.
\subsection{Linear saddle point equation}
In the large $N$ limit, both the average density of eigenvalues, and the density of $\overline{x}$ tend towards a continuous compact support density $\overline{\rho}(x)$.
In that limit, the resolvent is given by:
\begin{equation}
\omega(x) = T \, \int_{{\rm supp}\,\,\overline{\rho}}{\overline{\rho}(x')\,dx'\over x-x'}
\end{equation}
i.e.
\begin{equation}
\forall x\in{\rm supp}\,\,\overline{\rho}, \qquad
\overline{\rho}(x) = -{1\over 2i\pi T}(\omega(x+i0)-\omega(x-i0))
\end{equation}
and the saddle point equation \ref{sadlepointxbar}, becomes a linear functional equation:
\begin{equation}\label{sadlepointequrho}
\forall x\in{\rm supp}\,\,\overline{\rho}, \qquad
V'_h(x) = \omega(x+i0)+\omega(x-i0)
\end{equation}
The advantage of that equation, is that it is linear in $\omega$, and thus in $\overline{\rho}$.
The nonlinearity is hidden in ${\rm supp}\,\,\overline{\rho}$.
\subsubsection{Example: One cut}
If the support of $\overline{\rho}$ is a single interval:
\begin{equation}
{\rm supp}\,\,\overline{\rho} = [a,b]\quad , \quad a<b
\end{equation}
then, look for a solution of the form:
\begin{equation}
\omega(x) = {1\over 2}\left(V'_h(x) - M_h(x)\sqrt{(x-a)(x-b)}\right)
\end{equation}
The saddle point equation \ref{sadlepointequrho} implies that $M_h(x+i0)=M_h(x-i0)$, i.e. $M_h$ has no discontinuities,
and because of its large $x$ behaviour, as well as its behaviours near $\xi$, it must be a rational function of $x$, with a simple pole at $x=\xi$.
$M_h$, $a$ and $b$ are entirely determined by their behaviours near poles, i.e.:
\begin{equation}
\omega(x) \mathop\sim_{x\to\infty} {T\over x}
\end{equation}
\begin{equation}
\omega(x) \mathop\sim_{x\to\xi} {\rm regular}\quad \longrightarrow M_h(x)\mathop\sim_{x\to\xi} -{h\over x-\xi}
\end{equation}
Thus, one may write:
\begin{equation}
\omega(x) = {1\over 2}\left(V'(x) - M(x)\sqrt{(x-a)(x-b)} - {h\over x-\xi}\left(1-{\sqrt{(x-a)(x-b)}\over \sqrt{(\xi-a)(\xi-b)}}\right)\right)
\end{equation}
where $M(x)$ is now a polynomial (which still depends on $h$ and $T$ and the other parameters), it is such that:
\begin{equation}
M(x) = \mathop{\rm Pol}_{x\to\infty}\,\, {V'(x)\over \sqrt{(x-a)(x-b)}}
\end{equation}
The density is thus:
\begin{equation}
\overline{\rho}(x) ={1\over 2\pi T}M_h(x)\sqrt{(x-a)(b-x)}
\quad , \quad
{\rm supp}\,\,\overline{\rho} = [a,b]
\end{equation}
$$\begin{array}{r}
{\epsfxsize 12cm\epsffile{curve.eps}}
\end{array}$$
\subsubsection{Multi-cut solution}
Let us assume that the support of $\overline{\rho}$ is made of $s$ separated intervals:
\begin{equation}
{\rm supp}\,\,\overline{\rho} = \cup_{i=1}^s [a_i,b_i]
\end{equation}
then, for any sequence of integers $n_1,n_2,\dots, n_s$ such that $\sum_{i_1}^s n_i=n$, it is possible
to find a solution for the saddle point equation.
That solution obeys \ref{sadlepointequrho}, as well as the conditions:
\begin{equation}
\forall i=1,\dots,s
\quad , \quad
\int_{a_i}^{b_i} \rho(x) dx = T {n_i\over N}
\end{equation}
The solution of the saddle point equation can be described as follows:
let the polynomial $\sigma(x)$ be defined as:
\begin{equation}
\sigma(x):=\prod_{i=1}^s (x-a_i)(x-b_i)
\end{equation}
The solution of the saddle point equation \ref{sadlepointequrho}, is of the form:
\begin{equation}
\omega(x) = {1\over 2}\left(V'_h(x) - M_h(x)\sqrt{\sigma(x)}\right)
\end{equation}
where $M_h(x)$ is a rational function of $x$, with a simple pole at $x=\xi$.
$M_h$, and $\sigma(x)$ are entirely determined by their behaviours near poles, i.e.:
\begin{equation}
\omega(x) \mathop\sim_{x\to\infty} {T\over x}
\end{equation}
\begin{equation}
\omega(x) \mathop\sim_{x\to\xi} {\rm regular}\quad \longrightarrow M_h(x)\mathop\sim_{x\to\xi} -{h\over x-\xi}
\end{equation}
and by the conditions that:
\begin{equation}
\forall i=1,\dots,s \quad , \quad \int_{a_i}^{b_{i}} M_h(x)\sqrt{\sigma(x)} dx = 2i\pi T{n_i\over n}
\end{equation}
\subsection{Algebraic geometry: hyperelliptical curves}
Consider the curve given by:
\begin{equation}
\omega(x) = {1\over 2}\left(V_h'(x) - M_h(x)\sqrt{(x-a)(x-b)}\right)
\end{equation}
It has two sheets, i.e. for each $x$, there are two values of $\omega(x)$, depending on the choice of sign of the square-root.
- In the physical sheet (choice $+\sqrt{}$), it behaves near $\infty$ like $\omega(x)\sim T/x$
- In the second sheet (choice $-\sqrt{}$), it behaves near $\infty$ like $\omega(x)\sim V'_h(x)$
Since $\omega(x)$ is a complex valued, analytical function of a cmplex variable $x$,
the curve can be thought of as the embedding of a Riemann surface into ${\mathbb{C}}\times {\mathbb{C}}$.
I.e. we have a Riemann surface ${\cal E}$, with two (monovalued) functions defined on it:
$p\in{\cal E}\, , \,\, \to x(p)\in{\mathbb{C}}$, and $p\in{\cal E}\, , \,\, \to \omega(p)\in{\mathbb{C}}$.
For each $x$, there are two $p\in{\cal E}$ such that $x(p)=x$, and this is why there are two values of $\omega(x)$.
Each of the two sheets is homeomorphic to the complex plane, cut along the segments $[a_i,b_i]$, and the two sheets are glued together along the cuts.
The complex plane, plus its point at infinity, is the Riemann sphere.
Thus, our curve ${\cal E}$, is obtained by taking two Riemann spheres, glued together along $s$ circles.
It is a genus $s-1$ surface.
$$\begin{array}{r}
{\epsfxsize 14cm\epsffile{sheetg.eps}}
\end{array}$$
$$\begin{array}{r}
{\epsfysize 4cm\epsffile{surfpatate.eps}}
\end{array}$$
\subsection{Genus zero case (one cut)}
If the curve as genus zero, it is homeomorphic to the Riemann sphere ${\cal E}={\mathbb{C}}$.
One can always choose a rational parametrization:
\begin{equation}
x(p)={a+b\over 2}+\gamma(p+{1/p})
\quad , \quad \gamma={b-a\over 4}
\end{equation}
\begin{equation}
\sqrt{(x-a)(x-b)}=\gamma(p-1/p)
\end{equation}
so that $\omega$ is a rational function of $p$.
That representation maps the physical sheet onto the exterior of the unit circle, and the second sheet onto the interior of the unit circle.
The unit circle is the image of the two sides of the cut $[a,b]$, and the branchpoints $[a,b]$ are maped to $-1$ and $+1$.
Changing the sign of the square root is equivalent to changing $p\to 1/p$.
The branch points are of course the solutions of $dx/dp=0$, i.e. $dx(p)=0$:
\begin{equation}
dx(p) = \gamma\,\left(1-{1\over p^2}\right)\, dp
\quad , \quad
dx(p)=0\leftrightarrow p=\pm 1 \leftrightarrow x(p)=a,b
\end{equation}
There are two points at $\infty$, $p=\infty$ in the physical sheet, and $p=0$ in the second sheet.
$$\begin{array}{r}
{\epsfxsize 9cm\epsffile{sheetgenuszero.eps}}
\,\,
{\epsfxsize 7cm\epsffile{Egenuszero.eps}}
\end{array}$$
\bigskip
Since the resolvent $\omega(p)$ is a rational function of $p$, it is then entirely determined by its behaviour near its poles.
the poles are at $p=\infty$, $p=0$,
$p=p_\xi$ and $p=\overline{p}_\xi$ (the two points of ${\cal E}$ such that $x(p)=\xi$, such that $p_\xi$ is in the physical sheet, and $\overline{p}_\xi$ is in the second sheet):
The boundary conditions:
\begin{equation}\label{bndomgzero}
\left\{
\begin{array}{l}
\displaystyle \omega(p) \mathop\sim_{p\to\infty} {T\over x(p)} \cr
\displaystyle \omega(p) \mathop\sim_{p\to 0} V'(x(p))-{T\over x(p)}-{h\over x(p)} \cr
\displaystyle \omega(p) \mathop\sim_{p\to \overline{p}_\xi} -{h\over x(p)-\xi} \cr
\displaystyle \omega(p) \mathop\sim_{p\to p_\xi} {\rm regular} \cr
\end{array}
\right.
\end{equation}
\subsubsection{$T$ derivative}
Now, let us compute $\partial\omega(p)/\partial T$ at $x(p)$ fixed.
Eq. \ref{bndomgzero} becomes:
\begin{equation}
\left\{
\begin{array}{l}
\displaystyle {\partial \omega(p)\over \partial T} \mathop\sim_{p\to\infty} {1\over x(p)} \cr
\displaystyle {\partial \omega(p)\over \partial T} \mathop\sim_{p\to 0} -{1\over x(p)} \cr
\displaystyle {\partial \omega(p)\over \partial T} \mathop\sim_{p\to \overline{p}_\xi} {\rm regular} \cr
\displaystyle {\partial \omega(p)\over \partial T} \mathop\sim_{p\to p_\xi} {\rm regular} \cr
\end{array}
\right.
\end{equation}
Moreover, we know that $\omega(x)$ has a square-root behaviour near $a$ and $b$, in $\sqrt{(x-a)(x-b)}$, and $a$ and $b$ depend on $T$,
thus $\partial\omega/\partial T$ may behave in $((x-a)(x-b))^{-1/2}$ near $a$ and $b$, i.e. $\partial\omega/\partial T$ may have simple poles at $p=\pm 1$.
Finaly, $\partial\omega(p)/\partial T$, has simple poles at $p=1$ and $p=-1$, and vanishes at $p=0$ and $p=\infty$, the only possibility is:
\begin{equation}
\left.{\partial \omega(p)\over \partial T}\right|_{x(p)}={p\over \gamma (p^{2}-1)} = {1\over p}\, {dp\over dx}
\end{equation}
which is better written in terms of differential forms:
\begin{equation}
\left.{\partial \omega(p)\over \partial T}\right|_{x(p)}\, dx(p)= {dp\over p}=d\ln{p}
\end{equation}
the RHS is independent of the potential, it is universal.
With the notation \ref{defOm}, we have:
\begin{equation}
\Omega(p)dx(p)={dp\over p}
\quad , \quad
\Lambda(p)=\gamma p
\end{equation}
\subsubsection{$h$ derivative}
The $h$ derivative is computed in a very similar way.
\begin{equation}
\left\{
\begin{array}{l}
\displaystyle {\partial \omega(p)\over \partial h} \mathop\sim_{p\to\infty} O(p^{-2}) \cr
\displaystyle {\partial \omega(p)\over \partial h} \mathop\sim_{p\to 0} -{1\over x(p)} \cr
\displaystyle {\partial \omega(p)\over \partial h} \mathop\sim_{p\to \overline{p}_\xi} -{1\over x(p)-\xi} \cr
\displaystyle {\partial \omega(p)\over \partial h} \mathop\sim_{p\to p_\xi} {\rm regular} \cr
\end{array}
\right.
\end{equation}
implies that $\partial \omega/\partial h$ can have poles at $p=\pm 1$ and at $p=\overline{p}_\xi$, and vanishes at $p=0$.
The only possibility is:
\begin{equation}
\left.{\partial \omega(p)\over \partial h}\right|_{x(p)}={-p\,\overline{p}_\xi\over \gamma (p-\overline{p}_\xi)(p^{2}-1)}
\end{equation}
i.e.
\begin{equation}
\left.{\partial \omega(p)\over \partial h}\right|_{x(p)}\, dx(p) = {dp\over p}-{dp\over p-\overline{p}_\xi} = d\ln{p\over p-\overline{p}_\xi}
\end{equation}
which again is universal.
With the notation \ref{defH}, we have:
\begin{equation}
H(p,p_\xi)dx(p)={dp\over p}-{dp\over p-{1\over p_\xi}}
\quad , \quad
H(p_\xi) = \ln{\left({p_\xi\over p_\xi-\overline{p}_\xi}\right)} = -\ln{\left({1\over\gamma}\,{dx\over dp}(\xi)\right)}
\end{equation}
\subsection{Higher genus}
For general genus, the curve can be parametrized by $\theta$-functions.
Like rational functions for genus 0, $\theta$-functions are the building blocks of functions defined on a compact Riemann surface, and any such function is
entirely determined by its behaviour near its poles, as well as by its integrals around irreducible cycles.
All the previous paragraph can be extended to that case.
Let $\infty_+$ and $\infty_-$ be the points at infinity, i.e. the two poles of $x(p)$, with $\infty_+$ in the physical sheet and $\infty_-$ in the second sheet.
Let $p=p_\xi$ and $p=\overline{p}_\xi$ be the two points of ${\cal E}$ such that $x(p)=\xi$, and with $p_\xi$ in the physical sheet, and $\overline{p}_\xi$ in the second sheet.
The differential form $\omega(p) dx(p)$ is entirely determined by:
\begin{equation}\label{ompoles}
\left\{
\begin{array}{ll}
\displaystyle \omega(p)dx(p) \mathop\sim_{p\to \infty_+} T\,{dx(p)\over x(p)} & \displaystyle \quad , \quad \mathop{\rm Res\,}_{\infty_+} \omega(p)dx(p)=-T \cr
\displaystyle \omega(p)dx(p) \mathop\sim_{p\to \infty_-} dV(x(p)) - T{dx(p)\over x(p)}-h{dx(p)\over x(p)} & \displaystyle \quad , \quad \mathop{\rm Res\,}_{\infty_-} \omega(p)dx(p)=T+h \cr
\displaystyle \omega(p)dx(p) \mathop\sim_{p\to \overline{p}_\xi} -h{dx(p)\over x(p)-\xi} & \displaystyle \quad , \quad \mathop{\rm Res\,}_{\overline{p}_\xi} \omega(p)dx(p)=-h \cr
\displaystyle \omega(p)dx(p) \mathop\sim_{p\to p_\xi} {\rm regular} & \displaystyle \quad , \quad \mathop{\rm Res\,}_{p_\xi} \omega(p)dx(p)=0 \cr
\displaystyle \oint_{{\cal A}_i} \omega(p) dx(p) = T{n_i\over n} = {n_i\over N} & \cr
\end{array}
\right.
\end{equation}
Since $\partial\omega /\partial T,h $ can diverge at most like $(x-a_i)^{-1/2}$ near a branch point $a_i$, and $dx(p)$ has a zero at $a_i$, the differential form
$\partial \omega dx/\partial T,h$ has no pole at the branch points.
\subsection{Introduction to algebraic geometry}
We introduce some basic concepts of algebraic geometry. We refer the reader to \cite{Farkas, Fay} for instance.
\begin{theorem}
Given two points $q_1$ and $q_2$ on the Riemann surface ${\cal E}$, there exists a unique differential form $dS_{q_1,q_2}(p)$,
with only two simple poles, one at $p=q_1$ with residue $+1$ and one at $p=q_2$ with residue $-1$, and which is normalized on the ${\cal A}_i$ cycles, i.e.
\begin{equation}\label{defdS}
\left\{
\begin{array}{l}
\displaystyle \mathop{\rm Res\,}_{p\to q_1} dS_{q_1,q_2}(p)=+1 \cr
\displaystyle \mathop{\rm Res\,}_{p\to q_2} dS_{q_1,q_2}(p)=-1 \cr
\displaystyle \oint_{{\cal A}_i} dS_{q_1,q_2}(p) = 0 \cr
\end{array}
\right.
\end{equation}
$dS$ is called an ``abelian differential of the third kind''.
\end{theorem}
Starting from the behaviours near poles and irreducible cycles \ref{ompoles}, we easily find:
\begin{equation}
\Omega(p)dx(p)=\left.{\partial\omega(p) dx(p)\over \partial T}\right|_{x(p)} = -dS_{\infty_+,\infty_-}(p)
\end{equation}
\begin{equation}
H(p,p_\xi) dx(p) = \left.{\partial\omega(p) dx(p)\over \partial h}\right|_{x(p)} = -dS_{\overline{p}_\xi,\infty_-}(p) = dS_{p_\xi,\infty_+}(p)-d\ln{\left(x(p)-x(p_\xi)\right)}
\end{equation}
\begin{theorem}
On an algebraic curve of genus $g$, there exist exactly $g$ linearly independent ``holomorphic differential forms'' (i.e. with no poles), $du_i(p)$, $i=1,\dots, g$.
They can be chosen normalized as:
\begin{equation}
\oint_{{\cal A}_i} du_j(p)=\delta_{ij}
\end{equation}
\end{theorem}
For hyperelliptical surfaces, it is easy to see that if $L(x)$ is a polynomial of degree at most $g-1=s-2$, the differential form
${L(x)\over \sqrt{\prod_{i=1}^s (x-a_i)(x-b_i)}}dx$ is regular at $\infty$, at the branch points, and thus has no poles.
And there are $g$ linearly independent polynomials of degree at most $g-1$. The irreducible cycles ${\cal A}_i$ is a contour surrounding $[a_i,b_i]$ in the positive direction.
\begin{definition} The matrix of periods is defined by:
\begin{equation}
\tau_{ij}:=\oint_{{\cal B}_i} du_j(p)
\end{equation}
where the irreducible cycles ${\cal B}_i$ are chosen canonicaly conjugated to the ${\cal A}_i$, i.e. ${\cal A}_i\cap{\cal B}_j=\delta_{ij}$.
In our hyperelliptical case, we choose ${\cal B}_i$ as a contour crossing $[a_i,b_i]$ and $[a_s,b_s]$.
The matrix of periods is symmetric $\tau_{ij}=\tau_{ji}$, and its imaginary part is positive $\Im\tau_{ij}>0$.
It encodes the complex structure of the curve.
\end{definition}
The holomrphic forms naturaly define an embedding of the curve into ${\mathbb{C}}^g$:
\begin{definition}
Given a base point $q_0\in{\cal E}$, we define the Abel map:
\begin{eqnarray}
{\cal E} &\longrightarrow& {\mathbb{C}}^g \cr
p &\longrightarrow& {\vec u}(p) = (u_1(p),\dots,u_g(p)) \quad , \quad u_i(p):=\int_{q_0}^p du_i(p)
\end{eqnarray}
where the integration path is chosen so that it does not cross any ${\cal A}_i$ or ${\cal B}_i$.
\end{definition}
\begin{definition}
Given a symmetric matrix $\tau$ of dimension $g$, such that $\Im\tau_{ij}>0$, we define the $\theta$-function, from ${\mathbb{C}}^g\to {\mathbb{C}}$ by:
\begin{equation}\label{deftheta}
\theta(\vec{u},\tau) = \sum_{\vec{m}\in {\mathbb{Z}}^g} {\mathbf e}^{i\pi {\vec m}^t \tau\vec{m}}\,{\mathbf e}^{2i\pi {\vec m}^t\vec{u}}
\end{equation}
It is an even entire function.
For any $\vec{m}\in {\mathbb{Z}}^g$, it satisfies:
\begin{equation}
\theta(\vec{u}+\vec{m})=\theta(\vec{u})
\quad , \quad
\theta(\vec{u}+\tau\vec{m})={\mathbf e}^{-i\pi(2 {\vec m}^t\vec{u} + {\vec m}^t\tau \vec{m})}\, \theta(\vec{u})
\end{equation}
\end{definition}
\begin{definition} The theta function vanishes on a codimension $1$ submanifold of ${\mathbb{C}}^g$, in particular, it vanishes at the odd half periods:
\begin{equation}
\vec{z}={{\vec m}_1+\tau\, {\vec m}_2\over 2}
\,\, ,\,\,\,
{\vec m}_1\in {\mathbb{Z}}^g\, ,\,\, {\vec m}_2\in {\mathbb{Z}}^g \,\, , \,\,\, ({\vec m}_1^t{\vec m}_1)\in 2{\mathbb{Z}}+1
\,\,\,\longrightarrow\,\,
\theta(\vec{z})=0
\end{equation}
For a given such odd half-period, we define the characteristic $\vec{z}$ $\theta$-function:
\begin{equation}
\theta_{\vec{z}}(\vec{u}):={\mathbf e}^{i\pi m_2\vec{u}+}\,\theta(\vec{u}+\vec{z})
\end{equation}
so that:
\begin{equation}
\theta_{\vec{z}}(\vec{u}+\vec{m})= {\mathbf e}^{i\pi \vec{m}_2^t\vec{m}}\,\theta_{\vec{z}}(\vec{u})
\quad , \quad
\theta_{\vec{z}}(\vec{u}+\tau\vec{m})= {\mathbf e}^{-i\pi \vec{m}_1^t \vec{m}}\,{\mathbf e}^{-i\pi(2 {\vec m}^t\vec{u} + {\vec m}^t\tau \vec{m})}\, \theta_{\vec{z}}(\vec{u})
\end{equation}
and
\begin{equation}
\theta_{\vec{z}}(\vec{0})=0
\end{equation}
\end{definition}
\begin{definition}
Given two points $p,q$ in ${\cal E}$, as well as a basepoint $p_0\in{\cal E}$ and an odd half period $z$, we define the prime form $E(p,q)$:
\begin{equation}
E(p,q):={\theta_{\vec{z}}(\vec{u}(p)-\vec{u}(q))\over \sqrt{dh_{\vec{z}}(p) dh_{\vec{z}}(q)}}
\end{equation}
where $dh_{\vec{z}}(p)$ is the holomorphic form:
\begin{equation}
dh_{\vec{z}}(p):= \sum_{i=1}^g \left.{\partial \theta_{\vec{z}}(\vec{u})\over \partial u_i}\right|_{\vec{u}=\vec{0}}\, du_i(p)
\end{equation}
\end{definition}
\begin{theorem}
The abelian differentials can be written:
\begin{equation}
dS_{q_1,q_2}(p) = d\ln{E(p,q_1)\over E(p,q_2)}
\end{equation}
\end{theorem}
With these definitions, we have:
\begin{equation}
\Lambda(p) = \gamma\,{\theta_{\vec{z}}(\vec{u}(p)-\vec{u}(\infty_-))\over \theta_{\vec{z}}(\vec{u}(p)-\vec{u}(\infty_+))}
\quad , \quad
\gamma:=\mathop{\rm lim}_{p\to\infty_+} \,{x(p)\,\theta_{\vec{z}}(\vec{u}(p)-\vec{u}(\infty_+))\over \theta_{\vec{z}}(\vec{u}(\infty_+)-\vec{u}(\infty_-))}
\end{equation}
\begin{equation}
H(p_\xi)={\theta_{\vec{z}}(\vec{u}(p_\xi)-\vec{u}(\infty_-))\theta_{\vec{z}}(\vec{u}(\infty_+)-\vec{u}(\overline{p}_\xi))\over \theta_{\vec{z}}(\vec{u}(p_\xi)-\vec{u}(\overline{p}_\xi))\theta_{\vec{z}}(\vec{u}(\infty_+)-\vec{u}(\infty_-))}
= -\gamma\,{\theta_{\vec{z}}(\vec{u}(\infty_+)-\vec{u}(\infty_-))\over \theta_{\vec{z}}(\vec{u}(p_\xi)-\vec{u}(\infty_+))^2}\,{dh_{\vec{z}}(p_\xi)\over dx(p_\xi)}
\end{equation}
\section{Asymptotics of orthogonal polynomials}
\subsection{One-cut case}
In the one-cut case, (i.e. genus zero algebraic curve), and if $V$ is a real potential, there is only one dominant saddle point if $\xi\notin [a,b]$,
and two conjugated dominant saddle points if $x\in[a,b]$.
More generaly, there is a saddle point corresponding to each determination of $p_\xi$ such that $x(p_\xi)=\xi$.
i.e. $p_\xi$ and $\overline{p}_\xi=1/p_\xi$.
The dominant saddle point is the one such that $\Re(V_{\rm eff}(p_\xi)-V(\xi))$ is minimal.
The two cols have a contribution of the same order if:
\begin{equation}
\Re V_{\rm eff}(p_\xi) = \Re V_{\rm eff}(\overline{p}_\xi)
\end{equation}
i.e. if $\xi$ is such that:
\begin{equation}\label{defcutsVeff}
\Re \int_{\overline{p}_\xi}^{p_\xi} W(x)dx =0
\end{equation}
If the potential is real, it is easy to see that the set of points which satisfy \ref{defcutsVeff} is $[a,b]$,
in general, it is a curve in the complex plane, going from $a$ to $b$, we call it the cut $[a,b]$ (similar curves were studied in \cite{moore}).
Then we have:
\begin{itemize}
\item For $x\notin[a,b]$, we write $\xi={a+b\over 2} + \gamma (p_\xi+1/p_\xi)$, $\gamma={b-a\over 4}$:
\begin{equation}
p_n(\xi) \sim \,\sqrt{H(p_\xi)}\,\left(\Lambda(p_\xi)\right)^{n-N}\,{\mathbf e}^{-{N\over 2}(V_{\rm eff}(p_\xi)-V(\xi))} (1+O(1/N))
\end{equation}
i.e.
\begin{equation}
p_n(\xi) \sim \,\sqrt{\gamma\over x'(p_\xi)}\,\left(\gamma\, p_\xi\right)^{n-N}\,{\mathbf e}^{-{N\over 2}(V_{\rm eff}(p_\xi)-V(\xi))} (1+O(1/N))
\end{equation}
\item For $x\in[a,b]$, i.e. $p$ is on the unit circle $p={\mathbf e}^{i\phi}$, $\xi={a+b\over 2}+2\gamma\cos\phi$:
\begin{eqnarray}
p_n(\xi) &\sim& \,\sqrt{H(p_\xi)}\,\left(\Lambda(p_\xi)\right)^{n-N}\,{\mathbf e}^{-{N\over 2}(V_{\rm eff}(p_\xi)-V(\xi))} (1+O(1/N)) \cr
&& + \,\sqrt{H(\overline{p}_\xi)}\,\left(\Lambda(\overline{p}_\xi)\right)^{n-N}\,{\mathbf e}^{-{N\over 2}(V_{\rm eff}(\overline{p}_\xi)-V(\xi))} (1+O(1/N))
\end{eqnarray}
i.e.
\begin{equation}
p_n(\xi) \sim {\gamma^{n-N}\over \sqrt{2\sin\phi(\xi)}}\,2\cos{\left(N\pi\int_a^\xi \rho(x)dx - (n-N+{1\over 2}) \phi(\xi) + \alpha\right)} (1+O(1/N))
\end{equation}
i.e. we have an oscillatory behaviour
$$\begin{array}{r}
{\epsfxsize 10cm\epsffile{asymp.eps}}
\end{array}$$
\end{itemize}
\subsection{Multi-cut case}
\label{sectmulticutasymp}
In the multicut case, in addition to having saddle-points corresponding to both determinantions of $p_\xi$,
we have a saddle point for each filling fraction configuration $n_1,\dots, n_s$ with $\sum_{i=1}^s n_i=n$.
We write:
\begin{equation}
\epsilon_i = {n_i\over N}
\end{equation}
The saddle point corresponding to filling fractions which differ by a few units, contribute to the same order,
and thus cannot be neglected. One has to consider the sommation over filling fractions \cite{BDE}.
Thus, one has to consider the action of a saddle point as a function of the filling fractions.
We leave as an exercise for the reader to prove that the derivatives of $F$ are given by:
\begin{equation}
{\partial F\over \partial \epsilon_i} = -\oint_{{\cal B}_i} W(x)dx
\end{equation}
and:
\begin{equation}
{\partial^2 F\over \partial \epsilon_i\partial T} = -2i\pi (u_i(\infty_+)-u_i(\infty_-))
\end{equation}
\begin{equation}
{\partial^2 F\over \partial \epsilon_i\partial h} = -2i\pi (u_i(p_\xi)-u_i(\infty_+))
\end{equation}
\begin{equation}\label{dFtauij}
{\partial^2 F\over \partial \epsilon_i\partial \epsilon_j} = -2i\pi \tau_{ij}
\end{equation}
The last relation implies that $\Re F$ is a convex function of $\epsilon$, thus it has a unique minimum:
\begin{equation}
\vec\epsilon^*
\quad , \quad
\Re \left.{\partial F\over \partial \epsilon_i}\right|_{\vec\epsilon=\vec\epsilon^*} = 0
\end{equation}
We write:
\begin{equation}
\zeta_i := -{1\over 2i\pi} \left.{\partial F\over \partial \epsilon_i}\right|_{\vec\epsilon=\vec\epsilon^*}
\quad , \quad \zeta_i\in {\mathbb{R}}
\end{equation}
We thus have the Taylor expansion:
\begin{eqnarray}
F(T,h,\vec\epsilon)
&\sim& F(1,0,\vec\epsilon^*) -2i\pi \vec\zeta^t (\vec\epsilon-\vec\epsilon^*) +(T-1) {\partial F\over \partial T} + {h\over 2} (V_{\rm eff}(p_\xi)-V(\xi)) \cr
&& +{(T-1)^2\over 2} {\partial^2 F\over \partial T^2}-(T-1)h \ln{\Lambda(p_\xi)}-{h^2\over 2} \ln{H(p_\xi)}\cr
&& -2i\pi (\vec\epsilon-\vec\epsilon^*)^t \tau (\vec\epsilon-\vec\epsilon^*)
-2i\pi (T-1) (\vec\epsilon-\vec\epsilon^*)^t (\vec{u}(\infty_+)-\vec{u}(\infty_-)) \cr
&& -2i\pi h (\vec\epsilon-\vec\epsilon^*)^t (\vec{u}(p_\xi)-\vec{u}(\infty_+)) + \dots
\end{eqnarray}
Thus:
\begin{eqnarray}
Z
&\sim& \sum_I C_I {\mathbf e}^{-N^2 F(\{x\}_I)} \cr
&\sim& \sum_{p=p_\xi,\overline{p}_\xi}
{\mathbf e}^{-N^2 F(1,0,\vec\epsilon^*)}{\mathbf e}^{N^2\left(-(T-1) {\partial F\over \partial T} - {h\over 2} (V_{\rm eff}(p)-V(\xi)) -{(T-1)^2\over 2} {\partial^2 F\over \partial T^2}+(T-1)h \ln{\Lambda(p)}+{h^2\over 2} \ln{H(p)}\right)}\cr
&& \sum_{\vec{n}}
{\mathbf e}^{i\pi (\vec{n}-N\vec\epsilon^*)^t \tau (\vec{n}-N\vec\epsilon^*)}
{\mathbf e}^{2i\pi N \vec\zeta^t (\vec{n}-N\vec\epsilon^*)} \cr
&& \qquad {\mathbf e}^{2i\pi N(T-1) (\vec{n}-N\vec\epsilon^*)^t (\vec{u}(\infty_+)-\vec{u}(\infty_-)) }
{\mathbf e}^{2i\pi Nh (\vec{n}-N\vec\epsilon^*)^t (\vec{u}(p)-\vec{u}(\infty_+))} \cr
\end{eqnarray}
In that last sum, because of convexity, only values of $\vec{n}$ which don't differ from $N\vec\epsilon^*$ form more than a few units, contribute substantialy.
Therefore, up to a non perturbative error (exponentialy small with $N$), one can extend the sum over the $n_i$'s to the whole ${\mathbb{Z}}^g$, and recognize a $\theta$-function (see \ref{deftheta}):
\begin{eqnarray}
Z
&\sim& \sum_{p=p_\xi,\overline{p}_\xi}
{\mathbf e}^{-N^2 F(1,0,\vec\epsilon^*)}{\mathbf e}^{N^2\left((T-1) {\partial F\over \partial T} + {h\over 2} (V_{\rm eff}(p)-V(\xi)) +{(T-1)^2\over 2} {\partial^2 F\over \partial T^2}+(T-1)h \ln{\Lambda(p)}+{h^2\over 2} \ln{H(p)}\right)}\cr
&& {\mathbf e}^{i\pi N^2 \vec\epsilon^{*t} \tau \vec\epsilon^*}
{\mathbf e}^{-2i\pi N^2 \vec\zeta^t \vec\epsilon^*}
{\mathbf e}^{-2i\pi N^2(T-1) \vec\epsilon^{*t} (\vec{u}(\infty_+)-\vec{u}(\infty_-)) }
{\mathbf e}^{-2i\pi N^2 h \vec\epsilon^{*t} (\vec{u}(p)-\vec{u}(\infty_+))} \cr
&&
\theta(
N (\vec\zeta-\tau \vec\epsilon^*)
+N(T-1) (\vec{u}(\infty_+)-\vec{u}(\infty_-))
+Nh (\vec{u}(p)-\vec{u}(\infty_+))
,\tau) \cr
\end{eqnarray}
with $T-1={n-N\over N}$ and $h=0$ or $h=1/N$, we get the asymptotics:
\begin{eqnarray}\label{asympmulticut}
p_n(\xi) &\sim& \sum_{x(p)=\xi}
\sqrt{H(p)}\, (\Lambda(p))^{n-N}\,{\mathbf e}^{-{N\over 2} (V_{\rm eff}(p)-V(\xi))}\, {\mathbf e}^{-2i\pi N \vec\epsilon^{*t} (\vec{u}(p)-\vec{u}(\infty_+))} \cr
&&
{\theta(N (\vec\zeta-\tau \vec\epsilon^*) +(n-N) (\vec{u}(\infty_+)-\vec{u}(\infty_-)) +(\vec{u}(p)-\vec{u}(\infty_+)) ,\tau)
\over \theta(N (\vec\zeta-\tau \vec\epsilon^*) +(n-N) (\vec{u}(\infty_+)-\vec{u}(\infty_-)) ,\tau)}
\cr
\end{eqnarray}
Again, depending on $\xi$, we have to choose the determination of $p_\xi$ which has the minimum energy.
If we are on a cut, i.e. if condition \ref{defcutsVeff} holds, both determinations contribute.
To summarize, outside the cuts, the sum \ref{asympmulticut} reduces to only one term, and along the cuts, the sum \ref{asympmulticut} contains two terms.
\section{Conclusion}
We have shown how the asymptotics of orthogonal polynomials (a notion related to integrability) is deeply related to algebraic geometry.
This calculation can easily be extended to many generalizations, for multi-matrix models \cite{eynchain, eynchaint, BEHAMS, eynhabilit}, non-hermitean matrices ($\beta=1,4$) \cite{eynbetapol}, rational potentials \cite{BEHsemiclas}, ...
\subsection*{Aknowledgements}
The author wants to thank the organizer of the Les Houches summer school Applications of Random Matrices in Physics
June 6-25 2004.
|
{
"timestamp": "2005-03-22T13:59:08",
"yymm": "0503",
"arxiv_id": "math-ph/0503052",
"language": "en",
"url": "https://arxiv.org/abs/math-ph/0503052"
}
|
\section{Introduction}\label{I}
In the theory of non-equilibrium statistical mechanics, the entropy production is a crucial
quantity. Typical for non-equilibrium steady states is the (strict) positivity of the entropy production
which is accompanied by presence of currents and hence breakage of time-reversal symmetry.
In \cite{maes} the entropy production was introduced at the level of trajectories. The idea
is that even for a non-equilibrium system in the steady state, the space-time measure is
still a Gibbs measure and the asymmetric part under time reversal of the Hamiltonian of the space-time Gibbs
measure is the entropy production. Hence, in this formalism, the entropy production is
a trajectory-valued function which measures the degree of irreversibility.
The relative entropy density between the forward and the backward
process is then the {\em mean entropy production}, which is strictly positive if and only if
the process is reversible (i.e., in ``detailed balance'', or
``equilibrium'').
See also \cite{mr} for the relation between strictly
positive mean entropy production and reversibility, and \cite{JQQ} for
a recent account on entropy production in a broader context.
In this point of view, in order to {\em estimate} the entropy production, e.g., in order to
test the reversibility of the process, one needs a way to compute it from trajectories. This is quite similar
to the problem of estimating the entropy of a process. A basic approach
consists in approximating the measure by its empirical version \cite{shields}.
Another particularly useful and simple way of estimating entropy is
via the Ornstein-Weiss theorem \cite{shields,weiss}. The entropy is approximated by the
logarithm of the return time of the first $n$ symbols, divided by $n$. Similarly, relative entropy
density can be estimated using waiting times, see e.g. \cite{konto}.
In this paper we consider Gibbsian processes with values in a finite alphabet, and with
summable modulus of continuity.
We introduce an estimator of the entropy production based on a single trajectory
(we call it the hitting-time estimator) and an estimator based on two independent trajectories
(which we call the waiting-time estimator). For both estimators we obtain consistency and asymptotic
normality, with an asymptotic variance coinciding with that of the entropy production.
Moreover, for the waiting-time estimator we obtain a large deviation principle.
It turns out that its large deviation function has the same symmetry
as in the so-called fluctuation theorem \cite{galco,lebspo,maes}, and in fact coincides
with the large deviation function of the entropy production
itself in the region where it is finite. This shows that the estimator has also nice properties from
the physical point of view.
The basic technique we use is the exponential law with good control of the error for hitting and waiting
times \cite{miguel,miguelnew}.
This provides us with a precise control of the difference between the estimators and the entropy production.
The rest of the paper is organized as follows. In section 2 we introduce the entropy production in the spirit
of \cite{maes}, see also \cite{JQQ}. In section 3 we introduce the estimators, in section 4 we state their
fluctuation properties and section 5 is devoted to proofs.
\section{Context}\label{C}
We will consider a stationary process $\{X_n:n\in\mathbb Z\}$ taking values in a
finite set $A$.
A trajectory of this process, i.e., an element of $A^{\mathbb Z}$ will be denoted
by $\omega$.
The space of all trajectories is denoted by $\Omega=A^{\mathbb Z}$. For
$\omega\in\Omega$, and
$n\in\mathbb Z$, $\theta_n \omega$ is the trajectory defined by $(\theta_n
\omega)_k := \omega_{k+n}$.
A function $f:\Omega\to\mathbb R$ is called local if it depends only on finitely
many coordinates
of the trajectory.
A block of length $n$ is a sequence $x_1^n:=x_1\cdots x_n$ of elements of
$A$. The cylinder
$[x_1^n]$ based on $x_1^n$ is the set of $\omega\in\Omega$ such that
$\omega_j=x_j$ for
$j=1,\ldots,n$.
The distribution ${\mathbb P}$ of the process $\{X_n:n\in\mathbb Z\}$ is supposed to be
a translation invariant Gibbs measure with
translation invariant potential $U$. The associated ``energy per site''
$f_U$ is defined as usual:
$$
f_U(\omega):= \sum_{\Lambda\ni 0} \frac{U(\Lambda,\omega)}{|\Lambda|}
$$
where the sum runs over all finite subsets of $\mathbb Z$ (containing the
origin).
It is well-known that under mild assumptions \cite{Geo}
there exists a constant $K>0$ such that for all $x_1^n$, all
$\omega\in[x_1^n]$, we have the uniform estimate
\begin{equation}\label{gibbs}
K^{-1} \leq
\frac{{\mathbb P}([x_1^n])}{\exp(n P(f_U) + \sum_{j=0}^{n-1} f_U(\theta_j
\omega))}
\leq K
\end{equation}
where $P(f_U)$ is the ``pressure'' associated to $U$.
For a block $x_1^n$, its time reverse is denoted by $x_n^1=x_n
x_{n-1}\cdots x_1$.
Similarly, $X_1^n$ denotes the random block $X_1\cdots X_n$ whereas
$X_n^1$ denotes the random block $X_n\cdots X_1$.
For the definition of the entropy production of the process
$\{X_n:n\in\mathbb Z\}$, we follow
\cite{maes,mrv}.
We denote by ${\mathbb P}^{{\scriptscriptstyle R}}$ the distribution of the time-reversed process, i.e.,
the distribution
of $\{X_{-n}:n\in\mathbb Z\}$.
The entropy production of the process up to time $n$ is defined as
\begin{equation}\label{dracula}
{\dot{\mathbf S}}_n(X_1,\ldots,X_n):=\log\frac{{\mathbb P}([X_1^n])}{{\mathbb P}([X_n^1])}=
\log\frac{{\mathbb P}([X_1^n])}{{\mathbb P}^{{\scriptscriptstyle R}}([X_1^n])}\,\cdot
\end{equation}
This random variable is a measure of the irreversibility of the process up
to time $n$.
We recall that the relative entropy density $h({\mathbb Q}|{\mathbb P})$ between a
translation invariant probability
measure ${\mathbb Q}$ on $\Omega$ and ${\mathbb P}$ is the limit
$$
h({\mathbb Q}|{\mathbb P})=\lim_{n\rightarrow\infty}\frac{H_n({\mathbb Q}|{\mathbb P})}{n}
$$
where
$$
H_n({\mathbb Q}|{\mathbb P}):=\sum_{x_1^n\in A^n} {\mathbb Q}([x_1^n])
\log\frac{{\mathbb Q}([x_1^n])}{{\mathbb P}([x_1^n])}\,\cdot
$$
We have the following well-known properties \cite{Geo}:
$$
h({\mathbb Q}|{\mathbb P})=P(f_U)-\int f_U\ d{\mathbb Q} + s({\mathbb Q})
$$
where $s({\mathbb Q})$ is the entropy density of ${\mathbb Q}$. Moreover,
$h({\mathbb Q}|{\mathbb P})\geq 0$, with equality if and only if ${\mathbb Q}$ is an equilibrium
state for $U$ (variational principle).
Using \eqref{gibbs} and the Ergodic Theorem, it follows immediately that
\begin{equation}\label{chou}
\lim_{n\rightarrow\infty} \frac{{\dot{\mathbf S}}_n(X_1,\ldots, X_n)}{n}= h({\mathbb P}|{\mathbb P}^{{\scriptscriptstyle R}}):={\mathbf{MEP}}\quad
{\mathbb P}-\textup{almost surely}\,.
\end{equation}
This quantity is called the {\em mean entropy production}. It is equal to
$0$ if and only
if the process is reversible, i.e., the potential $U^{{\scriptscriptstyle R}}$ associated to
${\mathbb P}^{{\scriptscriptstyle R}}$ is physically
equivalent to the potential $U$.
We now precise the classes of potentials for which our results hold.
A first restriction is to assume that $f_U$ has a summable modulus of
continuity, i.e.,
\begin{equation}\label{avion}
\sum_{n\geq 1} \textup{var}_n f_U <\infty
\end{equation}
where
$$
\textup{var}_n f_U := \sup\{ |f_U(\omega)-f_U(\omega')|:
\omega_i=\omega'_i, \forall |i|\leq n\}\,.
$$
In particular this implies that ${\mathbb P}$ is the unique Gibbs measure (equilibrium state)
with potential $U$.
It is convenient to work with an $f_U$ which depends only on
``future'' coordinates, that is, only on $\omega_1,\omega_2,\ldots$.
It is indeed proved in \cite{CQ} that if $f_U$ satisfies
\eqref{avion}, then there exists a function
$f_U^+(\omega):=f_U^+(\omega_1,\omega_2,\ldots)$ which is physically
equivalent to $f_U$, i.e., which gives the same Gibbs measure as
$f_U$, and which has also summable variations. ``Physically
equivalent'' means there exists
a measurable function $\kappa=\kappa_U$ and a real constant $C=C_U$
such that $f_U^+ = f_U + \kappa - \kappa\circ \theta + C$. It is
easy to check that \eqref{gibbs} holds with $f_U^+$ in place of
$f_U$ by suitably modifying the constant $K$. Moreover, we can
simplify the notations by assuming that $P(f_U^+)=0$. If it is
not the case, replace $f_U^+$ by the physically equivalent potential
$f_U^+ - P(f_U^+)$. Recapitulating, we obtain that
there exists a constant $K'>0$ such that for all $x_1^n$, all
$\omega\in[x_1^n]$, we have the uniform estimate
\begin{equation}\label{gibbsbis}
K'^{-1} \leq
\frac{{\mathbb P}([x_1^n])}{\exp(\sum_{j=0}^{n-1} f_U^+(\theta_j \omega))}
\leq K'\,.
\end{equation}
Of course, the same estimate holds for ${\mathbb P}^{{\scriptscriptstyle R}}$ with the obvious
modifications. This immediately gives that there exists some constant
$\tilde{K}>0$ such that
\begin{equation}\label{porc}
-\tilde{K} \leq {\dot{\mathbf S}}_n - \sum_{j=0}^{n-1} [(f_U^+ -
f_{U^{{\scriptscriptstyle R}}}^+)\circ\theta_j] \leq \tilde{K}
\end{equation}
for all $n\geq 1$.
Using \eqref{chou} and the Ergodic Theorem, we deduce immediately that
$$
{\mathbf{MEP}} = \int (f_U^+ - f_{U^{{\scriptscriptstyle R}}}^+)\ d{\mathbb P}\,.
$$
The possibility of working with a ``one-sided'' potential physically
equivalent to the
``two-sided'' one is very important because it will allow us to apply
known results obtained by transfer-operator techniques.
The assumption \eqref{avion} also implies a ``strong mixing'' property
which is needed to
prove our results.
When dealing with central limit asymptotics, we will restrict
ourselves to potentials having exponentially decreasing modulus of
continuity, i.e.,
\begin{equation}\label{bateau}
\exists C>0, 0\leq \eta <1\quad\textup{such that}\quad
\textup{var}_n f_U \leq C \eta^n\quad\forall n\geq 1\,.
\end{equation}
This will allow us to use a result proved in \cite{PP}.
We will precise further these points at the appropriate places.
\begin{remark}
If we assume that
$$
\sum_{\Lambda: \min \Lambda =0} \textup{diam}(\Lambda)
\textup{var}(U(\Lambda,\cdot)) <\infty
$$
where
$\textup{var}(U(\Lambda,\cdot)):=\max(U(\Lambda,\cdot))-\min(U(\Lambda,\cdot))$
this implies \eqref{avion}, see \cite{CQ}.
\end{remark}
\section{Estimators of entropy production based on hitting and return times}\label{wr}
In this section we introduce two estimators based on a single trajectory
or
on two independent trajectories. To define them we have to introduce
hitting times.
The hitting time of a cylinder $[x_1^n]$ is defined as
$$
\T_{x_1^n}(\omega)
:=\inf\{k\geq 1: \theta_k \omega\in [x_1^n] \}\,.
$$
For the sake of convenience, we introduce the notations
$$
\T^+_n(\omega):=\T_{\omega_1^n}(\omega)
\quad\textup{and}\quad
\T^-_n(\omega):=\T_{\omega_n^1}(\omega)\,.
$$
The hitting-time estimator $\dot{{\mathcal S}}^{{\scriptscriptstyle H}}_n(\omega)$ of the entropy production is
defined as
$$
\dot{{\mathcal S}}^{{\scriptscriptstyle H}}_n(\omega):= \log\frac{\T^-_n(\omega)}{\T^+_n(\omega)}\,\cdot
$$
In words, this is the difference of the logarithms of the first time at which we observe
the first $n$ symbols in reversed order in the trajectory and the first return time of the
first $n$ symbols.
It will follow from our analysis that {\em typically},
$\T^-_n\gg \T^+_n$ if the process is not reversible.
Hence our hitting-time estimator of the entropy production will be
typically positive.
The waiting-time estimator $\dot{{\mathcal S}}^{{\scriptscriptstyle W}}_n(\omega,\omega')$ of the entropy production
is based on
two trajectories $\omega, \omega'$ chosen {\em independently} of one
another according to
${\mathbb P}$. We introduce the following convenient notations:
$$
{\mathbf W}^+_n(\omega,\omega'):=\T_{\omega_1^n}(\omega')\quad\textup{and}\quad
{\mathbf W}^-_n(\omega,\omega'):=\T_{\omega_n^1}(\omega')\,.
$$
The waiting-time estimator is then defined as
$$
\dot{{\mathcal S}}^{{\scriptscriptstyle W}}_n(\omega,\omega'):=
\log\frac{{\mathbf W}^-_n(\omega,\omega')}{{\mathbf W}^+_n(\omega,\omega')}\,\cdot
$$
The main motivation to introduce this alternative estimator is that we
will obtain
a better control of its large deviation properties.
\begin{remark}
We can define two other estimators based on the so-called matching
times \cite{konto}. They are in some sense the ``duals'' of the above
estimators.
To introduce the ``dual" of the hitting-time estimator, consider the first
$n$ symbols $x_1,\ldots x_n$ of the process and define
\[
\L^+_n = \min \{ k\leq n: \mbox{the word}\ x_1^k\ \mbox{does not reappear in} \ x_1^n\}
\]
and
\[
\L^-_n = \min \{ k\leq n: \mbox{the reversed word}\ x_k^1\ \mbox{does not reappear in} \ x_1^n\}
\]
Then the estimator of the entropy production dual to the hitting-time estimator is given by
$\log(\L^+_n/\L^+_-)$.
The advantage of these estimators is that they are
based on a trajectory of finite length $n$. However, all the asymptotic fluctuation properties
of these estimators can be derived from the ones of the present paper by the duality relations.
So we do not study them in detail in this paper.
\end{remark}
\section{Convergence and fluctuations of the estimators}\label{MR}
We now state our results on consistency and asymptotic normality for the
estimators we just introduced, as well as large deviation properties for estimators based on
two independent trajectories. Recall that ${\mathbf{MEP}}$ is the mean entropy production, see
\eqref{chou}.
\subsection{Almost-sure approximation and consistency}
The following theorem provides an almost-sure approximation of ${\dot{\mathbf S}}_n$, the
entropy production up to time $n$ (see \eqref{dracula}),
by both the return-time and the waiting-time estimators.
\begin{theorem}\label{thm1}
Assume that \eqref{avion} holds. Then there exists a constant $C=C({\mathbb P})>0$ such that
\begin{enumerate}
\item Eventually ${\mathbb P}$-almost surely
$$
-C\log n\leq
\dot{{\mathcal S}}^{{\scriptscriptstyle H}}_n - {\dot{\mathbf S}}_n
\leq C\log n\,;
$$
\item Eventually ${\mathbb P}\!\times\!{\mathbb P}$-almost surely
$$
-C\log n\leq
\dot{{\mathcal S}}^{{\scriptscriptstyle W}}_n - {\dot{\mathbf S}}_n
\leq C\log n\,.
$$
\end{enumerate}
\end{theorem}
Using the previous theorem and \eqref{chou}, we immediately obtain the
following corollary
establishing the consistency of our entropy production estimators.
\begin{corollary}
We have the following almost-sure convergences:
\begin{enumerate}
\item ${\mathbb P}$-almost surely
$$
\lim_{n\rightarrow\infty}\frac{\dot{{\mathcal S}}^{{\scriptscriptstyle H}}_n}{n}= {\mathbf{MEP}}\,;
$$
\item ${\mathbb P}\!\times\!{\mathbb P}$-almost surely
$$
\lim_{n\rightarrow\infty}\frac{\dot{{\mathcal S}}^{{\scriptscriptstyle W}}_n}{n}= {\mathbf{MEP}}\, .
$$
\end{enumerate}
\end{corollary}
\subsection{Asymptotic normality}
The expectation with respect to ${\mathbb P}$ is denoted by $\mathbb E$.
Let
\begin{equation}\label{variance}
\sigma^2:=\sum_{\ell\geq 1} \left[
\mathbb E((f_U^+ -f_{U^{{\scriptscriptstyle R}}}^+)\cdot (f_U^+ -f_{U^{{\scriptscriptstyle R}}}^+)\circ
\theta_\ell)-(\mathbb E(f_U^+ - f_{U^{{\scriptscriptstyle R}}}^+))^2\right]\,.
\end{equation}
It can be showed that $\sigma^2<\infty$ if \eqref{bateau} holds.
It is well-known that $\sigma^2>0$ unless $U$ is physically equivalent to
$U^{{\scriptscriptstyle R}}$,
i.e., $f_U^+ - f_{U^{{\scriptscriptstyle R}}}^+ $ is a co-boundary, which in turn is equivalent
with ${\mathbb P}={\mathbb P}^{{\scriptscriptstyle R}}$,
i.e., the process is reversible.
For more details on this, we refer to \cite{PP}.
\begin{theorem}\label{pouac}
Assume that \eqref{bateau} holds. Then
we have the following central limit asymptotics:
\begin{enumerate}
\item For the hitting-time estimator
$$
\frac{\dot{{\mathcal S}}^{{\scriptscriptstyle H}}_n - n {\mathbf{MEP}}}{\sqrt{n}}\to
\mathcal{N}(0,\sigma^2)\,,\textup{as}\;n\to\infty
$$
in ${\mathbb P}$-distribution.
\item For the waiting-time estimator
$$
\frac{\dot{{\mathcal S}}^{{\scriptscriptstyle W}}_n - n {\mathbf{MEP}}}{\sqrt{n}}\to
\mathcal{N}(0,\sigma^2)\,,\textup{as}\;n\to\infty
$$
in ${\mathbb P}\!\times\!{\mathbb P}$-distribution.
\end{enumerate}
Moreover,
\begin{equation}\label{varvar}
\lim_{n\rightarrow\infty}\frac{\textup{Var}(\dot{{\mathcal S}}^{{\scriptscriptstyle H}}_n)}{n}=\lim_{n\rightarrow\infty}\frac{\textup{Var}(\dot{{\mathcal S}}^{{\scriptscriptstyle W}}_n)}{n}=
\sigma^2
\end{equation}
where $\textup{Var}$ denotes the variance.
\end{theorem}
\begin{remark}
Using the results of \cite{KMS}, we could extend the previous theorem
to potentials with a modulus of continuity decreasing polynomially,
i.e., like $1/n^{\alpha}$ for $\alpha>0$ large enough.
\end{remark}
\subsection{Large deviations}
Our goal is to analyze the deviations of order one of $\dot{{\mathcal S}}^{{\scriptscriptstyle W}}_n/n$ around the mean
entropy production ${\mathbf{MEP}}$. To this end, we introduce the following ``free-energy-like'' function,
which is nothing but the scaled-cumulant generating function for the process $(\dot{{\mathcal S}}^{{\scriptscriptstyle W}}_n)$:
$$
\mathcal{W}_U(p):=\lim_{n\rightarrow\infty}\frac{1}{n}\log
\mathbb E_{{\mathbb P}\!\times\!{\mathbb P}}\left(e^{p\dot{{\mathcal S}}^{{\scriptscriptstyle W}}_n}\right)\, , \; p\in\mathbb R
$$
provided the limit exists.
On another hand, define the scaled cumulant generating function for the
process $({\dot{\mathbf S}}_n)$ as:
$$
\mathcal{E}_U(p):=\lim_{n\rightarrow\infty}\frac{1}{n}\log \mathbb E_{{\mathbb P}}\left(e^{p{\dot{\mathbf S}}_n}\right)\,
, p\in\mathbb R\,.
$$
It is easy to deduce from \eqref{gibbsbis} that
$$
\mathcal{E}_U(p)=P(-p f_{U^{{\scriptscriptstyle R}}}^+ + (1+p)f_U^+)\ ,\;\forall p\in\mathbb R\,.
$$
From this formula one immediately sees that
$$
\mathcal{E}_U(-1-p)
=
\mathcal{E}_{U^{{\scriptscriptstyle R}}}(p)\,.
$$
On another hand, it is obvious from the definition of ${\dot{\mathbf S}}_n$ that
$$
\mathcal{E}_U(p)
=
\mathcal{E}_{U^{{\scriptscriptstyle R}}}(p)\,.
$$
Hence
$$
\mathcal{E}_U(-1-p)
=
\mathcal{E}_{U}(p)
$$
which is a version of the Gallavotti-Cohen fluctuation theorem, see
\cite{galco}, \cite{lebspo}, \cite{maes}.
Notice that $\mathcal{E}_U\equiv 0$ if $U$ is physically equivalent
to $U^{{\scriptscriptstyle R}}$.
We now state a large deviation result for ${\dot{\mathbf S}}_n$.
Let ${\mathcal I}_U$ be the Legendre transform of ${\mathcal E}_U$, i.e.,
$$
{\mathcal I}_U(q)= \sup_{p\in\mathbb R} \left(pq - {\mathcal E}_U(p)\right) \,.
$$
Then we have
\begin{proposition}\label{macbeth}
Assume that \eqref{avion} holds and that the process $(X_n)$ is not reversible (i.e., that
$U$ is not physically equivalent to $U^{{\scriptscriptstyle R}}$). Then the function $p\mapsto {\mathcal E}_U(p)$
is continuously differentiable and strictly convex. Moreover,
there exists an open interval $(\underline{q}, \overline{q})$ such that, for every interval $J$
with $J\cap (\underline{q}, \overline{q})\neq \emptyset$
$$
\lim_{n\rightarrow\infty} \frac{1}{n}\log {\mathbb P}\left\{\frac{{\dot{\mathbf S}}_n(X_1,\ldots,X_n)}{n} \in J \right\}=
-\inf_{q\in J\cap (\underline{q}, \overline{q})} {\mathcal I}_U(q)\,.
$$
\end{proposition}
The interest of this result lies in its formulation adapted to our
context and convenient to state the next result, the main one of this
section. In essence such kind of result appears, e.g., in \cite{MV}.
\begin{theorem}\label{pouic}
If assumption \eqref{avion} holds
then we have
\begin{equation}
\mathcal{W}_U(p)=
\left\{
\begin{array}{l}
\mathcal{E}_U(p)\quad\textup{if}\;-1<p<1\\
+\infty \quad\textup{otherwise}\,.
\end{array}\label{boulgakov}
\right.
\end{equation}
In particular, if the process $(X_n)$ is not reversible (i.e., $U$ is not physically equivalent to $U^{{\scriptscriptstyle R}}$) then
$\dot{{\mathcal S}}^{{\scriptscriptstyle W}}_n$ and ${\dot{\mathbf S}}_n$ have the same large deviations in the
open interval $(c_-,c_+)$, with $c_-:=\lim_{p\to -1}\mathcal{E}_U'(p)<0$
and $c_+:=\lim_{p\to 1}\mathcal{E}_U'(p)>0$: For every interval $J$
with $J\cap (c_-, c_+)\neq \emptyset$
\begin{equation}\label{pelleas}
\lim_{n\rightarrow\infty} \frac{1}{n}\log {\mathbb P}\left\{\frac{\dot{{\mathcal S}}^{{\scriptscriptstyle W}}_n}{n} \in J \right\}=
-\inf_{q\in J\cap (c_-, c_+)} {\mathcal I}_U(q)\,.
\end{equation}
\end{theorem}
\bigskip
It is easy to check that ${\mathbf{MEP}}\in (c_-,c_+)$. Indeed ${\mathcal E}_U'(0)={\mathbf{MEP}}$ (one
uses differentiability and convexity to prove that).
The next proposition highlights the symmetry properties of $\mathcal{W}$.
We write explicitly the dependence of $\mathcal{W}$ on the potential $U$.
\begin{proposition}\label{symmetry}
Under assumption \eqref{avion} we have the following identities
\begin{enumerate}
\item For all $-1<p\leq 0$, we have
$$
\mathcal{W}_{U}(-1-p)=
\mathcal{W}_{U^{{\scriptscriptstyle R}}}(p) =
\mathcal{W}_U(p)=
\mathcal{W}_{U^{{\scriptscriptstyle R}}}(-1-p)\,.
$$
\item For all $p\in(-1,1)$, we have
$$
\mathcal{W}_{U}(p) =
\mathcal{W}_{U^{{\scriptscriptstyle R}}}(p)\,.
$$
\end{enumerate}
\end{proposition}
\begin{remark}
One may ask why we did not study the large deviations of $\dot{{\mathcal S}}^{{\scriptscriptstyle H}}_n$, the
hitting-time estimator. Indeed, the analysis of the corresponding
scaled cumulant generating function is made more complicated due
to the effect of ``too soon'' recurrent cylinders. We shall not
detail more on this. Following the approach of \cite{CGS}, we can
obtain a partial counterpart of Theorem \ref{pouic} for $\dot{{\mathcal S}}^{{\scriptscriptstyle H}}_n$ :
its scaled cumulant generating function coincides with
$\mathcal{E}_U(p)$ but only in an {\em implicit} interval
$[\tilde{c}_-,\tilde{c}_+]$, where $\tilde{c}_-<0$ and
$\tilde{c}_+>0$.
\end{remark}
\section{Proofs}\label{proofs}
\subsection{Key lemmas}
The following results are the main tools to derive our results.
\begin{keylemma}\label{MKL}
Assume that ${\mathbb P}$ is a translation invariant Gibbs measure such that
\eqref{avion} holds.
Then there exist strictly positive constants $c,C,\rho_1,\rho_2$, with
$\rho_1\leq \rho_2$,
such that for all $n\in\mathbb N$, all cylinders $[a_1^n]$ and all $t>0$
there exists $\rho(a_1^n)\in[\rho_1,\rho_2]$ such that
\begin{equation}\label{strong-approximation}
\Big\vert {\mathbb P}\{\T_{a_1^n}{\mathbb P}([a_1^n])>t\}- e^{-\rho(a_1^n)t}\Big\vert
\leq C e^{-c n} e^{-\rho(a_1^n)t}\,.
\end{equation}
\end{keylemma}
\begin{proof}
In \cite{miguel}, the author proved this result
under the assumption that the process is $\psi$-mixing.
Besides, it is proved in \cite{walters} that if $f_U^+$ has summable
variations, then the process $(X_n)$ is $\psi$-mixing. (This can
be read off the proof of Theorem 3.2 in \cite{walters}.)
\end{proof}
The next lemma will be crucial to control certain moments. This is a
rewriting of Lemma 9 in \cite{miguel}.
\begin{lemma}\label{tarte}
For all cylinder $[a_1^n]$, all $t$ such that $t \leq 1/2$, we have
$$
1-e^{-\rho_1 t} \leq {\mathbb P}\{\T_{a_1^n}{\mathbb P}([a_1^n])\leq t\}\leq 1-
e^{-\rho_2 t}
$$
where $\rho_1,\rho_2$ are the constants of Key-lemma \ref{MKL}.
\end{lemma}
We now state the analog to Key-lemma \ref{MKL} for return times.
To do so, we need to define the set of $n$-cylinders with ``internal
periodicity'' $k\leq n$:
$$
\mathcal{S}_k(n):=\{[a_1^n]: \min\{j\in\{1,...,n\}: [a_1^n]\cap
\theta_{j}[a_1^n]\neq\emptyset\}=k \}\,.
$$
Notice that the set of $n$-cylinders can be written as $\bigcup_{1\leq
p\leq n} \mathcal{S}_k(n)$.
\begin{keylemma}\label{MKLbis}
Assume that ${\mathbb P}$ is a translation invariant Gibbs measure such that
\eqref{avion} holds.
Then there exist strictly positive constants $c,c',C$
such that for any $n\in\mathbb N$, any $k\in\{1,...,n\}$,\
any cylinder $[a_1^n] \in \mathcal{S}_k(n)$, one has for all $t\geq k$
\begin{equation}
\Big\vert {\mathbb P}\big\{\omega:\T_{a_1^n}(\omega){\mathbb P}([a_1^n]) >t\big\vert\
[a_1^n]\big\}
- \zeta(a_1^n) \exp(-\zeta(a_1^n)t)\Big\vert
\leq
C \ e^{-c n}\ e^{-c' t}
\end{equation}
where $\zeta(a_1^n)$ is such that $|\ \zeta(a_1^n)-\rho(a_1^n)|\leq D e^{-c n}$,
for some $D>0$. The parameter $\rho(a_1^n)$ is defined in Key-lemma \ref{MKL}.
Moreover,
$$
{\mathbb P}\{\omega:\T_{a_1^n}(\omega)>t\ | \ [a_1^n]\}=1\quad \textup{for
all}\quad t<k\, .
$$
\end{keylemma}
\begin{proof}
This Key-lemma is a rewriting of \cite[Section 6]{miguelnew}. As for the
previous Key-lemma, the assumption is that the process is $\psi$-mixing.
\end{proof}
\subsection{Proof of Theorem \ref{thm1}}
Let us start with the proof of the second statement of the theorem. We
shall prove
that eventually ${\mathbb P}\!\times\!{\mathbb P}$-almost surely
\begin{equation}\label{pizza}
-C_1\log n\leq \log({\mathbf W}_n^+(\omega,\omega') {\mathbb P}([\omega_1^n])\leq \log C_1 +\log\log n
\end{equation}
for some $C_1>0$.
It will be clear that by the same reasoning we will also have
that eventually ${\mathbb P}\!\times\!{\mathbb P}$-almost surely
\begin{equation}
-C_2\log n\leq \log({\mathbf W}_n^{-}(\omega,\omega') {\mathbb P}([\omega_n^1])\leq
\log C_2 +\log\log n
\end{equation}
for some $C_2>0$.
Putting together these two results immediately gives the
statement 2 of the theorem.
We first prove the upper bound in \eqref{pizza}. We want to find a
summable upper-bound to
$$
{\mathbb P}\!\times\!{\mathbb P}\{\log({\mathbf W}_n^+ {\mathbb P}([x_1^n]))> \log t\}=
$$
\begin{equation}\label{split}
\sum_{x_1^n}{\mathbb P}([x_1^n])\
{\mathbb P}\left\{\log(\T_{x_1^n} {\mathbb P}([x_1^n]))> \log t \right\}
\end{equation}
where $t$ will be a suitable function of $n$.
We apply Key-lemma \ref{MKL} to get for all $t>0$
$$
{\mathbb P}\!\times\!{\mathbb P}\{\log({\mathbf W}_n^+ {\mathbb P}([x_1^n]))> \log t\}
\leq C e^{-cn}+ e^{-\rho_1 t}\, .
$$
Take $t=t_n= \log n^{\alpha_1}$, $\alpha_1>0$, to get
$$
{\mathbb P}\!\times\!{\mathbb P}\{\log({\mathbf W}_n^+ {\mathbb P}([x_1^n]))> \log\log n^\alpha\}
\leq C e^{-cn} + \frac{1}{n^{\rho_1 \alpha_1}}\,.
$$
By the Borel-Cantelli Lemma we get
$$
\log({\mathbf W}_n^+ {\mathbb P}([a_1^n]))\leq \log\log n^{\alpha_1}
$$
eventually ${\mathbb P}\!\times\!{\mathbb P}$-almost surely provided that
$\alpha_1\rho_1 >1$.
To obtain the lower bound in \eqref{pizza}, we have, by Key-lemma
\ref{MKL}
$$
{\mathbb P}\!\times\!{\mathbb P}\{\log({\mathbf W}^+_n {\mathbb P}([x_1^n]))\leq \log t\}
\leq C e^{-cn}+ 1-e^{-\rho_2 t}\leq C e^{-cn}+\rho_2 t
$$
for all $t>0$.
Choose $t=t_n=n^{-\alpha_2}$, $\alpha_2>1$ and apply the Borel-Cantelli
Lemma to get
$$
\log({\mathbf W}_n^+ {\mathbb P}([x_1^n]))> -\alpha_2\log n
$$
eventually ${\mathbb P}\!\times\!{\mathbb P}$-almost surely.
Let us now prove the first statement of the
theorem. The proof is very similar except we have to deal with
``bad'' cylinders and use Key-lemma \ref{MKLbis}. We will only establish
that eventually ${\mathbb P}$-almost
surely the inequality
\begin{equation}\label{bof}
-C_1\log n\leq \log(\T_n^+(\omega) {\mathbb P}([\omega_1^n])\leq \log C_1 +\log\log n
\end{equation}
for some $C_1>0$. The analogous inequality for $\T^-(\omega)$ is
obtained as above (i.e., using Key-lemma \ref{MKL}).
We have the decomposition
$$
{\mathbb P}\{\omega:\log(\T_n^+(\omega) {\mathbb P}([\omega_1^n]))> \log t\}=
$$
$$
\sum_{k=1}^n \sum_{x_1^n\in \mathcal{S}_k(n)}{\mathbb P}([x_1^n])\
{\mathbb P}\left\{\omega:\log(\T_{x_1^n}(\omega) {\mathbb P}([x_1^n])> \log t \ | \
[x_1^n]\right\}
$$
where $\mathcal{S}_k(n)$ is defined just before we state Key-lemma
\ref{MKLbis}.
For all $t\geq k {\mathbb P}([x_1^n])$ and $n$ large enough, we get using
Key-lemma \ref{MKLbis}
$$
{\mathbb P}\left\{\log(\T_{x_1^n} {\mathbb P}([x_1^n])> \log t \ | \ [x_1^n])\right\}\leq
(\rho_2 + D) e^{-\frac{\rho_1}{2}t} + C e^{-cn}
$$
where we used the fact that if $n$ is large enough,
$\rho_1/2 \leq \rho_1 - D e^{-c n} \leq \zeta(a_1^n) \leq \rho_2 +
D$. We now choose $t=t_n=\log n^{\alpha_1}$, $\alpha_1>0$. If $n$ is
large enough, then $t_n \geq k {\mathbb P}([a_1^n])$.
This is because we have the uniform estimate ${\mathbb P}([a_1^n])\leq
e^{-G n}$, for some $G>0$, since ${\mathbb P}$ is a Gibbs measure.
Hence we obtain
$$
{\mathbb P}\{\log(\T_n^+ {\mathbb P}([\omega_1^n]))> \log\log n^{\alpha_1}\} \leq
\frac{\rho_2 + D}{n^{\alpha_1 \rho_1/2}} + C e^{-cn}
$$
which is summable provided that $\alpha_1 \rho_1/2 >1$. The
Borel-Cantelli Lemma then gives the upper-bound in \eqref{bof}.
The lower-bound is obtained as for the waiting-time estimator but
using Key-lemma \ref{MKLbis}. \hfill $\qed$
\subsection{Proof of Theorem \ref{pouac}}
Let us prove the second statement of the theorem and that
$\lim_{n\rightarrow\infty}\frac{\textup{Var}(\dot{{\mathcal S}}^{{\scriptscriptstyle W}}_n)}{n}= \sigma^2$.
For this it is enough to prove that
\begin{equation}\label{bong}
\lim_{n\rightarrow\infty}\frac{1}{n}\int \left(\dot{{\mathcal S}}^{{\scriptscriptstyle W}}_n - {\dot{\mathbf S}}_n \right)^2 d{\mathbb P}\!\times\!{\mathbb P}=0\,.
\end{equation}
Indeed, proving \eqref{bong} implies, on one hand, that
$$
\lim_{n\rightarrow\infty}\frac{\textup{Var}({\dot{\mathbf S}}_n)}{n}=\lim_{n\rightarrow\infty}\frac{\textup{Var}(\dot{{\mathcal S}}^{{\scriptscriptstyle W}}_n)}{n}\,\cdot
$$
On the other hand, it also implies that
$(\dot{{\mathcal S}}^{{\scriptscriptstyle W}}_n-n{\mathbf{MEP}})/\sqrt{n}$ converges in law to the normal
$\mathcal{N}(0,\sigma^2)$
if, and only if, $({\dot{\mathbf S}}_n-n{\mathbf{MEP}})/\sqrt{n}$ converges in law to the same law.
Now it is obvious from \eqref{porc} that
$$
\frac{{\dot{\mathbf S}}_n - \sum_{j=0}^{n-1} [(f_U^+ -
f_{U^{{\scriptscriptstyle R}}}^+)\circ\theta_j]}{\sqrt{n}}\to 0\quad
{\mathbb P}-\textup{almost-surely}\,.
$$
By applying a result of \cite{PP}, we obtain that
$$
\frac{\sum_{j=0}^{n-1} [(f_U^+ -
f_{U^{{\scriptscriptstyle R}}}^+)\circ\theta_j] - n{\mathbf{MEP}} }{\sqrt{n}}\stackrel{\textup{in
law}}{\longrightarrow}
\mathcal{N}(0,\sigma^2)\,.
$$
Since we have the formula (see \cite{PP})
$$
\sigma^2=\lim_{n\rightarrow\infty} \frac{1}{n}\int \big(
\sum_{j=0}^{n-1} [(f_U^+ - f_{U^{{\scriptscriptstyle R}}}^+)\circ\theta_j] - n{\mathbf{MEP}}\big)^2\ d{\mathbb P}
$$
it is obvious by \eqref{porc} that
$$
\lim_{n\rightarrow\infty}\frac{\textup{Var}({\dot{\mathbf S}}_n)}{n}=\sigma^2\,.
$$
Therefore we have reduced the statements of the theorem about $\dot{{\mathcal S}}^{{\scriptscriptstyle W}}_n$ to
proving \eqref{bong}.
By definition we have
$$
\int \left(\dot{{\mathcal S}}^{{\scriptscriptstyle W}}_n - {\dot{\mathbf S}}_n \right)^2 d{\mathbb P}\!\times\!{\mathbb P}=
\sum_{x_{1}^{n}} {\mathbb P}([x_1^n]) \int \left[
\log(\T_{n}^{-}{\mathbb P}([x_n^1]))-\log(\T_{n}^{+}{\mathbb P}([x_1^n]))
\right]^2 d{\mathbb P}\ .
$$
Let us now prove that the integral in the rhs is bounded above by a positive
number
independent of $n$, implying immediately \eqref{bong}. To prove this
assertion,
it is sufficient to prove that
\begin{equation}\label{bing}
\int \left[\log(\T_{n}^{+}{\mathbb P}([x_1^n]))\right]^2 d{\mathbb P} \leq D_1,\quad
\int \left[\log(\T_{n}^{-}{\mathbb P}([x_n^1]))\right]^2 d{\mathbb P} \leq D_2\quad
\end{equation}
where $D_1, D_2>0$ are independent of $n$.
We only prove the first inequality since the other one is proved in
exactly the
same way.
We have the following identities:
$$
\int \left[\log(\T_{n}^{+}{\mathbb P}([x_1^n]))\right]^2 d{\mathbb P} = \int_0^\infty
{\mathbb P}\left([\log(\T_{n}^{+}{\mathbb P}([x_1^n]))]^2 >t \right) dt =
$$
$$
2 \int_1^\infty {\mathbb P}\left( \T_{n}^{+}{\mathbb P}([x_1^n]) >t \right)
\frac{\log t}{t}\ dt
+
2 \int_0^1 {\mathbb P}\left( \T_{n}^{+}{\mathbb P}([x_1^n]) <t \right)
\frac{-\log t}{t}\ dt=
$$
$$
2 \int_1^\infty {\mathbb P}\left( \T_{x_1^n}{\mathbb P}([x_1^n]) >t \right)
\frac{\log t}{t}\ dt
+
2 \int_0^1 {\mathbb P}\left( \T_{x_1^n}{\mathbb P}([x_1^n]) <t \right)
\frac{-\log t}{t}\ dt =: \textup{I}\, + \, \textup{II}\,.
$$
Now we use Key-lemma \ref{MKL} and get
$$
{\mathbb P}\left( \T_{x_1^n}{\mathbb P}([x_1^n]) >t \right) \leq (1+C) e^{-\rho_1 t},\;\forall n\geq 1\,.
$$
Therefore
$$
\textup{I}\leq 2(1+C) \int_1^\infty \frac{\log t}{t} \ e^{-\rho_1 t} \ dt=:D_1'<\infty\,.
$$
For the integral II, we have the following estimates
$$
\textup{II} = 2 \left(\int_0^{\frac{1}{2}} +\int_{\frac{1}{2}}^1
\right)
{\mathbb P}\left( \T_{x_1^n}{\mathbb P}([x_1^n]) <t \right)
\frac{-\log t}{t}\ dt \leq
$$
$$
\int_0^{\frac{1}{2}}\frac{-\log t}{t}\ (1-e^{-\rho_2 t})\ dt
+
\int_{\frac{1}{2}}^1 \frac{-\log t}{t}\ dt \leq
$$
$$
-\rho_2 \int_0^{\frac{1}{2}} \log t\ dt - \int_{\frac{1}{2}}^1 \frac{\log
t}{t}\ dt := D_1''<\infty
$$
where we used Lemma \ref{tarte} to bound the first integral.
This finishes the proof for the waiting-time estimator. Concerning the
hitting-time estimator, we leave the proof to the reader. It is very
similar to the previous one except that one has to use Key-lemma
\ref{MKLbis}. \hfill $\qed$
\subsection{Proof of Proposition \ref{macbeth}}
The proof is an application of G\"artner-Ellis theorem \cite{ellis}. In particular we have
to check that the function $p\mapsto \mathcal{E}_U(p)$ is continuously differentiable and strictly convex
under assumption \eqref{avion}.
The strict convexity follows from the assumption that
the process is not reversible. As already mentioned above, this amounts to
requiring that $U$ is not physically equivalent to $U^{{\scriptscriptstyle R}}$, i.e., that $f_U^+ - f_{U^{{\scriptscriptstyle R}}}^+$ is not a
co-boundary. The open interval $(\underline{q},\overline{q})$ is defined by
$\underline{q}=\inf_{q\in\mathbb R}=\lim_{p\to-\infty} {\mathcal E}_U'(p)$ and
$\overline{q}=\sup_{q\in\mathbb R}=\lim_{p\to+\infty} {\mathcal E}_U'(p)$. These limits exist by convexity
arguments.
We refer to \cite{israel} from which one can deduce these classical facts on differentiability and
convexity of the pressure function. \hfill $\qed$
\subsection{Proof of Theorem \ref{pouic}}
We prove formula \eqref{boulgakov}.
We first deal with $0<p<1$. The case $-1<p<0$ is obtained by a similar
reasoning, so we omit the proof. The case $p=0$ is trivial.
We observe that
$$
\mathbb E_{{\mathbb P}\!\times\!{\mathbb P}}\left(e^{p\dot{{\mathcal S}}^{{\scriptscriptstyle W}}_n}\right)=
\sum_{x_1^n} {\mathbb P}([x_1^n])^{p+1} {\mathbb P}([x_n^1])^{-p}\
\mathbb E_{\mathbb P}\left[\left(\frac{Y_n}{Z_n}\right)^p\right]
$$
where $Y_n:=\T_{x_n^1} {\mathbb P}([x_n^1])$,
$Z_n:=\T_{x_1^n} {\mathbb P}([x_1^n])$.
We then have
$$
\mathbb E_{\mathbb P}\left[\left(\frac{Y_n}{Z_n}\right)^p\right] =
\int_0^\infty dy \int_0^\infty dz\
\left(\frac{y}{z}\right)^p
\ {\mathbb P}\{Y_n\in dy,Z_n\in dz\}
$$
\begin{equation}
= \int_0^1 dy\int_0^1 dz\
\left(\frac{y}{z}\right)^p
\ {\mathbb P}\{Y_n\in dy,Z_n\in dz\}
+
\int_1^\infty dy \int_1^\infty dz\
\left(\frac{y}{z}\right)^p \ {\mathbb P}\{Y_n\in dy,Z_n\in dz\}
\label{cigare}
\end{equation}
We obtain the obvious upper bound
\begin{eqnarray}
\nonumber
\eqref{cigare} & \leq &
\int_0^1 dy\int_0^1 dz\
\frac{1}{z^p}
\ {\mathbb P}\{Y_n\in dy,Z_n\in dz\}
+
\int_1^\infty dy \int_1^\infty dz\
y^p \ {\mathbb P}\{Y_n\in dy,Z_n\in dz\}
\\
& \leq & \mathbb E_{{\mathbb P}}\left(\frac{1}{Z_n^p}\right) +\mathbb E_{{\mathbb P}}(Y_n^p)\,.
\label{briquet}
\end{eqnarray}
We get easily the lower bound
\begin{eqnarray}
\nonumber
\eqref{cigare} & \geq &
\int_1^\infty dy \int_1^\infty dz\
\frac{1}{z^p} \ {\mathbb P}\{Y_n\in dy,Z_n\in dz\}
\\
& \geq & \mathbb E_{{\mathbb P}}\left(\frac{1}{Z_n^p} {{\mathit 1} \!\!\>\!\! I} \{Z_n\geq 1\} \right)
\label{allu}
\end{eqnarray}
where ${{\mathit 1} \!\!\>\!\! I} \{\cdot\}$ denotes the indicator function.
Proving Theorem \ref{pouic} for $0<p<1$ is thus reduced to proving that
the rhs in \eqref{briquet}
is bounded above by a positive number independent of $n$, and that the rhs
in \eqref{allu} is bounded below
by a positive number independent of $n$.
Let us start with an upper bound for $\mathbb E_{{\mathbb P}}(Y_n^p)$. We have
$$
\mathbb E_{{\mathbb P}}(Y_n^p)= p \int_0^\infty y^{p-1} {\mathbb P}\{\T_{x_n^1}
{\mathbb P}([x_n^1])>y\}\ dy\,.
$$
By using Key-lemma \ref{MKL} with $a_1^n=x_n^1$, we obviously have
${\mathbb P}\{\T_{x_n^1}
{\mathbb P}([x_n^1])>y\}<A e^{-By}$ for some $A,B>0$.
Let us now upper-bound $\mathbb E_{{\mathbb P}}\left(\frac{1}{Z_n^p}\right)$.
We have
$$
\mathbb E_{{\mathbb P}}\left(\frac{1}{Z_n^p}\right)=
|p| \left(\int_0^{\frac{1}{2}}+\int_{\frac{1}{2}}^\infty\right)
z^{-|p|-1} {\mathbb P}\{\T_{x_1^n} {\mathbb P}([x_1^n])\leq z\}\ dz\,.
$$
The integral from $\frac{1}{2}$ to $\infty$ is bounded above by
$\int_{\frac{1}{2}}^\infty z^{-|p|-1} dz<\infty$.
To bound the other integral we use Lemma \ref{tarte}:
$$
\int_0^{\frac{1}{2}}z^{-|p|-1} {\mathbb P}\{\T_{x_1^n} {\mathbb P}([x_1^n])\leq z\}\
dz\leq \int_0^{\frac{1}{2}} \frac{1-e^{-\rho_2 z}}{z^{|p|+1}} dz
<\infty\,.
$$
We now estimate from below $\mathbb E_{{\mathbb P}}(Y_n^p {{\mathit 1} \!\!\>\!\! I} \{Y_n\geq 1\})$.
We have
$$
\mathbb E_{{\mathbb P}}(Y_n^p {{\mathit 1} \!\!\>\!\! I} \{Y_n\geq 1\})=
|p| \int_1^\infty y^{-|p|-1} {\mathbb P}\{\T_{x_n^1} {\mathbb P}([x_n^1])\leq y\}\
dy\,.
$$
By Key-lemma \ref{MKL} with $a_1^n=x_1^n$ we have
$$
{\mathbb P}\{\T_{x_n^1} {\mathbb P}([x_n^1])\leq y\} \geq 1-(1+C e^{-cn}) e^{-\rho_1
y}\,.
$$
Observe that $1-(1+C e^{-cn}) e^{-\rho_1 y} \geq 1-(1+C e^{-cn})
e^{-\rho_1}$
for all $y\geq 1$ and for all $n\geq 1$.
Therefore
$$
\mathbb E_{{\mathbb P}}(Y_n^p {{\mathit 1} \!\!\>\!\! I} \{Y_n\geq 1\}) \geq 1-(1+C e^{-cn}) e^{-\rho_1}>0
$$
provided that $n$ is large enough.
Recapitulating, we proved that for all $0<p<1$ and all $n$ large enough
$$
E^{-1}\leq \mathbb E_{\mathbb P}\left[\left(\frac{Y_n}{Z_n}\right)^p\right] \leq E
$$
for some $E>0$ independent of $n$ and $x_1^n$. Hence, for all $0<p<1$,
we get
$$
\lim_{n\rightarrow\infty}\frac{1}{n}\log \mathbb E_{{\mathbb P}\!\times\!{\mathbb P}}\left(e^{p\dot{{\mathcal S}}^{{\scriptscriptstyle W}}_n}\right)=
\lim_{n\rightarrow\infty}\frac{1}{n}\log \sum_{x_1^n} {\mathbb P}([x_1^n])^{p+1}
{\mathbb P}([x_n^1])^{-p}=
\mathcal{E}_U(p)\,.
$$
The last equality follows obviously from \eqref{gibbsbis} and
\eqref{porc}.
\bigskip
We now turn to the case $|p|\geq 1$. We only deal with the case
$p\geq 1$ since the case $p\leq -1$ is obtained by the same reasoning.
We have
\begin{eqnarray}
\mathbb E_{\mathbb P}\left[\left(\frac{Y_n}{Z_n}\right)^p\right] & \geq &
p \int_0^1 \frac{1}{y^{p+1}}\ {\mathbb P}\{\T_{x_1^n} {\mathbb P}([x_1^n])\leq y\}\ dy\\
& \geq &
p \int_0^1 \frac{1}{y^{p+1}}\ (1-(1+C e^{-cn}) e^{-\rho_1 y})\ dy\\
& = &+ \infty
\end{eqnarray}
for $n$ large enough and where we used Key-lemma \ref{MKL} to get the
second inequality.
To prove \eqref{pelleas}, we apply a variant of G\"artner-Ellis theorem found in
\cite{PS}. To this end, we use formula \eqref{boulgakov} and the differentiability/convexity
properties of the function $p\mapsto {\mathcal E}_U(p)$. We have to restrict to
the interval $(c_-,c_+)$ where ${\mathcal W}_U$ is finite and coincides with ${\mathcal E}_U$.
\hfill $\qed$
|
{
"timestamp": "2005-07-11T12:09:00",
"yymm": "0503",
"arxiv_id": "math-ph/0503071",
"language": "en",
"url": "https://arxiv.org/abs/math-ph/0503071"
}
|
\section{Introduction}
The idea of a generalized complex structure -- a concept which interpolates between complex and symplectic structures -- seems to provide a differential geometric language in which some of the structures of current interest in string theory fit very naturally. There is an associated notion of \emph{generalized K\"ahler manifold} which essentially consists of a pair of commuting generalized complex structures. A remarkable theorem of Gualtieri \cite{Gu} shows that it has an equivalent interpretation in standard geometric terms: a manifold with two complex structures $I_+$ and $I_-$; a metric $g$, Hermitian with respect to both; and connections $\nabla_+$ and $\nabla_-$ compatible with these structures but with skew torsion $db$ and $-db$ respectively for a $2$-form $b$. This so-called \emph{bihermitian structure} appeared in the physics literature as long ago as 1984 \cite{R} as a target space for the supersymmetric $\sigma$-model and in the pure mathematics literature more recently (\cite{AGG} for example) in the context of the integrability of the canonical almost complex structures defined by the Weyl tensor of a Riemannian four-manifold. The theory has suffered from a lack of interesting examples.
The first purpose of this paper is to use the generalized complex structure approach to find non-trivial explicit examples on ${\mathbf C}{\rm P}^2$ and ${\mathbf C}{\rm P}^1\times {\mathbf C}{\rm P}^1$. We use an approach to generalized K\"ahler structures of generic type which involves closed $2$-forms satisfying algebraic conditions. This is in principle much easier than trying to write down the differential-geometric data above. What we show is that every $SU(2)$-invariant K\"ahler metric on ${\mathbf C}{\rm P}^2$ or the Hirzebruch surface ${\mathbf F}_2$ generates naturally a generalized K\"ahler structure, where for ${\mathbf F}_2$ (which is diffeomorphic to $S^2\times S^2$), the complex structures $I_+,I_-$ are equivalent to ${\mathbf F}_0={\mathbf C}{\rm P}^1\times {\mathbf C}{\rm P}^1$.
The second part of the paper shows that a bihermitian structure on a $4$-manifold (where $I_+$ and $I_-$ define the same orientation) defines naturally a bihermitian structure on the moduli space of solutions to the anti-self-dual Yang-Mills equations, and this gives another (less explicit) source of examples.
What appears naturally in approaching these goals is the appearance of holomorphic Poisson structures, and in a way the main point of the paper is to bring this aspect into the foreground. It seems as if this type of differential geometry is related to complex Poisson manifolds in the way in which hyperk\"ahler metrics are adapted to complex symplectic manifolds. Yet our structures are more flexible -- like K\"ahler metrics they can be changed in the neighbourhood of a point. The link with Poisson geometry occurs in three interlinking ways:
\begin{itemize}
\item
a holomorphic Poisson structure defines a particular type of generalized complex structure (see \cite{Gu}),
\item
the skew form $g([I_+,I_-]X,Y)$ for the bihermitian metric is of type $(2,0)+(0,2)$ and defines a holomorphic Poisson structure for either complex structure $I_+$ or $I_-$ (in the four-dimensional case this was done in \cite{AGG}),
\item
a generalized complex structure $J:T\oplus T^*\rightarrow T\oplus T^*$ defines by restriction a homomorphism $\pi:T^*\rightarrow T$ which is a real Poisson structure (this has been noted by several authors, see \cite{AB}).
\end{itemize}
We may also remark that Gualtieri's deformation theorem \cite{Gu} showed that interesting deformations of complex manifolds as generalized complex manifolds require the existence of a holomorphic Poisson structure.
We address all three Poisson-related issues in the paper. The starting point for our examples is the generalized complex structure determined by a complex Poisson surface (namely, a surface with an anticanonical divisor) and we solve the equations for a second generalized complex structure which commutes with this one. When we study the moduli space of instantons we show that the holomorphic Poisson structures defined by $g([I_+,I_-]X,Y)$ are the canonical ones studied by Bottacin \cite{Bot1}. Finally we examine the symplectic leaves of the real Poisson structures $\pi_1,\pi_2$ on the moduli space.
The structure of the paper is as follows. We begin by studying generalized K\"ahler manifolds as a pair $J_1,J_2$ of commuting generalized complex structures, and we focus in particular on the case where each $J_1,J_2$ is the B-field transform of a symplectic structure -- determined by a closed form $\exp (B+i\omega)$ -- giving a convenient algebraic form for the commuting property. We then implement this to find the two examples. In the next section we introduce the bihermitian interpretation and prove that $g([I_+,I_-]X,Y)$ does actually define a holomorphic Poisson structure.
The following sections show how to introduce a bihermitian structure on the moduli space ${\mathcal M}$ of gauge-equivalence classes of solutions to the anti-self-dual Yang-Mills equations. At first glance this seems obvious -- we have two complex structures $I_+,I_-$ on $M$ and hence two complex structures on ${\mathcal M}$, since ${\mathcal M}$ is the moduli space of $I_+$- or $I_-$-stable bundles, and we have a natural ${\mathcal L}^2$ metric. This would be fine for a K\"ahler metric but not in the non-K\"ahler case. Here L\"ubke and Teleman \cite{LT} reveal the correct approach -- one chooses a different horizontal to the gauge orbits in order to define the metric on the quotient. In our case we have two complex structures and two horizontals and much of the manipulation and integration by parts which occurs in this paper is caused by this complication.
One aspect we do not get is a natural pair of commuting generalized complex structures on ${\mathcal M}$ -- we obtain the differential geometric data above, and an exact $3$-form $db$, but not a natural choice of $b$. We get a generalized K\"ahler structure only modulo a closed B-field on ${\mathcal M}$. This suggests that ${\mathcal M}$ is not, at least directly, a moduli space of objects defined solely by one of the commuting generalized complex structures on $M$, but there is clearly more to do here.
We give finally a quotient construction which also demonstrates the problem of making a generalized K\"ahler structure descend to the quotient. This procedure, analogous to the hyperk\"ahler quotient, could be adapted to yield the bihermitian metric on ${\mathcal M}$ in the case $M$ is a $K3$ or torus. Unfortunately we have not found a quotient construction for the instanton moduli space which works in full generality, but this might be possible by using framings on the anticanonical divisor.
\vskip .5cm
\noindent{{\bf Acknowledgements:}} The author wishes to thank M. Gualtieri, G. Cavalcanti and V. Apostolov for useful discussions.
\section{Generalized K\"ahler manifolds}
\subsection{Basic properties}
The notion of a generalized K\"ahler structure was introduced by M. Gualtieri in \cite{Gu}, in the context of the generalized complex structures defined by the author in \cite{Hit}. Recall that ``generalized geometry" consists essentially of replacing the tangent bundle $T$ of a manifold by $T\oplus T^*$ with its natural indefinite inner product $$(X+\xi,X+\xi)=-i_X\xi,$$ and the Lie bracket on sections of $T$ by the Courant bracket
$$[X+\xi, Y+\eta]=
[X,Y]+\mathcal{L}_{X}\eta -\mathcal{L}_{Y}\xi -\frac{1}{2}
d(i_{X}\eta -i_{Y}\xi)$$
on sections of $T\oplus T^*$. One then introduces additional structures on $T\oplus T^*$ compatible with these.
A \emph{generalized complex structure} is a complex structure $J$ on $T\oplus T^*$ such that $J$ is orthogonal with respect to the inner product and with the integrability condition that if $A,B$ are sections of $(T\oplus T^*)\otimes \mathbf{C}$ with $JA=iA,JB=iB$, then $J[A,B]=i[A,B]$ (using the Courant bracket). The standard examples are a complex manifold
where $$J_1=\pmatrix {I&0\cr
0& -I}$$
and a
symplectic manifold where $$J_2=\pmatrix {0&-\omega^{-1}\cr
\omega & 0}.$$
The $+i$ eigenspace of $J$ is spanned by $\{\dots, \partial/\partial z_j\dots, \dots, d\bar z_k,\dots\}$ in the first case and $\{\dots, \partial/\partial x_j- i \sum\omega_{jk}dx_k,\dots\}$ in the second.
Another example of a generalized complex manifold is a holomorphic Poisson manifold -- a complex manifold with a holomorphic bivector field
$$\sigma=\sum\sigma^{ij}\frac{\partial}{\partial z_i}\wedge\frac{\partial}{\partial z_j}$$
satisfying the condition $[\sigma,\sigma]=0$, using the Schouten bracket. This defines a generalized complex structure where the $+i$ eigenspace is
$$ E=\left[\dots, \frac{\partial}{\partial z_j},\dots, d\bar z_k+\sum_{\ell}\bar\sigma^{k\ell}\frac{\partial}{\partial \bar z_{\ell}},\dots \right],$$
and if $\sigma=0$ this gives a complex structure.
Gualtieri observed (see also \cite{AB}) that the real bivector defined by the upper triangular part of $J:T\oplus T^*\rightarrow T\oplus T^*$ is always a real Poisson structure. In the symplectic case this is the canonical Poisson structure and in the complex case it is zero. Both facts show that Poisson geometry plays a central role in this area, a feature we shall see more of later.
\vskip .25cm
The algebraic compatibility condition between $\omega$ and $I$ to give a K\"ahler manifold (i.e. that $\omega$ be of type $(1,1)$) can be expressed as $J_1J_2=J_2J_1$ and this is the basis of the definition of a \emph{generalized} K\"ahler structure:
\begin{definition} \label{GKdef} A \emph{generalized K\"ahler structure} on a manifold consists of two commuting generalized complex structures $J_1,J_2$ such that the quadratic form $(J_1J_2A,A)$ on $T\oplus T^*$ is definite.
\end{definition}
\vskip .25cm
At a point, a generalized complex structure can also be described by a form $\rho$: the $+i$ eigenspace bundle $E$ consists of the $A=X+\xi\in (T\oplus T^*)\otimes \mathbf{C}$ which satisfy $A\cdot \rho=i_X\rho+\xi\wedge\rho=0$. For the symplectic structure $\rho=\exp i\omega$, and for a complex structure with complex coordinates $z_1,\dots,z_n$ we take the $n$-form $\rho=dz_1\wedge dz_2\wedge \dots\wedge dz_n$. The structure is called even or odd according to whether $\rho$ is an even or odd form. The generic even case is the so-called B-field transform of a symplectic structure where
$$\rho=\exp \beta = \exp (B+i\omega)$$
and $B$ is an arbitrary $2$-form. The generalized complex structure defined by a holomorphic Poisson structure $\sigma$ is of this type if $\sigma$ defines a non-degenerate skew form on $(T^*)^{1,0}$; then $B+i\omega$ is its inverse.
If $\rho$ extends smoothly to a neighbourhood of the point, and is {\it closed}, then the integrability condition for a generalized complex structure holds.
\vskip .25cm
The following lemma is useful for finding generalized K\"ahler structures where both are of this generic even type (which requires the dimension of $M$ to be of the form $4k$). We shall return to this case periodically to see how the various structures emerge concretely.
\begin{lem} \label{commute} Let $\rho_1=\exp \beta_1, \rho_2=\exp\beta_2$ be closed forms defining generalized complex structures $J_1,J_2$ on a manifold of dimension $4k$. Suppose that
$$(\beta_1-\beta_2)^{k+1}=0=(\beta_1-\bar\beta_2)^{k+1}$$
and $(\beta_1-\beta_2)^{k}$ and $(\beta_1-\bar\beta_2)^{k}$ are non-vanishing. Then $J_1$ and $J_2$ commute.
\end{lem}
\begin{lemprf} Suppose that $(\beta_1-\beta_2)^{k+1}=0$ and $(\beta_1-\beta_2)^{k}$ is non-zero. Then the $2$-form $\beta_1-\beta_2$ has rank $2k$, i.e. the dimension of the space of vectors $X$ satisfying $i_X(\beta_1-\beta_2)=0$ is $2k$. Since $i_X 1+\xi\wedge 1=0$ if and only if $\xi=0$, this means that the space of solutions $A=X+\xi$ to
$$A\cdot \exp (\beta_1-\beta_2)=0=A\cdot 1$$
is $2k$-dimensional. Applying the invertible map $\exp \beta_2$, the same is true of solutions to
$$A\cdot \exp \beta_1=0=A\cdot \exp \beta_2.$$
This is the intersection $E_1\cap E_2$ of the two $+i$ eigenspaces. Repeating for $\beta_1-\bar\beta_2$ we get $E_1\cap \bar E_2$ to be $2k$-dimensional. These two bundles are common eigenspaces of $(J_1, J_2)$ corresponding to the eigenvalues $(i,i)$ and $(i,-i)$ respectively. Together with their conjugates they decompose $(T\oplus T^*)\otimes \mathbf{C}$ into a direct sum of common eigenspaces of $J_1,J_2$, thus $J_1J_2=J_2J_1$ on every element.
\end{lemprf}
We also need to address the definiteness of $(J_1J_2 A,A)$ in Definition 1. Let $V_+$ be the $-1$ eigenspace of $J_1J_2$ (the notation signifies $J_1=+J_2$ on $V_+$). This is
$$E_1\cap E_2\oplus \bar E_1\cap \bar E_2.$$
If $X$ is a vector in the $2k$-dimensional space defined by $i_X(\beta_1-\beta_2)=0$ then $A=X-i_X\beta_2$ satisfies $A\cdot \exp \beta_1=0=A\cdot \exp \beta_2$, i.e. $A\in E_1\cap E_2$. But then
\begin{equation}
(A+\bar A,A+\bar A)=i_X\beta_2(\bar X)+i_{\bar X}\bar\beta_2(X)=(\beta_2-\bar\beta_2)(X,\bar X)
\label{posit}
\end{equation}
so we need to have this form to be definite. Note that interchanging the roles of $\beta_1,\beta_2$, this is the same as $(\beta_1-\bar\beta_1)(X,\bar X)$ being definite.
\subsection{Hyperk\"ahler examples}
A hyperk\"ahler manifold $M$ of dimension $4k$ provides a simple example of a generalized K\"ahler manifold. Let $\omega_1,\omega_2,\omega_3$ be the three K\"ahler forms corresponding to the complex structures $I,J,K$ and set
$$\beta_1=\omega_1+\frac{i}{2}(\omega_2-\omega_3),\quad \beta_2=\frac{i}{2}(\omega_2+\omega_3).$$
Then $\beta_1-\beta_2=\omega_1-i\omega_3$ is a $J$-holomorphic symplectic $2$-form and so clearly satisfies the conditions of Lemma \ref{commute}. Similarly $\beta_1-\bar\beta_2= \omega_1+i\omega_2$ is holomorphic symplectic for $K$. The vectors $X$ satisfying $i_X(\beta_1-\beta_2)=0$ are the $(0,1)$ vectors for $J$, and $\beta_2-\bar\beta_2=i(\omega_2+\omega_3)$ whose $(1,1)$ part with respect to $J$ is $i\omega_2$. Thus
$$(\beta_2-\bar\beta_2)(X,\bar X)=i\omega_2(X,\bar X)$$
which is positive definite. Thus a hyperk\"ahler manifold satisfies all the conditions to be generalized K\"ahler.
\vskip .25cm
D. Joyce observed (see \cite{AGG}) that one can deform this example. Let $f$ be a smooth real function on $M$, and use the symplectic form $\omega_1$ to define a Hamiltonian vector field. Now integrate it to a one-parameter group of symplectic diffeomorphisms $F_t:M\rightarrow M$, so that $F_t^*\omega_1=\omega_1$. Define
$$\beta_1=\omega_1+\frac{i}{2}(\omega_2-F_t^*\omega_3),\quad \beta_2=\frac{i}{2}(\omega_2+F_t^*\omega_3),$$
and then
$$\beta_1-\beta_2=\omega_1-iF^*_t\omega_3=F^*_t(\omega_1-i\omega_3).$$
This is just the pull-back by a diffeomorphism of $\omega_1-i\omega_3$ so also satisfies the constraint of Lemma \ref{commute}. We also have
$\beta_1-\bar\beta_2= \omega_1+i\omega_2$ which is just the same as the hyperk\"ahler case, so both constraints hold. If $t$ is sufficiently small this will still give a positive definite metric.
This simple example at least shows the flexibility of the concept -- we can find a new structure from an arbitrary real function, somewhat analogous to the addition of $\partial\bar\partial f$ to a K\"ahler form. In the compact four-dimensional situation this type of structure restricts us to tori and K3 surfaces. We give next an explicit example on the projective plane.
\subsection {Example: the projective plane} \label{cp2}
The standard $SU(2)$ action on $\mathbf{C}^2$ extends to ${\mathbf C}{\rm P}^2$ and the invariant $2$-form $dz_1\wedge dz_2$ extends to a meromorphic form with a triple pole on the line at infinity. Its inverse $\partial/\partial z_1\wedge \partial/\partial z_2$ is a holomorphic Poisson structure with a triple zero on the line at infinity. We shall take the generalized complex structure $J_1$ to be defined by this, and seek an $SU(2)$-invariant generalized complex structure $J_2$ defined by $\exp (B+i\omega)$ in such a way that the pair define a generalized K\"ahler structure. On $\mathbf{C}^2$ the Poisson structure is non-degenerate, so the generalized complex structure on that open set is defined by the closed form $\rho_1=\exp dz_1dz_2$.
\vskip .25cm
We begin by parametrizing $\mathbf{C}^2\setminus \{0
\}$ by $\mathbf{R}^+ \times SU(2)$:
$$\pmatrix {z_1\cr
z_2}=\pmatrix {z_1& -\bar z_2\cr
z_2 & \bar z_1}\pmatrix {1\cr
0}=rA\pmatrix {1\cr
0}.$$
Then, with the left action, the entries of $A^{-1}dA=A^*dA$ are invariant $1$-forms. We calculate
$$A^*dA=-\frac{dr}{r}I+\frac{1}{r^2}\pmatrix {\bar z_1 dz_1+\bar z_2 dz_2& -\bar z_1 d\bar z_2+\bar z_2 d\bar z_1\cr
z_1 d z_2- z_2 d z_1 & z_1 d\bar z_1+ z_2 d \bar z_2}=\pmatrix {i\sigma_1& -\sigma_2-i\sigma_3\cr
\sigma_2+i\sigma_3 & -i\sigma_1}$$
where
\begin{eqnarray*}
v_1&=&r^{-1}dr+i\sigma_1=(\bar z_1 dz_1+\bar z_2 dz_2)/r^2\\
v_2&=&\sigma_2+i\sigma_3= (z_1 d z_2- z_2 d z_1)/r^2
\end{eqnarray*}
and these give a basis for the $(1,0)$-forms. We see that
$2dr=r(v_1+\bar v_1)$ so that
$$\partial r=rv_1/2,\quad \bar\partial r=r\bar v_1/2$$ and hence
$$\partial v_1=0,\quad \bar\partial v_1= dv_1=id\sigma_1=2i\sigma_2\sigma_3=-v_2\bar v_2.$$
Furthermore
$$\partial v_2=v_1v_2,\quad \bar\partial v_2=-\bar v_1 v_2,
\quad \bar\partial(v_1v_2)=v_1\bar v_1v_2.$$
\vskip .25cm
We look for invariant solutions to the generalized K\"ahler equations where
$$\rho_1=\exp \beta_1=\exp[dz_1dz_2]=\exp[r^2v_1v_2]$$
and $\rho_2=\exp \beta_2$ where
$$\beta_2=\sum_{i,j} H_{ij}v_i\bar v_j+ \lambda v_1v_2+\mu \bar v_1\bar v_2$$
(with $H_{ij},\lambda$ and $\mu$ functions of $r$) is a general invariant $2$-form.
The algebraic compatibility conditions from Lemma 1 are:
$$(\beta_2-\beta_1)^2=0=(\beta_2-\bar\beta_1)^2$$
which gives on subtraction
$$\beta_2(v_1v_2-\bar v_1\bar v_2)=0$$
or equivalently $\lambda=\mu$. We then get
$$0=\beta_2^2-2\beta_2\beta_1=(\sum H_{ij}v_i\bar v_j)^2+2\lambda^2v_1v_2\bar v_1\bar v_2-2\lambda r^2v_1v_2\bar v_1\bar v_2$$
or equivalently
\begin{equation}
\det H=\lambda(\lambda-r^2)
\label{detH}
\end{equation}
\vskip .5cm
We also know that $d\beta_2=0$ so that
\begin{eqnarray*}
\bar\partial (H_{ij}v_i\bar v_j)+\partial \lambda \bar v_1\bar v_2+\lambda \partial (\bar v_1\bar v_2)&=&0\\
\partial (H_{ij}v_i\bar v_j)+\bar\partial \lambda v_1 v_2+\lambda \bar\partial (v_1v_2)&=&0
\end{eqnarray*}
But $H$ and $\lambda$ are functions of $r$ and so from the first equation, expanding and collecting terms in $v_1\bar v_1\bar v_2$ we obtain
\begin{equation}
rH_{12}'+2H_{12}=r\lambda'-2\lambda
\label{H12}
\end{equation}
while collecting terms in $\bar v_1 v_2\bar v_2$ yields
\begin{equation}
rH_{22}'=2H_{11}
\label{H22}
\end{equation}
The second equation gives (\ref{H22}) again and also
\begin{equation}
rH_{21}'+2H_{21}=-r\lambda'+2\lambda
\label{H21}
\end{equation}
We can solve these by quadratures: from (\ref{H12}) we get
\begin{eqnarray*}
r^2H_{12}&=&\int^r (s^2\lambda' -2s\lambda)ds=r^2\lambda-4\int_a^r s\lambda ds\\
r^2H_{21}&=&-r^2\lambda+4\int_{a'}^r s\lambda ds.
\end{eqnarray*}
If we set
$$L(r)=\int_a^r s\lambda ds$$
then $\lambda=L'/r$ and
$$r^2H_{12}=rL'-4L,\quad
r^2H_{21}= -rL'+4L+b$$
and then $\det H=\lambda(\lambda-r^2)$ gives
$$H_{11}H_{22}=8\frac{LL'}{r^3}-16\frac{L^2}{r^4}-L'r+b\frac{L'}{r^3}-4b\frac{L}{r^4}.$$
Substituting $rH_{22}'=2H_{11}$ from (\ref{H22}) and integrating by parts leads to
\begin{equation}
H_{22}^2=16\frac{L^2}{r^4}+4b\frac{L}{r^4}-4L+c
\label{Hform}
\end{equation}
Thus an arbitrary complex function $L$ and three constants of integration $a,b,c$ give the general solution to the equations. Note for comparison that an $SU(2)$-invariant K\"ahler metric involves one \emph{real} function of $r$ -- the invariant K\"ahler potential.
\vskip .25cm
There is a lot of choice here but to produce an example let us take for simplicity $a=a'=0$ so that $H_{12}=-H_{21}$ and therefore $b=0$, and take $c=0$ so that
\begin{equation}
H_{22}^2=16\frac{L^2}{r^4}-4L
\label{examp1}
\end{equation}
Let $L$ be real, then so is $\lambda$ and $H_{12}$. If $L$ negative then $H_{22}^2$ is positive from (\ref{examp1}). This means that $H_{22}$ is real, and hence from (\ref{H22}) so is $H_{11}$. Choose the positive square root for $H_{22}$.
Now $\beta_2=\sum H_{ij}v_i\bar v_j+ \lambda v_1v_2+\mu \bar v_1\bar v_2$ and $\lambda$ and $H_{ij}$ are real and $H_{12}=-H_{21}$ so
\begin{equation}
\beta_2-\bar\beta_2= 2(H_{11}v_1\bar v_1+H_{22}v_2\bar v_2)
\label{ibeta2}
\end{equation}
and for this to be symplectic $H_{11}$ and $H_{22}$ must be non-zero. To get a generalized K\"ahler metric we need from (\ref{posit}) to have $(\beta_1-\bar\beta_1)(X,\bar X)$ definite on the space of vectors $X$ with $i_X(\beta_1-\beta_2)=0.$ If $\nu_1,\nu_2,\bar\nu_1,\bar\nu_2$ is the dual basis to $v_1,v_2,\bar v_1,\bar v_2$ then $X$ must be a linear combination of
\begin{equation}
\lambda\nu_1-H_{12}\bar\nu_1+H_{11}\bar \nu_2,\quad \lambda\nu_2-H_{22}\bar\nu_1-H_{12}\bar\nu_2.
\label{CPhol}
\end{equation}
Since $\beta_1-\bar\beta_1=r^2(v_1v_2-\bar v_1\bar v_2)$ this gives $(\beta_1-\bar\beta_1)(X,\bar X)$ relative to this basis as the Hermitian form
$$\pmatrix{2r^2\lambda H_{11}& \cr & 2r^2\lambda H_{22}}$$
so we also need $H_{11}$ to be positive.
\vskip .25cm
Notice now the point we have reached: $H_{11}$ and $H_{22}$ must be positive, which means that
\begin{equation}
H_{11}v_1\bar v_1+H_{22}v_2\bar v_2
\label{kform}
\end{equation}
is a positive definite Hermitian form. Moreover $rH_{22}'=2H_{11}$, and this implies that the form is K\"ahler. In fact if $\phi(r)$ satisfies $H_{22}=r\phi'/2$, this is the K\"ahler metric $i\partial\bar\partial \phi$, with $\phi$ as a K\"ahler potential.
Thus each $SU(2)$-invariant K\"ahler metric defines canonically, through the functions $L,\lambda$ and $H_{12}$ defined in terms of $H_{22}$, an $SU(2)$-invariant generalized K\"ahler metric on $\mathbf{C}^2\setminus \{0\}$.
\begin{prp} If the K\"ahler metric (\ref{kform}) extends to ${\mathbf C}{\rm P}^2$, so does the generalized K\"ahler structure.
\end{prp}
\begin{prf} Since $\beta_1^{-1}$ is a global holomorphic Poisson structure on ${\mathbf C}{\rm P}^2$, we know that the generalized complex structure $J_1$ extends to the whole of ${\mathbf C}{\rm P}^2$, so
we only need to check that $\beta_2$ also extends. We begin at $r=0$, the origin in $\mathbf{C}^2$. Clearly $r^2=z_1\bar z_1+z_2\bar z_2$ is smooth on $\mathbf{C}^2$. We shall use the fact that if $f(r)$ extends to a smooth function on a neighbourhood of the origin in $\mathbf{C}^2$ then $f(r)=f(0)+r^2f_1(r)$ where $f_1$ is also a smooth function.
If $g$ is the K\"ahler metric and $X=r\partial/\partial r$ the Euler vector field, then $g(X,X)=H_{11}$ is smooth on $\mathbf{C}^2$ and vanishes at the origin so
$H_{11}=r^2f_1$ for smooth $f_1>0$. The volume form of $g$ is $r^{-1}H_{11}H_{22}dr\sigma_1\sigma_2\sigma_3$ and comparing with the Euclidean volume $r^3dr\sigma_1\sigma_2\sigma_3$ we see that $H_{22}=r^2f_2$ for $f_2>0$ smooth.
Equation (\ref{examp1}) gives
$$L=\frac{r^4}{8}\left[1-\sqrt{1+(4H_{22}^2/r^4)}\right]=\frac{r^4}{8}\left[1-\sqrt{1+4f_2^2}\right]$$
and so $L=r^4 f_3$ for $f_3$ smooth.
By definition,
$\lambda=L'/r=4r^2f_3+r^3f_3'=r^2f_4$
since for any smooth $f(r)$
$$rf'=\sum x_i\frac{\partial f}{\partial x_i}$$
which is smooth. (In fact since this expression also vanishes at $0$ we have $rf'=r^2g$ for $g$ smooth.)
Since $r^2v_1v_2=dz_1dz_2$, this shows that the term $\lambda (v_1v_2+ \bar v_1\bar v_2)$ is smooth.
Now $r^2H_{12}=rL'-4L=r^5f_3'$ so $H_{12}=r^2(rf_3')=r^4f_5$ for smooth $f_5$, which means that $H_{12}v_1\bar v_2$ and $H_{21}v_2\bar v_1$ are smooth since $r^2v_1=\bar z_1 dz_1+\bar z_2 dz_2, r^2v_2= z_1 d z_2- z_2 d z_1$. Hence the form $\beta_2$ is smooth at the origin.
From (\ref{ibeta2}) the imaginary part of $\beta_2$ is nondegenerate at the origin since the K\"ahler metric is.
\vskip .25cm
As $r\rightarrow \infty$ we need to take homogeneous coordinates on ${\mathbf C}{\rm P}^2$ so that $\mathbf{C}^2$ is parametrized by
$[1,z_1,z_2]=[1/z_1,1,z_2/z_1]$, so we use local affine coordinates $w_1,w_2$ where for $z_1\ne 0$,
$$w_1=\frac{1}{z_1},\quad w_2=\frac{z_2}{z_1}.$$
The projective line at infinity is then $w_1=0$. In these coordinates we have
$$r^2=\frac{1+\vert w_2\vert^2}{\vert w_1\vert^2}$$
so $1/r^2$ is smooth and
\begin{equation}
v_1=\frac{\bar w_2dw_2}{1+\vert w_2\vert^2}-\frac{dw_1}{w_1},\quad v_2=\frac{\bar w_1dw_2}{w_1(1+\vert w_2\vert^2)}
\label{v1v2}
\end{equation}
Note here that
$$\frac{1}{r^2}v_1=\frac{\vert w_1\vert^2\bar w_2 dw_2}{(1+\vert w_2\vert^2)^2}-\frac{\bar w_1dw_1}{1+\vert w_2\vert^2}$$
is smooth at $r=\infty$, and similarly $v_2/r^2, v_1\bar v_2/r^2$ are smooth.
The coefficient of $dw_1d\bar w_1$ in $H_{11}v_1\bar v_1+H_{22}v_2\bar v_2$ is
$H_{11}/\vert w_1\vert^2$ so this is smooth and hence $r^2H_{11}=g_1$, a smooth function. Considering the coefficient of $dw_2d\bar w_2$ we see that $H_{22}$ is smooth.
Now
\begin{equation}
L=\frac{r^4}{8}\left[1-\sqrt{1+4H_{22}^2/r^4}\right]=-\frac{1}{2}\frac{H_{22}^2}{1+\sqrt{1+4H_{22}^2/r^4}}
\label{ell}
\end{equation}
which is smooth and
$g_1=r^2H_{11}=r^3H_{22}'/2$
so that differentiating (\ref{ell})
$\lambda={L'}/{r}=g_2/r^4$
where $g_2$ is smooth. This means from (\ref{v1v2}) that $\lambda(v_1\bar v_1+v_2\bar v_2)$ is smooth. Finally
$H_{12}=\lambda-{4}L/{r^2}=g_3/r^2$
where $g_3$ is smooth, and so $H_{12}v_1\bar v_2$ is smooth. Thus $\beta_2$ extends as $r\rightarrow \infty$.
The argument for $z_2\ne 0$ is the similar.
\end{prf}
\subsection{Example: the Hirzebruch surface ${\mathbf F_2}$}\label{f2}
We can apply the above formalism with different boundary conditions to the Hirzebruch surface ${\mathbf F}_2$. Recall that this is
$${\mathbf F}_2=P({\mathcal O}\oplus {\mathcal O}(-2))=P({\mathcal O}\oplus K)$$
since the canonical bundle $K$ of ${\mathbf C}{\rm P}^1$ is ${\mathcal O}(-2)$. The canonical symplectic form on $K$ extends to a meromorphic form $\beta_1$ on ${\mathbf F}_2$, and its inverse, a Poisson structure, defines the generalized complex structure $J_1$.
\vskip .25cm
On $K$ we take local coordinates $(w,z)\mapsto wdz$ where $z$ is an affine coordinate on ${\mathbf C}{\rm P}^1$. Then for each quadratic polynomial $q(z)$
$$q(z)\frac{d}{dz}$$
is a global holomorphic vector field on ${\mathbf C}{\rm P}^1$ so that
$$(w,z)\mapsto (w,wz,wz^2)$$
is a well defined map from $K$ to the cone $x_2^2=x_1x_3$ in $\mathbf{C}^3$. The map
$$(z_1,z_2)\mapsto (z_1^2,z_1z_2,z_2^2)$$
maps the quotient $\mathbf{C}^2/\pm 1$ isomorphically to this cone and the Hirzebruch surface is a compactification of the surface obtained by resolving the singularity at the origin of this cone. Our ansatz above for $\mathbf{C}^2\setminus\{0\}$ extends to the quotient which is $\mathbf{R}^+\times SO(3)$ since we were using left-invariant forms. We need to adapt in a different way to extend at $r\rightarrow 0$ which is a rational curve of self-intersection $-2$ and $r\rightarrow \infty$, a rational curve of self-intersection $+2$.
\vskip .25cm
To proceed as $r\rightarrow 0$ we change coordinates from $z_1,z_2$ to $w,z$:
$$w=z_1^2,\qquad z=z_2/z_1.$$
Then $dzdw=2dz_2dz_1$, so here we see that the standard $2$-form on $\mathbf{C}^2$ is a multiple of the canonical symplectic form on the holomorphic cotangent bundle. We find
$$r^2=\vert w\vert(1+\vert z\vert^2)$$
so in particular $r^4$ is smooth. Furthermore
\begin{equation}
2\frac{dr}{r}=\frac{1}{2}\left[\frac{dw}{w}+\frac{d\bar w}{\bar w}\right]+\frac{d(z\bar z)}{1+\vert z\vert^2}
\label{diffr}
\end{equation}
We also calculate
\begin{equation}
v_1=\frac{dw}{2w}+\frac{\bar z dz}{1+\vert z\vert^2},\quad v_2=\frac{wdz}{\vert w\vert (1+\vert z\vert^2)}
\label{v12}
\end{equation}
Thus $r^2v_1v_2$ and $r^4v_1\bar v_1$ are smooth, and
$$v_2\bar v_2=\frac{dz d\bar z}{(1+\vert z\vert^2)^2}$$
which is smooth.
\vskip .25cm
Suppose in this case that $H_{11}v_1\bar v_1+H_{22}v_2\bar v_2$ extends as a K\"ahler form. Then considering the coefficient of $dwd\bar w$, $H_{11}=r^4 f_1$ where $f_1>0$ is smooth and $H_{22}$ itself is smooth and positive. The reality conditions on $H_{ij}$ are the same as the ${\mathbf C}{\rm P}^2$ case and the constants of integration $a,a',b$ vanish as before but we now take $c$ in (\ref{Hform}) to be the limiting value $H_{22}^2(0)$. Since $H_{22}'>0$, $H_{22}^2-c>0$.
From (\ref{Hform}) we obtain
\begin{equation}
L=\frac{r^4}{8}\left[1-\sqrt{1+4(H_{22}^2-c)/r^4}\right].
\label{Lformula}
\end{equation}
We now use the familiar formula for a smooth function $f$
\begin{equation}
f(x)-f(x_0)=\sum_i(x-x_0)_i\int_0^1\frac{\partial f}{\partial x_i}(x_0+t(x-x_0))dt
\label{expand}
\end{equation}
where the coordinates $x_i$ are the real and imaginary parts of $z,w$ and we take $x_0=(z_0,0)$. From (\ref{diffr}) we calculate the derivatives
$$\frac{\partial H_{22}}{\partial w}=\frac{r}{4w}H_{22}'=\frac{1}{2w}H_{11}=\frac{r^4 f_1}{2w}=\frac{1}{2}\bar w(1+\vert z\vert^2)^2f_1$$
(since $rH_{22}'=2H_{11}$ and $H_{11}=r^4f_1$ for smooth $f_1$) and
$$\frac{\partial H_{22}}{\partial z}=\frac{r\bar z}{2(1+\vert z\vert^2)}H_{22}'=\frac{w\bar w f_1}{1+\vert z\vert^2}.$$
Putting these and their conjugates into (\ref{expand}) with $f=H_{22}$ we see that $H_{22}(x)-H_{22}(x_0)=w\bar w f_2$
for a smooth function $f_2$ and hence from the formula for $L$ above
$L=r^4f_3$ where $f_3$ is smooth. This gives
$$\lambda=\frac{L'}{r}=4r^2f_3+4r^2w\frac{\partial f_3}{\partial w}.$$ This is $r^2f_4$ where $f_4$ is smooth and so $\lambda(v_1v_2+\bar v_1\bar v_2)$ is smooth since $r^2v_1 v_2$ is smooth.
Now
\begin{equation}
H_{12}=\frac{L'}{r}-4\frac{L}{r^2}=r^3f_3'=4wr^2\frac{\partial f_3}{\partial w}=4w\vert w\vert (1+\vert z\vert^2)\frac{\partial f_3}{\partial w}
\label{h12}
\end{equation}
From (\ref{v12}) we see that $H_{12}v_1\bar v_2$ is smooth.
\vskip .25cm
In a neighbourhood of the curve $r=\infty$ we have coordinates $w'=1/w, z'=z$ and the calculations are very similar. In particular $1/r^4$ is smooth and $H_{22}$ is smooth and nonzero at infinity. Let $c'=\lim_{r\rightarrow\infty}H_{22}^2$. Then from (\ref{Lformula}) we have
$$L=-\frac{1}{4}(c'-c)+\frac{1}{r^4}g$$
where $g$ is smooth. This gives the required behaviour of $L$ and $\lambda$ for $\beta_2$ to extend to the curve at infinity.
\section{Bihermitian metrics}
\subsection{Generalized K\"ahler and bihermitian structures}
The generalized K\"ahler structures described above have a very concrete Riemannian description, owing to the following remarkable theorem of Gualtieri \cite{Gu}:
\begin{thm} \label{bi}
A generalized K\"ahler structure on a manifold $M^{2m}$ is equivalent to:
\begin{itemize}
\item
a Riemannian metric $g$
\item
two integrable complex structures $I_+,I_-$ compatible with the metric
\item
a $2$-form $b$ such that
$d^c_-\omega_-=db=-d^c_+\omega_+$
\end{itemize}
where $\omega_+,\omega_-$ are the two hermitian forms and $d^c=I^{-1}dI=i(\bar\partial-\partial)$.
\end{thm}
An equivalent description is to say that there are two connections $\nabla^+,\nabla^-$ which preserve the metric and the complex structures $I_+,I_-$ respectively and these are related to the Levi-Civita connection $\nabla$ by
\begin{equation}
\nabla^{\pm}=\nabla\pm\frac{1}{2}g^{-1}h
\label{del}
\end{equation}
where $h=db$ is of type $(2,1)+(1,2)$ with respect to both complex structures. In the K\"ahler case $I_+=I,I_-=-I$ and $b=0$.
This is the geometry introduced $20$ years ago in the physics literature \cite{R} and more recently studied by differential geometers in four dimensions as ``bihermitian metrics", as in \cite{AGG}.
\vskip.25cm
Following \cite{Gu}, to derive this data from the generalized K\"ahler structure one looks at the eigenspaces of $J_1J_2$. Since $J_1$ and $J_2$ commute, $(J_1J_2)^2=(-1)^2=1$. As before we choose $V_+$ to be the subbundle where $J_1=J_2$ and $V_-$ where $J_1=-J_2$. If the quadratic form $(J_1J_2A,A)$ is negative definite, the natural inner product on $T\oplus T^*$ is positive definite on $V_+$, and negative definite on the complementary eigenspace $V_-$. Since the signature of the quadratic form is $(2m,2m)$ each such space is $2m$-dimensional. Moreover since $T$ and $T^*$ are isotropic, $V_+\cap T=0=V_+\cap T^*$ and so $V_+$ is the graph of an invertible map from $T$ to $T^*$, i.e. a section $g+b$ of $T^*\otimes T^*$, where $g$ is the symmetric part and $b$ the skew-symmetric part. The bundle $V_+$ is preserved by $J_1$ and identified with $T$ by projection, and hence $J_1$ (or equivalently $J_2$) induces a complex structure $I_+$. Similarly on $V_-$, $J_1$ or $-J_2$ gives $T$ the complex structure $I_-$.
Conversely, as Gualtieri shows, given the bihermitian data above, the two commuting generalized complex structures are defined by
\begin{equation}
J_{1/2}=\frac{1}{2}\pmatrix{1&0\cr
b&1}\pmatrix{I_+\pm I_- & -(\omega_+^{-1}\pm\omega_-^{-1})\cr
\omega_+\pm \omega_- & -(I_+^*\pm I_-^*)}\pmatrix{1&0\cr
-b&1}
\label{J12}
\end{equation}
\vskip .25cm
Our standard examples are constructed from closed forms $\rho_1=\exp\beta_1,\rho_2=\exp\beta_2$, so we look next at how the bihermitian structure is encoded in these.
The identification of $T$ with $V_+$ can be written as $X\mapsto X+(g(X,-)+b(X,-))$. If $X$ is a $(1,0)$-vector with respect to $I_+$ then this is
$$X+\xi= X+i_X(b- i\omega_+)$$
where $\omega_+$ is the Hermitian form for $I_+$. If this lies in $E_1$, it annihilates $\exp \beta_1$, so $i_X\beta_1+ i_X(b- i\omega_+)=0$. Thus $\beta_1+b-i\omega_+$ is of type $(0,2)$, and similarly for $E_2$. Thus there are $(0,2)$-forms $\gamma_1,\gamma_2$ such that
$$\beta_1=-b+i\omega_+ +\gamma_1,\quad \beta_2=-b+i\omega_+ +\gamma_2.$$
Since $\beta_1,\beta_2$ are closed this means that $\gamma=\bar\beta_1-\bar\beta_2=\bar\gamma_1-\bar\gamma_2$ is a holomorphic $(2,0)$-form with respect to $I_+$.
The form $(\beta_1-\beta_2)$ defines the complex structure $I_+$ -- the $(1,0)$ vectors are the solutions to $i_X(\beta_1-\beta_2)=0$ and the metric on such $(1,0)$-vectors is given by $(\beta_1-\bar\beta_1)(X,\bar X)$.
Changing to $V_-$, the identification with $T$ is $X\mapsto X+(-g(X,-)+b(X,-))$ and then $\beta_1=-b+i\omega_- +\delta_1,\quad \bar\beta_2=-b+i\omega_- +\delta_2.$
where $\delta_1,\delta_2$ are $(0,2)$-forms with respect to $I_-$.
\vskip .25cm
In four real dimensions we now give the precise relationship between the bihermitian description and the generalized K\"ahler one. First note that $\omega_-^{1,1}$ is self-dual and type $(1,1)$ so there is a real smooth function $p$ such that
$$\omega_-^{1,1}=p\omega_+.$$
Moreover since $\omega_+^2=\omega_-^2$, $\vert p\vert \le 1$.
From above we have
\begin{eqnarray*}
\beta_1&=&-b+i\omega_++\gamma_1=-b+i\omega_-+\delta_1\\
\beta_2&=&-b+i\omega_++\gamma_2=-b-i\omega_-+\bar\delta_2
\end{eqnarray*}
where $\gamma_1,\gamma_2$ are $(0,2)$ with respect to $I_+$ and $\delta_1,\delta_2$ are $(0,2)$ with respect to $I_-$. We let $\bar\gamma=\gamma_1-\gamma_2$ be the closed $(0,2)$ form, non-vanishing since $\beta_1-\beta_2$ is non-zero from Lemma 1.
\begin{prp} \label{4formulas}In the terminology above,
\begin{itemize}
\item
$\beta_1=b+i\omega_+-(p-1)\bar\gamma/2$
\item
$\beta_2=b+i\omega_+-(p+1)\bar\gamma/2$
\item
$\omega_-=p\omega_++i(p^2-1)\bar\gamma/4-i(p^2-1)\gamma/4$
\end{itemize}
\end{prp}
\begin{prf} Since we are in two complex dimensions, there are functions $q_1,q_2$ such that the $(0,2)$ forms $\gamma_1,\gamma_2$ are given by $\gamma_1=q_1\bar\gamma, \gamma_2=q_2\bar\gamma$ and since
$$\beta_1-\beta_2=\gamma_1-\gamma_2=\bar\gamma$$
we have $q_1-q_2=1$. Similarly $\omega_-^{0,2}=r\bar\gamma$.
We have $\omega_-^2=\omega_+^2$ since this is the Riemannian volume form and
$\omega_-=p\omega_++r\gamma+\bar r\bar\gamma$ since it is self-dual, hence
$$\omega_+^2=\omega_-^2=(p\omega_++r\gamma+\bar r\bar\gamma)^2=p^2\omega_+^2+2\vert r\vert^2\gamma\bar\gamma$$
and so
\begin{equation}
(1-p^2)\omega_+^2=2\vert r\vert^2\gamma\bar\gamma.
\label{eqA}
\end{equation}
Also $i\omega_++\gamma_1=i\omega_-+\delta_1$ and $\delta_1^2=0$ since it is of type $(0,2)$ relative to $I_-$ so
$$0=(i\omega_++\gamma_1-i\omega_-)^2=(i\omega_++q_1\gamma-i[p\omega_++r\gamma+\bar r\bar\gamma])^2$$
and this gives
\begin{equation}
-(1-p)^2\omega_+^2=2i(q_1-ir)\bar r\gamma\bar\gamma.
\label{eqB}
\end{equation}
The same argument for $\delta_2$ gives
\begin{equation}
(1+p)^2\omega_+^2=2i(q_2+ir)\bar r\gamma\bar\gamma.
\label{eqC}
\end{equation}
From (\ref{eqA}),(\ref{eqB}),(\ref{eqC}) we obtain
$$q_1=\frac{2ir}{p+1},\quad q_2=\frac{2ir}{p-1}$$
and from $q_1-q_2=1$ it follows that $r=i(p^2-1)/4$ and hence
$q_1=-(p-1)/2$ and $q_2=-(p+1)/2$.
\end{prf}
\begin{rmk} The function $p$ (which figures prominently as the \emph{angle function} in the calculations of \cite{AGG}) can be read off from the $2$-forms $\beta_1,\beta_2$ using the above formulas. Recall that the imaginary part of $\beta$ must be symplectic to define a generalized complex structure. We calculate the two Liouville volume forms:
$$(\beta_1-\bar\beta_1)^2=(p-1)\gamma\bar\gamma\qquad (\beta_2-\bar\beta_2)^2=-(p+1)\gamma\bar\gamma.$$
\end{rmk}
\subsection{Examples}
Because of Theorem \ref{bi}, the constructions in (\ref{cp2}) and (\ref{f2}) using generalized complex structures furnish us with bihermitian metrics. We now write these down. The complex structures $I_+,I_-$ are determined by the respective $(0,2)$ forms $\beta_1-\beta_2$ and $\beta_1-\bar\beta_2$. It is straightforward to see that
\begin{eqnarray*}
\lambda(\beta_1-\beta_2)&=& (H_{12}v_1+H_{22}v_2+\lambda\bar v_1)(-H_{11}v_1+H_{12}v_2+\lambda \bar v_2)\\
\lambda(\beta_1-\bar\beta_2)&=& (H_{12}\bar v_1-H_{22}\bar v_2+\lambda v_1)(H_{11}\bar v_1+H_{12}\bar v_2+\lambda v_2)
\end{eqnarray*}
The metric is obtained from the Hermitian form $\beta_1-\bar\beta_1$ on $(1,0)$ vectors. Using the basis of $(0,1)$ forms for $I_+$ given by the decomposition of $\beta_1-\beta_2$ above this turns out to be diagonal and the metric itself written as
$$H_{11}\left[\frac{dr^2}{r^2-2\lambda + 2H_{12}}+\frac{r^2\sigma_1^2}{r^2-2\lambda-2H_{12}}\right]+H_{22}\left[\frac{r^2\sigma_2^2}{r^2-2\lambda + 2H_{12}}+\frac{r^2\sigma_3^2}{r^2-2\lambda-2H_{12}}\right].$$
\begin{rmk} If we replace the Poisson structure $\sigma=\partial/\partial z_1\wedge \partial/\partial z_2$ in our examples on ${\mathbf C}{\rm P}^2$ or $ {\mathbf F}_2$ by $t\sigma$, then as $t\rightarrow 0$ the limiting generalized complex structure $J_1$ arises from a complex structure and we should obtain simply a K\"ahler metric.
This is equivalent to replacing the $2$-form $\beta_1$ by $t^{-1}\beta_1$. The differential equations for $H_{ij}$ remain the same but the algebraic constraint $\det H=\lambda(\lambda-r^2)$ becomes $\det H=\lambda(\lambda-t^{-1}r^2)$. The metric then becomes
$$tH_{11}\left[\frac{dr^2}{r^2-2t\lambda + 2tH_{12}}+\frac{r^2\sigma_1^2}{r^2-2t\lambda-2tH_{12}}\right]+tH_{22}\left[\frac{r^2\sigma_2^2}{r^2-2t\lambda + 2tH_{12}}+\frac{r^2\sigma_3^2}{r^2-2t\lambda-2tH_{12}}\right]$$
and removing the overall factor of $t$ this tends to the K\"ahler metric $H_{11}v_1\bar v_1+H_{22}v_2\bar v_2$ we started our constructions with.
\end{rmk}
Concerning our examples of ${\mathbf C}{\rm P}^2$ and ${\mathbf F}_2$, one should be careful to distinguish the various complex structures. In each case we took a complex structure which had a holomorphic Poisson structure and used that to define a \emph{generalized} complex structure $J_1$. We then found a generalized complex structure $J_2$ commuting with it and reinterpreted the pair as a bihermitian metric with two integrable complex structures $I_+$ and $I_-$.
It is well-known that ${\mathbf C}{\rm P}^2$ has a unique complex structure so that all three complex structures are equivalent by a diffeomorphism in that case. However, all the Hirzebruch surfaces ${\mathbf F}_{2m}$ are diffeomorphic to $S^2\times S^2$. For $m> 0$ there is a unique holomorphic $SL(2,\mathbf{C})$ action which has two orbits of complex dimension one: a curve of self-intersection $+2m$ and one of $-2m$.
The complex structures $I_+,I_-$ that arose from our construction admit a holomorphic $SU(2)$ action and there are two spherical orbits of real dimension $2$ corresponding to $r=0$ and $r=\infty$. We shall show that the sphere $S_0$ given by $r=0$ is not holomorphic with respect to $I_+$.
Note first that the $2$-form $\beta_1=r^2v_1v_2$ vanishes on $S_0$ because $\beta_1$ has type $(2,0)$ in the ${\mathbf F}_2$ complex structure and $S_0$ is holomorphic. Since $\lambda=r^2f_4$, this means that $\lambda(v_1v_2+\bar v_1 \bar v_2)$ vanishes on $S_0$. But from (\ref{h12})
$$H_{12}=4w\vert w\vert (1+\vert z\vert^2)\frac{\partial f_3}{\partial w}$$
so that
$$H_{12}v_1\bar v_2=4\frac{\partial f_3}{\partial w}(\bar w dw d\bar z +\bar z\vert w\vert ^2dzd\bar z)$$
and this vanishes on $S_0$ since $w=0$ there. Thus, restricted to $S_0$, all the terms in
$\beta_1-\beta_2$ except $H_{11}v_1\bar v_1+H_{22}v_2 \bar v_2$ vanish,
and the latter is non-zero since it is the K\"ahler metric we started from.
However $\beta_1-\beta_2$ is a $(0,2)$-form in the complex structure $I_+$ and this must vanish on $S_0$ if it is a holomorphic curve.
\vskip .25cm
We conclude that, with the complex structure $I_+$ this must be the Hirzebruch surface ${\mathbf F}_0={\mathbf C}{\rm P}^1\times {\mathbf C}{\rm P}^1$.
\subsection{Holomorphic Poisson structures} \label {holp}
Apostolov et al. in \cite{AGG} considered the four-dimensional bihermitian case where $I_+$ and $I_-$ define the same orientation and proved that the subset on which $I_+=\pm I_-$ is an anticanonical divisor with respect to both complex structures. Now an anticanonical divisor is a holomorphic section of $\Lambda^2T^{1,0}$ -- a holomorphic bivector $\sigma$. Since $[\sigma,\sigma]$ is a holomorphic section of $\Lambda^3 T^{1,0}$, in two complex dimensions this
automatically vanishes and we have a Poisson structure. All compact surfaces with holomorphic Poisson structure have been listed by Bartocci and Macr\`\i \,\,using the classification of complex surfaces \cite{Bart}, so considering this list provides a basis for seeking compact bihermitian metrics in this dimension. Particular cases (overlooked in \cite{AGG}) are the projective bundle $P(1\oplus K)$ over any compact algebraic curve $C$, and the ``twisted' version $P(V)$ where
$$0\rightarrow K\rightarrow V\rightarrow 1\rightarrow 0$$
is the nontrivial extension in $H^1(C,K)\cong \mathbf{C}$. When $C={\mathbf C}{\rm P}^1$ these two surfaces are are ${\mathbf F}_2$ and ${\mathbf F}_0$ respectively.
We show now that the Poisson structure appears naturally in higher dimensions too.
\vskip .25cm
Let $M$ be a generalized K\"ahler manifold, now considered from the bihermitian point of view. Following \cite{AGG} we consider the $2$-form
$$S(X,Y)=g([I_+,I_-]X,Y).$$
Since
\begin{eqnarray*}
S(I_+X,I_+Y)&=&g(I_+I_-I_+X,I_+Y)-g(I_-I_+^2X,I_+Y)\\
&=&g(I_-I_+X,Y)+g(I_-X,I_+Y)\\
&=&=g([I_-,I_+]X,Y)=-S(X,Y)
\end{eqnarray*}
this form is of type $(2,0)+(0,2)$.
Pick the complex structure $I_+$. Using the antilinear isomorphism $T^{1,0}\cong (\bar T^*)^{0,1}$ provided by the hermitian metric, its $(0,2)$ part can be identified with a section $\sigma_+$ of the bundle $\Lambda^2T^{1,0}$.
\begin{prp} \label{biv} The bivector $\sigma_+$ is a holomorphic Poisson structure.
\end{prp}
\begin{prf} We shall first show that $\sigma_+$ is holomorphic, and then that its Schouten bracket vanishes.
Let $z_1,\dots,z_n$ be local holomorphic coordinates, then
$$\sigma_+=\sum (I_- dz_i,dz_j)\frac{\partial}{\partial z_i}\wedge\frac{\partial}{\partial z_j}$$
where we use the inner product on $1$-forms defined by the metric and the complex structure $I_-$ on $1$-forms. We need to show that the functions $(I_- dz_i,dz_j)$ are holomorphic. Now
\begin{equation}
\frac{\partial}{\partial \bar z_k}(I_- dz_i,dz_j)=((\nabla^+_{\bar k}I_-) dz_i,dz_j)+(I_-\nabla^+_{\bar k}dz_i,dz_j)+(I_- dz_i, \nabla^+_{\bar k}dz_j).
\label{holo}
\end{equation}
The Levi-Civita connection $\nabla$ has zero torsion so
$$0=d(dz_i)=\sum dz_k\wedge \nabla_k dz_i+\sum d\bar z_k \wedge \nabla_{\bar k}dz_i.$$
But from (\ref{del}) $\nabla^+=\nabla +H/2$ where $H=g^{-1}db$, so
\begin{equation}
0=\sum_k dz_k\wedge (\nabla^+_k-H_{k}/2) dz_i+\sum_k d\bar z_k \wedge (\nabla^+_{\bar k}-H_{\bar k}/2)dz_i.
\label{dbareq}
\end{equation}
Now $\nabla^+$ preserves $I_+$ so that $\nabla^+_k dz_i$ and $\nabla^+_{\bar k} dz_i$ are $(1,0)$-forms. However, since $H$ is of type $(2,1)+(1,2)$, $H_k(dz_i)$ has a $(0,1)$ component. Equating the $(1,1)$ component of (\ref{dbareq}) to zero, the two contributions of $H$ give
\begin{equation}
\nabla^+_{\bar k}dz_i=H_{\bar k}(dz_i)
\label{nab1}
\end{equation}
Now $I_-$ is preserved by $\nabla^-$ and from (\ref{del}) $\nabla^-=\nabla^+-H$, so
$$\nabla_{\bar k}^+ I_-=[H_{\bar k},I_-].$$
Using this and (\ref{nab1}) in (\ref{holo}) we obtain
$$\frac{\partial}{\partial \bar z_k}(I_- dz_i,dz_j)=([H_{\bar k},I_-] dz_i,dz_j)+(I_-H_{\bar k}(dz_i),dz_j)+(I_- dz_i, H_{\bar k}(dz_j))=0$$
and so $\sigma_+$ is holomorphic.
\vskip .25cm
To prove that $\sigma_+$ is Poisson we use (\ref{J12}) and the observation that the upper triangular part of $J_1$ is a real Poisson structure. This means that
$$[\omega_+^{-1}+\omega_-^{-1},\omega_+^{-1}+\omega_-^{-1}]=0.$$
Now since $\omega_+$ is of type $(1,1)$, $\omega_+^{-1}+\omega_-^{-1}=h+\sigma_++\bar\sigma_+$ where $h$ is a bivector of type $(1,1)$. Because $\sigma_+$ is holomorphic, $[h,\sigma_+]$ has no $(3,0)$ component and so the $(3,0)$ component of $0=[h+\sigma_++\bar\sigma_+,h+\sigma_++\bar\sigma_+]$ is just $[\sigma_+,\sigma_+]$. Hence $[\sigma_+,\sigma_+]=0$ and we have a holomorphic Poisson structure.
\end{prf}
\vskip .25cm
When the generalized K\"ahler structure is defined by $\rho_1=\exp \beta_1,\rho_2=\exp \beta_2$, as in Lemma 1, $\sigma_+$ has a direct interpretation. Recall that $\bar\beta_1-\bar\beta_2 =\gamma$ is a
non-degenerate holomorphic $2$-form with respect to $I_+$. Then
\begin{prp} \label{gprop} Let $\sigma_+:(T^{1,0})^*\rightarrow T^{1,0}$ be the holomorphic Poisson structure corresponding to the generalized K\"ahler structure given by $2$-forms $\beta_1,\beta_2$, and let $\gamma=\bar\beta_1-\bar\beta_2:T^{1,0}\rightarrow (T^{1,0})^*$ be the holomorphic $2$-form. Then
$$\sigma_+=2i\gamma^{-1}.$$
\end{prp}
\begin{prf} From (\ref{J12}) $\sigma_+$ is given by the upper-triangular part of $J_1$ evaluated on one-forms of type $(1,0)$ with respect to $I_+$. Since $\gamma$ is a non-degenerate $(2,0)$ form, any $(1,0)$ form can be written $i_X\gamma$ for a $(1,0)$-vector $X$. So we require to prove that if $X$ is a $(1,0)$ vector, then the $(1,0)$ component of
$J_1(i_X\gamma)$ is $2iX$. Now
\begin{eqnarray*}
J_1(i_X\gamma)&=&J_1(i_X(\bar\beta_1-\bar\beta_2))\\
&=& J_1(i_X\bar\beta_1-X+X-i_X\bar\beta_2)
\end{eqnarray*}
and by the definition of $J_1$,
\begin{equation}
J_1(i_X\bar\beta_1-X)=-i(i_X\bar\beta_1-X)
\label{first}
\end{equation}
The term $X-i_X\bar\beta_2$ is acted on as $-i$ by $J_2$ and we split it into components for the two $J_1$ eigenspaces:
$$X-i_X\bar\beta_2=Y-i_Y\bar\beta_2+Z-i_Z\bar\beta_2.$$
Since $Z-i_Z\bar\beta_2$ is in the $-i$-eigenspace of both $J_1$ and $J_2$, $Z$ is of type $(0,1)$. Since $X=Y+Z$, $X=Y^{1,0}$. Now
$$J_1(X-i_X\bar\beta_2)=i(Y-i_Y\bar\beta_2)-i(Z-i_Z\bar\beta_2)$$
and adding this to (\ref{first}), the upper triangular part of $J_1$ is given by
$$J_1(i_X(\bar\beta_1-\bar\beta_2))=2iX -2iZ$$
whose $(1,0)$ part is $2iX$.
\end{prf}
\begin{ex}
The examples of ${\mathbf C}{\rm P}^2$ and ${\mathbf F}_2$ were constructed by using $2$-forms $\beta_1,\beta_2$. Since $\beta_1$ had a pole on the curve at $r=\infty$ and $\beta_2$ was smooth everywhere, the Poisson structures $\sigma_+=2i(\bar\beta_1-\bar\beta_2)^{-1}$ and $\sigma_-=2i(\bar\beta_1-\beta_2)^{-1}$ vanish there.
\end{ex}
\section{Moduli spaces of instantons}
\subsection{Stability}
On a $4$-manifold with a Hermitian structure, the anti-self-dual (ASD) $2$-forms are the $(1,1)$-forms orthogonal to the Hermitian form. Thus on a generalized K\"ahler $4$-manifold, a connection with anti-self-dual curvature (an instanton) has curvature of type $(1,1)$ with respect to both complex structures $I_+,I_-$. In fact, where $I_+\ne \pm I_-$, anti-self-duality is equivalent to this condition.
The equations $d^c_-\omega_-=db=-d^c_+\omega_+$ imply that $$dd_{\pm}^c\omega_{\pm}=0$$
which means that the metric is a Gauduchon metric with respect to both complex structures. With a Gauduchon metric one defines the \emph{degree} of a holomorphic line bundle $L$ by
$$\mathop{\rm deg}\nolimits L=\frac{1}{2\pi}\int_M F\wedge\omega$$
where $F$ is the curvature form of a connection on $L$ defined by a Hermitian metric. Since a different choice of metric changes $F$ by $dd^cf$, the condition $dd^c\omega=0$ and integration by parts shows that the degree, a real number, is independent of the choice of Hermitian metric on $L$. It has the usual property of degree that if a holomorphic section of $L$ vanishes on a divisor $D$ then
$$\mathop{\rm deg}\nolimits L=\int_D\omega.$$
So line bundles with sections which vanish somewhere have positive degree.
\begin{rmk} Let us consider this non-K\"ahler degree for a bihermitian surface such that the Poisson structure vanishes on a divisor, like our examples of ${\mathbf C}{\rm P}^2$ and ${\mathbf C}{\rm P}^1\times {\mathbf C}{\rm P}^1$, and assume for convenience that the surface also carries a K\"ahler metric. The canonical bundle $K$ has no holomorphic sections since the product with the Poisson structure, a section of $K^*$, would give a holomorphic function with zeroes. This means $H^{2,0}(M)=0$ and so $H^2(M)$ is purely of type $(1,1)$.
Now suppose that one of the generalized complex structures is defined by $\exp \beta$ where $\beta$ is closed. We saw in (\ref{4formulas}) that
$\beta=-b+i\omega_++\gamma_1$ where $\gamma_1$ is of type $(0,2)$, so that the $(1,1)$ component of $\beta-\bar\beta$ is $2i\omega_+$. Thus the integral of $\omega_+$ over a holomorphic curve $C$, which is positive, is the same as the integral of the \emph{closed} form $(\beta-\bar\beta)/2i$. Let $W$ be the cohomology class of this form. Then we see that for every effective divisor $D$ on $M$, $WD>0$. Furthermore, $W$ is represented by the form
$$(\beta-\bar\beta)/2i=\omega_+-i(\gamma_1-\bar\gamma_1)/2$$
which is self-dual, hence $W^2>0$. It follows from Nakai's criterion that $W$ is the cohomology class of a K\"ahler metric.
Since the ample cone generates the whole of the cohomology, we see that the non-K\"ahler degree in this case agrees with the ordinary K\"ahler degree of some K\"ahler metric.
Observe also that $\beta-\bar\beta$ is also equal to $2i\omega_-+\delta_1-\bar\delta_1$ so that we obtain the same degree function on cohomology for $I_+$ and $I_-$.
\end{rmk}
\vskip .25cm
Using this definition of degree, one can define the slope of a subbundle, and from that the stability of a holomorphic bundle. The key theorem in the area, proved by Buchdahl \cite{Buch} for surfaces and Li and Yau \cite{LY} in the general case, is that a bundle is stable if and only if it has an irreducible ASD connection. A good reference for this is the book \cite{LT}.
From this we already see that the moduli space ${\mathcal M}$ of ASD connections on a generalized K\"ahler manifold has two complex structures, by virtue of being the moduli space of stable bundles for both $I_+$ and $I_-$.
We shall prove the following theorem:
\begin{thm} \label{GKmod} Let $M^4$ be a compact even generalized K\"ahler manifold. Then the smooth points of the moduli space of ASD connections on a principal $SU(k)$-bundle over $M$ carries a natural bihermitian metric such that $d^c_-\omega_-=H=-d^c_+\omega_+$ for some exact $3$-form $H$ of type $(2,1)+(1,2)$.
\end{thm}
From Gualtieri's theorem this has a generalized K\"ahler interpretation once we choose a $2$-form $b$ such that $db=H$.
\begin{rmk} In general, the moduli space of stable bundles may have singularities if the obstruction space $H^2(M,\mathop{\rm End}\nolimits_0 E)$ (where $\mathop{\rm End}\nolimits_0$ denotes trace-free endomorphisms) is non-vanishing. However, if the Poisson structure $s$ on $M$ is non-zero, then
$$s:H^0(M,\mathop{\rm End}\nolimits_0 E\otimes K)\rightarrow H^0(M,\mathop{\rm End}\nolimits_0 E)$$
is injective. But stable bundles are simple,so $H^0(M,\mathop{\rm End}\nolimits_0 E)=0$. We deduce that $H^0(M,\mathop{\rm End}\nolimits_0 E\otimes K)$, and hence also its Serre dual $H^2(M,\mathop{\rm End}\nolimits_0 E)$, must vanish, so the moduli space is smooth (see \cite{Bot1}).
This vanishing also gives us by Riemann-Roch the dimension of the $SU(k)$ moduli space
$$\dim_{\mathbf{C}}{\mathcal M}=2kc_2(E)-(k^2-1)\frac{1}{12}(c_1^2+c_2)(M).$$
The simplest case would be $k=2, c_2(E)=n$ for our examples ${\mathbf C}{\rm P}^2,{\mathbf F}_2$ (or any rational surface) where $\dim_{\mathbf{C}}{\mathcal M}=4n-3.$
\end{rmk}
\subsection{The metric on the moduli space}
In \cite{LT} the metric structure of the moduli space of instantons on a Gauduchon manifold is discussed. It differs in general from the Riemannian or K\"ahler case. In the Riemannian situation, the space of all connections is viewed as an infinite-dimensional affine space with group of translations $\Omega^1(M,\lie{g})$ and ${\mathcal L}^2$ metric
$$(a_1,a_2)=-\int_M\mathop{\rm tr}\nolimits(a_1\wedge \mathop{*\!}\nolimits a_2).$$
The solutions to the ASD equations form an infinite-dimensional submanifold with induced metric, and its quotient by the group of gauge transformations ${\mathcal G}$ is the moduli space, which acquires the quotient metric. To define this, one identifies the tangent space of the quotient at a point $[A]$ with the orthogonal complement to the tangent space of the gauge orbit at the connection $A$, with its restricted inner product. The orthogonal complement is identified with the bundle-valued $1$-forms $a\in \Omega^1(M,\lie{g})$ which satisfy the equation
\begin{equation}
d_A^*a\, (=-\mathop{*\!}\nolimits \mathop{d_A*\!}\nolimits a)=0
\label{horizont}
\end{equation}
As the authors of \cite{LT} point out, this metric in the Gauduchon case is not Hermitian with respect to the natural complex structure that the moduli space acquires through its identification with the moduli space of stable bundles. Instead of the orthogonality (\ref{horizont}), one takes a different horizontal subspace defined by
\begin{equation}
\omega \wedge d_A^ca=0.
\label{newhorizont}
\end{equation}
\begin{lem} \label{hori}$\omega \wedge d^c_A a = \mathop{d_A*\!}\nolimits a- d^c\omega \wedge a.$
\end{lem}
From this we see that when the metric is K\"ahler, $d^c\omega=0$, and so the two horizontality conditions coincide.
\begin{lemprf}
Note that for any $\psi\in \Omega^0(M,\lie{g})$,
\begin{equation}
d_A^c(\omega \wedge\mathop{\rm tr}\nolimits(a \psi))=d^c\omega\wedge\mathop{\rm tr}\nolimits(a\psi)+\omega\wedge\mathop{\rm tr}\nolimits(d^c_A a\psi)-\omega\wedge\mathop{\rm tr}\nolimits(a \wedge d_A^c\psi)
\label{exp1}
\end{equation}
and $d_A^c\psi=I^{-1}d_AI\psi=-Id_A\psi$, so that $$\omega\wedge\mathop{\rm tr}\nolimits(a\wedge d_A^c\psi)=-\omega\wedge\mathop{\rm tr}\nolimits(a\wedge Id_A\psi)=(a,d_A\psi)\omega^2=\mathop{\rm tr}\nolimits(\mathop{*\!}\nolimits a \wedge d_A\psi).$$
Integrating (\ref{exp1}) and using Stokes' theorem and the relation above, we get
$$\int_M[d^c\omega\wedge\mathop{\rm tr}\nolimits(a\psi) +\omega\wedge\mathop{\rm tr}\nolimits(d^c_A a\psi)-\mathop{\rm tr}\nolimits(\mathop{d_A*\!}\nolimits a \psi)]=0$$ so that
\begin{equation}
\omega \wedge d^c_A a = \mathop{d_A*\!}\nolimits a- d^c\omega \wedge a.
\label{newhorizont1}
\end{equation}
\end{lemprf}
With this choice of horizontal, the metric on the moduli space is Hermitian with Hermitian form
$$\tilde\omega(a_1,a_2)=\int_M\omega\wedge \mathop{\rm tr}\nolimits(a_1\wedge a_2).$$
It is shown in \cite{LT} that $\tilde\omega$ satisfies $dd^c\tilde\omega=0$.
The horizontal subspace (\ref{newhorizont}) defines a connection on the infinite-dimensional principal ${\mathcal G}$-bundle over the moduli space and its curvature turns out to be of type $(1,1)$ on ${\mathcal M}$ (see \cite{LT}). We shall make use of these facts later.
\vskip .25cm
In order to prove Theorem \ref{GKmod} we need first to show that the application of L\"ubke and Teleman's approach to the two complex structures $I_+$ and $I_-$ yields the same metric.
\vskip .25cm
The tangent space to the moduli space at a smooth point is the first cohomology of the complex:
$$\Omega^0(M,\lie{g})\stackrel{d_A}\longrightarrow \Omega^1(M,\lie{g})\stackrel{d^+_A}\longrightarrow \Omega_+^2(M,\lie{g})$$
where here the $+$ refers to projection onto the self-dual part. The metric is the induced inner product on the subspace of $\Omega^1(M,\lie{g})$ defined by the horizontality condition $\omega \wedge d^c_A a=0$. We shall write $[a]$ for the tangent vector to the moduli space represented by $a$.
In our case we have two such horizontality conditions $\omega_- \wedge d^c_-a=0$ and $\omega_+ \wedge d^c_+a=0$ (suppressing the subscript $A$ for clarity) and two representatives $a$ and $a+d_A\psi$ for the same tangent vector. We shall call these plus- and minus- horizontal respectively. We prove:
\begin{lem} Let $a$ and $a+d_A\psi$ satisfy
$$\omega_-\wedge d^c_-a=0,\quad \omega_+ \wedge d^c_+(a+d_A\psi)=0.$$
Then $(a,a)=(a+d_A\psi,a+d_A\psi)$.
\end{lem}
\begin{lemprf}
Since in our case $d^c_-\omega_-=db=h=-d^c_+\omega_+$ our two horizontality conditions are, from (\ref{newhorizont1})
$$ \mathop{d_A*\!}\nolimits a- h \wedge a=0\quad \mathop{d_A*\!}\nolimits\, (a+d_A\psi)+h \wedge (a+d_A\psi)=0$$ and so, eliminating $h\wedge a$,
$$2\mathop{d_A*\!}\nolimits a+d_A\mathop{*\!}\nolimits d_A\psi+h\wedge d_A\psi=0.$$
This gives on integration
$$\int_M[2\mathop{\rm tr}\nolimits(\mathop{d_A*\!}\nolimits a \psi)+\mathop{\rm tr}\nolimits(\mathop{d_A*\!}\nolimits d_A\psi \psi)+h\wedge\mathop{\rm tr}\nolimits(d_A\psi \psi)]=0.$$
But $\mathop{\rm tr}\nolimits(d_A\psi\psi)=d\mathop{\rm tr}\nolimits\psi^2/2$ so the last term is $d[(\mathop{\rm tr}\nolimits\psi^2)h/2]$ as $h$ is closed. By Stokes' theorem we get
$$2(a,d_A\psi)+(d_A\psi,d_A\psi)=0$$
and hence
$$(a+d_A\psi,a+d_A\psi)=(a,a)$$
as required.
\end{lemprf}
\subsection{The bihermitian structure}
So far, we have seen that
${\mathcal M}$ has two complex structures
and a metric, Hermitian with respect to both.
We now need to show that
$d^c_+\tilde\omega_+=H=-d^c_-\tilde \omega_-$ for an exact $3$-form $H$.
\vskip .25cm
Denote by ${\mathcal A}$ the affine space of all connections on the principal bundle, then a
tangent vector is given by $a\in \Omega^1(M,\lie{g})$
and for any $2$-form $\omega$,
$$\Omega(a_1,a_2)=\int_M\omega\wedge \mathop{\rm tr}\nolimits(a_1\wedge a_2)$$
is a closed and gauge-invariant $2$-form on ${\mathcal A}$. It is closed because it is translation-invariant on $\mathcal{A}$ (has ``constant coefficients").
We defined Hermitian forms $\tilde\omega_{\pm}$ on ${\mathcal M}$ by
$$\tilde\omega_{\pm}([a_1],[a_2])=\Omega_{\pm}(a_1,a_2)=\int_M\omega_{\pm}\wedge\mathop{\rm tr}\nolimits(a_1\wedge a_2)$$
where $a_1,a_2$ are plus/minus-{\it horizontal}. Now the formula for the exterior derivative of a $2$-form $\alpha$ is
$$3d\alpha(a_1,a_2,a_3)=a_1\cdot\alpha(a_2,a_3)-\alpha([a_1,a_2],a_3)+\mathrm {cyclic}$$
so, since $\Omega$ is closed
$$3d\tilde\omega([a_1],[a_2],[a_3])=-\int_M\omega\wedge\mathop{\rm tr}\nolimits([a_1,a_2]_V)\wedge a_3)+\mathrm {cyclic}$$
where $[a_1,a_2]_V$ is the vertical component of the Lie bracket of the two vector fields. By definition this is the curvature of the ${\mathcal G}$-connection. If $\theta(a_1,a_2)\in \Omega^0(M,\lie{g})$ is this curvature then $[a_1,a_2]_V=d_A\theta(a_1,a_2)$. Using Stokes' theorem
\begin{eqnarray*}
3d\tilde\omega([a_1],[a_2],[a_3])&=&-\int_M\omega\wedge\mathop{\rm tr}\nolimits(d_A\theta(a_1,a_2)\wedge a_3)+\mathrm {cyclic}\\
&=& \int_M d\omega\wedge \mathop{\rm tr}\nolimits(\theta(a_1,a_2)a_3)+ \int_M\omega\wedge\mathop{\rm tr}\nolimits(\theta(a_1,a_2)d_Aa_3)+\mathrm {cyclic}\\
&=&\int_M d\omega\wedge \mathop{\rm tr}\nolimits(\theta(a_1,a_2)a_3) +\mathrm {cyclic}
\end{eqnarray*}
since $d_Aa_3$ is anti-self-dual and $\omega$ is self-dual so $\omega\wedge d_Aa_3=0$.
\vskip .25cm
Now
$d^c\omega(a_1,a_2,a_3)=-d\omega(Ia_1,Ia_2,Ia_3)$
and from \cite{LT} the
curvature of the ${\mathcal G}$-bundle is of type $(1,1)$. This means that $\theta(Ia_2,Ia_3)=\theta(a_2,a_3)$
and so, for the structure $I_-$
\begin{equation}
d_-^c\tilde\omega([a_1],[a_2],[a_3])=\int_M d_-^c\omega_-\wedge \mathop{\rm tr}\nolimits(\theta_-(a_1,a_2)a_3)+\mathrm {cyclic}
\label{deec}
\end{equation}
with a similar equation for $I_+$.
\vskip .25cm
To proceed further we need more information about the curvature $\theta(a_1,a_2)$. On the affine space ${\mathcal A}$ the Lie bracket of $a_1$ and $a_2$ considered as vector fields is just
$a_1\cdot a_2-a_2\cdot a_1$ where $a\cdot b$ denotes the flat derivative of $b$ in the direction $a$. The horizontality condition imposes a constraint:
$$ \mathop{d_A*\!}\nolimits a_2- h \wedge a_2=0.$$
Differentiating the constraint in the direction $a_1$ gives
$$[a_1,\mathop{*\!}\nolimits a_2]+\mathop{d_A*\!}\nolimits a_1\cdot a_2-h\wedge a_1\cdot a_2=0.$$
The vertical component of the Lie bracket is $d_A\theta(a_1,a_2)$ which thus satisfies
\begin{equation}
\mathop{d_A*\!}\nolimits d_A\theta-h \wedge d_A\theta +2[a_1,\mathop{*\!}\nolimits a_2]=0.
\label{thetaeq}
\end{equation}
Define the second order operator $\Delta:\Omega^0(M,\lie{g})\rightarrow \Omega^4(M,\lie{g})$ by
$$\Delta\psi=d_A\mathop{*\!}\nolimits d_A\psi-h\wedge d_A\psi,$$ then its formal adjoint is
$$\Delta^*\psi=d_A\mathop{*\!}\nolimits d_A\psi+h\wedge d_A\psi$$
and we rewrite (\ref{thetaeq}) as
\begin{equation}
\Delta\theta(a_1,a_2)+2[a_1,\mathop{*\!}\nolimits a_2]=0
\label{thetaeqs}
\end{equation}
for plus-horizontal vector fields $a_i$. Let $b_i=a_i+d_A\psi_i$ be the minus-horizontal representatives of $[a_i]$. By minus-horizontality we have
$$0=d_A\mathop{*\!}\nolimits b_i+h\wedge b_i=\mathop{d_A*\!}\nolimits\, (a_i+d_A\psi_i)+h\wedge (a_i+d_A\psi_i)=\Delta^*\psi_i+\mathop{d_A*\!}\nolimits a_i +h\wedge a_i$$
and together with the plus-horizontality condition $ \mathop{d_A*\!}\nolimits a_i- h \wedge a_i=0$ we get
\begin{equation}
2h\wedge a_i=-\Delta^*\psi_i.
\label{hai}
\end{equation}
Since $d^c_-\omega_-=h$, each integrand on the right hand side of (\ref{deec}) is, from (\ref{hai}), of the form
$$h\wedge \mathop{\rm tr}\nolimits(\theta(a_1,a_2)a_3)=-\mathop{\rm tr}\nolimits(\theta(a_1,a_2)\Delta^*\psi_3/2).$$
Performing the integration and using Stokes' theorem, we obtain
$$-\int_M \mathop{\rm tr}\nolimits(\theta(a_1,a_2)\Delta^*\psi_3)/2=-\int_M \mathop{\rm tr}\nolimits(\Delta\theta(a_1,a_2)\psi_3)/2=\int_M \mathop{\rm tr}\nolimits([a_1,\mathop{*\!}\nolimits a_2]\psi_3)$$
from (\ref{thetaeqs}).
\vskip.25cm
Working with the curvature of the plus-connection we get a similar expression so that we have two formulae:
\begin{eqnarray*}
d_-^c\tilde\omega_-([a_1],[a_2],[a_3])&=&\int_M \mathop{\rm tr}\nolimits([a_1,\mathop{*\!}\nolimits a_2]\psi_3)+\mathrm {cyclic}\\
d_+^c\tilde\omega_+([a_1],[a_2],[a_3])&=&-\int_M \mathop{\rm tr}\nolimits([b_1,\mathop{*\!}\nolimits b_2]\psi_3)+\mathrm {cyclic}
\end{eqnarray*}
Thus to
obtain $d_-^c\tilde\omega_- =-d_+^c\tilde\omega_+$, using $b_i=a_i+d_A\psi_i$ in the above leads to the need to prove:
\begin{lem}
$$\int_M [\mathop{\rm tr}\nolimits([a_1,\mathop{*\!}\nolimits d_A\psi_2]\psi_3)+\mathop{\rm tr}\nolimits([d_A\psi_1,\mathop{*\!}\nolimits a_2]\psi_3)+\mathop{\rm tr}\nolimits([d_A\psi_1,\mathop{*\!}\nolimits d_A\psi_2]\psi_3)]+\mathrm {cyclic}=0.$$
\end{lem}
\begin{lemprf}
Picking out the integrand involving $a_1$ in the cyclic sum we have
\begin{eqnarray*}
\mathop{\rm tr}\nolimits([a_1,\mathop{*\!}\nolimits d_A\psi_2]\psi_3)+\mathop{\rm tr}\nolimits([d_A\psi_3,\mathop{*\!}\nolimits a_1]\psi_2)&=&\mathop{\rm tr}\nolimits(\mathop{*\!}\nolimits a_1\wedge([\psi_3, d_A\psi_2]+ [d_A\psi_3,\psi_2]))\\
&=&-\mathop{\rm tr}\nolimits(\mathop{*\!}\nolimits a_1\wedge d_A[\psi_2,\psi_3])
\end{eqnarray*}
and on integrating,
this is
\begin{eqnarray*}
-\int_M\mathop{\rm tr}\nolimits(\mathop{*\!}\nolimits a_1\wedge d_A[\psi_2,\psi_3])&=&-\int_M\mathop{\rm tr}\nolimits(\mathop{d_A*\!}\nolimits a_1[\psi_2,\psi_3])\\
&=& -\int_M h\wedge \mathop{\rm tr}\nolimits(a_1[\psi_2,\psi_3])\\
&=& \int_M\mathop{\rm tr}\nolimits(\Delta^*\psi_1[\psi_2,\psi_3])/2
\end{eqnarray*}
from (\ref{hai}). But from the definition of $\Delta^*$ this is
$$\frac{1}{2}\int_M\mathop{\rm tr}\nolimits(d_A\mathop{*\!}\nolimits d_A\psi_1[\psi_2,\psi_3])-\frac{1}{2}\int_M h\wedge \mathop{\rm tr}\nolimits(d_A\psi_1[\psi_2,\psi_3]).$$
The cyclic sum of the second term vanishes since
$$d\mathop{\rm tr}\nolimits(\psi_1[\psi_2,\psi_3])=\mathop{\rm tr}\nolimits(d_A\psi_1[\psi_2,\psi_3])+\mathrm {cyclic}$$
and $h$ is closed.
Using Stokes' theorem on the first and expanding, the cyclic sum gives
$$\frac{1}{2}\int_M\mathop{\rm tr}\nolimits(\mathop{*\!}\nolimits d_A\psi_1\wedge( [d_A\psi_2,\psi_3]+[\psi_2,d_A\psi_3])+\mathrm {cyclic}$$
which is
$$-\int_M\mathop{\rm tr}\nolimits([d_A\psi_1,\mathop{*\!}\nolimits d_A\psi_2]\psi_3)]+\mathrm {cyclic}$$
and this proves the lemma.
\end{lemprf}
\vskip .25cm
We finally need to show that $H$ is exact. One might expect that we simply define a $2$-form $\tilde b$ from the $2$-form $b$ on $M$ by
\begin{equation}
\tilde b([a_1],[a_2])=\int_M b\wedge \mathop{\rm tr}\nolimits(a_1\wedge a_2)
\label{btilde}
\end{equation}
to get $d\tilde b=d_-^c\tilde\omega_-$ but this does not hold. The equation for the exterior derivative of $\tilde b$ gives
$$3d\tilde b([a_1],[a_2],[a_3])=\int_M db\wedge \mathop{\rm tr}\nolimits(\theta(a_1,a_2)a_3)+ \int_M b\wedge\mathop{\rm tr}\nolimits(\theta(a_1,a_2)d_Aa_3)+\mathrm {cyclic}.$$
When we used this above with $\omega_+, \omega_-$ replacing $b$, the second term vanished because $d_Aa_3$ is anti-self-dual and $\omega_{\pm}$ are self-dual. This is not the case for a general $b$, and will only be true if $b$ is self-dual. We shall see in Section 5 a more general occurrence of this phenomenon. However we do have the following:
\begin{lem} Any $2$-form $b$ on a compact oriented four-manifold $M$ is the sum of a closed form and a self-dual form.
\end{lem}
\begin{lemprf} Use the non-degenerate pairing on $2$-forms
$$(\alpha,\beta)=\int_M\alpha\wedge \beta.$$
The annihilator of the self-dual forms $\Omega^2_+$ in this pairing is $\Omega^2_-$, and the annihilator of $\Omega^2_{closed}$ is $\Omega^2_{exact}$ so the annihilator of $\Omega^2_++\Omega^2_{closed}$ is the intersection of $\Omega^2_-$ and $\Omega^2_{exact}$. But if $\alpha$ is exact, then by Stokes' theorem
$$\int_M\alpha\wedge \alpha=0$$
and if $\alpha\in \Omega^2_-$
$$\int_M\alpha\wedge \alpha=-(\alpha,\alpha)$$
so if both hold then $\alpha=0$.
\end{lemprf}
It follows from this that $db=db_+$ where $b_+$ is self-dual, and then (\ref{btilde}) does define a form $\tilde b_+$ on the moduli space. It follows than that $d\tilde b_+=d_-^c\tilde\omega_-=-d_+^c\tilde\omega_+$.
\subsection{The Poisson structures on ${\mathcal M}$}
As we saw in Proposition \ref{biv}, a generalized K\"ahler structure defines a holomorphic Poisson structure for each of the complex structures $I_+,I_-$. We shall determine these on the instanton moduli space next.
On the moduli space of stable bundles over a Poisson surface $M$, there is a canonical holomorphic Poisson structure, defined by Bottacin in \cite{Bot1} as follows. The holomorphic tangent space at a bundle $E$ is the sheaf cohomology group $H^1(M,\mathop{\rm End}\nolimits E)$ and by Serre duality, the cotangent space is $H^1(M,\mathop{\rm End}\nolimits E \otimes K)$. The Poisson structure on $M$ is a holomorphic section $s$ of the anticanonical bundle $K^*$ and for $\alpha,\beta\in H^1(M,\mathop{\rm End}\nolimits E \otimes K)$, the Poisson structure $\sigma$ on the moduli space is defined by taking $\mathop{\rm tr}\nolimits(\alpha\beta)\in H^2(M,K^2)$, multiplying by $s\in H^0(M,K^*)$ to get
$$\sigma(\alpha,\beta)=s\mathop{\rm tr}\nolimits(\alpha\beta)\in H^2(M,K)\cong \mathbf{C}.$$
The definition is very simple, the difficult part of \cite{Bot1} is proving the vanishing of the Schouten bracket.
\begin{thm} \label{botta} Let $\sigma_+$ be the $I_+$ - Poisson structure defined by the generalized K\"ahler structure on ${\mathcal M}$. Then $\sigma_+/2$ is the canonical structure on the moduli space of $I_+$-stable bundles.
\end{thm}
\begin{prf} In the generalized K\"ahler setup, the Poisson structure $\sigma_+$ is defined by the $(0,2)$ part of $\omega_-$ under the antilinear identification $T^{1,0}\cong (\bar T^*)^{0,1}$ defined by the metric.
A tangent vector to ${\mathcal M}$ is defined by $a\in \Omega^1(M,\lie{g})$ satisfying $d^+_Aa=0$, and this implies that $a^{0,1}\in \Omega^{0,1}(M,\mathop{\rm End}\nolimits E)$ satisfies $\bar\partial_Aa^{0,1}=0\in \Omega^{0,2}(M,\mathop{\rm End}\nolimits E)$, which is the tangent vector in the holomorphic setting -- it is a Dolbeault representative for a class in $H^1(M,\mathop{\rm End}\nolimits E)$.
The conjugate $a^{1,0}=\overline{a^{0,1}}$ defines a complex cotangent vector by the linear form
$$b^{0,1}\mapsto \int_M\omega_+\wedge\mathop{\rm tr}\nolimits (a^{1,0}\wedge b^{0,1})$$
and this is the antilinear identification $T^{1,0}\cong (\bar T^*)^{0,1}$ on the moduli space. However $\omega_+\wedge a^{1,0}\in \Omega^{2,1}(M,\mathop{\rm End}\nolimits E)$ is not a Dolbeault representative for the Serre dual -- it is not $\bar\partial$-closed -- so to see concretely the canonical Poisson structure we must find a good representative $(2,1)$ form.
\vskip .25cm
Now from $d_A^+a=0$ we have
$\omega_+\wedge d_A(a^{1,0}+a^{0,1})=0$
and from the horizontality condition $\omega_+\wedge d^c_+a=0$, we obtain
$\omega_+\wedge d_A(a^{1,0}-a^{0,1})=0$
so putting them together
\begin{equation}
\omega_+\wedge\bar\partial_A a^{1,0}=0,\quad \omega_+\wedge\partial_A a^{0,1}=0
\label{infasd}
\end{equation}
\vskip .25cm
From Lemma \ref{hori} applied to $I_+$ and $I_-$ we have
$$\omega_{\pm} \wedge d_{\pm}^c a = \mathop{d_A*\!}\nolimits a- d_{\pm}^c\omega_{\pm} \wedge a$$
so that since $d^c_-\omega_-=-d^c_+\omega_+$,
$$\omega_{-} \wedge d_{-}^c a = \omega_{+} \wedge d_{+}^c a +2 d_+^c\omega_{+} \wedge a.$$
If $a=a^{1,0}+d_A\psi$ is minus-horizontal then this equation tells us that
$$0=\omega_{+} \wedge d_{+}^c d_A\psi+2d_+^c\omega_{+}\wedge (a^{1,0}+d_A\psi)$$
since $a^{1,0}$ is plus-horizontal. We rewrite this as
\begin{equation}
2i\omega_{+}\wedge \bar\partial_A\partial_A \psi+2i\bar\partial\omega_{+}\wedge (a^{1,0}+\partial_A\psi)-2i\partial\omega_+\wedge\bar\partial_A\psi=0
\label{goodeq}
\end{equation}
using the fact that $\omega_+\wedge F=0$ where $F$ is the curvature of the connection $A$.
This gives, using $\bar\partial\partial\omega_+=0$ and (\ref{infasd}),
\begin{equation}
\bar\partial_A[\omega_+\wedge (a^{1,0}+\partial_A\psi)+\psi\partial\omega_+]=0
\label{Dolrep}
\end{equation}
Here, then, we have a
$\bar\partial$-closed form, and it represents the dual of $[a^{0,1}]$ using the metric on ${\mathcal M}$ since, from Stokes' theorem,
$$\int_M[\omega_+\wedge \mathop{\rm tr}\nolimits((a^{1,0}+\partial_A\psi)\wedge b^{0,1})+\partial\omega_+\wedge\mathop{\rm tr}\nolimits(\psi b^{0,1})]= \int_M\omega_+\wedge \mathop{\rm tr}\nolimits(a^{1,0}\wedge b^{0,1})-\int_M\omega_+\wedge\mathop{\rm tr}\nolimits(\psi\partial_Ab^{0,1})$$
and the second term on the right hand side vanishes from (\ref{infasd}).
\vskip .25cm
Now where the Poisson structure $s$ on $M$ is non-vanishing we have a closed $2$-form $\beta_1-\bar\beta_2$ which from Proposition \ref{4formulas} can be expressed as
$2i\omega_+-(p-1)\bar\gamma/2+(p+1)\gamma/2.$ Since this is closed, and $\gamma$ is of type $(2,0)$,
$4id\omega_+=\partial p\wedge \bar\gamma-\bar\partial p\wedge \gamma$
and so
\begin{equation}
4i\partial\omega_+=-\bar\partial p\wedge \gamma
\label{domega}
\end{equation}
We can therefore rewrite the Dolbeault representative as
$$\omega_+\wedge (a^{1,0}+\partial_A\psi)+i\psi\bar\partial p\wedge \gamma/4.$$
The canonical Poisson structure is therefore obtained by integrating over $M$ the form
\begin{equation}
\mathop{\rm tr}\nolimits[s(\omega_+\wedge (a_1^{1,0}+\partial_A\psi_1)+i\psi_1\bar\partial p\wedge \gamma/4)\wedge(\omega_+\wedge (a_2^{1,0}+\partial_A\psi_2)+i\psi_2\bar\partial p\wedge \gamma/4)]
\label{integrand}
\end{equation}
\vskip .25cm
Take the product of the two expressions with an $\omega_+$ factor.
For $(1,0)$ forms $a,b$, at each point
$[s(\omega_+\wedge a)]\wedge\omega_+\wedge b$ is a skew form on $T^{1,0}$ with values in $\Lambda^4T^*$ depending on a Hermitian form and a $(2,0)$ form $\gamma$ (recall from Proposition \ref{gprop} that $s\gamma=2i$). By $SU(2)$ invariance this must be a multiple of $\bar\gamma\wedge a\wedge b$ and a simple calculation shows that
$$[s(\omega_+\wedge a)]\wedge\omega_+\wedge b=-i\frac{\omega_+^2}{\gamma\bar\gamma}\,\bar\gamma\wedge a\wedge b.$$
However from (\ref{eqA}) and $r=i(p^2-1)/4$ we see that
$$\frac{\omega_+^2}{\gamma\bar\gamma}=\frac{1}{8}(1-p^2).$$
But now from Proposition \ref{4formulas}, $\omega_-=p\omega_++i(p^2-1)\bar\gamma/4-i(p^2-1)\gamma/4$ and so
$$[s(\omega_+\wedge a)]\wedge\omega_+\wedge b=\frac{i}{2}\omega_-^{0,2}\wedge a\wedge b=\frac{i}{2}\omega_-\wedge a\wedge b$$
since $a$ and $b$ are of type $(1,0)$.
Thus the first two expressions contribute to the integral the term
\begin{equation}
\frac{1}{2}\int_M\omega_- \wedge\mathop{\rm tr}\nolimits(a_1^{1,0}+\partial_A\psi_1)\wedge (a_2^{1,0}+\partial_A\psi_2)
\label{two0}
\end{equation}
\vskip .25cm
The last two terms in (\ref{integrand}) give zero contribution because of the common $\bar\partial p$ factor. For the other terms, the relation $s\gamma=2i$ means that we are considering the integral of
\begin{equation}
-\mathop{\rm tr}\nolimits[\psi_1\bar\partial p\wedge\omega_+\wedge (a_2^{1,0}+\partial_A\psi_2)]/2+\mathop{\rm tr}\nolimits[\psi_2\bar\partial p\wedge\omega_+\wedge (a_1^{1,0}+\partial_A\psi_1)]/2.
\label{crossterm}
\end{equation}
Take the first expression.
This no longer contains the singular term $\gamma$ so we can integrate over the manifold and
using Stokes' theorem we get
\begin{equation}
\frac{1}{2}\int_M p\omega_+\wedge\mathop{\rm tr}\nolimits(\bar\partial_A\psi_1\wedge (a_2^{1,0}+\partial_A\psi_2))+p\mathop{\rm tr}\nolimits[\psi_1\bar\partial_A[\omega_+\wedge (a^{1,0}+\partial_A\psi)]
\label{integrate}
\end{equation}
Now from (\ref{Dolrep}) and (\ref{domega})
$$\bar\partial_A[\omega_+\wedge (a^{1,0}+\partial_A\psi)]=-\bar\partial_A(\psi\partial\omega_+)=\bar\partial_A\psi\wedge\bar\partial p\wedge \gamma/4i$$
Using this we can write (\ref{integrate}) as
\begin{equation}
\frac{1}{2}\int_M p\omega_+\mathop{\rm tr}\nolimits(\bar\partial_A\psi_1\wedge (a_2^{1,0}+\partial_A\psi_2))-\frac{i}{8}\int_M p\bar\partial p\wedge\mathop{\rm tr}\nolimits(\bar\partial_A\psi_1\psi_2)\wedge \gamma
\label{integrate1}
\end{equation}
Now $\omega_-^{1,1}=p\omega_+$ and the first term integrates a $(1,1)$ form against $p\omega_+$ so we write this as
\begin{equation}
\frac{1}{2}\int_M \omega_-\wedge\mathop{\rm tr}\nolimits(\bar\partial_A\psi_1\wedge (a_2^{1,0}+\partial_A\psi_2))
\label{oneone}
\end{equation}
From Proposition \ref{4formulas}, we have $$\omega_-^{2,0}=-i(p^2-1)\gamma/4$$ so the last term in (\ref{integrate1}) is
$$\frac{1}{4}\int_M \bar\partial \omega^{2,0}\wedge\mathop{\rm tr}\nolimits(\bar\partial_A\psi_1\psi_2)$$
which using Stokes' theorem gives
$$\frac{1}{4}\int_M \omega_-^{2,0}\wedge\mathop{\rm tr}\nolimits(\bar\partial_A\psi_1\bar\partial \psi_2)$$
which we write as
$$\frac{1}{4}\int_M \omega_-\wedge\mathop{\rm tr}\nolimits(\bar\partial_A\psi_1\bar\partial \psi_2).$$
In the full integral there is another contribution of this form from the second term in (\ref{crossterm}) and adding all
terms in (\ref{integrand}) we obtain
$$\frac{1}{2}\int_M\omega_-\wedge\mathop{\rm tr}\nolimits(a_1^{1,0}+d_A\psi_1)\wedge\mathop{\rm tr}\nolimits(a_2^{1,0}+d_A\psi_2).$$
Since $a_1^{1,0}+d_A\psi_1, a_2^{1,0}+d_A\psi_2$ are minus-horizontal representatives of $a_1^{1,0},a_2^{1,0}$ we see from the definition of $\tilde\omega_-$ that this is $\tilde\omega_-^{0,2}/2$ evaluated on those two vectors and hence is half the Poisson structure defined by the bihermitian metric.
\end{prf}
\subsection{The generalized K\"ahler structure}
As we have seen, the bihermitian structure of $M^4$ naturally induces a similar structure on the moduli space of instantons, but we only get a pair $J_1,J_2$ of commuting generalized complex structures by \emph{choosing} a $2$-form with $db=H$. In that respect $J_1,J_2$ are defined modulo a closed B-field but we can still extract some information about them. In particular the formula (\ref{J12}) shows that the real Poisson structures defined by $J_1$ and $J_2$, namely $\omega_+^{-1}\pm \omega_-^{-1}$, are unchanged by $b\mapsto b+B$. We shall determine the \emph{symplectic foliation} on ${\mathcal M}$ determined by these Poisson structures, which relates to the ``type" of the generalized complex structure as discussed by Gualtieri.
The symplectic foliation of a Poisson structure $
\pi$ is determined by the subspace of the cotangent bundle annihilated by $\pi:T^*\rightarrow T$. From (\ref{J12}), in our case
$$\mathop{\rm ker}\nolimits \pi_1=\mathop{\rm ker}\nolimits (I_++I_-),\quad \mathop{\rm ker}\nolimits \pi_2=\mathop{\rm ker}\nolimits (I_+-I_-)$$
where $I_+,I_-$ act on $T^*$.
Note that if $I_+a=I_-a$ then
$$[I_+,I_-]a=I_+I_-a-I_-I_+a=(I_+)^2a-(I_-)^2a=-a+a=0$$
so that $\mathop{\rm ker}\nolimits (I_+-I_-)\subset \mathop{\rm ker}\nolimits[I_+,I_-]$, and similarly if $I_+a=-I_-a$. It follows that if $I_+a=I_-a$, then $I_+(I_+a)=I_-(I_+a)$ since both sides are equal to $-a$. Thus $\mathop{\rm ker}\nolimits \pi_1$ and $\mathop{\rm ker}\nolimits \pi_2$ are complex subspaces of $\mathop{\rm ker}\nolimits[I_+,I_-]$ (with respect to either structure).
Now the kernel of $[I_+,I_-]$ is, from \ref{holp}, the kernel of the holomorphic Poisson structure $\sigma_+$ (or $\sigma_-$). But Theorem \ref{botta} tells us that this is the canonical Poisson structure on ${\mathcal M}$. Its kernel is easily determined (see \cite{Bot1}). Recall that the Poisson structure is defined, as a map from $(T^{1,0})^*$ to $T^{1,0}$, by the multiplication operation of the section $s$ of $K^*$:
$$s:H^1(M,\mathop{\rm End}\nolimits E\otimes K)\rightarrow H^1(M,\mathop{\rm End}\nolimits E).$$
If $D$ is the anticanonical divisor of $s$ then we have an exact sequence of sheaves
$$0\rightarrow{\mathcal O}_M(\mathop{\rm End}\nolimits E\otimes K)\stackrel{s}\rightarrow {\mathcal O}_M(\mathop{\rm End}\nolimits E)\rightarrow {\mathcal O}_D(\mathop{\rm End}\nolimits E)\rightarrow 0$$
and the above is part of the long exact cohomology sequence.
Since a stable bundle is simple, $H^0(M,\mathop{\rm End}\nolimits E)$ is just the scalars, so the map
$H^0(M,\mathop{\rm End}\nolimits E)\rightarrow H^0(D,\mathop{\rm End}\nolimits E)$ just maps to the scalars. Hence the
kernel of $\sigma_+$ is isomorphic from the exact sequence to $H^0(D,\mathop{\rm End}\nolimits_0 E)$ under the connecting homomorphism:
$$\delta_+: H^0(D,\mathop{\rm End}\nolimits E)\rightarrow H^1(M,\mathop{\rm End}\nolimits E\otimes K).$$
When $D$, an anticanonical divisor, is of multiplicity $1$ and smooth, it is an elliptic curve by the adjunction formula: $KD+D^2=2g-2$ implies $0=K(-K)+(-K)^2=2g-2$. Generically a holomorphic bundle on an elliptic curve is a sum of line bundles, and then the dimension of $H^0(D,\mathop{\rm End}\nolimits_0 E)$ is $k-1$ if $\mathop{\rm rk}\nolimits E=k$. Thus the real dimension of $\mathop{\rm ker}\nolimits[I_+,I_-]$ is at least $2(k-1)$.
\vskip .25cm
Now the divisor $D$ is, by definition, the subset of $M$ on which $I_+=\pm I_-$, say $I_+=I_-$.
Thus the complex structure of the bundle $E$ determined by its ASD connection is the \emph{same} on $D$ for $I_+$ and $I_-$. So the same holomorphic section $u$ of $\mathop{\rm End}\nolimits_0 E$ on $D$ maps complex linearly in two different ways to the cotangent space of ${\mathcal M}$. To study these maps we should really say that there are real isomorphisms
$$\alpha_{\pm}: H_{\pm}^1(M,\mathop{\rm End}\nolimits E\otimes K)\rightarrow T^*_{[A]}$$
such that $\alpha_{\pm}$ is $I_{\pm}$-complex linear.
\begin{prp} \label{residue} $\alpha_+\delta_+=\alpha_-\delta_-$
\end{prp}
\begin{prf} Recall how the connecting homomorphism is defined in Dolbeault terms, for the moment in the case where $D$ has multiplicity one: we have a holomorphic section $u$ of $\mathop{\rm End}\nolimits_0 E$ on $D$, and then extend using a partition of unity to a $C^{\infty}$ section $\tilde u$ on $M$. Then since $u$ is holomorphic on $D$, $\bar\partial \tilde u$ is divisible by $s$, the section of $K^*$ whose divisor is $D$. Then
$\delta(u)$ is represented by the $(2,1)$-form $s^{-1}\bar\partial_A\tilde u$.
Let $a\in T_{[A]}$ be a tangent vector to the moduli space, so $a\in\Omega^1(M,\mathop{\rm End}\nolimits E)$ and satisfies $d_A^+a=0$. So $\bar\partial_A a^{0,1}=0$ and we evaluate the cotangent vector $\delta_+(u)$ on $a$ to get
$$\int_M\mathop{\rm tr}\nolimits(s^{-1}\bar\partial_A\tilde u \wedge a).$$
But $s^{-1}=\gamma/2i$ so this is
$$\frac{1}{2i}\int_M\gamma\mathop{\rm tr}\nolimits(\bar\partial_A\tilde u\wedge a).$$
Away from the divisor $D$, we have
$$\bar\partial(\gamma\wedge\mathop{\rm tr}\nolimits(\tilde u a))=\gamma\wedge \mathop{\rm tr}\nolimits(\bar\partial_A\tilde u \wedge a)$$
since both $\gamma$ and $a$ are $\bar\partial$-closed. By Stokes' theorem the integral is reduced to an integral around the unit circle bundle of the normal bundle of $D$ and from there to an integral over $D$. In fact, if $\gamma$ has a simple pole along $D$ then locally
$$\gamma=f(z_1,z_2)\frac{dz_1\wedge dz_2}{z_1}$$
where $z_1=0$ is the equation of $D$. The holomorphic one-form $f(0,z_2)dz_2$ is then globally defined on $D$ -- the \emph{residue} $\gamma_0$ of the meromorphic $2$-form. This residue is the same for $I_+$ and $I_-$ (from Proposition \ref{4formulas} the meromorphic form for $I_-$ is $-2i\omega_+-(p-1)/2\gamma+(p+1)/2\bar\gamma$ and $p=-1$ on $D$).
Thus the integral becomes
$$\frac{1}{2i}\int_D\gamma_0\wedge \mathop{\rm tr}\nolimits(ua).$$
This is defined entirely in terms of the data on $D$ and so is the same for $I_+$ and $I_-$.
\vskip .25cm
When the divisor has multiplicity $d$, the section $u$ extends holomorphically to the $(d-1)$-fold formal neighbourhood of the curve and our $C^{\infty}$ extension must agree with this. The result remains true. (Note that the discussion of Poisson surfaces and moduli spaces via the residue is the point of view advanced in Khesin's work \cite{Kh}.)
\end{prf}
\vskip .25cm
\begin{cor} The two real Poisson structures $\pi_1,\pi_2$ defined by the generalized complex structures $J_1,J_2$ on the moduli space ${\mathcal M}$ of $SU(k)$ instantons have kernels of dimension $0$ and $\ge 2(k-1)$.
\end{cor}
\begin{prf}
We saw at the beginning of the Section that if $I_+a=I_-a$ then $[I_+,I_-]a=0$. Proposition \ref{residue} shows that $I_+$ and $I_-$ agree on the kernel of $[I_+,I_-]$, so that $\mathop{\rm ker}\nolimits(I_+-I_-)=\mathop{\rm ker}\nolimits[I_+,I_-]$.
Now $\mathop{\rm ker}\nolimits(I_+-I_-)$ is the kernel of Poisson structure $\pi_1$ say, which is isomorphic to $H^0(D,\mathop{\rm End}\nolimits_0 E)$ and has, as we have seen, at least $2(k-1)$ real dimensions. The other Poisson structure $\pi_2$ has kernel $\mathop{\rm ker}\nolimits(I_++I_-)$. But this also lies in the kernel of $[I_+,I_-]$ so $I_+a=I_-a$. With $I_+a=-I_-a$ this means $a=0$.
\end{prf}
\vskip .25cm
The generalized complex structure $J_2$ on ${\mathcal M}$ where the kernel of the Poisson structure is zero is therefore of the form $\exp(B+i\omega)$ and it is tempting to associate it to the generalized complex structure of symplectic type on $M^4$. However, as we have seen, there appears to be no way to naturally associate or even define these structures, since the $2$-form $b$ does not descend to the moduli space.
\subsection{Examples of symplectic leaves}
We saw in the previous section that the symplectic leaves of $\pi_1$ are the same as the the symplectic leaves of the canonical complex Poisson structure on ${\mathcal M}$.
The simplest example is to take ${\mathbf C}{\rm P}^2$ with the anticanonical divisor defined by a triple line : $D=3L$. The moduli space of stable rank $2$ bundles with $c_2=2$ has dimension $4\times 2-3=5$ and has a very concrete description. Such a bundle $E$ is trivial on a general projective line but jumps to ${\mathcal O}(1)\oplus {\mathcal O}(-1)$ on the lines which are tangent to a nonsingular conic $C_E$. The moduli space ${\mathcal M}$ is then just the space of non-singular conics, which is a homogeneous space of $PGL(3,\mathbf{C})$.
The subgroup preserving $L$ (the line at infinity say) is the affine group $A(2)$ and if it preserves the Poisson structure it fixes $dz_1\wedge dz_2$. Hence the $5$-dimensional unimodular affine group $SA(2)$ acts on ${\mathcal M}$ preserving the Poisson structure. The subgroup $G$ which fixes the conic $z_1z_2=a$ consists of the transformations $(z_1,z_2)\mapsto (\lambda z_1,\lambda^{-1}z_2)$ so for each $a$, the orbit of the conic under $SA(2)$ is isomorphic to the $4$-dimensional quotient $SA(2)/G$. These orbits are the generic symplectic leaves of the Poisson structure, and thus are homogeneous symplectic and hence isomorphic to coadjoint orbits.
In fact if $z\mapsto Az+b$ is in the Lie algebra of $SA(2)$ then $G$ is the stabilizer of the linear map $f(A,b)=A_{11}$ so that $SA(2)/G$ is the orbit of $f$ in the dual of the Lie algebra.
This deals with conics which meet $L$ in two points. The ones which are tangential to $L$ (i.e. the bundles for which $L$ is a jumping line) are parabolas: e.g. $z_1^2=z_2$. The identity component of the stabilizer of this is the one-dimensional group
$(z_1,z_2)\mapsto (z_1+c,2cz_1+z_2+c^2)$ and this stabilizes the linear map
$(A,b)\mapsto A_{21}+4b_1$, so we again have a coadjoint orbit.
\vskip .25cm
In general, the symplectic leaves are roughly given by the bundles $E$ on $M$ which restrict to the same bundle on the anticanonical divisor $D$. ``Roughly", because we are looking at equivalence classes and a stable bundle on $M$ may not restrict to a stable bundle on $D$, so there may not be a well-defined map from ${\mathcal M}$ to a Hausdorff moduli space. On the other hand this is the quotient space of a (singular) foliation so we don't expect that.
When $D$ is the triple line $3L$ in ${\mathbf C}{\rm P}^2$ there is an alternative way of describing these leaves. On a generic line $E$ is trivial and the sections along that line define the fibre of a vector bundle $F$ on the dual plane, outside the curve $J$ of jumping lines. If we take a section of $E$ on $L$ we can try and extend it to the first order neighbourhood of $L$. Since the normal bundle to $L$ is ${\mathcal O}(1)$ there is an exact sequence of sheaves for sections on the $n$-th order neighbourhood:
$$0\rightarrow {\mathcal O}(E(-n))\rightarrow {\mathcal O}^{(n)}(E)\rightarrow {\mathcal O}^{(n-1)}(E)\rightarrow 0.$$
Since $H^0({\mathbf C}{\rm P}^1, {\mathcal O}(-1))=H^1({\mathbf C}{\rm P}^1, {\mathcal O}(-1))=0$, any section has a unique extension to the first order neighbourhood: this defines a \emph{connection} on $F$. The extension to the second order neighbourhood is obstructed since $H^1({\mathbf C}{\rm P}^1, {\mathcal O}(-2))\cong \mathbf{C}$ and this obstruction is the \emph{curvature} of the connection (see \cite{Hurt} for details of this twistorial construction).
What it means is that if $L$ is not a jumping line, then $E$ restricted to $3L$ is essentially the curvature of the connection on $F$ at the point $\ell$ in the dual plane defined by the line $L$, and the symplectic leaves are obtained by fixing the equivalence class of the curvature at that point. The curvature acquires a double pole on $J$.
From this point of view, the case $k=2,c_1=0,c_2=2$ concerns an $SO(3,\mathbf{C})$-invariant connection on a rank $2$-bundle on the complement of a conic, and this is essentially the Levi-Civita connection of ${\mathbf R}{\rm P}^2$ complexified. This is an $O(2)$-connection which becomes an $SO(2)$ connection on $S^2$ with curvature
$$\frac{dz\wedge d\bar z}{(1+\vert z\vert^2)^2}.$$
So the bundle on $D$ is equivalent to the transform of the complexification of this by a projective transformation. If the dual conic is defined by the symmetric $3\times 3$ matrix $Q_{ij}$ and $x$ is a vector representing $\ell$ then the curvature is
$$\frac{(\det Q)^{2/3}}{Q(x,x)^2}.$$
The symplectic leaves are then given by the equation $\det Q=aQ(x,x)^3$ for varying $a$.
\section{A quotient construction}
It is well-known that the moduli space of instantons on a hyperk\"ahler $4$-manifold is hyperk\"ahler and this can be viewed as an example in infinite dimensions of a hyperk\"ahler quotient -- the quotient of the space of all connections by the action of the group of gauge transformations. One may ask if, instead of the painful integration by parts that we did in the previous sections, there is a cleaner way of viewing the definition of a generalized K\"ahler structure on ${\mathcal M}$. The problem is that such a quotient would have to encompass not only the hyperk\"ahler quotient but also the ordinary K\"ahler quotient, and in finite dimensions these are very different -- the dimension of the quotient in particular is different!
We offer next an example of a generalized K\"ahler quotient which could be adapted to replace the differential geometric arguments in the previous sections for the case of a torus or K3, and at least gives another reason why the calculations should hold. It also brings out in a natural way the frustrating feature that the $2$-form $b$ does not descend in general to the quotient.
\vskip .25cm
We suppose the generalized K\"ahler structure is even and is given by global forms
$\rho_1=\exp {\beta_1}, \rho_2=\exp{\beta_2}$
where $\beta_1,\beta_2$ are closed complex forms on a real manifold $M$ of dimension $4k$. This is the test situation we have been considering throughout this paper. From Lemma \ref{commute}, the compatibility ($J_1J_2=J_2J_1$) is equivalent to
$$(\beta_1-\beta_2)^{k+1}=0,\qquad (\beta_1-\bar\beta_2)^{k+1}=0.$$
Now suppose a Lie group $G$ acts, preserving the forms $\beta_1,\beta_2$, and giving complex moment maps $\mu_1,\mu_2$. To make a quotient, we would like to take the joint zero set of $\mu_1$ and $\mu_2$ and divide by the group $G$, but these are two \emph{complex} functions so if they were generic we would get as a quotient a manifold of dimension $\dim M-5\dim G$ instead of $\dim M-4\dim G$.
To avoid this, we need to assume that $\beta_1,\beta_2,\bar\beta_1,\bar\beta_2$ are linearly dependent over $\mathbf{R}$.
\begin{rmk} If we were trying to set up the moduli space of instantons as a quotient of the space of all connections on a K3 surface or a torus, the following lemma links the condition of linear dependence of the moment maps to the necessity to choose a self-dual $b$.
\begin{lem} If $\dim M=4$, then $\beta_1,\beta_2,\bar\beta_1,\bar\beta_2$ are linearly independent over $\mathbf{R}$ at each point if and only if $b$ is self-dual.
\end{lem}
\begin{lemprf} From Proposition \ref{4formulas} we have
\begin{eqnarray*}
\beta_1+\bar\beta_1&=&2b-(p-1)(\gamma+\bar\gamma)/2\\
\beta_2+\bar\beta_2&=&2b-(p+1)(\gamma+\bar\gamma)/2\\
-i(\beta_1-\bar\beta_1)&=&2\omega_+-(p-1)i(\gamma-\bar\gamma)/2\\
-i(\beta_2-\bar\beta_2)&=&2\omega_+-(p+1)i(\gamma-\bar\gamma)/2\\
\end{eqnarray*}
We can easily solve these for $b,\omega_+,\gamma+\bar\gamma,i(\gamma-\bar\gamma)$ in terms of the $\beta_i$.
If $b$ is self-dual, it is a real linear combination of $\omega_+,\gamma+\bar\gamma,i(\gamma-\bar\gamma)$ since $\gamma$ is of type $(2,0)$ relative to $I_+$, hence we get a linear relation amongst the left hand sides.
Conversely, a linear relation among the left hand sides will express $b$ in terms of $\omega_+,\gamma+\bar\gamma,i(\gamma-\bar\gamma)$ unless it is of the form
$$(\beta_1+\bar\beta_1)-(\beta_2+\bar\beta_2)+i\lambda(\beta_1-\bar\beta_1)+i\mu (\beta_2-\bar\beta_2)=0.$$
But the $(1,1)$ component of this is $2(\mu-\lambda)\omega_+$ so $\lambda=\mu$ and then the relation can be written
$$(1+i\lambda)(\beta_1-\bar\beta_2)+(1-i\lambda)(\bar\beta_1-\beta_2)=0$$
but $(\bar\beta_1-\beta_2)$ is of type $(2,0)$ relative to $I_-$ so this is impossible.
\end{lemprf}
We see that the condition for $b$ to define $\tilde b$ on the moduli space ${\mathcal M}$ is related to the linear dependence issue of the moment maps.
\end{rmk}
Returning to the general case, for each vector field $X$ from the Lie algebra of $G$ we have $i_X\beta_i=d\mu_i$ and so $\beta_i$ restricted to $\mu_1=0=\mu_2$ is annihilated by $X$, and invariant under the group and hence is the pullback of a form $\tilde\beta_i$ on the quotient, which is also closed.
In the bihermitian interpretation, $\bar\beta_1-\bar\beta_2$ is a non-degenerate $(2,0)$-form relative to $I_+$ -- a holomorphic symplectic form -- and the quotient can then be identified with the holomorphic symplectic quotient. In particular if the complex dimension of the quotient is $2m$ then $(\tilde\beta_1-\tilde\beta_2)^{m+1}=0$ and $(\tilde\beta_1-\tilde\beta_2)^{m}\ne 0$. Similarly $(\bar\beta_1-\beta_2)$ is $(2,0)$ with respect to $I_-$ and we get the same property for $(\tilde\beta_1-\bar{\tilde \beta_2})$. From Lemma \ref{commute} we have a generalized K\"ahler structure on the quotient.
\vskip .25cm
Note that in this generic case the Poisson structures on the quotient are non-degenerate.
|
{
"timestamp": "2005-03-21T18:32:38",
"yymm": "0503",
"arxiv_id": "math/0503432",
"language": "en",
"url": "https://arxiv.org/abs/math/0503432"
}
|
\section{Introduction}
\label{sec:level1}
One of the most exciting subject in theoretical nuclear physics is the
double beta decay, especially due to the neutrino-less ($0\nu\beta\beta$)
process
\cite{Hax,Fas1,Fas2,Suh}. Indeed, its discovery would answer a
fundamental question whether
neutrino is a Majorana or a Dirac particle. The theories devoted to the
description of this process suffer of the lack of reliable tests for the
nuclear matrix elements. O possibility to overcome such difficulties would
be to use the matrix elements which describe realistically the rate of
$2\nu\beta\beta$ decay. In this context many theoretical work have been
focussed on $2\nu\beta\beta$ process. Most formalisms are based on the
proton-neutron quasiparticle random phase approximation (pnQRPA)
which includes the particle-particle ($pp$) channel in the two body interaction.
Since such an interaction is not considered in the mean field equations the
approach fails at a critical value of the interaction strength, $g_{pp}$.
Before this value is reached, the Gamow-Teller transition amplitude
($M_{GT}$) is decreasing rapidly and after a short interval is
becoming equal to
zero. The experimental data for this amplitude is reached for a value of
$g_{pp}$ close to that one which vanishes $M_{GT}$ and also close to
the critical value.
Along the time, the instability of the pnQRPA ground state was considered
in different approaches. The first formalism devoted to this feature
includes anharmonicities through the boson expansion technique
\cite{Rad1,Rad2,Suh1,Grif}.
Another method is the renormalized pnQRPA procedure (pnQRRPA)
\cite{Toi} which keeps
the harmonic picture but the actual boson is renormalized by effects coming
from the terms of the commutators algebra, which are not taken into
account in the standard pnQRPA approach.
In a previous paper\cite{Rad3} we have proved that the pnQRRPA procedure does not
include the additional effects in an consistent way. Indeed, if the
commutators of two quasiparticle operators involves the average of
monopole terms then these terms should be considered also in the
commutators of the scattering terms. If one does so, new degrees of freedom
are switched on and a new pnQRPA boson can be defined. This contains,
besides the standard two quasiparticle operators, the proton-neutron
quasiparticles scattering terms. If the amplitude of the scattering term
is dominant comparing it to the other amplitudes, the
pnQRRPA phonon describes a new nuclear state.
The aim of this paper is to show that such a mode appears in a natural way
within a time dependent treatment. The present approach points out new
properties of the new proton-neutron collective mode. We use a schematic
many-body Hamiltonian which for a single j-shell is exactly solvable.
In this way the approximations might be judged by comparing the predictions of the actual model with the
corresponding exact results. Since the semi-classical treatment is the proper way
to determine the mean field, one expects that the present approach
is suitable to account for ground state correlations in a consistent way
and therefore some of the
drawbacks mentioned in a previous publication \cite{Rad3}, like the breaking down
of the fully renormalized RPA before the standard RPA breaks down, are removed.
To understand better the virtues of the present model we compare
its predictions with the results obtained in a renormalized RPA approach and
a boson expansion formalism. Since the semi-classical methods have, sometimes,
intuitive grounds we aim at obtaining a clear interpretation for the new
proton-neutron mode. It is well known that the breaking down of the RPA
approach is associated to a phase transition. In this respect the semi-classical
formalism is a suitable framework to define the nuclear phases which are bridged
by the Goldstone mode. Above arguments justify our option for a semi-classical treatment
and also sketch a set of expectations.
This project is achieved according to the following plan:
In Section II, we describe the model Hamiltonian. The main features of the
fully renormalized RPA approach, presented in a previous paper, are
briefly reviewed. A time dependent variational principle is formulated
in connection with a truncated quasiparticle Hamiltonian, in Section III. This Hamiltonian
is the term of the model Hamiltonian which determines the equations of
motion for the quasiparticle proton-neutron scattering terms, in the
de-coupling regime. The classical equations of motion
and their solutions are presented in Section IV.
The new $pn$ collective mode is
alternatively described through the renormalized RPA approach and boson expansion formalism
in Section V.
Numerical results are
analyzed in Section VI while the final conclusions are given in Section VII.
\section{The model Hamiltonian. Brief review of frn-RPA }
\label{sec:level2}
Since we are not going to describe realistically some experimental data
but to stress on some specific features of a heterogeneous many
nucleon system with proton-neutron interaction, we consider a schematic
Hamiltonian which is very often \cite{Sam,Rad4} used to study the
single and double beta Fermi transitions:
\begin{eqnarray}
H & = &
\sum_{jm}{(\varepsilon_{pj}-\lambda_{p})c^{\dag}_{pjm}c_{pjm}}
+\sum_{jm}{(\varepsilon_{nj}-\lambda_{n})c^{\dag}_{njm}c_{njm}}\nonumber\\
&&-\frac{G_{p}}{4}\sum_{jm,j'm'}{c^{\dag}_{pjm}
c^{\dag}_{\widetilde{pjm}}c_{\widetilde{pj'm'}}c_{pj'm'}}
-\frac{G_{n}}{4}\sum_{jm,j'm'}{c^{\dag}_{njm}c^{\dag}_{\widetilde{njm}}
c_{\widetilde{nj'm'}}c_{nj'm'}}\nonumber\\
&&+\chi\sum_{jm,j'm'}{c^{\dag}_{pjm}c_{njm}c^{\dag}_{nj'm'}c_{pj'm'}}
-\chi_{1}\sum_{jm,j'm'}{c^{\dag}_{pjm}c^{\dag}_{\widetilde{njm}}
c_{\widetilde{nj'm'}}c_{pj'm'}}.
\end{eqnarray}
$c^{\dag}_{\tau jm}(c_{\tau jm})$ denotes the creation (annihilation) of a
$\tau(=p,n)$ nucleon in a spherical shell model state $|\tau;nljm\rangle
=|\tau jm\rangle$ with $\tau$ taking the values $p$ for protons and $n$
for neutrons, respectively. The time reversed state corresponding to
$|\tau jm\rangle$ is $|\tau{\widetilde{jm}}\rangle=(-)^{j-m}|\tau j-m\rangle$
For what follows it is useful to introduce the quasiparticle ($qp$)
representation, defined by the Bogoliubov-Valatin (BV) transformation:
\begin{eqnarray}
a^{\dag}_{pjm}& = & U_{pj}c^{\dag}_{pjm}-V_{pj}c_{\widetilde{pjm}},\;\;
a_{pjm} = U_{pj}c_{pjm}-V^{*}_{pj}c^{\dag}_{\widetilde{pjm}}, \nonumber\\
a^{\dag}_{njm}& = & U_{nj}c^{\dag}_{njm}-V_{nj}c_{\widetilde{njm}},\;\;
a_{njm} = U_{nj}c_{njm}-V^{*}_{nj}c^{\dag}_{\widetilde{njm}} .
\end{eqnarray}
which quasi-diagonalizes the first four terms, i.e in the new
representation they are replaced by a set of independent quasiparticles of
energies:
\begin{equation}
E_{\tau}=\sqrt{(\epsilon_{\tau}-\lambda_{\tau})^{2}+\Delta_{\tau}^{2}}.
\end{equation}
In the new $qp$ representation, the model Hamiltonian, denoted by $H_q$,
describes a set of
independent quasiparticles, interacting among themselves through a two body
interaction determined by the images of the $\chi$ and $\chi_1$ terms
through the BV transformation.
Various many-body approaches have been tested by using not the $qp$ image
of $H$ but another Hamiltonian derived from $H$ by ignoring the
scattering $qp$ terms:
\begin{eqnarray}
B^{\dag}(jpn)&=&\sum_{m}a^{\dag}_{pjm}a_{njm}, \nonumber\\
B(jpn)& = &\sum_{m}a^{\dag}_{njm}a_{pjm}.
\end{eqnarray}
and restricting the space of single particle states to a single $j$-state.
Thus, the model Hamiltonian contains, besides the terms for the $qp$
independent motion, a two body term which is quadratic in the two
quasiparticle operators $A^{\dag}, A$:
\begin{equation}
A^{\dag}(jpn)=\sum_{m}a^{\dag}_{pjm}a^{\dag}_{\widetilde{njm}},
\;\;A(jpn)=(A^{\dag}(jpn))^{\dag}.
\end{equation}
In a previous publication\cite{Rad3}, we showed that going beyond the quasiparticle
random phase
approximation (pnQRPA) through a renormalization procedure, a new degree of
freedom is switched on, which results in having a renormalized pnQRPA boson
operator as a superposition of the operators $A^{\dag}(jpn), A(jpn)$ and
scattering terms $B^{\dag}(jpn),B(jpn)$. This picture differs from the
standard $pnQRRPA$ approach, where the boson operators involve only the
operators $A^{\dag}$ and $A$, and is conventionally called as fully
renormalized RPA ($frn-RPA$). Obviously, when the amplitudes of
scattering terms are dominant, one deals with a new kind of collective $pn$
excitation.
In order to define clearly the distinct features of the new proton-neutron
($pn$) mode revealed in the present paper a brief description of the
results obtained in a previous publication \cite{Rad3} is necessary.
The equations of motion associated to the many-body Hamiltonian, written in
terms of quasiparticle operators, are determined by the commutators algebra
of the two quasiparticle ($A^{\dag}, A$) and scattering $(B^{\dag},B)$
operators defined by eqs. (2.5) and (2.4) respectively.
Within the $frn-RPA$, the exact commutators are approximated as follows:
\begin{eqnarray}
\left[A(jpn),A^{\dag}(jpn)\right ] &=& C^{(1)}_{jpn},
\nonumber \\
\left[B(jpn),B^{\dag}(jpn)\right ] &=& C^{(2)}_{jpn},
\nonumber\\
\left[A(jpn),B^{\dag}(jpn)\right ] &=&\left[A(jpn),B(jpn)\right ] = 0.
\end{eqnarray}
The terms $C^{(1)}_{jpn}, C^{(2)}_{jpn}$ appearing in the r.h.s. of the
above equations are the averages of the corresponding exact commutators, on
the correlated ground state $|0>$:
\begin{equation}
C^{(1)}_{jpn}=\langle0|1-\hat{N}_{jn}-\hat{N}_{jp}|0\rangle,\: \:
C^{(2)}_{jpn}=\langle0|\hat{N}_{jn}-\hat{N}_{jp}|0\rangle.
\end{equation}
with $\hat{N}_{j\tau}$ standing for the $\tau$ (=p,n) quasiparticle number
operator in the shell j.
The normalized operators
\begin{eqnarray}
\bar{A}^{\dag}(jpn)=\frac{1}{\sqrt{C^{(1)}_{jpn}}}A^{\dag}(jpn),\:
\bar{A}(jpn)=\left(\bar{A}^{\dag}(jpn)\right)^{\dag},
\nonumber\\
\bar{B}^{\dag}(jpn)=\frac{1}{\sqrt{|C^{(2)}_{jpn}|}}B^{\dag}(jpn),\:
\bar{B}(jpn)=\left(\bar{B}^{\dag}(jpn)\right)^{\dag},
\end{eqnarray}
satisfy bosonic commutation relations and thereby their equations of
motion are linear:
\begin{equation}
\left[H_q,\left(\matrix{\bar{A}^{\dag}(jpn) \cr
\bar{A}(jpn) \cr
\bar{B}^{\dag}(jpn) \cr
\bar{A}(jpn)}\right)\right]
=\sum_{j,j^{\prime}}T^{j,j^{\prime}}\left(\matrix{\bar{A}^{\dag}(jpn) \cr
\bar{A}(jpn) \cr
\bar{B}^{\dag}(jpn) \cr
\bar{B}(jpn)}\right).
\end{equation}
The matrix $T^{j,j^{\prime}}$ depends on the U and V coefficients as well as
on the strengths $\chi, \chi_1$ of the two body interactions.
The $frn-RPA$ approach defines a linear combination of the basic operators
$\bar{A}^{\dag}(jpn), \bar{A}(jpn), \bar{B}^{\dag}(jpn), \bar{A}(jpn)$,
\begin{equation}
\Gamma^{\dag}=\sum_{j}\left[X(j)\bar{A}^{\dag}(jpn)+Z(j)D^{\dag}(jpn)
-Y(j)\bar{A}(jpn)-W(j)D(jpn)\right],
\end{equation}
so that the following commutation relations with its
hermitian conjugate operator and the model Hamiltonian hold:
\begin{eqnarray}
\left[\Gamma,\Gamma^{\dag}\right] &=& 1,
\\
\left[H_q,\Gamma^{\dag}\right] &=& \omega \Gamma^{\dag}.
\end{eqnarray}
The operators $D^{\dag}(jpn)$ are identical with ${\bar{B}}^{\dag}(jpn)$
or $\bar{B}(jpn)$ depending on whether the sign of $C^{(2)}_{jpn}$ is plus or
minus. The equation (2.12) provides a set of homogeneous equations- called
the $frn-RPA$ equations- for the amplitudes $X,Y,Z,W$:
\begin{eqnarray}
\left(\matrix{{\cal A} & {\cal B}\cr
-{\cal B} &-{\cal A}}\right)\left(\matrix{X\cr Z\cr Y\cr
W}\right) =\omega \left(\matrix{X\cr Z\cr Y\cr W}\right),
\end{eqnarray}
while the equation (2.11) yields the normalization equation
\begin{equation}
\sum_{j}(X^2(j)+Z^2(j)-Y^2(j)-W^2(j))=1.
\end{equation}
The $frn-RPA$ matrices depend on the renormalization constants $C^{(1)},
C^{(2)}$ which, at their turn, depend on the phonon amplitudes. Therefore,
the equations (2.12) and (2.7) should be self-consistently solved.
In ref. \cite{Rad3} the $frn-RPA$ equations have been solved both for a proton-neutron
dipole-dipole interaction, needed for the description of the
double beta Gamow-Teller decay and for a proton-neutron monopole-monopole interaction
used in the calculation of the rates of the double beta Fermi decay.
Equations obtained in the two cases have some common features which,
for what follows, are worth being enumerated.
\noindent
1) The dimension of the $frn-RPA$ matrix is twice as large as that of the
standard RPA and consequently new solutions show up.
\noindent
2) The solutions characterized by that the largest phonon amplitude is of
type Z define a new class of proton-neutron excitations.
\noindent
3) Due to the attractive character of the two body interaction in the
particle-particle ($pp$) channel, the lowest new state has an energy which
is smaller than the minimal absolute value of the relative energy of the
proton and neutron quasiparticle partner states, related by the operators
$B^{\dag}(jpn), B(jpn)$.
\noindent
4) For the N=Z nuclei, this minimal value is vanishing and therefore the
lowest mode becomes spurious or in other words saying a new symmetry
is open. The new symmetry corresponds to the restriction
$C^{(2)}_{jpn}=0$, i. e. the average of the third component of the
isospin operator is vanishing.
This means that the system is invariant to rotations around any axes in
the ($X,Y$) plane of the isospin space associated to the (jpn) orbits.
\noindent
5) Important quantitative effects are expected for heavy nuclei having the
proton and neutron Fermi energies lying far apart from each other.
\noindent
6) The presence of the additional states influences also the structure of
the states lying close to those predicted by the standard RPA. Indeed, the
actual normalization condition for the phonon amplitudes implies new
values for the X and Y weights. Consequently, the strengths for $\beta^-$
and $\beta^+$ transitions are shared by the "old"-lying close to the
standard RPA states- and the ``new'' states- for which the amplitudes Z
are dominant.
\noindent
7) The standard RPA approach is based on the quasi-boson approximation and
therefore it ignores some important dynamic effects (only the terms
$A^{\dag}A^{\dag}, A^{\dag}A, AA$ are considered in an approximative
manner) and moreover
the Pauli principle is violated.
By contrast, within the $frn-RPA$ all the terms of the model Hamiltonian are
taken into account. Also the Pauli principle is, to a certain extent,
restored. Due to this feature, large corrections to the double beta
transition amplitude as well as to the Ikeda sum rule are expected by
changing the RPA to the frn-RPA.
\noindent
8) The equations of motion for the $A^{\dag },A$ and $B^{\dag}, B$
operators are coupled by the terms $A^{\dag}B^{\dag}, A^{\dag}B,
AB^{\dag}, AB$ involved in the quasiparticle Hamiltonian. These terms are
multiplied by the factors $ U_pV_nU_{p^{\prime}}U_{n^{\prime}},
U_pV_nV_{p^{\prime}}V_{n^{\prime}}, V_pU_nU_{p^{\prime}}U_{n^{\prime}},
V_pU_nV_{p^{\prime}}V_{n^{\prime}}$ in the $ph-ph$ interaction (the
$\chi$ term)
and by $U_pU_nU_{p^{\prime}}V_{n^{\prime}},
U_pU_nV_{p^{\prime}}U_{n^{\prime}}, V_pV_nU_{p^{\prime}}V_{n^{\prime}},
V_pV_nV_{p^{\prime}}U_{n^{\prime}}$ in the $pp-hh$ interaction (the $\chi_1$
term), respectively. Note that the coupling terms change the number of
either proton or neutron quasiparticles by two units. The terms
bringing the main contribution to the equations of motion for the
operators $A^{\dag}, A$ commute with $ \hat{N}_{jp}-\hat{N}_{jn}$
but not with $ \hat{N}_{jp}+\hat{N}_{jn}$. By contrary the terms
having the dominant contribution to the equations of motion for the
operators $B^{\dag},B$ commute with $ \hat{N}_{jp}+\hat{N}_{jn}$
and not with $ \hat{N}_{jp}-\hat{N}_{jn}$. None of the two operators,
$ \hat{N}_{jp}-\hat{N}_{jn}$, $ \hat{N}_{jp}+\hat{N}_{jn}$, commutes with
the coupling terms.
Retaining from the $\chi$-interaction the $pp-hh$ terms (those multiplied
by $U_pU_nV_{p^{\prime}}V_{n^{\prime}}$) and from the $\chi_1$ interaction
only the $ph-ph$ terms (those proportional to
$U_pV_nU_{p^{\prime}}V_{n^\prime} $ ) the equations of motion for the
operators $B^{\dag},B$ are decoupled from those for $A^{\dag}$ and $A$.
One may conclude that the new mode is determined by a combined effect
coming from the $pp-hh$ and $ph-ph$ terms belonging to the $\chi$ and
$\chi_1$ interactions, respectively.
\noindent
9) In the particle representation the $frn-RPA$ phonon operator is a linear
superposition of $ph, hp, pp$ and $hh$ operators.
\noindent
10) In the limit of large $pp$ and negligible $ph$ interactions, the
amplitudes $Z$ can be analytically calculated. The result is that $Z$ is
proportional to either $U_pV_n$ or $V_pU_n$, depending on whether the sign
of $E_p-E_n$ is plus or minus, respectively. When this amplitude prevails
over the other ones, the corresponding mode describes a neutron-hole
proton-particle (or a proton-hole neutron-particle) excitation of the
mother nucleus $(N,Z)$. Therefore the state is associated to the $(N-1,Z+1)$
(or to the $(N+1,Z-1)$) nucleus. In this case the state might be reached by exciting the ground state
through the transition operator $c^{\dag}_pc_n$ (or $c^{\dag}_nc_p$), which is typical for the
$\beta^-$ ( or $\beta^+$) decay. Since the double beta decay is conceived as taking place through two
successive $\beta^-$ transitions, one expects that this process is also influenced by considering
this new state as an intermediate state characterizing the odd-odd neighboring nucleus.
\noindent
11) When the $pp$ interaction is small the amplitude Z is proportional to
$U_pU_n$ if $E_p>E_n$ or to $V_pV_n$ in the case $E_p<E_n$.
The new mode characterizes the nucleus $(N+1,Z+1)$ in the first case and
the nucleus ($N-1,Z-1$) in the second situation. The transition operators
which could excite
these states are obviously of the types $c^{\dag}_pc^{\dag}_n$ and $c_nc_p$, respectively.
Note that the restriction of the phonon operator to the scattering terms
resembles the standard RPA boson operator written in the particle
representation. This comparison has, however, only a formal value since in
the quasiparticle representation there is no Fermi energy and therefore
one cannot speak about quasiparticle-quasihole excitations. Similar
features are met in solid state physics for the description of electron
excitations in narrow energy bands, spin waves and
plasma oscillations \cite{Hub}. In
nuclear physics, the scattering terms have been also considered but not
for proton-neutron excitations. Indeed, using the thermal response theory,
Tanabe \cite{Tan} studied the charge conserving phonons in nuclear systems
at a finite temperature. It seems that the contribution of the scattering
terms to the charge conserving bosons, does not survive at vanishing
temperature \cite{Hat}. Moreover, the
dispersion relation for the mode energy cannot be obtained from a
linearized set of equations as it is required by the spirit of the RPA
approach.
\section{Semi-classical treatment}
\label{sec:level3}
As we already mentioned, the scope of the present paper is to study the
$pn$ mode caused by the quasiparticle scattering terms within a
semi-classical approach. In this formalism the renormalization condition
(2.6) is missing and therefore the harmonic motion of the new degrees of
freedom hinges on a more physical ground.
Moreover, we address the question whether this mode survives when the
non-scattering terms are switched off. Thus, it is worth to know if such a
mode appears only when the scattering terms accompany the two
quasipatricle operators or it might be determined by the scattering terms
alone.
From the brief presentation of the previous Section it is clear that the
mode does not appear within the RPA approach. Indeed, it occurred within
the $frn-RPA$ after a consistent renormalization was performed (i. e. not
only the operators $A^{\dag}, A$ where renormalized but also $B^{\dag}$
and $B$). If that mode is a signature of the higher RPA formalisms, then it
should also appear within the semi-classical formalism as well as in the
boson expansion framework. As we shall see the semi-classical approach is
able to predict the mode even in the harmonic approximation, the mode
being associated with the small oscillations of the system around a static
correlated ground state. Moreover, the semi-classical frame is expected to
allow us an intuitive interpretation of this new type of excitation.
We recall that the higher order corrections to the standard RPA approach are
frequently studied, with different purposes, using a single j case and
ignoring the scattering terms. The procedure has the advantage that the
resulting Hamiltonian is exactly solvable. Therefore the quality of
the adopted approximations may be tested by comparing the predictions
with the corresponding exact results.
To touch the goal of the present paper
we adopt a similar point of view.
Indeed, if the coupling terms (mentioned at the point 8 of the previous
section) are ignored, the equations of motion for the scattering operators are
decoupled. Moreover the motion of these operators is determined also by an
exactly solvable Hamiltonian, which reads:
\begin{equation}
H^{(q)}_{pn}=E^{'}_{p}{\hat N}_{p}+E^{'}_{n}{\hat N}_{n}+\lambda_{1}
B^{\dag}(pn)B(pn)+\lambda_{2}\big(B^{\dag2}(pn)+B^{2}(pn)\big),
\end{equation}
where the following notations have been used:
\begin{eqnarray}
E_p^{\prime}&=&E_p+(\chi_1-\chi)V_p^2,\nonumber\\
E_n^{\prime}&=&E_n+(\chi+\chi_1)V^2_p(V_n^2-U_n^2),\nonumber\\
\lambda_1&=&\chi(U_p^2U_n^2+V_p^2V_n^2)-\chi_1(U_p^2V_n^2+V_p^2U_n^2),
\nonumber\\
\lambda_2&=&-(\chi+\chi_1)U_pU_nV_pV_n,\\
{\hat N}_{\tau}&=&\sum_{m}a^{\dag}_{\tau jm}a_{\tau jm}.
\end{eqnarray}
Also, to simplify the notation we omitted the quantum number $j$ for
the operators $B^{\dag}(jpn), B(jpn)$ as well as for the $U, V$
coefficients and quasiparticle energies.
This model Hamiltonian will be studied within a time dependent variational
formalism. Therefore, some static and dynamic properties will be described
by solving the equations provided by the time dependent variational
principle (TDVP)\footnote{Throughout this paper the units of $\hbar=1$ are used}:
\begin{equation}
\delta \int_0^t {\langle\Psi|H^{(q)}_{pn}-i\frac{\partial}{\partial t'}|\Psi\rangle}
\,dt' =0.
\end{equation}
If the variational state $|\Psi\rangle$ spans the whole Hilbert space
describing the many-body system, solving the equation (3.4) is equivalent
to solving the time dependent Schroedinger equation, which would be a very
difficult task. In the present paper,
the trial function is taken as:
\begin{equation}
|\Psi\rangle = exp[zB^{\dag}(pn)-z^{*}B(pn)]|NT\; -T\rangle,
\end{equation}
where $|NTT_{3}\rangle$ denotes the common eigenfunction of the quasiparticle
total number (${\hat N}$), the quasiparticle isospin squared
(${\hat T}^{2})$, and its
z-axis projection ($T_{z}$) operators, respectively.
$z$ is a complex function of time and $z^*$, the corresponding complex conjugate function.
We justify this choice by the symmetry properties of the
model Hamiltonian. Indeed, let us note first that $H^{(q)}_{pn}$ commutes
with the
quasiparticle total number operator. Moreover, it can be written in terms
of the quasiparticle total number operator and generators of the SU(2) isospin algebra
\begin{eqnarray}
\tau_{+1} & = & -\frac{1}{\sqrt 2}B^{\dag}(pn),\nonumber\\
\tau_{-1} & = & \frac{1}{\sqrt 2}B(pn),\nonumber\\
\tau_{0} & = & \frac{1}{2}(\hat N_{p}-\hat N_{n}).
\end{eqnarray}
Due to this property of $H^{(q)}_{pn}$, the function
$|\Psi\rangle$, which is a coherent state for the SU(2) group, is the
most suitable for a semi-classical treatment.
Before closing this section we would like to write the trial function
in a form which suits better the further purposes. Using the Cambel Hausdorff
factorization\cite{Kir} for the exponential function, as explained in Appendix A,
one obtains:
\begin{eqnarray}
|\Psi\rangle & = & {\cal N} e^{\alpha B^{\dag}(pn)}|NT\;-T\rangle,
\nonumber\\
{\cal N}& = &(1+\alpha^{*}\alpha)^{-T}.
\end{eqnarray}
where $\alpha$ depends on the polar coordinates, $z=\rho e^{i\varphi}$:
\begin{equation}
\alpha=\tan(\rho)e^{i\varphi}.
\end{equation}
\section{Equations of motion}
\label{sec:level4}
In order to write the equations of motion provided by the TDVP (3.4),
we need the matrix element of $H^{(q)}_{pn}$ as well as of the time derivative operator,
$\frac{\partial}{\partial t}$. These can be evaluated by direct calculation, using the
expressions (3.5) when the average of $H^{(q)}_{pn}$ is considered and (3.7) for the
classical action. The result is:
\begin{eqnarray}
\langle\Psi|H^{(q)}_{pn}|\Psi\rangle& =& -T(E_{p}^{\prime}-E_{n}^{\prime}+2\lambda_{1})
+\frac{N}{2}(E_{p}^{\prime}+E_{n}^{\prime})+2T(E_{p}^{\prime}-E_{n}^{\prime}+2\lambda_{1})
\frac{\alpha^{*}\alpha}{1+\alpha^{*}\alpha}\nonumber\\
&&+2T(2T-1)\left[\lambda_{1}\frac{\alpha^{*}\alpha}
{(1+\alpha^{*}\alpha)^{2}}+\lambda_{2}\frac{\alpha^{*2}+
\alpha^{2}}{(1+\alpha^{*}\alpha)^{2}}\right],
\nonumber\\
\langle\Psi|\frac{\partial}{\partial t}|\Psi \rangle &=&
T\frac{\alpha^{*}\stackrel{\bullet}{\alpha}-\stackrel{\bullet}{\alpha}^{*}
\alpha}{1+\alpha^{*}\alpha}.
\end{eqnarray}
Considering $\alpha,\alpha^*$ as classical phase space coordinates, the
TDVP equation (3.4) yields the following classical equations of motion,
describing the nuclear system:
\begin{eqnarray}
\frac{\partial{\cal H}}{\partial\alpha}&=&-2i
\frac{T\stackrel{\bullet}{\alpha}^{*}}{(1+\alpha^{*}\alpha)^{2}}, \nonumber\\
\frac{\partial{\cal H}}{\partial \alpha^{*}}&=&2i
\frac{T\stackrel{\bullet}{\alpha}}{(1+\alpha^{*}\alpha)^{2}}.
\end{eqnarray}
Here ${\cal H}$ denotes the classical energy function:
\begin{equation}
{\cal H}=\langle\Psi|H^{(q)}_{pn}|\Psi\rangle.
\end{equation}
In order to quantize the classical trajectories satisfying the equations
(4.2) as well as to have an one to one correspondence between the classical
and quantal behaviors of the nucleon system, it is convenient to chose
those conjugate variables which bring the equations of motion in a
canonical Hamilton form. A possible choice of the coordinates with the
above mentioned property is
\begin{eqnarray}
r&=&\frac{2T}{1+\alpha^{*}\alpha},\\
\psi&=&-\frac{1}{2i}(\ln \alpha-\ln \alpha^{*})=-\varphi.
\end{eqnarray}
Indeed, in the new variables the classical equations read:
\begin{eqnarray}
\frac{\partial{\cal H}}{\partial r}&=&-\stackrel{\bullet}{\psi},\nonumber\\
\frac{\partial{\cal H}}{\partial \psi}&=&\stackrel{\bullet}{r}.
\end{eqnarray}
with the classical energy:
\begin{eqnarray}
{\cal H}&=&T(E_{p}^{\prime}-E_{n}^{\prime}+2\lambda_{1})+\frac{N}{2}
(E_{p}^{\prime}+E_{n}^{\prime})
\nonumber\\
&&-(E_{p}^{\prime}-E_{n}^{\prime}+2\lambda_{1})r+\frac{2T-1}{2T}r(2T-r)
(\lambda_{1}+2\lambda_{2}\cos2\psi).
\end{eqnarray}
Note that $r$ has the significance of a generalized coordinate while $\psi$
that of generalized linear momentum.
Due to the generalized momentum $\psi $, the equations motion are not
linear and therefore analytical solutions are not obtainable.
The equations can however be approximatively solved if they are linearized around the
minimum point of the energy function:
\begin{equation}
\stackrel{\circ}{r}=T\left[1-\frac{E_{p}^{\prime}-E_{n}^{\prime}+
2\lambda_{1}}{(2T-1)(\lambda_{1}-2\lambda_{2})}\right],
\;\;
\stackrel{\circ}{\psi}=\frac{\pi}{2}.
\end{equation}
In order that the minimum exits, it is necessary that the generalized
coordinates satisfy a
consistency condition, required by the definition range of $r$:
\begin{equation}
0\leq\stackrel{\circ}{r}\leq2T.
\end{equation}
By means of (4.8), this provides a constraint for the strengths of the two
body interactions.
The linearized equations, written in terms of the deviations
\begin{equation}
q=r-\stackrel{\circ}{r},\; p=\psi-\stackrel{\circ}{\psi},
\end{equation}
are of harmonic type:
\begin{eqnarray}
-\stackrel{\bullet}{p}=-2\frac{2T-1}{2T}(\lambda_{1}-2\lambda_{2})q,
\nonumber\\
\stackrel{\bullet}{q}=4\frac{2T-1}{T}\stackrel{\circ}{r}
(2T-\stackrel{\circ}{r})\lambda_{2}p.
\end{eqnarray}
These describe a harmonic motion for the conjugate coordinates, with the
angular frequency:
\begin{equation}
\omega=2\frac{2T-1}{T}[-\lambda_{2}(\lambda_{1}-2\lambda_{2})\stackrel{\circ}
{r}(2T-\stackrel{\circ}{r})]^{\frac{1}{2}}.
\end{equation}
The condition that $\omega$ is a real quantity brings an additional constraint for
the strength parameters $\chi,\chi_1$:
\begin{equation}
\chi\geq\chi_1\left(\frac{U_pV_n-V_pU_n}{U_pU_n+V_pV_n}\right)^2.
\end{equation}
As we said already before, the schematic model has the advantage, over the
realistic formalisms, that allows us to compare the approximative solutions
with the exact one. For the particular Hamiltonian used in the present paper,
the exact eigenvalues can be obtained by diagonalization in the basis
{$|NTM\rangle$}. Indeed, in this basis the model Hamiltonian has the
following non-vanishing matrix elements:
\begin{eqnarray}
\langle NTM|H^{(q)}_{pn}|NTM\rangle &=&\frac{1}{2}(E_p^{\prime}+E_n^{\prime})N
+(E_p^{\prime}-E_n^{\prime})M+
\lambda_1(T+M)(T-M+1),\nonumber\\
\langle NTM+2|H^{(q)}_{pn}|NTM\rangle &=&\lambda_2\left[(T-M-1)(T-M)(T+M+1)(T+M+2)
\right]^{\frac{1}{2}},
\nonumber\\
\langle NTM|H^{(q)}_{pn}|NTM+2\rangle &=&\langle NTM+2|H^{(q)}_{pn}|NTM\rangle.
\end{eqnarray}
\section{The renormalized RPA and boson expansion}
\label{sec:level5}
Within the RPA approach, the renormalization of the quasiparticle mean
field due to the two quasiparticle interactions is usually ignored.
Therefore the Hamiltonian considered is:
\begin{equation}
H_{qp}=E_{p}{\hat N}_{p}+E_{n}{\hat N}_{n}+\lambda_{1}
B^{\dag}(pn)B(pn)+\lambda_{2}\big(B^{\dag2}(pn)+B^{2}(pn)\big),
\end{equation}
The operators $B^{\dag},B$ satisfy the commutation relation:
\begin{equation}
[B(pn),B^{\dag}(pn)]={\hat N}_n-{\hat N}_p.
\end{equation}
If the r.h. side of the above equation is replaced by its
average on the ground state,
\begin{equation}
C=\langle 0|{\hat N}_n-{\hat N}_p|0\rangle
\end{equation}
which is to be determined, then the operators
$B,B^{\dag}$ become bosons, after the following renormalization
\begin{equation}
{\widetilde {B}}^{\dag}(pn)=\frac{1}{\sqrt{C}}B^{\dag}(pn),\;
{\widetilde {B}}(pn)=\frac{1}{\sqrt{C}}B(pn),
\end{equation}
if $C$ is positive, while for negative $C$ the renormalized operators are:
\begin{equation}
{\widetilde {B}}^{\dag}(pn)=\frac{1}{\sqrt{|C|}}B(pn),\;
{\widetilde {B}}(pn)=\frac{1}{\sqrt{|C|}}B^{\dag}(pn).
\end{equation}
Suppose, for the time being, that $C>0$. If that is not the case
the corresponding calculations can be worked out in a similar way.
The equations of motion for the renormalized operators are:
\begin{eqnarray}
\left[H_{qp},{\widetilde {B}}^{\dag}(pn)\right]& = & (E_p-E_n+\lambda_1C)
{\widetilde{B}}^{\dag}(pn) +2\lambda_2C{\widetilde{B}}(pn),
\nonumber \\
\left[ H_{qp},{\widetilde{B}}(pn)\right]& =& -2\lambda_2C{\widetilde{B}}^{\dag}(pn)
-(E_p-E_n+\lambda_1C){\widetilde{B}}(pn) .
\end{eqnarray}
Since the equations are linear in ${\widetilde {B}}^{\dag}(pn)$ and
${\widetilde{B}}(pn)$, one can define the phonon operator
\begin{equation}
\Gamma^{\dag}=X\widetilde {B}^{\dag}(pn)-Y\widetilde {B}(pn),
\end{equation}
with the amplitudes determined such that the following equations are
fulfilled:
\begin{eqnarray}
\left[H_{qp},\Gamma^{\dag}\right]&=&\omega\Gamma^{\dag},\nonumber\\
\left[\Gamma,\Gamma^{\dag}\right]&=&1.
\end{eqnarray}
The first equation provides the dispersion equation for the mode energy
\begin{equation}
\omega=\left[(E_p-E_n+\lambda_1C)^2-4\lambda_2^2C^2\right]^{\frac{1}{2}},
\end{equation}
while the second one the normalization relation for phonon amplitudes:
\begin{equation}
X^2-Y^2=1.
\end{equation}
The renormalized RPA vacuum is defined by
\begin{equation}
\Gamma|0\rangle=0.
\end{equation}
The solution of the above equation is:
\begin{equation}
|0\rangle=e^{-\frac{1}{8}(\frac{Y}{X})^2}e^{\frac{Y}{2X}{\widetilde
{B}}^2} |NT,-T\rangle.
\end{equation}
Then the renormalization constant $C$ can be exactly evaluated:
\begin{equation}
C=2T-2+\frac{2}{X^2}.
\end{equation}
Since $T\ge1$, the constant C is always positive.
The equations of motion allow us to express the amplitude Y in terms of X:
\begin{equation}
Y=\frac{1}{2\lambda C}[\omega-(E_p-E_n+\lambda_1C)]X,
\end{equation}
which together with the normalization condition (5.10) determines fully the
amplitudes X and Y in terms of C and $\omega$. Inserting the result for $X$
into the equation (5.13),
one obtains an equation for C as a function of $\omega$.
This and eq.(5.9) form a set of two nonlinear equations for the unknowns
$\omega$ and $C$.
As we mentioned before, another way to improve the RPA treatment is to use
the boson expansion concept. Through this procedure, the $SU(2)$
algebra, with the fermionic generators $\tau_{\pm 1}, \tau_0$ defined by
eq.(3.6), is mapped to
a boson $SU(2)$ algebra, generated by $\hat {T}_{\pm 1}, \hat {T_0}$.
Denoting by $b^+,b$ a pair of boson operators, the
$SU(2)$ algebra generators $\hat {T}_{\pm 1},\hat {T}_0$ can be constructed
as function of $b^+$ and b. The resulting expressions are conventionally
called as the boson expansion of the fermionic generators, respectively.
There are three distinct boson mappings for the fermionic SU(2) algebra found
by Holstein-Primakoff \cite{Hol}, Dyson \cite{Dy} and one of the present
authors (A. A. R.)\cite{Rad5}, respectively.
For the present purpose here we use the Holstein-Primakoff (HP) expansion:
\begin{eqnarray}
\hat{T}_{+1}&=&-\sqrt{T}b^+\left(1-\frac{b^+b}{2T}\right)^{\frac{1}{2}},
\nonumber\\
\hat{T}_{-1}&=&\sqrt{T}\left(1-\frac{b^+b}{2T}\right)^{\frac{1}{2}}b,
\nonumber\\
\hat{T}_0&=&b^+b-T.
\end{eqnarray}
By a direct calculation it can be checked that, by this mapping, to the
operator $\tau ^2$ it corresponds a C-number:
\begin{equation}
\hat{T}^2=T(T+1).
\end{equation}
The fermion Hamiltonian $H_{qp}$ commutes with the quasiparticle
total number and the same is true for the generators $\tau_{\pm 1},\tau_0$.
Therefore the image of the quasiparticle total number operator through
the HP mapping is invariant against any rotation in the isospin space and
consequently, according to the above equation, is a C-number.
Apart from an additive constant, the image of $H_{qp}$ through the HP
boson expansion is:
\begin{equation}
H^{(b)}_{qp}=(E_p-E_n)\hat{T}_0-2\lambda_1\hat{T}_{+1}\hat {T}_{-1}
+2\lambda_2(\hat {T}^2_{+1}+\hat {T}^2_{-1}).
\end{equation}
Making use of eqs. (5.15), the boson mapping of $H_{qp}$ is a infinite
series in the bosons $b^+, b$, due to the square root operators.
Expanding the square root operators and truncating the result at the
second order in bosons, the boson Hamiltonian becomes:
\begin{equation}
H^{(b)}_{qp;2}=(E_p-E_n+2\lambda_1T)b^+b+2\lambda_2T({b^+}^2+b^2).
\end{equation}
For a limited range of the interaction strength, this Hamiltonian can be
diagonalized through a canonical transformation:
\begin{eqnarray}
b^+&=&UB^++VB,
\nonumber\\
b&=&UB+VB^+,
\nonumber\\
1&=&U^2-V^2.
\end{eqnarray}
The restriction that the "dangerous" terms have a vanishing strength
yields the expression for the transformation coefficients and the
coefficient, $\omega_1$, of the diagonal term $B^+B$:
\begin{eqnarray}
\left(\matrix{U\cr V}\right)&=&\frac{1}{\sqrt{2}}\left[\mp
1+\frac{|E_p-E_n|}{\sqrt{(E_p-E_n+2\lambda_1T)^2 -16\lambda_2^2T^2}}
\right]^{\frac{1}{2}},
\nonumber\\
\omega_1&=&\left[(E_p-E_n+2\lambda_1T)^2
-16\lambda_2^2T^2\right]^{\frac{1}{2}}.
\end{eqnarray}
Comparing the expressions of $\omega_1$ (5.20) and $\omega$ (5.9),
one sees that the two energies are identical for the limiting case of
$X=1$, which is met when $\lambda_2=0$ (see eqs. (5.9) and (5.14)).
At this stage it is worthwhile to make the following remarks:
a) When the HP boson expansion of the model Hamiltonian is truncated at
the second
order terms in bosons, the quasiparticle total number operator is no
longer a C number. Therefore the contribution of this term should have
been considered in a consistent manner.
Moreover the truncation is justified only for large values of the total
isospin T.
b) The same inconsistency appears
in the calculation of the renormalization constant C. Indeed the
expression (5.14) is exact and therefore includes all contributions coming
from the infinite boson series of the correlated ground state, given by
(5.14).
c) Since the boson mapping (5.15) is an unitary transformation, the
exact eigenvalues of $H_{qp}$ are reproduced by diagonalizing the boson
expanded Hamiltonian $H^{(b)}_{qp}$.
For the second order truncated Hamiltonian, the canonical transformation
breaks down at a critical value of the attractive interaction strength.
However the diagonalization procedure is able to find the eigenvalues for
any strength of the attractive interaction. The resulting energies exhibits a phase transition
(the first derivative has a jump) at the critical value of the strength.
If the second branch of the energy curve could also be approximated by an
harmonic mode, describing small oscillations of the classical system around
a stationary state, this is still an open question \cite{Rad4}.
d) The HP boson representation provides for the harmonic
mode the interpretation of an wobbling motion of the system around the
total isospin. e) The HP boson expansion is justified (in the sense that
some eigenvalues of the truncated Hamiltonian are close to the
corresponding exact ones) when the rotation axis in the isospin space is
close to the quantization axis (z axis), which is usually taken as the axis
to which the maximum ``moment of inertia'' corresponds. If the angle between the
rotation axis and z-axis is large the harmonic energy may collapse.
In this case the quantization axis should be chosen as one of the X and Y axes
depending on the magnitude of the strength of the $\tau_x^2$ and
$\tau_y^2$ terms from the quasiparticle Hamiltonian $H^{(q)}_{pn}$.
In this case the boson representation suitable for the low order
description should be of Dyson type \cite{Rad6}. The harmonic approximation
for the new representation describes also a wobbling motion
of a frequency equal to the square root of the product of the inverse of the non-maximal
moments of inertia normalized to the inverse of the maximal moment of inertia.
\section{Numerical results}
\label{sec:level6}
The formalism described in the previous sections, has been applied to the
case $j=\frac{19}{2}$. On the proton level, 6 protons are distributed
while in the neutron level, 14 neutrons. Alike nucleons interact with each
other through pairing forces whose strength are $G_p=0.2 {\rm MeV}$ and $G_n=0.4$
MeV. From the pairing equations it results the following expression for
the quasiparticle energy:
\begin{equation}
E_{\tau}=\frac{1}{2}G_{\tau}\Omega,\; \Omega=\frac{2j+1}{2}.
\end{equation}
With the data specified above the result for the quasiparticle energy is:
\begin{equation}
E_p=2\; {\rm MeV},\;\;E_n=1\;\;{\rm MeV}.
\end{equation}
According to our previous study, the renormalized RPA ground state
involves a small number of quasiparticles. For example, for a small
strength of the particle-particle interaction, the quasiparticle
total number is about 2 while for large values of the above mentioned
strength the number may reach the value 4. Due to this behavior of the
correlated ground state we considered for the isospin carried by the
quasiparticles in the ground state, alternatively the values 1 and 2.
Although these values vary with increasing the particle particle strength
we kept them constant.
The numerical analysis refers to the dependence of the energy $\omega$
of the new nuclear mode, on the strengths of the $ph$ and $pp$ monopole
interactions, $\chi, \chi_1$. Aiming at showing how good is the
semi-classical approach for this new type of pn excitation, we calculated
also the
exact eigenvalues of the model Hamiltonian, by diagonalizing the associated
matrix (4.14) within the basis $|NTM\rangle$.
The results are shown in Fig. 1 and Fig. 2.
From Fig. 1, one notices that the harmonic mode collapses for a critical
value of the attractive interaction $\chi_1$.
This critical value is certainly depending on the repulsive interaction
strength. The larger is that strength the larger the critical value.
In Fig. 1, we have also plotted the normalized energy for the first excited
state. There are intervals for $\chi_1$ where the energy of the harmonic
mode approximates reasonably well the exact excitation energy. Moreover, for
two values of the strength parameter, the exact solutions are precisely
reproduced. For the case T=2, the two energies, exact and $\omega$, are
the same for $\chi_1=0$ and $\chi_1=0.3$ for $\chi=1$ and $\chi=0.5$
respectively, but the curves are going apart for the first part of interval
and then converge to an intersection point close to the critical value.
The peculiar feature of $\omega$ as a function of
$\chi_1$, which distinguishes it from the standard RPA modes,
consists of its non-monotonic behavior with respect to the increase of
the strength of the $pp$ interaction. The reason is that in the common
cases the mean field is constant when the two body interaction is varied,
while here by changing $\chi_1$ we change also the minimum point for energy
and therefore another mean field is obtained. It is interesting to notice
that although the ph interaction, the $\chi$ term, is kept constant,
the change of the mean field is equivalent to an increase of the effective
$ph$ interaction until $\omega$ reaches the maximum value from where the
attractive component of the two body interaction prevails.
In Fig. 2, the energies $\omega$ and the normalized
energy of the first excited state are shown as function of $\chi$, the
strength parameter of the $ph$ interaction. Both energies are
monotonically increasing with the increase of the interaction strength.
In contrast to what happens in the case of $\chi_1$ dependence, here
the change of the mean field by changing the energy minimum does not change
the repulsive character of the $\chi$ interaction. The agreement between
$\omega$ and the exact energy of the first excited state is reasonable good.
In Fig. 3. the energies characterizing the harmonic mode predicted by the
renormalized RPA and semi-classical method are plotted as function of
$\chi_1$. Also, the exact energy of the first excited state is presented.
Although they have different trends, the semi-classical and renormalized
RPA energies are not far from each other for $\chi<0.45$. At the
critical value $\chi$=0.57 the energy yielded by the
renormalized RPA is going very fast to zero. This behavior is specific
to the present model where only the scattering terms are considered.
Indeed, if the phonon operator includes both the two quasiparticle and
scattering terms, the corresponding mode collapses for larger $\chi_1$.
In the semi-classical treatment this happens only for very large $\chi_1$
since the static ground state is changed by increasing $\chi_1$.
The result obtained with the truncated HP boson expanded Hamiltonian
(see eq. (5.20)) is very close to the result shown in Fig. 3 for the
renormalized RPA procedure.
Comparing the results from Fig. 1a and Fig. 3, we remark on
the following features.
While the renormalized RPA energy collapses at a relatively small value of
$\chi_1$, the mode energy predicted by the semi-classical formalism
vanishes for a very large $\chi_1$, far beyond the realistic value, which is
$\chi_1=\chi$. This feature is a consequence of
changing the static ground state with $\chi_1$.
The energy behavior provided by the semi-classical method is also
different from that predicted by the standard renormalized pnQRPA
(see for example ref. 9) where the mode energy is a monotonic function of
$\chi_1$ and goes asymptotically to zero.
In this context we recall that the $frn-RPA$ breaks down \cite{Rad3}
before the standard RPA does, and that happened due to the fact that the
lowest $frn-RPA$ energy is that associated with the new collective mode.
From the present calculations one sees that {\it this is not true within the
semi-classical approach and therefore including the scattering terms in
the expression of the phonon operator does not prevent the treatment of
the many-body system
for a realistic value of the $pp$-interaction strength.}
The vanishing energies for the new mode, shown in Figs. 1a and 3 suggest
that a phase transition occurs according to the corresponding formalisms.
As we already mentioned this is clearly revealed if one diagonalizes the
Hamiltonian given be eq. (5.18) \cite{Sam,Rad4}. In the renormalized RPA
procedure the new phase is determined by a new minimum of the classical
energy associated to $H^{(b)}_{pn;2}$, reflecting the fact that the
$\lambda_2$ term is the dominant one for these values of $\chi_1$.
In the full-line and dotted-line curves of Fig. 1a, the corresponding energies
also vanish at certain critical values which result in having again a phase transition.
This is reflected in the
curve obtained by exact calculations, by the fact that the energy is minimum for the critical strength.
The increasing branch shown by the exact calculations (corresponding to
the second nuclear phase) might be semi-classically described by changing
the trial function, involved in the time dependent variational equations,
by rotating it (in the isospin space) with an angle which corresponds to the
orientation of the axis of maximum ``moment of inertia''.
It is remarkable that the far intersection points of the curves obtained
by semi-classical and exact calculations respectively, are lying close to
the critical values of the semi-classical description. Also, the first
intersection point is not far from the critical value of the
renormalized RPA treatment. In the classical treatment this feature is
well known \cite{Ghe}. Indeed, in the above quoted reference it is shown,
for a triaxial rotor cranked on an arbitrarily oriented axis, that for certain
critical values of the strength parameters, the period of the harmonic orbits
is equal to the period characterizing the motion on the closed exact orbit.
\section{Conclusions}
\label{sec:level7}
The main result of this paper refers to the existence of an harmonic mode
determined by the scattering quasiparticle terms, which are usually
neglected in the standard RPA approach.
The new mode is described within a time dependent variational formalism with an exactly solvable
many-body Hamiltonian. The variational state is a coherent state for the underlying symmetry
group, which is the SU(2) group.
A pair of classical canonical conjugate coordinates, which bring the equations of
motion to the Hamilton form, is found. The classical energy has an interesting structure.
It is quadratic in coordinate but highly non-linear in the conjugate momentum.
Therefore one finds first the stationary point which minimizes the energy, and then linearizes the
equations of motion around the minimum point in the classical phase space. The solution for the linearized equations is
harmonic and its time period determines the energy of the new mode. Despite the fact the classical system has an harmonic
motion, the mode does not exist in the standard RPA approach. In this sense one may say that the present description corresponds to a
''renormalized RPA''. However as we have seen, by comparing the corresponding predictions, the renormalization involved in
the semi-classical description is completely different from the renormalization described in Section V as well
as from the boson expansion method.
It is known the fact that the topological structure of the energy surface depends on the strength parameters involved in the model Hamiltonian.
Thus, in the parameters space one can define several regions, each of them corresponding to a distinct nuclear phase.
Having this in mind, we studied the behavior of the new mode energy when the strength parameter for the $pp$
interaction ($\chi_1$) is varied. A particular feature for the semi-classical description is that the energy is not monotonic
decreasing function of $\chi_1$, but it increases in the first part of the interval, reflecting that here the $ph$ two body interaction prevails,
reaches a maximum value, then decreases and finally vanishes. This property is caused by that for each $\chi_1$
a new ground state is determined. This aspect is missing in both the renormalized RPA and boson expansion procedures.
Since the model Hamiltonian resembles the triaxial rotor which was semi-classically studied by one of the present authors (A. A. R)
in refs. \cite{Rad6,Ghe}, the interpretation of the new mode is imported from there. Thus, the new mode describes a wobbling motion around
a given total isospin.
The vanishing energy is a sign for a phase transition. In the first phase the rotation axis, in the isospin space
lies close to the z-axis, which has the maximum moment of inertia in the region of small $\chi_1$, while for $\chi_1$
larger than the critical value (where the energy vanishes), the rotation axis lies closer to the (X,Y) plane in the isospin space.
While the first phase may be described by a HP boson expansion formalism, for the second phase the Dyson boson representation is the proper one [20].
In the semi-classical approach, the new phase might be described by changing the trial function associated to the first phase,
through a rotation which brings the z-axis to the actual axis of maximal ``moment of inertia''.
The occurrence of the phase transition can be noticed also in the curve showing the exact first excitation energy as function of $\chi_1$.
Indeed at the critical value of $\chi_1$, this curve exhibits a minimum.
Another critical values of $\chi_1$ are those where the mode energy is equal to the exact excitation energy produced by the diagonalization procedure.
For these values the linearization does not affect at all the period of the exact closed classical orbit. As a matter of fact, for $\chi_1$ lying close to
these points the linearization are best justified. It is interesting to notice that these values of $\chi_1$
lie however close to the values where the phase transitions in the
semi-classical treatment (the far intersection point) and the renormalized RPA approach (the near intersection point) take place.
This observation allows us to conclude that
the semi-classical approach works very well for the values of $\chi_1$ where the renormalized RPA breaks down and that the interval where
the linearization procedure does not work, ending with the critical value where the semi-classical energy vanishes, is very narrow.
The energy of the new mode vanishes for a value of $\chi_1$ which is far beyond the physical value ($\chi_1=\chi$).
In this way the drawback of the $frn-RPA$, of breaking down earlier than the standard RPA does, is removed.
How could the new state be populated? We identified the transition operators which could excite the new state from the ground state.
The conclusion is that these state can be seen either in a $\beta^-$ (or $\beta^+$) decay or in a deuteron transfer reaction experiment.
The coupling of this mode to other collective states will be studied in
a subsequent paper using a realistic interaction and a large model space
for the single particle motion.
\vskip0.5cm
{\bf Acknowledgement}
\vskip0.5cm
\noindent
One of us (B. C.) wants to thank Prof. Amand Faessler for hospitality
in Institute of Theoretical Physics of Tuebingen University where
a part of this work was performed. A. A. R. thanks Prof. Faessler for
reading the manuscript and valuable remarks concerning the structure of
the variational function.
\section{Appendix A}
\label{sec:level8}
Here to derive the factorization of the trial function
$|\Psi\rangle$. To this purpose we address the following more general
question. Which are the t-functions $A(t), B(t), C(t)$ satisfying the
equation
\begin{equation}
e^{t[(z\sum_{m}a_{pm}^{\dag}a_{nm}-z^{*}\sum_{m}a_{nm}^{\dag}a_{pm})]}
=e^{A(t)a_{p}^{\dag}a_{n}}e^{C(t)(\hat N_{p}
-\hat N_{n})}e^{B(t)a_{n}^{\dag}a_{p}},
\end{equation}
with the initial conditions
\begin{equation}
A(0)=B(0)=C(0)=0
\end{equation}
and t a real parameter. Once we solve this problem
the needed factorization is obtain from (4.1) for $t=1$.
Taking the first derivative of the eq.(6.1), with respect to t, and
identifying the coefficients of the similar operators one obtains the
following system of differential equations for the three unknown functions,
$A(t), B(t), C(t)$ :
\begin{eqnarray}
\stackrel{\bullet}{z} &=& \stackrel{\bullet}{A}-2A
\stackrel{\bullet}{C}-\stackrel{\bullet}{B}A^{2}e^{-2C(t)},\nonumber\\
0&=& \stackrel{\bullet}{C}+\stackrel{\bullet}{B}Ae^{-2C(t)},\nonumber\\
-\stackrel{\bullet}{z}^* &= &\stackrel{\bullet}{B}e^{-2C(t)}.
\end{eqnarray}
Eliminating the functions $B, C$ from these equations, one obtains the following equation for
$A(t)$.
\begin{equation}
\stackrel{\bullet}{z}=\stackrel{\bullet}{A}-A^{2}\stackrel{\bullet}{z}^*
\end{equation}
which admits the solution:
\begin{equation}
A(t)=\tan(\rho t)e^{i\varphi}.
\end{equation}
Here the polar coordinates $(\rho,\varphi)$($z=\rho e^{i\varphi}$) have been used.
Inserting the result for $A(t)$ in the eq. (6.3), the equations for the remaining
functions can be easily integrated. The result is:
\begin{eqnarray}
C(t)&=&-\ln \big( \cos(\rho t) \big)\nonumber\\
B(t)&=&\tan(\rho t) e^{-i\varphi}.
\end{eqnarray}
For the sake of simplifying the writing, hereafter the following
notation will be used:
\begin{equation}
\alpha=A(1)
\end{equation}
Using these results the trial function can be written as:
\begin{eqnarray}
|\Psi\rangle & = & e^{-2C(1)T}e^{A(1)B^{\dag}(pn)}|NT\;-T
\rangle\equiv {\cal N} e^{A(1)B^{\dag}(pn)}|NT\;-T\rangle
\end{eqnarray}
where $\cal N$ denotes the normalization factor:
\begin{equation}
{\cal N} = e^{-2C(1)T}=e^{2 \ln(\cos \rho)T}={(1+|\alpha|^2)}^{-T}.
\end{equation}
\nopagebreak
|
{
"timestamp": "2005-03-14T12:04:03",
"yymm": "0503",
"arxiv_id": "nucl-th/0503041",
"language": "en",
"url": "https://arxiv.org/abs/nucl-th/0503041"
}
|
\section{Introduction.}
Recently, a number of analysts \cite{A-M-N, B-V, M-M} have studied
various generalized notions of derivations in the context of
Banach algebras. There are some applications in the other fields
of study \cite{H-L-S}. Such maps have been extensively studied in
pure algebra; cf. \cite{A-R, BRE, HVA}.
Let, throughout the paper, $A$ denote a Banach algebra (not
necessarily unital) and let $M$ be a Banach right $A$-module.
A linear mapping $d : A \to A$ is called a derivation if $d(ab)=
d(a)b + ad(b)\quad(a, b\in A)$. If $a\in A$ and we define $d_{a}$
by $d_{a}(x)=ax-xa \quad (x\in A)$. Then $d_{a}$ is a derivation
and such derivation is called inner.
A linear mapping $\delta : M\to M$ is called a generalized
derivation if there exists a derivation $d : A \to A$ such that
$\delta(xa)=\delta(x)a + xd(a)\quad(x\in M, a\in A)$. For
convenience, we say that such a generalized derivation $\delta$
is a $d$-derivation. In general, the derivation $d : A\to A$ is
not unique and it may happen that $\delta$ (resp. $d$) is bounded
but $d$ (resp. $\delta$) is not bounded. For instance, assume that
the action of $A$ on $M$ is trivial, i.e $MA=\{0\}$. Then every
linear mapping $\delta : M\to M$ is a $d$-derivation for each
derivation $d$ on $A$.
Our notion is a generalization of both concepts of a generalized
derivation (cf. \cite{BRE, HVA}) and of a multiplier (cf.
\cite{DAL}) on an algebra (see also \cite{MOS2}). For seeing this,
regard the algebra as a module over itself. The authors in
\cite{A-M-N} investigated the generalized derivations on Hilbert
$C^*$-modules and showed that these maps may appear as the
infinitesimal generators of dynamical systems.
\begin{example} Let $M$ be a right Hilbert $C^{*}$-module over a $C^*$-algebra $A$ of compact operators acting on a
Hilbert space (see \cite{LAN} for more details on Hilbert
$C^*$-modules). By Theorem 4 of \cite{B-G}, $M$ has an
orthonormal basis so that each element $x$ of $M$ can be
expressed as
$x=\displaystyle{\sum_{\lambda}}v_{\lambda}<v_{\lambda},x>$. If
$d$ is a derivation on $A$, then the mapping $\delta : M\to M$
defined by $\delta(x)=\displaystyle{\sum
_{\lambda}}v_{\lambda}d(<v_{\lambda},x>)$ is a $d$-derivation
since
\begin{eqnarray*}
\delta(xa)&=&\delta\left(\displaystyle{\sum_{\lambda}}v_{\lambda}<v_{\lambda},xa>\right)\\
&=&\displaystyle{\sum_{\lambda}}v_{\lambda}d(<v_{\lambda},x>a)\\
&=&\displaystyle{\sum_{\lambda}}v_{\lambda}d(<v_{\lambda},x>)a+\displaystyle{\sum_{\lambda}}v_{\lambda}<v_{\lambda},x>d(a)\\
&=&\delta(x)a + xd(a).
\end{eqnarray*}
\end{example}
The set ${\mathcal B}(M)$ of all bounded module maps on $M$ is a
Banach algebra and $M$ is a Banach ${\mathcal B}(M)-A$-bimodule
equipped with $T.x=T(x)\quad(x\in M, T\in {\mathcal B}(M))$, since
we have $T.(xa)=T(xa)=T(x)a=(T.x)a$ and $\|T.xa\|
\leq\|T\|\;\|x\|\;\|a\|$, for all $a \in A, x\in M, T\in {\mathcal
B}(M)$.
We call $\delta : M\to M$ a generalized inner derivation if there
exist $a\in A$ and $T\in {\mathcal B}(M)$ such that
$\delta(x)=T.x - xa = T(x)- xa$. Mathieu in \cite{MAT} called a
map $\delta : A\to A$ a generalized inner derivation if
$\delta(x)=bx-xa$ for some $a, b\in A$. If we consider $A$ as a
right $A$-module in a natural way, and take $T(x)=bx$, then our
definition covers the notion of Mathieu.
In this paper we deal with the derivations on the triangular
Banach algebras of the form ${\mathcal T}=\left(\begin{array}{cc}
{\mathcal B}(M) & M\\ 0 & A \end{array}\right)$. Such algebras
were introduced by Forrest and Marcoux \cite{F-M1} that in turn
are motivated by work of Gilfeather and Smith in \cite{G-S}
(these algebras have been also investigated by Y. Zhang who
called them module extension Banach algebras \cite{ZHA}). Among
some facts on generalized derivations, we investigate the
relation between generalized derivations on $M$ and derivations
on ${\mathcal T}$. In particular, we show that the generalized
first cohomology group of $M$ is isomorphic to the first
cohomology group of ${\mathcal T}$.
\section{Main Results.}
If we consider $A$ as an $A$-module in a natural way then we have
the following lemma about generalized derivations on $A$.
\begin{lemma}
A linear mapping $\delta : A \to A$ is a generalized derivation if
and only if there exist a derivation $d : A\to A$ and a module map
$\varphi : A\to A$ such that $\delta=d+\varphi$.\end{lemma}
\begin{proof} Suppose $\delta$ be a generalized derivation on $A$,
then there exists a derivation $d$ on $A$ such that $\delta$ is a
$d$-derivation. Put $\varphi=\delta-d$. Then for each $a, x\in A$
we have
$$\varphi(xa)=\delta(xa)-d(xa)=\delta(x)a+xd(a)-(d(x)a+xd(a))=(\delta(x)-d(x))a=\varphi(x)a$$
Thus $\varphi$ is a module map and $\delta=d+\varphi$.
Conversely, let $d$ be a derivation on $A$, $\varphi$ be a module
map on $A$ and put $\delta=d+\varphi$. Then clearly $\delta$ is a
linear map and
$$\delta(xa)=d(xa)+\varphi(xa)=d(x)a+xd(a)+\varphi(x)a=(d(x)+\varphi(x))a+xd(a)=\delta(x)a+xd(a)$$
for all $a, x\in A$. Therefore $\delta$ is a
$d$-derivation.\end{proof}
The next two results concern the boundedness of a generalized derivation.
\begin{theorem} Let $A$ have a bounded left approximate identity $\{e_{\alpha}\}_{\alpha\in I}$
and let $\delta$ be a $d$-derivation on $A$. Then $\delta$ is
bounded if and only if $d$ is bounded. \end{theorem}
\begin{proof} First we show that every module map on $A$ is
bounded. Suppose that $\varphi$ is a module map on $A$ and let
$\{a_{n}\}$ is a sequence in $A$ converging to zero in the norm
topology. By a consequence of Cohen Factorization Theorem (see
Corollary 11.12 of \cite{B-D}) there exist a sequence $\{b_{n}\}$
and an element $c$ in $A$ such that $b_{n}\to 0$ and
$a_{n}=cb_{n},\quad (n\in {\mathbb N})$. Then
$\varphi(a_{n})=\varphi(cb_{n})=\varphi(c)b_{n}\to 0$. Thus by the
closed graph theorem, $\varphi$ is bounded.
Now let $\delta$ be a $d$-derivation. By Lemma 2.1,
$\delta=d+\varphi$ for some module map $\varphi$ on $A$.
Therefore $\delta$ is bounded if and only if $d$ is
bounded.\end{proof}
\begin{corollary} Every generalized derivation on a $C^{*}$-algebra is bounded.\end{corollary}
\begin{proof} Every derivation on a $C^*$-algebra is automatically continuous; cf. \cite{J-S}.\end{proof}
Let $\varphi : A\to A$ be a homomorphism (algebra morphism). A
linear mapping $T : M\to M$ is called a $\varphi$-morphism if
$T(xa)=T(x)\varphi(a)\quad(a\in A, x\in M)$. If $\varphi$ is a
isomorphism and T is a bijective mapping then we say T to be a
$\varphi$-isomorphism. An $id_{A}$-morphism is a module map (module
morphism). Here $id_{A}$ denotes the identity operator on $A$.
\begin{proposition} Suppose $\delta$ is a bounded $d$-derivation on $M$ and $d$ is bounded. Then $T=\exp(\delta)$ is
a bi-continuous $\exp(d)$-isomorphism.\end{proposition}
\begin{proof} Using induction one can easily show that
$$\delta^{(n)}(xa)=\sum_{r=0}^{n}(_{r}^{n})\delta^{(n-r)}(x)d^{(r)}(a).$$
For each $a\in A, x\in M$ we have
\begin{eqnarray*}
T(xa)&=&\exp(\delta)(xa)\\
&=&\sum_{n=0}^{\infty}\frac{1}{n!}\delta^{(n)}(xa)\\
&=&\sum_{n=0}^{\infty}\frac{1}{n!}\sum_{r=0}^{n}(_{r}^{n})\delta^{(n-r)}(x)d^{(r)}(a)\\
&=&\sum_{n=0}^{\infty}\sum_{r=0}^{n}(\frac{1}{(n-r)!} \delta^{(n-r)}(x)(\frac{1}{r!}d^{(r)}(a))\\
&=&(\sum_{n=0}^{\infty}\frac{1}{n!}\delta^{(n)}(x))(\sum_{n=0}^{\infty}\frac{1}{n!}d^{(n)}(a))\\
&=&\exp(\delta)(x)\exp(d)(a)
\end{eqnarray*}
The operators $\exp(\delta), \exp(d)$ are invertible in the
Banach algebras of bounded operators on $M$ and $A$,
respectively. Hence $T$ is an $\exp(d)$-isomorphism.\end{proof}
\begin{proposition}
Let $\delta$ be a bounded generalized derivation on $M$. Then
$\delta$ is a generalized inner derivation if and only if there
exists an inner derivation $d_{a}$ on $A$ such that $\delta$ is
$d_{a}$-derivation. \end{proposition}
\begin{proof} Let $\delta$ be a generalized inner derivation. Then there exist $a\in A$ and $T\in
{\mathcal B}(M)$ such that $\delta(xa)=T(x)-xa\quad (x\in M)$. We
have $\delta(x)b+xd_{a}(b)=(T(x)-xa)b+xab-xba
=T(x)b-xba=T(xb)-(xb)a=\delta(xb) \quad (b \in A, x \in M)$.
Hence $\delta$ is a $d_{a}$-derivation.
Conversely, suppose $\delta$ is a $d_{a}$-derivation for some
$a\in A$. Define $T : M\to M$ by $T(x)=\delta(x)+xa$. Then T is
linear, bounded and
$T(xb)=\delta(xb)+(xb)a=(\delta(x)b+xd_{a}(b))+xba=\delta(x)b+xab-xba+xba=(\delta(x)+xa)b=T(x)b$.
It follows that $T\in {\mathcal B}(M)$ and
$\delta(x)=(\delta(x)+xa)-xa=T(x)-xa$. Therefore $\delta$ is a
generalized inner derivation.\end{proof}
The linear spaces of all bounded generalized derivations and
generalized inner derivations on $M$ are denoted by $GZ^{1}(M,M)$
and $GN^{1}(M,M)$, respectively. We call the quotient space
$GH^{1}(M,M)=GZ^{1}(M,M)/GN^{1}(M,M)$ the generalized first
cohomology group of $M$.
\begin{corollary}
$GH^{1}(M,M)=0$ whenever $H^{1}(A,A)=0$
\end{corollary}
\begin{proof} Let $\delta : M\to M$ be a generalized derivation.
Then there exists a derivation $d : A \to A$ such that $\delta$ is
a $d$-derivation. Due to $H^{1}(A,A)=0$, we deduce that $d$ is
inner and, by Proposition 2.5, so is $\delta$. Hence
$GH^{1}(M,M)=0$.
\end{proof}
Using some ideas of \cite{F-M1, MOS1}, we give the following
notion:
\begin{definition}{\rm ${\mathcal T}=\{\left(\begin{array}{cc}T & x \\0 & a \end{array}\right);
T\in {\mathcal B}(M), x\in M, a\in A \}$ equipped with the usual
$2\times 2$ matrix addition and formal multiplication and with
the norm $\|\left(\begin{array}{cc} T & x \\ 0 & a
\\\end{array}\right)\|=\|T\|+\|x\|+\|a\|$ is a Banach algebra. We
call this algebra the triangular Banach algebra associated to M.}
\end{definition}
The following two theorems give some interesting relations
between generalized derivations on $M$ and derivations on
${\mathcal T}$.
Let $\delta$ be a bounded $d$-derivation on $M$. We define
$\Delta_{\delta} : {\mathcal B}(M)\to {\mathcal B}(M)$ by
$\Delta_{\delta}(T)=\delta T-T\delta$. Then $\Delta_{\delta}$ is
clearly a derivation on ${\mathcal B}(M)$.
\begin{theorem}
Let $\delta$ be a bounded $d$-derivation on $M$ and $d$ be
bounded. Then the map $D^{\delta} : {\mathcal T} \to {\mathcal T}$
defined by $D^{\delta}\left(\begin{array}{cc} T & x \\ 0 & a
\end{array}\right)=\left(
\begin{array}{cc} \Delta_{\delta}(T) & \delta(x) \\ 0 & d(a)\end{array}\right)$ is a bounded derivation on ${\mathcal T}$.
Also $\delta$ is a generalized inner derivation if and only if $D^{\delta}$ is an inner derivation. \end{theorem}
\begin{proof} It is clear that $D^{\delta}$ is linear. For any $T_{1},T_{2}\in {\mathcal B} (M), x_{1},x_{2}\in M, a_{1},a_{2}\in A$ we have
\begin{eqnarray*}
&&D^{\delta}(\left(\begin{array}{cc}T_{1} & x_{1} \\ 0 &
a_{1}\end{array}\right)\left(\begin{array}{cc} T_{2} & x_{2} \\ 0 & a_{2} \end{array}\right))=
D^{\delta}\left(\begin{array}{cc} T_{1}T_{2} & T_{1}.x_{2}+x_{1}a_{2} \\
0 & a_{1}a_{2} \end{array}\right)\\
&=&\left(\begin{array}{cc} \Delta_{\delta}(T_{1}T_{2}) &
\delta(T_{1}.x_{2}+x_{1}a_{2}) \\ 0 & d(a_{1}a_{2})\end{array}
\right )\\
&=&\left(\begin{array}{cc} \Delta_{\delta}(T_{1}T_{2}) &
\delta(T_{1}(x_{2}))+ \delta(x_{1})a_{2}+x_{1}d(a_{2})\\ 0 &
a_{1}d(a_{2})+d(a_{1})a_{2}\end{array}\right )\\
&=&\left(\begin{array}{cc}
T_{1}\Delta_{\delta}(T_{2})+\Delta_{\delta}(T_{1})T_{2} &
T_{1}.\delta(x_{2})+x_{1}d(a_{2})+(\delta
T_{1}-T_{1}\delta)(x_{2})+\delta(x_{1})a_{2} \\ 0 & a_{1}d(a_{2})+d(a_{1})a_{2}\end{array}\right )\\
&=&\left(\begin{array}{cc} T_{1} & x_{1} \\ 0 & a_{1}
\end{array}\right)\left(\begin{array}{cc} \Delta_{\delta}(T_{2}) & \delta(x_{2})
\\ 0 & d(a_{2})\end{array}\right) + \left(\begin{array}{cc} \Delta_{\delta}
(T_{1}) & \delta(x_{1}) \\ 0 & d(a_{1}) \end{array}\right)\left(\begin{array}{cc} T_{2} & x_{2} \\ 0 & a_{2}\end{array}\right)\\
&=&\left(\begin{array}{cc} T_{1} & x_{1} \\ 0 & a_{1}
\end{array}\right)D^{\delta}\left(\begin{array}{cc} T_{2} & x_{2}
\\ 0 & a_{2}\end{array}\right)+D^{\delta}(\left(\begin{array}{cc} T_{1} & x_{1} \\ 0 & a_{1}\end{array}\right))\left(
\begin{array}{cc} T_{2} & x_{2} \\ 0 & a_{2} \end{array}\right)
\end{eqnarray*}
Thus $D^{\delta}$ is a derivation on ${\mathcal T}$. Due to
$\|\left(\begin{array}{cc} \Delta_{\delta}(T) & \delta(x) \\
0 & d(a)
\end{array}\right)\|=\| \Delta_{\delta}(T)\|+\|\delta(x)\|+\|d(a)\|\leq
\max \{\|\Delta_{\delta}\|,\|\delta\|,\|d\|\}\|\left(\begin{array}{cc} T & x \\
0 & a \end{array}\right)\|$, we infer that $D^{\delta}$ is
bounded.
Now suppose that $\delta$ is a generalized inner derivation. Then
there exist $a\in A$ and $T\in {\mathcal B}(M)$ such that
$\delta(x)=T(x)-xa \quad(x\in M)$. For all $S\in {\mathcal
B}(M), b\in A$ and $y\in M $ we have
\begin{eqnarray*}
D_{\left(\begin{array}{cc} T & 0
\\ 0 & a \end{array}\right)}\left(\begin{array}{cc} S & y \\ 0
& b \end{array}\right) &:=& \left(\begin{array}{cc} T & 0 \\ 0 &
a \end{array} \right)\left(\begin{array}{cc} S & y \\ 0 & b
\end{array}\right)-\left(\begin{array}{cc} S & y \\ 0 & b\end{array}\right)\left(\begin{array}{cc} T & 0 \\ 0 & a
\end{array} \right)\\
&=&\left(\begin{array}{cc} TS-ST & T.y-ya \\ 0 & ab-ba
\end{array}\right)\\
&=& \left(\begin{array}{cc} \Delta_{\delta}(S) & \delta(y)
\\0 & d_{a}(b)\end{array}\right)\\
&=&D^{\delta}\left(\begin{array}{cc} S & y \\ 0 & b
\end{array}\right). \end{eqnarray*}
Hence $D^{\delta}=D_{\left(\begin{array}{cc} T & 0 \\ 0 & a
\end{array} \right)}$ and so $D^{\delta}$ is an inner derivation.
Conversely, let $\delta$ be a bounded $d$-derivation such that
the associated derivation $D^{\delta}$ be an inner derivation,
say $D^{\delta} = D_{\left(\begin{array}{cc} T_0 & x_0
\\ 0 & a_0 \end{array}\right)}$. Then for each $T \in {\mathcal B}(M), x \in
M,a \in A$ we have
\begin{eqnarray}\label{inner}
\left(\begin{array}{cc} \Delta_\delta(T) & \delta(x) \\ 0 & d(a)
\end{array} \right)
&=& D^{\delta}(\left(\begin{array}{cc} T & x \\ 0 & a \end{array} \right))\nonumber \\
&=& D_{\left(\begin{array}{cc} T_0 & x_0
\\ 0 & a_0 \end{array}\right)}\left(\begin{array}{cc} T & x \\ 0
& a \end{array}\right)\nonumber \\
&=& \left(\begin{array}{cc} T_0T-TT_0 & T_0(x)+x_0a-T(x_0)-xa_0 \\
0 & a_0a-aa_0 \end{array}
\right)\nonumber \\
&=&\left(\begin{array}{cc} T_0T-TT_0 & T_0(x)+x_0a-T(x_0)-xa_0
\\0 & d_{a_0}(a)
\end{array}\right)
\end{eqnarray}
Hence $d=d_{a_0}$ is inner. Putting $a=0$ and $T=0$ in
(\ref{inner}) we conclude that $\delta(x) = T_0(x) - xa_0 \quad
(x \in M)$. Hence $\delta$ is a generalized inner derivation.
\end{proof}
The converse of the above theorem is true in the unital case.
\begin{theorem} Let $A$ be unital and ${\mathcal T}$ be the triangular Banach algebra associated to a unital Banach
right $A$-module $M$. Assume that $D : {\mathcal T} \to {\mathcal
T}$ is a bounded derivation. Then there exist $m_{0}\in M$, a
bounded derivation $d : A\to A$ and a bounded $d$-derivation
$\delta : M\to M$ such that
$$D\left(\begin{array}{cc} T & x \\ 0 & a \end{array}\right)=\left(\begin{array}{cc} \Delta_{\delta}(T)&
\delta(x)+m_{0}a-T.m_{0} \\
0 & d(a) \end{array}\right)$$ Moreover, $D$ is inner if and only
if $\delta$ is a generalized inner derivation.\end{theorem}
\begin{proof} We use some ideas of Proposition 2.1 of
\cite{F-M1}. By simple computation one can verify that
(i) $D\left(\begin{array}{cc} 0 & 0 \\ 0 & 1_{A}
\end{array}\right)=\left(\begin{array}{cc} 0 & m_{0} \\ 0 & 0
\end{array}\right)$ for some $m_{0}\in M$;
(ii) $D\left(\begin{array}{cc} 0 & 0 \\ 0 & a \end{array}\right)=\left(\begin{array}{cc} 0 & m_{0}a \\
0 & d(a) \end{array}\right)$ for some bounded derivation $d$ on
$A$;
(iii) $D\left(\begin{array}{cc} 0 & x \\ 0 & 0 \end{array}\right)=\left(\begin{array}{cc} 0 & \delta(x) \\
0 & 0 \end{array}\right)$ for some linear mapping $\delta$ on $M$;
(iv) $D\left(\begin{array}{cc} T & 0 \\ 0 & 0
\end{array}\right)=\left(\begin{array}{cc}
\Delta_{\delta}(T) & -T.m_{0} \\ 0 & 0 \end{array}\right)$;
and finally $D\left(\begin{array}{cc} T & x \\ 0 & a
\end{array}\right)=\left(\begin{array}{cc} \Delta_{\delta}(T) &
\delta(x)+m_{0}a-T.m_{0} \\ 0 & d(a)\end {array}\right)$.
We have
\begin{eqnarray*}
\left(\begin{array}{cc} 0 & \delta(xa) \\ 0 & 0
\end{array}\right)&=&D(\left(
\begin{array}{cc} 0 & xa \\ 0 & 0 \end{array}\right))=D(\left(\begin{array}{cc}
0 & x \\ 0 & 0 \end{array}\right)\left(\begin{array}{cc} 0 & 0 \\ 0 & a \end{array}\right))\\
&=&\left( \begin{array}{cc} 0 & x \\ 0 & 0 \end{array}\right)D(\left(\begin{array}{cc} 0 & 0 \\
0 & a \end{array}\right))+D(\left(\begin{array}{cc} 0 & x \\ 0 & 0 \end{array}\right))\left(
\begin{array}{cc} 0 & 0 \\ 0 & a \end{array}\right)\\
&=&\left(\begin{array}{cc} 0 & x \\ 0 & 0\end{array}\right)\left(\begin{array}{cc} 0& m_0a \\
0 & d(a) \end{array}\right)+\left(\begin{array}{cc} 0 & \delta(x) \\
0 & 0 \end{array}\right)\left( \begin{array}{cc}
0 & 0 \\ 0 & a \end{array}\right)\\
&=&\left(\begin{array}{cc} 0 & \delta(x)a+xd(a) \\ 0 & 0
\end{array}\right)
\end{eqnarray*}
Thus $\delta(xa)=\delta(x)a+xd(a)$ and so $\delta$ is a
$d$-derivation.
It is clear that $D$ is inner if and only if $d$ is inner and,
using Proposition 2.5, the latter holds if and only if $\delta$ is
a generalized inner derivation.\end{proof}
\begin{theorem} Let $A$ be a unital Banach algebra, $M$ be a unital Banach
right $A$-module and ${\mathcal T}=\left(\begin{array}{cc}
{\mathcal B}(M) & M\\ 0 & A \end{array}\right)$. Then
$H^{1}({\mathcal T},{\mathcal T})\cong GH^{1}(M,M)$\end{theorem}
\begin{proof} Let $\Psi : GZ^{1}(M,M)\to H^{1}({\mathcal T},{\mathcal T})$ be defined by
$$\Psi(\delta)=[D^{\delta}]$$ where $[D^{\delta}]$ represents the equivalence class of
$D^{\delta}$ in $H^{1}({\mathcal T},{\mathcal T})$. Clearly
$\Psi$ is linear. We shall show that $\Psi$ is surjective. To end
this, assume that $D$ is a bounded derivation on ${\mathcal T}$.
Let $\delta$, $d$, $\Delta_{\delta}$ and $m_{0}\in M$ be as in
the Theorem 2.9. Then
\begin{eqnarray*}
(D-D^{\delta})\left(\begin{array}{cc}
T & x \\ 0 & a \end{array}\right)&=&\left(\begin{array}{cc} \Delta_{\delta}(T) & \delta(x)+m_{0}a-T.m_{0}\\
0 & d(a) \end{array}\right) -\left(\begin{array}{cc}
\Delta_{\delta}(T) & \delta(x)\\ 0 & d(a)
\end{array}\right)\\
&=&\left(\begin{array}{cc} 0 & m_{0}a-T.m_{0}\\ 0 & 0
\end{array}\right)\\
&=&D_{\left(\begin{array}{cc} 0 & -m_{0} \\ 0 & 0
\end{array}\right)}\left(
\begin{array}{cc} T & x \\ 0 & a \end{array}\right).
\end{eqnarray*}
So $[D]=[D^{\delta}]=\Psi(\delta)$ and thus $\Psi$ is
surjective. Therefore $H^{1}({\mathcal T},{\mathcal T})\cong
GZ^{1}(M,M)/Ker(\Psi)$.
Note that $\delta\in Ker(\Psi)$ if and only if $D^{\delta}$ is
inner derivation on ${\mathcal T}$. Hence
$Ker(\Psi)=GN^{1}(M,M)$, by Theorem 2.8. Thus $H^{1}({\mathcal
T},{\mathcal T})\cong GH^{1}(M,M)$.
\end{proof}
\begin{example} Suppose that $A$ is unital and $M=A$. Then ${\mathcal B}(A)=A$ and
so $GH^{1}(A,A)\cong H^{1}(\left(\begin{array}{cc}A&A\\0&A
\end{array}\right ), \left(\begin{array}{cc}A&A\\0&A
\end{array}\right ))=H^1(A,A)$, by Proposition 4.4 of
\cite{F-M2}. In particular, every generalized derivation on a
unital commutative semisimple Banach algebra \cite{S-W}, a unital
simple $C^*$-algebra \cite{SAK}, or a von Neumann algebra
\cite{KAD} is generalized inner.\end{example}
We have investigated the interrelation between generalized
derivations on a Banach algebra and its ordinary derivations. We
also studied generalized derivations on a Banach module in virtue
of derivations on its associated triangular Banach algebra. Thus,
we established a link between two interesting research areas:
Banach algebras and triangular algebras.
\textbf{Acknowledgment.} The authors sincerely thank the referee for
valuable suggestions and comments.
|
{
"timestamp": "2006-06-17T13:38:40",
"yymm": "0503",
"arxiv_id": "math/0503618",
"language": "en",
"url": "https://arxiv.org/abs/math/0503618"
}
|
\section{Introduction}
The search for discrete quantum phase-space quasiprobability distribution functions is a subject of continuous and growing interest in
the literature \cite{wooters,gapi1,cohendet,gapi2,opat,voros,gama,zhang,luis,haki,muk,wooters2,vourdas}. The possibility of representing
quantum systems characterized by a finite-dimensional state space by such discrete quasidistributions lays the ground for interesting
developments and fruitful applications on quantum computation and quantum information theory \cite{r1s3,r2s3,r3s3,r4s3,r5s3,r6s3,r7s3,
r8s3}. It is well known that, as a well established counterpart to the discrete case, a huge variety of quasiprobabilty distribution
functions can be defined upon continuous phase-space \cite{lee}. In this sense, the Cahill-Glauber (CG) approach \cite{cahill} to the
subject has proved to be a powerful mapping technique that provides a general class of quasiprobability distribution functions, where
the Wigner, Glauber-Sudarshan and Husimi functions appear as particular cases. Therefore, it might be considered as a wide-range
phase-space approach to quantum mechanics regarding degrees of freedom with classical counterparts.
The aim of this paper is to present a discrete extension of the CG approach. Such extension is not obtained from that approach but,
instead, properly constructed out of the finite dimensional context. Furthermore, this {\sl ab initio} construction inherently embodies
the discrete analogues of the desired properties of the CG formalism. In particular, discrete Wigner, Husimi and Glauber-Sudarshan
quasiprobability distribution functions are obtained. Thus, besides the theoretical interest of its own, such extension has direct
applications in quantum information processing, quantum tomography and quantum teleportation, which are explored in a following work
\cite{nois}.
This work is organized as follows: In the next section we briefly outline the CG approach, setting the stage for section III, where our
proposal for a discrete extension of the CG mapping kernel is presented. In section IV basic properties of the mapping technique are
discussed, and the continuum limit is carried out on section V. Finally, section VI contains our summary and conclusions. Also,
important calculations are detailed in the Appendix.
\section{The Cahill-Glauber Mapping Kernel}
For the sake of clarity, in what follows we will briefly review the central ideas which constitute the core of the CG approach, and
that will be properly generalized in the following sections. Basically, the cornerstone of the formalism is the mapping kernel
(hereafter $\hbar = 1$)
\begin{equation}
\label{CG1}
{\bf T}^{(s)}(q,p) = \int_{-\infty}^{\infty} \frac{dp^{\prime} dq^{\prime}}{2\pi} \exp \left[ -i p^{\prime}(q-\mathbf{Q}) \right]
\exp \left[ i q^{\prime} (p-\mathbf{P}) \right] \exp \left( - \frac{i}{2} p^{\prime} q^{\prime} \right) \exp \left[ \frac{s}{4}(q^{\prime 2}
+ p^{\prime 2}) \right] \; ,
\end{equation}
which is responsible for the mapping of bounded operators on the continuous phase-space, being $s$ a complex variable satisfying the
condition $|s| \leq 1$. Here, the momentum and coordinate operators obey the Weyl-Heisenberg commutation relation $[ {\bf Q}, {\bf P}]
= i {\bf 1}$. Since the above expression explicitly depends on $s$, this parameter labels an infinite family of mapping kernels. Each
mapping kernel can be seen as the double Fourier transform of the displacement generators multiplied by a phase factor $\exp \left[ (i/2)
p^{\prime} q^{\prime} \right]$ and by the folding function $\exp \left[ (s/4)(q^{\prime 2} + p^{\prime 2}) \right]$. For purposes which will
become evident later, we write the mapping kernel as
\begin{equation}
\label{II}
{\bf T}^{(s)}(q,p) = \int_{-\infty}^{\infty} \frac{dp^{\prime} dq^{\prime}}{2 \pi} \exp \left[ -i p^{\prime}(q-\mathbf{Q}) \right]
\exp \left[ i q^{\prime}(p-\mathbf{P}) \right] \exp \left( - \frac{i}{2} p^{\prime} q^{\prime} \right) (\langle 0 | q^{\prime},p^{\prime}
\rangle)^{-s} \; ,
\end{equation}
where $| q^{\prime}, p^{\prime} \rangle$ is a coherent state.
The mapping of a given operator is achieved by the trace operation $\mathcal{O}^{(s)}(q,p) = \mbox{${\rm Tr}$} [ {\bf T}^{(s)}(q,p) {\bf O}]$, being
$\mathcal{O}^{(s)}(q,p)$ the function which represents ${\bf O}$ in the associated usual phase-space. The mapping is one-to-one, and the
operator is reobtained from its associated function by
\begin{displaymath}
{\bf O} = \int_{-\infty}^{\infty} \frac{dpdq}{2\pi} \, \mathcal{O}^{(s)}(q,p) {\bf T}^{(-s)}(q,p) \; .
\end{displaymath}
It is clear that, for each operator, there is an infinite family of associated functions labeled by $s$. In particular, the phase-space
representatives of the density operator are referred to as quasiprobability distributions functions and have, obviously,
distinguishable importance \cite{lee}. One of the great virtues of the Cahill-Glauber approach is that three special and important
types of quasidistributions, namely the Glauber-Sudarshan $(s=1)$, Wigner $(s=0)$ and Husimi functions $(s=-1)$, are particular cases.
Each of these functions have been extensively explored and reviewed in the literature \cite{schleich}.
A particular mapping kernel, characterized by a given parameter $s$, can be expressed in terms of another mapping kernel with a
different parameter value. The same holds true to the functions associated with a given operator. In fact, the procedure in the latter
case can be easily shown to be the same as in the former. That is, we may discuss only the relation between the mapping kernels,
knowing that equivalent relations are observed by the associated functions. In this way, the connection between the two mapping kernels
is seen to be given by the trace of the product
\begin{eqnarray}
\label{fold0.5}
\mbox{${\rm Tr}$} [ {\bf T}^{(s_1)}(q_{1},p_{1}) {\bf T}^{(s_{2})}(q_{2},p_{2}) ] &=& \int_{-\infty}^{\infty} \frac{dq dp}{2\pi} \exp \left\{ i \left[
q (p_{1}-p_{2}) - p (q_{1}-q_{2}) \right] \right\} \stackrel{(\langle 0 |q,p \rangle)^{-(s_{1}+s_{2})}}{\overbrace{\exp \left[ \frac{s_{1}+s_{2}}{4}
( q^{2}+p^{2} ) \right]}} \\
\label{fold1}
&=& \frac{-2}{s_{1}+s_{2}} \exp \left\{ \frac{2}{s_{1}+s_{2}} \left[ (p_{1}-p_{2})^{2} + (q_1-q_2)^{2} \right] \right\} \qquad
\mbox{${\rm Re}$} (s_{1} + s_{2}) < 0 \; .
\end{eqnarray}
We immediately recognize the important role played by the last exponential function in (\ref{fold0.5}), since, if the condition
$\mbox{${\rm Re}$} (s_{1}+s_{2}) < 0$ is not observed, the trace gives a divergent result. Thus, that condition imposes a constraint that defines a
hierarchy. That is, on continuous phase space there is a hierarchical structure of mapping kernels allowing one to express a given
phase-space function in terms of a Gaussian smoothing of another, and, as such, inverse relations do {\em not} exist. In other words,
{\em the Gaussian folding hierarchical structure observed by the quasidistributions has its roots in the functional form of} $\langle 0 |
q,p \rangle$. We stress this particular point as the discrete equivalent to equation (\ref{fold0.5}) {\em does not} imply a hierarchical
relation.
\section{The discrete mapping kernel}
\subsection{Preliminaries}
\subsubsection{Operator bases}
Long ago Schwinger proposed the following set of operators to act as a basis on an operator space
\begin{displaymath}
{\bf S}(\eta,\xi) = \frac{1}{\sqrt{N}} {\bf U}^{\eta} {\bf V}^{\xi} \exp \left( \frac{\pi i}{N} \eta \xi \right) \; ,
\end{displaymath}
where the ${\bf U}$'s and ${\bf V}$'s are the so-called Schwinger unitary operators \cite{schw}, $N$ is the dimension of the associated
state space and the indices $\{ \eta,\xi \}$ run on any complete set of residues mod$(N)$; in particular we choose the closed interval
$\left[ -\ell,\ell \right]$, with $\ell = (N-1)/2$. For simplicity, we shall restrict ourselves to the odd $N$ case. Even dimensionalities,
for the purposes of this paper, can also be dealt with simply by working on non-symmetrized intervals.
The set $\{ {\bf S}(\eta,\xi) \}_{\eta,\xi = -\ell,\ldots,\ell}$ spans a complete and orthonormal basis on the $N^{2}$ space of linear
operators acting on finite complex vectorial spaces, in the sense that, as the trace operation stands as the inner product on operator
spaces, any linear operator can be written as
\begin{equation}
\label{1}
{\bf O} = \sum_{\eta,\xi = -\ell}^{\ell} \mbox{${\rm Tr}$} \left[ {\bf S}^{\dagger}(\eta,\xi) {\bf O} \right] {\bf S}(\eta,\xi) \; .
\end{equation}
The fundamental result
\begin{displaymath}
\mbox{${\rm Tr}$} \left[ {\bf S}^{\dagger}(\mu,\nu) {\bf S}(\eta,\xi) \right] = \delta_{\eta,\mu}^{[N]} \, \delta_{\xi,\nu}^{[N]}
\end{displaymath}
ensures that this decomposition is unique. The superscript $[N]$ on the Kroenecker deltas denotes that they are different from zero
whenever their indices are mod$(N)$ congruent. The Schwinger basis elements also obey the property \cite{gama}
\begin{equation}
\label{2}
{\bf S}^{\dagger}(\eta,\xi) = {\bf S}(-\eta,-\xi) \; .
\end{equation}
\subsubsection{Discrete Coherent States}
The Schwinger operator bases elements also act as displacement operators on a particular reference state to form discrete coherent
states as \cite{gapi2,gama}
\begin{equation}
\label{cohe}
| \eta,\xi \rangle = \sqrt{N} {\bf S}(\eta,-\xi) |0,0 \rangle \; ,
\end{equation}
where the reference state is written by means of the Jacobi $\vartheta_{3}$-function (whose explicit form is shown in Appendix A) as
\begin{equation}
\label{vacuo}
|0,0 \rangle = \frac{1}{\mathcal{N}} \sum_{\gamma = - \ell}^{\ell} \vartheta_{3} \left( 2a \gamma | 2ia \right) | u_{\gamma} \rangle \; ,
\end{equation}
where $\{ | u_{\gamma} \rangle \}_{\gamma = - \ell,\ldots,\ell}$ are the eigenstates of the unitary operator ${\bf U}$,
\begin{displaymath}
{\mathcal{N}}^{2} = \frac{1}{2 \sqrt{a}} \left[ \vartheta_{3}(0|i a) \vartheta_{3}(0|4ia) + \vartheta_{4}(0|ia) \vartheta_{2}(0|4ia) \right]
\end{displaymath}
is the normalization constant, and $a=(2N)^{-1}$. Due to the properties of the $\vartheta_{3}$-function, the reference state above is
preserved under the action of the Fourier operator \cite{gama,mehta}
\begin{displaymath}
\mbox{\boldmath $\mathfrak{F}$} |0,0 \rangle = |0,0 \rangle \; ,
\end{displaymath}
where
\begin{displaymath}
\mbox{\boldmath $\mathfrak{F}$} = \sum_{\gamma = -\ell}^{\ell} |v_{\gamma} \rangle \langle u_{\gamma} | \; ,
\end{displaymath}
and $\{ | v_{\gamma} \rangle \}_{\gamma = - \ell,\ldots,\ell}$ are the eigenstates of ${\bf V}$, with $\langle u_{\mu} | v_{\gamma} \rangle = \exp
[ (2 \pi i /N) \mu \gamma ]$. Parity of the $\vartheta_{3}$-function also ensures that $\langle u_{\kappa } | 0,0 \rangle = \langle u_{-\kappa} |
0,0 \rangle$, from which it follows
\begin{equation}
\label{par}
\langle 0,0 | \mu,\nu \rangle = \langle 0,0 | -\mu,-\nu \rangle \; .
\end{equation}
There are, of course, a number different recipes of discrete coherent states, some of them also in connection with $\vartheta$-functions, for instance \cite{zhang,voros}.
\subsection{The extended mapping kernel}
Now let us define the extended mapping kernel as
\begin{displaymath}
{\bf S}^{(s)}(\eta,\xi) = {\bf S}(\eta,\xi) \left[ \mathcal{K}(\eta,\xi) \right]^{-s}
\end{displaymath}
where $s$ is a complex number satisfying $\left| s \right| \leq 1$, and $\mathcal{K}(\eta,\xi)= \langle 0,0| \eta,\xi \rangle$ denotes the
overlap of coherent states explicitly calculated in Appendix A. The set $\{ {\bf S}(\eta,\xi) \}_{\eta,\xi = -\ell,\ldots,\ell}$ itself
spans a complete and orthogonal basis on operator space. Nevertheless, we can go back to decomposition (\ref{1}), use equation
(\ref{2}), and introduce convenient factors to get
\begin{displaymath}
{\bf O} = \sum_{\eta,\xi = -\ell}^{\ell} \mbox{${\rm Tr}$} \left[ {\bf S}(-\eta,-\xi) {\bf O} \right] {\bf S}(\eta,\xi)
{\underbrace{[ \mathcal{K}(-\eta,-\xi)]^{s} [\mathcal{K}(\eta,\xi)]^{-s}}_{1}} \; ,
\end{displaymath}
where equation (\ref{par}) has been used. Conveniently grouping the terms the new decomposition reads
\begin{equation}
\label{novdec}
{\bf O} = \sum_{\eta,\xi = -\ell}^{\ell} \mbox{${\rm Tr}$} \left[ {\bf S}^{(-s)}(-\eta,-\xi) {\bf O} \right] {\bf S}^{(s)}(\eta,\xi) \; .
\end{equation}
Now, introducing the double Fourier transform of ${\bf S}^{(s)}(\eta,\xi)$, i.e.
\begin{displaymath}
{\bf T}^{(s)}(\eta,\xi) = \frac{1}{\sqrt{N}} \sum_{\mu,\nu = -\ell}^{\ell} {\bf S}^{(s)}(\eta,\xi) \exp \left[ - \frac{2\pi i}{N}
(\eta \mu + \xi \nu ) \right] \; ,
\end{displaymath}
and its Fourier inverse, we can, after a few steps, write equation (\ref{novdec}) as
\begin{equation}
\label{decomp}
{\bf O} = \frac{1}{N} \sum_{\mu,\nu = -\ell}^{\ell} \mathcal{O}^{(-s)}(\mu,\nu) {\bf T}^{(s)}(\mu,\nu) \; ,
\end{equation}
with $\mathcal{O}^{(-s)}(\mu,\nu) = \mbox{${\rm Tr}$} \left[ {\bf T}^{(-s)}(\mu,\nu) {\bf O} \right]$, defining a one-to-one mapping between operators and
functions defined on a discrete phase-space $\{\mu ,\nu \}$, where explicitly
\begin{equation}
\label{novabase}
{\bf T}^{(s)}(\mu,\nu) = \frac{1}{N} \sum_{\eta,\xi = -\ell}^{\ell} {\bf U}^{\eta} {\bf V}^{\xi} \exp \left[ -\frac{2\pi i}{N}
( \eta \mu + \xi \nu ) \right] \exp \left( \frac{\pi i}{N} \eta \xi \right) [ \mathcal{K}(\eta,\xi) ]^{-s} \; ,
\end{equation}
and $\mathcal{K}(\eta,\xi)$ can be shown to be a sum of products of Jacobi $\vartheta$-functions (as seen in Appendix A),
\begin{eqnarray}
\label{fold}
\mathcal{K}(\eta,\xi) &=& \frac{1}{4 \sqrt{a} \mathcal{N}^{2}} \left\{ \vartheta_{3} (a \eta | ia) \vartheta_{3} (a \xi | ia) +
\vartheta_{3} (a \eta | ia) \vartheta_{4} (a \xi | ia) \exp (i \pi \eta) \right. \nonumber \\
& & + \left. \vartheta_{4} (a \eta | ia) \vartheta_{3} (a \xi | ia) \exp (i \pi \xi) + \vartheta_{4} (a \eta | ia) \vartheta_{4}
(a \xi | ia) \exp \left[ i \pi (\eta + \xi + N) \right] \right\} \; .
\end{eqnarray}
The new kernel, written as in equation (\ref{novabase}), allows us to conclude that the above sum of products of $\vartheta$-functions
plays, in the discrete phase-space, the role reserved to the Gaussians in the continuous case.
\section{Properties}
\subsection{Basic general properties}
From the properties of the mapping kernel it is straightforward to obtain general properties of the associated functions in
phase-space. We observe that {\em all} following properties correctly generalize the continuous CG ones. First we note that
\begin{equation}
\label{prop1}
\textrm{(i)} \left[ {\bf T}^{(s)}(\mu,\nu) \right]^{\dagger} = {\bf T}^{(s^{\ast})}(\mu,\nu) \; ,
\end{equation}
implying that the mapping kernel is Hermitian for real values of the parameter $s$. As a direct consequence, the phase-space
representatives of Hermitian operators are real.
Direct calculations also show that
\begin{eqnarray}
\label{prop2}
\textrm{(ii)} &\! \! \!& \frac{1}{N} \sum_{\mu,\nu = -\ell}^{\ell} {\bf T}^{(s)}(\mu,\nu) = {\bf 1} \; , \\
\textrm{(iii)} &\! \! \!& \mbox{${\rm Tr}$} \left[ {\bf T}^{(s)}(\mu,\nu) \right] = 1 \; , \\
\textrm{(iv)} &\! \! \!& \mbox{${\rm Tr}$} \left[ {\bf T}^{(s)}(\mu,\nu) {\bf T}^{(-s)}(\mu^{\prime},\nu^{\prime}) \right] = N
\delta_{\mu,\mu^{\prime}}^{[N]} \delta_{\nu,\nu^{\prime}}^{[N]} \; .
\end{eqnarray}
The property (iv) is a crucial one from which expression (\ref{decomp}) could be immediately obtained. From this property also follows
the general result
\begin{displaymath}
\mbox{${\rm Tr}$} ( {\bf AB} ) = \frac{1}{N} \sum_{\mu,\nu = - \ell}^{\ell} \mathcal{A}^{(s)}(\mu,\nu) \mathcal{B}^{(-s)}(\mu,\nu) \; .
\end{displaymath}
In fact, property (iv) is a particular case of the general expression
\begin{equation}
\label{gentr}
\mbox{${\rm Tr}$} \left[ {\bf T}^{(s)}(\mu,\nu) {\bf T}^{(t)}(\mu^{\prime},\nu^{\prime}) \right] = \frac{1}{N} \sum_{\eta,\xi = - \ell}^{\ell}
\exp \left\{ \frac{2 \pi i}{N} \left[ \eta (\mu^{\prime} - \mu) + \xi (\nu^{\prime} - \nu) \right] \right\} [ \mathcal{K}(\eta,\xi) ]^{-(t+s)} \; ,
\end{equation}
which is the counterpart of equation (\ref{fold0.5}). It must be stressed that this expression is {\em always} well defined, even for
$\mbox{${\rm Re}$} (t+s) < 0$, as $\mathcal{K}(\eta,\xi)\neq 0$.
\subsection{Particular cases}
There are three important particular cases to be discussed:
\begin{itemize}
\item[i)\hspace{.8 em}(s=0)] In such a case it is easy to see that
\begin{equation}
\label{DWW}
{\bf T}^{(0)}(\mu,\nu) = {\bf G}(\mu,\nu) \; ,
\end{equation}
where ${\bf G}(\mu,\nu)$ is the mapping kernel introduced by Galetti and Piza, which is a discrete generalization of the Weyl-Wigner
mapping kernel \cite{gapi1,gapi2,ruga,ruga2}. In that case, being $\mbox{\boldmath $\rho$}$ the density operator, equation (\ref{decomp}) would read, for
${\bf O} = \mbox{\boldmath $\rho$}$,
\begin{displaymath}
\mbox{\boldmath $\rho$} = \frac{1}{N} \sum_{\mu,\nu = -\ell}^{\ell} \mathcal{W}(\mu,\nu) {\bf G}(\mu,\nu) \; ,
\end{displaymath}
with $\mathcal{W}(\mu,\nu) = \mbox{${\rm Tr}$} \left[ {\bf G}(\mu,\nu) \mbox{\boldmath $\rho$} \right]$ a discrete Wigner function.
\item[ii)\hspace{.8 em}(s=-1)] A fundamental property of our mapping kernel is
\begin{equation}
\label{fund}
{\bf T}^{(-1)}(\mu,\nu) = | \mu,\nu \rangle \langle \mu,\nu | \; ,
\end{equation}
which can be proved decomposing the coherent state projector in the Schwinger operator basis as (using equations (\ref{1}) and
(\ref{2}))
\begin{displaymath}
| \mu,\nu \rangle \langle \mu,\nu | = \frac{1}{N} \sum_{\eta,\xi = -\ell}^{\ell} {\bf U}^{\eta} {\bf V}^{\xi} \exp \left( \frac{i \pi}{N}
\eta \xi \right) \mbox{${\rm Tr}$} \left[ | \mu,\nu \rangle \langle \mu,\nu | {\bf V}^{-\xi} {\bf U}^{-\eta} \exp \left( - \frac{i \pi}{N} \eta \xi \right) \right]
\end{displaymath}
which, by applying the definition of the coherent states, equation (\ref{cohe}), reads
\begin{displaymath}
| \mu,\nu \rangle \langle \mu,\nu | = \frac{1}{N} \sum_{\eta,\xi = -\ell}^{\ell} {\bf U}^{\eta} {\bf V}^{\xi} \langle 0,0 | {\bf V}^{\nu}
{\bf U}^{-\mu} {\bf V}^{-\xi} {\bf U}^{-\eta} {\bf U}^{\mu} {\bf V}^{-\nu} | 0,0 \rangle \; ,
\end{displaymath}
and using the Weyl commutation relation, ${\bf U}^{\alpha} {\bf V}^{\beta} = \exp [- (2 \pi i/N) \alpha \beta] {\bf V}^{\beta}
{\bf U}^{\alpha}$,
\begin{displaymath}
|\mu ,\nu \rangle \langle \mu ,\nu |=\frac{1}{N}\sum_{\eta ,\xi=-\ell}^{\ell}
\mathbf{U}^{\eta} \mathbf{V}^{\xi} \exp \left[ -\frac{2\pi i}{N}(\eta \mu +\xi \nu )
\right] \exp \left( \frac{\pi i}{N}\eta \xi \right) \langle 0,0|\eta ,\xi \rangle \; ,
\end{displaymath}
where in the last step parity of $\langle 0,0 | \eta,\xi \rangle$ with respect to $\eta$ was used. This proves our assertion. As a consequence,
the phase-space decomposition of the density operator, associated with this particular value of the parameter $s$, reads
\begin{displaymath}
\label{glauber}
\mbox{\boldmath $\rho$} = \frac{1}{N} \sum_{\mu,\nu = -\ell}^{\ell} P(\mu,\nu) | \mu,\nu \rangle \langle \mu,\nu |
\end{displaymath}
allowing us to identify $P(\mu ,\nu )$ as a discrete Glauber-Sudarshan distribution.
\item[iii)\hspace{.8 em} (s=1)] In this case we may write
\begin{displaymath}
\mbox{\boldmath $\rho$} = \frac{1}{N} \sum_{\mu,\nu = -\ell}^{\ell} \mbox{${\rm Tr}$} \left[ {\bf T}^{(-1)}(\mu,\nu) \mbox{\boldmath $\rho$} \right] {\bf T}^{(1)}(\mu,\nu) \; ,
\end{displaymath}
which is simply
\begin{displaymath}
\mbox{\boldmath $\rho$} = \frac{1}{N} \sum_{\mu,\nu = -\ell}^{\ell} \mathcal{H}(\mu,\nu) {\bf T}^{(1)}(\mu,\nu) \; .
\end{displaymath}
By definition $\mathcal{H}(\mu,\nu) = \langle \mu,\nu | \mbox{\boldmath $\rho$} | \mu,\nu \rangle$ is positive definite, and it can be identified as a discrete
Husimi function.
\end{itemize}
As any operator can be decomposed by the use of expression (\ref{decomp}), it follows that we are allowed to write
\begin{displaymath}
{\bf T}^{(-1)}(\mu,\nu) = \frac{1}{N} \sum_{\sigma,\lambda = -\ell}^{\ell} \mbox{${\rm Tr}$} \left[ {\bf T}^{(-0)}(\sigma,\lambda ) {\bf T}^{(-1)}
(\mu,\nu) \right] {\bf T}^{(0)}(\sigma,\lambda) \; ,
\end{displaymath}
where the minus signal was kept only for clarity. We then use equation (\ref{gentr}) to write explicitly
\begin{equation}
\label{ffold}
\mbox{${\rm Tr}$} \left[ {\bf T}^{(0)}(\sigma,\lambda) {\bf T}^{(-1)}(\mu,\nu) \right] = \frac{1}{N} \sum_{\eta,\xi = -\ell}^{\ell} \exp \left\{
\frac{2 \pi i}{N} [\eta (\mu - \sigma) + \xi (\nu -\lambda)] \right] \mathcal{K}(\eta,\xi) \; ,
\end{equation}
that is, the discrete Fourier transform of the $\mathcal{K}(\eta,\xi)$ is the folding function. The above result can also be written in
the compact form
\begin{displaymath}
\mbox{${\rm Tr}$} \left[ {\bf T}^{(0)}(\sigma,\lambda) {\bf T}^{(-1)}(\mu,\nu) \right] = \langle \mu,\nu | {\bf G}(\sigma,\lambda) | \mu,\nu \rangle \; ,
\end{displaymath}
which is precisely the Wigner function associated with a coherent state $| \mu,\nu \rangle$. We therefore have
\begin{equation}
\label{hier1}
{\bf T}^{(-1)}(\mu,\nu) = \frac{1}{N} \sum_{\sigma,\lambda = -\ell}^{\ell} \langle \mu,\nu | {\bf G}(\sigma,\lambda) | \mu,\nu \rangle
{\bf T}^{(0)}(\sigma,\lambda) \; .
\end{equation}
In the same form we now decompose ${\bf T}^{(0)}(\mu,\nu)$ as
\begin{displaymath}
{\bf T}^{(0)}(\mu,\nu) = \frac{1}{N} \sum_{\sigma,\lambda = -\ell}^{\ell} \mbox{${\rm Tr}$} \left[ {\bf T}^{(0)}(\sigma,\lambda)
{\bf T}^{(-1)}(\mu,\nu) \right] {\bf T}^{(1)}(\sigma,\lambda) \; ,
\end{displaymath}
which allows us to use once again the above result for the trace and write
\begin{equation}
\label{hier2}
{\bf T}^{(0)}(\mu,\nu) = \frac{1}{N} \sum_{\sigma,\lambda = -\ell}^{\ell} \langle \mu,\nu | {\bf G}(\sigma,\lambda) | \mu,\nu \rangle
{\bf T}^{(1)}(\sigma,\lambda) \; .
\end{equation}
Multiplying both equations (\ref{hier1}) and (\ref{hier2}) by the density operator $\mbox{\boldmath $\rho$}$ and taking the trace, we are led to the
suggestive results
\begin{eqnarray}
\mathcal{H}(\mu,\nu) &=& \frac{1}{N} \sum_{\sigma,\lambda = -\ell}^{\ell} \langle \mu,\nu | {\bf G}(\sigma,\lambda) | \mu,\nu \rangle
\mathcal{W}(\sigma,\lambda) \; , \nonumber \\
\mathcal{W}(\mu,\nu) &=& \frac{1}{N} \sum_{\sigma,\lambda = -\ell}^{\ell} \langle \mu,\nu | {\bf G}(\sigma,\lambda) | \mu,\nu \rangle
P(\sigma,\lambda) \; , \nonumber
\end{eqnarray}
which are the discrete counterparts of the well known Gaussian smoothing that occurs in the continuous case, in agreement with the
hierarchy present in that context.
It must be stressed, however, that, opposed to the continuous case, it is now possible to write
\begin{eqnarray}
{\bf T}^{(0)}(\mu,\nu) &=& \frac{1}{N} \sum_{\sigma,\lambda = -\ell}^{\ell} \Lambda (\mu - \sigma, \nu - \lambda) {\bf T}^{(-1)}
(\sigma,\lambda) \nonumber \\
{\bf T}^{(1)}(\mu,\nu) &=& \frac{1}{N} \sum_{\sigma,\lambda = -\ell}^{\ell} \Lambda (\mu - \sigma, \nu -\lambda) {\bf T}^{(0)}
(\sigma,\lambda) \; , \nonumber
\end{eqnarray}
where
\begin{equation}
\label{invfold}
\Lambda (\mu-\sigma,\nu-\lambda) = \frac{1}{N} \sum_{\eta,\xi = -\ell}^{\ell} \exp \left\{ \frac{2 \pi i}{N} [\eta (\mu-\sigma)+
\xi (\nu-\lambda)] \right\} [ \mathcal{K}(\eta,\xi) ]^{-1} \; ,
\end{equation}
which, at least in principle, can always be calculated (we remind again that $\mathcal{K}(\eta,\xi)$ is finite and different from zero).
Also very illustrative is the result that follows from the decomposition
\begin{equation}
\label{fold2}
{\bf T}^{(-1)}(\mu,\nu) = \frac{1}{N} \sum_{\sigma,\lambda = -\ell}^{\ell} \mbox{${\rm Tr}$} \left[ {\bf T}^{(-1)}(\sigma,\lambda) {\bf T}^{(-1)}
(\mu,\nu) \right] {\bf T}^{(1)}(\sigma,\lambda) \; .
\end{equation}
With
\begin{displaymath}
\mbox{${\rm Tr}$} \left[ {\bf T}^{(-1)}(\sigma,\lambda) {\bf T}^{(-1)}(\mu,\nu) \right] = \frac{1}{N} \sum_{\eta,\xi = -\ell}^{\ell} \exp \left\{
\frac{2 \pi i}{N} [\eta (\mu -\sigma) + \xi (\nu -\lambda) ] \right\} [\mathcal{K}(\eta,\xi)]^{2} \; ,
\end{displaymath}
which can be shown to be $| \langle \mu,\nu | \sigma,\lambda \rangle |^{2}$, we have
\begin{equation}
\label{fold3}
{\bf T}^{(-1)}(\mu,\nu) = \frac{1}{N} \sum_{\sigma,\lambda = -\ell}^{\ell} | \langle \mu,\nu | \sigma,\lambda \rangle |^{2} \, {\bf T}^{(1)}
(\sigma,\lambda) \; .
\end{equation}
Thus $| \langle \mu,\nu | \sigma,\lambda \rangle |^{2}$ itself, which is the Husimi function associated with the discrete coherent-state
$| \mu,\nu \rangle$, acts here as the smoothing function.
\section{Continuum limit}
Following the procedure detailed in both \cite{ruga,ruga2}, the continuum limit of the mapping kernel (\ref{novabase}) is reached as
follows: we introduce the scaling parameter $\epsilon = (2 \pi /N )^{1/2}$, which will become infinitesimal as $N \rightarrow \infty$, and
the two Hermitian operators
\begin{equation}
\label{29}
{\bf P} = \sum_{\mu = -\ell}^{\ell} \mu \epsilon p_{0} | v_{\mu} \rangle \langle v_{\mu} | \qquad {\bf Q} = \sum_{\mu^{\prime} = -\ell}^{\ell}
\mu^{\prime} \epsilon q_{0} | u_{\mu^{\prime}} \rangle \langle u_{\mu^{\prime}} | \; ,
\end{equation}
constructed out of the projectors of the eigenstates of ${\bf U}$ and ${\bf V}$. The parameters $p_{0}$ and $q_{0}$, with $p_{0} q_{0}
= \hbar = 1$, are chosen to be real, carrying units of momentum and position, respectively, while $\epsilon p_{0}$ and $\epsilon q_{0}$
are the distance between successive eigenvalues of the ${\bf P}$ and ${\bf Q}$ operators. Then, rewriting the Schwinger operators as
\begin{equation}
\label{28}
{\bf V} = \exp \left( \frac{i \epsilon {\bf P}}{p_{0}} \right) \qquad {\bf U} = \exp \left( \frac{i \epsilon {\bf Q}}{q_{0}} \right) \; ,
\end{equation}
and performing the change of variables $q^{\prime} = - q_{0} \epsilon \xi$, $p^{\prime} = p_{0} \epsilon \eta$, $p = p_{0} \epsilon \nu$
and $q = q_{0} \epsilon \mu$, we obtain
\begin{displaymath}
{\bf T}^{(s)}(q,p) = \sum_{q^{\prime} = - q_{0} \epsilon \ell}^{q_{0} \epsilon \ell} \sum_{p^{\prime} = - p_{0} \epsilon \ell }^{p_{0}
\epsilon \ell} \frac{\Delta q^{\prime} \Delta p^{\prime}}{2 \pi} \exp \left[ - i p^{\prime} (q-{\bf Q}) \right] \exp \left[ i q^{\prime}
(p-{\bf P}) \right] [\mathcal{K}(p^{\prime}/p_{0} \epsilon, - q^{\prime}/q_{0} \epsilon)]^{-s} \exp \left( - \frac{i}{2} q^{\prime}
p^{\prime} \right) \; .
\end{displaymath}
As $N \rightarrow \infty$, it follows that $\Delta q^{\prime} \rightarrow dq^{\prime}$ and $\Delta p^{\prime} \rightarrow dp^{\prime}$.
Since the continuum limit of the discrete coherent-states has been already discussed in \cite{gapi2,gama}, it is clear that the term
$[ \mathcal{K}(p^{\prime}/p_{0} \epsilon, - q^{\prime}/q_{0} \epsilon)]^{-s}$, which is even, will go to $(\langle 0 |q^{\prime},p^{\prime}
\rangle )^{-s}$. Therefore we end up with
\begin{displaymath}
{\bf T}^{(s)}(q,p) = \int_{-\infty}^{\infty} \frac{dp^{\prime} dq^{\prime}}{2 \pi} \exp \left[ - i p^{\prime}(q-{\bf Q}) \right] \exp
\left[ i q^{\prime}(p-{\bf P}) \right] \exp \left( - \frac{i}{2} p^{\prime} q^{\prime} \right) (\langle 0 | q^{\prime},p^{\prime} \rangle )^{-s} \; ,
\end{displaymath}
which is exactly the mapping kernel (\ref{II}) of Cahill and Glauber.
\section{Concluding Remarks}
The results obtained here show a genuine discrete mathematical structure which closely parallels the one of Cahill and Glauber. This
was achieved pursuing the lines proposed in \cite{gapi1}, which makes use of the discrete Fourier transform of the Schwinger operator
basis, to deal with the discrete phase-space problem. Now, we stress that expression (\ref{II}) is as simple as it is important, since
it clarifies the role of the coherent states overlap within the CG approach. By its turn, the discrete coherent states proposed in
\cite{gapi2,zhang,gama,voros} provide a natural path for a discrete extension of the CG formalism, while the properties of these states
have played a crucial role as they led, for example to the basic equation (\ref{fund}). Thus, the coherent states overlap can be seen
as the link between the discrete and continuous approaches. Furthermore, the continuum limit presented in section V ensures that the CG
mapping scheme is correctly recovered through a limiting procedure which is mathematically consistent \cite{ruga2,bar1,bar2,bar3}.
It is worth mentioning that Opatrn\'{y} {\em et al} \cite{opat} have pursued a goal similar to ours. Although both approaches share
virtues, our formalism presents mathematical features that allow us to achieve farther reaching results. It is precisely the correct
choice for the reference state, and the mathematical procedure adopted here, that lead to the obtention of such a wide set of important
results.
The use of Schwinger operators is crucial if one is concerned with the problem of ordering. As they are unitary shift operators,
equation (\ref{novabase}) makes it clear that the associated expansion is necessarily linked to a particular ordering of ${\bf U}$ and
${\bf V}$, which can be directly connected to the ${\bf Q}$ and ${\bf P}$ ordering of the continuous case.
Concerning the role of the Jacobi Theta functions in the discrete phase-space context -- they are implicit in the $\mathcal{K}
(\eta,\xi)$ term --, comparison with the usual CG results makes it evident that the Gaussian (or anti-Gaussian) terms, which are present
in the continuous case, are here replaced by the sum of products of $\vartheta$-functions (\ref{fold}), and its Fourier transform
(\ref{ffold}); both play here the role of the smoothing functions.
It is always important to emphasize the discrete case's peculiar features that do not have correspondence in the continuum. A plain
example of these is expressed by the well-behaved function given by equation (\ref{invfold}), whose continuum limit clearly diverges,
as the hierarchical structure presented in (\ref{fold1}) would imply. The finite character of the discrete scenario prevents such a
behaviour since, even if some terms in equation (\ref{invfold}) might become large for large $N$, they remain always finite due to the
behaviour of the $\vartheta$-functions. This allows one -- to give a extreme example -- to express, in the discrete scenario, the
Glauber-Sudarshan function in terms of the Husimi function.
Finally, it is worth mentioning that the mathematical formalism developed here opens new possibilities of investigations in quantum
tomography and quantum teleportation. These considerations are under
current research and will be published elsewhere (for instance, see reference \cite{nois}).
\section*{Acknowledgments}
This work has been supported by Funda\c{c}\~{a}o de Amparo \`{a} Pesquisa do Estado de S\~{a}o Paulo (FAPESP), Brazil, project nos.
03/13488-0 (MR), 01/11209-0 (MAM), and 00/15084-5 (MAM and MR). DG acknowledges partial financial support from the Conselho Nacional
de Desenvolvimento Cient\'{\i}fico e Tecnol\'{o}gico (CNPq), Brazil.
|
{
"timestamp": "2005-05-30T21:01:30",
"yymm": "0503",
"arxiv_id": "quant-ph/0503054",
"language": "en",
"url": "https://arxiv.org/abs/quant-ph/0503054"
}
|
\section{Introduction}\label{sec:intro}
Often the experimentalist needs a low-noise preamplifier for the
analysis of low-frequency components (below 10 Hz) from a 50 \ohm\
source. The desired amplifier chiefly exhibits low residual flicker
and high thermal stability, besides low white noise. Thermal
stability without need for temperature control is a desirable feature.
In fact the problem with temperature control, worse than complexity,
is that in a nonstabilized environment thermal gradients fluctuate,
and in turn low-frequency noise is taken in. A low-noise amplifier
may be regarded as an old subject, nonetheless innovation in analysis
methods and in available parts provides insight and new design. The
application we initially had in mind is the postdetection preamplifier
for phase noise measurements~\cite{rubiola02rsi}. Yet, there resulted
a versatile general-purpose scheme useful in experimental electronics
and physics.
\section{Design Strategy}\label{sec:strategy}
The choice of the input stage determines the success of a precision
amplifier. This issue involves the choice of appropriate devices and
of the topology.
Available low-noise devices are the junction field-effect transistor
(JFET) and the bipolar transistor (BJT), either as part of an
operational amplifier or as a stand-alone component. The white noise
of these devices is well
understood~\cite{van.der.ziel:noise-ssdc,van.der.ziel:fluctuations,netzer81pieee,erdi81jssc}.
Conversely, flicker noise is still elusive and relies upon models, the
most accredited of which are due to McWhorter~\cite{mcwhorter57} and
Hooge~\cite{hooge69pla}, or on smart narrow-domain analyses, like~\cite{green85jpd-1,green85jpd-2,jamaldeen99jap}, rather than on a unified theory. Even worse,
aging and thermal drift chiefly depend on proprietary technologies,
thus scientific literature ends up to be of scarce usefulness. The
JFET is appealing because of the inherently low white noise. The
noise temperature can be as low as a fraction of a degree Kelvin.
Unfortunately, the low noise of the JFET derives from low input
current, hence a high input resistance (some M\ohm) is necessary. The
JFET noise voltage is hardly lower than 5 \unit{nV/\sqrt{Hz}}, some
five to six times higher than the thermal noise of a 50 \ohm\ resistor
($\sqrt{4kTR}=0.89$ \unit{nV/\sqrt{Hz}}). The JFET is therefore
discarded in favor of the BJT\@.
A feedback scheme, in which the gain is determined by a resistive
network, is necessary for gain accuracy and flatness over frequency.
Besides the well known differential stage, a single-transistor
configuration is possible (Ref.~\cite{motchenbacher:low-noise:1ed},
page 123), in which the input is connected to the base and the
feedback to the emitter. This configuration was popular in early
audio hi-fi amplifiers. The advantage of the single-transistor scheme
is that noise power is half the noise of a differential stage. On the
other hand, in a dc-coupled circuit thermal effects are difficult to
compensate without reintroducing noise, while thermal compensation of
the differential stage is guaranteed by the symmetry of the
base-emitter junctions. Hence we opt for the differential pair.
\begin{table}
\begin{sideways}
\begin{minipage}{0.88\textheight}
\caption{\label{tab:opa}%
\vrule width0pt height2.5ex depth2ex
Selection of some low-noise BJT amplifiers.}
\centering
\begin{tabular}{|c|cccc|c|c|}\hline
& OP27\footnotemark[1]
& LT1028\footnotemark[1]
& MAT02\footnotemark[2]
& MAT03\footnotemark[2]
& \parbox{12ex}{unit}
& \parbox{12ex}{{\footnotesize MAT03}\\
measured\footnotemark[3]%
\vrule width0pt height0ex depth0.5ex}
\\\hline
\vrule width0pt height2.5ex depth0ex
WHITE NOISE&&&&&&\\
noise voltage\footnotemark[4] $\sqrt{h_{0,v}}$
& 3 & 0.9 & 0.9 & 0.7 & \unit{nV/\sqrt{Hz}} & 0.8 \\
noise current\footnotemark[4] $\sqrt{h_{0,i}}$
& 0.4 & 1 & 0.9 & 1.4~\footnotemark[5]& \unit{pA/\sqrt{Hz}} & 1.2\\
noise power $2\sqrt{h_{0,v}h_{0,i}}$
&$2.4{\times}10^{-21}$
&$1.8{\times}10^{-21}$
&$1.6{\times}10^{-21}$
& $2.0{\times}10^{-21}$
& \unit{W/Hz}
& $1.9{\times}10^{-21}$\\
noise temperature $T_w$
& 174 & 130 & 117 & 142 & K & 139 \\
optimum resistance $R_{b,w}$
& 7500 & 900 & 1000 & 500 & \ohm & 667 \\
$2{\times}50$\ohm-input noise
& 3.3 & 1.55 & 1.55 & 1.5 & \unit{nV/\sqrt{Hz}} & 1.5~%
\footnotemark[6]\\\hline
\vrule width0pt height2.5ex depth0ex
FLICKER NOISE&&&&&&\\
noise voltage\footnotemark[4] $\sqrt{h_{-1,v}}$
& 4.3 & 1.7 & 1.6 & 1.2 & \unit{nV/\sqrt{Hz}} & (~$0.4$~)%
\footnotemark[7] \\
noise current\footnotemark[4] $\sqrt{h_{-1,i}}$
& 4.7 & 16 & 1.6 &n.\,a.& \unit{pA/\sqrt{Hz}} & 11 \\
noise power $2\sqrt{h_{-1,v}h_{-1,i}}$
& $4.1{\times}10^{-20}$
& $5.3{\times}10^{-20}$
& $5.1{\times}10^{-21}$
& -- & \unit{W/Hz}
& (\ldots)\footnotemark[8] \\
1-Hz noise temperature $T_f$
& 2950 & 3850 & 370 & -- & K & (\ldots)\footnotemark[8] \\
optimum resistance $R_{b,f}$
& 910 & 106 & 1000 & -- & \ohm & (\ldots)\footnotemark[8] \\
$2{\times}50$\ohm-input noise
& 4.3 & 2.3 & 1.6 & -- & \unit{nV/\sqrt{Hz}} & 1.1~%
\footnotemark[6] \\\hline
\vrule width0pt height2.5ex depth0ex
THERMAL DRIFT
& 200 & 250 & 100 & 300 & nV/K & --
\\\hline
\end{tabular}
\footnotetext[1]{Low-noise operational amplifier.}
\footnotetext[2]{Matched-transistor pair.
MAT02 is \textsc{npn}, MAT03 is \textsc{pnp}.
Data refer to the pair, biased at $I_C=1$ mA.}
\footnotetext[3]{Some MAT03 samples measured in our laboratory.
See Sec.~\protect\ref{sec:frontend}}
\footnotetext[4]{Power-law model of the spectrum, voltage or current,
$S(f)=h_0+h_{-1}f^{-1}+h_{-2}f^{-2}+\ldots$}
\footnotetext[5]{Obtained from the total noise with 100 k\ohm\
input resistance.}
\footnotetext[6]{Measured on the complete amplifier
(Sec.~\protect\ref{sec:results}), independently
of the measurement of the above $S_v$ and $S_i$.}
\footnotetext[7]{Derives from the noise current through $r_{bb'}$.
See Sec.~\protect\ref{sec:results}.}
\footnotetext[8]{Can not be compared to other data because voltage
and current are correlated. See Sec.~\protect\ref{sec:results}.}
\end{minipage}
\end{sideways}
\end{table}
Table~\ref{tab:opa} compares a selection of low-noise bipolar
amplifiers. The first columns are based on the specifications
available on the web
sites~\cite{www.analog-devices,www.linear-technology}. The right-hand
column derives from our measurements, discussed in
Secs.~\ref{sec:frontend} and \ref{sec:results}. Noise is described in
terms of a pair of random sources, voltage and current, which are
assumed independent. This refers to the Rothe-Dahlke
model~\cite{rothe56ire}. Nonetheless, a correlation factor arises in
measurements, due to the distributed base resistance $r_{bb'}$.
Whether and how $r_{bb'}$ is accounted for in the specifications is
often unclear. The noise spectra are approximated with the power law
$S(f)=\sum_{\alpha}h_\alpha f^\alpha$. This model, commonly used in
the domain of time and frequency, fits to the observations and
provides simple rules of transformation of spectra into two-sample
(Allan) variance $\sigma_y(\tau)$. This variance is an effective way
to describe the stability of a quantity $y$ as a function of the
measurement time $\tau$, avoiding the divergence problem of the
$f^\alpha$ processes in which $\alpha\le-1$.
References~\cite{rutman78pieee} and \cite{rubiola01im} provide the
background on this subject, and application to operational amplifiers.
The noise power spectrum $2\sqrt{h_vh_i}$ is the minimum noise of the
device, i.e., the noise that we expect when the input is connected to
a cold (0~K) resistor of value $R_b=\sqrt{h_{v}/h_{i}}$, still under
the assumption that voltage and current are uncorrelated. When the
input resistance takes the optimum value $R_b$, voltage and current
contributions to noise are equal. The optimum resistance is $R_{b,w}$
for white noise and $R_{b,f}$ for flicker. Denoting by $f_{c}$ the
corner frequency at which flicker noise is equal to white noise, thus
$f_{c,v}$ for voltage and $f_{c,i}$ for current, it holds that
$R_{b,w}/R_{b,f}=\sqrt{f_{c,i}/f_{c,v}}$. Interestingly, with most
bipolar operational amplifiers we find
$f_{c,i}/f_{c,v}\approx50{-}80$, hence $R_{b,w}/R_{b,f}\approx7{-9}$.
Whereas we have no explanation for this result, the lower value of the
flicker optimum resistance is a fortunate outcome. The equivalent
temperature is the noise power spectrum divided by the Boltzmann
constant $k=1.38{\times}10^{-23}$ J/K\@. A crucial parameter of
Table~\ref{tab:opa} is the total noise when each input is connected to
a 50~\ohm\ resistor at room temperature. This calculated value
includes noise voltage and current, and the thermal noise of the two
resistors. In a complete amplifier two resistors are needed, at the
input and in the feedback circuit.
Still from Table~\ref{tab:opa}, the transistor pairs show lower noise
than the operational amplifiers, although the PNP pair is only
partially documented. Experience indicates that PNP transistors are
not as good as NPN ones to most extents, but exhibit lower noise. In
other domains, frequency multipliers and radio-frequency oscillators
make use of PNP transistors for critical application because of the
lower flicker noise. Encouraged by this fact, we tried a differential
amplifier design based on the MAT03, after independent measurement of
some samples.
\section{Input Stage}\label{sec:frontend}
\begin{figure}[t]
\centering\includegraphics[scale=1]{measure-mat}
\caption{Noise measurement of a transistor pair. For clarity,
the distributed base resistance $r_{bb'}$ is extracted from the
transistors.}
\label{fig:measure-mat}
\end{figure}
The typical noise spectrum of the MAT03, reported in the data sheet,
shows an anomalous slope at low frequencies (0.1--1 Hz), significantly
different from $f^{-1}$. This is particularly visible at low
collector current (10--100 $\mu$A), but also noticeable at $I_C=1$
mA\@. We suspect that the typical spectrum reflects the temperature
fluctuation of the environment through the temperature coefficient of
the offset voltage $V_{OS}$ rather than providing information on the
flicker noise inherent in the transistor pair. The measurement of a
spectrum from 0.1 Hz takes some 5 min. At that time scale, in a
normal laboratory environment the dominant fluctuation is a drift. If
the drift is linear, $v(t)=ct$ starting at $t=0$, the Fourier
transform is $V(\omega)=j\pi c\delta(\omega)-c/\omega^2$. Dropping
off the term $\delta(\omega)$, which is a dc term not visible in a
log-log scale, the power spectrum density, i.e., the squared Fourier
transform, is
\begin{equation}
\label{eq:f-drift}
S_v(\omega)=\frac{c^2}{\omega^4} \qquad\mbox{or}\qquad
S_v(f)=\frac{(2\pi)^4c^2}{f^4}~~.
\end{equation}
A parabolic drift---seldom encountered in practice---has a spectrum
proportional to $f^{-6}$, while a smoothly walking drift tends to be
of the $f^{-5}$ type. As a consequence, a thermal drift can be
mistaken for a random process of slope $f^{4}$ to $f^{5}$, which may
hide the inherent $f^{-1}$ noise of the device. For this reason, the
test circuit (Fig.~\ref{fig:measure-mat}) must be enclosed in an
appropriate environment. We used, with similar results, a Dewar flask
coupled to the environment via a heat exchanger, and a metal box
mounted on a heat sink that has a mass of 1 kg and a thermal
resistance of 0.6 K/W\@. These odd layouts provide passive
temperature stabilization through a time constant and by eliminating
convection, and evacuate the small amount of heat (200 mW) dissipated
by the circuit.
\begin{figure}[t]
\centering\includegraphics[scale=0.8,angle=0]{f695}
\caption{Typical spectrum of the noise voltage.}
\label{fig:f695}
\end{figure}
Due to the low value of $r_{bb'}$ (15--20 \ohm) the current
measurement can be made independent of voltage noise, but not vice
versa. Thus, we first measure the noise current setting
$R_B=8$~k\ohm, which is limited by the offset current; then we measure
the noise voltage setting $R_B=10$~\ohm. A technical difficulty is
that at 1 Hz and below most spectrum analyzers---including our
one---must be coupled in dc, hence high offset stability is needed in
order to prevent saturation of the analyzer. The measured spectra are
$S_i(f)=1.45{\times}10^{-24}+1.2{\times}10^{-22}f^{-1}$ \unit{A^2/Hz}
(i.e., 1.2\unit{pA/\sqrt{Hz}} white, and 11\unit{pA/\sqrt{Hz}}
flicker), and $S_v(f)=10^{-18}+1.8{\times}10^{-19}f^{-1}$
\unit{V^2/Hz} (i.e., 1\unit{nV/\sqrt{Hz}} white, and
425\unit{pV/\sqrt{Hz}} flicker). The current spectrum is the inherent
noise current of the differential pair. Conversely, with the voltage
spectrum (Fig.~\ref{fig:f695}) we must account for the effect of $R_B$
and $r_{bb'}$. With our test circuit, the expected white noise is
$h_{0,v}=4kTR+2qI_BR\simeq1.7{\times}10^{-20}R$ \unit{V^2/Hz}, which
is the sum of thermal noise and the shot noise of the base current
$I_B$. $R=2(R_B+r_{bb'})$ is the equivalent base resistance, while
the shot noise of the collector current is neglected. Assuming
$r_{bb'}=16$~\ohm\ (from the data sheet), the estimated noise is
$h_{0,v}\simeq9{\times}10^{19}$ \unit{V^2/Hz}. This is in agreement
with the measured value of $10^{-18}$ \unit{V^2/Hz}. Then, we observe
the effect of the current flickering on the test circuit is
$R^2h_{-1,i}\simeq1.6{\times}10^{-19}$ \unit{V^2/Hz}. The latter is
close to the measured value $1.8{\times}10^{-19}$ \unit{V^2/Hz}.
Hence, the observed voltage flickering derives from the current noise
through the external resistors $R_B$ and the internal distributed
resistance $r_{bb'}$ of the transistors. Voltage and current are
therefore highly correlated. As a further consequence, the product
$2\sqrt{h_{-1,v}h_{-1,i}}$ is not the minimum noise power, and the
ratio $\sqrt{h_{-1,v}/h_{-1,i}}$ is not the optimum resistance. The
corresponding places in Table~\ref{tab:opa} are left blank. Due to
the measurement uncertainty, we can only state that a true independent
voltage flickering, if any, is not greater than $4{\times}10^{-20}$
\unit{A^2/Hz}. The same uncertainty affects the optimum resistance
$R_{b,f}$, which is close to zero.
The measured white noise is in agreement with the data sheet. On the
other hand, our measurements of flicker noise are made in such unusual
conditions that the results should not be considered in contradiction
with the specifications, as the specifications reflect the the
low-frequency behavior of the device in a normal environment.
\section{Implementation and Results}\label{sec:results}
\begin{figure}[t]
\centering\includegraphics[scale=1]{scheme}
\caption{Scheme of the low-noise amplifier.}
\label{fig:scheme}
\end{figure}
Figure~\ref{fig:scheme} shows the scheme of the complete amplifier,
inspired to the ``super low-noise amplifier'' proposed in Fig.~3a of
the MAT03 data sheet. The NPN version is also discussed in
Ref.~\cite{franco:opa} (p.~344). The original circuit makes use of
three differential pairs connected in parallel, as it is designed for
the lowest white noise with low impedance sources ($\ll50$~\ohm), like
coil microphones. In our case, using more than one differential pair
would increase the flicker because of current noise.
The collector current $I_C=1.05$ mA results as a trade-off between
white noise, which is lower at high $I_C$, dc stability, which is
better at low dissipated power, flicker, and practical convenience.
The gain of the differential pair is $g_mR_C=205$, where
$g_m=I_C/V_T=41$~mA/V is the transistor transconductance, and $R_C=5$
k\ohm\ is the collector resistance. The overall gain is
$1+R_G/R_B\simeq500$. Hence the gain of the OP27 is of 2.5, which
guarantees the closed-loop stability (here, oscillation-free
operation). If a lower gain is needed, the gain of the differential
stage must be lowered by inserting $R_A$. The trick is that the
midpoint of $R_A$ is a ground for the dynamic signal, hence the
equivalent collector resistance that sets the gain is $R_C$ in
parallel to $\frac{1}{2}R_G$. The bias current source is a cascode
Wilson scheme, which includes a light emitting diode (LED) that
provides some temperature compensation.
The stability of the collector resistors $R_C$ is a crucial point
because the voltage across them is of 5~V\@. If each of these
resistors has a temperature coefficient of $10^{-6}$/K, in the worst
case there results a temperature coefficient of 10 $\mu$V/K at the
differential output, which is equivalent to an input thermal drift of
50~nV/K\@. This is 1/6 of the thermal coefficient of the differential
pair. In addition, absolute accuracy is important in order to match
the collector currents. This is necessary to take the full benefit
from the symmetry of the transistor pair.
\begin{figure}[t]
\centering\includegraphics[width=\textwidth]{franck-ampli-small}
\caption{Prototype of the low-noise amplifier.}
\label{fig:prototype}
\end{figure}
Two equal amplifiers are assembled on a printed circuit board, and
inserted in a $10{\times}10{\times}2.8$ \unit{cm^3}, 4 mm thick
aluminum box (Fig.~\ref{fig:prototype}). The box provides thermal coupling to the environment
with a suitable time constant, and prevents fluctuations due to
convection. $LC$ filters, of the type commonly used in HF/VHF
circuits, are inserted in series to the power supply, in addition to
the usual bypass capacitors. For best stability, and also for
mechanical compatibility with our equipment, input and output
connector are of the SMA type. Input cables should not PTFE-insulated
because of piezoelectricity (see the review
paper~\cite{fukada00uffc}).
\begin{figure}[t]
\centering\includegraphics[scale=0.8]{f691}
\caption{Residual noise of the complete amplifier,
input terminated to a 50~\ohm\ resistor.}
\label{fig:f691}
\end{figure}
Figure~\ref{fig:f691} shows the noise spectrum of one prototype input
terminated to a 50~\ohm\ resistor. The measured noise is
$\sqrt{h_0}=1.5$ \unit{nV/\sqrt{Hz}} (white) and $\sqrt{h_{-1}}=1.1$
\unit{nV/\sqrt{Hz}} (flicker). The corner frequency at which the
white and flicker noise are equal is $f_c=0.5$ Hz. Converting the
flicker noise into two-sample (Allan) deviation, we get
$\sigma_v(\tau)=1.3$ nV, independent of the measurement time $\tau$.
Finally, we made a simple experiment aimed to explain in practical
terms the importance of a proper mechanical assembly. We first
removed the Al cover, exposing the circuit to the air flow of the
room, yet in a quiet environment, far from doors, fans, etc., and then
we replaced the cover with a sheet of plain paper (80 \unit{g/m^2}).
The low-frequency spectrum (Fig.~\ref{fig:f694}) is
$5{\times}10^{-19}f^{-5}$ \unit{V^2/Hz} in the first case, and about
$1.6{\times}10^{-19}f^{-4}$ \unit{V^2/Hz} in the second case. This
indicates the presence of an irregular drift, smoothed by the paper
protection. Interestingly, Hashiguchi~\cite{sikula03arw} reports
on thermal effects with the same slope and similar cutoff frequencies,
observed on a low-noise JFET amplifier for high impedance sources.
\begin{figure}[t]
\centering\includegraphics[scale=0.8]{f694}
\caption{Thermal effects on the amplifier.}
\label{fig:f694}
\end{figure}
\def\bibfile#1{/Users/rubiola/Documents?workocs/bib/#1}
\bibliographystyle{amsalpha}
|
{
"timestamp": "2005-03-01T15:06:51",
"yymm": "0503",
"arxiv_id": "physics/0503012",
"language": "en",
"url": "https://arxiv.org/abs/physics/0503012"
}
|
\section{Introduction}
In this paper we study some class of division algebras over
a Laurent series field with arbitrary residue field. Namely, we study division algebras which satisfy the
following condition:
there exists a section $\bar{D} \hookrightarrow D$ of the residue homomorphism
$D\rightarrow \bar{D}$,
where $D$ is a central division algebra over a complete discrete valued field $F=k((t))$. We say that these division algebras are splittable.
If $char k=0$, all such division algebras are tame and therefore belong to the group of tame division algebras, which was carefully studied in the papers \cite{JW} and \cite{PY} even in a much more general situation of a henselian field $F$ of arbitrary characteristic. So, we consider mostly wild division algebras.
An extensive analysis of the wild
division algebras of degree $p$ over a field $F$ with complete discrete rank 1
valuation with $char (\bar{F})= p$ was given by Saltman in \cite{Sa} (Tignol
in \cite{Ti} analyzed more general case of the defectless division algebras
of degree $p$ over a fild $F$ with Henselian valuation). Here we study splittable division algebras of arbitrary index.
This class (which is not a subgroup in $Br (F)$) contains a class of good splittable division algebras (see the definition in section 2), which posess several beautiful properties. In particular, we prove a decomposition theorem for such algebras. This theorem is a
generalization of the decomposition theorems for tame division algebras given
by Jacob and Wadsworth in \cite{JW}.
For arbitrary splittable division algebras we give only several assorted results, and the study of this class is far from to be complete. Nevertheless, we investigate here technical tools, which are important for the study of such algebras, and prove a relation between the level and a higher order level for some splittable division algebras (see section 6). We hope this technique will be applied to the study of the cyclisity question for certain division algebras od degree $p^k$.
As an application we get several results, which are partly well known (see proposition \ref{cyclisity}) and party not. In particular, we get the positive answer on the following
conjecture: the exponent of $A$ is equal to its index for any division algebra
$A$ over a $C_2$-field $F= F_1((t_2))$, where $F_1$ is a $C_1$-field.
Here is a brief overview of this paper.
In section 2 we give a definition of splittable and good splittable division algebras and prove that all tame division algebras over $F=k((t))$ are good splittable.
Section 3 contains the most important technical tools for the study of splittable division algebras. We define a notion of $\delta$-maps and investigate a theory of $\delta$-maps for such algebras. In this section we define also the notion of a local height, which is a possible generalization of Saltman's level.
In section 4 we prove the period-index conjecture metioned above. This section contains also a small history of the question known to the author. We note that the proof does not use all the results from section 3.
In section 5 we study good splittable division algebras and prove the decomposition theorem.
In section 6 we reprove some results of Saltman about semiramified division algebras of index $p$ over $F$ using the technique from section 3. Then we define a notion of a higher order level and prove several general properties of splittable division algebras satisfying the following condition: $Z(\bar{D})/\bar{F}$ is a simple extension. At the end of section we put several open questions.
We use the notation of \cite{JW}. We always denote by $D$ a division algebra
finite dimensional over its center $F=k((t))= Z(D)$. Recall that any Henselian
valuation on $F$ has a unique extension to a valuation on $D$. We denote the valuation on $F$ by $v$ and its unique extension on $D$ by $w$.
Given a valuation $w$ on $D$, we denote by $\Gamma_{D}$ its value group, by
$V_D$ its valuation ring, by $M_D$ its maximal ideal and by $\bar{D}=
V_D/M_D$ its residue division ring.
By \cite{S}, p.21 one has the fundamental inequality
$$
[D:F]\ge |\Gamma_D:\Gamma_E| \cdot [\bar{D}:\bar{F}].
$$
$D$ is called defectless over $F$ if equality holds and defective otherwise.
It is known that $D$ is defectless if it has a discrete valuation of rank 1.
Jacob and Wadsworth in \cite{JW} introduced the basic homomorphism
$$
\theta_D:\Gamma_D/\Gamma_F\rightarrow Gal(Z(\bar{D})/\bar{F})
$$
induced by conjugation by elements of $D$. They showed that $\theta_D$ is
surjective and $Z(\bar{D})$ is the compositum of an abelian Galois and a
purely inseparable extension of $\bar{F}$.
We say $D$ is tame division algebra if $char (\bar{F})= 0$ or $char
(\bar{F})= q\ne 0$, $D$ is defectless over $F$, $Z(\bar{D})$ is separable
over $\bar{F}$, and $q{\not |}|ker(\theta_D)|$. We say $D$ is wild division
algebra if it is non tame.
We call a division algebra $D$ {\it inertially split} if $Z(\bar{D})$ is
separable over $\bar{F}$, the map $\theta_D$ is an isomorphism, and $D$ is
defectless over $F$.
\bigskip
{\bf Acknowledgements}
I am grateful to Professor A. N. Parshin, Professor E.-W. Zink, and M.
Grabitz for useful discussions and attention to my work. I am very grateful to Professor A.Wadsworth for
carefully reading my paper and for showing me a mistake in the very first version of this paper and to Professor V.I.Yanchevskii for valuable discussions during his visit in Berlin. Finally, I thank my wife Olga for her support and encouragement.
\section{Cohen's theorem}
Recall one definition from \cite{Zh}.
\begin{defi}
A division algebra $D$ is said to be splittable if there is a homomorphism
$\bar{D} \hookrightarrow {\cal O}_D\subset D$
that is a section of the map ${\cal O}_D \rightarrow \bar{D}$.
\end{defi}
There is a natural question if there exists a generalization of Cohen's
theorem, i.e. is any
central division algebra splittable or not. It is not true if a division
algebra is not finite dimensional
over its centre, as Dubrovin's example in \cite{Zh} shows. It is not true also
for some finite dimensional
division algebras, as the example to theorem 2.7. in \cite{Sa} shows.
But it is true
for tame division
algebras over complete discrete valued fields. This easily follows
from results of Jacob and Wadsworth \cite{JW} (compare with \cite{Zh}, Th.1).
\begin{th}
\label{Cohen}
Let $(F,v)$ be a valued field which is complete with respect to a discrete
rank 1 valuation $v$. Suppose $char F= char \bar{F}$. Let $D$ be a tame
division
algebra with $Z(D)= F$ and $[D:F]< \infty$.
Then there exists a section $\bar{D}\hookrightarrow D$ of the residue
homomorphism $D\rightarrow \bar{D}$.
\end{th}
{\bf Proof.} Since $F$ is a complete field, $F$ is a Henselian field and $v$
extends uniquely to a valuation $w$ on $D$. Since $D$ is tame,
$Z(\bar{D})/\overline{Z(D)}$ is a cyclic Galois extension.
There exists an
inertial lift $Z$ of $Z(\bar{D})$ over $F$, $Z$ is Galois over $F$,
and by classical Cohen's theorem there exists a section
$\tilde{Z}(\bar{D})\hookrightarrow Z$.
Consider the centraliser $C= C_D(Z)$ of $Z$ in $D$. Then we have $\bar{C}=
\bar{D}$.
Indeed, by Double Centraliser Theorem we have $[D:F]= [C:F][Z:F]$ and
$[Z:F]= |Gal (Z(\bar{D})/\bar{F})|$. By \cite{JW}, prop.1.7 a
homomorphism $\theta_D: \Gamma_D/\Gamma_F\rightarrow Gal(Z(\bar{D})/\bar{F})$
is surjective, so for any parameter $z$ we have
$\theta_D(w(z))= \sigma$, where $<\sigma >
= Gal (Z(\bar{D})/\bar{F})$. It is clear that $z\notin
C$. Now let $u_1,\ldots ,u_{[C:F]}$ be a $F$-basis of $C$. It is easy to see
that the elements $u_j, zu_j, \ldots , z^{n-1}u_j
$, $j= 1,\ldots ,[C:F]$, where $n= ord (\sigma )$,
the order of $\sigma $, are linearly independent, so form a basis for $D$ over
$F$. Since
$$
w(F\langle zu_j, \ldots , z^{n-1}u_j,
j= 1,\ldots ,[C:F]\rangle )\cap \Gamma_C= 0,
$$
where $F\langle zu_j, \ldots , z^{n-1}u_j,
j= 1,\ldots ,[C:F]\rangle$ denote a vector space in $D$ over $F$ generated
by elements $u_jz^i$,
this implies that for any element $x\in D$ with
$w(x)= 0$ we can find elements $r_1, \ldots r_{[C:F]}\in F$ such that $x=
r_1u_1+\ldots +r_{[C:F]}u_{[C:F]}\mbox{\quad mod \quad} M_D$. Hence
$\bar{C}= \bar{D}$.
Note that $C$ is an unramified division algebra. Indeed, by \cite{JW}, th.2.8,
th.2.9 $C$ contains a copy of the inertial lift of a maximal separable
subfield in $\bar C$, say $\tilde C$. Then the centralizer $C_C(\tilde{C})$
must be a totally ramified division algebra, i.e. it is trivial and $\tilde C$
is a maximal subfield. So, $C$ must be unramified.
Fix an embedding $i: \bar F \hookrightarrow F$. It can be extended to the
embedding $i':\bar Z \hookrightarrow Z$, $i'|_{\bar F}=i$ by Hensel lemma. Now
consider the algebra $A= \bar{C}\otimes_{\bar Z}Z(C)$. It is easy to see
that $A$ is an
unramified division algebra with $\bar{A}= \bar{C}= \bar{D}$.
Therefore by \cite{Az}, Th.31, $A\cong C$; so there exists a section
$\bar{D}\hookrightarrow C$.
The theorem is proved.\\
$\Box$\\
Later we will see that much more can be said about good splittable algebras:
\begin{defi}
\label{goodsplit}
A division algebra $D$ is called good splittable if there exists a section
$s:\bar D\hookrightarrow
D$ compatible with an embedding $i:\overline{Z(D)}\hookrightarrow Z(D)$, i.e.
$s(\overline{Z(D)})=i(\overline{Z(D)})\subset Z(D)$.
\end{defi}
It's easy to see that all tame division algebras are good splittable, because
by Hensel lemma any embedding $\overline{Z(D)}\hookrightarrow Z(D)$ can be
uniquely extended to any separable extension of $Z(D)$.
It is interesting to know what kind of splittable division algebras are good
splittable.
By theorem 3.9. in \cite{Sa} even a splittable division algebra $D$ of degree $p=char D$ is not a good splittable algebra if the level of $D$ (the notion of level we will recall in section 3, see remark to lemma \ref{svva}) is divisible by $p$.
Nevertheless, it is an open question whether it is true or not, for example, for division algebras with $\bar{D}=Z(\bar{D})$ such that $\bar{D}/\bar{F}$ is a simple extension and the local height (see the definition in the same remark) is not divisible by $p$.
We will discuss this question in section 6.
\section{Delta-maps of splittable algebras}
In this section we develop some ideas from \cite{Zh}, where some properties of
$\delta$-maps for special kind of local skew fields were studied. Technical
properties of $\delta$-maps play the main role in all our results. Here we
will give a list of these properties.
Let $D$ be a finite dimensional
division algebra over a complete valued field $F= k((t))$. Let $w$ be a
unique extension of the valuation $v$ to $D$. We will denote by $z$ any
parameter of $D$, i.e. any element with $\langle w(z)\rangle = \Gamma_D$.
Consider the ring ${\mbox{\dbl Z}} \langle\alpha, \delta \rangle$ of noncommutative
polinomials in
two variables. Define the map
$$
\sigma :{\mbox{\dbl Z}} \langle\alpha , \sigma \rangle\rightarrow {\mbox{\dbl Z}} \langle\alpha
,\delta , \delta_i; i\ge
1\rangle ,
$$
$$
\sigma (\alpha^{a_1}\delta^{b_1}\ldots \alpha^{a_n}\delta^{b_n})=
\alpha^{a_1}\delta_{b_1}\ldots \delta_{b_{n-1}}\alpha^{a_n-1}\delta^{b_n},
$$
where $a_1, b_n\ge 0$, $a_i, b_j\ge 1$, $i>1$, $j<n$ for every word in ${\mbox{\dbl Z}}
\langle\alpha , \delta \rangle$.
Let $S_i^k\in {\mbox{\dbl Z}} \langle\alpha ,\delta \rangle$, $i\ge k$, $i\ge 1$ be
polynomials given
by the following formula:
$$
S_i^k= \sum_{\tau\in S_i/G}\tau (\underbrace{\alpha\ldots
\alpha}_{i-k}\underbrace{\delta\ldots \delta}_{k}),
$$
where $S_i$ is a permutation group and $G$ is an isotropy subgroup.
\begin{lemma}{(\cite{Zh}, lemma 2)}
The polynomials $S_i^k$ satisfy the following property:
$$
S_i^i= \delta^i, \mbox{\quad} S_i^0= \alpha^i, \mbox{\quad}
S_{i+1}^{k+1}= \alpha S_i^{k+1}+\delta S_i^k $$
\end{lemma}
For any splittable division algebra can be defined a notion of $\delta$-maps:
\begin{prop}{(\cite{Zh}, prop. 1,2)}
\label{ooo}
Let $D$ be a splittable division algebra.
Fix some parameter $z$ and some embedding $u: \bar{D} \hookrightarrow D$.
Then
$D$ is isomorphic to a division algebra $\bar{D}((z))$,
which is defined to be the vector space of series with multiplication defined
by the formula
$$
zaz^{-1}= \alpha (a)+\delta_1(a)z+\delta_2(a)z^2+\ldots ,\mbox{\quad}
a\in \bar{D},
$$
where $\alpha :\bar{D}\rightarrow \bar{D}$ is an
automorphism and $\delta_i:\bar{D}\rightarrow \bar{D}$ are linear maps such
that the map $\delta_i$ satisfy the identity
$$\delta_i(ab)= \sum_{k= 0}^i\sigma(\delta^{i-k}\alpha)
(a)\sigma(S_i^k\alpha)
(b),\mbox{\quad} a,b\in \bar D
$$
\end{prop}
{\bf Remark} Note that the values $\sigma (S_i^k\alpha )$ and $\sigma
(\delta^{i-k}\alpha )$ belong to the subring ${\mbox{\dbl Z}} \langle\alpha , \delta_i,
i\ge
1\rangle$, so the formula is well defined.
Note that $\delta$-maps depend on the choice of a parameter and an embedding.
The automorphism $\alpha$, as it easy to see, depend only on the choice of a
parameter.
In the proposition we identify $\bar{D}$ with $u(\bar{D})$.
\begin{corol}{(\cite{Zh}, corol. 1)}
\label{formuly}
Suppose $\alpha= Id$. Then
$$
\delta_i(ab)=
\delta_i(a)b+
\sum_{k= 1}^{i}\delta_{i-k}(a)\sum_{(j_1,\ldots ,j_l)}
C_{i-k+1}^l\delta_{j_1}\ldots \delta_{j_l}(b),
$$
where $\delta_0=\alpha$ and the second sum is taken over all the vectors
$(j_1,\ldots ,j_l)$ such that $0< l\le min\{i-k+1, k\}$, $j_m\ge 1$, $\sum
j_m= k$.
\end{corol}
Further we will need even more general definition.
\begin{defi}
\label{maps}
In the situation of proposition \ref{ooo}
let us define maps ${}_m^{(z,u)}\delta_i: \bar{D}\rightarrow \bar{D}$,
$m\in{\mbox{\dbl Z}}$,
$i\in {\mbox{\dbl N}}$ as follows.
$$
z^maz^{-m}= u({}^{(z)}\alpha^m(\bar{a}))+u({}_m^{(z,u)}\delta_1(\bar{a}))z+
u({}_m^{(z,u)}\delta_2(\bar{a}))z^2+\ldots ,\mbox{\quad}
a\in u(\bar{D}).
$$
If $m= 0$, put ${}_m^{(z,u)}\delta_i= 0$.
\end{defi}
Note that ${}^{(z)}\alpha |_{Z(\bar{D})}$ does not depend on the choice of $z$.
Note that if ${}^{(z)}\alpha= id$, then ${}_m^{(z,u)}\delta_i= 0$ for
$m= p^k$, where $k$
is sufficiently large, $k$ depends on $i$. Moreover, ${}_m^{(z,u)}\delta_i=
{}_{m+p^k}^{(z,u)}\delta_i$ for $k$ sufficiently large. We will use also the
following
notation:
$$
{}_m^{(z,u)}\tilde{\delta_i}= {}_{-m}^{(z,u)}\delta_i, \mbox{\quad}
{}_{1}^{(z,u)}\delta_i= {}^{(z,u)}\delta_i
$$
Sometimes, we will write ${}_m\delta_i$ instead of
${}_m^{(z,u)}\delta_i$ and ${}_m^{(z,u)}\delta_i(a)$ instead of
$u({}_m^{(z,u)}\delta_i(\bar{a}))$ whenever the context is clear.
Immediately from the definition follows
\begin{lemma}
\label{triviall}
In the situation of definition \ref{maps}
we have
(i) for $|m|>1$
$$
{}_m^{(z,u)}\delta_i(a)={}^{(z)}\alpha^{sign(m)}
({}_{sign(m)(|m|-1)}^{(z,u)}\delta_i(a))+
{}_{sign(m)}^{(z,u)}\delta_i({}^{(z)}\alpha^{sign(m)(|m|-1)}(a))+
$$
$$
\sum_{j=1}^{i-1}{}_{sign(m)}^{(z,u)}\delta_j({}_{sign(m)(|m|-1)}^{(z,u)}\delta_{i-j}(a)),
$$
where $sign(m)=m/|m|$, $a\in \bar{D}$;
(ii) for any $m\ne 0$
$$
{}^{(z)}\alpha^{-m} ({}_{m}^{(z,u)}\delta_i)+
{}_{-m}^{(z,u)}\delta_i({}^{(z)}\alpha^{m})+
\sum_{j=1}^{i-1}{}_{-m}^{(z,u)}\delta_j({}_{m}^{(z,u)}\delta_{i-j})=0
$$
\end{lemma}
\begin{prop}
\label{flyii}
For fixed $z,u$ from proposition \ref{ooo} we have
(i) The maps ${}_m^{(z,u)}\delta_i$ satisfy the following identities:
$$
{}_m{\delta_i}(ab)=
{}_m{\delta_i}(a)\alpha^{i+m}(b)+\alpha^m(a){}_m{\delta_i}(b)+
\sum_{k= 1}^{i-1}{}_m{\delta_{i-k}}(a)
{}_{i-k+m}{\delta_{k}}(b)
$$
(ii) Suppose $\alpha = id$. Then
the maps ${}_m^{(z,u)}\delta_i$ satisfy the following identities:
$$
{}_m{\delta_i}(ab)=
{}_m{\delta_i}(a)b+a{}_m{\delta_i}(b)+
\sum_{k= 1}^{i-1}{}_m{\delta_{i-k}}(a)\sum_{(j_1,\ldots ,j_l)}
C_{i-k+m}^l\delta_{j_1}\ldots
{\delta_{j_l}}(b)
$$
where the second sum is taken over all the vectors
$(j_1,\ldots ,j_l)$ such that $0< l\le min\{i-k+m, k\}$, $j_m\ge 1$, $\sum
j_m= k$; $C_j^k= 0$ if $j= 0$, and $C_j^k= C_{j+p^q}^k$ for $q>>0$ if
$j\le 0$.
\end{prop}
{\bf Proof.} For any $a,b\in\bar{D}$ we have
$$
\alpha^m(ab)z^{m}+{}_m{\delta_1}(ab)z^{m+1}+{}_m{\delta_2}(ab)z^{m+2}+\ldots=
z^{m}(ab)=
$$
\begin{equation}
\label{(*)}
(\alpha^m(a)z^{m}+{}_m{\delta_1}(a)z^{m+1}+{}_m{\delta_2}(a)z^{m+2}+\ldots )b
\end{equation}
If we represent the right-hand side of (\ref{(*)}) as a series with coeffitients
shifted to the left and then compare the corresponding coeffitients on the
left-hand side and right-hand side, we get some formulas for
${}_m\delta_i(ab)$. We have to prove that these formulas are the same as in
our proposition.
Let
$$z^{i+m-k}b= \alpha^{i+m-k}(b)z^{i+m-k}+\ldots +x'_kz^{i+m}+\ldots $$
and
$$(\alpha^m(a)z^{m}+{}_m{\delta_1}(a)z^{m+1}+{}_m{\delta_2}(a)z^{m+2}+\ldots )b
= \alpha^m(ab)z^m+y_{m+1}z^{m+1}+y_{m+2}z^{m+2}+\ldots
$$
Then we have
$$
y_{i+m}= \alpha^m(a)x'_i+\sum_{k= 0}^{i-1}{}_m\delta_{i-k}(a)x'_k
$$
In the proof of \cite{Zh}, prop.2 we have shown that
$$
z^{i+1-k}b= \alpha^{i+1-k}(b)z^{i+1-k}+\ldots +\sigma (S_i^k\alpha
)(b)z^{i+1}+\ldots
$$
Hence $x'_k= \sigma (S_{i+m-1}^k\alpha )(b)$ for $k< i$. It is easy to
see that $x'_i= {}_m\delta_i(b)$, $x'_0= \alpha^{i+m}(b)$ and $\sigma
(S_{i+m-1}^k\alpha ) = {}_{i+m-k}\delta_k$, which proves
(i).
For $\alpha= id$, by corollary \ref{formuly},
$$
\sigma
(S_{i+m-1}^k\alpha )(b)= \sum_{(j_1,\ldots ,j_l)}
C_{i-k+m}^l\delta_{j_1}\ldots
{\delta_{j_l}}(b),
$$
where $l, j_1, \ldots ,j_l$ were defined in our proposition. This proves
(ii).\\
The proposition is proved.\\
$\Box$
\begin{lemma}{(\cite{Zh}, lemma 3 )}
\label{ozamene}
In the situation of proposition \ref{ooo}
suppose ${}^{(z,u)}_i\delta_j$ is the first map such that
${}^{(z,u)}_i\delta_j(a)\ne 0$ for given $a\in \bar{D}$, $i\in {\mbox{\dbl Z}} \backslash
\{0\}$, i.e.
${}^{(z,u)}_i\delta_1(a)=\ldots= {}^{(z,u)}_i\delta_{j-1}(a)= 0$,
${}^{(z,u)}_i\delta_j(a)\ne 0$ (so we have a map $i\mapsto j(i)$). Then
(i) for $z'= z+u(b)z^{q+1}$, $b\in \bar{D}$ we have
${}^{(z')}\alpha^i (a)={}^{(z)}\alpha^i (a)$,
${}^{(z',u)}_i\delta_{k}(a)={}^{(z,u)}_i\delta_{k}(a)$ for $k<q$ and
$$
{}^{(z',u)}_i\delta_{q}(a)= {}^{(z,u)}_i\delta_{q}(a)
+b'{}^{(z)}\alpha^{q+i}(a)-
{}^{(z)}\alpha^i (a)b',
$$
where $b'=\sum_{k=0}^{i-1}{}^{(z)}\alpha^k(b)$.
(ii) Suppose ${}^{(z)}\alpha^n|_{Z(\bar{D})}= id$, $n\ge 1$, $a\in
Z(\bar{D})$ and \\
${}^{(z,u)}_1\delta_1({}^{(z)}\alpha^k(a))=\ldots=
{}^{(z,u)}_1\delta_{j-1}({}^{(z)}\alpha^k(a))= 0$ for any $k$.
Then for $z'= z+u(b)z^{q+1}$, $b\in \bar{D}$ we have
${}^{(z')}\alpha^i (a)={}^{(z)}\alpha^i (a)$, ${}^{(z',u)}_i\delta_{k}(a)=
{}^{(z,u)}_i\delta_{k}(a)$ for $k<q+j$ and
$$
{}^{(z',u)}_i\delta_{q+j}(a)={}^{(z,u)}_i\delta_{q+j}(a)+
b'{}^{(z)}\alpha^q({}^{(z,u)}_i\delta_{j}(a))-{}^{(z,u)}_i\delta_{j}(a){}^{(z)}\alpha^j(b')+
$$
$$
b'\sum_{k=
1}^{q}{}^{(z)}\alpha^{q-k}({}^{(z,u)}\delta_{j}({}^{(z)}\alpha^{k+i-1}(a))) -
{}^{(z,u)}_i\delta_{j}(a)\sum_{k= 0}^{j-1}{}^{(z)}\alpha^{k}(b),
$$
where $b'=\sum_{k=0}^{i-1}{}^{(z)}\alpha^k(b)$, if $n|q$ or ${}^{(z)}\alpha
(a)=a$. \\
In particular, if ${}^{(z)}\alpha= id$ and $(i,p)=1$, then
$$
{}^{(z',u)}_i\delta_{q+j}(a)={}^{(z,u)}_i\delta_{q+j}(a)+
(q-j){}^{(z,u)}_i\delta_{j}(a)b
$$
(iii) for $z'= u(b)z$, $b\in Z(\bar{D})$, $b\ne 0$ we have
${}^{(z')}\alpha (a)={}^{(z)}\alpha (a)$,
${}^{(z',u)}\delta_{k}(a)={}^{(z,u)}\delta_{k}(a)$ for $k<j$ and
$$
{}^{(z',u)}\delta_{j}(a)={}^{(z,u)}\delta_{j}(a){}^{(z)}\alpha (b^{-1})\cdots
{}^{(z)}\alpha^j(b^{-1})
$$
if $i=1$.
\end{lemma}
{\bf Proof.}
(i)
We have
$$
{z'}^ia{z'}^{-i}= (1+b'z^q+\ldots )z^iaz^{-i}(1+b'z^q+\ldots )^{-1}=
(z^iaz^{-i}+b'z^qz^iaz^{-i}+\ldots )(1-b'z^q+\ldots )=
$$
$$
(z^iaz^{-i}-z^iaz^{-i}b'z^q+\ldots +b'z^qz^iaz^{-i}-\ldots )=
$$
$$
(z^iaz^{-i}-[{}^{(z)}\alpha^i (a)+
{}^{(z,u)}_i\delta_j(a)z^j+\ldots ]b'z^q+b'z^q[{}^{(z)}\alpha^i
(a)+{}^{(z,u)}_i\delta_j(a)z^j+\ldots ]+\ldots
)=
$$
$$
(z^iaz^{-i}-[{}^{(z)}\alpha^i
(a)b'+{}^{(z,u)}_i\delta_j(a){}^{(z)}\alpha^j(b')z^j+\ldots ]z^q+
b'{}^{(z)}\alpha^{q+i}(a)z^q+
\ldots )=
$$
$$
(z^iaz^{-i}+(-{}^{(z)}\alpha^i (a)b'+b'{}^{(z)}\alpha^{q+i}(a))z^q+
\ldots )=
{}^{(z)}\alpha^i (a)+\ldots + {}^{(z,u)}_i\delta_{q-1}(a)z'^{q-1}+
$$
$$
({}^{(z,u)}_i\delta_{q}(a) +
b'{}^{(z)}\alpha^{q+i}(a)-{}^{(z)}\alpha^i (a)b')z'^q+ \ldots
$$
(ii)
Put $c=z'^iz^{-i}-1-b'z^{q+i}$. So, $w(c)>q+i$. Note that
$c{}^{(z)}\alpha^k(a)={}^{(z)}\alpha^k(a)c$, since $n|q$ or ${}^{(z)}\alpha
(a)=a$ and $a\in Z(\bar{D})$.
We have
$$
z'^iaz'^{-i}= (1+b'z^q+c)z^iaz^{-i}(1+b'z^q+c)^{-1}=
(z^iaz^{-i}+b'z^qz^iaz^{-i}+cz^iaz^{-i})(1+b'z^q+c)^{-1}
=
$$
$$
({}^{(z)}\alpha^i (a)+{}^{(z,u)}_i\delta_{j}(a)z^j+\ldots +
{}^{(z,u)}_i\delta_{q+j}(a)z^{q+j}+\ldots +
b'z^q({}^{(z)}\alpha^i (a)+{}^{(z,u)}_i\delta_{j}(a)z^j+\ldots
))(1+b'z^q+c)^{-1}=
$$
$$
({}^{(z)}\alpha^i (a)+b'{}^{(z)}\alpha^{q+i}(a)z^q+
{}^{(z)}\alpha^{i}(a)c+
{}^{(z,u)}_i\delta_{j}(a)z^j+\ldots
+{}^{(z,u)}_i\delta_{q+j}(a)z^{q+j}+\ldots +
$$
$$
b'\sum_{k=
1}^{q}({}^{(z)}\alpha^{q-k}({}^{(z,u)}\delta_{j}({}^{(z)}\alpha^{k+i-1}(a) ) ))
z^{q+j}+
b'({}^{(z)}\alpha^{q}({}^{(z,u)}_i\delta_{j}(a)))
z^{q+j}
+\ldots )(1+b'z^q+c)^{-1}=
$$
$$
{}^{(z)}\alpha^i (a)+[{}^{(z,u)}_i\delta_{j}(a)z^j+\ldots +
{}^{(z,u)}_i\delta_{q+j}(a)z^{q+j}+\ldots +
b'\sum_{k=
1}^{q}({}^{(z)}\alpha^{q-k}({}^{(z,u)}\delta_{j}({}^{(z)}\alpha^{k+i-1}(a) ) ))
z^{q+j}+
$$
$$
b'({}^{(z)}\alpha^{q}({}^{(z,u)}_i\delta_{j}(a)))
z^{q+j}
+\ldots )](1-b'z^q-c+\ldots )=
$$
$$
{}^{(z)}\alpha^i (a)+{}^{(z,u)}_i\delta_{j}(a)z^j+\ldots +
{}^{(z,u)}_i\delta_{q+j}(a)z^{q+j}+\ldots +
b'\sum_{k=
1}^{q}({}^{(z)}\alpha^{q-k}({}^{(z,u)}\delta_{j}({}^{(z)}\alpha^{k+i-1}(a) ) ))
z^{q+j}+
$$
$$
b'({}^{(z)}\alpha^{q}({}^{(z,u)}_i\delta_{j}(a))
z^{q+j}
+\ldots
-{}^{(z,u)}_i\delta_{j}(a){}^{(z)}\alpha^{j}(b')z^{q+j}+\ldots =
$$
$$
{}^{(z)}\alpha^i (a)+\ldots +{}^{(z,u)}_i\delta_{q+j-1}(a)z'^{q+j-1}+
({}^{(z,u)}_i\delta_{q+j}(a)+b'{}^{(z)}\alpha^q({}^{(z,u)}_i\delta_{j}(a))-
{}^{(z,u)}_i\delta_{j}(a){}^{(z)}\alpha^j(b')
$$
$$
+b'\sum_{k=
1}^{q}{}^{(z)}\alpha^{q-k}({}^{(z,u)}\delta_{j}({}^{(z)}\alpha^{k+i-1}(a))) -
{}^{(z,u)}_i\delta_{j}(a)\sum_{k= 0}^{j-1}{}^{(z)}\alpha^{k}(b))z'^{q+j}
+\ldots ,
$$
since $z'^j= z^j+\sum_{k= 0}^{j-1}{}^{(z)}\alpha^{k}(b)z^{q+j}+\ldots $.
(iii)
We have
$$
z'az'^{-1}= bzaz^{-1}b^{-1}=
{}^{(z)}\alpha (a)+ b{}^{(z,u)}\delta_{j}(a){}^{(z)}\alpha^j(b^{-1})z^j+
\ldots=
$$
$$
{}^{(z)}\alpha (a)+ {}^{(z,u)}\delta_{j}(a){}^{(z)}\alpha (b^{-1})\ldots
{}^{(z)}\alpha^j(b^{-1})z'^j+ \ldots ,
$$
since ${}^{(z')}\alpha |_{Z(\bar{D})}={}^{(z)}\alpha |_{Z(\bar{D})}$.\\
$\Box$
\begin{corol}
\label{ozamene3}
In the situation of lemma \ref{ozamene} we have
$$
j=w(xu(a)x^{-1}-u(a)),
$$
where $x\in D$ is any element with $w(x)=i$, if $a\in Z(\bar{D})$, $\alpha
(a)=a$ and $(i,p)=1$, where $p=char D$.
If $i=1$, we will denote $j$ by
$j(u,a)$ or by $i(u,a)$.
\end{corol}
{\bf Proof.} Since for some parameter $z$ we have $x=b(1+x_1z+\ldots )z^i$,
where $b, x_k\in u(\bar{D})$, the proof is easily follows from the proof of
(ii) in lemma \ref{ozamene}.\\
$\Box$
In the sequel we will need the following definition.
\begin{defi}
Let
$(\alpha ,\beta )$ be endomorphisms of a division algebra $D$. A map
$\delta :$ $D\rightarrow D'$, where $D\subset D'$ are algebras, is called a
$(\alpha ,\beta )$-derivation if it is linear and satisfy the following
identity
$$
\delta (ab)= \delta (a)\alpha (b)+\beta (a)\delta (b)
$$
where $a,b\in D$.\\
We will say that $(\alpha ,1)$-derivation is an
$\alpha$-derivation.
\end{defi}
\begin{lemma}{(cf. \cite{Zh}, lemma 4)}
\label{lemma2}
Let $\delta$ be an $(\alpha ,\beta )$-derivation of an arbitrary division
algebra $D$ such that
$\alpha ,\beta$ preserve $Z(D)$ and $\alpha|_{Z(D)}\ne \beta|_{Z(D)}$.
Then $\delta$ is an inner derivation, i.e. there exists $d\in D$ such
that
$$
\delta (a)= d\alpha (a)-\beta (a)d
$$
for all $a\in D$.
\end{lemma}
{\bf Proof.}
Put $d= \delta (a)(a^{\alpha}-a^{\beta})^{-1}$, where $a\in Z(D)$ is any
element such
that $\alpha (a)\ne \beta (a)$. Put $\delta_{in}(x)=
d\alpha (x)-\beta (x)d$. We claim that $\delta= \delta_{in}$. Indeed,
consider the map $\bar{\delta}= \delta -\delta_{in}$. It is an
$(\alpha ,\beta )$-derivation. Take arbitrary $b\in D$. Then
$\bar{\delta}(ab)= \bar{\delta}(ba)$. But we have
$$\bar{\delta}(ab)= \bar{\delta}(a)\alpha (b)+\beta (a)\bar{\delta}(b)=
\beta (a)\bar{\delta}(b),$$
and
$$\bar{\delta}(ba)= \bar{\delta}(b)\alpha (a)+\beta (b)\bar{\delta}(a)=
\alpha (a)\bar{\delta}(b)$$
Therefore, $\bar{\delta}(b)= 0$ for any $b$.\\
$\Box$
\begin{prop}{(cf. \cite{Zh}, lemma 10)}
\label{X}
Let $D$ be a splittable division algebra. Let $n=Gal
(Z(\bar{D})/\overline{Z(D)})$.
There exists a
parameter $z'$ such that
$$
{}^{(z',u)}_m\delta_j=0
$$
if $n\not | j$.
\end{prop}
{\bf Proof.} Since for $n=1$ there is nothing to prove, we will assume that
$n>1$. Let $z$ be some fixed parameter. By \cite{JW}, prop. 1.7
${}^{(z)}\alpha |_{Z(\bar{D})}$ has order $n$.
By proposition \ref{flyii}, ${}^{(z,u)}\delta_1$ is a
$({}^{(z)}\alpha^2,{}^{(z)}\alpha
)$-derivation.
Since $n>1$, ${}^{(z)}\alpha^2|_{Z(\bar{D})}\ne {}^{(z)}\alpha
|_{Z(\bar{D})}$. Therefore, by lemma \ref{lemma2},
${}^{(z,u)}\delta_1$ is an inner derivation and
${}^{(z,u)}\delta_1(a)=
d{}^{(z)}\alpha^2(a)- {}^{(z)}\alpha (a)d$, $a\in \bar{D}$.
Put $z_1= z-u(d)z^2$. By lemma \ref{ozamene}, (i) we have for any $a\in
\bar{D}$
${}^{(z_1,u)}\delta_1(a)=0$ and ${}^{(z)}\alpha (a)={}^{(z_1)}\alpha (a)$. So,
${}^{(z_1,u)}\delta_1=0$ and ${}^{(z)}\alpha ={}^{(z_1)}\alpha$.
By proposition \ref{flyii}, ${}^{(z_1,u)}\delta_2$ is a
$({}^{(z_1)}\alpha^3,{}^{(z_1)}\alpha )$-derivation.
If $n\ne 2$ then it is inner and we can apply lemma
\ref{ozamene}.
By induction we get that there exists a parameter $z_{n-1}$ such
that
${}^{(z_{n-1},u)}\delta_j=0$ for $j< n$ and ${}^{(z)}\alpha
={}^{(z_{n-1})}\alpha$. It is easy to see that then
${}^{(z_{n-1},u)}_m\delta_j=0$ for $j< n$ and all $m\in{\mbox{\dbl Z}}$.
Note that ${}^{(z_{n-1},u)}\delta_n$ is a
$({}^{(z_{n-1})}\alpha^{n+1},{}^{(z_{n-1})}\alpha )=
({}^{(z_{n-1})}\alpha ,{}^{(z_{n-1})}\alpha
)$-derivation, i.e. ${{}^{(z_{n-1},u)}\delta_{n}}{}^{(z_{n-1})}\alpha^{-1}$ is
a derivation.
Note that ${{}^{(z_{n-1},u)}\delta_{n+1}}$ is a
$({}^{(z_{n-1})}\alpha^2,{}^{(z_{n-1})}\alpha )$-derivation.
This follows by proposition \ref{flyii}, since ${}^{(z_{n-1},u)}_m\delta_j=0$
for $j< n$ and all $m\in{\mbox{\dbl Z}}$.
So, by lemma \ref{lemma2}, ${}^{(z_{n-1},u)}\delta_{n+1}$ is an inner
derivation.
Using lemma \ref{ozamene}, (i) with $
z_{n+1}= z_{n-1}+bz_{n-1}^{n+2}$ for an appropriate $b$, we have
${}^{(z_{n+1},u)}\delta_j=0$ for $j< n+2$, $n\not |j$ and ${}^{(z)}\alpha
={}^{(z_{n+1})}\alpha$. Moreover, ${}^{(z_{n+1},u)}_m\delta_j=0$ for $j< n+2$,
$n\not |j$ and all $m\in{\mbox{\dbl Z}}$. This easily follows from lemma \ref{triviall}.
By induction we can assume that there exists a
parameter $z_k$ such that
${}^{(z_{k},u)}_m\delta_j=0$ for $j< k+1$, $n\not |j$ and all $m\in{\mbox{\dbl Z}}$, and
${}^{(z)}\alpha ={}^{(z_{k})}\alpha$.
So, by proposition \ref{flyii}, if $n\not | k+1$, then
${{}^{(z_{k},u)}\delta_{k+1}}$ is an inner
$({}^{(z_{k})}\alpha^{k+2},{}^{(z_{k})}\alpha )$-derivation. And
if $n | k+1$, we can apply the same arguments and conclude that
${}^{(z_{k},u)}\delta_{k+2}$
is a $({}^{(z_{k})}\alpha^{k+2},{}^{(z_{k})}\alpha )$-derivation.
Therefore, by lemma \ref{ozamene} there exists a parameter $z_{k+1}=
z_k+bz_k^{k+2}$
($z_k+bz_k^{k+3}$ if $n | k+1$) such that
${}^{(z_{k+1},u)}_m\delta_j=0$ for $j< k+2$, $n\not |j$ and all $m\in{\mbox{\dbl Z}}$, and
${}^{(z)}\alpha ={}^{(z_{k+1})}\alpha$ (or ${}^{(z_{k+1},u)}_m\delta_j=0$ for
$j< k+3$, $n\not |j$ and all $m\in{\mbox{\dbl Z}}$, and ${}^{(z)}\alpha
={}^{(z_{k+1})}\alpha$ if $n | k+1$).
Since $z_{l+1}=
(1+b_lz_l^{k_l})z_l$ for every $l$, the sequence ${\{ z_l\}}_{l=
1}^{\infty}$ converges
in $D$, which completes the proof of the proposition.\\
$\Box$
\begin{lemma}
\label{(5)}
Let $D$ be a splittable division algebra
as in proposition \ref{ooo}, of characteristic $p>0$.
Let $t\in Z(\bar{D})$ be an element such that $\alpha (t)=t$.
Let $j=i(u,t)$ be the minimal positive integer such that
${}^{(z,u)}\delta_j|_{{\mbox{\sdbl F}}_p(t)}\ne
0$ (see corollary \ref{ozamene3}), and we assume $j<\infty$. Then the maps ${}^{(z,u)}_n\delta_m$, $kj\le
m<(k+1)j$,
$k\in \{1,\ldots ,p-1 \}$ satisfy the following properties:
i) there exist elements $c_{n,m,k}\in\bar{D}$ such that
$$
{}^{(z,u)}_n\delta_m|_{{\mbox{\sdbl F}}_p(t)}=c_{n,m,1}\delta +\ldots +c_{n,m,k}\delta^k,
$$
where $\delta :{\mbox{\dbl F}}_p(t)\rightarrow {\mbox{\dbl F}}_p(t)$ is a derivation such that
$\delta (t)= 1$, and
$$c_{n,kj,k}= (k!)^{-1}{}^{(z,u)}_n\delta_j(t){}^{(z,u)}_{n+j}\delta_j(t)\ldots
{}^{(z,u)}_{n+(k-1)j}\delta_j(t).
$$
ii) Let $\zeta =ord ({}^{(z)}\alpha |_{Z(\bar{D})})$. Then $\zeta |j$ and \\
$c_{n,kj,k}\ne 0$ if $(n, j )=1$ and
${}^{(z)}\alpha ({}^{(z,u)}\delta_j(t))\ne {}^{(z,u)}\delta_j(t)$;\\
$c_{n,kj,k}\ne 0$ if ${}^{(z)}\alpha ({}^{(z,u)}\delta_j(t))= {}^{(z,u)}\delta_j(t)$ and
$n , (n+j) , \ldots , (n+(k-1)j) \ne 0\mbox{\quad mod\quad}p$.
If ${}^{(z)}\alpha =id$, then $c_{n,kj,k}\ne 0$ iff
$n , (n+j) , \ldots , (n+(k-1)j) \ne 0\mbox{\quad
mod\quad}p$.
\end{lemma}
{\bf Proof.} i) The proof is by induction on $k$. Let $a,b\in {\mbox{\dbl F}}_p(t)$. For
$k= 1$, by proposition \ref{flyii}, (ii) we have
$$
{}_n\delta_m(ab)= {}_n\delta_m(a)b+a{}_n\delta_m(b)
$$
because all the maps $\delta_q$, $q<j$ are equal to zero on ${\mbox{\dbl F}}_p(t)$.
Hence, ${}_n\delta_m$ is a derivation on ${\mbox{\dbl F}}_p(t)$, ${}_n\delta_m|_{{\mbox{\sdbl F}}_p(t)}=
c_{n,m,1}\delta$ and $c_{n,j,1}=
{}_n\delta_j(t)$.
For arbitrary $k$, by proposition \ref{flyii}, (i) and by the induction
hypothesis we have
$$
{}_n\delta_m(t^q)= q{}_n\delta_m(t)t^{q-1}+{}_n\delta_j(t)(\sum_{l=
0}^{q-2}(c_{n+j,m-j,1}\delta +\ldots
+c_{n+j,m-j,k-1}\delta^{k-1})(t^{q-1-l})t^l)+
$$
\begin{equation}
\label{(**)}
\ldots +{}_n\delta_{m-j}(t)(\sum_{l= 0}^{q-2}(c_{m-j+n,m-s,1}\delta
)(t^{q-1-l})t^l).
\end{equation}
Therefore, ${}_n\delta_m(t^p)= 0$,
because $k\le p-1$ and $\sum_{l=
0}^{p-2}\delta^i(t^{p-1-l})t^l= 0$ for $i\le p-2$. Hence,
${}_n\delta_m|_{{\mbox{\sdbl F}}_p(t)}= c_{n,m,1}\delta +\ldots +c_{n,m,p-1}\delta^{p-1}$
and we
only have to show that $c_{n,m,q}= 0$ for $q>k$.
Using (\ref{(**)}) we can calculate $c_{n,m,j}$. We have
$$
c_{n,m,1}= {}_n\delta_m(t);
$$
$$
c_{n,m,2}= \frac{1}{2!}({}_n\delta_m(t^2)-2c_{n,m,1}t)= \frac{1}{2}
({}_n\delta_j(t)(c_{n+j,m-j,1}\delta (t))+\ldots
+{}_n\delta_s(t)(c_{s+n,m-s,1}\delta (t)))
$$
$$
\ldots
$$
$$
c_{n,m,q}= \frac{1}{q!}({}_n\delta_j(t)(\sum_{l=
0}^{q-2}c_{n+j, m-j, q-1}\delta^{q-1}(t^{q-1-l})t^l)+ \ldots
$$
$$
+{}_n\delta_{m-(q-1)j}(t)(\sum_{l= 0}^{q-2}c_{m+n-(q-1)j, (q-1)j,
q-1}\delta^{q-1}
(t^{q-1-l})t^l))
$$
\begin{equation}
\label{recurrent}
= \frac{1}{q}({}_n\delta_j(t)c_{n+j, m-j, q-1}+ \ldots
+{}_n\delta_{m-(q-1)j}(t)c_{m+n-(q-1)j, (q-1)j, q-1})
\end{equation}
Hence, $c_{n, m, k+1}= \ldots = c_{n, m, p-1}= 0$ and
$$c_{n, kj, k}= q^{-1}
{}_n\delta_j(t)c_{n+j, kj-j, k-1}=
(k!)^{-1}{}^{(z,u)}_n\delta_j(t){}^{(z,u)}_{n+j}\delta_j(t)\ldots {}^{(z,u)}_{n+(k-1)j}\delta_j(t).
$$
ii) Let us prove first that $\zeta$ divide $i$. For, if $i$ is not divisible by $\zeta$, we have, by proposition \ref{flyii},
$$
{}^{(z,u)}\delta_j(tx)={}^{(z,u)}\delta_j(t){}^{(z)}\alpha^{j+1}(x)+{}^{(z)}\alpha
(t)
{}^{(z,u)}\delta_j(x)={}^{(z,u)}\delta_j(xt)=
$$
$$
{}^{(z,u)}\delta_j(x){}^{(z)}\alpha^{j+1}
(t) +{}^{(z)}\alpha (x){}^{(z,u)}\delta_j(t),
$$
where $x\in Z(\bar{D})$, $\alpha (x)\ne x$. But then
${}^{(z)}\alpha^{j+1}(x)={}^{(z)}\alpha (x)$, a contradiction.
If ${}^{(z)}\alpha =id$, the same arguments show that
${}^{(z,u)}\delta_j(t)\in Z(\bar{D})$.
If $x\in \bar{D}$ is an arbitrary element, this formulae shows ${}^{(z)}\alpha^{j}$ is an inner automorphism $ad({}^{(z,u)}\delta_j(t)^{-1})$. Therefore,
${}^{(z)}\alpha^{j}({}^{(z,u)}\delta_j(t))={}^{(z,u)}\delta_j(t)$.
Assume ${}^{(z)}\alpha ({}^{(z,u)}\delta_j(t))\ne {}^{(z,u)}\delta_j(t)$. It's clear then that
$$
{}^{(z,u)}_{n+qj}\delta_j(t)=\sum_{l=0}^{n+qj-1}{}^{(z)}\alpha^l ({}^{(z,u)}\delta_j(t))\ne 0
$$
if $(n, j )=1$. So, $c_{n,kj,k}\ne 0$ by (i) in this case.
If ${}^{(z)}\alpha ({}^{(z,u)}\delta_j(t))= {}^{(z,u)}\delta_j(t)$, then
${}^{(z,u)}_{n+qj}\delta_j(t)=(n+qj){}^{(z,u)}\delta_j(t)\ne 0$ iff $p$ does not divide
$(n+qj)$. So, by (i) $c_{n,kj,k}\ne 0$ in this case iff $n , (n+j) , \ldots , (n+(k-1)j) \ne 0\mbox{\quad mod\quad}p$.
The lemma is proved.\\
$\Box$
\begin{lemma}
\label{ppp}
Let $D$ be a splittable division algebra as in lemma \ref{(5)}. Let $s\in
Z(\bar{D})$ be an element such that $\alpha (s)=s$.
Let $i=i(u,s)$ be the minimal positive integer such that
${}^{(z,u)}\delta_i(s)\ne 0$ (see corollary \ref{ozamene3}).
If $p|i$, then for any positive integral $k$ there exists a map
${}^{(z,u)}\delta_{j(k)}$
such that ${}^{(z,u)}\delta_{j(k)}(s^{p^k})\ne
0$.
\end{lemma}
{\bf Proof.} We claim that ${}^{(z,u)}\delta_{p^qi}$ is the first map such that
${}^{(z,u)}\delta_{p^qi}|_{{\mbox{\sdbl F}}_p(s^{p^q})}\ne 0$. The proof is by induction on
$q$. For
$q= 0$, there is nothing to prove. For arbitrary $q$, put $t=
s^{p^{q-1}}$. By proposition \ref{flyii} we have
$$
\delta_{p^qi}(t^p)= \delta_{p^{q-1}i}(t)\sum_{r=
0}^{p-2}{}_{1+p^{q-1}i}\delta_{p^{q-1}i(p-1)}(t^{p-1-r})t^r+ \sum_{l=
p^{q-1}i+1}^{p^qi-1}\delta_l(t)\sum_{r=
0}^{p-2}{}_{1+l}\delta_{p^qi-l}(t^{p-1-r})t^r
$$
By induction and lemma \ref{(5)}, ${}_{1+l}\delta_{p^qi-l}|_{{\mbox{\sdbl F}}_p(t)}=
c_{1+l,p^qi-l,1}\delta +\ldots +c_{1+l,p^qi-1,p-2}\delta^{p-2}$ for
$l>p^{q-1}i$. Therefore,
$\sum_{r=
0}^{p-2}{}_{1+l}\delta_{p^qi-l}(t^{p-1-r})t^r= 0$. By lemma \ref{(5)}, (ii),
${}_{1+p^{q-1}i}\delta_{p^{q-1}i(p-1)}|_{{\mbox{\sdbl F}}_p(t)}=
c_{1+p^{q-1}i, p^{q-1}i(p-1), 1}\delta +\ldots
+c_{1+p^{q-1}i, p^{q-1}i(p-1), p-1}\delta^{p-1}$ with
$c_{1+p^{q-1}i, p^{q-1}i(p-1), p-1}\ne 0$. Hence, $\delta_{p^qi}(t^p)=
-c_{1+p^{q-1}i, p^{q-1}i(p-1), p-1}\delta_{p^{q-1}i}(t)\ne 0$.
The same arguments show that ${}^{(z,u)}\delta_j(t^p)= 0$ for $j<p^qi$. So,
${}^{(z,u)}\delta_{p^qi}$ is the first non-zero map on ${\mbox{\dbl F}}_p(s^{p^q})$. \\
$\Box$\\
\begin{lemma}
\label{svva}
Let $D$ be a splittable division algebra. Let $z$ be a fixed parameter and
${}^{(z)}\alpha =id$, let $u$ be some fixed embedding
$u:\bar{D}\hookrightarrow D$.
Let ${}^{(z,u)}\delta_{i}$, $i\in {\mbox{\dbl N}}\cup\infty$ be the first non-zero map on
$\bar{D}$. Assume $(i,p)=1$, where $p=char D$.
Let ${}^{(z,u)}\delta_j$, $j>i$, $j\in {\mbox{\dbl N}}\cup\infty$ be the first map such
that
${}^{(z,u)}\delta_j\ne
0$ if $j$ is not divisible by $i$ and ${}^{(z,u)}\delta_j\ne
c_{j/i}{}^{(z,u)}\delta_i^{j/i}$ for some
$c_{j/i}\in \bar{D}$ otherwise. Then
a) for $k< p=char D$ (arbitrary $k$ if $char D=0$) we have
${}^{(z,u)}\delta_{ki}=
c_{k}{}^{(z,u)}\delta_i^{k}$, where
\begin{equation}
\label{(188)}
c_{k}= \frac{(i+1)\ldots (i(k-1)+1)}{k!},
\end{equation}
if $ki<j$.
b) if condition (\ref{(188)}) is satisfied for any $k$ with $ki<j$, then
${}^{(z,u)}_{-i}\delta_{q}=0$ for $i<q<j$ and ${}^{(z,u)}_{-i}\delta_{j}$ is a
derivation.
\end{lemma}
{\bf Remark.} We will call the number $i(u,z)=\min_{a\in
\bar{D}}\{w(zu(a)z^{-1}-u(a))\}$ defined in this lemma {\it a local height}.
The number $i=i(z,u)$ in lemma coinside with the level of $D$ defined in
\cite{Sa} if $D$ has index $p=char D$ and $D$ is splittable. As it follows
from lemmas \ref{ozamene}, \ref{ozamene2} (see below), $i(z,u)$ does not
depend on $z,u$ in this case. Corollary \ref{ozamene3} completes then the
proof that it coinside with the level defined by Saltman in the case $D$ is
splittable. This number will play an important role in this work. It was one
of the important parameters in \cite{Zh}. Recall the definition of {\it
level}: $h(D)=\min \{w(ab-ba)-w(a)-w(b)\}$.
{\bf Proof.} If we compare coefficients in formulae for $\delta_{ki}(ab)$ from
proposition \ref{flyii} with coefficients in formulae for $\delta_i^k(ab)$
multiplied by $c_k$, we must have
$$
c_kk=((k-1)i+1)c_{k-1},
$$
where from follows a).
>From the other hand side, if ${}_{-i}\delta_q$, $q>i$ is the first nonzero
map after
${}_{-i}\delta_i$, it must be a derivation by proposition \ref{flyii}, (i).
Note that in characterictic zero case this can happens only if $q\ge j$, because
a map $c\delta_i^k$ can not be a derivation if $k>1$, which proves b) in
this case.
Since the maps $\delta_q$ are uniquely defined, by lemma \ref{triviall}, by
the maps ${\tilde{\delta}}_l$, $l\le q$, and the maps ${\tilde{\delta}}_q$ are
uniquely defined by the maps
${}_{-i}{{\delta}}_l$, $l\le q$, and ${}_{-i}{{\delta}}_q$ are linear
combinations
of ${{\delta}}_l$, $l\le q$ with integer coefficients, we see that b) holds in
arbitrary characteristic.\\
$\Box$
{\bf Remark.} So we see that the maps ${}_i\delta_q$ in this lemma satisfy the
same identities as $\delta_{q/i}$. This can be thought of as a possible
reduction from level $i$ to level $1$.
\begin{defi}
\label{lemdef}
Let $D$ be a splittable division algebra. Let $u$ be some fixed embedding
$u:\bar{D}\hookrightarrow D$.
Let $s\in Z(\bar{D})$ be an element such that $\alpha (s)=s$.
Let $i=i(u,s)$ be the minimal positive integer such that
${}^{(z,u)}\delta_i(s)\ne 0$ ( corollary \ref{ozamene3} shows that $i$ does
not depend on
$z$). Assume $(i,p)=1$, where $p=char D$.
Define
$$
d(u,s)=\max_{z}\{w(z^{-i}u(s)z^i-u(s)-u({}^{(z,u)}_{-i}\delta_i(s))z^i)\} \in
{\mbox{\dbl N}}\cup\infty ,
$$
\end{defi}
As we can see from lemma \ref{svva} b), $d(u,s)$ can be interpreted under some conditions as the number $j$ there. So, this definition was motivated by this lemma.
\begin{lemma}
\label{vtorinv}
In the definition above for $p=char D>0$ and
${}^{(z)}\alpha |_{Z(\bar{D})}=id$ we have
i) ${d(u,s)}=2i \mbox{\quad mod\quad} p$ if $d(u,s)<\infty$;
ii) If ${}^{(z,u)}_{-i}\delta_i(s)\ne 0$, the map
${}^{(z,u)}_{-i}\delta_{d(u,s)+(p-1)i}$ is the first map such that
${}^{(z,u)}_{-i}\delta_{d(u,s)+(p-1)i}(s^p)\ne 0$ for any parameter $z$. In
particular, if $d(u,s)=\infty$, $[u(s^p), z^i]=0$.
\end{lemma}
{\bf Proof.}
(ii) Let ${}^{(z,u)}_{-i}\delta_{\kappa}$ be the first map such that
${}^{(z,u)}_{-i}\delta_{\kappa}(s^p)\ne 0$. By corollary \ref{ozamene3}
$\kappa$ does not depend on $z$. By the same reason,
${}^{(z,u)}_{-i}\delta_i$ is the first map such that
${}^{(z,u)}_{-i}\delta_i(s)\ne 0$ for any $z$.
Put $w:= d(u,s)+(p-1)i$ and fix $u,z$. By proposition \ref{flyii} we have
$$
{}_{-i}{{\delta}}_w(s^p)=
{}_{-i}{{\delta}}_{d(u,s)}(s)\sum_{q= 0}^{p-2}
{}_{d(u,s)-i}{{\delta}}_{(p-1)i}(s^{p-1-q})s^q+
$$
$$
\sum_{k= d(u,s)+1}^{w-1} {}_{-i}{{\delta}}_k(s)\sum_{q= 0}^{p-2}
{}_{k-i}{{\delta}}_{w-k}(s^{p-1-q})s^q
$$
By lemma \ref{(5)}, ${}_{k-i}{{\delta}}_{w-k}|_{{\mbox{\sdbl F}}_p(s)}=
c_{k-i,w-k,1}\delta
+\ldots +c_{k-i,w-k,p-2}\delta^{p-2}$ for $w-k<(p-1)i$ and
${}_{d(u,s)-i}{{\delta}}_{(p-1)i}|_{{\mbox{\sdbl F}}_p(s)}= c_{d(u,s)-i,(p-1)i,1}\delta
+\ldots
+c_{d(u,s)-i,(p-1)i,p-1}\delta^{p-1}$ with $c_{d(u,s)-i,(p-1)i,p-1}\ne 0$ if $d(u,s)-i=i \mbox{\quad mod\quad} p$. Indeed, as we have shown in the proof of lemma \ref{(5)}, (ii),
the order $n$ of the automorphism ${}^{(z)}\alpha$ on ${}^{(z,u)}\delta_i(s)$ must divide $i$, so $(n,p)=1$. Now we have two possibilities: $n{\not |}d(u,s)$ and $n|d(u,s)$.
In the first case we can repeat the arguments to the first assertion in lemma \ref{(5)}, (ii) to show that $c_{d(u,s)-i,(p-1)i,p-1}\ne 0$. In the second case we have
${}_{d(u,s)-i+qi}\delta_i(s)= (d(u,s)-i+qi)/i{}_i\delta_i(s)\ne 0$ if $d(u,s)-i+qi$ is not divided by $p$. So, by lemma \ref{(5)}, (i) $c_{d(u,s)-i,(p-1)i,p-1}\ne 0$ iff $d(u,s)-i=i \mbox{\quad mod\quad} p$ in this case.
Hence,
$$
{}_{-i}{{\delta}}_w(s^p)=
-{}_{-i}{{\delta}}_{d(u,s)}(s)c_{d(u,s)-i,(p-1)i,p-1}\ne 0
$$
if $d(u,s)-i=i \mbox{\quad mod\quad} p$.
This also shows that ${}_{-i}{{\delta}}_w$ is the {\it first} map
such that ${}_{-i}{{\delta}}_w|_{{\mbox{\sdbl F}}_p(s^p)}\ne 0$ if $d(u,s)-i=i \mbox{\quad
mod\quad} p$.
i) By Skolem-Noether theorem there exists a parameter $z'$ in $D$ such that
${}^{(z')}\alpha =id$. Put
$$d'(u,z',s)= w(z'^{-j}u(s)z'^{j}-u(s)-u({}^{(z',u)}_{-i}\delta_i(s))z'^i).$$
Since ${}^{(z')}\alpha =id$, the map
${}^{(z',u)}\delta_i$ is the first map such that
${}^{(z',u)}_{-i}\delta_i(s)\ne 0$. If
$d'(u,z',s)\ne 2i \mbox{\quad mod\quad} p$, we can find a parameter $z''$ such
that
$d'(u,z'',s)>d'(u,z',s)$ using lemma \ref{ozamene}, (ii). Continuing this
procedure, we find a parameter $z$ such that $d'(u,z,s)=2i \mbox{\quad
mod\quad} p$ or $d'(u,z,s)=\infty$.
Using arguments from ii) we get that the map
${}^{(z,u)}_{-i}\delta_{d'(u,z,s)+(p-1)i}$ is the first map such that
${}^{(z,u)}_{-i}\delta_{d'(u,z,s)+(p-1)i}(s^p)\ne 0$ for the parameter $z$. As
it was noted in the beginning of the proof, the number $\kappa =d'(u,z,s)+(p-1)i$ does not depend on the parameter.
Since $d'(u,z,s)\le d(u,s)$, we get $d'(u,z,s)= d(u,s)$. For, otherwise we can repeat the arguments from (ii) and conclude that ${}^{(z,u)}_{-i}\delta_{d(u,s)+(p-1)i}(s^p)= 0$, a contradiction.
The lemma is proved.\\
$\Box$
It would be interesting to know more about a behaviour of
${}^{(z,u)}_m\delta_j$ with respect to the embedding $u$. We will give an
answer in one special case, namely, when $\bar{D}=Z(\bar{D})$ and
$Z(\bar{D})/\overline{Z(D)}$ is a simple extension.
\begin{lemma}
\label{simple}
Let $D$ be a division algebra such that $char D=p>0$,
$\bar{D}=Z(\bar{D})$, $Z(\bar{D})$ is not perfect and
$Z(\bar{D})/\overline{Z(D)}$ is a simple extension (so, $D$ is splittable).
Let $\bar{u}$ be a primitive element of the extension
$Z(\bar{D})/\overline{Z(D)}$ such that $\bar{u}\notin (Z(\bar{D}))^p$ and let
$u$ be any lift of $\bar{u}$ in $D$.
Then there exists an embedding
$u:\bar{D}\hookrightarrow D$ such that $u(\bar{u})=u$ and any map
${}^{(z,u)}_m\delta_j$ is uniqely defined by the values
${}^{(z,u)}_m\delta_j(u^q)$ or, equivalently, by the values
${}^{(z,u)}_l\delta_k(u)$, $k\le j$.
In particular, if ${}^{(z,u)}_m\delta_k(u)=0$ for $k\le j$, then
${}^{(z,u)}_m\delta_j=0$.
\end{lemma}
{\bf Proof.} Consider a field $Z(D)(u)$. It is a complete discrete valued
field as a finite extension of $Z(D)$. By classical Cohen theorem, there
exists an embedding $\overline{Z(D)(u)}=\bar{D}\hookrightarrow Z(D)(u)\subset
D$.
By \cite{Co}, lemmas 11,12 the embedding is completely defined by a $p$-basis
$\Gamma$ of the field $\overline{Z(D)(u)}$. Namely, for any lift $G$ of a
given $p$-basis $\Gamma$ there exists an embedding $s$ such that $G\subset
s(\overline{Z(D)(u)})$.
Let's show that there exists a $p$-basis $\Gamma$ of the field $\bar{D}$ such
that $\bar{u}\in \Gamma$ and $\Gamma\ni \gamma\in \overline{Z(D)}$ if
$\gamma\ne \bar{u}$.
Consider a set of all non-void sets $\Gamma'$ of elements $\gamma_{\tau}\in
\bar{D}$ satisfying the following property:\\
A) $\bar{u}\in\Gamma'$, $\Gamma' \ni \gamma\in \overline{Z(D)}$ if $\gamma\ne
\bar{u}$ and $[{\bar{D}}^p(\gamma_1,\ldots ,\gamma_r):\bar{D}^p]=p^r$ for any
$r$ distinct elements of $\Gamma'$.
This set is not void, since it contains the set $\Gamma'=\{\bar{u}\}$. By
Zorn's lemma,
there exists a maximal set $\Gamma$ satisfying A). Then
$\bar{D}=\bar{D}^p(\Gamma )$. Indeed, since $\overline{Z(D)}^p(\bar{u})\subset
\bar{D}^p(\Gamma )$, it suffice to show that any element from
$\overline{Z(D)}$ lies in $\bar{D}^p(\Gamma )$. Suppose $a\in
\overline{Z(D)}$, $a\notin \bar{D}^p(\Gamma )$. Then the set $\Gamma'=\{a\cup
\Gamma\}$ satisfy A), a contradiction with maximality of $\Gamma$.
Now, we can take a lift of $\Gamma$ in the following way. We take $u$ as a
lift of $\bar{u}$, and we take lifts of all other elements in $Z(D)$. This
lift defines an embedding $u:\bar{D}\hookrightarrow D$.
Let us show that any map ${}^{(z,u)}_m\delta_j$ (for some fixed $z$) is
uniqely defined by the values ${}^{(z,u)}_l\delta_k(u)$, $k\le j$. We have
$u(\bar{D})=u(\overline{Z(D)})(u)$ and any element $a\in u(\bar{D})$ can be
represented as a polynomial in finite number of elements from $\Gamma$ with
coefficients from $u(\bar{D})^{p^k}$ for any $k>0$.
Note that for any $j$ there exists $k>0$ such that for any $b\in
\overline{Z(D)}^{p^k}$ ${}^{(z,u)}_l\delta_q(b)=0$ for all $q\le j$ and all
$l$. Indeed, assume ${}^{(z,u)}_1\delta_q(b)\ne 0$ for some $q\le j$, $b\in
\overline{Z(D)}^{p^k}$ and ${}^{(z,u)}_l\delta_s(c)= 0$ for all $l$, all
$c\in \overline{Z(D)}^{p^k}$ and all $s<q$. Then, since
${}^{(z)}\alpha |_{\overline{Z(D)}}=id$ and by proposition \ref{flyii},
${}^{(z,u)}_l\delta_s(b^p)= 0$ for all $b\in \overline{Z(D)}^{p^k}$, all $l$
and all $s\le q$.
Now, since $u(\bar{D})^{p^k}=u(\overline{Z(D)})^{p^k}(u^{p^k})$, any element
$a\in u(\bar{D})$ can be represented as a polynomial in finite number of
elements from $\Gamma$ with coefficients from $u(\overline{Z(D)})^{p^k}$.
Since all elements except $u$ in $\Gamma$ belong to the center $Z(D)$, the
value of ${}^{(z,u)}_m\delta_j(a)$
is uniqely determined by the values ${}^{(z,u)}_m\delta_j(u^l)$ that are
uniqely defined, by proposition \ref{flyii}, by the values
${}^{(z,u)}_l\delta_k(u)$, $k\le j$. \\
$\Box$
{\bf Remark} In the case $Z(\bar{D})$ perfect field there is only one
embedding $u$, which is compatible with the embedding
$\overline{Z(D)}\hookrightarrow Z(D)$. So, the assertion of lemma is easy in
this case.
\begin{lemma}{(cf. \cite{Zh}, lemma 8)}
\label{ozamene2}
In the situation of lemma \ref{simple}
suppose
${}^{(z,u)}_m\delta_1=\ldots = {}^{(z,u)}_m\delta_{j-1}= 0$,
${}^{(z,u)}_m\delta_j\ne 0$. Let $n$ be the order of ${}^{(z)}\alpha$. Then
(i) for $u'= u+bz^q$, $b\in u(\bar{D})$, $n|q$ we have
${}^{(z,u')}_m\delta_l={}^{(z,u)}_m\delta_l$, $l<q$ and
$${}^{(z,u')}_m\delta_q (\bar{u})= {}^{(z,u)}_m\delta_q
(\bar{u})+{}^{(z)}\alpha^m
(\bar{b})
-\frac{\partial}{\partial \bar{u}}({}^{(z)}\alpha^m (\bar{u}))\bar{b},$$
where the derivative is taken in the field $\bar{D}=\bar{D}^p(\Gamma )$.
(ii) Suppose ${}^{(z)}\alpha = id$. Then for $u'= u+bz^q$, $b\in
u(\bar{D})$ we have
${}^{(z,u')}_m\delta_l={}^{(z,u)}_m\delta_l$, $l<q+j$ and
$${}^{(z,u')}_m\delta_{q+j}(\bar{u})= {}^{(z,u)}_m\delta_{q+j}(\bar{u})+
{}^{(z,u)}_m\delta_{j}(\bar{b})
-\frac{\partial }{\partial \bar{u}}({}^{(z,u)}_m\delta_j (\bar{u}))\bar{b}, $$
where the derivative is taken in the field $\bar{D}=\bar{D}^p(\Gamma )$.
(iii) Suppose ${}^{(z)}\alpha= id$. Let $\bar{u'}\in \bar{D}$ be any
primitive element of the extension $\bar{D}/\overline{Z(D)}$ satisfying the
conditions of lemma
\ref{simple}, and let $u'\in D$ be any lift of $\bar{u'}$.
Then we have
${}^{(z,u')}_m\delta_l={}^{(z,u)}_m\delta_l$, $l<j$ and
$$
{}^{(z,u')}_m\delta_{j}(\bar{u'})= {}^{(z,u)}_m\delta_{j}(\bar{u})
\frac{\partial }{\partial \bar{u}}(\bar{u'}),
$$
where the derivative is taken in the field $\bar{D}=\bar{D}^p(\Gamma )$.
\end{lemma}
{\bf Proof.}
First of all, let's note that there exists $k\in {\mbox{\dbl N}}$ such that for any $a\in
\overline{Z(D)}^{p^k}$ holds $u(a)-u'(a)=0 \mbox{\quad mod\quad} M_D^{q+1}$,
where $u'$ is any another embedding, $q\in {\mbox{\dbl N}}$ is any given number.
Indeed, assume for any $c\in \overline{Z(D)}^{p^s}$ holds $u(c)-u'(c)=0
\mbox{\quad mod\quad} M_D^{l}$, i.e.
$u(c)=u'(c)+c_lz^l+\ldots$, where $c_l\in u'(\bar{D})$. Then
$u(c^p)=(u(c))^p=
(u'(c))^p+pu'(c)^{p-1}c_lz^l+\ldots$, so
$u(c^p)-u'(c^p)=0 \mbox{\quad mod\quad} M_D^{l+1}$.
>From this immediately follows that $u(a)-u'(a)=0 \mbox{\quad mod\quad}
M_D^{q}$ for any $a\in \bar{D}$ if $u'$ is defined by the element $u'=u+bz^q$,
because $u(\bar{u})-u'(\bar{u})=bz^q$. Moreover, if we represent $a$ as some
polynomial $P(\gamma_1,\ldots ,\gamma_r, \bar{u})$ with coefficients from
$\overline{Z(D)}^{p^k}$, then it is clear that
$$[u(a)-u'(a)]z^{-q}=-\frac{\partial}{\partial \bar{u}}(P(\gamma_1,\ldots
,\gamma_r, \bar{u}))
\bar{b} \mbox{\quad mod\quad} M_D
$$
if $n|q$,
since $u(\gamma_l)=u'(\gamma_l)$ for any $l$ and $z^quz^{-q}=u \mbox{\quad
mod\quad} M_D$. It is also clear that the derivative can be taken even in the
field $\bar{D}^p(\Gamma )$.
So, we have \\
(i)
$$
z^mu'z^{-m}= z^m(u+bz^q)z^{-m}=
u({}^{(z)}\alpha^m (\bar{u}))+u({}^{(z,u)}_m\delta_j(\bar{u}))z^j +\ldots +
(u({}^{(z)}\alpha^m(\bar{b}))
$$
$$
+u({}^{(z,u)}_m\delta_j(\bar{b}))z^j+ \ldots )z^q=
u({}^{(z)}\alpha^m(\bar{u}))+\ldots +(u({}^{(z,u)}\delta_q (\bar{u}))+
u({}^{(z)}\alpha^m(\bar{b})))z^q+\ldots =
$$
$$
u'({}^{(z)}\alpha^m (\bar{u}))+\ldots +(u'({}^{(z,u)}_m\delta_q
(\bar{u}))+u'({}^{(z)}\alpha^m(\bar{b}))-
u'(\frac{\partial}{\partial \bar{u}}{}^{(z)}\alpha^m
(\bar{u})\bar{b}))z^q+\ldots ,
$$
(ii)
We have
$$
z^mu'z^{-m}= z^m(u+bz^q)z^{-m}=
u(\bar{u})+u({}^{(z,u)}_m\delta_j (\bar{u}))z^j+\ldots +(u(\bar{b})+
u({}^{(z,u)}_m\delta_j(\bar{b}))z^j+ \ldots )z^q=
$$
$$
u(\bar{u})+u({}^{(z,u)}_m\delta_j (\bar{u}))z^j+\ldots
+(u({}^{(z,u)}_m\delta_q
(\bar{u}))+u(\bar{b}))z^q+u({}^{(z,u)}_m\delta_{q+1}(\bar{u}))
z^{q+1}+\ldots
$$
$$
+u({}^{(z,u)}_m\delta_{q+j-1}(\bar{u}))z^{q+j-1}+(u({}^{(z,u)}_m\delta_{q+j}(\bar{u}))+
u({}^{(z,u)}_m\delta_j (\bar{b})))z^{q+j}+\ldots=
$$
$$
u'(\bar{u})+u'({}^{(z,u)}_m\delta_j (\bar{u}))z^j+\ldots
+u'({}^{(z,u)}_m\delta_{q+j-1}(\bar{u}))z^{q+j-1}+
(u'({}^{(z,u)}_m\delta_{q+j}(\bar{u}))+u'({}^{(z,u)}_m\delta_j (\bar{b}))-
$$
$$
u'(\frac{\partial}{\partial \bar{u}}({}^{(z,u)}_m\delta_j
(\bar{u}))\bar{b}))z^{q+j}+\ldots
$$
(iii) Assume $u'=u(\bar{u'})+a_1z+\ldots$, where $a_i\in u(\bar{D})$. Since,
by proposition \ref{flyii}, the map ${}^{(z,u)}_m\delta_j$ is a derivation,
we have
$$
z^mu'z^{-m}= [u(\bar{u'})+u({}^{(z,u)}_m\delta_j(\bar{u'}))z^j+\ldots ] +
[a_1+u({}^{(z,u)}_m\delta_j(a_1)z^j+\ldots ]z+\ldots =
$$
$$
u'+u({}^{(z,u)}_m\delta_j(\bar{u'}))z^j+\ldots
=u'+u({}^{(z,u)}_m\delta_j(\bar{u})
\frac{\partial }{\partial \bar{u}}(\bar{u'}))z^j+\ldots =
u'+u'({}^{(z,u)}_m\delta_j(\bar{u})
\frac{\partial }{\partial \bar{u}}(\bar{u'}))z^j+\ldots
$$
$\Box$
\section{The period-index problem}
In this section we will prove the following theorem.
\begin{th}
\label{gipoteza}
The following conjecture: the exponent of $A$ is equal to its index for any
division algebra $A$ over a $C_2$-field $F$ has the positive answer for
$F= F_1((t))$, where $F_1$ is a $C_1$-field.
\end{th}
Recall that a field $F$ is called a {\it $C_i$-field } if any homogeneous form
$f(x_1, \ldots ,x_n)$
of degree $d$ in $n>d^i$ variables with coefficients in $F$ has a non-trivial
zero.
Some basic properties of $C_i$-fields see, for example, in \cite{PY}.
This conjecture was proposed by M. Artin and was solved for some another
examples of the field $F$ by many authors. As it is known for me, the positive
answer for all division algebras of index $ind A= 2^a3^b$ was given in
\cite{PY}, for division algebras over the field $F=k((X))((Y))$, where $k$ is
a perfect field of characteristic $p\ne 0$ such that $\dim_{{\mbox{\sdbl F}}_p}k/\wp
(k)=1$, was given by Tignol in the Appendix in \cite{AJ} (we include this case though $F$ may not be a $C_2$-field), for division
algebras of index prime to the characterictic of $F$, where $F$ is a function
field of a surface, was given in \cite{DJ}. I propose, the positive answer was
also known for division algebras over $F=F_1((t))$ of characteristic 0. We
will give the prove of the theorem above in any characteristic.
{\bf Proof.} 1) Recall that any extension of a $C_1$-field is simple.
Indeed, suppose $E= \bar{F}(u_1, \ldots , u_r)$. Consider the field $K=
\bar{F}(u_1^p, \ldots , u_r^p)$.
By Tsen's theorem, $K$ and $E$ are $C_1$-fields. So, the form
$x_1^p+x_2^pu_1+\ldots +
x_p^pu_1^{p-1}+x_{p+1}^pu_2$ has a non-trivial zero in $E$. But $x_i^p\in K$
and elements
$1, u_1,\ldots , u_1^{p-1}, u_2$ are linearly independent over $K$, a
contradiction.
2) Assume the theorem is known in the prime exponent case. We deduce the
theorem by ascending induction on $e=exp A$. If $e$ is not a prime number,
then write $e=lm$. By assumption $A^{\otimes m}$ can be split by a field
extension $F\subset F'$ of degree $l$. This implies that $A_{F'}$ has
exponent dividing $m$. Note that $F'$ is also a Laurent series field. By the
induction hypothesis applied to the pair $(F', A_{F'})$, there exists a field
extension $F'\subset L$ of degree dividing $m$ splitting $A_{F'}$. Therefore
$A$ is split by the extension $F\subset L$ of degree dividing $lm$ and we
conclude the theorem.
3) So, let $exp A= l$ be a prime number. By the basic properties of the
exponent and the index (see, e.g. \cite{PY}) we have then $ind A= l^k$ for
some natural $k$.
Suppose $(l,p=char F)=1$.
It is known that the conjecture is true for all division algebras of index
$ind A= 2^a3^b$, so we can assume $l\ne 2,3$. We can assume $F$ contains the
group $\mu_l$ of $l$-roots of unity, because
$[F(\mu_l):F]<l$ and we can reduce the problem to the algebra $A\otimes_F
F(\mu_l)$. Then by the Merkuriev-Suslin theorem
$A$ is similar to the tensor product of symbol-algebras of index $l$.
To conclude the statement of the corollary it is sufficient to prove that
every two symbol algebras $A_1, A_2$ contain $F$-isomorphic maximal subfields.
Since every division algebra over a $C_1$-field is trivial and every field
extension is simple, every symbol-algebra of index $l$ over $F$ is
splittable. Since $(l,p)=1$, it is good splittable and its residue field is a
cyclic Galois extension of $\bar F$.
So, if $z_i$ is a parameter from proposition \ref{X} for algebra $A_i$, then
$z_i$ acts on $\bar{A_i}$ as a Galois automorphism and $z_i^l\in F$. We have
$v(z_i^l)=1$ ($v$ is the valuation on $F$).
Let us show that $A_1$ contains a $l$-root of any element $u$ in $F$ with
$v(u)\ne 0$. So, $A_1$ will contain a subfield isomorphic to $F(z_2)$. Since
for any element $1+b$, $v(b)>0$ there exists
a $l$-root $(1+b)^{1/l}\in F$, it is sufficient to prove that $A_1$ contains
any $l$-root of elements $ct$, $c\in u(\bar{F})$, where $u$ is some fixed
embedding $u:\bar{A_1}\hookrightarrow A_1$.
Assume $z_1^l=c_1t$, $c_1\in u(\bar{F})$. Note that for any element $b\in
u(\bar{A_1})$ we have $(bz_1)^l=u(N_{\bar{A_1}/\bar{F}}(b))z_1^l$. But the
norm map $N_{\bar{A_1}/\bar{F}}$ is surjective, since $\bar F$ is a
$C_1$-field (see, e.g. \cite{PY}, 3.4.2), so there exists $b$ such that
$(bz)^l=ct$.
4) Suppose now $exp A=p$. Then $ind A=p^k$.
By Albert's theorem (in \cite{Al}) there exists a field $F'=F(u_1^{1/p},\ldots
,u_k^{1/p})$ which splits $A$. Using the same arguments as in 1) one can show
that every such a field has maximum two generators, say $F'=F(u_1^{1/p},
u_2^{1/p})$. Therefore, $ind A\le p^2$. If $ind A=p$, there is nothing to
prove, so we assume $ind A=p^2$ and $F'$ is a maximal subfield in $A$.
5) Suppose $F_1$ is a perfect field.
By Albert's theorem, $A\cong A_1\otimes_F A_2$, where $A_1,A_2$ are cyclic
algebras of degree $p$, $A_1=(L_1/F,\sigma_1, u_1)$, $A_2=(L_2/F, \sigma_2,
u_2)$. Since $F_1$ is perfect, $\bar{A_1}/\bar{F}$,
$\bar{A_2}/\bar{F}$ are Galois extensions. So, $A_1, A_2$ are good splittable.
Let us show that $A_1, A_2$ have common splitting field of degree $p$ over
$F$.
This leads to a contradiction.
By proposition \ref{X} there exist parameters $z_1\in A_1$, $z_2\in A_2$ such that
they act on $\bar{A_1}$, $\bar{A_2}$ as Galois automorphisms. Note that then
$z_1^p, z_2^p\in F$. Let us show that $F(z_1)$ splits $A_2$.
Consider the centralizer $D=C_A(F(z_1))$. Consider the element
$t_1=z_2z_1^{-1}$. We have $t_1^p\in F$, $w(t_1)=0$, where
$w$ denote the unique extension of the valuation $v$ on $F$. Since
$\bar{D}/\overline{Z(D)}$ is a Galois extension, there exists an element
$b_1\in F$ such that $w(t_1-b_1)>0$. Since $(t_1-b_1)^p\in F$, there exists
natural $k_1$ such that
$w((t_1-b_1)z_1^{-k_1})=0$. Denote $t_2=(t_1-b_1)z_1^{-k_1}$. We have again
$t_2^p\in F$. Repeating this arguments and using the completeness of $D\subset
A$ we get \\
$z_2=t_1z_1=(t_2z_1^{k_1}+b_1)z_1=\ldots =b_1z_1+b_2z_1^{k_1+1}+\ldots$,\\
so, $z_2\in F(z_1)=Z(D)$.
6) Suppose $F_1$ is not perfect.
Since $F'$ is generated by two elements over $F$, it contains all $p$-roots of
$F$. Then, every two elements $u,z\in F$ such that $z^{1/p}\notin F(u^{1/p})$,
where
$z^{1/p}, u^{1/p}\in F'$, also generate $F'$ over $F$. This follows from the
same arguments as in 1), 4).
Now take $u\in F_1\backslash F_1^p$, $z=u+t$. It's clear that $p$-roots of
these elements generate $F'$ over $F$. Moreover, the fields $F(u^{1/p}),
F(z^{1/p})$ are {\it "unramified"} over $F$, i.e.
$[\overline{F(u^{1/p})}:\bar{F}]=p=[F(u^{1/p}):F]$,
$[\overline{F(z^{1/p})}:\bar{F}]=p$. Denote $u_1=u^{1/p}$,
$u_2=z^{1/p}$ in $F'$. Then by Albert's theorem, $A\cong A_1\otimes_F A_2$,
where $A_1,A_2$ are cyclic algebras of degree $p$, $A_1=(L_1/F,\sigma_1, u)$,
$A_2=(L_2/F, \sigma_2, z)$.
Concider the centralizer $D=C_A(F(u_1))$. Suppose $\bar{D}/\overline{Z(D)}$ is
a separable extension. Then there exist a lift $u:\bar{D}\hookrightarrow D$ of
arbitrary embedding $u':\overline{F(u_1)}\hookrightarrow F(u_1)$. Consider the
embedding $u'=u_1$ defined in lemma \ref{simple}.
Since $F(u_1)/F$ is a purely inseparable extension, $u'$ is a good embedding,
so $u$ is a good embedding of $\bar{D}=\bar{A}$ in $D\subset A$. So, we get
$A$ is a good splittable algebra, and $u(\bar{A})$ contain a purely
inseparable over $F$ element. But this is a contradiction with lemma
\ref{ppp}. So, $\bar{A}/\bar{F}$ can not contain a separable subextension,
because in this case $\bar{D}/\overline{Z(D)}$ must be a separable extension.
Now we can use, for shorteness, lemmas A.4., A.6. of Tignol in Appendix to the
paper
\cite{AJ}. These lemmas show that a tensor product $A_1\otimes A_2$ of any two
symbols
$A_1, A_2$ is similar either to a single symbol in $Br(F)$ (in which case we are done) or to a
product of two symbols of level zero. Recall that, by Saltman's results in
\cite{Sa}, every division
algebra of level zero is tame, which means in our case that the residue
division algebra is a separable extension over $\bar{F}$. A notion of level
was already discussed above in remark to lemma \ref{svva}.
So, assume $A\sim D_1\otimes D_2$, where $D_1, D_2$ are tame division algebras of
degree $p$ over $F$. We can assume $A$ and $D_1\otimes D_2$ are division algebras, so
$A\cong D_1\otimes D_2$. Since $D_1, D_2$ are tame,
we conclude $\bar{A}$ must contain a separable element, a contradiction.
The theorem is proved. \\
$\Box$
\section{Good splittable algebras}
In this section we prove a decomposition theorem for good splittable division
algebras. This theorem shows how the studying of good splittable division
algebras can be reduced to the studying of division algebras with simple
described structure. So, good splittable algebras are the most easy and good
algebras to study.
\begin{lemma}
\label{goodspl}
Let $D$ be a good splittable division algebra, $F=Z(D)$, and let
$Z(\bar{D})=\bar{F}(s)$ be a purely inseparable over $\bar{F}$ field of degree
$p=char D>0$. Let $u:\bar{D}\hookrightarrow D$ be a good embedding.
Then there exists a parameter $z$ such that
${}^{(z,u)}_{-i}\delta_j=0$ for $j>i$, where $i=i(z,u)$ is a local height, and
$u({}^{(z,u)}\delta_i(s))=x$,
where $x\in Z(D)$. Moreover, $(i,p)=1$.
\end{lemma}
{\bf Proof.} Since $Z(\bar{D})/\bar{F}$ is a purely inseparable extension,
${}^{(z)}\alpha |_{Z(\bar{D})}=id$ for any parameter $z$. By Skolem-Noether
theorem there exists a parameter $z$ in $D$ such that ${}^{(z)}\alpha =id$.
Suppose
${}^{(z,u)}\delta_i(s)=0$, where $i=i(z,u)$. Then
${}^{(z,u)}\delta_i|_{Z(\bar{D})}=0$, since $u$ is a good embedding and
$Z(\bar{D})/\bar{F}$ is a simple extension. So, ${}^{(z,u)}\delta_i$ is an
inner derivation by Scolem-Noether theorem, and by lemma \ref{ozamene}, (i)
there exists a parameter $z'$ such that ${}^{(z',u)}\delta_i=0$,
${}^{(z')}\alpha =id$.
So, we can assume ${}^{(z,u)}\delta_i(s)\ne 0$ for some parameter $z$. Since
$s^p\in Z(D)$, by lemma \ref{ppp} we have $(i,p)=1$.
Since ${}^{(z,u)}\delta_i$ is a derivation,
${}^{(z,u)}\delta_i(s)\in Z(\bar{D})$ (see the arguments in lemma \ref{(5)}, (ii)).
Since $(i,p)=1$, there exists $k$ such that $p| (1-ki)$. So,
by lemma \ref{ozamene}, (iii), for the parameter
$z'=({}^{(z,u)}\delta_i(s))^k$ we have ${}^{(z')}\alpha =id$,
${}^{(z',u)}\delta_i(s)\in \bar{F}$, i.e. $u({}^{(z',u)}\delta_i(s))\in Z(D)$.
Since $s^p\in Z(D)$, by lemma \ref{vtorinv} we must have $d(u,s)=\infty$. In
the proof of lemma \ref{vtorinv}, (i) was shown that $d(u,s)=d'(u,z,s)$ for
some parameter $z$, and the construction of this element uses lemma
\ref{ozamene}, (ii), so it preserves the initial values of ${}^{(z')}\alpha$,
${}^{(z',u)}\delta_i$. So, ${}^{(z,u)}_{-i}\delta_j=0$ for $j>i$ and the lemma
is proved.\\
$\Box$
\begin{prop}
\label{razlozhenie}
Let $D$ be a splittable division algebra. Then
we have
$D\cong D_1\otimes_F D_2$, where $D_1, D_2$ are splittable division algebras
such that
$D_1$ is an inertially split algebra.
If $D$ is a good splittable division algebra, then
$Z(\bar{D_2})/\bar{F}$ is
a purely inseparable extension and $D_2$ is a good splittable algebra ($D_1$
or $D_2$ may be trivial).
So, $D\sim A\otimes_FB\otimes_FD_2$, where $A$ is a cyclic division algebra
and $B$ is an unramified division algebra.
\end{prop}
{\bf Proof.} If $char D=0$, the proposition is obvious, so we assume $char
D>0$.
By \cite{P}, p.261, $D\cong D_1\otimes_F\ldots \otimes_FD_k$,
where $[D:F]= p_1^{r_1}\ldots p_k^{r_k}$ and $[D_i:F]= p_i^{r_i}$. Let
$p_2= p$. Since $D_i$ are defectless over $F$, $D_1,D_3,\ldots
D_k$ are inertially split. Therefore, by theorem \ref{Cohen} the algebra
$B=D_1\otimes D_3\otimes
\ldots \otimes D_k$ is good splittable.
Assume first that $D$ is good splittable.
By proposition 1.7. in \cite{JW}, if $s\in Z(\bar{D})$ is an element such that
$\alpha (s)=s$, then this element is a purely inseparable element over
$\bar{F}$. So, if $D$ is a good splittable division algebra, then
by lemma \ref{ppp} $D_2$ is either inertially split or $Z(\bar{D_2})/\bar{F}$
is a purely inseparable extension. For, otherwise there exists an element
$s\in Z(\bar{D_2})\subset Z(\bar{D})$ as above and by proposition \ref{X} $p|i(u,s)$ for any embedding $u$. If $u$ is a good embedding, then $s^{p^k}\in Z(D)$ for some $k$, a contradiction.
So, we assume below $Z(\bar{D_2})/\bar{F}$ is a purely inseparable extension.
Now, we have (see, e.g. th.1 in \cite{Mor}) $\bar{D}\cong
\bar{D_2}\otimes_{\bar{F}}\bar{B}$ and so $u(\bar{D})\cong
u(\bar{D_2})\otimes_{u(\bar{F})}u(\bar{B})$, where $u$ is a good embedding.
So, $E=u(Z(\bar{D_2}))$ is a purely inseparable field over $u(\bar{F})\subset
Z(D)$.
Consider the field $E'=u(K)\otimes_{u(\bar{F})}F$, where $K$ is a maximal
separable subfield in $\bar{B}$. This is an inertial lift of $K$ in $D$.
Consider the centralizer
$C=C_D(E')\cong D_2\otimes_FE'$. Let $M$ be a maximal subfield in $\bar{D_2}$.
Note that $u(\bar{D_2})\subset C$, so $L\subset C$, where
$L=u(M)F$ is the composit of $u(M)$ and $F$, and $E\subset L$.
Note that $[L:F]=ind D_2=ind C$. The field $L$ splits $C$ by dimension
arguments. So, it must split $D_2$, since $([E':F],p)=1$, and $D_2$ is a
$p$-algebra. Therefore, $L$ is isomorphic to a maximal subfield in $D_2$, so
$D_2$ contain a copy of purely inseparable "unramified" subfield, whose
residue field is isomorphic to $Z(\bar{D_2})$. Therefore,
$D_2$ is a god splittable algebra. For, the centralizer of this field is an
unramified division algebra, so by theorem \ref{Cohen} is splittable. So,
$D_2$ is good splittable if the purely inseparable field is good splittable.
But it is good splittable since it contains a subfield isomorphic to
$u(Z(\bar{D_2}))$ by the construction. (Another way to see it is to use
arguments from lemma \ref{simple} to show that there exists an appropriate
$p$-basis).
Let $D$ be a splittable algebra. Then the same arguments as in the previous
paragraph show that $L$ is isomorphic to a maximal subfield in $D_2$ (it is
not important that $Z(\bar{D_2})/\bar{F}$ may be not a purely inseparable
extension).
Now, the composit $EF\subset L$, $EF\ne L$, since every element from $E$
commute with $u(\bar{D_2})$, where $u$ is some fixed embedding. So we must
have $\overline{C_{D_2}(EF)=\bar{D_2}}$ and $C_{D_2}(EF)$ is an unramified
division algebra. Therefore, $D_2$ is splittable division algebra.
Decomposition theorems \cite{JW}, Thm. 5.6-5.15
complete the proof.\\
$\Box$
This proposition shows that the study of splittable division algebras can be
reduced to the study of splittable $p$-algebras. So, below in this section and
in the next section we will deal with $p$-algebras only.
\begin{prop}
\label{555}
Let $D$ be a good splittable division algebra such that
$Z(\bar{D})/\overline{Z(D)}$ is
a purely inseparable extension. Then $D\cong D_1\otimes_{Z(D)}D_2$, where
$D_1$ is an unramified division algebra and $D_2$ is a good splittable
division algebra such
that $\bar{D_2}$ is a field, $\bar{D_2}/\overline{Z(D)}$ is a purely
inseparable extension, $[\bar{D_2}:\overline{Z(D)}]=
[\Gamma_{D_2}:\Gamma_{Z(D)}]$.
\end{prop}
{\bf Proof.} The proof is by induction on the degree
$[Z(\bar{D}):\overline{Z(D)}]$.
Assume $[Z(\bar{D}):\overline{Z(D)}]=p$. Let ${}^{(z,u)}\delta_i$ be the map
from lemma \ref{goodspl}. Then ${}^{(z,u)}\delta_i^p$ is a derivation trivial
on the centre $Z(D)$, hence by Scolem-Noether theorem it is an inner
derivation.
We claim that $z^p\in Z(D)$.
We have
$$
z^{-i}az^i= a+{}_{-i}\delta_i(a)z^i, \mbox{\quad} a\in u(\bar{D})
$$
Therefore,
$$
z^{-pi}az^{pi}= a+{}_{-i}\delta_i^p(a)z^{pi}, \mbox{\quad} a\in u(\bar{D})
$$
and
$$
z^{pi}az^{-pi}= a+\delta'_1(a)z^{pi}+{\delta'_1}^2(a)z^{2pi}+\ldots ,
$$
where $\delta'_1= (-1){}_{-i}\delta_i^p=i^p\delta_i^p$. So,
$$
z^paz^{-p}=
a+\frac{1}{i}\delta'_1(a)z^{pi}+c_2\frac{1}{i^2}{\delta'_1}^2(a)z^{2pi}+\ldots
,
$$
where $c_k$ are given by (\ref{(188)}) in lemma \ref{svva}. So, $z^p\in
Z(D)$ iff $\delta_i^p= 0$. Suppose $\delta_i^p\ne 0$. Consider an element
$Y\in Z(D)$, $w(Y)>0$. Let
$$
Y= a_1z^p+\ldots , \mbox{\quad } a_1\in u(\bar{D}).
$$
First note that
$$
Y= a_1z^p+a_2z^{2p}+a_3z^{3p}+\ldots , \mbox{\quad} a_i\in u(\bar{D})
$$
Indeed, $Y$ must satisfy $[Y,s]= 0$, where $s$ is a generator of
$u(Z(\bar{D}))$ over $u(\bar{F})$. Since $s\in u(Z(\bar{D}))$ and
$w([z^k,s])=k+i$ if $(k,p)=1$ and $w([z^k,s])=\infty$ otherwise, we then have
$[z^{i_k},s]= 0$ for every $k$, where
$$
Y= \sum_{k= 1}^{\infty}a_kz^{i_k}
$$
Therefore, $p|i_k$.
Then, $Y$ must satisfy $Ya= aY$ for any $a\in u(\bar{D})$. Therefore,
$a_1,\ldots a_i\in u(Z(\bar{D}))$ and we must have
$$
aa_{i+1}-a_{i+1}a= a_1\delta'_1(a)/i
$$
and
$$
aa_{2i+1}-a_{2i+1}a= a_i\delta'_1(a)+a_1c_2{\delta'_1}^2(a).
$$
Since $\Delta (a)= aa_{2i+1}-a_{2i+1}a$ is an inner derivation, we get
${\delta'_1}^2= \delta$, where $\delta$ is a derivation, which is a
contradiction if $\delta\ne 0$ and $char D\ne 2$. In the last case we can use
the same arguments with
$a_{3i+1}$. Therefore, ${\delta'_1}^2= \delta= 0$ and $\delta'_1= 0$,
and $z^p\in Z(D)$.
Consider the algebra $W=u(Z(\bar{D}))((z))$. Since $z^p\in Z(D)$ and
$u(\bar{F})\subset Z(D)$, we have $Z(W)=u(\bar{F})((z^p))=F$. So, $D\cong
W\otimes_FC_D(W)$ by Double Centralizer theorem. It is clear that $C_D(W)$ is
an unramified division algebra.
Now suppose the proposition is proved for
$[Z(\bar{D}):\overline{Z(D)}]=p^{k-1}$. By Albert's theorem (th.13 in
\cite{Al}) $D_2$ then is a cyclic algebra as a product of cyclic subalgebras
$D_i$, where $\bar{D_i}/\bar{F}$ is a simple purely inseparable extension and
$D_i$ is a good splittable algebra.
Assume $[Z(\bar{D}):\overline{Z(D)}]=p^{k}$.
For a good embedding there exists a lift $\tilde{K}$ of a subfield
$\overline{Z(D)}\subset K\subset Z(\bar{D})$ such that the extension
$K/\overline{Z(D)}$ has degree $p$, i.e. $\bar{\tilde{K}}= K$,
$\Gamma_{\tilde{K}}= \Gamma_{Z(D_2)}$, $u(K)\subset \tilde{K}$,
$\tilde{K}/Z(D)$ is a purely inseparable extension of degree $p$. By the
induction hypothesis the centralizer $C_D(\tilde{K})\cong
A_1\otimes_{\tilde{K}}A_2$, where $A_2$ is a cyclic division algebra and
$\bar{A_2}$ is a field. Note that $\bar{A_2}=Z(\bar{D})$.
By theorem 6 in \cite{Al} we can assume $A_2=(L/\tilde{K},\sigma ,a)$, where
$a$ generate $\tilde{K}$ over $Z(D)$. So, $A_2$ contains a maximal purely
inseparable Kummer subfield $E=\tilde{K}(y)$ with $y^{p^{k-1}}=a$, so
$E=Z(D)(y)$. By theorem 3 in \cite{Al} $L=L_0\times \tilde{K}$, where $L_0$ is
cyclic of degree $p^{k-1}$ over $Z(D)$ and $yx_0=\sigma (x_0)y$, where $x_0\in
L_0$.
Consider the centralizer $B=C_D(L_0)$. We claim $B\cong B_1\otimes_{L_0} B_2$,
where $B_2$ is a cyclic division algebra of degree $p$ and $B_2$ contains
$\tilde{K}$.
Note that
$B$ contains $Z(D)(a)=\tilde{K}$ and $A_1$. If $\tilde{K}L_0=L$ is
"unramified" over $L_0$, then we apply the arguments for the first step of our
induction to the algebra $B$. By construction, $B_2$ then will contain $L$, so
$\tilde{K}$. Suppose $L$ is totally ramified over $L_0$ and let $z$ be a
parameter of $L$, i.e. an element with the least possible positive mean of
valuation on $L$. Since $L$ is purely inseparable over $L_0$, $z^p$ is a
parameter of $L_0$.
We have $W:=C_B(L)= C_D(L)\cong A_1\otimes_{\tilde{K}}L$ is an unramified
division algebra. Consider an embedding $u':\bar{L}=\bar{L_0}\hookrightarrow
L_0$. As it was shown in the proof of theorem \ref{Cohen} there is a lift
$\tilde{u'}$ of $u'$, $\tilde{u'}:
\bar{W}\hookrightarrow W$. Now consider the subalgebra
$W'=\tilde{u'}(\bar{W})((z^p))$. We have $Z(W')=u'(\bar{L})((z^p))=L_0$, so
$W'$ is an unramified subalgebra in $B$. By Double Centralizer theorem,
$B\cong W'\otimes_{L_0}C_B(W')$, where $C_B(W')$ is a division algebra of
degree $p$ and contains $L_0(z)=L$, so it contains $\tilde{K}$ and it is
cyclic by Albert's theorem (th.12 in \cite{Al}).
Now we can word by word repeat the arguments in the proof of theorem 12 in
\cite{Al} to show that there exists a cyclic Galois extension $L'$ of $L_0$
which is cyclic Galois over $Z(D)$, and $y$ acts as a Galois automorphism on
$L'/Z(D)$ which generates $Gal(L'/Z(D))$. So, there is the cyclic subalgebra
$D_2=(L'/Z(D), ad(y), y^{p^k})$ in $D$. Note that $A_2\subset D_2$, and $A_2$
is known to be a good splittable algebra with
$[\bar{A_2}:\overline{Z(A_2)}]=[\Gamma_{A_2}:\Gamma_{Z(A_2)}]$. Since
$\bar{A_2}=\bar{D_2}$ and $Z(A_2)=\tilde{K}$ is a purely inseparable extension
of $Z(D)$, $D_2$ is a good splittable algebra such that $\bar{D_2}$ a field
and $[\bar{D_2}:\overline{Z(D)}]=[\Gamma_{D_2}:\Gamma_{Z(D)}]$. By Double
Centralizer theorem $D\cong D_1\otimes_{Z(D)}D_2$, where $D_1=C_D(D_2)$ must
be an unramified division algebra, which completes the proof.\\
$\Box$
Combining all results in this section, we get the following theorem.
\begin{th}
\label{itog}
Let $D$ be a finite dimensional good splittable central division algebra over
a field $F=k((t))$.
If $char ({F})= p>0$,
then $D\cong D_1\otimes_{F}D_2\otimes_{F}A_1\otimes_{F}\ldots
\otimes_{F}A_m$,
where $A_i$ are cyclic division algebras such that
$[\bar{A_i}:\overline{Z(D)}]=
[\Gamma_{A_i}:\Gamma_{Z(D)}]$ and $\bar{A_i}/\overline{Z(D)}$ are simple
purely inseparable field extensions, $D_1$ is an inertially split division
algebra, $(ind (D_1), p)= 1$, $D_2$ is an unramified division algebra ($D_1,
D_2, A_i$ may be trivial).
If $char F=0$, then $D$ is an inertially split division algebra.
\end{th}
\section{Splittability and good splittability}
In this section we collect some assorted results about a relation between splittable and good splittable division algebras and about splittable division algebras.
We consider here only division algebras with the following property:
$Z(\bar{D})/\overline{Z(D)}$ is a simple extension.
\begin{prop}
\label{cyclisity}
Let $D$ be a central division algebra over $F$ of $char D=p>0$ such that
$Z(\bar{D})=\bar{D}$
and $[Z(\bar{D}):\bar{F}]=p$.
Then $D$ is a splittable algebra and the local height $i=i(u,z)$ (in the
situatuion when it is defined, i.e. when $\alpha =id$) does not depend on $u$ and $z$.
It is a good splittable algebra if $(i,p)=1$. If $p|i$, then there exists a parameter $z$ such that $z^p\in Z(D)$ and any "unramified" maximal subfield is
cyclic Galois.
So, in both cases $D$ is a cyclic division algebra of degree $p$.
\end{prop}
{\bf Proof.} Since $\bar{D}/\bar{F}$ is a simple extension, we have
$[\bar{D}:\bar{F}]=[\Gamma_D:\Gamma_F]$. Indeed,
consider the fields $E=F(s)$ and $E'=F(z)$, where $s$ is any element such that
$\bar{s}$ is a primitive element of the extension $\bar{D}/\bar{F}$ and $z$ is
any parameter of $D$. Then $[\bar{D}:\bar{F}]\le [E:F]\le
[D:F]^{1/2}=([\bar{D}:\bar{F}][\Gamma_D:\Gamma_F])^{1/2}$, so
$[\bar{D}:\bar{F}]\le [\Gamma_D:\Gamma_F]$. From another hand side,
$[\Gamma_D:\Gamma_F]\le [E':F]\le
([\bar{D}:\bar{F}][\Gamma_D:\Gamma_F])^{1/2}$, so
$[\bar{D}:\bar{F}]=[\Gamma_D:\Gamma_F]$.
So, $D$ is splittable division algebra of degree $p$.
If $Z(\bar{D})/\bar{F}$ is a separable extension, then $D$ is a good
splittable algebra by theorem \ref{Cohen}. So, we assume it is a purely
inseparable extension, $Z(\bar{D})=\bar{F}(\bar{u})$. For any lift $u$ of the
element $\bar{u}$
let $u$ be an embedding constructed in lemma
\ref{simple}, i.e. ${}^{(z,u)}\delta_j$ is defined by the values
${}^{(z,u)}\delta_j(u^k)$ for any $j$. By corollary \ref{ozamene3} the local
height $i(u,z)$ does not depend on $z$, and by lemma \ref{ozamene2} $i(u,z)$ does not
depend on $u$.
For arbitrary embedding $u'$,
since ${}^{(z,u')}\delta_{i(u',z)}$ is a derivation and $\bar{D}/\bar{F}$ is a simple extension, ${}^{(z,u')}\delta_{i(u',z)}$ is completely defined by a value at $\bar{u}$.
Therefore, $i(u',z)=w(zu'(\bar{u})z^{-1}-u'(\bar{u}))$ and $i(u',z)$ is completely defined by the lift $u'(\bar{u})$. But arbitrary lift of $\bar{u}$ defines an embedding, on which we have proved $i$ does not depend. So, $i(u,z)$ does not depend on $z$ and $u$.
Now assume $p|i$.
Using lemma \ref{ozamene}, we can assume without loss of generality that
${}^{(z,u)}\delta_j=0$ if $j$ is not divisible by $p$.
Indeed, if ${}^{(z,u)}\delta_j\ne 0$, then we apply lemma \ref{ozamene}, (ii)
to show that there exists a parameter $z_j$ such that
${}^{(z_j,u)}\delta_j(u)=0$ and ${}^{(z_j,u)}\delta_k={}^{(z,u)}\delta_k$ for
$k<j$,
${}^{(z_j)}\alpha =id$. Since ${}^{(z_j,u)}\delta_j$ is a derivation by
proposition
\ref{flyii} and by induction (similar arguments was already used in the proof
of proposition \ref{X}),
and since it
is defined by the values on $u^k$, so by the values on $u$, we have
${}^{(z,u)}\delta_j= 0$.
Since for $j_1>j_2$ we have $w(z_{j_1}-z_{j_2})> j_1-i$, the sequence
$\{z_j\}$ convereges to a parameter $z'$, which satisfies our condition.
So, there exists the subalgebra $A=u(\bar{D})((z^p))$. Let's show that
$Z(D)\subset A$.
Note that every element $a\in D$ can be written as $a=a_0+a_1z+\ldots
+a_{p-1}z^{p-1}$, where $a_i\in A$. Note that $z^kAz^{-k}\subset A$ for every
$k$. So, if $a\in Z(D)$, then $za_jz^{-1}=a_j$ and $ua_jz^ju^{-1}=a_jz^j$ for
every $j$. For $j>0$ we have $a_jz^j=\sum_k a_{jk}z^{kp+j}$, so by corollary
\ref{ozamene3} $ua_jz^ju^{-1}\ne a_jz^j$.
Therefore, $a=a_0\in A$.
Since $A\ne D$, $A$ must be commutative, so $z^p\in Z(D)$. Moreover, $A/Z(D)$
is cyclic Galois. Since the arguments work for arbitrary lift $u$ of the
element $\bar{u}$, arbitrary "unramified" maximal subfield in $D$ must be
Galois over $F$.
Now let $(i,p)=1$.
Using lemma \ref{ozamene}, (iii) we can find a parameter $z$ and a primitive
element $s\in \bar{D}$
such that ${}^{(z,u)}\delta_i(s)=sc$, where $c\in \bar{F}$. Indeed, since
$(i,p)=1$, there exists $k$ such that $1-ki$ is divisible by $p$. So, by lemma
\ref{ozamene}, (iii) for a parameter
$z'=u({}^{(z,u)}\delta_i(\bar{u})^k)z$ we have ${}^{(z',u)}\delta_i(\bar{u})\in
\bar{F}$, so by lemma \ref{ozamene2}, (iii)
${}^{(z',u)}\delta_i(s)=1$, where
$s=\bar{u}{}^{(z',u)}\delta_i(\bar{u})^{-1}$.
Now, there exists $k_1$ such that $-ik_1-1$ is divisible by $p$, so for
$z''=s^{k_1}z'$ we have ${}^{(z'',u)}\delta_i(s)=sc$, where $c=s^{-ik_1-1}\in
\bar{F}$. It is easy to see that, since $s=\bar{u}a$, where $a\in
\bar{F}$, the map ${}^{(z,u)}\delta_j$ is uniquely defined also by
${}^{(z,u)}\delta_j(s^k)$, so by ${}^{(z,u)}_m\delta_l(s)$ for $l\le j$.
So, we assume without loss of generality that $s=\bar{u}$, $z=z''$.
Using lemma \ref{ozamene2}, (ii) we can find a converge sequence $\{u_j\}$,
$u_j\in D$, $j\ge i$ such that $u_{j+1}=u_j+b_jz^{j+1-i}$, $u_i=u$, $b_j\in
u_j(\bar{D})$ (here $u_j$ is an embedding defined by $u_j$, see lemma
\ref{simple}) and
${}^{(z,u_j)}_m\delta_k(\bar{u})\bar{u}^{-1}\in \bar{F}$ for all $k\le j$ and
all $m$.
Indeed, suppose it is true for $j\ge i$. Let
${}^{(z,u_j)}_m\delta_{j+1}(\bar{u})=a_0+\ldots a_{p-1}\bar{u}^{p-1}$, $a_k\in
\bar{F}$. Since
${}^{(z,u_j)}_m\delta_i={}^{(z,u)}_m\delta_i=m{}^{(z,u)}\delta_i$, we have
$$
{}^{(z,u_j)}_m\delta_i(a_k\bar{u}^k)-\frac{\partial }{\partial
\bar{u}}({}^{(z,u_j)}_m\delta_i(\bar{u}))a_k\bar{u}^k= (k-1)mca_k\bar{u}^k.
$$
So, $u_{j+1}=u_j-u_j(\sum_{k, k\ne
1}(k-1)^{-1}m^{-1}c^{-1}a_k\bar{u}^k)z^{j+1-i}$ will satisfy our condition.
We will denote by $u$ now a limit of the sequence $\{u_j\}$. Using induction
and proposition \ref{flyii} one can easily show that
${}^{(z,u)}_m\delta_j(\bar{u}^k)\bar{u}^{-k}\in \bar{F}$ for any integer $k$.
So, there is the subalgebra $A=u(\bar{F})((z))$ in $D$. Using similar
arguments as in the case $p|i$, one can show that $A$ contains $Z(D)$. Since
$A\ne D$, it must be commutative, so $u^p\in Z(D)$. Then $u$ is a good
embedding, which completes the proof. \\
$\Box$
Let $D$ be a splittable division algebra and let $Z(\bar{D})/\overline{Z(D)}$
be a purely inseparable extension. As it was shown in the proof of lemma
\ref{goodspl}, then there exists a parameter $z$ in $D$ such that
${}^{(z,u)}\delta_{i}|_{Z(\bar{D})}\ne 0$, where $i=i(u,z)$ is a local height.
Though $D$ may be not a good splittable algebra, the arguments from there are
valid for every splittable algebra. We will call such a parameter {\it an
appropriate parameter}, and the number $i(u)=\max_zi(u,z)=i(u,z)$ for an
appropriate parameter {\it a semilocal height}. Let's prove the following
simple lemma.
\begin{lemma}
\label{simple2}
Let $D$ be a splittable central division $p$-algebra over $F$, where $p=char
D>0$, and let $Z(\bar{D})=\bar{F}(s)$ be a simple extension over $\bar{F}$.
Then
i) there exists an embedding $u$ such that
${}^{(z,u)}_l\delta_j|_{Z(\bar{D})}$ is defined by the values
${}^{(z,u)}_l\delta_j(s^k)$ for any $j,l,z$ (as in lemma \ref{simple});
ii) $[Z(\bar{D}):\bar{F}]=[\Gamma_{D}:\Gamma_{F}]$;
iii) if $\alpha |_{Z(\bar{D})}\ne id$ or $i(u)$ is divisible by $p$, then
there exists a subalgebra $A=u(\bar{D})((z))$ for some appropriate parameter
$z$ such that $Z(D)\subset Z(A)$. Moreover, $Z(A)$ is a cyclic Galois
extension over $Z(D)$.
\end{lemma}
{\bf Proof.} i)
For arbitrary embedding $u$ consider the field $E=u(Z(\bar{D}))F\subset D$
and the centralizer $W=C_D(E)$. We have $\bar{W}=\bar{D}$ and so
$Z(\bar{W})=\bar{E}$. Therefore, $W$ must be an unramified division algebra,
and
by theorem \ref{Cohen} there exists a lift on $\bar{W}$ of arbitrary
embedding
$\bar{E}\hookrightarrow E$. Now we can take an embedding defined by the
element $s$ as in
lemma \ref{simple}. It's lift will be desired embedding. We will denote this
embedding also by $s$.
ii) By proposition 1.7. in \cite{JW} the basic homomorphism $\theta_D$ (see
introduction) is surjective. So, it is sufficient to prove the assertion only
for the centralizer $C_D(K)$, where $K$ is a lift of a Galois part of the
extension $Z(\bar{D})/\bar{F}$. So, we will assume below $Z(\bar{D})/\bar{F}$
is a purely inseparable extension.
Consider a maximal separable subfield $M$ in $\bar{D}$, and let $M'$ be a
separable part of the extension $M/\bar{F}$. By \cite{JW}, th.2.8, th.2.9.
there exists an inertial lift of $M'$ in $D$, say $\tilde{M}$. Consider the
centralizer $B=C_D(\tilde{M})$. Then $\bar{B}$ is a field. Our assertion will
be proved if we show it for $B$, since $[\tilde{M}:F]=ind (\bar{D})$ and
$[D:F]=ind(\bar{D})^2[Z(\bar{D}):\bar{F}][\Gamma_{D}:\Gamma_{F}]$.
Since $\bar{B}/\overline{Z(B)}$ is a simple extension, we can repeat the
arguments from the beginning of proposition \ref{cyclisity}.
iii) If $\alpha |_{Z(\bar{D})}\ne id$, consider the parameter $z$ from
proposition \ref{X}. Then, clearly, $A=u(\bar{D})((z))$ will be a subalgebra
with the center $K$, which is an inertial lift of a Galois part of the
extension $Z(\bar{D})/\bar{F}$.
Assume $\alpha |_{Z(\bar{D})}= id$ and $i(u)$ is divisible by $p$. Let $z$ be
an appropriate parameter.
Using lemma \ref{ozamene}, we can prove that ${}^{(z,u)}\delta_j=0$ if $j$ is
not divisible by $p$.
Indeed, let ${}^{(z,u)}\delta_j\ne 0$ be the first map with this property for
$(j,p)=1$. If ${}^{(z,u)}\delta_j|_{Z(\bar{D})}=0$, then we apply lemma
\ref{ozamene}, (i) to
show that there exists a parameter $z_j$ such that
${}^{(z_j,u)}\delta_j=0$ and ${}^{(z_j,u)}\delta_k={}^{(z,u)}\delta_k$ for
$k<j$,
${}^{(z_j)}\alpha =id$, since ${}^{(z,u)}\delta_j$ is a derivation by
proposition
\ref{flyii} and by induction (similar arguments was already used in the proof
of proposition \ref{X}) and so it is an inner derivation by Scolem-Noether
theorem.
If ${}^{(z,u)}\delta_j|_{Z(\bar{D})}\ne 0$, then we apply lemma
\ref{ozamene}, (ii) to show that there exists a parameter $z_j$ such that
${}^{(z_j,u)}\delta_j(s)=0$ and ${}^{(z_j,u)}\delta_k={}^{(z,u)}\delta_k$ for
$k<j$,
${}^{(z_j)}\alpha =id$. Since ${}^{(z_j,u)}\delta_j$ is a derivation
and since
its restriction on $Z(\bar{D})$
is defined by the values on $s^k$, so by the values on $s$, we have
${}^{(z,u)}\delta_j|_{Z(\bar{D})}= 0$,
and we reduce the problem to the previous case.
Since for $j_1>j_2$ we have $w(z_{j_1}-z_{j_2})> j_1-i$, the sequence
$\{z_j\}$ convereges to a parameter $z'$, which satisfies our condition.
Therefore, there exists a subalgebra $A=u(\bar{D})((z'))$ in $D$. Using the
same arguments as in proposition \ref{cyclisity} one can show that
$Z(D)\subset Z(A)$ Since $z'$ preserves $A$, it preserves the centre $Z(A)$
>From the other hand side, it acts nontrivially on it. So, $Z(A)$ is a cyclic
Galois extension of degree $p$, and $ad(z')$ generates its Galois group. \\
$\Box$
This lemma shows that the study of splittable $p$-algebras over $F$ can be
reduced to the study of splittable $p$-algebras with a purely inseparable
extension $Z(\bar{D})/\bar{F}$ and $(i(u),p)=1$.
\begin{defi}
\label{ivariant}
Let $D$ be a splittable division $p$-algebra with a purely inseparable
extension $Z(\bar{D})/\bar{F}$.
For any element $a\in \bar{D}$ define the number
$$
d_D(a)= \max_{u,z} w(z^{-i(u,a)}u(a)z^{i(u,a)}-
u(a)-u({}^{(z,u)}_{-i(u,a)}\delta_{i(u,a)}(a))z^{i(u,a)})\in {\mbox{\dbl N}}\cup\infty ,
$$
where parameters $z$ are taken from the set of appropriate parameters and
$i(u,a)$ was defined in corollary \ref{ozamene3}.
\end{defi}
It seems that the number $d_D(a)$ will play the role of a higher order level
in a splittable division algebra. We will see that it codes a part of
information about a division algebra.
\begin{lemma}
\label{predvarit}
Let $D$ be a splittable division $p$-algebra, $p>2$, with a purely inseparable
simple extension
$Z(\bar{D})/\bar{F}$, let $u$ be some fixed embedding
$u:\bar{D}\hookrightarrow D$.
Suppose $Z(\bar{D})=\bar{F}(a)$ and
$(i(u,a),p)=1$. Suppose $d(u,a)\le 2i(u,a)$.
Let $z$ be a parameter such that ${}^{(z,u)}\delta_{i(u,a)}
({}^{(z,u)}_{-i(u,a)}\delta_{i(u,a)}(a))=0$, ${}^{(z)}\alpha =id$ and
${}^{(z,u)}_{-i(u,a)}\delta_{q}|_{{\mbox{\sdbl F}}_p(a)}=0$ for
$i(u,a)<q<d(u,a)$.
Put $j(k):=i(u,a^{p^k})$.
Suppose for every $k\ge 1$
a parameter $z_k$ such that
${}^{(z_k,u)}_{-j(k)}\delta_{r}|_{{\mbox{\sdbl F}}_p(a^{p^k})}=0$ for
$j(k)<r<d(u,a^{p^k})$ satisfy a condition
${}^{(z_k,u)}\delta_{i(u,a)}={}^{(z,u)}\delta_{i(u,a)}$, ${}^{(z_k)}\alpha ={}^{(z)}\alpha$.
Suppose for every
$k\ge 1$ we have
$d(u,a^{p^k})-j(k)=d(u,a)-j(0)$.
Then
the maps ${}^{(z,u)}_{w+(p-1-r)j(k)}\delta_{\zeta}$, $rj(k)<\zeta\le (r-1)j(k)+d(u,a^{p^k})$,
$r\in \{1, \ldots , p-1\}$, $k\ge 0$
satisfy the following properties:
$$
{}^{(z,u)}_{w+(p-1-r)j(k)}\delta_{\zeta}|_{{\mbox{\sdbl F}}_p(a^{p^k})}=c_{w+(p-1-r)j(k),\zeta ,1}\delta +\ldots +
c_{w+(p-1-r)j(k),\zeta ,r}\delta^{r},
$$
where the
derivation $\delta$ was defined in lemma \ref{(5)}, $c_{w+(p-1-r)j(k),\zeta ,r}\in Z(\bar{D})$,
$c_{w+(p-1-r)j(k),\zeta ,r}\ne 0$ only if $\zeta = (r-1)j(k)+d(u,a^{p^k})$.
Moreover, $c_{w+(p-1-r)j(k), (r-1)j(k)+d(u,a^{p^k}),r}\ne 0$ if
$w=i(u,a) \mbox{\quad mod\quad}p$;
$$c_{w+(p-1-r)j(k),(r-1)j(k)+d(u,a^{p^k}), r}=r!c_{w+(p-r)j(k),(r-2)j(k)+d(u,a^{p^k}), r-1}{}_{w+(p-1-r)j(k)}\delta_{j(k)}(a^{p^k}),$$
and ${}^{(z_k,u)}_{w+(p-2)j(k)}\delta_{d(u,a^{p^k})}(a^{p^k})=
{}^{(z_k,u)}_{-j(k)}\delta_{d(u,a^{p^k})}(a^{p^k})$.
\end{lemma}
{\bf Proof.} The proof is similar to the proof of lemma \ref{(5)}, (i). It is by induction on $r$ simultaneously for all $k\ge 0$.
For $r=1$, using lemma \ref{triviall} and induction, one can easily show that
${}^{(z_k,u)}\delta_{q}(a^{p^k})=-(j(k))^{-1}{}^{(z_k,u)}_{-j(k)}\delta_{q}(a^{p^k})$ for $j(k)\le q< d(u,a^{p^k})$ (we assume here $z_0=z$).
By lemma \ref{vtorinv}, (i) we have $d(u,a)-i(u,a)=i(u,a) \mbox{\quad mod\quad}p$. So, by lemma \ref{vtorinv}, (ii) and by induction we have $j(k)=j(0) \mbox{\quad mod\quad}p$.
So, ${}^{(z_k,u)}_{w+(p-2)j(k)}\delta_{q}|_{{\mbox{\sdbl F}}_p(a^{p^k})}=0$ if $j(k)< q< d(u,a^{p^k})$ and
${}^{(z_k,u)}_{w+(p-2)j(k)}\delta_{q}|_{{\mbox{\sdbl F}}_p(a^{p^k})}\ne 0$ only if $q=d(u,a^{p^k})$.
Since ${}^{(z_k,u)}_{-j(k)}\delta_{j(k)}|_{{\mbox{\sdbl F}}_p(a^{p^k})}$ is a derivation and
since, by proposition \ref{flyii}, (i), the map
${}^{(z_k,u)}_{-j(k)}\delta_{d(u,a^{p^k})}|_{{\mbox{\sdbl F}}_p(a^{p^k})}$ must be a derivation, we have
${}^{(z_k,u)}_{w+(p-2)j(k)}\delta_{d(u,a^{p^k})}(a^{p^k})\in Z(\bar{D})$. For, as it was shown in the proof of lemma \ref{(5)}, (ii)
for any derivation $\delta$ we have $\delta (b)\in Z(\bar{D})$ for any $b\in Z(\bar{D})$. Since
${}^{(z_k,u)}_{w+(p-2)j(k)}\delta_{d(u,a^{p^k})}(a^{p^k})= q_1{}^{(z_k,u)}_{-j(k)}\delta_{d(u,a^{p^k})}(a^{p^k})+
q_2{}^{(z_k,u)}_{m}\delta_{j(0)}({}^{(z_k,u)}_{-j(k)}\delta_{j(k)}(a^{p^k}))$ for some integer $q_1,q_2,m$, we have proved our assertion.
So, $c_{w+(p-2)j(k),d(u,a^{p^k}), 1}\in Z(\bar{D})$.
If $w=j(0) \mbox{\quad mod\quad}p$, then
${}^{(z_k,u)}_{w+(p-2)j(k)}\delta_{d(u,a^{p^k})}(a^{p^k})={}^{(z_k,u)}_{-j(k)}\delta_{d(u,a^{p^k})}(a^{p^k})$, since $w+(p-1)j(0)=0\mbox{\quad mod\quad}p$ and $char D> 2$. So, we have $c_{w+(p-2)j(k),d(u,a^{p^k}), 1}\ne 0$.
Put now $t=a^{p^k}$.
For arbitrary $r$ by proposition \ref{flyii}, (i) we have
$$
{}^{(z_k,u)}_{w+(p-1-r)j(k)}\delta_{\zeta}(t^q)=
q{}_{w+(p-1-r)j(k)}\delta_{\zeta}(t)t^{q-1}+
$$
$$
{}_{w+(p-1-r)j(k)}\delta_{j(k)}(t)
\sum_{l= 0}^{q-2}{}_{w+(p-r)j(k)}\delta_{\zeta -j(k)}(t^{q-1-l})t^l+
$$
$$
{}_{w+(p-1-r)j(k)}\delta_{d(u,t)}(t)
\sum_{l= 0}^{q-2}{}_{w+(p-1-r)j(k)+d(u,t)}\delta_{\zeta -d(u,t)}(t^{q-1-l})t^l+
$$
$$
\sum_{i=d(u,t)+1}^{\zeta -1} {}_{w+(p-1-r)j(k)}\delta_{i}(t)
\sum_{l= 0}^{q-2}{}_{w+(p-1-r)j(k)+i}\delta_{\zeta -i}(t^{q-1-l})t^l.
$$
Using the same arguments as in the proof of lemma \ref{(5)},(i) we see that ${}_{w+(p-1-r)j(k)}\delta_{\zeta}(t^p)=0$ and
${}_{w+(p-1-r)j(k)}\delta_{\zeta}|_{{\mbox{\sdbl F}}_p(t)}=c_{w+(p-1-r)j(k),\zeta ,1}\delta +\ldots +
c_{w+(p-1-r)j(k),\zeta ,p-1}\delta^{p-1}$. To show that $c_{w+(p-1-r)j(k),\zeta ,i}=0$ for $i>r$ it suffice, by formulae (\ref{recurrent}) in lemma \ref{(5)}, to show that all the maps in the formula above are represented in the form $c_1\delta +\ldots +c_{r-1}\delta^{r-1}$. Let us show it in details.
Since $\zeta -d(u,t)-1<(r-1)j(k)$, by lemma \ref{(5)}, (ii)
${}_{m}\delta_{\zeta -i}|_{{\mbox{\sdbl F}}_p(t)}=c_{m,\zeta -i,1}\delta +\ldots +c_{m,\zeta -i,r-2}\delta^{r-2}$ for any $i>d(u,t)$.
If $w=j(0) \mbox{\quad mod\quad}p$, then
$w+(p-1-r)j(k)+d(u,t)+(r-2)j(k)=0 \mbox{\quad mod\quad}p$. Since
$\zeta -d(u,t)\le (r-1)j(k)$, by lemma \ref{(5)}, (ii) we have
${}_{w+(p-1-r)j(k)+d(u,t)}\delta_{\zeta -d(u,t)}|_{{\mbox{\sdbl F}}_p(t)}=c_{w+(p-1-r)j(k)+d(u,t),\zeta -d(u,t),1}\delta +\ldots +c_{w+(p-1-r)j(k)+d(u,t),\zeta -d(u,t),r-2}\delta^{r-2}$.
If $w\ne j(0) \mbox{\quad mod\quad}p$, then by the same reason we have
${}_{w+(p-1-r)j(k)+d(u,t)}\delta_{\zeta -d(u,t)}|_{{\mbox{\sdbl F}}_p(t)}=c_{w+(p-1-r)j(k)+d(u,t),\zeta -d(u,t),1}\delta +\ldots +c_{w+(p-1-r)j(k)+d(u,t),\zeta -d(u,t),r-1}\delta^{r-1}$ and by lemma \ref{(5)}, (i) $c_{w+(p-1-r)j(k)+d(u,t),\zeta -d(u,t),r-1}\in Z(\bar{D})$ as a product of elements from $Z(\bar{D})$.
At last, by the induction hypothesis
${}_{w+(p-r)j(k)}\delta_{\zeta -j(k)}|_{{\mbox{\sdbl F}}_p(t)}=c_{w+(p-r)j(k),\zeta -j(k),1}\delta +\ldots +c_{w+(p-r)j(k),\zeta -j(k),r-1}\delta^{r-1}$ and $c_{w+(p-r)j(k),\zeta -j(k),r-1}\ne 0$ only if $\zeta -j(k)=(r-2)j(k)+d(u,t)$, and $c_{w+(p-r)j(k),\zeta -j(k),r-1}\in Z(\bar{D})$. Since ${}_{w+(p-1-r)j(k)}\delta_{j(k)}(t)\in Z(\bar{D})$, by formulae
(\ref{recurrent}) we get $c_{w+(p-1-r)j(k),\zeta ,r}\in Z(\bar{D})$ and if $w=j(0) \mbox{\quad mod\quad}p$, then
$c_{w+(p-1-r)j(k),\zeta ,r}\ne 0$ iff $\zeta =(r-1)j(k)+d(u,t)$,
$$c_{w+(p-1-r)j(k), (r-1)j(k)+d(u,t),r}=r!c_{w+(p-r)j(k),(r-2)j(k)+d(u,t), r-1}{}_{w+(p-1-r)j(k)}\delta_{j(k)}(t)\ne 0.$$
The lemma is proved.\\
$\Box$
\begin{lemma}
\label{predvarit2}
Let $D$ be a division algebra as in lemma \ref{predvarit}.
Suppose $d(u,a)\le 2i(u,a)$ and $char D>2$.
Then for every $k$ there exists a parameter
$z_k$ such that
${}^{(z_k,u)}_{-j(k)}\delta_{r}|_{{\mbox{\sdbl F}}_p(a^{p^k})}=0$ for
$j(k)<r<d(u,a^{p^k})$ and ${}^{(z_k)}\alpha ={}^{(z)}\alpha$,
${}^{(z_k,u)}\delta_{j(l)}={}^{(z,u)}\delta_{j(l)}$ for all $l\le k$ (we use here the notation defined in lemma \ref{predvarit}).
Moreover, for every
$k\ge 1$ we have
$d(u,a^{p^k})-j(k)=d(u,a)-j(0)$ and
$${}^{(z_{k},u)}_{-j(k)}\delta_{d(u,a^{p^k})}(a^{p^k})=-{}^{(z_{k-1},u)}_{-j(k-1)}
\delta_{d(u,a^{p^{k-1}})}(a^{p^k})c_{d(u,t)-j(k-1), j(k)-j(k-1),p-1},$$
where
$c_{d(u,t)-j(k-1), j(k)-j(k-1),p-1}$ is defined in lemma \ref{predvarit}.
\end{lemma}
{\bf Proof.} The proof is by induction on $k$. By lemma \ref{vtorinv}
$d(u,a)=2j(0)\mbox{\quad mod \quad }p$ and $j(1)=d(u,a)+(p-1)j(0)$. So, by the induction hypothesis we can assume for arbitrary $k$ that $d(u,a^{p^{k-1}})=2j(0)\mbox{\quad mod \quad }p$ and $j(k-1)=j(0)\mbox{\quad mod \quad }p$, and $j(k)=d(u,a^{p^{k-1}})+(p-1)j(k-1)$.
For the convinience we can start with a parameter $z=z_0$, which satisfy the conditions of lemma \ref{predvarit}. Indeed, taking an appropriate parameter $z$ and changing it by a parameter $u(c)z$ for an appropriate $c\in Z(\bar{D})$ (as in the proof of proposition \ref{cyclisity}), we can assume that ${}^{(z,u)}_{-j(0)}\delta_{j(0)}(a)\in Z(\bar{D})^p$. Now, using arguments from the proof of lemma \ref{vtorinv}, (i), we can find such a parameter $z_0$.
The idea of the proof is the following. We prove first that
${}^{(z_{k-1},u)}_{-j(k-1)}\delta_{j(k)+d(u,a)-j(0)}(a^{p^k})\ne 0$. Then we prove that there exists a parameter $z_k$ such that ${}^{(z_{k},u)}_{-j(k)}\delta_{\zeta}(a^{p^k})= 0$ for $j(k)<\zeta <j(k)+d(u,a)-j(0)$ and ${}^{(z_{k},u)}_{-j(k)}\delta_{j(k)+d(u,a)-j(0)}
(a^{p^k})\ne 0$. It will be shown that $z_k$ satisfy the conditions of lemma.
So, assume $j(k)\le \zeta \le j(k)+d(u,a)-j(0)=j(k)+d(u,a^{p^{k-1}})-j(k-1)$. Put
$t=a^{p^{k-1}}$.
By proposition \ref{flyii}, (i) we have
$$
{}^{(z_{k-1},u)}_{-j(k-1)}\delta_{\zeta}(t^p)=
$$
$$
{}^{(z_{k-1},u)}_{-j(k-1)}\delta_{d(u,t)}(t)
\sum_{l= 0}^{p-2}{}^{(z_{k-1},u)}_{d(u,t)-j(k-1)}\delta_{\zeta -d(u,t)}(t^{p-1-l})t^l+\ldots +
$$
$$
{}^{(z_{k-1},u)}_{-j(k-1)}\delta_{\zeta -(p-1)j(k-1)}(t)
\sum_{l= 0}^{p-2}{}^{(z_{k-1},u)}_{\zeta -pj(k-1)}\delta_{(p-1)j(k-1)}(t^{p-1-l})t^l+
$$
$$
\sum_{i=\zeta -(p-1)j(k-1)+1}^{\zeta -1} {}^{(z_{k-1},u)}_{-j(k-1)}\delta_{i}(t)
\sum_{l= 0}^{p-2}{}^{(z_{k-1},u)}_{i-j(k-1)}\delta_{\zeta -i}(t^{q-1-l})t^l.
$$
By lemma \ref{(5)}, (i) in the last sum
${}^{(z_{k-1},u)}_{i-j(k-1)}\delta_{\zeta -i}|_{{\mbox{\sdbl F}}_p(t)}=c_{i-j(k-1),\zeta -i,1}\delta +\ldots +c_{i-j(k-1),\zeta -i,p-2}\delta^{p-2}$, since $\zeta -i<(p-1)j(k-1)$. So, this sum is equal to zero.
By lemma \ref{(5)}, (ii) we have ${}^{(z_{k-1},u)}_{\zeta -pj(k-1)}
\delta_{(p-1)j(k-1)}|_{{\mbox{\sdbl F}}_p(t)}= c_{\zeta -pj(k-1),(p-1)j(k-1),1}\delta +\ldots +
c_{\zeta -pj(k-1),(p-1)j(k-1),p-1}\delta^{p-1}$ and
$c_{\zeta -pj(k-1),(p-1)j(k-1),p-1}\ne 0$ iff $\zeta =j(k-1)=j(0)\mbox{\quad mod\quad}p$.
By lemma \ref{(5)}, (i) we have
${}^{(z_{k-1},u)}_{m}\delta_{q}|_{{\mbox{\sdbl F}}_p(t)}=c_{m,q,1}\delta +\ldots +c_{m,q,p-1}\delta^{p-1}$ for $(p-1)j(k-1)<q<(p-1)j(k-1)+d(u,a)-j(0)$, and by lemma \ref{predvarit} $c_{m,q,p-1}=0$.
By lemma \ref{predvarit} we have
${}^{(z_{k-1},u)}_{d(u,t)-j(k-1)}\delta_{\zeta -d(u,t)}|_{{\mbox{\sdbl F}}_p(t)}=
c_{d(u,t)-j(k-1), \zeta -d(u,t), 1}\delta +\ldots + c_{d(u,t)-j(k-1), \zeta -d(u,t), p-1}\delta^{p-1}$ with $c_{d(u,t)-j(k-1), \zeta -d(u,t), p-1}\ne 0$ if $\zeta -d(u,t)=j(0)$.
So, we have the following picture: ${}^{(z_{k-1},u)}_{-j(k-1)}\delta_{\zeta}(t^p)\ne 0$ only if $\zeta =j(0)\mbox{\quad mod \quad }p$ or if $\zeta =j(k)+d(u,a)-j(0)$. In the last case $${}^{(z_{k-1},u)}_{-j(k-1)}\delta_{\zeta}(t^p)=-{}^{(z_{k-1},u)}_{-j(k-1)}
\delta_{d(u,t)}(t^p)c_{d(u,t)-j(k-1), j(k)-j(k-1),p-1},$$
where
$c_{d(u,t)-j(k-1), j(k)-j(k-1),p-1}$ can be calculated using lemma \ref{predvarit}.
Let's show that there exists a parameter $z_{k}$ such that
${}^{(z_{k},u)}_{-j(k-1)}\delta_{\zeta}(t^p)=0$ for $j(k)<\zeta <j(k)+ d(u,a)-j(0)$.
By lemma \ref{ozamene}, (ii) there exists a change of parameters $z_{k-1}\mapsto z'=z_{k-1}+bz_{k-1}^{p+1}$ such that ${}^{(z',u)}_{-j(k-1)}\delta_{j(k)+p}(t^p)=0$.
It suffice to prove that any such a change of parameters as in lemma \ref{ozamene}, (ii) with $p|q$ changes only the values of maps ${}_{-j(k-1)}\delta_{\zeta}$ with
$\zeta =j(0)\mbox{\quad mod \quad }p$. For, if it is true, we can make several changes and kill all nonzero maps ${}^{(z_{k-1},u)}_{-j(k-1)}\delta_{\zeta}$ with $j(k)<\zeta <j(k)+ d(u,a)-j(0)$, since they are derivations and therefore are completely defined by their values at $t^p$.
To prove it, we can use the calculations in the proof of lemma \ref{ozamene}, (ii). Since
$d(u,a)-j(0)\le j(0)$, it is easy to see that for a change $z\mapsto z'=z+bz^{kp+1}$, $p>2$ we have there
$$
z'^{-j(k-1)}t^pz'^{j(k-1)}=t^p+{}^{(z,u)}_{-j(k-1)}\delta_{j(k)}(t^p)z^{j(k)}+\ldots +
{}^{(z,u)}_{-j(k-1)}\delta_{j(k)+j(0)}(t^p)z^{j(k)+j(0)}+\ldots .
$$
Since $z'=z+bz^{kp+1}$, any power $z^{l}$ can be expressed as a series in $z'$, all powers of which are equal to $l$ modulo $p$. So, this change will change only maps with right indexes equal to $j(k)$ modulo $p$.
Since ${}^{(z_{k-1},u)}_{-j(k-1)}\delta_{\zeta}(t^p)\ne 0$ only if $\zeta =j(0)\mbox{\quad mod \quad }p$ for $\zeta <j(k)+d(u,a)-j(0)$, our assertion is proved.
So, there exists a parameter $z_k$ we have: ${}^{(z_k,u)}_{-j(k-1)}\delta_{\zeta}(t^p)\ne 0$ only if $\zeta =j(k)+d(u,a)-j(0)$ or $\zeta =j(k)$. Since $z_k$ was constructed as a sequence of changes as in lemma \ref{ozamene}, (ii), we have ${}^{(z_k)}\alpha ={}^{(z_{k-1})}\alpha$ and
${}^{(z_k,u)}\delta_{j(q)}={}^{(z_{k-1},u)}\delta_{j(q)}$ for any $q\le k$.
At last, let's prove that ${}^{(z_k,u)}_{-j(k)}\delta_{\zeta}(t^p)\ne 0$ only if $\zeta =j(k)+d(u,a)-j(0)$ or $\zeta =j(k)$. But this follows immediately from the definition of these maps, since $j(k)=j(k-1)\mbox{\quad mod \quad}p$, $d(u,a)-j(0)\le j(0)$ and $char D>2$. In particular,
${}^{(z_k,u)}_{-j(k)}\delta_{j(k)}(t^p)={}^{(z_k,u)}_{-j(k-1)}\delta_{j(k)}(t^p)$,
${}^{(z_k,u)}_{-j(k)}\delta_{j(k)+d(u,a)-j(0)}(t^p)={}^{(z_k,u)}_{-j(k-1)}\delta_{j(k)+d(u,a)-j(0)}(t^p)$.
The lemma is proved.\\
$\Box$
Now we can prove the following theorem.
\begin{th}
\label{posledn}
Let $D$ be a division $p$-algebra of $char D=p>2$ with the center $Z(D)=F$. Suppose $Z(\bar{D})=\bar{D}$ and $\bar{D}/\bar{F}$ is a simple purely inseparable extension,
$\bar{D}=\bar{F}(a)$. Suppose that the semilocal height $i(u)$, which does not depend on the embedding $u$ in this case, is not divisible by $p$.
Then $d_D(a)>i(u)$.
\end{th}
{\bf Proof.} By lemma \ref{simple2}, (ii) $[\bar{D}:\bar{F}]=[\Gamma_D:\Gamma_F]$. So, the field $F(\tilde{a})$, where $\tilde{a}$ is a lift of $a$, is a maximal "unramified" subfield and therefore $D$ is a splittable division algebra. Obviously, $\alpha =id$.
Since ${}^{(z,u)}\delta_{i(u,z)}$ is a derivation and $\bar{D}/\bar{F}$ is a simple extension, ${}^{(z,u)}\delta_{i(u,z)}$ is completely defined by a value at $a$. So, by lemma \ref{ozamene} $i(u,z)$ does not depend on $z$ and $i(u,z)=i(u)$.
Therefore, $i(u)=w(zu(a)z^{-1}-u(a))$ and $i(u)$ is completely defined by the lift $u(a)$. From the other hand side,
any lift $\tilde{a}$ of $a$ defines, by lemma \ref{simple}, an embedding $\tilde{a}$, and by lemma
\ref{ozamene2} $i(\tilde{a})$ does not depend on $\tilde{a}$. So, $i(u)$ does not depend on $u$.
The idea of the proof is following. We consider linear spaces which are the images of the maps ${}^{(z,u)}\delta_{j(k)}|_{\bar{F}(a^{p^k})}$ in $\bar{D}$, where $j(k)$ were defined in lemma \ref{predvarit2} and $z,u$ are fixed. We show that every such spase has zero intersection with each other if $d_D(a)\le i(u)$. Then we show that this contradicts with the fact that $u(a)$ generate a finite dimensional space over $F$.
So, assume $d_D(a)\le i(u)$.
To calculate the spaces ${}^{(z,u)}\delta_{j(k)}({\bar{F}(a^{p^k})})\in \bar{D}$ we use lemmas \ref{vtorinv}, \ref{predvarit} and \ref{predvarit2}. We fix a parameter $z$ defined in lemma \ref{predvarit}. By lemmas \ref{simple}, \ref{ozamene2}, (iii) we can find a primitive element $\bar{u}\in \bar{D}$ of the extension $\bar{D}/\bar{F}$ such that ${}^{(z,u)}\delta_{j(0)}(\bar{u})=1$, where $u$ is an embedding defined in lemma \ref{simple} for some lift $u$ of the element $\bar{u}$. Using lemma \ref{ozamene}, (ii) we can find an embedding $u$ such that ${}^{(z,u)}\delta_{d(u,\bar{u})}(\bar{u})\notin {}^{(z,u)}\delta_{j(0)}(\bar{D})$. We fix this embedding. From lemmas \ref{ozamene}, \ref{ozamene2} immediately follows that $d(u,\bar{u})=d_D(\bar{u})=d_D(a)$. So, we assume without loss of generality $a=\bar{u}$.
Put $J(k):={}^{(z,u)}\delta_{j(k)}(a^{p^k})$. Put $A(k):={}^{(z,u)}\delta_{j(k)}(\bar{F}(a^{p^k}))$, $A'(k):=\bar{F}(a^{p^{k+1}})\cdot a^{p^k(p-1)}J(k)$.
We have $A(k)=\oplus_{q=0}^{p-2}\bar{F}(a^{p^{k+1}})\cdot a^{p^kq}J(k)$ and
$\bar{D}\cdot J(k)=A(k)\oplus A'(k)$ as ${\mbox{\dbl F}}_p$-linear spaces.
From lemma \ref{vtorinv} follows that
$${}^{(z,u)}\delta_{j(k)}(a^{p^k})={}^{(z_k,u)}\delta_{j(k)}(a^{p^k})=
q{}^{(z_{k-1},u)}_{-j(k-1)}\delta_{d(u,a^{p^{k-1}})}(a^{p^{k-1}})c_{d(u,a^{p^{k-1}})-j(k-1), (p-1)j(k-1), p-1},$$
where $q\in {\mbox{\dbl F}}_p^*$, $z_k$ were defined in lemma \ref{predvarit}, $c_{d(u,a^{p^{k-1}})-j(k-1), (p-1)j(k-1), p-1}$ is calculated in lemma \ref{(5)}, (i) and it is not equal to zero by lemma \ref{(5)}, (ii), and ${}^{(z_{k-1},u)}_{-j(k-1)}\delta_{d(u,a^{p^{k-1}})}(a^{p^{k-1}})$ is calculated in lemma \ref{predvarit2}.
By lemma \ref{predvarit2} we have ${}^{(z_{k-1},u)}_{-j(k-1)}\delta_{d(u,a^{p^{k-1}})}
(a^{p^{k-1}})=-j(k-1){}^{(z,u)}\delta_{d(u,a^{p^{k-1}})}(a^{p^{k-1}})$.
Combining all these calculation together and using induction, we get $J(k)=q_kJ(k-1)^pJ(1)=\tilde{q_k}J(1)^{p^{k-1}+p^{k-2}+\ldots +1}$ for $k\ge 1$, where $q_k\in {\mbox{\dbl F}}_p$.
Therefore, there is the following filtration
$$
\bar{F}\subset \ldots \subset \bar{F}(a^{p^{k+1}})J(k+1)\subset \bar{F}(a^{p^k})J(k)\subset \ldots \subset \bar{D},
$$
and for every $k\ge 1$ we have $\bar{F}(a^{p^k})\cdot J(k)\subset A'(k-1)$. So, $A(k)\cap A(k_1)=\{0\}$ if $k\ne k_1$.
Now consider an element $b\in F$ such that $\bar{b}=a^{p^l}$ for some $l>0$. We assume $l$ is a minimal possible integer. It exists, because $D$ is a finite dimensional algebra over $F$. Let
$b=u(a^{p^l})+b_1z+\ldots $, where $b_k\in u(\bar{D})$. Put $I:=\min \{w(zb_kz^{k-1}-b_kz^k)\}$ (we assume here that $b_0=u(a^{p^l})$). Note that $I<\infty$, since
by lemma \ref{predvarit2} $j(l)<\infty$, i.e. ${}^{(z,u)}\delta_{j(l)}(a^{p^l})\ne 0$. Now we must have
$$
zbz^{-1}=\sum_{k=0}^{\infty}zb_kz^{k-1}= b+\sum_r {}^{(z,u)}\delta_{j(r)}(b_{q_r})z^I+\ldots =b,
$$
where $b_{q_r}\in \bar{F}(a^{p^r})$ and $b_{q_r}\notin \bar{F}(a^{p^{r+1}})$. So, $\sum_r {}^{(z,u)}\delta_{j(r)}(b_{q_r})=0$, but it is impossible, since $A(k)\cap A(k_1)=\{0\}$ if $k\ne k_1$, a contradiction.
The theorem is proved. \\
$\Box$
{\bf Remark.} It would be interesting to know the answer on the following questions.
i) Suppose $D$ is a division algebra as in the theorem \ref{posledn}. Does there exist a pair $(z,u)$ such that all nonzero maps ${}^{(z,u)}\delta_q$ satisfy the property $i(u)|q$? If it is true, there is a subalgebra $D'\subset D$ with $[D:D']<\infty$ and $D'$ has level 1 (see remark before lemma \ref{vtorinv}). So, we can reduce studying of $D$ to the algebra of level 1.
ii) Is it true that $D$ is a good splittable algebra, i.e. cyclic? Probably, it is possible to apply our technique to give an answer to this question at least in the case of level 1.
|
{
"timestamp": "2005-03-28T16:54:44",
"yymm": "0503",
"arxiv_id": "math/0503637",
"language": "en",
"url": "https://arxiv.org/abs/math/0503637"
}
|
\section{Introduction}
For the free Bose gas with Dirichlet and Neumann boundary conditions, Bose-Einstein Condensation (BEC) is rigorously treated in \cite{LP,LW}. For the mean-field Bose gas with periodic boundary conditions BEC is rigorously proved and a detailed analysis of the thermodynamic limit is given in \cite{FV}. The proof is based on bounds on the correlation functions for equilibrium states, given in terms of the correlation inequalities \cite{FV2,FV3}. The subtle point in this proof is the analysis of the singularity around the zero-mode.\\
If one considers attractive boundary conditions instead of periodic boundary conditions, the problem changes drastically. The free Bose gas with attractive boundary conditions is extensively studied in \cite{R,LW}. Due to the gap in the one-dimensional one-body problem, one has Bose-Einstein Condensation in all dimensions $\nu\geq 1$. An important result is the fact that the condensation is a surface effect. In \cite{VVZ} it is computed that the condensate is localized at a logarithmic distance from the boundary.\\
The subject of this note is to proceed with the imperfect Bose gas with attractive boundary conditions, i.e.\ the free Bose gas with attractive boundary conditions plus a mean field term. The first problem that occurs is to express this mean field term in momentum space diagonalizing the kinetic energy (the free Bose gas part). As the spectrum of the latter one has two strictly negative eigenvalues, say $\epsilon_L(0)$ and $\epsilon_L(1)$, separated from the rest of the spectrum, then one can discuss the corresponding number operators $N_0$ and $N_1$ as being added to the total number operator in the interaction or not. In section \ref{interact_term} we argue why they should not be present. The argument is essentially based on the fact that we want a space homogeneous mean field term. The model is defined in \ref{hamiltonian}.\\
In section \ref{condensatie} we give a completely rigorous proof of the occurence of Bose-Einstein Condensation for the imperfect Bose gas with attractive boundary conditions. We perform all details only in dimension $\nu =1$. From dimension $\nu\geq 2$ on, the proof becomes technically more tedious. In particular, because of the fact that the condensate is located near the boundaries, for higher dimensions the thermodynamic limit for hypercubic boxes is not very suitable nor realistic and should in stead be taken with increasing absorbing balls. But we leave this extra exercise for a later occasion in which we consider the problem of the shape-dependence.\\
In the one-dimensional case we remark that the condensation is equally distributed over the two negative energy levels. The condensation is localized in the same area as for the free Bose gas with attractive boundary conditions, see \cite{R}.
\section{The Model}
\subsection{Attractive Boundary conditions}
If one considers a free gas of bosons in an interval $\left[-L/2,L/2\right]$ of length $L$, then the energy levels are determined by the one-dimensional Schr\"odinger equation (with units $\frac{\hbar^2}{2m}=1$)
\begin{equation*}
-\Delta \phi = \epsilon_L \phi^L ,
\end{equation*}
with boundary
conditions:
$$\left\{
\begin{array}{lll}
\left(\displaystyle\frac{d\phi}{dx}-\sigma\phi\right)_{x=-L/2} & = & 0 ,\\
\left(\displaystyle\frac{d\phi}{dx}+\sigma\phi\right)_{x=L/2} & =
& 0 ,
\end{array} \right.$$
where $\sigma<0$.\\
If one considers these attractive boundary conditions, the spectrum consists of two negative eigenvalues tending to the same limit $-\sigma^2$ (when $L\rightarrow \infty$) and an infinite number of positive
eigenvalues (for $L|\sigma|>2$): $\epsilon_L(k)$ for $k=0,1,2,\ldots$, where
$$\epsilon_L(0) < \epsilon_L(1) <
0 < \epsilon_L(2) < \epsilon_L(3) < \ldots ,$$
$$\epsilon_L(0)=-\sigma^2- O(\mathrm{e}^{-L|\sigma|}) ,$$
$$\epsilon_L(1)=-\sigma^2+O(\mathrm{e}^{-L|\sigma|}) ,$$
\begin{equation}\label{spect}
k\geq 2:\ \ \left(\frac{(k-1)\pi}{L}\right)^2 < \epsilon_L(k) <
\left(\frac{k\pi}{L}\right)^2 .
\end{equation}
The corresponding eigenfunctions $\{\phi_k^L\}_{k \in\mathbbm{N}}$ are given by
\begin{eqnarray}
\phi_0^L(x) & = &
\sqrt{\frac{2}{L}}\left(1+\frac{\sinh(L|\sigma|)}{L|\sigma|}\right)^{-1/2}
\cosh(-|\sigma|x) ,\nonumber\\ \phi_1^L(x) & = &
\sqrt{\frac{2}{L}}\left(-1+\frac{\sinh(L|\sigma|)}{L|\sigma|}\right)^{-1/2}
\sinh(-|\sigma|x) ,\nonumber\\ \phi_k^L(x)& = & \left\{
\begin{array}{ll}
\sqrt{\displaystyle\frac{2}{L}}\left(1+\displaystyle\frac{\sin
(\sqrt{\epsilon_L(k)}L)}
{\sqrt{\epsilon_L(k)}L}\right)^{-1/2}\cos(\sqrt{\epsilon_L(k)}x) ,
\qquad & \mbox{for $k$ \ even} ,\\
\sqrt{\displaystyle\frac{2}{L}}\left(1-\displaystyle\frac{\sin
(\sqrt{\epsilon_L(k)}L)}
{\sqrt{\epsilon_L(k)}L}\right)^{-1/2}\sin(\sqrt{\epsilon_L(k)}x) ,
\qquad & \mbox{for $k$ odd} .
\end{array}\right.\nonumber
\end{eqnarray}
\subsection{Hamiltonian}\label{hamiltonian}
We consider a one-dimensional system of identical bosons on an interval $[-\frac{L}{2},\frac{L}{2}]\subset\mathbbm{R}$ with attractive boundary conditions.
The model is specified by the local Hamiltonians $H_{L, MF}^{\sigma}$ on the boson Fock space $\mathcal{F}_{L, B}$:
\begin{equation}\label{ham}
H_{L, MF}^{\sigma} = T_{L}^{\sigma} + \frac{\l}{2}\frac{\tilde{N}_L^2}{L}
\end{equation}
where $T_{L}^{\sigma}$ is the kinetic energy operator with the $\epsilon_L(k)$ the eigenvalues (\ref{spect}) of the free Laplacian with attractive boundary conditions:
\begin{equation*}
T_{L}^{\sigma} = \sum_{k\in\mathbbm{N}}\epsilon_L(k) a_{k}^\ast a_{k}
\end{equation*}
The operators $a_k^\ast=a^\ast(\phi_L^k)$ and $a_k=a(\phi_L^k)$ are the Bose creation and annihilation operators with the testfunctions $\phi_L^k$ the above eigenfunctions of the free Laplacian with attractive boundary conditions. The total particle number operator is denoted by $N_L=\sum_{k\in\mathbbm{N}}N_k = \sum_{k\in\mathbbm{N}} a_{k}^\ast a_{k}$, and the particle number operator corresponding to the positive spectrum by $\tilde{N}_L$:
\begin{equation*}
\tilde{N}_{L} = \sum_{k=2}^{\infty}a_{k}^\ast a_{k}
\end{equation*}
We consider a positive coupling constant $\l \in \mathbbm{R}^+$ for the sake of thermodynamic stability, see \cite{FV}.
\subsection{The Interaction Term}\label{interact_term}
Remark that the interaction in the Hamiltonian $H_{L, MF}^{\sigma}$ is not of the usual form $\frac{\l}{2}\frac{N_L^2}{L}$, with
\begin{eqnarray}
N_L & = & \int_{-L/2}^{L/2} \, dx \, a^\ast(x)a(x)\label{number}\\
& = & \sum_{k\in\mathbbm{N}}a_k^\ast a_k,\nonumber
\end{eqnarray}
but of the form $\frac{\l}{2}\frac{\tilde{N}_L^2}{L}$, where
\begin{equation*}
\tilde{N}_L = \sum_{k=2}^\infty a_k^\ast a_k.
\end{equation*}
The reason for choosing $\tilde{N}_L$ in stead of $N_L$ is the breaking of the spatial translation invariance in the two terms with $k=0$ and $k=1$. Moreover the latter two terms yield also gauge symmetry breaking under the effect of space translations. Using the straightforward computation
\begin{equation*}
\cosh(|\sigma|(x+a)) = \cosh(|\sigma|x)\mathrm{e}^{-|\sigma|a}+\mathrm{e}^{|\sigma|x}\sinh(|\sigma|a)
\end{equation*}
for $a\in\mathbbm{R}$, we get (for large $L$):
\begin{eqnarray*}
\tau_a (a_0^\ast) & \approx & a_0^\ast \mathrm{e}^{-|\sigma|a}+(a_0^\ast - a_1)\sinh(|\sigma|a)\\
\tau_a (a_0) & \approx & a_0 \mathrm{e}^{-|\sigma|a}+(a_0 - a_1^\ast)\sinh(|\sigma|a)
\end{eqnarray*}
where $\tau_a$ is the translation automorphism over the distance $a\in\mathbbm{R}$.
One gets a similar expression for $\tau_a (a_1^\sharp)$.\\
In particular, the terms $\tau_a (N_0)$ and $\tau_a (N_1)$ are not translation invariant and diverge exponentially for $|a|\rightarrow\infty$ as
\begin{equation*}
\tau_a (a_0^\ast a_0) \approx \mathrm{e}^{2|\sigma||a|}a_0^\ast a_0
\end{equation*}
moreover, these terms break the gauge symmetry.\\
Remark also that on the other hand the particle number operator $\tilde{N}_L$ is a good local approximation of $N_L$ (\ref{number}) for all translation invariant states, such that the interaction term $\frac{\l}{2}\frac{\tilde{N}_L^2}{L}$ is the appropriate mean field term.
\section{Bounds on the correlation function and condensation}\label{condensatie}
Our aim is to find the equilibrium states of the system in the grand canonical ensemble. The equilibrium state $\omega_L$ at inverse temperature $\b$ is characterized by the following correlation inequality for all $L$, see \cite{FV2,FV3}
\begin{equation}\label{corr_ineq}
\b \omega_L(X^\ast [H_{L, MF}^{\sigma}-\mu_L N_L,X]) \geq \omega_L(X^\ast X)\ln \frac{\omega_L(X^\ast X)}{\omega_L(X X^\ast)}
\end{equation}
for all local observables $X$, where $\omega_{L}(\cdot)$ is the grand canonical equilibrium state at chemical potential $\mu_L$ and inverse temperature $\b$:
\begin{equation*}
\omega_{L}(X) = \frac{\Tr_{\mathcal{F}_{L, B}}X\exp\{-\b (H_{L, MF}^\sigma -\mu_L N_L)\}}{\Tr_{\mathcal{F}_{L, B}}\exp\{-\b H_{L, MF}^\sigma\}}
\end{equation*}
with $\mathcal{F}_{L, B}$ the boson Fock space over $\mathcal{L}^2([-L/2,L/2])$.\\
Concerning the thermodynamic limit ($L\rightarrow\infty$), we perform this limit keeping the total density $\rho$ constant. Therefore the chemical potential $\mu_L$ is now determined by the particle density $\rho$ and is the solution of the particle density equation: for each given density $\rho$ we have
\begin{equation}\label{part_density}
\rho = \frac{\omega_L(N_L)}{L}
\end{equation}
From the correlation inequality (\ref{corr_ineq}) follows immediately the inequality
\begin{equation}\label{corr_ineq2}
\omega_L\left(\big[X^\ast,[H_{L, MF}^\sigma -\mu_L N_L , X]\right]\big)\geq 0 .
\end{equation}
In this section we focus on the proof of the condensation for the model $H_{L, MF}^{\sigma}$ (\ref{ham}). In this proof we need bounds which we derive from the correlation inequality (\ref{corr_ineq}) for some specific observables $X$'s. Due to the special character of the spectrum (\ref{spect}), it is necessary to distinguish between products of creation and annihilation operators $a^\sharp_k$ in the $0$- or $1$-mode, and those in the $k$-mode (with $k\geq 2$).\\
The first Lemma is valid for all $k\in\mathbbm{N}$.
\begin{lemma}\label{Nj_Nk}
If $j \neq k_i$ $(i = 1,\ldots ,m)$, $k_i \neq k_{i'}$ for $i \neq i'$ and
$m,n_i \in \mathbbm{N}_0$ for $i = 1,\ldots ,m$, then
\begin{eqnarray}
\lefteqn{\mathrm{e}^{\b (\epsilon_L(j)-\epsilon_L(k_1))}\omega_L\left(N_{j}
(N_{k_1}+1)^{n_1} (N_{k_2})^{n_2} \ldots (N_{k_m})^{n_m}\right)}\nonumber\\
& = & \omega_L\left((N_{j}+1)(N_{k_1})^{n_1} (N_{k_2})^{n_2} \ldots
(N_{k_m})^{n_m}\right)\label{lemma1}
\end{eqnarray}
\end{lemma}
\textit{Proof:}
The proof follows from the correlation inequality (\ref{corr_ineq}) by taking $X$
successively equal to
\begin{equation*}
a_{j}^\ast a_{k_1}\left((N_{k_1})^{n_1-1} (N_{k_2})^{n_2} \ldots
(N_{k_m})^{n_m}\right)^{1/2}
\end{equation*}
and
\begin{equation*}
a_{k_1}^\ast a_{j}\left((N_{k_1})^{n_1-1} (N_{k_2})^{n_2} \ldots
(N_{k_m})^{n_m}\right)^{1/2}
\end{equation*}
\hfill $\square$\\
For the chemical potential $\mu_L$, we find the same upperbound as in the case of the free Bose gas with attractive boundary conditions.
\begin{lemma}\label{mu}
For $k=0,1$, one has
\begin{equation*}
\mu_L \leq \epsilon_L(k)
\end{equation*}
and
\begin{equation}\label{mu_expr}
\mu = \lim_{L\rightarrow\infty}\mu_L \leq -\sigma^2
\end{equation}
\end{lemma}
\textit{Proof:}
Take $X = a_{k}^\ast$ where $k=0,1$ in the inequality (\ref{corr_ineq2}).\\
The second inequality (\ref{mu_expr}) is obtained by taking the thermodynamic limit $L$ tending to infinity.
\hfill $\square$
\begin{lemma}\label{Nk_B}
For each $k=0,1$ and $\mu_{L}<-\sigma^2$, we have:
\begin{equation}
\omega_{L}(N_{k}) = \frac{1}{\mathrm{e}^{\b(\epsilon_L(k)-\mu_L)}-1}
\end{equation}
\end{lemma}
\textit{Proof:}
The result follows from the inequality (\ref{corr_ineq}) by taking $X$ successively equal to $a_{k}$ and $a_{k}^\ast$ with $k=0,1$.
\hfill $\square$
\begin{lemma}\label{corr_ineq_Nk}
For each $n\in\mathbbm{N}$ and $k\geq 2$, we have
\begin{equation*}
\b\omega_{L}\left(-\epsilon_{L}(k)N_{k}^{n+1}+\mu_{L} N_{k}^{n+1}-\l \frac{\tilde{N}_{L}}{L}N_{k}^{n+1} +\frac{\l}{2}\frac{N_{k}^{n+1}}{L}\right)\geq \omega_L(N_{k}^{n+1})\ln \frac{\omega_L(N_{k}^{n+1})}{\omega_L((N_{k}+1)^{n+1})}
\end{equation*}
\end{lemma}
\textit{Proof:} The result follows from the correlation inequality (\ref{corr_ineq}) with $X=a_k N_k^{n/2}$.
\hfill $\square$\\ \\
In order to prove condensation, we need a convenient upperbound for $\omega_{L}(N_{k})$ for all $k\in\mathbbm{N}$. This bound is derived in the following Lemma.
\begin{lemma}\label{Nk}
For each $k\geq 2$ we have
\begin{equation*}
\omega_{L}(N_{k}) \leq \frac{1}{\mathrm{e}^{c_{k}(L)}-1}
\end{equation*}
where
\begin{equation}\label{ck}
c_{k}(L) = \b\left(\epsilon_{L}(k)+\sigma^2-\frac{\l}{2L}-o(\mathrm{e}^{-L|\sigma|})\right)
\end{equation}
\end{lemma}
\textit{Proof:}
By Lemma \ref{corr_ineq_Nk} with $n=0$:
\begin{equation*}
\omega_{L}(N_{k})\ln\frac{\omega_{L}(N_{k})}{\omega_{L}(N_{k})+1} \leq \b\omega_L\left(-\epsilon_{L}(k)N_{k} + \mu_{L} N_{k} - \l\frac{\tilde{N}_{L}}{L}N_{k} + \frac{\l}{2}\frac{N_{k}}{L}\right)
\end{equation*}
From Lemma \ref{mu} we know that $\mu_L \leq -\sigma^2 - o(\mathrm{e}^{-L|\sigma|})$. It is also easy to see that $\omega_L(\tilde{N}_L N_k) \geq 0$. This leads to
\begin{equation*}
\omega_{L}(N_{k})\ln\frac{\omega_{L}(N_{k})}{\omega_{L}(N_{k})+1} \leq -\b\left(\epsilon_{L}(k) + \sigma^2 + o(\mathrm{e}^{-L|\sigma|}) + \frac{\l}{2L}\right)\omega_\L(N_k)
\end{equation*}
which gives us immediately the result.
\hfill $\square$\\ \\
Using the results of the previous Lemma's, we are now ready to prove the existence of condensation in the two lowest energy levels.
\begin{theorem}\label{condensation}
Let $\rho_{cond}$ be equal to
\begin{equation*}
\rho_{cond} = \lim_{L\rightarrow\infty} \frac{1}{L}\omega_L\left(N_0 + N_1\right).
\end{equation*}
Then
\begin{equation}\label{condensation_expr}
\rho_{cond} \geq \rho - \frac{1}{\pi}\int_{0}^{\infty}\, dk \frac{1}{\mathrm{e}^{\b(k^2+\sigma^2)}-1}
\end{equation}
where $\rho_{cond}$ is the density of the condensate. The condensate density is localized in the $2$ lowest energy levels.
\end{theorem}
\textit{Proof:}
From the definition of the particle density $\rho$ (\ref{part_density}), we have
\begin{equation*}
\frac{1}{L}\omega_L\left(N_0 + N_1\right) = \rho - \frac{1}{L} \sum_{k=2}^\infty\omega_{L}(N_k)
\end{equation*}
By using the estimate of Lemma \ref{Nk}, one gets
\begin{equation*}
\frac{1}{L}\omega_L\left(N_0 + N_1\right) \geq \rho - \frac{1}{L} \sum_{k=2}^\infty \frac{1}{\mathrm{e}^{c_{k}(L)}-1}
\end{equation*}
with the $c_k(L)$'s as in (\ref{ck}).\\
Taking the thermodynamic limit $L\rightarrow\infty$ gives us the result (\ref{condensation_expr}).
\hfill $\square$\\ \\
Clearly (\ref{condensation_expr}) shows condensation. Indeed, remark that the integral is convergent for all $\sigma\neq 0$ and that it decreases for $\b$ increasing. Hence for $\rho$ large enough or for $\b$ large enough, it follows that the condensate density $\rho_{cond}$ is strictly positive.\\
Finally we derive a result about the type of condensation. We prove that the condensate density is realized in both the two lowest energy modes, with equal weight in the thermodynamic limit.
\begin{theorem}
\begin{itemize}
\item[(i)]The condensate is equally distributed on the two lowest energy levels.
\item[(ii)]From this, one can compute the asymptotics of the chemical potential $\mu_L$ for large $L$:
\begin{equation}\label{mu_asymp}
\mu_L = -\sigma^2 - \frac{2}{\b\rho_{cond} L} + o(L^{-1})
\end{equation}
\end{itemize}
\end{theorem}
\textit{Proof:}
\begin{itemize}
\item[(i)] From Lemma \ref{Nj_Nk}, with $m=1$, $j=1$ and $k=0$, we get
\begin{equation*}
\left(\mathrm{e}^{\b(\epsilon_L(1)-\epsilon_L(0))}-1\right)\omega_L(N_1 N_0) + \mathrm{e}^{\b(\epsilon_L(1)-\epsilon_L(0))}\omega_L(N_1) = \omega_L(N_0)
\end{equation*}
Using the spectral properties
\begin{eqnarray*}
\epsilon_L(0) & = & -\sigma^2- O(\mathrm{e}^{-L|\sigma|})\\
\epsilon_L(1) & = & -\sigma^2+O(\mathrm{e}^{-L|\sigma|})
\end{eqnarray*}
such that
\begin{equation*}
\epsilon_L(1)-\epsilon_L(0)\approx o(\mathrm{e}^{-L|\sigma|}),
\end{equation*}
then
\begin{equation*}
\frac{\omega_\L(N_0)}{L} = \frac{\omega_\L(N_1)}{L} + o(\mathrm{e}^{-L|\sigma|}).
\end{equation*}
Taking the thermodynamic limit $L\rightarrow\infty$, one gets (i).
\item[(ii)] From Lemma \ref{Nk_B} and since we know that the condensate is equally distributed on the two lowest energy levels, we have
\begin{equation*}
\lim_{L\rightarrow\infty}\frac{\omega_L(N_{0})}{L} = \lim_{L\rightarrow\infty} \frac{1}{L}\frac{1}{\mathrm{e}^{\b(\epsilon_L(0)-\mu_L)}-1} = \frac{\rho_{cond}}{2}
\end{equation*}
Series expansion of this expression with respect to the quantities $\epsilon_L(0)-\mu_L$ and $\epsilon_L(0)=-\sigma^2- O(\mathrm{e}^{-L|\sigma|})$, gives us the asymptotics of $\mu_L$.
\end{itemize}
\hfill $\square$
|
{
"timestamp": "2005-03-29T13:17:36",
"yymm": "0503",
"arxiv_id": "math-ph/0503068",
"language": "en",
"url": "https://arxiv.org/abs/math-ph/0503068"
}
|
\section{Introduction}
There are countless experiments which demonstrate the wave behaviour of light. Two typical experiments are the two-slit and Mach-Zehnder arrangements. That such experiments demonstrate the wave behaviour of light, even where the light is feeble\footnote{By feeble light we mean light of such low intensity that on average only one photon at a time is in the apparatus.} \cite{T09}, is not in dispute. What is questionable is the experimental evidence for the particle behaviour of light.
To avoid later misunderstanding of the essential point of this article, it is necessary for me to make clear that I use the term `particle behaviour' to refer to the description prior to the final detected result but not to the character of the final detected result. This is a more restrictive usage than is usual in the literature where the term `particle behaviour' also encompasses the character of the final detected experimental result. I also use the term `particle behaviour' in two context dependent ways: In the context of Bohr's principle of complementarity I use the term `particle behaviour' to refer to the description of the experiment in terms of the complementary particle concept (understanding that according to Bohr the particle concept, along with other complementary concepts, is an abstraction to aid thought to which physical reality cannot be attached). In the context of the causal interpretation I take the term `particle behaviour' to be synonymous with `particle ontology'. Similar considerations apply to the term `wave behaviour', but the distinction here is not so crucial since a main point of this article is to demonstrate that a final detected result showing a particle character does not force a particle description or particle ontology prior to the final detected result.
More recent and interesting experiments concerning particle-wave duality and complementarity have been suggested and subsequently performed. Ghose {\it et al} \cite{GHOSE91} proposed an experiment involving tunneling between two closely spaced prisms which has since been carried out by Mizobuchi {\it et al} \cite{MIZ92} (although the statistical results of the experiment have been questioned by \cite{UNNIK, GHOSE99, BRIDA04}). Later, Brida {\it et al} \cite{BRIDA04} realized an experiment suggested by Ghose \cite{GHOSE99} in which tunneling at a twin prism arrangement is replaced by birefringence. Also of interest is Afshar's experiment \cite{AFSHAR04}. All of these experiments use light and aim to disprove or generalize\footnote{Brida {\it et al} view the observation of simultaneous particle and wave behaviour as demonstrating a need to generalize complementarity in the sense of Wootters and Zurek \cite{WZ}and Greenberger and Yasin \cite{GY}. I have argued that the generalization in fact completely contradicts complementarity and is the antithesis of Bohr's teachings \cite{KPW}. See section 6 for further discussion of this point.} complementarity (whereas GRA's aim was to confirm complementarity) by claiming to have demonstrated particle and wave behaviour in the same experiment. In all of these experiments, the final detection result is attributed by the authors to which-path information and, therefore, to particle behaviour (according to the usual criteria accepted in the literature), but the experiments are so arranged that the light undergoes a process (tunneling in the case of Mizobuchi {\it et al}'s experiment, birefringence in Brida {\it et al}'s experiment, and interference in Afshar's experiment) which the authors claim necessarily represents wave behaviour. Hence, they claim to observe wave and particle behaviour in the same experiment. We do not agree with them for the same reasons that we do not agree with GRA's claim to have proved complementarity, a claim we will argue against in this article. Generally, we take the view that complementarity is so imprecise that it can neither be proved nor disproved. We will elaborate further on this in the rest of the article with regard to the GRA experiments, but we will also briefly describe and comment further on Mizobuchi {\it et al}'s, Brida {\it et al}'s and Afshar's experiments in section \ref{CGBAS}. We have chosen to focus on the GRA experiments in this article because they were the first to introduce a gating system for producing genuine single photon states and because their experiments lend themselves to illustrating important features of CIEM. Further, the detailed treatment of this experiment serves as a model that can be easily adapted to the later experiments, thereby providing arguments against the claims of observing simultaneous wave and particle behaviour in these experiments. The quantum eraser experiment of Kim {\it et al} \cite{KIM00} is a variant of the Wheeler delayed-choice idea \cite{WHR78, K05}. The use of particle-wave duality and complementarity in this experiment seems to imply that a measurement performed in the present effects the outcome of an earlier measurement. This now raises the further issue of the present effecting the past, which is surely unacceptable. We will also give a brief description and comment on this experiment in section \ref{CGBAS}.
Experimental evidence for the particle behaviour of light is mainly of two forms: which-path experiments and the photoelectric effect (also the Compton effect). A closer look at each of these shows that neither unambiguously demonstrate particle behaviour. In the case of the photoelectric effect it is well known that a semiclassical description can be given in which the light is treated as a classical electromagnetic field and only the atom is treated quantum mechanically \cite{W26}. A weakness of this counter example is that semiclassical radiation theory is known not to be fully consistent with experiment and fails in those cases where light exhibits nonclassical properties (as in some experiments which involve second-order coherence). Further, it is not clear that a semiclassical model of the photoelectric effect can explain the experimental fact that the photon is absorbed in a time of the order of $10^{-9}\;\mathrm{s}$ (\cite{VW76}, p. 10). Indeed, it was just this feature of the photoelectric effect that seemed to require that a photon be a localized particle prior to absorption, and is perhaps the reason why the photoelectric effect is commonly regarded as evidence for the particle behaviour of light. A more convincing argument against the photoelectric effect as evidence of particle behaviour is the provision of a fully quantum mechanical model of the photoelectric effect based on the causal interpretation of the electromagnetic field (CIEM) \cite{K85, K87, K94}. In CIEM, light is modeled as a real vector field; there are no photon particles\footnote{From here on we will use the term `photon' very loosly to refer to a quantum of energy which may or may not be spread out over large regions of space with a value of $\hbar\omega$ for a Fock state or with a value an average around $\hbar\omega$ for a wave packet.}. The field has the property of being nonlocal, meaning that an interaction at one point in the field can change the field at points beyond $ct$. The CIEM model of the photoelectric effect is of the nonlocal absorption of a photon by a localized atom. The photon prior to absorption may be spread over large regions of space. The fact that the absorption is nonlocal explains the experimental result that the absorption of the photon takes place in a time of the order of $10^{-9}\;\mathrm{s}$. We are not forced to accept that the photon must be localized prior to absorption. We conclude that the photoelectric effect cannot be regarded as conclusive evidence for the particle behaviour of light. We note that the Compton effect, also commonly accepted as evidence for the particle behaviour of light, can also be modeled by CIEM (\cite{K94}, p. 343), so that this also cannot be taken as evidence for the particle behaviour of light. To be clear, we are not claiming that the final detected results of the photoelectric and the Compton effect do not have a particle character (they clearly do). What we claim is that a particle description prior to the final result, whether from the perspective of complementarity or from the perspective of an ontology, is not forced upon us. This is because the particle character of the experimental results can be explained in terms of a wave model.
Let us now turn to which-path experiments. In a typical which-path experiment light has a choice of two paths. Determining
which-path the light actually took is considered as proof of particle behaviour. As Bohr showed in response to Einstein's famous
which-path two-slit experiment, if the path is determined with certainty, interference is lost \cite{BR59A}. Consider a which-path two-slit experiment in which we determine the path by closing one of the holes (obviously losing interference). Although crude, it is conceptually equivalent to Einstein's experiment. The point is, that even when we close the hole and are certain which-path the light took, this does not rule out a wave model. This argument holds even in more refined which-path two-slit experiments. We may conclude that in such experiments the which-path criteria for particle behaviour is somewhat arbitrary.
There is an aspect of the two-slit experiment that seems to be universally overlooked and that we wish to draw attention to. Einstein's aim in his which-path two-slit experiment was to obtain the path of an individual photon and still retain an interference pattern, thereby experimentally detecting particle and wave behaviour in the same experiment\footnote{Actually, Einstein considered Bohr's principle of complementarity and quantum mechanics to be synonymous. By experimentally contradicting complementarity Einstein wanted to demonstrate that quantum mechanics is incomplete (\cite{JAM74}, p. 127). We have argued elsewhere that Bohr's principle of complementarity and quantum mechanics are not synonymous (\cite{K05}, p. 299).}. This is contrary to Bohr's principle of complementarity which requires mutually exclusive experimental arrangements for complementary concepts \cite{BR59A, JAM74, BR28}. As we have said, Bohr was able to show that a certain determination of the photon path would destroy the interference pattern. Bohr's response was almost universally accepted and complementarity was saved. But consider this: Forget path determination and consider a two-slit experiment in which an interference pattern is formed. This interference pattern is built up of a large number of individual photoelectric detections (or some similar process in a photographic emulsion). If the photoelectric effect is accepted as evidence of the particle behaviour of light, then is not particle and wave behaviour observed in the same experiment?
We now turn to another which-path experiment which uses a beam-splitter. This will be our main focus in this article because we consider GRA's version of this experiment, which uses an atomic cascade and a gating system to produce a near ideal single photon state, as perhaps the best experimental attempt to demonstrate the particle behaviour of light \cite{G86, AG86}. In a wave model, light is split into two beams at the beam-splitter. In a particle model, each photon must choose one and only one path. Thus, using feeble light (one photon at a time) a particle model predicts perfect anticoincidence, whereas some coincidences are expected in a wave model. GRA therefore took perfect anticoincidence as the signature of particle behaviour. GRA quantified this feature in terms of the degree of second-order coherence. Semiclassical radiation theory predicts $g^{(2)}\geq 1$. As we shall see, quantum mechanical coherent or chaotic states give results in the classical regime. This is to be expected, as neither chaotic nor coherent light exhibits nonclassical behaviour. For number states on the other hand, perfect anticoincidence is expected, so that $g^{(2)}=0$.
Photoelectric detectors are placed in each output arm of the beam-splitter. For a detection to take place there must be enough energy to ionize an atom in the detector. For classical light, and quantum mechanical chaotic or coherent light, there is always some probability that more than one photon is present after the beam-splitter however feeble the light, and this entails the possibility of coincidences. But, for a single photon state there is enough energy to ionize only a single atom in one and only one output arm of the beam-splitter, so that perfect anticoincidence is predicted.
The novelty of the GRA experiments is the use of an atomic cascade and a gating system, which we describe below, in order to produce near ideal single photon states. Their results gave a value of $g^{(2)}$ much less than $1$ and confirmed the expected anticoincidence. GRA interpreted their results to be a conclusive demonstration of the particle behaviour of light.
But, underlying the assertion that anticoincidence is a signature for particle behaviour is the assumption that the photoelectric detection process (or any other atomic absorption process) is local. This implies that the photon is a localized particle before absorption by the detecting atom. But, we saw above that the quantum theory does not rule out nonlocal absorption in the photoelectric effect (nor, more generally, in any atomic absorption process). In fact, no model of light as photon particles that is consistent with the quantum theory has ever been developed\footnote{Ghose {\it et al} have developed a particle interpretation of bosons \cite{GHOSE93, GHOSE96}, including the photon \cite{GHOSE01}, based on the Kemmer-Duffin formalism \cite{KEMMER39}. It is to be emphasized that this formalism, which allows an interpretation of bosons as particles, applies in the approximation that the energies are below the threshold for pair production. We maintain that the full theory does not allow a particle ontology. Since the particle ontology of the approximation stands in contradiction to the ontology of the full field theory (since particle and wave concepts are mutually exclusive), we maintain that the particle ontology of the approximate theory cannot have physical significance (Ghose {\it et al} do not address this issue). A further point is this: As Ghose himself points out, reference (\cite{GHOSE96}, p. 1448), for the boson particle interpretation to be consistent negative energy solutions must be interpreted as antiparticles moving backwards in time. In this case, an EPR correlated particle-antiparticle pair would exhibit the pathological feature of a nonlocal connection between the present and the past (we note that this particular criticism does not apply to the electromagnetic field). For more details on this and related approaches see reference \cite{WS2005}.}. On the other hand, CIEM models light as a nonlocal field. Atomic absorption processes, including the photoelectric effect, are modeled as the nonlocal absorption of a photon. CIEM has been shown to be fully consistent with the quantum theory \cite{K94}. Our main purpose in this article is to provide a model that explains perfect anticoincidence that does not treat photons as particles. By showing that anticoincidence experiments do not rule out a wave model we prove that GRA's experiment cannot be viewed as conclusive evidence for particle behaviour of light.
The wave behaviour of light has been confirmed a countless number of times for chaotic or coherent sources. Following Einstein's 1905 explanation of the photoelectric effect \cite{E1905} in which the idea of photon particles was first invoked, the question was raised as to whether or not, in very low intensity experiments, single photons alone in the apparatus can produce interference. Numerous experiments using feeble light followed \cite{T09}. With a few exceptions the conclusion was reached that single photons can interfere with themselves. In such experiments the energy flux $\cal{E}$ is calculated and the number of photons per unit area per unit time is calculated using ${\cal E}/ \hbar \omega$. $\cal{E}$ is reduced to such low levels that it is more probable than not that only one photon is present in the apparatus at any one time. However, the probability that more than one photon is present remains, so that the single photon nature of these experiments can be questioned. By building a Mach-Zehnder interferometer around their which-path apparatus GRA were able to confirm that the near ideal single photon state produced the expected interference. Although no surprise, GRA's experiment is perhaps the first experiment to confirm the interference of single photons. The wave nature of light is not disputed and it is obvious how in CIEM interference is obtained given that light is modeled as a field (always). We will nevertheless outline the CIEM treatment of the Mach-Zehnder interferometer given in detail in reference \cite{K05}.
CIEM is a hidden variable theory. There is a large literature on hidden variable theories and we direct the interested reader to the three articles cited in reference \cite{HVTHR}. Two of these, one old one new, are surveys of hidden variable theories and include a comprehensive list of references. We also refer the reader to two interesting Ph.D thesis in the area of hidden variable theories
\cite{SC2005, WS2005}.
In the next sections we describe GRA's two experiments focusing on theoretical derivations, and then go on to give the CIEM model of these experiments, focusing on the which-path experiment.
\section{The GRA experiments}
The following description of the GRA experiments is based mainly on reference \cite{G86}. The experiments use the radiative cascade of calcium $4p^2\;^1S_0\rightarrow4s4p\;^1P_1\rightarrow4s^2\;^1S_0$ described in reference \cite{AGR81}. The first cascade to the intermediate state yields a photon $\nu_1$ of wavelength $551.3\;\mathrm{nm}$. The intermediate state, with lifetime $\tau=4.7\;\mathrm{ns}$, decays according to the usual atomic decay law for the lifetime of a state (\cite{BJ89}, p. 538):
\begin{equation}
P(t)=1-e^{-t/\tau}, \label{DL}
\end{equation}
where $P(t)$ is the probability of decay in time $t$. The second cascade photon $\nu_2$ has wavelength $422.7\;\mathrm{nm}$. The $\nu_2$ photon, according to the decay law, is emitted with near certainty within the time $\omega=2\tau=9.8\;\mathrm{ns}$ of emission of the first $\nu_1$ photon. The number of $\nu_1$ photons per second, $N_1$, is counted by photomultiplier $PM_1$, and each $\nu_1$ photon triggers a gate of duration $\omega$. Because the probability of decay within gate $\omega$ is high, there is a high probability that the $\nu_2$ partner of $\nu_1$ enters the beam-splitter. For low count rates we can be nearly certain that there is only one $\nu_2$ photon in the beam-splitter arrangement within the gate time $\omega$. In this way a near ideal single photon state is produced.
\begin{figure}[tb]
\unitlength=1in
\begin{picture}(6,3)
\put(.3,0){\scalebox{3}{\includegraphics{GRAfig1new.eps}}}
\put(1.7, .2){Figure 1. GRA's which-path experiment}
\end{picture}
\end{figure}
\section{GRA's which-path experiment}
Refer to figure 1. The photomultipliers $PM_t$ and $PM_r$ count the number of transmitted and reflected $\nu_2$ photons per second, and photomultiplier $PM_c$ counts the number of coincidences per second. These count rates are given by $N_t$, $N_r$ and $N_c$ respectively. The counts are taken over a large number of gates with a total run time $T$ of about 5 hours. The probabilities for single and coincidence counts are given by
\begin{equation}
p_t=\frac{N_t}{N_1},\;\;\;\;\;\;\;\;p_r=\frac{N_r}{N_1},\;\;\;\;\;\;\;\;p_c=\frac{N_c}{N_1}.
\end{equation}
The classical and quantum mechanical predictions for the coincidence counts are very different.
In their experiment, GRA measured the quantity $\alpha$, which they defined as \cite{G86}
\begin{equation}
\alpha=\frac{\mbox{\it \small COINCIDENCE PROBABILITY}}{\mbox{\it \small ACCIDENTAL COINCIDENCE PROBABILITY}}=\frac{p_c}{p_t p_r}=\frac{N_1N_c}{N_t N_r}. \label{alpha}
\end{equation}
Both classically and quantum mechanically, the quantity $\alpha$ is a special case of the degree of second-order coherence. Classically, $g^{(2)}_c$ is defined by (\cite{L73}, p. 111)
\begin{equation}
g^{(2)}_c(\mbox{{ \boldmath{$\mit r$}}}_1t_1,\mbox{{ \boldmath{$\mit r$}}}_2 t_2; \mbox{{ \boldmath{$\mit r$}}}_2 t_2, \mbox{{ \boldmath{$\mit r$}}}_1t_1)=\frac{\langle \mbox{\boldmath $E$}^*(\mbox{{ \boldmath{$\mit r$}}}_1t_1)\mbox{\boldmath $E$}^*(\mbox{{ \boldmath{$\mit r$}}}_2t_2)\mbox{\boldmath $E$}(\mbox{{ \boldmath{$\mit r$}}}_2t_2)\mbox{\boldmath $E$}(\mbox{{ \boldmath{$\mit r$}}}_1t_1)\rangle}{\langle|\mbox{\boldmath $E$}(\mbox{{ \boldmath{$\mit r$}}}_1t_1)|^2 \rangle \langle|\mbox{\boldmath $E$}(\mbox{{ \boldmath{$\mit r$}}}_2t_2)|^2\rangle},
\end{equation}
where $\mbox{\boldmath $E$}$ is the electric field vector. For $\mbox{{ \boldmath{$\mit r$}}}_1=\mbox{{ \boldmath{$\mit r$}}}_2$ and $t_1=t_2$, $g^{(2)}_c$ reduces to
\begin{equation}
g^{(2)}_c=\frac{\langle(\mbox{\boldmath $E$}^*\mbox{\boldmath $E$})^2\rangle}{\langle\mbox{\boldmath $E$}^*\mbox{\boldmath $E$}\rangle \langle\mbox{\boldmath $E$}^*\mbox{\boldmath $E$}\rangle}=\frac{\langle I^2\rangle}{\langle I \rangle^2}, \label{DSOC}
\end{equation}
where $I$ is the intensity. We will see in the next subsection that $\alpha=g^{(2)}_c$. Similar definitions apply in quantum mechanics (\cite{L73}, p. 219):
\begin{equation}
g^{(2)}(\mbox{{ \boldmath{$\mit r$}}}_1t_1,\mbox{{ \boldmath{$\mit r$}}}_2 t_2; \mbox{{ \boldmath{$\mit r$}}}_2 t_2, \mbox{{ \boldmath{$\mit r$}}}_1t_1)=\frac{\langle \mbox{\boldmath $\hat E$}^-(\mbox{{ \boldmath{$\mit r$}}}_1t_1)
\mbox{\boldmath $\hat E$}^-(\mbox{{ \boldmath{$\mit r$}}}_2t_2)\mbox{\boldmath $\hat E$}^+(\mbox{{ \boldmath{$\mit r$}}}_2t_2)\mbox{\boldmath $\hat E$}^+(\mbox{{ \boldmath{$\mit r$}}}_1t_1)\rangle}{\langle \mbox{\boldmath $\hat E$}^-(\mbox{{ \boldmath{$\mit r$}}}_1t_1)\mbox{\boldmath $\hat E$}^+(\mbox{{ \boldmath{$\mit r$}}}_1t_1) \rangle \langle\mbox{\boldmath $\hat E$}^-(\mbox{{ \boldmath{$\mit r$}}}_2t_2)\mbox{\boldmath $\hat E$}^+(\mbox{{ \boldmath{$\mit r$}}}_2t_2)
\rangle}, \label{G2}
\end{equation}
where the $\mbox{\boldmath $\hat E$}$'s are quantum mechanical operators defined by
\begin{equation}
\mbox{\boldmath $\hat E$}^+(\mbox{{ \boldmath{$\mit r$}}} t)=\frac{i}{V^{\frac{1}{2}}}\sum_{k\mu}\sqrt{\frac{\hbar k c}{2}}\mbox{\boldmath $\hat\varepsilon$}_{k\mu}\hat{a}_{k\mu} e^{i(\mbox{{\scriptsize\boldmath{$k$}}}.\mbox{{\scriptsize\boldmath{$x$}}}-\omega_k t)}, \;\;\;
\mbox{\boldmath $\hat E$}^-(\mbox{{ \boldmath{$\mit r$}}} t) =-\frac{i}{V^{\frac{1}{2}}}\sum_{k\mu}\sqrt{\frac{\hbar k c}{2}}\mbox{\boldmath $\hat\varepsilon$}_{k\mu}\hat{a}_{k\mu}^{\dagger} e^{-i(\mbox{{\scriptsize\boldmath{$k$}}}.\mbox{{\scriptsize\boldmath{$x$}}}-\omega_k t)}. \label{EPEM}
\end{equation}
By substituting eq. (\ref{EPEM}) into eq. (\ref{G2}) with $\mbox{{ \boldmath{$\mit r$}}}_1=\mbox{{ \boldmath{$\mit r$}}}_2$ and $t_1=t_2$ and considering only a single mode and a single polarization direction, eq. (\ref{G2}) reduces to
\begin{equation}
g^{(2)}=\frac{\langle a_{2}^{\dagger}a_2 a_{1}^{\dagger}a_1\rangle}{\langle a_{1}^{\dagger}a_1\rangle \langle a_{2}^{\dagger}a_2\rangle}.\label{G2qm}
\end{equation}
For a single mode and single polarization direction, the quantum mechanical operator for the magnitude of the intensity (\cite{L73}, p. 184; \cite{K05}, p. 304) reduces to
\begin{equation}
\hat{I}_1=\frac{\hbar k c^2 }{V}a^{\dagger}_1a_1.
\end{equation}
Multiplying the numerator and the denominator of eq. (\ref{G2qm}) by $(\hbar k c^2/V)^2$, we can write $g^{(2)}$ in terms of the expectation value of the intensity operator:
\begin{equation}
g^{(2)}=\frac{\langle I_1 I_2\rangle}{\langle I_1 \rangle \langle I_2\rangle}.\label{G2qmI}
\end{equation}
Again, we will see in the next subsection that this is equivalent to GRA's $\alpha$.
In the following subsections we calculate the classical prediction for $g^{(2)}$ using semiclassical radiation theory and compare this with the quantum mechanical predictions for $g^{(2)}$ for a number state, a coherent state, and a chaotic state.
\subsection{$g_c^{(2)}$ for a classical field}
We now calculate the classical prediction for the various probabilities. The intensity of the $n^{th}$ gate is given by the time average of the instantaneous intensity $I(t)$:
\begin{equation}
i_n=\frac{1}{\omega}\int^{t_n+\omega}_{t_n} I(t)\;dt.
\end{equation}
Although the electromagnetic field is treated classically, the photoelectric detection is treated quantum mechanically. This semiclassical radiation theory gives the probability for a detection as proportional to the intensity and to time (\cite{L73}, p. 183 and p. 185; \cite{M76} p. 31 and p. 40) (as is the case quantum mechanically). The probabilities for singles counts during the $n^{th}$ gate are, therefore,
\begin{equation}
p_{tn}=\alpha_t i_n\omega,\;\;\;\;\;\;\;\;\;\;\;p_{rn}=\alpha_r i_n\omega,
\end{equation}
where $\alpha_t$ and $\alpha_r$ are the global detection efficiencies. The intensity averaged over all the gates is
\begin{equation}
\langle i_n \rangle =\frac{1}{N_1 T}\sum^{N_1 T}_{n=1} i_n,
\end{equation}
where $N_1T$ is the total number of counts in $PM_1$, which is equal to the total number of gates. So, the overall probability for singles counts becomes
\begin{equation}
p_t=\alpha_t\omega\langle i_n\rangle,\;\;\;\;\;\;\;\;\;\;p_r=\alpha_r\omega\langle i_n \rangle.\label{PtPr}
\end{equation}
During a single gate, the probability of a detection in one arm is statistically independent of detection in the other arm. Therefore, the probability of a coincidence count during a single gate is given as the product of the probabilities of detection in each arm:
\begin{equation}
p_{cn}=\alpha_t\alpha_r\omega^2 i_n^2.
\end{equation}
The probability of a coincidence count averaged over all the gates becomes
\begin{equation}
p_c=\alpha_t\alpha_r\omega^2\langle i_n^2\rangle.\label{Pc}
\end{equation}
If the coincidences are purely accidental, then the probabilities $p_t$ and $p_r$ over the ensemble of all gates are statistically independent, so that the accidental coincidence probability is given by the product
\begin{equation}
p_t p_r=\alpha_t\alpha_r\omega^2\langle i_n\rangle^2. \label{Ptr}
\end{equation}
This represents the minimum classical probability of coincidence. These averages satisfy the inequality (\cite{BS73}, p. 185, inequality no. 4)
\begin{equation}
\langle i_n^2 \rangle \geq \langle i_n \rangle^2, \label{PcPtPr}
\end{equation}
from which it follows, by using eq.'s (\ref{Pc}) and (\ref{Ptr}), that
\begin{equation}
p_c \geq p_t p_r.
\end{equation}
In terms of $\alpha$, eq. (\ref{alpha}), we can also write the inequality (\ref{PcPtPr}) as
\begin{equation}
\alpha\geq 1. \label{G2c}
\end{equation}
Substituting eqs. (\ref{Pc}) and (\ref{Ptr}) into eq. (\ref{alpha}) gives
\begin{equation}
\alpha=\frac{\langle i_n^2 \rangle}{\langle i_n \rangle^2}, \end{equation}
which is equal to the classical second-order coherence function $g_c^{(2)}$ given in eq. (\ref{DSOC}).
\subsection{Quantum mechanical $g^{(2)}$ for a number state, a coherent state and a chaotic state}
In quantum mechanics, the same reasoning as for the classical case leads to the same expressions for the probabilities $p_t$, $p_r$ and $p_c$, and for $\alpha$. The difference is that the classical averages of the intensities are replaced by quantum mechanical expectation values of the intensity operator. Thus
\begin{equation}
\alpha=\frac{p_c}{p_t p_r}=\frac{\alpha_t\alpha_r\omega^2\langle I_{\alpha} I_{\beta} \rangle}{\alpha_t\omega\langle I_{\alpha}\rangle\alpha_r\omega\langle I_{\beta}\rangle}=\frac{\langle I_{\alpha} I_{\beta} \rangle}{\langle I_{\alpha}\rangle\langle I_{\beta}\rangle}=\frac{\langle b^{\dag}_{\alpha} b_{\alpha} b^{\dag}_{\beta} b_{\beta} \rangle}{\langle b^{\dag}_{\alpha} b_{\alpha}\rangle\langle b^{\dag}_{\beta} b_{\beta}\rangle}. \label{alphaqm}
\end{equation}
The subscripts $\alpha$ and $\beta$ refer to the horizontal and vertical beams that emerge after the first beam-splitter. We see that $\alpha$ is equal to $g^{(2)}$, eq. (\ref{G2qm}) or eq. (\ref{G2qmI}), in the quantum case also.
To calculate $g^{(2)}$ we first consider the theoretical treatment of a single beam-splitter. By now a two input approach to the
beam-splitter is almost universally accepted even when one of the inputs is the vacuum\footnote{In passing, we mention that Caves \cite{C80} uses a two input approach in connection with the search for gravitational waves using a Michelson interferometer. He suggests, as one of two possible explanations, that vacuum fluctuations due to a vacuum input are responsible for the `standard quantum limit' which places a limit on the accuracy of any measurement of the position of a free mass.} (e.g. \cite{OHM87}), but some workers still use a single input (\cite{L73}, p. 222\footnote{Here the beam-splitter is described as part of the Hanbury-Brown and Twiss experiment.}; \cite{SZ97}, p. 494\footnote{Here the beam-splitter is used as part of an atomic interferometer.}). The two input approach leads to an elegant mathematical description of the action of a beam-splitter in terms of a unitary $2\times 2$ transformation matrix which has the form of a rotation matrix \cite{CST}. Here we will use a use a single input approach since this greatly simplifies the mathematical treatment of the GRA experiments in terms of CIEM, and since it gives the same results as the two input approach for the quantities we are interested in (expectation values of the number operator, coincidence counts, and interference terms). Further, both approaches lead to essentially the same physical model of the GRA experiments in terms of CIEM.
\begin{figure}[tb]
\unitlength=1in
\begin{picture}(6, 2.6)
\put(.8,-0.1){\scalebox{3}{\includegraphics{GRAfig2new.eps}}}
\put(1.5, 0){Figure 2. Input and output destruction operators.\label{INOUTBS}}
\end{picture}
\end{figure}
The single input and two output annihilation and creation operators are related as follows:
\begin{equation}
a=t^{*}_{{\alpha}{\alpha}}b_{\alpha}+r^{*}_{{\alpha}{\beta}}b_{\beta},\;\;\;\;\;\;\;\;\;\;\;a^{\dagger}_{\alpha}=t_{{\alpha}{\alpha}}b^{\dagger}_{\alpha}+r_{{\alpha}{\beta}}b^{\dagger}_{\beta}. \label{cao}
\end{equation}
The $b$'s satisfy the usual commutation relation $[b_{\alpha}, b^{\dagger}_{\alpha}]=[b_{\beta}, b^{\dagger}_{\beta}]=1$ while any combination of $b_{\alpha}$ and $b_{\beta}$ or their conjugates commute. To preserve the commutator $[a, a^{\dagger}]=1$, we must have
\begin{equation}
|t_{{\alpha}{\alpha}}|^2+|r_{{\alpha}{\beta}}|^2=t^2+r^2=1, \label{TR}
\end{equation}
with $|t_{{\alpha}{\alpha}}|^2=t^2$ and $|r_{{\alpha}{\beta}}|^2=r^2$. Using eq.'s (\ref{cao}) and (\ref{TR}) we may proceed to calculate $g^{(2)}$ for various quantum states. We begin with the number state $| n \rangle$,
\begin{equation}
| n \rangle=\frac{(a^{\dagger}_{\alpha})^n}{(n!)^{\frac{1}{2}}}| 0 \rangle=\frac{(t_{{\alpha}{\alpha}}b^{\dagger}_{\alpha}+r_{{\alpha}{\beta}}b^{\dagger}_{\beta})^n}{(n!)^{\frac{1}{2}}}| 0 \rangle.
\end{equation}
Use of the binomial theorem to expand the brackets gives
\begin{eqnarray}
| n \rangle&=&\frac{1}{(n!)^{\frac{1}{2}}}\left[ \left( \begin{array}{c}n\\0 \end{array}\right) (t_{{\alpha}{\alpha}}b^{\dagger}_{\alpha})^n +\left( \begin{array}{c}n\\1 \end{array}\right) (t_{{\alpha}{\alpha}} b^{\dagger}_{\alpha})^{(n-1)} (r_{{\alpha}{\beta}}b^{\dagger}_{\beta})^1\right. \nonumber\\
& &+ \left( \begin{array}{c}n\\2 \end{array}\right) (t_{{\alpha}{\alpha}} b^{\dagger}_{\alpha})^{(n-2)} (r_{{\alpha}{\beta}}b^{\dagger}_{\beta})^2+......
+\left( \begin{array}{c}n\\n-1 \end{array}\right) (t_{{\alpha}{\alpha}} b^{\dagger}_{\alpha})^1 (r_{{\alpha}{\beta}}b^{\dagger}_{\beta})^{(n-1)}\nonumber\\
&&\left. +\left( \begin{array}{c}n\\n \end{array}\right) (r_{{\alpha}{\beta}} b^{\dagger}_{\beta})^n \right]| 0 \rangle.
\end{eqnarray}
With this expression for $|n\rangle$ we can evaluate the expectation value for the number of photons in the horizontal arm, $\langle n| b^{\dagger}_{\alpha} b_{\alpha} |n\rangle$, by multiplying out the brackets, noting that cross-terms are zero, and evaluating the action of the number operator on the various number states. After a number of rearrangement steps we arrive at
\begin{eqnarray}
\langle b^{\dagger}_{\alpha} b_{\alpha}\rangle=\langle n| b^{\dagger}_{\alpha} b_{\alpha} |n\rangle&=&nt^2\left[ t^{2(n-1)}+
t^{2(n-2)}r^2 \frac{(n-1)!}{(n-2)!}+ t^{2(n-3)}r^4\frac{(n-1)!}{(n-3)!2!}\right.\nonumber\\
&&\left.+t^{2(n-4)}r^6\frac{(n-1)!}{(n-4)!3!}+......+r^{2(n-1)}\right].
\end{eqnarray}
We recognize the series in the square brackets as the binomial expansion for $(t^2+r^2)^{n-1}=1$, and we get
\begin{equation}
\langle b^{\dagger}_{\alpha} b_{\alpha} \rangle=nt^2. \label{bhbh}
\end{equation}
By the same procedure as above we also get the expectation value for the number of photons in the vertical beam,
\begin{equation}
\langle b^{\dagger}_{\beta} b_{\beta}\rangle=\langle n| b^{\dagger}_{\beta} b_{\beta} |n\rangle=nr^2, \label{bVbV}
\end{equation}
and the expectation value for the number of coincidences,
\begin{equation}
\langle b^{\dagger}_{\alpha} b_{\alpha}b^{\dagger}_{\beta} b_{\beta} \rangle=\langle n| b^{\dagger}_{\alpha} b_{\alpha}b^{\dagger}_{\beta} b_{\beta} |n\rangle=n(n-1)r^2t^2.\label{bHbHbVbV}
\end{equation}
Substituting the above expectation values into eq. (\ref{G2qm}) gives the second-order coherence function for a number state,
\begin{equation}
g^{(2)}=\frac{n(n-1)r^2t^2}{nt^2nr^2}=\frac{(n-1)}{n},\;\;\;\;\;\;n\geq 2.
\end{equation}
For $n=0,1$ $g^{(2)}$=0. We see that a single photon input shows perfect anticorrelation, contrary to the classical result for $g^{(2)}_c$, eq. (\ref{G2c}).
Next we consider the coherent state
\begin{equation}
|\alpha\rangle=e^{-|\alpha|^2/2}\sum_n \frac{\alpha^n}{(n!)^{\frac{1}{2}}}|n\rangle.
\end{equation}
The expectation value in the horizontal arm is
\begin{equation}
\langle b^{\dagger}_{\alpha} b_{\alpha}\rangle=\langle\alpha| b^{\dagger}_{\alpha}b_{\alpha}|\alpha\rangle=e^{-|\alpha|^2}\sum_{n=0} \frac{|\alpha|^{2n}}{n!}\langle n| b^{\dagger}_{\alpha}b_{\alpha}|n\rangle +e^{-|\alpha|^2}\sum_{n'}\sum_{\stackrel{\mbox{\scriptsize$n$}}{\!\!\!\!\!\!\!\!\!\!\!n\neq n'}}
\frac{(\alpha^{*})^{n'}}{(n'!)^{\frac{1}{2}}}\frac{\alpha^{n}}{(n!)^{\frac{1}{2}}}\langle n'| b^{\dagger}_{\alpha}b_{\alpha}|n\rangle.
\end{equation}
The second term consisting of cross terms is zero. After substituting eq. (\ref{bhbh}) into the above, we get
\begin{equation}
\langle b^{\dagger}_{\alpha}b_{\alpha}\rangle=t^2 e^{-|\alpha|^2}\sum_{n=0} \frac{|\alpha|^{2n}}{n!}n=t^2 e^{-|\alpha|^2}|\alpha|^2\sum_{n=0} \frac{|\alpha|^{2n}}{n!}=t^2 e^{-|\alpha|^2}|\alpha|^2e^{|\alpha|^2}
=t^2 |\alpha|^2.
\end{equation}
In a similar way, we calculate the expectation value of the number operator in the vertical beam to be
\begin{equation}
\langle b^{\dagger}_{\beta} b_{\beta} \rangle=\langle\alpha| b^{\dagger}_{\beta}b_{\beta}|\alpha\rangle=r^2 |\alpha|^2,
\end{equation}
and the expectation value for coincidence counts to be
\begin{equation}
\langle b^{\dagger}_{\alpha} b_{\alpha}b^{\dagger}_{\beta} b_{\beta} \rangle=\langle\alpha| b^{\dagger}_{\alpha}b_{\alpha}b^{\dagger}_{\beta}b_{\beta}|\alpha\rangle=t^2r^2 |\alpha|^4.
\end{equation}
Substituting the above expectation values into eq. (\ref{G2qm}) gives the second-order coherence function for a coherent state as
\begin{equation}
g^{(2)}=\frac{t^2r^2 |\alpha|^4}{t^2 |\alpha|^2r^2 |\alpha|^2}=1.
\end{equation}
This corresponds to the minimum classical value for $g^{(2)}$ so that measurement of the degree of second order coherence cannot distinguish between classical and coherent light.
Lastly, we consider chaotic light. In quantum mechanics, chaotic light is a mixture of number states and is represented by the density operator (\cite{L73}, p. 158)
\begin{equation}
\rho=\sum_n P_n|n\rangle \langle n|.
\end{equation}
For light in thermal equilibrium, let $P_n$ be the probability of occurance of a number state $|n\rangle$ with energy $E_n=n\hbar\omega$. The probability $P_n$ is given by the Boltzmann distribution law applied to discrete quantum states (\cite{L73}, p. 8),
\begin{equation}
P_n=\frac{e^{-n\hbar\omega/kT}}{\sum^{\infty}_{n=0}e^{-n\hbar\omega/kT}}=(1-e^{-\hbar\omega/kT})\sum_n e^{-n\hbar\omega/kT},
\end{equation}
where $k$ is Boltzmann's constant, and $T$ is the temperature in degrees Kelvin. The expectation value of the horizontal beam number operator is
\begin{eqnarray}
\langle b^{\dagger}_{\alpha} b_{\alpha} \rangle&=&{\mathrm Tr}(\rho b^{\dag}_{\alpha} b_{\alpha})=\sum_{n'}\langle n' |\rho b^{\dagger}_{\alpha} b_{\alpha}| n' \rangle=\sum_{n'} \sum_n(1-U)U^n \langle n'| n \rangle\langle n|b^{\dagger}_{\alpha} b_{\alpha}| n' \rangle \nonumber\\
&= &(1-U)\sum_n U^n \langle n| b^{\dagger}_{\alpha} b_{\alpha}| n \rangle,
\end{eqnarray}
with $U=\exp(-\hbar\omega/kT)$. Substituting the expectation value (\ref{bhbh}), and rearranging gives
\begin{equation}
\langle b^{\dagger}_{\alpha} b_{\alpha}\rangle=t^2 \frac{U}{1-U}.
\end{equation}
Using the other expectation values for the number state as above, we easily get the results
\begin{equation}
\langle b^{\dagger}_{\beta} b_{\beta} \rangle=r^2\frac{U}{1-U},\;\;\;\;\;\;\langle b^{\dagger}_{\alpha} b_{\alpha}b^{\dagger}_{\beta} b_{\beta} \rangle=t^2r^2 \frac{2U^2}{(1-U)^2}.
\end{equation}
Substituting the above into eq. (\ref{G2qm}) gives the degree of second-order coherence for a chaotic state
\begin{equation}
g^{(2)}=2
\end{equation}
Like the result with the coherent state this value lies in the classical range.
\subsection{Comparison of theoretical and experimental results}
GRA's arrangement, figure 1, gives the degree of second-order coherence $g^{(2)}$ directly by measurement of $N_t$, $N_r$ and $N_c$ and use of eq. (\ref{alpha}). A value of $g^{(2)}\geq 1$ would agree with classical mechanics while a zero value would confirm quantum mechanics. In practice, experimental error prevents an exact zero value. Therefore, before comparing experimental and theoretical results, we first derive, following GRA \cite{G86}, a practical quantum mechanical prediction.
\begin{figure}[tb]
\unitlength=1in
\begin{picture}(6,3)
\put(0.7,0.5){\scalebox{2.5}{\includegraphics{GRAfig3.eps}}}
\put(1.1, .1){Figure 3. Plot of the function $g^{(2)}(N\omega)$ with $f(w)=0.9$. \label{G2qmexp}}
\end{picture}
\end{figure}
Let $N$ be the number of decays per second in the window of photomultiplier $PM_1$ of efficiency $\epsilon_1$. Then, $N_1=\epsilon_1 N$ is the number of $\nu_1$ photons detected per second by $PM_1$. From the atomic decay law (\ref{DL}), the probability $P_2$ of a $\nu_2$ photon partner of a $\nu_1$ photon entering the beam-splitter during a gate $\omega$ triggered by $\nu_1$ is $1-\exp(-\omega/\tau)$. Because of the angular correlation between $\nu_1$ and $\nu_2$, the probability $P_2$ is increased by a factor $a$ slightly greater than $1$ \cite{F73}. This probability is denoted by $f(\omega)=a[1-\exp(-\omega/\tau)]$, and is a number close to $1$ in GRA's experiment.The probability $P_2$ is also increased by accidental $\nu_2$'s. These are $\nu_2$ photons that enter the beam-splitter whose $\nu_1$ partners do not trigger a gate $\omega$. Once a $\nu_1$ photon has triggered a gate, the $\nu_1$ photons resulting from $N\omega$ decays during the gate $\omega$ cannot trigger another decay. Hence, their $N\omega$ $\nu_2$ partners are the accidental $\nu_2$ photons. Since $N\omega$ is the number of accidental $\nu_2$'s entering the beam-splitter during gate $\omega$, then $N_1N\omega$ is the number of accidental $\nu_2$'s entering the beam-splitter per second. The probability of an accidental $\nu_2$ photon entering the beam-splitter is therefore $N_1N\omega/N_1=N\omega$. Thus,
\begin{equation}
P_2=f(\omega)+N\omega=\frac{N_2}{N_1},
\end{equation}
where
\begin{equation}
N_2=N_1[f(\omega)+N\omega]
\end{equation}
is the number of $\nu_2$ photons that enter the beam-splitter per second. Now, define $\epsilon_t$ and $\epsilon_r$ to be the efficiencies of $PM_t$ and $PM_r$, respectively. These efficiencies include the reflection and transmission coefficients, the collection solid angle, and the detector efficiency. The number $N_t$ of $v_2$ photons transmitted is $N_t=\epsilon_tN_2$, while the number reflected is $\epsilon_r N_2$. Then, the probabilities of detecting a transmitted $v_2$ photon in $PM_t$ and a reflected $v_2$ in $PM_r$ are
\begin{equation}
p_t=\frac{N_t}{N_1}=\frac{\epsilon_tN_2}{N_1} =\epsilon_t\left[f(\omega)+N\omega\right], \;\;\;\;\;\;\;\;\;\;\\
p_r=\frac{N_r}{N_1}=\frac{\epsilon_r N_2}{N_1} =\epsilon_r\left[ f(\omega)+N\omega\right].
\end{equation}
Since $p_t$ and $p_r$ are statistically independent classically, the probability of a coincidence count becomes
\begin{equation}
p_c=p_tp_r=\epsilon_t\epsilon_r\left[ f(\omega)+N\omega\right]^2
=\epsilon_t\epsilon_r\left[ f(\omega)^2+2N\omega f(\omega) +N^2\omega^2\right].\label{PC}
\end{equation}
The term $f(\omega)^2$ suggests a repeated detection of the same photon. Since this is not possible, $f(\omega)^2$ is set equal to zero. Thus, substituting $f(\omega)^2=0$ into eq. (\ref{PC}) gives the quantum mechanical experimental expression for $p_c$. Substituting $p_t$, $p_r$ and $p_c$ into eq. (\ref{alpha}) gives:
\begin{equation}
g^{(2)}(N\omega)=\frac{2N\omega f(\omega) + N^2\omega^2}{[f(\omega)+N\omega]^2}.
\end{equation}
A plot of this function is given in figure 3. It is noticeable that as the erroneous $N\omega$ $\nu_2$ photon count increases compared to $f(\omega)$ the value of $g^{(2)}$ approaches the classical minimum value. GRA's experimental results closely agree with the plot of figure 3, and therefore confirm the quantum mechanical anticorrelation of the two beams.
\section{GRA's Interference Experiment}
\begin{figure}[tb]
\unitlength=1in
\begin{picture}(6,3.5)
\put(.6,.9){\scalebox{3}{\includegraphics{GRAfig4new.eps}}}
\put(0.5, .4){Figure 4. GRA's interference experiment. The experiment uses the same novel gating}
\put(0.5, .2){system (not shown) to produce a near ideal single photon state as in GRA's which-path,} \put(0.5, 0){experiment.\label{GRAmz}}
\end{picture}
\end{figure}
In the second interference experiment, GRA built a Mach-Zehnder interferometer around the first beam-splitter as shown in figure 4. Quantum mechanics predicts that each beam is oppositely modulated and that the fringe visibility of each beam as a function of path difference (or of a phase shift produced by a phase shifter) is $1$. In the experiment, interference fringes with visibility greater than 98\% were observed. Although the interference is expected, this is perhaps the first experiment to demonstrate interference for a genuine single photon state, as GRA themselves have emphasized.
\section{GRA's experiments according to CIEM}
GRA concluded from their results that in a which-path measurement a photon does not split at the beam-splitter and therefore chooses only one path, but, in a one-photon-at-a-time interference experiment a photon splits at the beam-splitter and interferes with itself to produce an interference pattern. They view this result as experimental confirmation of particle-wave duality, and hence, of Bohr's principle of complementarity.
Without doubt, GRA's experiments with the novel and ingenious gating system constitutes an important experimental confirmation of quantum mechanics for genuine single photon states. But, by providing a detailed wave model of both experiments, we want to show that GRA's experiments cannot be regarded as confirmation of particle-wave duality, and hence, nor of Bohr's principle of complementarity.
We refer the reader to reference \cite{K87}, but particularly reference \cite{K94} for details of CIEM. Before proceeding we first give an outline of CIEM as given in reference (\cite{K05}, p. 300).
\subsection{Outline of CIEM}
In what follows we use the radiation gauge in which the divergence of the vector potential is zero $\nabla.\mbox{\boldmath{$A$}}(\mbox{\boldmath $x$},t)=0$, and the scalar potential is also zero $\phi(\mbox{\boldmath $x$},t) = 0$. In this gauge the electromagnetic field has only two transverse components. Heavyside-Lorentz units are used throughout.
Second quantization is effected by treating the field $\mbox{\boldmath{$A$}}(\mbox{\boldmath $x$},t)$ and its conjugate momentum $\mbox{{ \boldmath{$\mit \Pi$}}}(\mbox{\boldmath $x$},t)$ as operators satisfying the equal-time commutation relations. This procedure is equivalent to introducing a field Schr\"{o}dinger equation
\begin{equation}
\int {\cal H} ( \mbox{\boldmath{$A$}}', \mbox{{ \boldmath{$\mit \Pi$}}}') {\mit \Phi}[\Ab,t]\; d\mbox{\boldmath $x$}'= i \hbar \frac{\partial {\mit \Phi}[\Ab,t]}{\partial t},\label{SE}
\end{equation}
where the Hamiltonian density operator ${\cal H}$ is obtained from the classical Hamiltonian density of the electromagnetic field,
\begin{equation}
{\cal H} =\frac{1}{2}(\mbox{\boldmath{$E$}}^{2}+\mbox{\boldmath{$B$}}^{2})=
\frac{1}{2}[c^{2}\mbox{{ \boldmath{$\mit \Pi$}}}^{2}+(\nabla\times\mbox{\boldmath{$A$}})^2 ], \label{H}
\end{equation}
by the operator replacement $\mbox{{ \boldmath{$\mit \Pi$}}}\rightarrow -i\hbar\, \delta'/\delta' \mbox{\boldmath{$A$}}$. $\mbox{\boldmath{$A$}}'$ is shorthand for $\mbox{\boldmath{$A$}}(\mbox{\boldmath $x$}',t)$. In earlier articles \cite{K05, K94} $\delta'/\delta' \mbox{\boldmath{$A$}}$ (without the prime) was defined as the variational derivative \footnote{\label{FD} For a scalar function $\phi$ the variational or functional derivative is defined as $\frac{\delta'}{\delta' \phi}=\frac{\partial}{\partial\phi}-\Sigma_i\left(\frac{\partial}{\partial\left(\frac{\partial \phi}{\partial x_i}\right)}\right)$ (\cite{SHFF68}, p. 494). For a vector function $\mbox{\boldmath{$A$}}$ we have defined it to be $\frac{\delta}{\delta \mbox{{\scriptsize\boldmath{$A$}}}}=\frac{\delta}{\delta A_{x}}{\mbox{{\boldmath{$\mit i$}}}}+\frac{\delta}{\delta A_{y}}{\mbox{{\boldmath{$\mit j$}}}}+\frac{\delta}{\delta A_{z}}{\mbox{{\boldmath{$\mit k$}}}}$, where each component is defined in the same as for the scalar function.}. This definition leads to the equal-time commutation relations
\[
[A_{i}(\mbox{\boldmath $x$},t), \mathit{\Pi}_{j}(\mbox{\boldmath $x$}',t)]=-\frac{1}{c}[A_{i}(\mbox{\boldmath $x$},t), E_{j}(\mbox{\boldmath $x$}',t)]= i\hbar \delta_{ij}\delta^3( \mbox{\boldmath $x$} - {\mbox{\boldmath $x$}'}).
\]
Unfortunately, these commutation relations are known to be inconsistent both with Gauss's law in free space, $\nabla.\mbox{\boldmath{$E$}}=0$, and the Coulomb gauge condition, $\nabla.\mbox{\boldmath{$A$}}=0$, since it follows from these that either of the two left-hand-side terms are zero, whereas the divergence of the delta function $\delta^3( \mbox{\boldmath $x$} - {\mbox{\boldmath $x$}'})$ is not zero. We noted this inconsistency in our original development of CIEM \cite{K94}, but justified this simplification by noting that it leads to the correct equations of motion. This justification, however, has recently been criticized by Struyve in reference \cite{WS2005}, p. 88\footnote{We would like to thank one of the referees for pointing out this reference and for re-emphasizing this inconsistency.}. As is well known, the commutation relations that are consistent with $\nabla.\mbox{\boldmath{$E$}}=0$ and $\nabla.\mbox{\boldmath{$A$}}=0$ are
\begin{equation}
[A_{i}(\mbox{\boldmath $x$},t), \mathit{\Pi}_{j}(\mbox{\boldmath $x$}',t)]=i\hbar \delta_{ij}^{tr}( \mbox{\boldmath $x$} - \mbox{\boldmath $x$}') \label{CCB}
\end{equation}
where $ \delta_{ij}^{tr}( \mbox{\boldmath $x$} - \mbox{\boldmath $x$}')$ is the transverse delta function defined by \cite{BJD, LHR}
\[
\delta_{ij}^{tr}(\mbox{\boldmath $x$}-\mbox{\boldmath $x$}')= \frac{1}{(2 \pi)^3} \int e^{i{\mbox{{\boldmath{$\mit k$}}}}.(\mbox{\boldmath $x$}-\mbox{\boldmath $x$}')}
\left ( \delta_{ij} - \frac{k_i k_j}{\mbox{{\boldmath{$\mit k$}}}^{2}} \right ) d^3{\mbox{{\boldmath{$\mit k$}}}}=\left(\delta_{ij} - \frac{\partial_i \partial_j}{\nabla^2}\right)\delta^3(\mbox{\boldmath $x$}-\mbox{\boldmath $x$}')
\]
We can establish consistency with the correct equal-time commutation relations, eq. (\ref{CCB}), by modifying the definition of the momentum operator as follows:
\[
\mathit{\Pi}_i=-i\hbar\frac{\delta}{\delta A_i}=-i\hbar\left( \frac{\delta'}{\delta' A_i}-\sum_k\frac{\partial_i\partial_k}{\nabla^2}\frac{\delta'}{\delta' A_k}\right),
\]
where $\delta'/\delta' A_k$ is the usual functional derivative defined in footnote \ref{FD}.
We note that the definition of the normal mode momentum operator given in the original article in which CIEM is developed \cite{K94} is consistent with the correct commutation relations, eq. (\ref{CCB}), and does not need modification.
The solution of the field Schr\"{o}dinger equation is the wave functional ${\mit \Phi}[\Ab,t]$. The square of the modulus of the wave functional $|{\mit \Phi}[\Ab,t]|^2$ gives the probability density for a given field configuration $\mbox{\boldmath{$A$}}(\mbox{\boldmath $x$},t)$. This suggests that we take $\mbox{\boldmath{$A$}}(\mbox{\boldmath $x$},t)$ as a beable. Thus, as we have already said, the basic ontology is that of a field; there are no photon particles.
We substitute ${\mit \Phi}=R[\mbox{\boldmath{$A$}},t]\exp(iS[\mbox{\boldmath{$A$}},t]/\hbar)$, where $R[\mbox{\boldmath{$A$}},t]$ and $S[\mbox{\boldmath{$A$}},t]$ are two real functionals which codetermine one another, into the field Schr\"{o}dinger equation. Then, differentiating, rearranging and equating imaginary terms gives a continuity equation:
\begin{equation}
\frac{\partial R^{2}}{\partial t} + c^{2} \int \frac{\delta}{\delta \mbox{\boldmath{$A$}}'}
\left(R^{2}\frac{\delta S}{\delta \mbox{\boldmath{$A$}}'} \right) \; d\mbox{\boldmath $x$}' = 0.
\end{equation}
The continuity equation is interpreted as expressing conservation of probability in function space. Equating real terms gives a Hamilton-Jacobi type equation:
\begin{equation}
\frac{\partial S}{\partial t}+\frac{1}{2}\int\left(\frac{\delta S}{\delta
\mbox{\boldmath{$A$}}'}\right)^{2} c^{2}+(\nabla\times\mbox{\boldmath{$A$}}')^{2}+\left(-\frac{\hbar^2
c^{2}}{R}\frac{\delta^{2} R}{\delta \mbox{\boldmath{$A$}}'^{2}} \right) d\mbox{\boldmath $x$}'= 0. \label{HJ1}
\end{equation}
This Hamilton-Jacobi equation differs from its classical counterpart by the extra classical term
\begin{equation}
Q =-\frac{1}{2}\int\frac{\hbar^{2} c^{2}}{R}
\frac{\delta^{2} R}{\delta \mbox{\boldmath{$A$}}'^{2}}\;d\mbox{\boldmath $x$}',
\end{equation}
which we call the field quantum potential.
By analogy with classical Hamilton-Jacobi theory we define the total energy and momentum
conjugate to the field as
\begin{equation}
E = -\frac{\partial S[\mbox{\boldmath{$A$}}]}{\partial t},\;\;\;\;\;\mbox{{ \boldmath{$\mit \Pi$}}}=\frac{\delta S[\mbox{\boldmath{$A$}}]}{\delta \mbox{\boldmath{$A$}}}.
\end{equation}
In addition to the beables $\mbox{\boldmath{$A$}}(\mbox{\boldmath $x$},t)$ and $\mbox{{ \boldmath{$\mit \Pi$}}}(\mbox{\boldmath $x$},t)$, we can define other field beables: the electric field, the magnetic induction, the energy and energy density, the momentum and momentum density, the intensity, etc. Formulae for these beables are obtained by replacing $\mbox{{ \boldmath{$\mit \Pi$}}}$ by $\delta S/\delta \mbox{\boldmath{$A$}}$ in the classical formula.
Thus, we can picture an electromagnetic field as a field in the classical sense, but with the additional
property of nonlocality. That the field is inherently nonlocal, meaning that an interaction at
one point in the field instantaneously influences the field at all other points, can be seen in
two ways: First, by using Euler's method of finite differences a functional can be
approximated as a function of infinitely many variables:
${\mit \Phi}[\Ab,t]\rightarrow{\mit \Phi}(\mbox{\boldmath{$A$}}_1,\mbox{\boldmath{$A$}}_2,\ldots,t)$. Comparison with a many-body wavefunction
$\psi(\mbox{\boldmath $x$}_1,\mbox{\boldmath $x$}_2,...,t)$ reveals the nonlocality. The second way is from the equation of motion
of $\mbox{\boldmath{$A$}}(\mbox{\boldmath $x$},t)$, i.e., the free field wave equation. This is obtained by taking the functional
derivative of the Hamilton-Jacobi equation, (\ref{HJ1}):
\begin{equation}
\nabla^{2}\mbox{\boldmath{$A$}}-\frac{1}{c^{2}}\frac{\partial^{2}\mbox{\boldmath{$A$}}}{\partial t^{2}}= \frac{\delta
Q}{\delta\mbox{\boldmath{$A$}}}.
\end{equation}
In general $\delta Q/\delta\mbox{\boldmath{$A$}}$ will involve an integral over space in which the
integrand contains $\mbox{\boldmath{$A$}}(\mbox{\boldmath $x$},t)$. This means that the way that $\mbox{\boldmath{$A$}}(\mbox{\boldmath $x$},t)$ changes with time at
one point depends on $\mbox{\boldmath{$A$}}(\mbox{\boldmath $x$},t)$ at all other points, hence the inherent nonlocality.
\subsection{Normal mode coordinates\label{NMC}}
To proceed it is mathematically easier to expand $\mbox{\boldmath{$A$}}(\mbox{\boldmath $x$},t)$ and $\mbox{{ \boldmath{$\mit \Pi$}}}(\mbox{\boldmath $x$},t)$ as Fourier series
\begin{equation}
\mbox{\boldmath{$A$}}(\mbox{\boldmath $x$},t)=\frac{1}{V^{\frac{1}{2}}}\sum_{k\mu}\ekq_{k\mu}(t)e^{i\kb.\xb},\;\;\;\;\;\;\;\;\;\;
\mbox{{ \boldmath{$\mit \Pi$}}}(\mbox{\boldmath $x$},t)=\frac{1}{V^{\frac{1}{2}}}\sum_{k\mu}\mbox{\boldmath $\hat\varepsilon$}_{k\mu}\pi_{k\mu}(t) e^{-i\kb.\xb}, \label{AFS}
\end{equation}
where the field is assumed to be enclosed in a large volume $V=L^3$. The wavenumber $k$ runs from $-\infty$ to $+\infty$ and $\mu=1,2$ is the polarization index. For $\mbox{\boldmath{$A$}}(\mbox{\boldmath $x$},t)$ to be a real function we must have
\begin{equation}
\mbox{\boldmath $\hat\varepsilon$}_{-k\mu}q_{-k\mu}=\ekq_{k\mu}^{*}.\label{QMP}
\end{equation}
Substituting eq.'s (\ref{H}) and (\ref{AFS}) into eq. (\ref{SE}) gives the Schr\"{o}dingier equation in terms of the normal mode coordinates $q_{k\mu}$:
\begin{equation}
\frac{1}{2}\sum_{k\mu}\left(-\hbar^{2}c^{2}\frac{\partial^2{\mit \Phi}}{\partial
q_{k\mu}^{*}\partialq_{k\mu} }+\kappa^{2}\qksq_{k\mu}{\mit \Phi}\right)=
i\hbar\frac{\partial{\mit \Phi}}{\partial t}. \label{SEN}
\end{equation}
The solution ${\mit \Phi}( q_{k\mu},t)$ is an ordinary
function of all the normal mode coordinates and this simplifies proceedings.
We substitute ${\mit \Phi}=R(q_{k\mu},t)\exp[iS(q_{k\mu},t)/\hbar]$, where $R(q_{k\mu},t)$ and $S(q_{k\mu},t)$ are real functions which codetermine one another, into eq. (\ref{SEN}). Then, differentiating, rearranging and equating real terms gives the continuity equation in terms of normal modes:
\begin{equation}
\frac{\partial R^2}{\partial t}+\sum_{k\mu}\left[\frac{c^2}{2}\frac{\partial}{\partial
q_{k\mu}}\left(R^2\frac{\partial S}{\partialq_{k\mu}^{*}}\right)+
\frac{c^2}{2}\frac{\partial}{\partialq_{k\mu}^{*}}\left(R^2\frac{\partial S}{\partialq_{k\mu}}
\right) \right]=0.
\end{equation}
Equating imaginary terms gives the Hamilton-Jacobi equation in terms of normal modes:
\begin{equation}
\frac{\partial S}{\partial t}+\sum_{k\mu}\left[\frac{c^2}{2}\frac{\partial
S}{\partialq_{k\mu}^{*}}\frac{\partial S}{\partial q_{k\mu}}+\frac{\kappa^{2}}{2}\qksq_{k\mu}
+\left(-\frac{\hbar^{2} c^{2}}{2R}\frac{\partial^{2}R}
{\partialq_{k\mu}^{*}\partialq_{k\mu}}\right)\right]=0. \label{HJ2}
\end{equation}
The term
\begin{equation}
Q = -\sum_{k\mu}\frac{\hbar^{2} c^{2}}{2R}\frac{\partial^{2}R} {\partialq_{k\mu}^{*}\partialq_{k\mu}} \label{QP}
\end{equation}
is the field quantum potential. Again, by analogy with classical Hamilton-Jacobi theory we define the total energy and the conjugate momenta as
\begin{equation}
E=-\frac{\partial S}{\partial t},\;\;\;\;\;\pi_{k\mu}=\frac{\partial S}{\partialq_{k\mu}},\;\;\;\;\;
\pi_{k\mu}^{*}=\frac{\partial
S}{\partialq_{k\mu}^{*}}.
\end{equation}
The square of the modulus of the wave function $|{\mit \Phi}( q_{k\mu},t)|^2$ is the probability density
for each $q_{k\mu}(t)$ to take a particular value at time $t$. Substituting a particular set of values of $q_{k\mu}(t)$ at time $t$ into eq. (\ref{AFS}) gives a particular field configuration at time $t$, as before. Substituting the initial values of $q_{k\mu}(t)$ gives the initial field configuration.
The normalized ground state solution of the Schr\"{o}dinger equation is given by
\begin{equation}
{\mit \Phi}_0=N e^{-\sum_{k\mu}(\kappa/2\hbar c)q_{k\mu}^{*}q_{k\mu}}e^{-\sum_{k}i\kappa ct/2},
\end{equation}
with $N= \prod_{k=1}^{\infty}(k/\hbar c \pi)^{\frac{1}{2}}$\footnote{The normalization factor $N$ is found by substituting $q_{k\mu}^{*}=f_{k\mu}+ig_{k\mu}$ and its conjugate into ${\mit \Phi}_0$ and using the normalization condition $\int_{-\infty}^{\infty} |{\mit \Phi}_0|^2df_{k\mu}dg_{k\mu}=1$, with $df_{k\mu}\equiv df_{k_11}df_{k_12}df_{k_21}\ldots$, and similarly for $dg_{k\mu}$.}. Higher excited states are obtained by the action of the creation operator $a^{\dag}_{k\mu}$:
\begin{equation}
{\mit \Phi}_{n_{k\mu}}=\frac{(a_{k\mu}^{\dag})^{n_{k\mu}}}{\sqrt{n_{k\mu}!}}{\mit \Phi}_{0}e^{- in_{k\mu}\kappa ct}.
\end{equation}
For a normalized ground state, the higher excited states remain normalized. For ease of writing we will not include the normalization factor $N$ in most expressions, but normalization of states will be assumed when calculating expectation values.
Again, the formula for the field beables are obtained by replacing the conjugate momenta $\pi_{k\mu}$ and $\pi_{k\mu}^{*}$ by $\partial S/\partialq_{k\mu}$ and $\partial S/\partialq_{k\mu}^{*}$ in the corresponding classical formula. The following is a list of formulae for the beables:\\ \mbox{}\\
The vector potential $\mbox{\boldmath{$A$}}(\mbox{\boldmath $x$},t)$ is given in eq. (\ref{AFS}). The electric field is
\begin{equation}
\mbox{\boldmath $E$}(\mbox{\boldmath $x$},t)=-c\mbox{{ \boldmath{$\mit \Pi$}}}(\mbox{\boldmath $x$},t)=-\frac{1}{c}\frac{\partial \mbox{\boldmath{$A$}}}{\partial t}= - \frac{c}{V^{\frac{1}{2}}}\sum_{k\mu}\mbox{\boldmath $\hat\varepsilon$}_{k\mu}\frac{\partial S}{\partialq_{k\mu}}e^{-i\kb.\xb}. \label{PEX}
\end{equation}
The magnetic induction is
\begin{equation}
\mbox{\boldmath $B$}(\mbox{\boldmath $x$},t)=\nabla\times\mbox{\boldmath{$A$}}(\mbox{\boldmath $x$},t)=\frac{i}{V^{\frac{1}{2}}}\sum_{k\mu}(\mbox{\boldmath $k$}\times\mbox{\boldmath $\hat\varepsilon$}_{k\mu})q_{k\mu}(t)e^{i\kb.\xb}. \label{BEX}
\end{equation}
We may also define the energy density, which includes the quantum potential density (see reference \cite{K94}), but we will not write these here as we will not need them. The total energy is found by integrating the energy density over $V$ to get
\begin{equation}
E=-\frac{\partial S}{\partial t}=\sum_{k\mu}\left[\frac{c^{2}}{2} \frac{\partial
S}{\partialq_{k\mu}^{*}}\frac{\partial S}{\partialq_{k\mu}}+\frac{\kappa^{2}}{2}\qksq_{k\mu}
+\left(-\frac{\hbar^{2}c^{2}}{2R}\frac{\partial^{2}R} {\partial
q_{k\mu}^{*}\partialq_{k\mu}} \right)\right].
\end{equation}
The intensity is equal to momentum density multiplied by $c^2$:
\begin{equation}
\mbox{\boldmath $I$}(\mbox{\boldmath $x$},t)=c^2\mbox{\boldmath${\cal G}$}= \frac{-ic^2}{V}\sum_{k\mu}\sum_{k'\mu'}\left[ \mbox{\boldmath $\hat\varepsilon$}_{k'\mu'}\times(\mbox{\boldmath $k$}\times\mbox{\boldmath $\hat\varepsilon$}_{k\mu})\frac{\partial S}{\partial q_{k'\mu'}}q_{k\mu}e^{i(\kb-\kbp).\xb} \right]. \label{I}
\end{equation}
We have adopted the classical definition of intensity in which the intensity is equal to the Poynting vector (in Heavyside-Lorentz units), i.e., $\mbox{\boldmath $I$}=c(\mbox{\boldmath $E$}\times\mbox{\boldmath $B$})$. The definition leads to a moderately simple formula for the intensity beable. We note that the definition above contains a zero point intensity. But, because $\mbox{\boldmath $I$}$ is a vector (whereas energy is not) the contributions to the zero point intensity from individual waves with wave vector $\mbox{\boldmath $k$}$ cancel each other because of symmetry; for each $\mbox{\boldmath $k$}$ there is another $\mbox{\boldmath $k$}$ pointing in the opposite direction. The above, however, is not the definition normally used in quantum optics. This is probably because, although it leads to a simple formula for the intensity beable, it leads to a very cumbersome expression for the intensity operator in terms of the creation and annihilation operators:
\begin{eqnarray}
&&\mbox{{ \boldmath{$\mbox{\boldmath $\hat I$}$}}}
=\frac{-\hbar c^2}{4V}\sum_{k\mu}\sum_{k'\mu'}\left[ \frac{k}{k'}\mbox{\boldmath $\hat\varepsilon$}_{k\mu}\times(\mbox{\boldmath $k$}'\times\mbox{\boldmath $\hat\varepsilon$}_{k'\mu'})- \frac{k'}{k}(\mbox{\boldmath $k$}\times\mbox{\boldmath $\hat\varepsilon$}_{k\mu})\times\mbox{\boldmath $\hat\varepsilon$}_{k'\mu'} \right] \nonumber\\
&&\times\left[ \hat{a}_{k\mu}\hat{a}_{k'\mu'}e^{i(\mbox{{\scriptsize\boldmath{$k$}}}+\mbox{{\scriptsize\boldmath{$k'$}}}).\mbox{{\scriptsize\boldmath{$x$}}}} - \hat{a}_{k\mu}\hat{a}^\dag_{k'\mu'}e^{i(\kb-\kbp).\xb}- \hat{a}^\dag_{k\mu}\hat{a}_{k'\mu'}e^{-i(\kb-\kbp).\xb} +\hat{a}^\dag_{k\mu}\hat{a}^\dag_{k'\mu'}e^{-i(\mbox{{\scriptsize\boldmath{$k$}}}+\mbox{{\scriptsize\boldmath{$k'$}}}).\mbox{{\scriptsize\boldmath{$x$}}}} \right]. \label{Ipan}
\end{eqnarray}
In quantum optics the intensity operator is defined instead as $\mbox{\boldmath $\hat I$}
=c(\mbox{{ \boldmath{$\hat{E}^+$}}}\times \mbox{{ \boldmath{$\hat{B}^-$}}} - \mbox{{ \boldmath{$\hat{B}^-$}}}\times \mbox{{ \boldmath{$\hat{E}^+$}}})$, and leads to a much simpler expression in terms of creation and annihilation operators
\begin{equation}
\mbox{\boldmath $\hat I$}= \frac{\hbar c^2}{V}\sum_{k\mu}\sum_{k'\mu'}\hat{\mbox{\boldmath $k$}}\sqrt{kk'} \hat{a}^\dag_{k\mu}\hat{a}_{k'\mu'}e^{i(\mbox{{\scriptsize\boldmath{$k'$}}}-\mbox{{\scriptsize\boldmath{$k$}}}).\mbox{{\scriptsize\boldmath{$x$}}}}. \label{Iqo}
\end{equation}
This definition is justified because it is proportional to the dominant term in the interaction Hamiltonian for the photoelectric effect upon which instruments that measure intensity are based. We note that the two forms of the intensity operator lead to identical expectation values and perhaps further justifies the simpler definition of the intensity operator.
From the above we see that objects such as $q_{k\mu}$, $\pi_{k\mu}$, etc., regarded as time independent operators in the Schr\"{o}dinger picture of the usual interpretation, become functions of time in CIEM.
For a given state ${\mit \Phi}(q_{k\mu},t)$ of the field we determine the beables by first finding $\partial S/\partialq_{k\mu}$ and its complex conjugate using the formula
\begin{equation}
S=\left(\frac{\hbar}{2i}\right)\ln\left(\frac{{\mit \Phi}}{{\mit \Phi}^*}\right).\label{FlaS}
\end{equation}
This gives the beables as functions of the $q_{k\mu}(t)$ and $q_{k\mu}^{*}(t)$. The beables can then be obtained in terms of the initial values by solving the equations of motion for $q_{k\mu}(t)$ and $q_{k\mu}^{*}(t)$. There are two alternative but equivalent forms of the equations of motion. The first follows from the classical formula
\begin{equation}
\pi_{k\mu} = \frac{\partial{\cal L}}{\partial\left(\frac{d q_{k\mu}}{d t} \right)} =\frac{1}{c^2}\frac{dq_{k\mu}^{*}}{d t},
\end{equation}
where ${\cal L}$ is the Lagrangian density of the electromagnetic field, by replacing $\pi_{k\mu}$ by $\partial S/\partialq_{k\mu}$. This gives the equations of motion as
\begin{equation}
\frac{1}{c^2}\frac{dq_{k\mu}^{*}(t)}{d t} =\frac{\partial S}{\partialq_{k\mu}(t)}.\label{EQMG}
\end{equation}
The second form of the equations of motion for $q_{k\mu}$ is obtained by differentiating the Hamilton Jacobi equation (\ref{HJ2}) by $q_{k\mu}^{*}$. This gives the wave equations
\begin{equation}
\frac{1}{c^2}\frac{d^{2}q_{k\mu}^{*}}{d t^{2}}+\kappa^{2}q_{k\mu}^{*}=-\frac{\partial
Q}{\partialq_{k\mu}}. \label{WEQ}
\end{equation}
The corresponding equations for $q_{k\mu}$ are the complex conjugates of the above. These equations of motion differ from the classical free field wave equation by the derivative of the quantum potential. From this it follows that where the quantum potential is zero or small the quantum field behaves like a classical field. In applications we will obviously choose to solve the simpler eq. ({\ref{EQMG}}).
We conclude with a few words to clarify our model. The electromagnetic field beables are $\mbox{\boldmath $E$}(\mbox{\boldmath $x$},t)$ and $\mbox{\boldmath $B$}(\mbox{\boldmath $x$},t)$ and are objectively existing entities in real space. The state ${\mit \Phi}=R\exp[iS/\hbar]$ is made up of the $R$ and $S$ functionals. By thinking in terms of the approximation of a functional as a function of infinitely many variables or in term of normal mode coordinates we can picture $R$ and $S$ as connecting the field coordinates and shaping the behaviour of the field through the equations of motion
(\ref{EQMG}) or (\ref{WEQ}), but the $R$ and $S$ beables (and hence the state ${\mit \Phi}$) are not the electromagnetic field itself. The $R$ and $S$ beables co-determine one another and the motion of the field can be determined from either one without reference to the other. This is reflected in the two possible forms of the equations of motion.
\subsection{GRA's which-path experiment according to CIEM}
Refer to figure 1. To keep the mathematics simple we assume (a) a symmetrical beam-splitter so that the reflection and transmission coefficients are equal and given by $r=t=1/\sqrt{2}$, (b) a $\pi/2$ phase shift upon reflection, and (c) no phase shift upon transmission. With this in mind, the state of the photon after the beam-splitter but before the mirrors and phase shifter is
\begin{equation}
{\mit \Phi}_{I}=\frac{1}{\sqrt{2}}\left( {\mit \Phi}_{\alpha}+i{\mit \Phi}_{\beta}\right), \label{PHRI}
\end{equation}
where ${\mit \Phi}_{\alpha}$ and ${\mit \Phi}_{\beta}$ are solutions of the normal mode Schr\"{o}dinger equation and are given by
\begin{eqnarray}
{\mit \Phi}_{\alpha}(q_{k\mu},t)& =& \left(\frac{2\kappa_\alpha}{\hbar c}\right)^{\frac{1}{2}}
\alpha_{k_\alpha\mu_\alpha}^{*}{\mit \Phi}_{0}e^{-i\kappa_\alpha ct}, \;\;\;\;\;\;\;\;\;
{\mit \Phi}_{\beta}(q_{k\mu},t) = \left(\frac{2\kappa_\beta}{\hbar c}\right)^{\frac{1}{2}}
\beta_{k_\beta\mu_\beta}^{*}{\mit \Phi}_{0}e^{-i\kappa_\beta ct},\nonumber\\
{\mit \Phi}_0(q_{k\mu},t)&=&N e^{-\sum_{k\mu}(\kappa/2\hbar c)q_{k\mu}^{*}q_{k\mu}}e^{-\sum_{k}i\kappa ct/2}.
\end{eqnarray}
The magnitudes of the $k$-vectors are equal, i.e., $k_\alpha=k_\beta=k_0$. The $\alpha_{k_\alpha\mu_\alpha}$ normal mode coordinates represent the horizontal beam and the $\beta_{k_\beta\mu_\beta}$ coordinates represent the vertical beam. It is clear that the single photon input state ${\mit \Phi}_i(q_{k\mu},t) =(2\kappa_0/(\hbar c))^{\frac{1}{2}}
q_{k_0\mu_0}^{*}(t){\mit \Phi}_{0}e^{-i\kappa_0 ct} $ is split by the beam-splitter into two beams. This remains true irrespective of whether a subsequent measurement is a which-path measurement or it is the observation of interference. The mathematical description is unique.
In CIEM the normal mode coordinates are regarded as functions of time and represent an actually existing electromagnetic field. The modulus squared of the wavefunction is a probability density from which the probabilities for the normal modes to have particular values are found. The totality of these probabilities gives the probability for a particular field configuration. Thus, the ontology is that of a field; there are no photon particles. In fact, for a number state the most probable field configuration is one or more plane waves, which, in general, are nonlocal (\cite{K94}, p. 326). As we mentioned earlier, in CIEM we use the term photon to refer to a quantum of energy $\hbar\omega$ (or an average about this value for a wave packet) without in any way implying particle properties.
To find the equations of motion for the normal mode coordinates we first find $S$ from ${\mit \Phi}_{I}=R(q_{k\mu},t)\exp(iS(q_{k\mu}, t))$ and then substitute into
\begin{equation}
\frac{1}{c^2}\frac{dq_{k\mu}^{*}(t)}{d t} =\frac{\partial S}{\partialq_{k\mu}(t)}.
\end{equation}
This gives the equations of motion
\begin{eqnarray}
\frac{d\alpha_{k_\alpha\mu_\alpha}^{*}}{dt}&=&c^2\frac{\partial S}{\partial\alpha_{k_\alpha\mu_\alpha}}=\frac{\hbar c^2}{2}\frac{i}{\left(\alpha_{k_\alpha\mu_\alpha}-i\beta_{k_\beta\mu_\beta} \right)}, \label{EQMa} \\
\frac{d\beta_{k_\beta\mu_\beta}^{*}}{dt} &=&c^2\frac{\partial S}{\partial\beta_{k_\beta\mu_\beta}} =\frac{\hbar c^2}{2}\frac{1}{\left(\alpha_{k_\alpha\mu_\alpha}-i\beta_{k_\beta\mu_\beta} \right)}, \label{EQMb}\\
\frac{dq_{k\mu}^{*}}{dt} &=&c^2\frac{\partial S}{\partialq_{k\mu}} =0,\;\;\;\; \mathrm{for}\; k\neq \pm k_{\alpha}, \pm k_{\beta}. \label{EQMq}
\end{eqnarray}
Eqs. (\ref{EQMa}) and (\ref{EQMb}) are coupled differential equations and the coupling indicates that the two beams are nonlocally connected. The solutions are
\begin{equation}
\alpha_{k_\alpha\mu_\alpha}^{*}(t)=\alpha_0 e^{i(\omega_\alpha t+\sigma_0)}, \;\;\;
\beta_{k_\beta\mu_\beta}^{*}(t)=\beta_0 e^{i(\omega_\beta t+\tau_0)}, \;\;\;
q_{k\mu}^{*}(t)= q_{k\mu 0}e^{i\zeta_{k\mu 0}}\;\mathrm{for}\;k\neq \pm k_{\alpha}, \pm k_{\beta}, \label{QKS}
\end{equation}
where $\sigma_0$ and $\tau_0$ are integration constants corresponding to the initial phases, and $\alpha_0$ and $\beta_0$ are constant initial amplitudes. The omega's, $\omega_\alpha=\hbar c^2/4\alpha_0^2$ and $\omega_{\beta}=\hbar c^2/4\beta_0^2$, are nonclassical frequencies which depend on the amplitudes $\alpha_0$ and $\beta_0$. The vector potential, electric intensity, magnetic induction and intensity beables are given by the formulae
\begin{eqnarray}
\mbox{\boldmath{$A$}}(\mbox{\boldmath $x$},t)&=&\frac{1}{V^{\frac{1}{2}}}\sum_{k\mu}\ekq_{k\mu}(t)e^{i\kb.\xb},\nonumber\\
\mbox{\boldmath $E$}(\mbox{\boldmath $x$},t)&=&-c\mbox{{ \boldmath{$\mit \Pi$}}}(\mbox{\boldmath $x$},t)=-\frac{1}{c}\frac{\partial \mbox{\boldmath{$A$}}}{\partial t}= - \frac{c}{V^{\frac{1}{2}}}\sum_{k\mu}\mbox{\boldmath $\hat\varepsilon$}_{k\mu}\frac{\partial S}{\partialq_{k\mu}}e^{-i\kb.\xb}, \nonumber\\
\mbox{\boldmath $B$}(\mbox{\boldmath $x$},t)&=&\nabla\times\mbox{\boldmath{$A$}}(\mbox{\boldmath $x$},t)=\frac{i}{V^{\frac{1}{2}}}\sum_{k\mu}(\mbox{\boldmath $k$}\times\mbox{\boldmath $\hat\varepsilon$}_{k\mu})q_{k\mu}(t)e^{i\kb.\xb}, \nonumber\\
\mbox{\boldmath $I$}(\mbox{\boldmath $x$},t)&=&c^2\mbox{\boldmath${\cal G}$}= \frac{-ic^2}{V}\sum_{k\mu}\sum_{k'\mu'}\left[ \mbox{\boldmath $\hat\varepsilon$}_{k'\mu'}\times(\mbox{\boldmath $k$}\times\mbox{\boldmath $\hat\varepsilon$}_{k\mu})\frac{\partial S}{\partial q_{k'\mu'}}q_{k\mu}e^{i(\kb-\kbp).\xb} \right].
\end{eqnarray}
Substituting equations (\ref{EQMa}) to (\ref{QKS}) into the above formulae gives the field beables associated with the state ${\mit \Phi}_I$:
\begin{eqnarray}
\mbox{\boldmath{$A$}}_I(x,t)&=&\frac{2}{V^{\frac{1}{2}}}\left(\mbox{\boldmath $\hat\varepsilon$}_{k_\alpha\mu_\alpha}\alpha_0\cos{\mit \Theta}_{\alpha}+\mbox{\boldmath $\hat\varepsilon$}_{k_\beta\mu_\beta}\beta_0\cos{\mit \Theta}_{\beta}
\right) +\frac{\mbox{\boldmath $u$}_I(\mbox{\boldmath $x$})}{V^{\frac{1}{2}}},\nonumber \\
\mbox{\boldmath $E$}_I(\mbox{\boldmath $x$},t)&=&\frac{-\hbar c}{2V^{\frac{1}{2}}}\left(\frac{\mbox{\boldmath $\hat\varepsilon$}_{k_\alpha\mu_\alpha}}{\alpha_0}\sin{\mit \Theta}_{\alpha}+\frac{\mbox{\boldmath $\hat\varepsilon$}_{k_\beta\mu_\beta}}{\beta_0}\sin{\mit \Theta}_{\beta} \right), \label{EI}\\
\mbox{\boldmath $B$}_I(\mbox{\boldmath $x$},t) &=&\frac{-2}{V^{\frac{1}{2}}}\left[(\mbox{\boldmath $k$}_\alpha\times\mbox{\boldmath $\hat\varepsilon$}_{k_\alpha\mu_\alpha})\alpha_0\sin{\mit \Theta}_{\alpha} + (\mbox{\boldmath $k$}_\beta\times\mbox{\boldmath $\hat\varepsilon$}_{k_\beta\mu_\beta})\beta_0 \sin {\mit \Theta}_{\beta}\right] + \frac{\mbox{\boldmath $v$}_I(\mbox{\boldmath $x$})}{V^{\frac{1}{2}}}, \nonumber\\
\mbox{\boldmath $I$}_I(\mbox{\boldmath $x$},t)&=&\frac{\hbar c^2}{2V}\left(\mbox{\boldmath $k$}_\alpha+\mbox{\boldmath $k$}_\beta-\mbox{\boldmath $k$}_\alpha\cos2{\mit \Theta}_{\alpha} -\mbox{\boldmath $k$}_\beta\cos 2{\mit \Theta}_{\beta}\right)-\frac{\mbox{\boldmath $f$}_I(\mbox{\boldmath $x$})\mbox{\boldmath $g$}_I(\mbox{\boldmath $x$},t)}{V},
\end{eqnarray}
with ${\mit \Theta}_{\alpha}=\mbox{\boldmath $k$}_\alpha.\mbox{\boldmath $x$}-\omega_{\alpha} t-\sigma_0$ and ${\mit \Theta}_{\beta}=\mbox{\boldmath $k$}_\beta.\mbox{\boldmath $x$}-\omega_{\beta} t-\tau_0$, and
\begin{eqnarray}
\mbox{\boldmath $u$}_I(\mbox{\boldmath $x$})&=&\!\!\!\!\!\sum_{\stackrel{\scriptstyle{k\mu}}{k\neq \pm k_{\alpha},\pm k_{\beta}}}\!\!\!\!\!\mbox{\boldmath $\hat\varepsilon$}_{k\mu}\qke^{i\kb.\xb},
\;\;\;\;\;\mbox{\boldmath $v$}_I(\mbox{\boldmath $x$})=\nabla\times\mbox{\boldmath $u$}_I(\mbox{\boldmath $x$})=i\!\!\!\!\!\sum_{\stackrel{\scriptstyle{k\mu}}{k\neq \pm k_{\alpha},\pm k_{\beta}}}\!\!\!\!\!(\mbox{\boldmath $k$}\times\mbox{\boldmath $\hat\varepsilon$}_{k\mu})\qke^{i\kb.\xb}, \label{VXI}\\
\mbox{\boldmath $f$}_I(\mbox{\boldmath $x$})&=&i\hbar c^2\!\!\!\!\!\sum_{\stackrel{\scriptstyle{k\mu}}{k\neq \pm k_{\alpha},\pm k_{\beta}}}\!\!\!\!\!\mbox{\boldmath $\hat\varepsilon$}_{k_\alpha\mu_\alpha}\times(\mbox{\boldmath $k$}\times\mbox{\boldmath $\hat\varepsilon$}_{k\mu})\qke^{i\kb.\xb},\;\;\;\;\;
\mbox{\boldmath $g$}_I(\mbox{\boldmath $x$},t)=\sin {\mit \Theta}_{\alpha}+\sin {\mit \Theta}_{\beta}.
\end{eqnarray}
Complementarity is not a direct interpretation of the mathematical formalism, so that the uniqueness of the mathematical description is not reflected in the duality of complementary concepts. The ontology of CIEM, on the other hand, is a direct interpretation of the elements of the mathematical formalism. The beables above therefore reflect the splitting of the state ${\mit \Phi}_i$ into two beams. In other words, the photon always splits at the beam-splitter irrespective of the nature of any planned future measurement.
Quantum mechanics predicts that in a which-path measurement a photon will be detected in only one path. Feeble light experiments of the past have confirmed this prediction indirectly, while GRA's which-path experiment provides direct confirmation. Our CIEM model must therefore explain how a photon is detected in only one path, even though the photon must split at the beam-splitter. To see how this comes about we outline the interaction of the electromagnetic field in state ${\mit \Phi}_I$ with the photomultipliers. For mathematical simplicity we model the photomultipliers $PM_t$ and $PM_r$ as hydrogen atoms. We assume that the incident photon has sufficient energy to ionize one of the hydrogen atoms.
The treatment we give here is a short summary of a more detailed outline given in reference (\cite{K05}, p. 310). The initial state of the field before interaction with the hydrogen atom is given by eq. (\ref{PHRI}). The initial state of the hydrogen atom is
\begin{equation}
u_i(\mbox{\boldmath $x$},t)=\frac{1}{\sqrt{\pi a^3}}e^{-r/a}e^{-iE_{ei}t/\hbar},
\end{equation}
where $a=4\pi\hbar^2/\mu e^2$ is the Bohr magneton. With the initial state ${\mit \Phi}_{I_{k\mu}i}(q_{k\mu},\mbox{\boldmath $x$},t)={\mit \Phi}_{I_{k\mu}}(q_{k\mu},t)u_i(\mbox{\boldmath $x$},t)$, the Schr\"{o}dinger equation
\begin{equation}
i\hbar\frac{\partial {\mit \Phi}}{\partial t}= (H_R+H_A+H_I){\mit \Phi}
\end{equation}
can be solved using standard perturbation theory. $H_R$, $H_A$ and $H_I$ are the free radiation, free atomic, and interaction Hamiltonians, respectively, and are given by
\begin{equation}
H_R = \sum_{k\mu}\left(a^{\dag}_{k\mu}a_{k\mu}+\frac{1}{2}\right)\hbar\omega_k, \;\;\;\;\; H_A = \frac{-\hbar^2}{2\mu}\nabla^2+V(\mbox{\boldmath $x$}), \;\;\;\;\;H_I = \frac{i\hbar e}{\mu c}\left(\frac{\hbar c}{2V}\right)^{\frac{1}{2}} \sum_{k\mu}\frac{1}{\sqrt{k}}a_{k\mu}e^{i\kb.\xb}\mbox{\boldmath $\hat\varepsilon$}_{k\mu}.\nabla,
\end{equation}
with $\omega_k=kc$ and $\mu=m_em_n/(m_e+m_n)$ is the reduced mass. The final solution is
\begin{equation}
{\mit \Phi}={\mit \Phi}_{I_{k\mu}i}(q_{k\mu},\mbox{\boldmath $x$},t)-\frac{{\mit \Phi}_0(q_{k\mu},t)}{V}\sum_n \eta_{0n}(t)\mbox{\boldmath $\hat\varepsilon$}_{k_0\mu_0}.\mbox{\boldmath $k$}_{en}\frac{1}{\sqrt{V}}e^{i\left(\mbox{{\scriptsize\boldmath{$k$}}}_{en}.\mbox{{\scriptsize\boldmath{$x$}}}-E_{en}t/\hbar\right)}, \label{FSOL}
\end{equation}
with
\begin{equation}
\eta_{0n}(t)=\left(\frac{e}{\mu c}\right)\sqrt{\frac{\hbar c}{2V}}\left[\frac{(i-e^{i\phi})}{\sqrt{2 k_0}}\right]\left[\frac{\hbar}{\sqrt{V\pi a^3}} \frac{8\pi a^3}{(1+a^2 k_{en}^2)^2} \right]\left(\frac{1- e^{iE_{0n,I_{k\mu}i}t/\hbar}}{E_{0n,I_{k\mu}i}}\right).
\end{equation}
$E_{0n,I_{k\mu}i}$ is given by
\begin{equation}
E_{0n,I_{k\mu}i}=E_0+E_{en}-E_{I_{k\mu}}-E_{ei}.
\end{equation}
Eq. (\ref{FSOL}) clearly shows that one entire photon is absorbed. This is further emphasized by the integral
\begin{equation}
\sum_{k\mu}\frac{1}{\sqrt{k}}\int{\mit \Phi}^{*}_{N_{k\mu}}a_{k\mu}{\mit \Phi}_{I_{k\mu}}\;dq_{k\mu}
= \frac{1}{\sqrt{2k_0}}(i-e^{i\phi})\int {\mit \Phi}^{*}_{N_{k\mu}}{\mit \Phi}_0\;dq_{k\mu}= \frac{1}{\sqrt{2k_0}}(i-e^{i\phi})\delta_{N_{k\mu}0}\delta_{kk_0}\delta{\mu\mu_0},
\end{equation}
which is part of the matrix element $H_{N_{k\mu}n,I_{k\mu}i}$ used in obtaining the final solution. This term shows that if the interaction takes place at all then an entire electromagnetic quantum must be absorbed by the hydrogen atom.
The initial state ${\mit \Phi}_{I_{k\mu}}$ represents a single photon divided between the two beams, but in the interaction with an atom positioned in one of the beams, the entire photon must be absorbed. Given that the interferometer arms can be of arbitrary length such absorption must in general be nonlocal. In this way we can explain why a photon that always divides at the beam-splitter nevertheless registers in only one path. The fact that this wave model exists prevents GRA's which-path experiment from being regarded as confirmation of the particle behaviour of light.
\subsection{GRA's interference experiment according to CIEM}
Refer to figure 4. Using the same phase and amplitude changes as in the previous section, and tracing the development of the two beams after $BM_2$, we arrive at the wavefunction
\begin{equation}
{\mit \Phi}_{II}=-\frac{1}{2}{\mit \Phi}_{c}(1+e^{i\phi})+\frac{i}{2}{\mit \Phi}_d(1-e^{i\phi}).
\end{equation}
By following a similar procedure to that of region I, we can find the $S$ corresponding to ${\mit \Phi}_{II}$ and hence set up and solve the equations of motion. Using these solutions the beables for region II are found to be
\begin{eqnarray}
\mbox{\boldmath{$A$}}_{II}(\mbox{\boldmath $x$},t)&=&\frac{2}{V^{\frac{1}{2}}}\left( \mbox{\boldmath $\hat\varepsilon$}_{k_c\mu_c} c_0 \cos{\mit \Theta}_c +\mbox{\boldmath $\hat\varepsilon$}_{k_d\mu_d} d_0\cos {\mit \Theta}_d\right)+\frac{\mbox{\boldmath $u$}_{II}(\mbox{\boldmath $x$})}{V^{\frac{1}{2}}},\nonumber\\
\mbox{\boldmath $E$}_{II}(\mbox{\boldmath $x$},t)&=&\frac{-\hbar c}{2V^{\frac{1}{2}}}\left(\frac{\mbox{\boldmath $\hat\varepsilon$}_{k_\alpha\mu_\alpha}}{c_0}(1+\cos\phi)\sin{\mit \Theta}_c + \frac{\mbox{\boldmath $\hat\varepsilon$}_{k_d\mu_d}}{d_0}(1-\cos\phi)\sin{\mit \Theta}_d \right),\nonumber\\
\mbox{\boldmath $B$}_{II}(\mbox{\boldmath $x$},t) &=&\frac{-2}{V^{\frac{1}{2}}}\left[(\mbox{\boldmath $k$}_c\times\mbox{\boldmath $\hat\varepsilon$}_{k_c\mu_c})c_0\sin{\mit \Theta}_c + (\mbox{\boldmath $k$}_d\times\mbox{\boldmath $\hat\varepsilon$}_{k_d\mu_d})d_0 \sin{\mit \Theta}_d\right] +\frac{\mbox{\boldmath $v$}_{II}(\mbox{\boldmath $x$})}{V^{\frac{1}{2}}}, \nonumber\\
\mbox{\boldmath $I$}_{II}(\mbox{\boldmath $x$},t)&=&\frac{\hbar c^2}{2V}\left[\mbox{\boldmath $k$}_c(1+\cos\phi)+\mbox{\boldmath $k$}_d(1-\cos\phi) -\mbox{\boldmath $k$}_c(1+\cos\phi)\cos2{\mit \Theta}_c+ \mbox{\boldmath $k$}_d(1-\cos\phi)\cos 2{\mit \Theta}_d)\right] \nonumber\\
&&-\frac{\mbox{\boldmath $f$}_{II}(\mbox{\boldmath $x$})\mbox{\boldmath $g$}_{II}(\mbox{\boldmath $x$},t)}{V}, \label{INTII}
\end{eqnarray}
with
\begin{eqnarray}
\mbox{\boldmath $u$}_{II}(\mbox{\boldmath $x$})&=&\!\!\!\!\!\sum_{\stackrel{\scriptstyle{k\mu}}{k\neq \pm k_c,\pm k_d}}\!\!\!\!\!\mbox{\boldmath $\hat\varepsilon$}_{k\mu}\qke^{i\kb.\xb},\;\;\;\;\;
\mbox{\boldmath $v$}_{II}(\mbox{\boldmath $x$})=i\!\!\!\!\!\sum_{\stackrel{\scriptstyle{k\mu}}{k\neq \pm k_c,\pm k_d}}\!\!\!\!\!(\mbox{\boldmath $k$}\times\mbox{\boldmath $\hat\varepsilon$}_{k\mu})\qke^{i\kb.\xb}=\nabla\times \mbox{\boldmath $u$}_{II}(\mbox{\boldmath $x$}), \label{VXII}\\
\mbox{\boldmath $f$}_{II}(\mbox{\boldmath $x$})&=&\frac{i\hbar c^2}{V}\!\!\!\!\!\sum_{\stackrel{\scriptstyle{k\mu}}{k\neq \pm k_c,\pm k_d}}\!\!\!\!\!\mbox{\boldmath $\hat\varepsilon$}_{k_0\mu_0}\times(\mbox{\boldmath $k$}\times\mbox{\boldmath $\hat\varepsilon$}_{k\mu})\qke^{i\kb.\xb},\; \mbox{\boldmath $g$}_{II}(\mbox{\boldmath $x$},t)=(1+\cos\phi)\sin{\mit \Theta}_c \\ \nonumber
&&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+(1-\cos\phi)\sin{\mit \Theta}_d,
\end{eqnarray}
and with ${\mit \Theta}_c=\mbox{\boldmath $k$}_c.\mbox{\boldmath $x$}-\omega_c t-\chi_0$ and ${\mit \Theta}_d=\mbox{\boldmath $k$}_d.\mbox{\boldmath $x$}-\omega_d t -\xi_0$.
The wavefunction and the beables clearly show interference. For example, for $\phi=0$ the $d$-beam is extinguished and for $\phi=\pi$ the $c$-beam is extinguished by interference.
\section{Comments on some other recent experimental tests of complementarity\label{CGBAS}}
In the proposed experiment of Ghose {\it et al} \cite{GHOSE91}, light is incident on a prism at an angle greater than the critical angle and hence undergoes total internal reflection. A second prism placed less than a wavelength from the first allows light to tunnel into the transmitted channel. Quantum mechanics predicts perfect anticoincidence. This is interpreted by Ghose {\it et al}, as is usual, as which-path information and hence as particle behaviour. Transmitted photons necessarily tunnel through the gap between the prisms, a phenomenon which the authors interpret as wave behaviour. In this way, the authors claim that wave and particle behaviour are observed in the same experiment in contradiction to Bohr's principle of complementarity. This experiment has since been performed by Mizobuchi {\it et al} \cite{MIZ92} using a GRA single photon source, but as we mentioned earlier, the statistical accuracy of their results has been questioned in references \cite{UNNIK, GHOSE99, BRIDA04}.
To resolve the technical difficulties with Mizobuchi {\it et al}'s experiment, Brida {\it et al}, following a suggested experiment by Ghose \cite{GHOSE99} and also employing the GRA single photon source, used a birefringent crystal to split a light beam into two beams (the ordinary and the extraordinary beams) instead of using tunneling between two closely spaced prisms. They interpreted the birefringent splitting as wave behaviour, while the perfect anticoincidence they observed they interpreted as particle behaviour. Again, the claim is the observation of wave and particle behaviour in the same experiment in contradiction of complementarity.
Afshar's experiment is of the two-slit type. He first observes interference a short distance in front of the slits and determines the position of the dark fringes. He then replaces the screen with a wire grid such that the grid wires coincide with the dark fringes. A lens is placed after the grid to form an image of the two slits. The images showed no loss of sharpness or intensity as compared to the image of the two slits without the grid in position. Afshar concluded that there was interference prior to formation of the image which he interpretes as wave behaviour. He assumes that the images of the slits are formed by photons coming from the slit on the same side as the image. He then interpretes image formation as providing path information, and hence particle behaviour. Ashar concludes that particle and wave behaviour is observed in the same experiment in contradiction of complementarity.
We do not agree that these experiments either disprove Bohr's principle of complementarity, or, as argued by Brida {\it et al},that they can be viewed as a generalization of Bohr's principle of complementarity. Our reasons follow.
As for the GRA experiments, all the above experiments can be explained using CIEM, i.e., they can be explained entirely in terms of a wave model. One is therefore not forced to conclude that these experiments require a generalization of Bohr's principle of complementarity (a generalization first suggested by Wootters and Zurek \cite{WZ} as mentioned in the introduction), a generalization which is severely flawed, as mentioned in the introduction. We will comment further below.
Arguments from the perspective of complementarity can be put to show that these experiments do not disprove complementarity. Let us first consider the experiments of Mizobuchi {\it et al} and Brida {\it et al}. Bohr emphasized that only the final experimental result (pointer reading) has physical significance and that an experiment should be viewed as a whole, not further analyzable \cite{BR59A, BR28}. We recall the statement of Wheeler, `No phenomenon is a phenomenon until it is an observed phenomenon' (\cite{WHR78}, p 14). In these two experiments, the observed results are anticoincidence detections which the above authors and advocates of complementarity or its variants can reasonably and unambiguously attribute to particle behaviour. The wave behaviour is not detected. It is therefore perfectly consistent for a Bohrian to maintain that the experiments unambiguously define a particle model even if this is counter-intuitive. The Afshar experiment avoids this criticism because the presence of the wire grid physically detects the interference. But, the Afshar experiment still fails because of the first point above, namely that CIEM provides a wave model of image formation by a large series of single photon detections.
Another point to consider is that the mutually exclusive wave and particle complementary concepts are not related to the mathematical formalism of the quantum theory. In this way they differ from complementary concepts such as position and momentum or the components of angular momentum which are not mutually exclusive classical concepts and are represented in the mathematical formalism of the quantum theory by Heisenberg uncertainty relations. In this case, what is called wave or particle behaviour in a given experiment is somewhat arbitrary. Apart from other points, this arbitrariness is an important reason why we feel complementarity can neither be proved nor disproved.
We now comment on a widely accepted generalization of complementarity by Wootters and Zurek in their influential article \cite{WZ}. This generalization admits partial wave and partial particle behaviour in the same experiment. Based on this generalization Wootters and Zurek \cite{WZ}, and later Yasin and Greenberger \cite{GY}, cast particle-wave duality in mathematical form. We have argued in earlier articles \cite{KPW} that far from being a generalization of complementarity, this approach in fact contradicts complementarity. From the mathematical perspective, these mathematical relations are constructs appended to the formalism of the quantum theory but not derived from it. As a measure of coherence they can be thought of as useful heuristic rules, but for the reasons we will give, can be attributed no more fundamental significance than this. For detailed arguments against this generalization we refer the reader to reference \cite{KPW} and restrict ourselves here to briefly emphasizing aspects of complementarity which demonstrate our point of view.
In his explanations of his principle of complementarity \cite{BR59A, JAM74,BR28}, Bohr repeatedly emphasized the mutual exclusiveness of complementary concepts, and the requirement of mutually exclusive experimental arrangements for their correct use or definition. He further emphasized that complementary concepts are abstractions to aid thought, and cannot be attributed physical reality. It seems to the present author that Bohr was concerned to provide a framework for the correct use of classical language or concepts. Thus, for the same physical object to be both a wave and a particle is, quite simply, a contradiction of definitions. This, the present author believes, is what led Bohr to emphasize that complementary concepts could not be attributed physical reality. By insisting on mutually exclusive experimental arrangements for the realization of complementary concepts, Bohr, in the authors view, allowed for the use of classical language/concepts in a way that avoids contradiction. It is for these reasons that we regard the Wootters and Zurek generalization of complementarity in terms of partial particle behaviour/knowledge and partial wave beaviour/knowledge as the complete antithesis of Bohr's principle of complementarity. Even apart from Bohr's teachings, what can it mean for a physical object to be partially a wave and partially a particle? Above, we made a distinction between particle and wave complementary concepts and other pairs of complementary concepts that Bohr did not make. Our arguments here need not apply to complementary concepts such as position and momentum, which classically are not mutually exclusive concepts. We note two things: First, the Wootters and Zurek generalization of complementarity is in terms of wave and particle concepts. Second, from the point of view of interpretation, particle and wave complementary concepts are the most fundamental, and lie at the heart of the interpretational issues of the quantum theory.
The experiment of Kim {\it et al} concerns both complementarity and the Wheeler delayed-choice issue, but its significance goes beyond these issues. The results of this experiment appear to suggest that a present measurement affects a past measurement. The Wheeler delayed-choice experiments indicate that a present measurement either creates or changes the past history leading to a particular result (there are subtle differences between Wheeler's and Bohr's position which are discussed in reference \cite{K05} section 1). The Kim {\it et al} and Wheeler delayed-choice experiment differ in that the past history is not actually observed in Wheeler's experiment, whereas in Kim et al's experiment it is the result of an actual past measurement that is changed by a measurement in the present. We will leave a detailed discussion of this experiment for a later article, but make one observation. The experiment uses a pair of correlated photons produced by the process of spontaneous parametric down conversion. By detecting the photon partner {\it after} the first photon is detected, the earlier measured wave or particle behaviour of the first photon is determined. What seems to have been left out of the Kim {\it et al} analysis is that once the first photon is detected and the state of the EPR partner changes accordingly, thereafter, the EPR correlation is broken. Hence, any measurement performed on the second photon can have no effect on its partner. This is a firm prediction of quantum mechanics. Nevertheless, the strange result in which a present measurement appears to determine the outcome of an earlier measurement needs explanation. Other articles relating to this issue can be found in reference \cite{QE}.
\section{Conclusion}
Their ingenious gating system allowed GRA to test, perhaps for the first time, quantum mechanical predictions for a single photon state. Interference is confirmed in the obvious way. The which-path predictions are also confirmed; the photon is detected in only one path. What we have shown though, is that a wave model (CIEM) can explain this result. It cannot therefore be concluded that the detection of the photon on one path confirms particle behaviour. In a particle model, the photon takes one path at the beam-splitter and is detected in that path, whereas in our wave model the photon splits at the beam-splitter, is nonlocally absorbed, and is again detected in only one path. Since the which-path measurement does not confirm particle behaviour, Bohr's principle of complementarity is also not confirmed, contrary to what is claimed by GRA. We conclude then, that GRA's experiments do not confirm complementarity. We may further add that if complementary is accepted, Wheeler's delayed-choice experiments lead to very strange conclusions: either history is changed at the time of measurement, or history is created at the time of measurement \cite{K05, BDH85}. CIEM, on the other hand, explains Wheeler's delayed-choice experiments in a unique and causal way.
\section{Acknowledgements}
I would like to thank Dr H.V. Mweene and Mr Y. Banda for proof reading my article. I would also like to thank the Dean of Natural Sciences, Dr S.F. Banda, for granting me a short study leave to complete this article.
|
{
"timestamp": "2006-08-29T17:54:33",
"yymm": "0503",
"arxiv_id": "quant-ph/0503201",
"language": "en",
"url": "https://arxiv.org/abs/quant-ph/0503201"
}
|
\section{Background and notation}
We begin by defining the class of matrix function algebras we will
study in this paper. From directed graphs, they arise as the left
regular representation of the directed graphs with $n$ vertices and
$n$ edges connecting each successive vertex in turn, to form a
single loop, or $n$-cycle. We will use the notation
$\mathcal{T}^{+}(\mathcal{C}_n)$ for these algebras, where $n$ is
the length of the cycle in the algebra.
We can view $\mathcal{T}^{+}(\mathcal{C}_n)$ as a matrix function
algebra of the form \[ \begin{bmatrix} f_{1,1}(z^n) & z f_{1,2}(z^n)
& z^2 f_{1,3}(z^n) & \cdots & z^{n-1}f_{1,n}(z^n) \\ z^{n-1}
f_{2,1}(z^n)
& f_{2,2}(z^n) & z f_{2,3}(z^n) & \cdots & z^{n-2}f_{2,n}(z^n) \\
z^{n-2}f_{3,1}(z^n) & z^{n-1}f_{3,2}(z^n) & f_{3,3}(z^n) & \cdots &
z^{n-3}f_{3,n}(z^n) \\ \vdots & \vdots & \vdots & \ddots & \vdots
\\ zf_{n,1}(z^n) & z^2 f_{n,2}(z^n) & z^{3}f_{n,3}(z^n) &
\cdots & f_{n,n}(z^n) \end{bmatrix} \] where $f_{i,j} \in
A(\mathbb{D})$ for all $ 1\leq i,j \leq n$.
These algebras inherit a matricial norm from the matricial norm on
$A(\mathbb{D})$ as $\mathcal{T}^{+}(\mathcal{C}_n)$ can be viewed as
sitting inside $M_n\otimes A(\mathbb{D})$. We will denote by
$A(z^n)$ the algebra $\{ f(z^n): f \in A(\mathbb{D}) \}.$ Notice
that $A(z^n)$ is a subalgebra of $A(\mathbb{D})$ for all $n$.
In what follows, $A$ will always denote an operator algebra, and by
representation we mean a continuous representation of $A$ as an
algebra of operators acting on a Hilbert space $\mathcal{H}$. We
will denote the elementary matrices with a 1 in the $i$-th diagonal
spot by $e_{ii}$. The notation $x = [x_{ij}]$ will denote an $n
\times n$ matrix, where the $i$-$j$ entry is $x_{ij}$. We will write
$\ell(i,j)$ for the formula $|i-j|(\mod{n})$. So that the above
matrix form of $\mathcal{T}^{+}(\mathcal{C}_n)$ can be written as
\[ \{[z^{\ell(i,j)}f_{i,j}(z^n)]: f_{i,j} \in A(\mathbb{D}) \mbox{
for all } 1 \leq i,j \leq n\} . \]
Lastly, for $1 \leq i \leq n-1$ define $Z_i \in
\mathcal{T}^{+}(\mathcal{C}_n)$ as the matrix with $z$ in the
$i$-$(i+1)$ position and zeroes elsewhere. Define $Z_n \in
\mathcal{T}^{+}(\mathcal{C}_n)$ as the matrix with $z$ in the
$n$-$1$ position and zeroes everywhere else. It is not hard to see
that $\mathcal{T}^{+}(\mathcal{C}_n)$ is generated by the set $\{
e_{ii}, Z_i: 1 \leq i \leq n \}$. This shorthand will be used later
when dealing with specific matrices.
\section{Noncommutative point derivations}
Some authors take the definition that follows as the definition of
derivation, see Chapter 9 in \cite{Paul:2002} for example. We use
this notation since we wanted to emphasize the connection between
the derivation and the particular representation. This particular
definition also emphasizes the connections with point derivations
from \cite{Browder:1969} which we exploit in later sections.
\begin{defn} Let $\pi: A \rightarrow B(\mathcal{H})$ be a
representation of $A$. We say that a continuous linear map $D: A
\rightarrow B(\mathcal{H})$ is a {\em point derivation at $\pi$} if
$D(ab) = D(a) \pi(b) + \pi(a) D(b)$ for all $a,b \in A$.
\end{defn}
Of course the function $D(a) = 0$ is a derivation. We refer to this
derivation as the trivial, or zero, derivation. We begin by
identifying a special class of derivations, of which the trivial
derivation is a special case.
\begin{defn} For $\pi:A \rightarrow B(\mathcal{H})$ a
representation of the operator algebra $A$ and for $X \in
B(\mathcal{H})$ we define the function $\delta_X : A \rightarrow
B(\mathcal{H})$ by $\delta_X(a) = \pi(a)X-X\pi(a)$ for all $a \in
A$.\end{defn}
Linearity of $\delta_X$ is obvious. If we let $\{ a_n \}$ be a
sequence in $A$, then $ \lim (\pi(a)X - X \pi(a)) = \pi(\lim a_n)X -
X \pi(\lim a_n) = \delta_{X}(\lim a_n)$ and hence $\delta_X$ is
continuous. We can also see this by noting that $\| \delta_X \| \leq
2 \|\pi \| \|X \|$.
\begin{prop} If $\pi: A \rightarrow B(\mathcal{H})$ is a
representation and $X \in B(\mathcal{H})$ then the function
$\delta_X$ is a continuous derivation at $\pi$. \end{prop}
\begin{proof} It remains to show only that $\delta_X$ is a
derivation. Let $a,b \in A$, then \begin{align*} \delta_X(ab) & =
\pi(ab) X -X \pi(ab) \\ & = \pi(a)\pi(b)X - \pi(a) X \pi(b) + \pi(a)
X \pi(b) - X \pi(a) \pi(b) \\ & = \pi(a) \delta_X(b) + \delta_X(a)
\pi(b).\end{align*}\end{proof}
\begin{defn} Letting $\pi: A \rightarrow B(\mathcal{H})$ be a
representation of the operator algebra $A$ we say that a derivation
at $\pi$ of the form $\delta_X$ is an {\em inner derivation at
$\pi$}.\end{defn}
If the range of $\pi$ is $\mathbb{C}$ non-trivial inner derivations
do not arise. However, in a noncommutative setting they often do.
\begin{prop} Assume that $\pi: A \rightarrow B(\mathcal{H})$ is a
representation such that $ \mathop{\mathrm{ran}} \pi$ is not isomorphic to
$\mathbb{C}$. There exists $X \in B(\mathcal{H})$ such that $
\delta_X \not\equiv 0$. \end{prop}
\begin{proof} It is well known that $B(\mathcal{H})' = \mathbb{C}$.
Since, $\mathop{\mathrm{ran}} \pi$ is not isomorphic to $\mathbb{C}$ there exists $a
\in A$ such that $ \pi(a) \not\in B(\mathcal{H})'$. In other words,
there is $X \in B(\mathcal{H})$ such that $X\pi(a) \neq \pi(a) X$.
It follows that $ \delta_X$ is nontrivial.\end{proof}
What follows is a theorem that, in certain cases, will allow us to
distinguish the inner derivations from other derivations.
\begin{thm}\label{inner} Let $\pi: A \rightarrow B(\mathbb{C}^{nk})$ be a
representation such that \[ \mathop{\mathrm{ran}} \pi \cong \oplus_{i=1}^n M_k \]
where $k \geq 1$ and $n$ is finite. A derivation $D$ at $\pi$ is
inner if and only if $D|_{\ker \pi} \equiv 0 $. \end{thm}
\begin{proof} Assume first, that $D$ is inner at $\pi$. Then for
$a \in \ker \pi$, $D(a) = \pi(a) X- X \pi(a)$ for some $X \in
B(\mathcal{H})$. Now $\pi(a) = 0$ and hence $ D(a) = 0$. As $a$
was an arbitrary element of the kernel the forward direction
follows.
Now suppose that $\ker \pi$ is in the kernel of $D$. We define a
map $\widehat{D}: \mathop{\mathrm{ran}} \pi \rightarrow B(\mathcal{H})$ by $
\widehat{D} (\pi(a)) = D(a)$. If $\pi(a) = \pi(b)$, then $ a-b \in
\ker \pi$ and hence $D(a-b) = 0$. It follows, by linearity of $D$,
that $D(a) = D(b)$ and hence the map $\widehat{D}$ is well defined.
Next, \begin{align*} \widehat{D} (\pi(a) \pi(b)) &= \widehat{D}(
\pi(ab)) \\ &= D(ab) \\ &= D(a) \pi(b) + \pi(a) D(b) \\ & =
\widehat{D}(\pi(a)) \pi(b) + \pi(a) \widehat{D}(\pi(b)).
\end{align*} It follows that $\widehat{D}$ defines a derivation on
$\mathop{\mathrm{ran}} \pi$. Further, since $\mathop{\mathrm{ran}} \pi$ is finite dimensional it
follows that $\widehat{D}$ is continuous.
Notice that if $ n=1$ then as $M_k$ is simple every $M_k$ valued
derivation is inner, \cite{Kadison-Ringrose:1997}. If $n > 1$ we
can use exact sequences of cohomology groups, see
\cite{Johnson:1972} to see that a continuous
$B(\mathbb{C}^{nk})$-valued derivation on \[ \oplus_{i=1}^n M_k \]
is inner. Hence there is $X \in M_{nk}$ such that
$\widehat{D}(\pi(a)) = \pi(a) X - X \pi(a)$. Since
$\widehat{D}(\pi(a)) = D(a)$ the result now follows.
\end{proof}
The next two propositions give us a short method of checking whether
non-inner derivations can occur at $\pi$. For an ideal $M$, we
denote by $M^2$ the algebraic ideal generated by elements of the
form $bc$ such that $b, c \in M$. We will denote the norm closure
of the ideal $M^2$ by $ \overline{M^2}$.
\begin{prop}\label{kernelsquared} If $\ker \pi = \overline{(\ker \pi)^2}$ for a
representation $\pi: A \rightarrow B(\mathcal{H})$, then for a
continuous derivation $D$ at $\pi$, $D|_{\ker \pi} \equiv 0$.
\end{prop}
\begin{proof} Let $a = bc$ where $b,c \in \ker \pi$. Then, \begin{align*}
D(a) &= D(bc) \\ &= \pi(b) D(c) + D(b) \pi(c) \\ & = 0 D(c) + D(b) 0
\\ & = 0. \end{align*} Since $(\ker \pi)^2$ is the ideal generated by
elements of the form $bc$ where $b,c \in \ker \pi$ it follows that
$D|_{(\ker \pi)^2} \equiv 0 $. Now, continuity of $D$ yields the
result.
\end{proof}
\begin{prop}\label{approximateidentity} If the kernel of the
representation $\pi: A \rightarrow B(\mathcal{H})$ has a bounded
left (right) approximate identity then any continuous derivation $D$
at $\pi$ is identically zero on $\ker \pi$.
\end{prop}
\begin{proof} Let $\{ e_{\lambda} \}$ be a bounded left (right)
approximate identity in $\ker \pi$. Then for any $f \in \ker \pi$
we know that $ \lim e_{\lambda} f = f$. But notice that $
e_{\lambda} f \in (\ker \pi)^2$ and hence $D(e_{\lambda} f) = 0$ for
all $ \lambda$. As $D$ is continuous it follows that $D(f)= 0$. As
$f$ was arbitrary the result follows.\end{proof}
\begin{cor} If $A$ is a $C^*$-algebra and $\pi$ is a
$*$-representation then every derivation at $\pi$ is identically
zero on $\ker \pi$.
\end{cor}
\begin{proof} It is well known that the kernel of a
$*$-representation is a $*$-ideal. Further every $*$-ideal in a
$C^*$-algebra is a $C^*$-algebra and hence has an approximate
identity. The result now follows.
\end{proof}
\section{Point derivations on $\mathcal{T}^{+}(\mathcal{C}_n)$}
\begin{defn} For $\lambda \in \overline{\mathbb{D}}$ we define the
representation $\varphi_{\lambda}: \mathcal{T}^{+}(\mathcal{C}_n)
\rightarrow M_n$ by \[ \varphi_{\lambda} (
[z^{\ell(i,j)}f_{i,j}(z^n)]) = [
{\lambda}^{\ell(i,j)}f_{i,j}({\lambda}^n)]. \]
\end{defn}
Notice that for $\lambda \neq 0$ the range of $\varphi_{\lambda}$ is
isomorphic to $M_n$. It follows that $\ker(\varphi_{\lambda})$ is a
maximal ideal of type $\lambda$, see \cite{Alaimia:1999}. In
contrast, the range of $\varphi_0$ is the diagonal matrices in
$M_n$. It is not the case that the kernel of $ \varphi_0$ is a
maximal ideal.
The representations of the form $\varphi_{\lambda}$ are enough to
ensure semisimplicity of $\mathcal{T}^{+}(\mathcal{C}_n)$ a well
known result for certain graph operator algebras, see
\cite{Davidson-Katsoulis:2004}, or \cite{Jury-Kribs:2004}, and
semicrossed products, see \cite{Pet:1988}. We include the proof in
this context for completeness, and since it is not difficult.
\begin{prop} The algebras $\mathcal{T}^{+}(\mathcal{C}_n)$ are semisimple.
\end{prop}
\begin{proof} Let \[ a = [z^{\ell(i,j)}f_{i,j}(z^n)]. \] Assume that $
\varphi_{\lambda} (a) = 0$ for all $0 < |\lambda| < 1$. Then in
particular, $f_{i,j}(\lambda^n) = 0$ for all $ 0< |\lambda| < 1$.
But since $ f_{i,j}(z^n)$ is analytic in $\mathbb{D}$ and
identically zero on a set containing a limit point in $\mathbb{D}$
then $f_{i,j}(z^n) \equiv 0$ for all $i,j$. Hence $a = (0)$ and the
result follows. \end{proof}
We now define another important class of representations.
\begin{defn} For $1 \leq i \leq n$ define the representation
$\varphi_{i,0}: \mathcal{T}^{+}(\mathcal{C}_n) \rightarrow
\mathbb{C}$ by \[ \varphi_{i,0} ( [z^{\ell(i,j)}f_{i,j}(z^n)]) =
f_{i,i}(0).
\]\end{defn}
For these representations, the range is $\mathbb{C}$ and hence the
kernels give rise to maximal ideals which, in the notation of
\cite{Alaimia:1999}, are of type $0$. Notice also, that since the
range is $\mathbb{C}$ there will be no inner derivations at
$\varphi_{0,i}$ for all $1 \leq i \leq n$. More is actually true.
\begin{prop} For $n \geq 2$, there is no nontrivial point derivation
at $\varphi_{0,i}: \mathcal{T}^{+}(\mathcal{C}_n) \rightarrow M_n$,
where $1 \leq i \leq n$.\end{prop}
\begin{proof} We will prove the result for $i = 1$, the general case
proceeds in a similar fashion. A simple calculation tells us that \[
\ker \varphi_{0,1} = \left\{
\begin{bmatrix} z^nf_{1,1}(z^n) & z
f_{1,2}(z^n) & z^2 f_{1,3}(z^n) & \cdots & z^{n-1}f_{1,n}(z^n) \\
z^{n-1} f_{2,1}(z^n) & f_{2,2}(z^n) & z f_{2,3}(z^n) & \cdots &
z^{n-2}f_{2,n}(z^n) \\ z^{n-2}f_{3,1}(z^n) & z^{n-1}f_{3,2}(z^n) &
f_{3,3}(z^n) & \cdots & z^{n-3}f_{3,n}(z^n) \\ \vdots & \vdots &
\vdots & \ddots & \vdots
\\ zf_{n,1}(z^n) & z^2 f_{n,2}(z^n) & z^{3}f_{n,3}(z^n) &
\cdots & f_{n,n}(z^n) \end{bmatrix} \right\} \] where $ f_{i,j} \in
A(\mathbb{D}) $ for all $i,j$. Multiplying two general elements of
$\ker \varphi_{0,1}$ together one can verify that $\ker
\varphi_{0,1} = (\ker \varphi_{0,1})^2$. Using Proposition
\ref{kernelsquared} together with Theorem \ref{inner} we know that
every derivation at $\varphi_{0,1}$ is inner. But since $\mathop{\mathrm{ran}}
\varphi_{0,i} = \mathbb{C}$ any inner derivation is the zero
derivation. The result now follows.\end{proof}
Unlike the previous class of representations, the representations
$\varphi_{\lambda}$ give rise to derivations which are inner at $
\varphi_{\lambda}$. It is the derivations which are not inner at $
\varphi_{\lambda}$ which interest us so we now look at what values
of $ \lambda$ give rise to derivations which are not inner.
\begin{thm} For $|\lambda|<1$ there exist non-inner derivations at
$\varphi_{\lambda}: \mathcal{T}^{+}(\mathcal{C}_n) \rightarrow M_n$.
\end{thm}
\begin{proof} We begin by noticing that the map $F: A(\mathbb{D})
\rightarrow \mathbb{C}$ given by $ F(f) = f'(\lambda)$ is a
continuous linear functional and hence completely continuous
\cite[Corollary 2.2.3]{Eff-Ruan:2000}. In particular we know that
the matricial map $F_{n, \lambda}: M_n \otimes A(\mathbb{D})
\rightarrow M_n$, given by $F_{n, \lambda}([f_{ij}]) =
[f'_{ij}(\lambda)]$ is continuous. Now as
$\mathcal{T}^{+}(\mathcal{C}_n)$ is a subalgebra of $M_n \otimes
A(\mathbb{D})$ we know that $F_{n, \lambda}$ restricted to
$\mathcal{T}^{+}(\mathcal{C}_n)$ yields a continuous linear map.
We need only show that $F_{n, \lambda}$ is a non-inner derivation at
$\varphi_{\lambda}$. To see that it is a derivation we will look at
$F_n$ applied to $M_n \otimes A(\mathbb{D})$. In particular, choose
two elements $f = [f_{ij}],g = [g_{ij}] \in M_n \otimes
A(\mathbb{D})$. Now notice that
\begin{align*} F_{n, \lambda}(fg) & = F_{n, \lambda} \left[
\sum_{j=1}^n f_{ij}g_{jk} \right] \\ & = \sum_{j=1}^n [ f'_{i,j}(
\lambda)g_{jk}(\lambda) +
f_{ij}(\lambda) g'_{jk}(\lambda)] \\
& = \left( \sum_{j=1}^n [ f'_{i,j}( \lambda)g_{jk}(\lambda)]\right)
+ \left( \sum_{j=1}^n [f_{ij}(\lambda) g'_{jk}(\lambda)] \right)
\\ & = [f'_{ij}(\lambda)][g_{ij}(\lambda)] +
[f_{ij}(\lambda)][g'_{ij}(\lambda)] \\ & = F_{n,
\lambda}(f)\varphi_{\lambda}(g) + \varphi_{\lambda}(f)F_{n,
\lambda}(g). \end{align*} Restricting to
$\mathcal{T}^{+}(\mathcal{C}_n)$ will not affect the derivation
property and hence $F_{n,\lambda}$ yields a derivation at $
\varphi_{\lambda}$.
Recall the definition of $Z_i$ as the matrix with a z in the
$i$-$(i+1)$ position for $ 1 \leq i \leq n-1$, or the $n$-$1$
position for $i=n$ and zeroes elsewhere. For $\lambda = 0$ we see
that $F_{n, 0}$ is not inner since $F_{n, 0}(Z_i) \neq 0$ for all
$i$ and yet $\varphi_{0}(Z_i) = 0$, applying Theorem \ref{inner}
verifies the result.
For $0 < |\lambda| <1$ let $f = z- (\lambda)^n$. Notice that $f$ is
an analytic function such that $f'(\lambda^n) \neq 0$ and yet
$f(\lambda^n) = 0$. Now let $\tilde{f}$ be the element of
$\mathcal{T}^{+}(\mathcal{C}_n)$ given by $[z^{\ell(i,j)}f(z^n)]$.
Notice that $\tilde{f} \in \ker \varphi_{\lambda}$. However, $F_{n,
\lambda}(\tilde{f}) = [(\lambda)^{\ell(i,j)}n \lambda^{n-1}] \neq
0$. The result now follows as in the case of $\lambda =
0$.\end{proof}
In the special case of point derivations at $\varphi_0$ we are able
to show more. In analogy with a description of certain homology
groups for the quiver algebras corresponding to a single vertex and
countable edges in \cite{Pop:1998a}, we now show a certain amount of
uniqueness for derivations at $ \varphi_0$.
\begin{prop} Let $D$ be a point derivation at $\varphi_0.$ Then $D$
can be written as $D_0+D_1$ where $D_0$ is inner at $ \varphi_0$,
$D_1$ is a point derivation at $ \varphi_0$, and $D_1(a) \neq 0$
guarantees that $a \in \ker \varphi_0$. Further, $D_1$ is uniquely
determined by the numbers $D_1(Z_i)$ with $ 1 \leq i \leq n$
\end{prop}
\begin{proof} Let $D: \mathcal{T}^{+}(\mathcal{C}_n) \rightarrow M_n$ be a point
derivation at $\varphi_0$. Notice that $\mathop{\mathrm{ran}} \varphi_0$ is finite
dimensional and hence $\ker \varphi_0$ has a Banach space complement
in $\mathcal{T}^{+}(\mathcal{C}_n)$ which we will denote by $ ( \ker
\varphi_0)^c$. Further, every $ a \in
\mathcal{T}^{+}(\mathcal{C}_n)$ can be written uniquely as $x_a +
y_a$ where $ x_a \in (\ker \varphi_0)^c$ and $ y_a \in \ker
\varphi_0$. Now there exist $ \lambda _i$ such that $ x_a =
\displaystyle{ \sum_{i=1}^n \lambda_ie_{ii} }$ where $e_{ii}$ is the
elementary matrix with $1$ in the $i$-$i$ position and zero
everywhere else.
We claim that if $ a ,b \in \mathcal{T}^{+}(\mathcal{C}_n)$ then,
with respect to the decomposition above, $ \varphi_0(x_ax_b) = 0 $
if and only if $x_a x_b = 0$. Writing $x_a = \sum_{i=1}^n
\lambda_ie_{ii}$ and $x_b = \sum_{i=1}^n \mu_ie_{ii}$ then,
\[ x_a x_b = \sum_{i=1}^n \lambda_i \mu_i e_{ii} \] and the claim
follows.
Define the map $D_1: \mathcal{T}^{+}(\mathcal{C}_n) \rightarrow M_n$
by letting $D_1(x_a + y_a) = D(y_a)$ with respect to the above
decomposition. We will use the claim in the previous paragraph to
show that $D_1$ is a derivation at $\varphi_{0}$. Linearity, and
continuity are clear. We need only establish the derivation
property. Now
\begin{align*} D_1(ab) & = D_1((x_a + y_a)(x_b+y_b)) \\ &= D_1(x_ax_b
+ y_ax_b + x_ay_b + y_ay_b) \\ & = D(y_ax_b) + D(x_ay_b) \\ &=
D(y_a) \varphi_0(x_b) + \varphi_0(y_a)D(x_b) + D(x_a)\varphi_0(y_b)
+ \varphi_0(x_a)D(y_b)
\\ & = D(y_a) \varphi_0(x_b) + \varphi_0(x_a)D(y_b) \\ &= D(y_a) \varphi_0 (x_b +
y_b) + \varphi_0(x_a) D_1(x_b + y_b) \\ & = D_1(x_a + y_a)
\varphi_0(x_b + y_b) + \varphi_0(x_a + y_a) D_1(x_b + y_b) \\ &=
D_1(a) \varphi_0(b) + \varphi_0(a) D_1(b)
\end{align*} and hence $ D_1$ is a derivation at $ \varphi_0$.
Notice that $D_0 = D-D_1$ is an inner derivation since $D-D_1|_{\ker
\varphi_0} = 0$. It follows that every point derivation at $
\varphi_0$ can be written as an inner derivation and a derivation
which sends $ (\ker \varphi_0)^c$ to zero.
Notice that each derivation of the form $D_1$ is uniquely determined
by the value on $\ker \varphi_0 \setminus \overline{(\ker
\varphi_0)^2}$. A technical calculation shows us that the set $\ker
\varphi_0 \setminus \overline{(\ker \varphi_0)^2}$ is given by \[ \{
\lambda_i Z_i: 1 \leq i \leq n \}. \] The result now follows.
\end{proof}
The previous result relies on a nice decomposition of every element
of $\mathcal{T}^{+}(\mathcal{C}_n)$ which is invariant under
derivations. Although we expect a similar result for the point
derivations at $ \varphi_{\lambda}$ for all $0 < |\lambda|< 1$ we
have not been able to prove such a result.
\begin{thm}\label{inneratt} For $\lambda \in \mathbb{T}$ every
derivation at $\varphi_{\lambda}: A \rightarrow M_n$ is
inner.\end{thm}
\begin{proof} We will show that $\ker \varphi_{\lambda}$ has a
bounded approximate identity and then apply Proposition
\ref{approximateidentity} and Theorem \ref{inner}.
We let $ \pi_n: A(\mathbb{D}) \rightarrow A(\mathbb{D})$ be the
contractive homomorphism induced by sending $ z \mapsto z^n$. Denote
the range of this map by $A(z^n)$ which matches our previous
definition of $A(z^n)$. Further $\pi_n$ is a contractive
isomorphism onto $A(z^n)$. (We are not making any claims about
contractivity of the reverse map). Notice that \[ \pi_n (\{ f \in
A(\mathbb{D}): f(\lambda) = 0 \}) \subseteq \{ g \in A(z^n):
g(\lambda^{\frac{1}{n}}) = 0 \}.\] Further, since $ | \lambda | = 1$
we know that there is a uniformly bounded net, see \cite[Section
1.6]{Browder:1969}, \[ \{ f_{\iota} \} \subseteq \{ f \in
A(\mathbb{D}): f(\lambda) = 0 \}\] such that $ f_{\iota} g
\rightarrow g$ for all
\[ g \in \{ f \in A(\mathbb{D}): f(\lambda) = 0 \}.\] Notice that $
\| \pi_n(f_{\iota}) \| \leq \| f_{\iota} \|$ and hence $ \{
\pi_n(f_{\iota}) \} $ is a bounded net in $\{ g \in A(z^n):
g(\lambda^{\frac{1}{n}}) = 0 \}$. Now if $g(\lambda^{\frac{1}{n}})
= 0$ and $g \in A(z^n)$ then \[ h = \pi_n^{-1}(g) \in \{ f \in
A(\mathbb{D}): f(\lambda) = 0 \}.\] It follows that $ f_{\iota} h
\rightarrow h$. Now $ \pi_n(f_{\iota} h) \rightarrow g$ and hence
the ideal \[ \{ g \in A(z^n): g(\lambda^{\frac{1}{n}}) = 0 \}\] has
a bounded approximate identity.
We define the net $\{ F_{\iota} \}$ to be the diagonal matrices with
$\pi_n(f_{\iota})$ along the diagonals. Now $ \{ F_{\iota} \}$ is a
bounded net as $\{ f_{\iota} \}$ is. Further, $F_{\iota} \in \ker
\varphi_{\lambda}$ for all $\iota$. It is not hard to see that $\{
F_{\iota} \}$ is an approximate identity in $\ker
\varphi_{\lambda}$.\end{proof}
We will use this theorem to show the main result in this paper, that
every $\mathcal{T}^{+}(\mathcal{C}_n)$-valued derivation is inner.
\section{Derivations on $\mathcal{T}^{+}(\mathcal{C}_n)$}
We begin with an elementary lemma relating derivations and point
derivations.
\begin{lem} Let $D: A \rightarrow A$ be a continuous derivation on
the operator algebra $A$. For a representation $\pi: A \rightarrow
B(\mathcal{H})$, the map $\pi \circ D : A \rightarrow
B(\mathcal{H})$ is a continuous derivation at $\pi$.\end{lem}
\begin{proof} Since $ \pi \circ D$ is a composition of continuous
linear maps, it follows that $ \pi \circ D$ is a continuous linear
map. Now let $a, b \in A$. Then \begin{align*} \pi \circ D (ab) &
= \pi ( D(a)b + aD(b)) \\ & = \pi(D(a))\pi(b) + \pi(a) \pi(D(b)) \\
&= \pi \circ D (a) \pi(b) + \pi(a) \pi \circ D (b). \end{align*} It
follows that $\pi \circ D$ is a derivation at $\pi$.\end{proof}
\begin{defn} Let $\pi: A \rightarrow B(\mathcal{H})$ be a
representation and $D: A \rightarrow A$ be a continuous derivation.
We say that $D$ is {\em locally inner at $\pi$} if $\pi \circ D$ is
inner at $\pi$.\end{defn}
We are now in a position to tackle the main theorem of this paper.
Showing that every $\mathcal{T}^{+}(\mathcal{C}_n)$-valued
derivation on $\mathcal{T}^{+}(\mathcal{C}_n)$ is inner will be a
simple corollary.
\begin{thm}\label{locallyinner} Let $D: \mathcal{T}^{+}(\mathcal{C}_n)
\rightarrow \mathcal{T}^{+}(\mathcal{C}_n)$ be a continuous
derivation which is locally inner at $\varphi_{\lambda}$ for all
$\lambda \in \mathbb{T}$, then $D$ is inner.\end{thm}
\begin{proof} For $\lambda \in \mathbb{T}$ let
$D_{\lambda}:= \varphi_{\lambda} \circ D$. Then, by hypothesis,
$D_{\lambda}(a)$ can be written as $ X_{\lambda}
\varphi_{\lambda}(a) - \varphi_{\lambda}(a) X_{\lambda}$ for all $a
\in \mathcal{T}^{+}(\mathcal{C}_n)$.
Notice that as $e_{ii} \in \mathcal{T}^{+}(\mathcal{C}_n)$ it
follows that $D(e_{ii}) \in \mathcal{T}^{+}(\mathcal{C}_n)$. By
hypothesis $D_{\lambda}(e_{ii}) = X_{\lambda}e_{ii} -
e_{ii}X_{\lambda}$. Now $X_{\lambda}e_{ii} - e_{ii}X_{\lambda}$ is
the matrix with $(-X_{\lambda})_{ij} $ in the $i$-$j$ position, for
$i\neq 0$, $(X_{\lambda})_{ji}$ in the $j$-$i$ position for $i \neq
j$ and 0 elsewhere. In particular, the matrix $Y$ with $0$ on the
diagonal such that $\varphi_{\lambda}(Y_{ij}) = (X_{\lambda})_{ij}$
off the diagonal for all $\lambda \in \mathbb{T}$ is an element of $
\mathcal{T}^{+}(\mathcal{C}_n)$.
Recall the definition of the matrices $Z_i$. Now $D_{\lambda}(Z_i) =
\lambda(X_{\lambda} e_{i,i+1} - e_{i,i+1}X_{\lambda})$ for all $
\lambda \in \mathbb{T}$ where we define $e_{n, n+1}$ to mean
$e_{n,1}$. Now the $i$-$(i+1)$ entry of $D_{\lambda}(Z_i)$ is $
\lambda (-X_{i+1,i+1}(\lambda) + X_{i,i}(\lambda))$, and hence for
all $i$, $X_{i+1,i+1}(\lambda) - X_{i,i}(\lambda)$ defines an
element of $\mathcal{T}^{+}(\mathcal{C}_n)$, call it $X_i$. Now
define a diagonal matrix $X'$ by letting the $i$-$i$ entry be $X_i -
X_1$.
Define $X = Y + X'$ which is in $\mathcal{T}^{+}(\mathcal{C}_n)$.
Further, $D_{\lambda}(e_{ii}) = \varphi_{\lambda}(e_iiX-Xe_ii)$ and
$D_{\lambda}(Z_i) = \varphi_{\lambda}(Z_iX-XZ_i)$. It follows that
for any $ a \in \mathcal{T}^{+}(\mathcal{C}_n)$, $ D_{\lambda}(a) =
\varphi_{\lambda}(aX-Xa)$ for all $ \lambda \in \mathbb{T}$. Now,
every element of $a \in \mathcal{T}^{+}(\mathcal{C}_n)$ is uniquely
determined by the values of $ \varphi_{\lambda}(a) $ for $ \lambda
\in \mathbb{T}$. It follows that $D(a) = aX-Xa$ and the result is
established.
\end{proof}
\begin{cor}\label{graphinner} Every derivation $D: \mathcal{T}^{+}(\mathcal{C}_n)
\rightarrow \mathcal{T}^{+}(\mathcal{C}_n)$ is inner. \end{cor}
\begin{proof} First, since $\mathcal{T}^{+}(\mathcal{C}_n)$ is semisimple we
know, \cite{Johnson-Sinclair:1968} that every derivation is
automatically continuous. We know from Theorem \ref{inneratt} that
every continuous derivation at $ \varphi_{\lambda}$ is inner. In
particular, $D_{\lambda}$ is continuous and locally inner on
$\mathbb{T}$.
\end{proof}
We remark that, as $A(\mathbb{D})$ is a special case of
$\mathcal{T}^{+}(\mathcal{C}_n)$, the above proof is an alternate
approach to the fact there are no nontrivial derivations on
$A(\mathbb{D})$. It would be interesting to know if the above
result can be extended to $\mathcal{L}_{\mathcal{C}_n}$ which is the
matrix function algebra as $\mathcal{T}^{+}(\mathcal{C}_n)$ with
$A(z^n)$ replaced by $H^{\infty}(z^n)$.
|
{
"timestamp": "2006-08-22T18:33:58",
"yymm": "0503",
"arxiv_id": "math/0503643",
"language": "en",
"url": "https://arxiv.org/abs/math/0503643"
}
|
\section{Introduction}
The detection of gravitational waves (GW) from astrophysical
sources is one of the most outstanding problems in experimental
gravitation today. Large laser interferometric gravitational wave
detectors like the LIGO, VIRGO, LISA, TAMA 300, GEO 600 and AIGO
are potentially opening a new window for the study of a vast and
rich variety of nonlinear curvature phenomena.
In recent works \cite{JVD96} we have analyzed the Fourier
transform (FT) of the Doppler shifted GW signal from a pulsar with
the use of the Plane Wave Expansion in Spherical Harmonics
(PWESH). Spherical-harmonic multipole expansions are used
throughout theoretical physics. The expansion of a plane wave in
spherical harmonics has a variety of applications not only in
quantum mechanics and electromagnetic theory \cite{MWIEEE}, but
also in many other areas. A number of researchers have used
spherical-harmonic expansions for a variety of problems in general
relativity, including problems where nonlinearity shows
up\cite{KThorne80}. The basis states in the PWESH expansion form a
complete set and facilitate such a study. It also turns out that
the consequent analysis of the Fourier Transform (FT) of the GW
signal from a pulsar has a very interesting and convenient
development in terms of the resulting spherical Bessel,
generalized hypergeometric function, the Gamma functions and the
Legendre functions. Both rotational and orbital motions of the
Earth and spindown of the pulsar can be considered in this
analysis which happens to have a nice analytic representation for
the GW signal in terms of the above special functions. The signal
can then be studied as a function of a variety of different
parameters associated with both the GW pulsar signal as well as
the orbital and rotational parameters. The numerical analysis of
this analytical expression for the signal offers a challenge for
fast and high performance parallel computation. The plane wave
expansion approach was also used by Bruce Allen and Adrian C.
Ottewill \cite{AO96} in their study of the correlation of GW
signals from ground-based GW detectors. They use the correlation
to search for anisotropies from stochastic background in terms of
the $l, m$ multipole moments. Our PWESH formalism enables a
similar study. Recent studies of the Cosmic Microwave Background
Explorer have raised the interesting question of the study of very
large multipole moments with angular momentum $l$ and its
projection $m$ going up to very large values of $l\sim1000$. Such
problems warrant an intensive analytic study supplemented by
numerical and parallel computation.
Since our FT depends on the Bessel function, a computational issue
arises due to large values of the index or order $n$ of the
function. In the GW form of the pulsar, the Doppler shifted
orbiting motion gives rise to Bessel functions $J_{n}(\frac{2 \pi
f_0 A \sin \theta}{c})$, where $\frac{2 \pi f_0 A \sin\theta}{c}$
is large for non-negligible angle $\theta$ as is shown in the
following section. Even for $\sin{\theta}\sim\frac{1}{1000}$, the
argument is large necessitating the consideration of large values
of $n$. The motivation of this work, is to extend the analysis in
Watson \cite{Watson} for large index, argument and overlapping
situations. Meissel \cite{M1} has made derivations for large order
Bessel functions both when the argument is smaller than the order
and vice versa. The asymptotics of these large order Bessel
functions are tricky in the sense that one runs into so-called
``transition" regions where such expansions fail. These regions
are values of the function when the argument is close to the given
order. As an application, we will address the phenomenological
situation of GW signal analysis of large order $n$ (which does
arise with combinations of $l$ and $m$) and supplement the
related computations with the presently derived results in a
forthcoming paper.
Captures of stellar-mass compact objects (CO) by massive black
holes are important capture sources for the Laser Interferometer
Space Antenna (LISA), the space based GW detector due to be
launched in about a decade\cite{PM}. Higher Harmonics of the
orbital frequency of the COs arise in the post Newtonian (PN)
capture GW model forms and contribute considerably to the total
signal to noise (S/N) ratio of the waveform. The GW form can be
decomposed into gravitational multipole moments which are treated
in the Fourier analysis of Keplerian eccentric orbits. The
radiation depends strongly on the orbital eccentricity $e$, and
Bessel functions $J_{n}(ne)$ are a natural consequence of the
analysis.
The calculation of partial derivatives of the potential scattering
phase shifts which often contain Bessel and Legendre functions of
large order angular momentum $l$, with respect to angular momentum
arise in a variety of scattering problems in atomic, molecular and
nuclear physics. In particular, large values of $l$ can arise in
rainbow, glory and orbit scattering. The analysis in our paper should help provide suitable approximations for large order and/or argument for the Bessel functions that arise in such problems.
\section{Fourier Transform of the GW signal}
The FT for the GW Doppler shifted pulsar signal \cite{JVD96} is
given as follows:
\begin{eqnarray}
\widetilde{h}(f)=S_{n l m}(\omega _{0},\omega
_{orb},T_{rE},n,l,m,A,R,k,\alpha ,\theta ,\phi)= \nonumber
\end{eqnarray}
\\
\begin{eqnarray}
{\sum_{n=-\infty}^{\infty}\sum_{l=0}^{\infty}\sum_{m=-l}^{l}\psi_0
\psi_1 \psi_2 \psi_3 \psi_4}
\end{eqnarray}
where
\begin{equation}
\psi_0(n,l,m,\alpha,\theta,\phi)= 4\pi i^{l}Y_{l m}(\theta ,\phi
)N_{l m}P_{l }^{m}(\cos \alpha )
\end{equation}
\begin{eqnarray}
\psi_1(n,\theta, \phi, T_{rE}, f_0,
A)=T_{rE}\sqrt{\frac{\pi}{2}}e^{-i\frac{2\pi f_{0}A}{c}\sin \theta
\cos \phi } i^{n}e^{-in\phi }\nonumber
\end{eqnarray}
\begin{eqnarray}
\times J_{n}\left(\frac{2\pi f_{0}A\sin \theta }{c}\right)
\end{eqnarray}
\begin{equation}
\psi_2(l,\omega_{orb}, \omega_{r}, n, m, R)=\left\{\frac{1-e^{i\pi
(l -B_{orb})R}}{1-e^{i\pi (l -B_{orb})}} \right\}
\frac{e^{-iB_{orb}\frac{\pi}{2}}}{2^{2l}}
\end{equation}
\begin{eqnarray}
\psi_3(k,l,m,n,\omega_{orb}, \omega_r)=k^{l+\frac{1}{2}}\nonumber\\
\times \frac{\Gamma \left(l +1\right) }{\Gamma \left(l
+\frac{3}{2}\right)\Gamma \left(\frac{l
+B_{orb}+2}{2}\right)\Gamma \left(\frac{l -B_{orb}+2}{2}\right)}
\end{eqnarray}
\begin{eqnarray}
\psi_4(k,l,m,n,\omega_{orb}, \omega_r)=_1F_{3}(l +1;l+\frac{3}{2}, \nonumber \\
\frac{l+B_{orb}+2}{2},\frac{l-B_{orb}+2}{2};\frac{-k^{2}}{16})
\end{eqnarray}
The angle $\alpha$ is the co-latitude detector angle and angles
$\theta$, $\phi$ are associated with the pulsar source. Here
$\omega_0=2\pi f_0$, $\omega_{orb}=\frac{2\pi}{T_{orb}}$
($T_{orb}=365$ days, $T_{rE} = 1$ day),
$B_{orb}=2\left(\frac{\omega-\omega_0}{\omega_r}+\frac{m}{2}+\frac{n
\omega_{orb}}{\omega_{rot}}\right)$, $k=\frac{4\pi f_0 R_E
\sin(\alpha)}{c}$ ($R_E$ is the radius of Earth, $c$ is the
velocity of light) and $A=1.5 \times 10^{11}$ meters is the
sun-earth distance.
\section{Extensions of Meissel's and Steepest Descent Expansions}
The Bessel function, of the type, $J_{\nu}(x)$ obeys the following
differential equation \cite{Watson},
\begin{equation}
z^2\frac{d^2J_{\nu}(\nu z)}{d z^2}+z\frac{dJ_{\nu}(\nu z)}{d
z}+\nu^2(1-z^2)J_{\nu}(\nu z)=0
\end{equation}
where the argument $x$ is parameterized by $\nu z$. If a function
$u(z)$ is introduced such that
\begin{equation}
J_{\nu}(\nu z)=\frac{\nu^{\nu}}{\Gamma (\nu+1)} \exp
\left\{\int_{}^{z} u(z) dz \right \}
\end{equation}
where $u(z)$ is a series in descending powers of $\nu$,
\begin{eqnarray}
u(z)=\nu u_0 + u_1 + \frac{u_2}{\nu} + \frac{u_3}{\nu^2}+
\frac{u_4}{\nu^3}+\frac{u_5}{\nu^4}+\frac{u_6}{\nu^5}+\frac{u_7}{\nu^6}\nonumber\\
+\frac{u_8}{\nu^7}+\frac{u_9}{\nu^8}+...
\end{eqnarray}
Substitution of this series and equation (8) in the differential
equation (7) yields the following expressions for $u_i(z)$,
$i=0...5$,
\begin{eqnarray}
u_0=\frac{\sqrt{1-z^2}}{z}, u_1=\frac{z}{2(1-z^2)},
u_2=-\frac{4z+z^2}{8(1-z^2)^{5/2}} \nonumber\\
u_3=\frac{4z+10z^3+z^5}{8(1-z^2)^{4}},
u_4=-\frac{64z+560z^3+456z^5+25z^7}{128(1-z^2)^{11/2}} \nonumber\\
u_5=\frac{16z+368z^3+924z^5+347z^7+13z^9}{32(1-z^2)^{7}} \nonumber
\end{eqnarray}
Hence, by integrating $u_i$, and substituting in Equation (8) we
arrive at Meissel's \textit{First} expansion \cite{M1}, which is
valid for the case when the argument is less than the order $\nu$.
We do not list $u_6,u_7,u_8$ and $u_9$ as one can obtain these
straightforwardly from their respective integrals shown below.
These results are expressed as,
\begin{equation}
J_{\nu}(\nu z)=\frac{(\nu z)^{\nu} \exp (\nu
\sqrt{1-z^2})\exp(-V_{\nu})}{e^{\nu}\Gamma (\nu+1)
(1-z^2)^{1/4}[1+\sqrt{1-z^2}]^{\nu}}
\end{equation}
where,
\begin{equation}
V_{\nu}=V_1+V_2+V_3+V_4+V_5+V_6+V_7+V_8+...
\end{equation}
and,
\begin{eqnarray}
V_1=\frac{1}{24\nu}\left(\frac{2+3z^2}{(1-z^2)^{3/2}}-2\right),V_2=-\frac{4z^2+z^4}{16\nu^2(1-z^2)^3}\nonumber
\end{eqnarray}
\begin{eqnarray}
V_3=-\frac{1}{5760\nu^3}\left(\frac{16-1512z^2-3654z^4-375z^6}{(1-z^2)^{9/2}}-16\right)\nonumber
\end{eqnarray}
\begin{eqnarray}
V_4=-\frac{32z^2+288z^4+232z^6+13z^8}{128\nu^{4}(1-z^2)^6}\nonumber
\end{eqnarray}
\begin{eqnarray}
V_5=-\frac{1}{322560\nu^5(1-z^2)^{15/2}}(67599\,{z}^{10}+1914210\,{z}^{8}\nonumber\\
+4744640\,{z}^{6}+1891200\,{z}^{4}+78720\,{z}^{2}+256)+\frac{1}{1260\nu^5}\nonumber
\end{eqnarray}
\begin{eqnarray}
V_6=\frac{z^2}{192(1-{z}^{2})^{9}{\nu}^{6}}(48+2580{z}^{2}+14884{z}^{4}\nonumber\\
+17493{z}^{6}+4242{z}^{8}+103{z}^{10})\nonumber
\end{eqnarray}
\begin{eqnarray}
V_7=-\frac{(1-z^2)^{-21/2}}{3440640\nu^7}(881664{z}^{2}+99783936{z}^{4}\nonumber
\end{eqnarray}
\begin{eqnarray}
+1135145088{z}^{6}+2884531440{z}^{8}+1965889800{z}^{10}\nonumber\\
+318291750{z}^{12}+5635995{z}^{14}-2048)-\frac{1}{1680\nu^7}\nonumber
\end{eqnarray}
\begin{eqnarray}
V_8={\frac{z^2}{4096(1-{z}^{2})^{12}{\nu}^{8}}}(1024+248320{z}^{2}+5095936{z}^{4}\nonumber\\
+24059968{z}^{6}+34280896{z}^{8}+15252048{z}^{10}\nonumber\\
+1765936{z}^{12}+23797{z}^{14})\nonumber
\end{eqnarray}
Hence we have actually increased Meissel's analysis by two orders.
Using symbolic packages these orders were computed and higher
terms should pose no problem if the application requires higher
accuracy.
For the case when the argument is larger than the index, Meissel
used the parametrization $z=\sec{\beta}$ \cite{M1}, and we shall
term it as his \textit{Second} expansion. Hence,
\begin{equation}
J_{\nu}(\nu \sec{\beta})=\sqrt{\frac{2 \cot{\beta}}{\nu \pi}}
e^{-P_{\nu}}\cos\left(Q_{\nu}-\frac{1}{4}\pi \right)
\end{equation}
where $P_{\nu}$ is given as,
\begin{equation}
P_{\nu}=P_1+P_2+P_3+P_4+...
\end{equation}
where
\begin{eqnarray}
P_{1}=\frac{\cot^{6}\beta}{16\nu^2}\left(4\sec^2{\beta}+\sec^4{\beta}\right)\nonumber
\end{eqnarray}
\begin{eqnarray}
P_{2}=-\frac{\cot^{12}\beta}{128\nu^4}(32\sec^2{\beta}+288\sec^4{\beta}+232\sec^6{\beta}\nonumber\\
+13\sec^8{\beta})\nonumber
\end{eqnarray}
\begin{eqnarray}
P_{3}=\frac{\cot^{18}\beta}{192\nu^6}(48\sec^2{\beta}+2580\sec^4{\beta}+14884\sec^6{\beta}\nonumber\\
+17493\sec^8{\beta}+4242\sec^{10}{\beta}+103\sec^{12}{\beta})\nonumber
\end{eqnarray}
\begin{eqnarray}
P_{4}=\frac{\cot^{24}\beta\sec^2{\beta}}{4096\nu^8}(1024+248320\sec^2{\beta}+5095936\sec^4{\beta}\nonumber
\end{eqnarray}
\begin{eqnarray}
+24059968\sec^6{\beta}+34280896\sec^8{\beta}+15252048\sec^{10}{\beta}\nonumber\\
+1765936\sec^{12}{\beta}+23797\sec^{14}{\beta})\nonumber
\end{eqnarray}
and $Q_{\nu}$ is given as,
\begin{equation}
Q_{\nu}=Q_1+Q_2+Q_3+Q_4+...
\end{equation}
and,
\begin{eqnarray}
Q_1=\nu(\tan{\beta}-\beta)-\frac{\cot^{3}\beta}{24\nu}\left(2+3\sec^2{\beta}\right)\nonumber
\end{eqnarray}
\begin{eqnarray}
Q_2=-\frac{\cot^{9}\beta}{5760\nu^3}(16-1512\sec^2{\beta}-3654\sec^4{\beta}\nonumber\\
-375\sec^6{\beta})\nonumber
\end{eqnarray}
\begin{eqnarray}
Q_3=-\frac{\cot^{15}\beta}{322560\nu^5}(256+78720\sec^2{\beta}+1891200\sec^4{\beta}\nonumber\\
+4744640\sec^6{\beta}+1914210\sec^{8}{\beta}+67599\sec^{10}{\beta})\nonumber
\end{eqnarray}
\begin{eqnarray}
Q_4=-\frac{\cot^{21}\beta}{3440640\nu^7}(881664\sec^2{\beta}+99783936\sec^4{\beta}\nonumber
\end{eqnarray}
\begin{eqnarray}
+1135145088\sec^6{\beta}+2884531440\sec^8{\beta}+1965889800\sec^{10}{\beta}\nonumber\\
+318291750\sec^{12}{\beta}+5635995\sec^{14}{\beta}-2048)\nonumber
\end{eqnarray}
It should be remarked that we disagree with Meissel's result for
$P_3$ in the last four terms. However, we obtain perfect agreement
with the rest of his results \cite{M1}. We have improved on his
result by using $V_7$, $V_8$ to obtain $P_4$ and $Q_4$. Hence, we
have increased the accuracy of this expansion by at least one
order from Meissel's earlier result. Again, higher order results
are easily obtainable and are available if needed.
In Figures 1 and 2 we have plotted these expansions in the regions
they are expected to fail. These are the so called ``transition"
regions, where each expansion approaches a singularity (as the
order equals the argument). For the computationally motivated (we
can compute exact values of Bessel functions with ease) case of
the $\nu=300$, we note the following. Fig.1 indicates the onset of
breakdown in the \textit{First} expansion for argument values
around and larger than 290. Similarly, Figure 2, indicates a
similar breakdown starting around the values 300 and persisting
till 310. Hence, the values outside these regions of breakdown or
transition regions are well covered by Meissel's expansions.
However, the issue as to deal with these regions need to be
addressed via separate methods, which will be addressed in more
detail in Section IV. The CPU time for these approximations was
less than 0.01 seconds per value on a 2.4 GHz Pentium IV processor
running MAPLE version 9. The ``exact" MAPLE solver took somewhere
between 0.03 to 0.08 seconds to compute each value. Clearly, there
is a lot more computational speed in using a few terms present in
these expansions. As an application, it should be noted that
values of this order are applicable to the Peters-Mathews model of
gravitational radiation from binary inspiralling stars \cite{PM}.
\begin{figure}
\centering \epsfig{file=Fig1_Meissel_First.eps,width= 2.5 in,
height= 2.5 in, angle=270} \caption{Meissel's \textit{First}
expansion and actual Bessel function graphed for argument $x$ and
order $\nu=300$ near the transition region. Solid line indicates
actual Bessel function values and circles indicate values given by
the expansion.} \label{fig_three}
\end{figure}
\begin{figure}
\centering \epsfig{file=Fig2_Meissel_Second.eps,width= 2.5 in,
height= 2.5 in, angle=270} \caption{Meissel's \textit{Second}
expansion and actual Bessel function graphed for argument $x$ and
order $\nu=300$ near the transition region. Solid line indicates
actual Bessel function values and circles indicate values given by
the expansion.} \label{fig_two}
\end{figure}
For the case when the argument equals the index, we extend
Meissel's \textit{Third} expansion \cite{M1} by two orders as
follows:
\begin{equation}
J_{n}(n)\sim\frac{1}{\pi}\sum_{m=0}^{\infty} \lambda_{m}
\Gamma\left(\frac{2m}{3} + \frac{4}{3}
\right)\left(\frac{6}{n}\right)^{\frac{2}{3}m +
\frac{1}{3}}\cos\pi (\frac{m}{3} + \frac{1}{6})
\end{equation}
where the terms, $\lambda_m$ ($m=0,1,2,..7$), are given by,
\begin{eqnarray}
\lambda_0=1,\lambda_1=\frac{1}{60},\lambda_2=\frac{1}{1400},\lambda_3=\frac{1}{25200},\nonumber
\end{eqnarray}
\begin{eqnarray}
\lambda_4=\frac{43}{17248000},\lambda_5=\frac{1213}{7207200000},\lambda_6=\frac{681563}{5721073600000},\nonumber
\end{eqnarray}
\begin{eqnarray}
\lambda_7=\frac{63319}{726485760000000}
\end{eqnarray}
We observe that inclusion of the higher order terms leads to 10
decimal accuracy compared to actual values of large order Bessel
functions.
The method of steepest descents was employed by Debye in
\cite{DBY}. For the case when the argument is less than the order,
he obtained,
\begin{equation}
J_{\nu}(\nu sech(\alpha))\sim
\frac{e^{\nu(\tanh{\alpha}-\alpha)}}{\sqrt{2 \pi
\nu\tanh{\alpha}}}
\sum_{m=0}^{\infty}\frac{\Gamma(m+\frac{1}{2})}{\Gamma(\frac{1}{2})}\frac{A_m}{(\frac{1}{2}\nu
\tanh{\alpha})^{m}}
\end{equation}
where,
\begin{eqnarray}
A_0=1, A_1=\frac{1}{8}-\frac{5}{24}\coth^2{\alpha} \nonumber
\end{eqnarray}
\begin{eqnarray}
A_2=\frac{3}{128}-\frac{77}{576}\coth^2{\alpha}+\frac{385}{3456}\coth^4{\alpha}\nonumber
\end{eqnarray}
\begin{eqnarray}
A_3=\frac{5}{1024}-\frac{1521}{25600}\coth^2{\alpha}+\frac{17017}{138240}\coth^4{\alpha}\nonumber\\
-\frac{17017}{248832}\coth^6{\alpha}\nonumber
\end{eqnarray}
\begin{eqnarray}
A_4=\frac{11513}{92897280}-\frac{21023}{9953280}\coth^2{\alpha}+\frac{138919}{19906560}\coth^4{\alpha}\nonumber\\
-\frac{49049}{5971968}\coth^6{\alpha}+\frac{230945}{71663616}\coth^8{\alpha}\nonumber
\end{eqnarray}
Following this method, we have computed two higher orders $A_3$
and $A_4$, using symbolic computation.
For the case when the argument is larger than the order, Debye
obtains the following expansion:
\begin{eqnarray}
J_{\nu}(\nu\sec{\beta})\sim\sqrt{\frac{2}{\pi\nu\tan{\beta}}}[\cos\left(\nu\tan{\beta}-\nu\beta-\frac{1}{4}\beta\right)\nonumber
\end{eqnarray}
\begin{eqnarray}
\times\sum_{m=0}^{\infty}(-1)^{m}\frac{\Gamma(m+\frac{1}{2})}{\Gamma(\frac{1}{2})}\frac{A_{2m}}{(\frac{1}{2}\nu\tanh{\alpha})^{2m}}+\sin(\nu\tan{\beta}\nonumber
\end{eqnarray}
\begin{eqnarray}
-\nu\beta-\frac{1}{4}\beta)\sum_{m=0}^{\infty}(-1)^{m}\frac{\Gamma(2m+\frac{3}{2})}{\Gamma(\frac{1}{2})}\frac{A_{2m+1}}{(\frac{1}{2}\nu
\tanh{\alpha})^{2m+1}}]
\end{eqnarray}
where,
\begin{eqnarray}
A_0=1, A_1=\frac{1}{8}+\frac{5}{24}\cot^2{\beta}\nonumber
\end{eqnarray}
\begin{eqnarray}
A_2=\frac{3}{128}+\frac{77}{576}\cot^2{\beta}+\frac{385}{3456}\cot^4{\beta}\nonumber
\end{eqnarray}
\begin{eqnarray}
A_3=\frac{5}{1024}+\frac{1521}{25600}\cot^2{\beta}+\frac{17017}{138240}\cot^4{\beta}\nonumber\\
+\frac{17017}{248832}\cot^6{\beta}\nonumber
\end{eqnarray}
\begin{eqnarray}
A_4=\frac{11513}{92897280}+\frac{21023}{9953280}\cot^2{\beta}+\frac{138919}{19906560}\cot^4{\beta}\nonumber\\
+\frac{49049}{5971968}\cot^6{\beta}+\frac{230945}{71663616}\cot^8{\beta}\nonumber
\end{eqnarray}
Again, we have extended Debye's result by two higher orders by
obtaining $A_3$ and $A_4$. However, due to the nature of this
method we could not obtain reliable results that spanned in a
generally predictable direction. Accuracy was limited to the
region of the stationary phase as expected and hence, we recommend
Meissel's expansions to be more reliable (except of course in the
``transition" region) than the method of steepest descent.
\section{Transitional regions: Contour Integration and extension of $\epsilon$ expansion}
To address the issues related to computation for large order
Bessel functions in the transition regions we present two methods
that are geared to work in such domains. Firstly, we present the
results by Watson, \cite{Watson}. For the case of the argument
being less than the order, he obtained via use of contour
integration,
\begin{eqnarray}
J_{\nu}(\nu sech(\alpha))= \frac{\tanh{\alpha}}{\pi
\sqrt{3}}\exp\left[\nu\left(\tanh{\alpha}+\frac{1}{3}\tanh^3{\alpha}-\alpha\right)\right]\nonumber\\
\times
K_{\frac{1}{3}}\left(\frac{1}{3}\nu\tanh^3{\alpha}\right)\nonumber
\end{eqnarray}
\begin{eqnarray}
+3\theta_{1}\nu^{-1}\exp[\nu(\tanh{\alpha}-\alpha)]
\end{eqnarray}
where $\theta_1<1$. Similarly, for the case when the argument is
greater than the order, he derived the following:
\begin{eqnarray}
J_{\nu}(\nu
\sec{\beta})=\frac{1}{3}\tan{\beta}\cos\left[\nu\left(\tan{\beta}-\frac{1}{3}\tan^3{\beta}-\beta\right)\right]\times\nonumber
\end{eqnarray}
\begin{eqnarray}
\left(J_{-\frac{1}{3}}+J_{\frac{1}{3}}\right)+3^{-\frac{1}{2}}\tan{\beta}\sin\left[\nu\left(\tan{\beta}-\frac{1}{3}\tan^3{\beta}-\beta\right)\right]\times\nonumber\\
\left(J_{-\frac{1}{3}}-J_{\frac{1}{3}}\right)\nonumber
\end{eqnarray}
\begin{eqnarray}
+24\theta_{2}\nu^{-1}
\end{eqnarray}
where $\theta_2<1$ and the argument for the Bessel functions
$J_{\pm\frac{1}{3}}$ is $\frac{1}{3}\tan^{3}{\beta}$. The great
advantage of these formulae is that they have error bounds given.
However, these extensions are not trivial as this involves solving
extensions to Airy-type integrals, for which we do not presently
have closed form answers. The other issue with these formulae is
that they are themselves given in fractional Bessel function form
which would pose computational problems once the arguments
involved are large.
On the other hand, Debye \cite{DBY}, introduced, what we will term
as ``$\epsilon$ expansion". The idea is motivated by introducing a
small parameter $\epsilon$, such that $\nu=z(1-\epsilon)$, where
$\nu$ denotes the order and $z$ is the argument of the Bessel
function.
\begin{equation}
J_{\nu}(z)\sim\frac{1}{3\pi}\sum_{m=0}^{\infty} B_{m}(\epsilon z)
\sin\frac{1}{3}(m+1)\pi\cdot
\frac{\Gamma(\frac{1}{3}m+\frac{1}{3})}{(\frac{1}{6}z)^{\frac{1}{3}(m+1)}}
\end{equation}
We have extended this analysis by 5 orders and the terms
$B_m(\epsilon z), m=0,1,2,..15$, are given as,
\begin{eqnarray}
B_0(\epsilon z)=1, B_1(\epsilon z)=\epsilon z, B_3(\epsilon
z)=\frac{1}{6}\epsilon^3 z^3 -\frac{1}{15} \epsilon z \nonumber
\end{eqnarray}
\begin{eqnarray}
B_4(\epsilon z)=\frac{1}{24}\epsilon^4 z^4 -\frac{1}{24}
\epsilon^2 z^2 + \frac{1}{280}\nonumber
\end{eqnarray}
\begin{eqnarray}
B_6(\epsilon z)={\frac {1}{720}}\,{z}^{6}{\epsilon}^{6}-{\frac
{7}{1440}}\,{z}^{4}{ \epsilon}^{4}+{\frac
{1}{288}}\,{z}^{2}{\epsilon}^{2}-{\frac {1}{3600} }
\end{eqnarray}
\begin{eqnarray}
B_7(\epsilon z)={\frac {1}{5040}}\,{z}^{7}{\epsilon}^{7}-{\frac
{1}{900}}\,{z}^{5}{ \epsilon}^{5}+{\frac
{19}{12600}}\,{z}^{3}{\epsilon}^{3}-{\frac {13}{
31500}}\,z\epsilon \nonumber
\end{eqnarray}
\begin{eqnarray}
B_{9}(\epsilon z)={\frac
{1}{362880}}\,{z}^{9}{\epsilon}^{9}-{\frac {1}{30240}}\,{z}^{7}
{\epsilon}^{7}+{\frac
{71}{604800}}\,{z}^{5}{\epsilon}^{5}\nonumber\\
-{\frac{121}{907200}}\,{z}^{3}{\epsilon}^{3}+{\frac{7939}{232848000}}\,z\epsilon\nonumber
\end{eqnarray}
\begin{eqnarray}
B_{10}(\epsilon z)={\frac
{1}{3628800}}\,{z}^{10}{\epsilon}^{10}-{\frac {11}{2419200}}\,{
z}^{8}{\epsilon}^{8}+{\frac
{143}{6048000}}\,{z}^{6}{\epsilon}^{6}\nonumber\\
-{ \frac{803}{18144000}}\,{z}^{4}{\epsilon}^{4}+{\frac
{43}{1728000}}\,{ z}^{2}{\epsilon}^{2}-{\frac
{1213}{655200000}}\nonumber
\end{eqnarray}
\begin{eqnarray}
B_{12}(\epsilon z)={\frac
{1}{479001600}}\,{z}^{12}{\epsilon}^{12}-{\frac{13}{217728000}}\,{z}^{10}{\epsilon}^{10}+\nonumber\\
{\frac
{299}{508032000}}\,{z}^{8}{\epsilon}^{8}-{\frac{377}{155520000}}\,{z}^{6}{\epsilon}^{6}+{\frac{337207}{83825280000}}\,{z}^{4}{\epsilon}^{4}\nonumber\\
-{\frac{59503}{27941760000}}\,{z}^{2}{\epsilon}^{2}+{\frac{151439}{977961600000}}\nonumber
\end{eqnarray}
\begin{eqnarray}
B_{13}(\epsilon z)={\frac
{1}{6227020800}}\,{z}^{13}{\epsilon}^{13}-{\frac{1}{171072000}}\,{z}^{11}{\epsilon}^{11}+\nonumber\\
{\frac{11}{145152000}}\,{z}^{9}{\epsilon}^{9}-{\frac{47}{108864000}}\,{z}^{7}{\epsilon}^{7}+{\frac{25853}{23950080000}}\,{z}^{5}{\epsilon}^{5}\nonumber\\
-{\frac{266303}{259459200000}}\,{z}^{3}{\epsilon}^{3}+\frac{169039}{698544000000}\,z\epsilon\nonumber
\end{eqnarray}
\begin{eqnarray}
B_{15}(\epsilon z)={\frac
{1}{1307674368000}}\,{z}^{15}{\epsilon}^{15}-{\frac{1}{23351328000}}\,{z}^{13}{\epsilon}^{13}\nonumber\\
+{\frac{113}{125737920000}}\,{z}^{11}{\epsilon}^{11}-{\frac{17}{1905120000}}\,{z}^{9}{\epsilon}^{9}\nonumber\\
+{\frac{76841}{1760330880000}}\,{z}^{7}{\epsilon}^{7}-{\frac{37021}{371498400000}}\,{z}^{5}{\epsilon}^{5}\nonumber\\
+{\frac{5141933}{57210753600000}}\,{z}^{3}{\epsilon}^{3}-{\frac{16720141}{810485676000000}}\,z\epsilon\nonumber
\end{eqnarray}
Terms $B_{3m-1}$, $m=1,2...$ do not contribute in eqn. (21) due
to the periodicity of the sine function. With symbolic
computation, we are able to generate higher orders if needed.
To illustrate the applicability and issues of both these methods
to the transition region, we present Figures 3 and 4, which are
plotted for the problematic regions (when the order is $\nu=300$)
in Figures 1 and 2. Both methods show remarkable ability in
capturing the functions in the domains of interest. In Figure 3,
the $\epsilon$ expansion starts working at values at 286 and
Watson's formula works to even a larger domain. Similarly, in
Figure 4, both the methods indicate success in regions where
Meissel's expansions fail. This starts at values of the argument,
and works up to $x=316$ for the $\epsilon$ expansion whereas,
again, the domain of Watson's formula is much greater. The reasons
for lesser range of the $\epsilon$ expansion can be attributed to
the fact that it is a power series compared to Watson's formula
which actually depends on fractional Bessel functions themselves.
Further, the $\epsilon$ expansion depends crucially on the size of
the parameter, which is connected with the order one is working
with. However, the reason why we will persist with this method is
that it will be more applicable when the argument of the Bessel
function is quite large.
\begin{figure}
\centering \epsfig{file=Fig3_eps_Watson.eps,width= 2.5 in, height=
2.5 in, angle=270} \caption{Comparison of $\epsilon$ expansion and
Watson's formulae for argument $x<300$ and order $\nu=300$.Solid
line indicates exact Bessel function values, diamonds represent
$\epsilon$ expansion and circles indicate values given by Watson's
formula.} \label{fig_three}
\end{figure}
\begin{figure}
\centering \epsfig{file=Fig4_eps_Watson.eps,width= 2.5 in, height=
2.5 in, angle=270} \caption{Comparison of $\epsilon$ expansion and
Watson's formula in the transition region for argument $x>300$ and
order $\nu=300$. Solid line indicates exact Bessel function
values, diamonds represent $\epsilon$ expansion and circles
indicate values given by Watson's formula.} \label{fig_four}
\end{figure}
To illustrate the type of values a GW pulsar FT would require, we
present Figures 5 and 6. Here, we choose a very large order (yet
realistic phenomenologically) for the Bessel function, which is 1
million. Also, in such a scenario, we would be looking at values
greater than one million, hence Meissel's second expansion along
with the appropriate Watson's formula (eq. 20) will be put to use.
We were not able to make exact comparison, obviously due to
massive computer times required. In this regard, the problem of
``exact" Bessel functions presents a genuine challenge to SHARCNET
(Shared Hierarchical Academic Research Cluster Network) and HPC in
general. In Figure 5, we observe strong evidence that the proposed
asymptotic expansions are appropriate for GW signal analysis.
Here, we note the transition region starting at values of the
argument at 1,000,000 and going up to 1,000,200. In this region,
both the $\epsilon$ expansion and Watson's formula almost coincide
with each other. As usual, the $\epsilon$ expansion breaks down
earlier, however, all three methods coincide in a certain region
indicating that we have consistent methods that work for values
relevant to GW analysis. Meissel's expansion is fairly easy to
implement computationally and indicates good stability for rather
large values of the argument. This is illustrated in Figure 6,
where we plot this expansion for values ranging from 1,000,200 to
32,500,000, which are relevant for GW phenomenology. This appears
as a black band and is a continuous function which indicates
oscillations tightly bunched together. It is noteworthy that the
method is stable and shows consistent behaviour over an extreme
range of values for the argument. The CPU time consumed by each of
the points, on the average took less than 0.01 seconds on MAPLE.
The Bessel utility in MAPLE crashed repeatedly after 15-30 minutes
on the same system described above. It should be remarked that
Watson's formula lacks in this capacity as it depends on
fractional Bessel functions itself, which will provide
computational challenge for such values. A detailed analysis
regarding computational advantage over exact computation will be
addressed in a later work. It is aimed to not only address the
question of GW analysis but will deal with general computational
issues regarding large order Bessel functions.
\begin{figure}
\centering \epsfig{file=Fig5_million_eps_fail.eps,width= 2.5 in,
height= 2.5 in, angle=270} \caption{Comparison of Meissel's
\textit{Second} expansion, $\epsilon$ expansion and Watson's
formulae for argument $x>1,000,000$ and order $\nu=1,000,000$.
Solid line indicates Meissel's \textit{Second} expansion values,
diamonds represent $\epsilon$ expansion and circles indicate
values given by Watson's formula.} \label{fig_five}
\end{figure}
\begin{figure}
\centering \epsfig{file=Fig6_million_Messel_success.eps,width= 2.5
in, height= 2.5 in, angle=270} \caption{Plot of Meissel's
\textit{Second} expansion for argument, $x$ ranging from 1,000,200
to 32,500,000 for order 1,000,000.} \label{fig_six}
\end{figure}
\section{Conclusion}
In this present work, we have given an extended asymptotic
analysis for the large order and argument Bessel functions. This
analytically improves the earlier pioneering works of Meissel,
Airey, Debye and Watson. These extensions should be of possible
use not only in GW signal analysis, but also in a variety of
problems in Engineering and the Sciences where the ubiquitous
Bessel functions are encountered.
\section*{Acknowledgments}
We are deeply grateful to SHARCNET for valuable grant support that
made this study feasible. We are also indebted to Drs. Nico Temme
(CWI, Amsterdam), Walter Gautschi (Purdue U.), D.G.C. McKeon (U.
Western Ontario), Tom Prince (JPL, Pasadena) and the referee for
valuable suggestions.
|
{
"timestamp": "2005-03-15T07:47:17",
"yymm": "0503",
"arxiv_id": "math-ph/0503037",
"language": "en",
"url": "https://arxiv.org/abs/math-ph/0503037"
}
|
\section*{}
The concept of financial log-periodicity is based on the appealing assumption
that the financial dynamics is governed by phenomena analogous to criticality
in the statistical physics sense (Sornette et al. 1996,
Feigenbaum and Freund 1996).
Criticality implies a scale invariance which, for a properly defined
function $F(x)$ characterizing the system, means that
\begin{equation}
F(\lambda x) = \gamma F(x).
\label{eq:F}
\end{equation}
A constant $\gamma$ describes how the properties of the system
change when it is rescaled by the factor $\lambda$.
The general solution to this equation reads:
\begin{equation}
F(x) = x^{\alpha} P(\ln(x)/\ln(\lambda)),
\label{eq:FP}
\end{equation}
where the first term represents a standard power-law that is characteristic
of continuous scale-invariance with the critical exponent
$\alpha = \ln(\gamma) / \ln(\lambda)$ and $P$ denotes a periodic function
of period one. This general solution can be interpreted in terms of
discrete scale invariance. It is due to the second term that the conventional
dominating scaling acquires a correction that is periodic in $\ln(x)$ and
may account for the zig-zag character of financial dynamics.
It demands however that if $x = \vert T - T_c \vert$, where $T$ denotes
the ordinary time labeling the original price time series, represents
a distance to the critical point $T_c$, the resulting spacing between the
corresponding consecutive repeatable structures at $x_n$ seen in the linear
scale follow a geometric contraction according to the relation
$(x_{n+1}-x_n) / (x_{n+2}-x_{n+1}) = \lambda$.
The critical points correspond to the accumulation of such oscillations and,
in the context of the financial dynamics, it is this effect that potentially
can be used for prediction.
An extremely important related element, for
a proper interpretation and handling of the financial patterns as well as for
consistency of the theory, is that such log-periodic oscillations manifest
their action self-similarly through various time scales
(Dro\.zd\.z et al. 1999). This applies both to the log-periodically
accelerating bubble market phase as well as to the
log-periodically decelerating anti-bubble phase. Furthermore, more and more
evidence is collected that the preferred scaling factor $\lambda \approx 2$
and is common to all the scales and markets (Dro\.zd\.z et al. 2003).
These two elements,
self-similarity and universality of the $\lambda$, set very valuable
and in fact crucial constraints on possible forms of the analytic
representations of the market trends and oscillation patterns,
including the future ones as well.
A specific form of the periodic function $P$ in Eq.~\ref{eq:FP} is as yet
not provided by any first principles which opens room for certain, seemingly
mathematically unrigorous assignments of patterns.
This, on the other hand, allows to correct for frequent market 'imprecisions'
when relating its real behavior versus the theory.
Very helpful in this respect is the requirement of self-similarity which
greatly clarifies the significance of a given pattern and allows to determine
on what time scale it operates.
Since in the corresponding methodology the oscillation structure carries
the most relevant information about the market dynamics, for transparency
of this presentation, we use the first term of its Fourier expansion,
\begin{equation}
P(\ln(x)/\ln(\lambda)) = A + B \cos({\omega \over 2\pi} \ln(x) + \phi).
\label{eq:FPE}
\end{equation}
This implies that $\omega = 2\pi / \ln(\lambda)$. Already such a simple
parametrization allows to properly reflect the contraction of oscillations,
especially on the larger time scales. On the smaller time scales just
replacing
the {\it cosine} by its modulus often, even quantitatively in addition,
describes departures of the market amplitude from its average trend.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=6cm, angle=270]{Figure1a.eps}
\includegraphics[width=6cm, angle=270]{Figure1b.eps}
\caption{(a) Logarithm of the Standard $\&$ Poor's 500 index
since 1800 (http://www.globalfindata.com). The thick solid line displays its optimal
log-periodic representation with $\lambda = 2$. The thin solid line
represents the inflation corrected S$\&$P500 expressed in 2004 US$\$$.
It significantly shifts the third minimum to the early 1980s and improves
agreement with the theoretical representation. (b) Logarithm of the S$\&$P500
from 1997 till the end of 2002, which corresponds to the magnification of the
small rectangle in (a). The solid lines illustrate the log-periodic
accelerating and decelerating representations with $\lambda = 2$, modulus
of the cosine used in Eq.~(3), and a common $T_c = 1.9.2000$.}
\end{center}
\end{figure}
One particularly relevant and special, for several reasons,
example is shown in Fig.~1.
The upper panel (a) illustrates a nearly optimal log-periodic representation
of the S$\&$P500 data over the most extended time-period of the recorded
stock market activity as dated since 1800. As already pointed out
(Dro\.zd\.z et al. 2003) this development signals in around 2025 a transition
of the S$\&$P500 to a globally declining phase as measured in the contemporary
terms. The magnification of the small rectangle covering the period
1997-2002 is displayed in the lower panel (b) of the same Fig.~1. It thus
illustrates the nature of the stock market evolution on a much smaller time
scale of resolution. An impressive log-periodicity with the same $\lambda=2$
on both sides of the transition date (September 1, 2000) can be seen.
The next stock market top from the perspective of the largest time scale
(Fig.~1a) can be estimated to occur in around the years 2010-2011.
In the spirit of the log-periodicity its neighborhood is to be accompanied
by the smaller time scale oscillations - similar in character to those in
Fig.~1b.
Of course, when going far away from those large scale transition points
such pure log-periodic structures - representative to the one level lower
time scale - must get dissolved.
A particularly interesting related question then is what characteristics are
to
govern the stock market dynamics in the transition period when going from
2000 to 2010. The most natural and straightforward way is to view this process
as schematically is indicated in Fig.~2.
This whole period is thus covered by the two main components represented by
the thin lines and the market dynamics is driven by the superposition of
of these two components whose phases, slopes and weights are adjusted such
that the overall global market trend up to now is reproduced.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=6cm, angle=270]{Figure2.eps}
\caption{A hypothetical log-periodic scenario,
represented by the thick solid line,
for the S$\&$P500 development until 2010. This solid line is obtained by
summing up the two $\lambda = 2$ components (thin lines):
log-periodically decelerating since 1.9.2000
and log-periodically accelerating toward 1.9.2010.}
\end{center}
\end{figure}
In this scenario, close to the two large-scale transition points
(September 2000 and, as provisionally estimated here based on Fig.~1a,
September 2010) the market is driven, as needed,
essentially by the single log-periodic components,
decelerating and accelerating one, correspondingly.
More complicated is the situation in the middle of this time interval where
the two components contribute comparably. Most interestingly, it indicates
that
the period of the stock market stagnation may extend even into the year 2008,
before it seriously starts rising.
It also demonstrates a possible mechanism that generates modulation structures
responsible for the apparent higher order corrections
(Johansen and Sornette 1999) to Eq.~(\ref{eq:FPE}).
The changes in the frequency relations observed in the transition period
between the bear and the bull market phases originate here from the
interference between the two components, both of the simple form as prescribed
by Eq.~(\ref{eq:FPE}) and with the same $\lambda = 2$.
Of course, similar effects of interference may occur on the whole
hierarchy of different time scales.
There is one more element that from time to time takes place
in the financial dynamics and whose identification appears
relevant for a proper interpretation of the financial patterns with
the same universal value of the preferred scaling factor $\lambda$.
This is the phenomenon of a "super-bubble" (Dro\.zd\.z et al. 2003)
which is a local bubble, itself evolving log-periodically,
superimposed on top of a long-term bubble. Two such spectacular examples
are provided by the Nasdaq in the first quarter of 2000 and by the gold price
in the beginning of 1981 (Dro\.zd\.z et al. 2003).
\begin{figure}[ht]
\begin{center}
\includegraphics[width=6cm, angle=270]{Figure3.eps}
\caption{The New York traded oil futures since 1998 and the
corresponding log-periodic $\lambda = 2$ representation in terms of Eq.~(3).}
\end{center}
\end{figure}
In connection with this second case it is important to remember that the same
value of $\lambda$ as for the stock market turns out appropriate.
That such its value may be characteristic to the whole commodities market
as well,
is shown in Fig.~3 which displays the New York traded oil futures versus
the best log-periodic $(\lambda = 2)$ representation. In fact, this scenario
has been drawn by the authors on September 15, 2004, insisting on using
$\lambda = 2$, even though one local minimum (in the beginning of
2004) in the corresponding sequence did not look very convincing.
Designed this way it was indicating a continuation of the increase
until the end of October and then a more serious reverse of the trend.
Subsequent development of the oil futures provides further arguments in favor
of this way of handling the financial log-periodicity.
\section*{References}
\begin{description}
\item[]Dro\.zd\.z S, Ruf F, Speth J, W\'ojcik M (1999)~
Imprints of log-periodic self-similarity in the stock market.
Eur. Phys. J. B 10:589-593
\item[]Dro\.zd\.z S, Gr\"ummer F, Ruf F, Speth J (2003)~
Log-periodic self-similarity: an emerging financial law?
Physica A 324:174-182
\item[]Feigenbaum JA, Freund PGO (1996)~
Discrete scale invariance in stock markets before crashes.
Int. J. Mod. Phys. B 10:3737-3745
\item[]Johansen A, Sornette D (1999)~
Financial "anti-bubbles": Log-periodicity in gold and Nikkei collapses.
Int. J. Mod. Phys. C 10:563-575
\item[]Sornette D, Johansen A, Bouchaud J.-P (1996)~
Stock market crashes, precursors and replicas.
J. Physique (France) 6:167-175
\end{description}
\end{document}
|
{
"timestamp": "2005-03-01T21:03:07",
"yymm": "0503",
"arxiv_id": "physics/0503006",
"language": "en",
"url": "https://arxiv.org/abs/physics/0503006"
}
|
\section{\label{sec:intro}Introduction}
Fluid flow with interfaces and free surfaces is common in nature and in many engineering
applications. Such interfacial flows which typically involve multiple scales remain a formidable non-linear
problem rich in physics and continue to pose challenges to experimentalists and theoreticians
alike~\cite{eggers97}. Numerical simulation of multiphase flows is challenging
as the shape and location of the interfaces must be computed in conjunction with the solution
of the flow field~\cite{hyman84,scardovelli99}. Computational methods based on the
lattice Boltzmann equation (LBE) for simulating complex emergent physical phenomena have
attracted much attention in recent years~\cite{chen98,succi02}. The LBE simulates multiphase
flows by incorporating interfacial physics at scales smaller than macroscopic scales. Phase
segregation and interfacial fluid dynamics can be simulated by incorporating inter-particle
potentials~\cite{shan93,shan94}, concepts based on free energy~\cite{swift95,swift96}
or kinetic theory of dense fluids~\cite{he98,he99,he02}.
The formulation of the standard LBE is based on the Cartesian coordinate system and does not
take into account axial symmetry that may exist. Numerous multiphase
flow situations exist where the fluid dynamics can be approximated as axisymmetric~\cite{sussman96,eggers97}.
Examples include head-on collision of drops, normal drop impingement on solid surfaces and
Rayleigh instability of cylindrical liquid columns. Currently, full three-dimensional (3D)
calculations have to be carried out for problems which may be approximated as axisymmetric~\cite{he99a,inamuro03,premnath04}.
In 3D computations, computational considerations restrict
the numerical resolution that may be employed and the physics may not be well resolved.
For example, in breakup of drops into satellite droplets the size of the droplets may be such that the 3D
grids may not resolve them. To improve the
computational efficiency of the LBE for axisymmetric multiphase flows, we propose an axisymmetric LB
model in this paper. The approach consists of adding source terms to the two-dimensional (2D) Cartesian LBE
model based on the kinetic theory of dense fluids for multiphase flows~\cite{he98,he99}. This
approach is similar in spirit to the idea proposed in~\cite{halliday01} to solve single-phase axisymmetric
flows. However, multiphase flow problems involve additional complexity as a result of interfacial
physics involved, i.e. the surface tension forces and the need to track the interfaces. In this case,
the accuracy of the numerical discretization of the source terms representing interfacial physics
also becomes an important consideration.
This paper is organized as follows. In Section \ref{sec:axismodel}, the axisymmetric LBE multiphase model is
described. Then, in Section \ref{sec:axismodelc}, its extension to simulate axisymmetric multiphase flows
with reduced compressibility effects is described. The computational methodology adopted is also discussed in this
section. In Section \ref{sec:results}, the axisymmetric model is applied to benchmark problems to evaluate its accuracy.
Finally, the paper closes with summary in Section \ref{sec:summary}.
\section{\label{sec:axismodel}Axisymmetric LBE Multiphase Flow Model}
To simulate axisymmetric multiphase flows, axisymmetric contributions of the order parameter, and
inertial, viscous and surface tension forces may be introduced to the standard 2D LBE. The source
terms, which will be shown to be spatially and temporally dependent, are determined by performing
a Chapman-Enskog multiscale analysis in such a way that the macroscopic mass and momentum
equations for multiphase flows are recovered self-consistently. The introduction of source terms
makes it necessary to calculate additional spatial gradients when compared to those in the standard
LBE. While this approach is developed for a specific LBE multiphase flow model based on kinetic theory
of dense fluids~\cite{he98,he99}, it can be readily extended to other LBE multiphase flow models.
The governing continuum equations of isothermal multiphase flow~\cite{nadiga96,zou99} in the cylindrical
coordinate system when the axisymmetric assumption is employed are
\begin{equation}
\partial_t \rho + \frac{1}{r} \partial_r \left( \rho r u_r \right) +
\partial_z \left( \rho u_z \right) = 0,
\label{eq:axiscont}
\end{equation}
\begin{equation}
\rho\left( \partial_t u_r + u_r \partial_r u_r + u_z \partial_z u_r \right)=
-\partial_r P + F_{s,r}+F_{ext,r}+\frac{1}{r}\partial_r \left( r \Pi_{rr} \right)+
\partial_z \left( \Pi_{rz} \right),
\label{eq:axismomr}
\end{equation}
\begin{equation}
\rho\left( \partial_t u_z + u_r \partial_r u_z + u_z \partial_z u_z \right)=
-\partial_z P + F_{s,z}+F_{ext,z}+\frac{1}{r}\partial_r \left( r \Pi_{zr} \right)+
\partial_z \left( \Pi_{zz} \right),
\label{eq:axismomz}
\end{equation}
where $\rho$ is the density and $u_r$ and $u_z$ are the radial and axial components of velocity.
These equations are derived from kinetic theory that incorporates intermolecular interactions forces
which are modeled as a function of density following the work of van der Waals~\cite{rowlinson}. The exclusion volume
effect of Enskog~\cite{chapman} is also incorporated to account for increase in collision probability due to the
increase in the density of non-ideal fluids. These features naturally give rise to surface tension and phase
segregation effects. The other
variables which appear in the above equations will now be described. $\Pi_{rr}$, $\Pi_{rz}$, $\Pi_{zz}$ are
the components of the viscous stress tensor and are given by
\begin{eqnarray}
\Pi_{rr}&=&2\mu \partial_r u_r, \\
\Pi_{rz}&=& \Pi_{zr}=\mu \left( \partial_z u_r + \partial_r u_z \right),\\
\Pi_{zz}&=&2\mu \partial_z u_z,
\end{eqnarray}
where $\mu$ is the dynamic viscosity. $F_{s,r}$ and $F_{s,z}$ are the axial and radial components respectively
of the surface tension force, which are given by~\cite{zou99}
\begin{eqnarray}
F_{s,r}&=& \kappa \rho \partial_r \left[\frac{1}{r} \partial_r (r\partial_r\rho)+\partial_z(\partial_z\rho) \right],
\label{eq:surfr}\\
F_{s,z}&=& \kappa \rho \partial_z \left[\frac{1}{r} \partial_r (r\partial_r\rho)+\partial_z(\partial_z\rho) \right],
\label{eq:surfz}
\end{eqnarray}
where $\kappa$ controls the strength of the surface tension force. This parameter is
related to the surface tension of the fluid, $\sigma$, through the density gradient across the interface by the equation~\cite{evans79}
\begin{equation}
\sigma = \kappa \int \left( \frac{\partial \rho}{\partial n} \right)^2 dn.
\label{eq:sigmakappa}
\end{equation}
Thus, the surface tension is a function of both the parameter $\kappa$ and the density profile across the interface.
The terms $F_{ext,r}$ and $F_{ext,z}$ in Eqs. (\ref{eq:axismomr}) and (\ref{eq:axismomz}) respectively
are the radial and axial components of external forces such as gravity.
The pressure, $P$, is related to density through the Carnahan-Starling-van der Waals equation of state (EOS)~\cite{carnahan69}
\begin{equation}
P=\rho R T \left\{ \frac{1+\gamma+\gamma^2-\gamma^3}{(1-\gamma)^3} \right\} - a\rho^2,
\label{eq:axiseos}
\end{equation}
where $\gamma=b\rho/4$. The parameter $a$ is related to the intermolecular pair-wise potential and $b$
to the effective diameter of the molecule, $d$, and the mass of a single molecule, $m$, by
$b=2\pi d^3/3m$. $R$ is a gas constant and $T$ is the temperature.
The Carnahan-Starling EOS has a \emph{supernodal} $P-1/\rho-T$ curve, i.e., $dP/d\rho<0$, for certain range of values of $\rho$,
when the state fluid temperature is below its critical value. This unstable part of the curve is the driving
mechanism responsible for keeping the phases of fluids segregated and for maintaining a self-generated
sharp interface.
We now modify the standard LBE in such a way that it effectively yields the axisymmetric multiphase flow equations,
Eqs. (\ref{eq:axiscont})-
(\ref{eq:axiseos}), in a self-consistent way. To facilitate this,
we employ the following coordinate transformation, illustrated in Fig.~\ref{fig:schemaxis}, which allows the governing equations to be
represented in a Cartesian-like coordinate system, i.e. $(x,y)$:
\begin{figure*}
\includegraphics{fig1
\caption{\label{fig:schemaxis}
Schematic of arrangement of coordinate system in axisymmetric multiphase flow ($(r,z)$ and $(y,x)$
coordinate directions are shown).}
\end{figure*}
\begin{equation}
(r,z) \rightarrow (y,x),
\end{equation}
\begin{equation}
(u_r,u_z) \rightarrow (u_y,u_x).
\end{equation}
Assuming summation convention for repeated subscript indices, Eqs. (\ref{eq:axiscont})-(\ref{eq:surfz}) may be
transformed to
\begin{equation}
\partial_t \rho + \partial_k \left( \rho u_k \right)=-\frac{\rho u_y}{y},
\label{eq:axiscont1}
\end{equation}
\begin{equation}
\rho \left( \partial_t u_i+ u_k\partial_k u_i \right)=-\partial_i P + F_{s,i}+F_{ext,i}+
\partial_k \left[ \mu\left( \partial_k u_i+\partial_i u_k \right) \right]+F_{ax,i},
\label{eq:axismom1}
\end{equation}
where
\begin{equation}
F_{s,i}=\kappa \rho \partial_i \nabla^2 \rho
\label{eq:forcemp}
\end{equation}
and $i,j,k\in \left\{ x,y \right\}$. The right hand side (RHS) in Eq. (\ref{eq:axiscont1}), $-\rho u_y/y$, is the
additional term in the continuity equation that arises from axisymmetry. The corresponding term
for the momentum equation, Eq. (\ref{eq:axismom1}), is
\begin{equation}
F_{ax,i}=\frac{\mu}{y}\left[\partial_y u_i+\partial_i u_y \right]+
\kappa \rho \partial_i \left( \frac{1}{y}\partial_y \rho \right).
\end{equation}
To recover Eqs. (\ref{eq:axiscont1}) and (\ref{eq:axismom1}), we introduce two additional source terms,
$S_{\alpha}^{'}$ and $S_{\alpha}^{''}$, to the standard 2D Cartesian LBE which has $\Omega_{\alpha}$ as its collision term and
a source term for the internal and external forces, $S_{\alpha}$. These unknown additional terms, representing the
axisymmetric mass and momentum contributions respectively, are to be determined so that the macroscopic
behavior of the proposed LBE corresponds to axisymmetric multiphase flow. Thus, we propose the
following LBE
\begin{eqnarray}
f_{\alpha}( \mbox{\boldmath$x$}+\mbox{\boldmath$e$}_{\alpha}\delta_t,t+\delta_t )-f_{\alpha}( \mbox{\boldmath$x$},t )&=&
\frac{1}{2}\left[\Omega_{\alpha}|_{(x,t)}+
\Omega_{\alpha}|_{(x+e_{\alpha}\delta_t,t+\delta_t)}
\right]+\nonumber\\
& &
\frac{1}{2}\left[S_{\alpha}|_{(x,t)}+
S_{\alpha}|_{(x+e_{\alpha}\delta_t,t+\delta_t)}
\right]\delta_t+\nonumber\\
& &
\frac{1}{2}\left[S_{\alpha}^{'}|_{(x,t)}+
S_{\alpha}^{'}|_{(x+e_{\alpha}\delta_t,t+\delta_t)}
\right]\delta_t+\nonumber\\
& &
\frac{1}{2}\left[S_{\alpha}^{''}|_{(x,t)}+
S_{\alpha}^{''}|_{(x+e_{\alpha}\delta_t,t+\delta_t)}
\right]\delta_t,
\label{eq:axislbe}
\end{eqnarray}
where $f_{\alpha}$ is the discrete single-particle distribution function, corresponding to the particle velocity,
$\mbox{\boldmath$e$}_{\alpha}$, where $\alpha$ is the velocity direction. The Cartesian component of the particle velocity, $c$,
is given by $c=\delta_x/\delta_t$, where $\delta_x$ is the lattice spacing and $\delta_t$ is the time step corresponding to the
two-dimensional, nine-velocity model(D2Q9)~\cite{qian92} shown in Fig.~\ref{fig:schemaxis}.
Here, the collision term is given by the BGK approximation~\cite{bhatnagar54}
\begin{equation}
\Omega_{\alpha}=-\frac{f_{\alpha}-f_{\alpha}^{eq}}{\tau}, \quad \tau=\frac{\lambda}{\delta_t},
\end{equation}
where $\lambda$ is the relaxation time due to collisions, $\delta_t$ is the time step and $f_{\alpha}^{eq}$ is
the truncated discrete form of the Maxwellian
\begin{equation}
f_{\alpha}^{eq}\equiv f_{\alpha}^{eq,M}(\rho,\mbox{\boldmath$u$})=
\omega_{\alpha}\left\{
1+\frac{\mbox{\boldmath$e$}_{\alpha} \cdotp \mbox{\boldmath$u$}}{RT}+
\frac{\left( \mbox{\boldmath$e$}_{\alpha} \cdotp \mbox{\boldmath$u$} \right)^2}{2(RT)^2}-
\frac{1}{2}\frac{\mbox{\boldmath$u$} \cdotp \mbox{\boldmath$u$}}{RT}
\right\},
\label{eq:trunceq}
\end{equation}
where $R$ is the gas constant, $T$ is the temperature and $w_{\alpha}$ is the weighting coefficients in the
Gauss-Hermite quadrature to represent the kinetic moment integrals of the distribution functions exactly~\cite{he97}.
For isothermal flows,
the factor $RT$ is related to the particle speed $c$ as $RT=1/3c^2$.
The term in Eq. (\ref{eq:axislbe})
\begin{equation}
S_{\alpha}=\frac{(e_{\alpha j}-u_j)(F_j+F_{ext,j})}{\rho RT}f_{\alpha}^{eq,M}(\rho,\mbox{\boldmath$u$})
\label{eq:sourcemp}
\end{equation}
represents the effect of internal and external forcing terms on the change in the distribution function.
The internal force term gives rise to surface tension and phase segregation effects which are given by
\begin{equation}
F_j=-\partial_j \psi + F_{s,j},
\label{eq:forceint}
\end{equation}
where the function $\psi=P-\rho RT$ is the non-ideal part of the equation of state given in Eq. (\ref{eq:axiseos}).
The first two terms on the RHS of Eq. (\ref{eq:axislbe}) corresponds to those presented by He \emph{et al.} (1998).
As mentioned above, the last two terms, $S_{\alpha}^{'}$ and $S_{\alpha}^{''}$, in this equation is
to be selected such that its behavior in the continuum limit would simulate the influence of the non-Cartesian-like
terms in Eqs. (\ref{eq:axiscont1}) and (\ref{eq:axismom1}) in a self-consistent way. Since
the zeroth kinetic moment of the term $f_{\alpha}^{eq,M}(\rho,0)$ is involved in the derivation of the macroscopic
mass conservation equation from the LBE, the source term $S_{\alpha}^{'}$ in Eq. (\ref{eq:axislbe}) is proposed to be equal to
$f_{\alpha}^{eq,M}(\rho,0)$ multiplied by an unknown $m^{'}$ and normalized by the density $\rho$. The other
source term $S_{\alpha}^{''}$ is proposed analogous to
the source term in Eq.(\ref{eq:sourcemp}). Thus, we propose
\begin{eqnarray}
S_{\alpha}^{'}&=&\frac{f_{\alpha}^{eq,M}(\rho,0)}{\rho}m^{'}\label{eq:sourcea1},\\
S_{\alpha}^{''}&=&\frac{(e_{\alpha j}-u_j)F_j^{''}}{\rho RT}f_{\alpha}^{eq,M}(\rho,\mbox{\boldmath$u$}).
\label{eq:sourcea2}
\end{eqnarray}
Here the unknowns, $m^{'}$ and $F_j^{''}$, in the above two equations can be determined through Chapman-Enskog
analysis as will be shown later. It must be stressed that all terms, including the collision term,
on the RHS are discretized by the application of the trapezoidal rule, since it has been argued that at least
a second-order treatment of the source terms is necessary for simulation of multiphase flow~\cite{he98,he99}.
The macroscopic fields are given by
\begin{eqnarray}
\rho&=&\sum_{\alpha} f_{\alpha},\\
\rho u_i&=&\sum_{\alpha} f_{\alpha} e_{\alpha i}.
\label{eq:amacrofields}
\end{eqnarray}
In this model, the order parameter is the density, $\rho$, which distinguishes the different phases in the flow.
Equation (\ref{eq:axislbe}) is implicit in time. To remove implicitness in this equation we introduce
a transformation following the procedure described by He and others~\cite{he98,he98a}, whereby
\begin{equation}
\bar{f}_{\alpha}=f_{\alpha}-\frac{1}{2}\Omega_{\alpha}-
\frac{1}{2}\left(S_{\alpha}+S_{\alpha}^{'}+S_{\alpha}^{''}\right)\delta_t
\label{eq:implicittr}
\end{equation}
in Eq. (\ref{eq:axislbe}), so that we obtain
\begin{equation}
\bar{f}_{\alpha}( \mbox{\boldmath$x$}+\mbox{\boldmath$e$}_{\alpha}\delta_t,t+\delta_t )-
\bar{f}_{\alpha}( \mbox{\boldmath$x$},t )=
\bar{\Omega}_{\alpha}|_{(x,t)}+\frac{\tau}{\tau+1/2}\left[ S_{\alpha}+S_{\alpha}^{'}+
S_{\alpha}^{''}\right]|_{(x,t)}\delta_t,
\label{eq:axislbee}
\end{equation}
where
\begin{equation}
\bar{\Omega}_{\alpha}=-\frac{\bar{f}_{\alpha}-f_{\alpha}^{eq}}{\tau+1/2}.
\label{eq:axisbgk}
\end{equation}
Thus, $\bar{f}_{\alpha}$ is the transformed distribution function that removes implicitness in the
proposed LBE, Eq. (\ref{eq:axislbe}), which describes the evolution of the $f_{\alpha}$ distribution function.
The following constraints on the equilibrium distribution and the various
source terms~\cite{luo00,guo02} are imposed from their definition:
\begin{equation}
\sum_{\alpha} f_{\alpha}^{eq}=\rho, \quad \sum_{\alpha} f_{\alpha}^{eq} e_{\alpha i}=\rho u_i, \quad
\sum_{\alpha} f_{\alpha}^{eq} e_{\alpha i} e_{\alpha j}=\rho RT \delta_{ij}+\rho u_i u_j, \nonumber
\end{equation}
\begin{equation}
\sum_{\alpha} f_{\alpha}^{eq} e_{\alpha i} e_{\alpha j} e_{\alpha k}=\rho (RT)^2
\left( u_i \delta_{jk}+u_j \delta_{ki}+u_k \delta_{ij}\right),
\end{equation}
\begin{equation}
\sum_{\alpha} S_{\alpha}=0, \quad \sum_{\alpha} S_{\alpha} e_{\alpha i} = F_i, \quad
\sum_{\alpha} S_{\alpha} e_{\alpha i} e_{\alpha j}=(F_i+F_{ext,i}) u_j+(F_j+F_{ext,j}) u_i,
\end{equation}
\begin{equation}
\sum_{\alpha} S_{\alpha}^{'}=m^{'}, \quad \sum_{\alpha} S_{\alpha}^{'} e_{\alpha i} = 0, \quad
\sum_{\alpha} S_{\alpha}^{'} e_{\alpha i} e_{\alpha j}=m^{'}RT\delta_{ij},
\end{equation}
\begin{equation}
\sum_{\alpha} S_{\alpha}^{''}=0, \quad \sum_{\alpha} S_{\alpha}^{''} e_{\alpha i} = F_i^{''}, \quad
\sum_{\alpha} S_{\alpha}^{''} e_{\alpha i} e_{\alpha j}=(F_i^{''} u_j+F_j^{''} u_i).
\end{equation}
Then the following relationships are obtained between the transformed distribution function and the macroscopic
fields, which also include the curvature effects resulting from axial symmetry:
\begin{eqnarray}
\rho&=&\sum_{\alpha} \bar{f}_{\alpha}+\frac{1}{2}m^{'}\delta_t \label{eq:dens1},\\
\rho u_i&=&\sum_{\alpha} \bar{f}_{\alpha} e_{\alpha i}+\frac{1}{2}(F_i+F_{ext,i}+F_i^{''})\delta_t.
\label{eq:densvel1}
\end{eqnarray}
Now, to establish the unknowns $m^{'}$ and $F_j^{''}$ in the above formulation, the Chapman-Enskog multiscale
analysis is performed~\cite{chapman}. Introducing the expansions~\cite{he97a}
\begin{eqnarray}
\bar{f}_{\alpha}( \mbox{\boldmath$x$}+\mbox{\boldmath$e$}_{\alpha}\delta_t,t+\delta_t )&=&
\sum_{\alpha=0}^{\infty} D_{t_n}\bar{f}_{\alpha}(\mbox{\boldmath$x$},t),\\
D_{t_n} &\equiv& \partial_{t_n}+e_{\alpha k} \partial_k,\\
f_{\alpha}&=&\sum_{\alpha=0}^{\infty}\epsilon^n f_{\alpha}^{(n)},\\
\partial_t&=&\sum_{\alpha=0}^{\infty}\epsilon^n \partial_{t_n},
\end{eqnarray}
where $\epsilon=\delta_t$ in Eq. (\ref{eq:axislbee}) and using Eq. (\ref{eq:implicittr}) to transform $\bar{f}_\alpha$ back to
$f_\alpha$, the following equations are obtained in
the consecutive order of the parameter $\epsilon$:
\begin{eqnarray}
O(\epsilon^0): f_{\alpha}^{(0)}&=&f_{\alpha}^{eq}\label{eq:order0},\\
O(\epsilon^1): D_{t_0} f_{\alpha}^{(0)}&=&-\frac{1}{\tau} f_{\alpha}^{(1)}+S_{\alpha}+
S_{\alpha}^{'}+S_{\alpha}^{''}\label{eq:order1},\\
O(\epsilon^2): \partial_{t_1} f_{\alpha}^{(0)}+ D_{t_0} f_{\alpha}^{(1)}&=&-\frac{1}{\tau} f_{\alpha}^{(2)}.
\label{eq:order2}
\end{eqnarray}
Now, invoking the Chapman-Enskog ansatz
\begin{equation}
\sum_{\alpha}
\left( \begin{array}{c}
1 \\
e_{\alpha i}
\end{array} \right)f_{\alpha}^{(0)}=
\left( \begin{array}{c}
\rho \\
\rho u_i
\end{array} \right),
\sum_{\alpha}
\left( \begin{array}{c}
1 \\
e_{\alpha i}
\end{array} \right)f_{\alpha}^{(n)}=
\left( \begin{array}{c}
0 \\
0
\end{array} \right),n \geq 1
\end{equation}
and performing $\sum_{\alpha}(\cdotp)$ on Eqs. (\ref{eq:order1}) and (\ref{eq:order2}), we obtain
\begin{eqnarray}
\partial_{t_0} \rho + \partial_k (\rho u_k)&=& m^{'} \label{eq:axisc1},\\
\partial_{t_1} \rho &=& 0,\label{eq:axisc2}
\end{eqnarray}
respectively. Combining the first- and second- order results given by Eqs. (\ref{eq:axisc1}) and (\ref{eq:axisc2})
and considering $\partial_t=\partial_{t_0}+\epsilon \partial_{t_1}$, we get
\begin{equation}
\partial_t \rho + \partial_k (\rho u_k)= m^{'} \label{eq:axisc}.\\
\end{equation}
Comparing this equation and Eq. (\ref{eq:axiscont1}), the unknown $m^{'}$ is
obtained as
\begin{equation}
m^{'}=-\frac{\rho u_y}{y}. \label{eq:mvalue}\\
\end{equation}
This is the axisymmetric contribution to the Cartesian form of the equation for the order parameter,
i.e., density characterizing the different phases of the flow. Taking the first kinetic moment,
$\sum_{\alpha}e_{\alpha i}(\cdotp)$, of Eqs. (\ref{eq:order1}) and (\ref{eq:order2}), respectively, we get
\begin{eqnarray}
\partial_{t_0} (\rho u_i) + \partial_k (\rho u_i u_k)&=& -\partial_i (\rho RT)+F_i+F_{ext,i}+F_i^{''}, \label{eq:axism1}\\
\partial_{t_1} (\rho u_i) + \partial_k \Pi_{ij}^{(1)}&=& 0, \label{eq:axism2}
\end{eqnarray}
where
\begin{equation}
\Pi_{ij}^{(1)}=\sum_{\alpha} f_{\alpha}^{(1)}e_{\alpha i} e_{\alpha j}.
\label{eq:visct1}
\end{equation}
Employing the expression
for $f_{\alpha}^{(1)}$ from Eq. (\ref{eq:order1}) in Eq. (\ref{eq:visct1}), together with the summational
constraints given above, and neglecting terms of the order $O(Ma^3)$ or higher, we get
\begin{equation}
\Pi_{ij}^{(1)}=-\tau RT \rho (\partial_j u_i+\partial_i u_j).
\label{eq:visct2}
\end{equation}
Equation (\ref{eq:axism2}) then simplifies to
\begin{equation}
\partial_{t_1} (\rho u_i) = \partial_j \left( \tau RT \rho (\partial_j u_i+\partial_i u_j) \right) \label{eq:axism2n}.
\end{equation}
Combining Eqs. (\ref{eq:axism1}) and (\ref{eq:axism2n}), we get
\begin{equation}
\partial_t (\rho u_i) + \partial_k (\rho u_i u_k)= -\partial_i (\rho RT)+F_i+F_{ext,i}+F_i^{''}+
\partial_j \left( \tau \delta_t RT \rho (\partial_j u_i+\partial_i u_j) \right),
\end{equation}
or substituting for $F_i$ from Eq. (\ref{eq:forceint}), we obtain
\begin{equation}
\partial_t (\rho u_i) + \partial_k (\rho u_i u_k)= -\partial_i P +F_{s,i}+F_{ext,i}+F_i^{''}+
\partial_j \left( \tau \delta_t RT \rho (\partial_j u_i+\partial_i u_j) \right).
\label{eq:axism3}
\end{equation}
Using Eqs. (\ref{eq:axisc}) and (\ref{eq:axism3}), this can be simplified to
\begin{eqnarray}
\rho \left( \partial_t u_i + u_k \partial_k u_i \right)-\frac{\rho u_i u_y}{y}&=&
-\partial_i P +F_{s,i}+F_{ext,i}+F_i^{''}+\nonumber\\
& & \partial_j \left( \tau \delta_t RT \rho (\partial_j u_i+\partial_i u_j) \right).
\label{eq:axism4}
\end{eqnarray}
Comparing Eqs. (\ref{eq:axismom1}) and (\ref{eq:axism4}), we obtain the other unknown
$F_i^{''}$ where
\begin{equation}
F_i^{''}=F_{ax,i}-\frac{\rho u_i u_y}{y}=\frac{\mu}{y}\left[ \partial_y u_i + \partial_i u_y \right]+
\kappa \rho \partial_i \left(\frac{1}{y}\partial_y \rho \right)-
\frac{\rho u_i u_y}{y}.
\label{eq:fvalue}
\end{equation}
This is the axisymmetric contribution to the Cartesian form of the equation for the momentum, where
the first, second and the third terms on the RHS correspond to the viscous, surface tension and inertial force contributions,
respectively. The dynamic viscosity is related to the relaxation
time for collisions by $\mu=\rho \tau \delta_t RT = \rho \lambda c_s^2$, where $c_s^2=1/3c^2$. The set of equations
corresponding to the axisymmetric LBE multiphase flow model is given by Eqs. (\ref{eq:axislbee}) and (\ref{eq:axisbgk}) together with
Eqs. (\ref{eq:sourcemp}), (\ref{eq:sourcea1}) and (\ref{eq:sourcea2}), (\ref{eq:dens1}) and (\ref{eq:densvel1}), and
(\ref{eq:mvalue}) and (\ref{eq:fvalue}). In general, this multiphase model and that proposed by He and others~\cite{he98}
face difficulties for fluids far from the critical point and/or in the presence of external forces. This
difficulty is related to the calculation of the intermolecular force in Eq.(\ref{eq:forceint}), involving
the computation of $\partial_j\psi$ which can become quite large across interfaces. Unless this term
is accurately computed, the model may become unstable because of numerical errors~\cite{he99a,he04}. Hence, an
improved treatment of this term is necessary. This will now be described.
\section{\label{sec:axismodelc}Axisymmetric LBE Multiphase Flow Model with Reduced Compressibility Effects}
He and co-workers~\cite{he99} have proposed that through a suitable transformation of the
distribution function, $f_{\alpha}$, which involves invoking the incompressibility condition of the fluid,
and employing a new distribution function for capturing the interface, the difficulty with handling the
intermolecular force term, $\partial_j\psi$, can be reduced. We apply this idea to the axisymmetric
model developed in the previous section. We replace
the distribution function $f_{\alpha}$ by another distribution function $g_{\alpha}$ through
the transformation~\cite{he99}
\begin{equation}
g_{\alpha}=f_{\alpha}RT+\psi(\rho)\frac{f_{\alpha}^{eq,M}(\rho,0)}{\rho}.
\label{eq:transg}
\end{equation}
The effect of this transformation will be discussed in greater detail below. By considering the fluid to be incompressible, i.e.
\begin{equation}
\frac{d}{dt}\psi(\rho)=\left( \partial_t+u_k \partial_k \right)\psi (\rho)=0,
\end{equation}
and using the transformation Eqs. (\ref{eq:transg}) and (\ref{eq:implicittr}), Eq. (\ref{eq:axislbee}) is replaced by
\begin{equation}
\bar{g}_{\alpha}( \mbox{\boldmath$x$}+\mbox{\boldmath$e$}_{\alpha}\delta_t,t+\delta_t )-
\bar{g}_{\alpha}( \mbox{\boldmath$x$},t )=
\bar{\Omega}_{g\alpha}|_{(x,t)}+\frac{\tau}{\tau+1/2}\left[ S_{g\alpha}+S_{g\alpha}^{'}+
S_{g\alpha}^{''}\right]|_{(x,t)}\delta_t,
\label{eq:axislbeg}
\end{equation}
where
\begin{equation}
\bar{\Omega}_{g\alpha}=-\frac{\bar{g}_{\alpha}-g_{\alpha}^{eq}}{\tau+1/2},
\label{eq:axisbgkg}
\end{equation}
and
\begin{equation}
g_{\alpha}^{eq}=f_{\alpha}^{eq}RT+\psi(\rho)\frac{f_{\alpha}^{eq,M}(\rho,0)}{\rho}.
\end{equation}
The corresponding source terms become
\begin{eqnarray}
S_{g\alpha}&=&(e_{\alpha j}-u_j)\times \nonumber\\
& & \left[
(F_j+F_{ext,j})\frac{f_{\alpha}^{eq,M}(\rho,\mbox{\boldmath$u$})}{\rho}-
\left(
\frac{f_{\alpha}^{eq,M}(\rho,\mbox{\boldmath$u$})}{\rho}-\frac{f_{\alpha}^{eq,M}(\rho,0)}{\rho}
\right)\partial_j \psi(\rho)
\right],
\label{eq:srcrefined}
\end{eqnarray}
\begin{equation}
S_{g\alpha}^{'}=S_{\alpha}^{'}RT=\frac{f_{\alpha}^{eq,M}(\rho,0)}{\rho}\left( -\frac{\rho u_y}{y} \right)RT,
\end{equation}
\begin{equation}
S_{g\alpha}^{''}=S_{\alpha}^{''}RT=(e_j-u_j)F_j^{''}\frac{f_{\alpha}^{eq,M}(\rho,\mbox{\boldmath$u$})}{\rho}.
\label{eq:sourceg2}
\end{equation}
The term $\partial_j \psi$ in Eq. (\ref{eq:srcrefined}) is multiplied by the factor
$\left( f_{\alpha}^{eq,M}(\rho,\mbox{\boldmath$u$})/\rho-f_{\alpha}^{eq,M}(\rho,0)/\rho \right)$. This factor,
from the definition of the equilibrium distribution function, $f_{\alpha}^{eq}$, in Eq. (\ref{eq:trunceq})
is proportional to the Mach number and thus becomes smaller in the incompressible limit. Hence, it alleviates
the difficulties associated with the calculation of the $\partial_j \psi$, a major source of numerical
instability with the original model~\cite{he98}.
Thus, Eqs. (\ref{eq:axislbeg})-(\ref{eq:sourceg2}) are found to be numerically more stable compared to
Eq. (\ref{eq:axislbee}) supplemented with Eqs. (\ref{eq:sourcemp}),(\ref{eq:sourcea1}) and (\ref{eq:sourcea2}).
In this new framework, we still need to introduce
an order parameter to capture interfaces. Here, we employ a function, $\phi$, referred to henceforth
as the index function, in place of the density, as the order parameter to distinguish the phases in the flow.
The evolution equation of the distribution function whose emergent dynamics govern the index function has
to be able to maintain phase segregation and mass conservation. To do this, we employ
Eq. (\ref{eq:axislbee}) together with Eqs. (\ref{eq:sourcemp}),(\ref{eq:sourcea1}) and (\ref{eq:sourcea2}) by keeping
the term involving $\partial_j \psi$ and $m^{'}$, while the rest of the terms may be dropped as they
play no role in mass conservation. In addition, the density is replaced by the index function in these
equations. Hence, the evolution of the distribution function for the index function is given by
\begin{equation}
\bar{f}_{\alpha}( \mbox{\boldmath$x$}+\mbox{\boldmath$e$}_{\alpha}\delta_t,t+\delta_t )-
\bar{f}_{\alpha}( \mbox{\boldmath$x$},t )=
\bar{\Omega}_{f\alpha}|_{(x,t)}+\frac{\tau}{\tau+1/2}\left[ S_{f\alpha}+
S_{f\alpha}^{'}\right]|_{(x,t)}\delta_t,
\label{eq:axislbef}
\end{equation}
where the collision and the source terms are given by
\begin{equation}
\bar{\Omega}_{f\alpha}=-\frac{\bar{f}_{\alpha}-\frac{\phi}{\rho}f_{\alpha}^{eq}}{\tau+1/2},
\label{eq:axisbgkf}
\end{equation}
\begin{equation}
S_{f\alpha}=\frac{(e_j-u_j)(-\partial_j \psi(\phi))}{\rho RT}f_{\alpha}^{eq,M}(\rho,\mbox{\boldmath$u$}),
\label{eq:srciref}
\end{equation}
\begin{equation}
S_{f\alpha}^{'}=\frac{\phi}{\rho}S_{\alpha}^{'}=
\frac{f_{\alpha}^{eq,M}(\rho,0)}{\rho}\left( -\frac{\phi u_y}{y} \right).
\end{equation}
The hydrodynamic variables such as pressure and fluid velocity can be
obtained by taking appropriate kinetic moments of the distribution function $g_{\alpha}$, i.e.
\begin{eqnarray}
P&=&\sum_{\alpha} \bar{g}_{\alpha}-\frac{1}{2}u_j\partial_j \psi(\rho)+\frac{1}{2}m^{'}RT\delta_t,\label{eq:axpres}\\
\rho RT u_i &=& \sum_{\alpha} \bar{g}_{\alpha} e_{\alpha i}+\frac{1}{2}\left( F_{s,i}+F_{ext,i} \right)\delta_t+
\frac{1}{2} F_i^{''}\delta_t.
\end{eqnarray}
This follows from the definition of $\bar{g}_{\alpha}$ given in Eq. (\ref{eq:transg}) and also includes curvature effects.
The index function is obtained from the distribution function $\bar{f}_{\alpha}$ by taking the zeroth
kinetic moment, i.e.
\begin{equation}
\phi=\sum_{\alpha}\bar{f}_{\alpha}+\frac{1}{2}\frac{\phi}{\rho}m^{'}\delta_t.
\end{equation}
The terms $m^{'}$ and $F_i^{''}$ are given in Eqs. (\ref{eq:mvalue}) and (\ref{eq:fvalue}),
respectively. The density is obtained from the index function through linear interpolation, i.e.
\begin{equation}
\rho(\phi)= \rho_L+\frac{\phi-\phi_L}{\phi_H-\phi_L}(\rho_H-\rho_L),
\label{eq:rintp}
\end{equation}
where $\rho_L$ and $\rho_H$ are the densities of the light and heavy fluids, respectively, and $\phi_L$ and
$\phi_H$ refer to the minimum and maximum values of the index function, respectively. These limits of the index
function are determined from Maxwell's equal area construction~\cite{rowlinson} applied to the function
$\psi(\phi)+\phi RT$.
Thus, the axisymmetric LBE multiphase flow model with reduced compressibility effects corresponds
to Eqs. (\ref{eq:axislbeg})-(\ref{eq:rintp}).
The relaxation time for collisions is related to the viscosity of the fluid using the same
expression as derived in the previous section. If the kinematic viscosity of the light fluid, $\nu_L$,
is different from that of the heavy fluid, $\nu_H$, its value at any point in the fluid is obtained
from the index function through linear interpolation, i.e.
\begin{equation}
\nu(\phi)= \nu+\frac{\phi-\phi_L}{\phi_H-\phi_L}(\nu-\nu_L).
\label{eq:vintp}
\end{equation}
It may be seen that the model requires the calculation of spatial gradients in Eqs. (\ref{eq:srcrefined}) and (\ref{eq:srciref})
and of the Laplacian in Eq. (\ref{eq:forcemp}).
Since maintaining
accuracy as well as isotropy is important for the surface tension terms, they are calculated
by employing a fourth-order finite-difference scheme for the gradient and a second-order scheme
for the Laplacian, given respectively by
\begin{equation}
\partial_i \varpi=\frac{1}{36\delta_x}\sum_{\alpha=1}^{8}\left[
8\varpi(\mbox{\boldmath$x$}+\mbox{\boldmath$e$}_{\alpha i}\delta_t)-
\varpi(\mbox{\boldmath$x$}+2\mbox{\boldmath$e$}_{\alpha i}\delta_t)
\right] \left( \frac{e_{\alpha i}}{c} \right)+O(\delta_t^4),
\end{equation}
and
\begin{equation}
\nabla^2 \varpi \equiv \partial_i \partial_i \varpi = \frac{1}{3\delta_x^2}
\sum_{\alpha=1}^8\left[
\varpi(\mbox{\boldmath$x$}+\mbox{\boldmath$e$}_{\alpha i}\delta_t)-
\varpi(\mbox{\boldmath$x$})
\right]+O(\delta_x^2),
\end{equation}
for any function $\varpi$. Notice that these discretizations are both based on the lattice based stencil, instead of the standard
stencil based on the coordinate directions.
In addition, in the application of this model, the implementation of boundary
conditions plays an important role. In particular, along the axisymmetric line, i.e. $y=0$, specular
reflection boundary conditions are employed for the distribution functions. For the two-dimensional,
nine velocity (D2Q9) model shown in the inset of Fig.~\ref{fig:schemaxis}, we set
$\bar{f}_2=\bar{f}_4$, $\bar{f}_5=\bar{f}_8$,
$\bar{f}_6=\bar{f}_7$ and $\bar{g}_2=\bar{g}_4$, $\bar{g}_5=\bar{g}_8$ and $\bar{g}_6=\bar{g}_7$ for
the distribution functions after the streaming step. For macroscopic conditions, along this line,
$u_y=\partial_y(\cdotp)=0$, through which the singular source terms of type $1/y(\cdotp)$ in the model
can be appropriately treated. On the other hand, boundary conditions along the other lines are similar to
those for the standard LBE.
\section{\label{sec:results}Results and Discussion}
In the rest of this paper, unless otherwise specified, the results are presented in lattice units, i.e. the velocities
are scaled by the particle velocity $c$, the distance by the minimum lattice spacing $\delta_x$ and time by $c/\delta_x$.
All other quantities are scaled as appropriate combinations of these basic units.
First, the axisymmetric LBE multiphase flow models are applied to verify the well-known
Laplace-Young relation for an axisymmetric drop. According to this relation,
$\Delta P=2 \sigma/R_d$, where $\Delta P$ is the difference
between the pressure inside and outside of a drop, $\sigma$ is the surface tension and $R_d$ is the drop radius.
For different choices of the surface tension parameter, $\kappa$, the surface tension values are obtained
from Eq. (\ref{eq:sigmakappa}) by the replacing density in Eq. (\ref{eq:surfr}) and (\ref{eq:surfz})
by the index function. To obtain the normal gradient used in Eq. (\ref{eq:sigmakappa}), a physical
configuration consisting of a liquid and a gas layer is set up. Once equilibrium is reached, the density
gradient may be computed and hence the surface tension. Having obtained the relationship between the surface
tension $\sigma$, and the parameter $\kappa$, axisymmetric drops of four different radii, $R_d=40, 50, 60$
and $70$, are set up in a domain discretized by $201\times 101$ lattice sites. Periodic
boundaries are considered in the $x$ direction and an open boundary condition is considered along the boundary
that is parallel to the axisymmetric boundary. By considering three different values of
$\kappa$, $0.05, 1.0$ and $0.15$, the pressure difference across the drops is determined.
Figure~\ref{fig:laplaceyoung1} shows a comparison of the pressure difference across the
interface of the drops computed using the axisymmetric
model developed in Section \ref{sec:axismodelc} and that predicted by
the Laplace-Young relation. It is found that the computed results are in good agreement with the theoretical values, with
\begin{figure}
\begin{center}
\includegraphics{fig2.eps}
\caption{\label{fig:laplaceyoung1}Pressure difference across axisymmetric drops as a function of radius for different
values of the surface tension parameter $\kappa$; Comparison of computed results using the axisymmetric LBE model versus
theoretical prediction based on the Laplace-Young relation. Quantities are in lattice units.}
\end{center}
\end{figure}
a maximum relative error of about $3\%$.
Another important test problem is that of an oscillating axisymmetric drop immersed
in a gas. Since current versions of the LBE simulate a relatively viscous fluid, it is
appropriate to compare the oscillation frequency with that of Miller and Scriven (1968)~\nocite{miller68}.
In contrast to earlier analytical solutions on
drop oscillations, this work considers viscous dissipation effects in the boundary layer at the interface.
According to~\cite{miller68}, the frequency for the $n^{th}$ mode of oscillation for a drop is given by
\begin{equation}
\omega_{n}=\omega_{n}^{*}-\frac{1}{2} \alpha \omega_{n}^{*\frac{1}{2}}+\frac{1}{4}\alpha^2,
\label{eq:msperiod}
\end{equation}
where $\omega_{n}$ is the angular response frequency, and $\omega_{n}^{*}$ is Lamb's natural resonance
frequency expressed as~\cite{lamb}
\begin{equation}
\left(\omega_{n}^{*}\right)^{2}=
\frac{n(n+1)(n-1)(n+2)}{R_d^3 \left[ n\rho_{g}+(n+1)\rho_{l} \right]} \sigma.
\end{equation}
$R_d$ is the equilibrium radius of the drop, $\sigma$ is the interfacial surface tension, and $\rho_{l}$ and
$\rho_{g}$ are the densities of the two fluids. The parameter $\alpha$ is given by
\begin{equation}
\alpha =
\frac{(2n+1)^2 (\mu_{l} \mu_{g} \rho_{l} \rho_{g})^{\frac{1}{2}}}
{2^{\frac{1}{2}} R_d \left[ n\rho_{g}+(n+1)\rho_{l} \right]
\left[ (\mu_{l} \rho_{l})^{\frac{1}{2}}+ (\mu_{g} \rho_{g})^{\frac{1}{2}} \right]},
\end{equation}
where $\mu_{l}$ and $\mu_{g}$ are the dynamic viscosity of the two liquids. The subscripts $g$ and $l$
refer to the ambient gas and liquid phases, respectively. We consider the second mode of oscillation and
analytical expressions for the time period are presented in Eq. (\ref{eq:msperiod}).
The initial computational setup consists of a prolate spheroid of minimum ($R_{min}$) and maximum ($R_{max}$)
radii of $40$ and $55$,
respectively, placed in the center of the domain discretized by $201 \times 101$ lattice sites. We consider
the surface tension parameters: $\kappa=0.2$, and the density of the gas and the drop to be $\rho_g=0.1$ and $\rho_l=0.4$,
respectively. The kinematic viscosity of both the gas and the drop are considered to be the same and given by
$\nu_g=\nu_l=1.6667\times 10^{-2}$. Figure~\ref{fig:oscchem} shows the configurations of an oscillating drop at different times
computed using the standard axisymmetric model with these conditions.
\begin{figure}
\begin{center}
\includegraphics{fig3.EPS}
\caption{\label{fig:oscchem}Configurations of an oscillating drop as a function of time;
$R_{min}=40$, $R_{max}=55$,
$\rho_g=0.1$, $\rho_l=0.4$, $\nu_l=\nu_g=1.6667\times 10^{-2}$. Quantities are in lattice units.}
\end{center}
\end{figure}
The drop changes from a
prolate shape at $t=2000$ to oblate shape at $t=16000$. Such shape changes continue till the drop reaches its
equilibrium spherical shape.
Figure \ref{fig:dropint1} shows the temporal evolution of the interface locations of the oscillating drop with the conditions above
for two different surface tension parameter: $\kappa=0.02$ and $0.08$.
\begin{figure}
\begin{center}
\includegraphics{fig4.eps}
\caption{\label{fig:dropint1}Interface location of an oscillating drop as a function of time for
two values of the surface tension
parameter $\kappa$; $R_{min}=40$, $R_{max}=55$, $\rho_g=0.1$, $\rho_l=0.4$, $\nu_l=\nu_g=1.6667\times 10^{-2}$. Quantities are in lattice units.}
\end{center}
\end{figure}
It is expected that increasing the surface tension
will reduce the time period of oscillations. The computed ($T_{LBE}$) and analytical ($T_{anal}$) time periods, where
$T_{anal}=2\pi/\omega_2$, when
$\kappa=0.02$ are $29483$ and $29448$ respectively. As $\kappa$ is increased to $0.08$, $T_{LBE}$ and $T_{anal}$ become
$14388$ and $14313$ respectively. It may be seen that the computed and analytical values agree well, the
difference being less than $1\%$. Also, the time period decreases as $\kappa$ is increased, which is consistent with
expectations.
Consider next the effect of changing the drop size on the time period of oscillations.
Figure \ref{fig:dropint2} shows the interface locations of an oscillating drop as a function of time for the following two initial sizes:
$R_{min}=30$ and $R_{max}=45$; $R_{min}=40$, $R_{max}=55$. Reducing the drop size reduces its time period.
\begin{figure}
\begin{center}
\includegraphics{fig5.eps}
\caption{\label{fig:dropint2}Interface location of an oscillating drop as a function of time for two drop sizes;
$\rho_g=0.1$, $\rho_l=0.4$, $\nu_l=\nu_g=1.6667\times 10^{-2}$, $\kappa=0.02$. Quantities are in lattice units.}
\end{center}
\end{figure}
The computed time period of the larger drop is equal to $29483$, while that for the smaller drop is $20118$.
Comparison of the computed time periods with the analytical solution shows that they agree within $1\%$ for these cases.
Next, consider three different kinematic viscosities of the liquid: $\nu_l=1.6667\times 10^{-2}, 3.3333\times 10^{-2}$ and
$5.0\times 10^{-2}$.
Figure \ref{fig:dropint3} shows the effect of drop viscosity on the temporal evolution of the interface locations of the drop.
\begin{figure}
\begin{center}
\includegraphics{fig6.eps}
\caption{\label{fig:dropint3}Interface location of an oscillating drop as a function of time for different kinematic viscosities $\nu_l$;
$R_{min}=40$, $R_{max}=55$, $\rho_g=0.1$, $\rho_l=0.4$, $\kappa=0.02$. Quantities are in lattice units.}
\end{center}
\end{figure}
It is found that as the kinematic viscosity is increased the time period increases moderately which
is consistent with the analytical solution. The computed time periods at these
viscosities are $29483$, $31030$ and $32925$, while the analytical values are $29448$, $30597$ and $31318$, respectively, with
a maximum error within $5.1\%$.
The third test problem considered here is that of the break-up of a cylindrical liquid column into drops, a fascinating
problem of long standing theoretical and practical interest. In a seminal work,
Rayleigh (1878)~\nocite{rayleigh78} showed through a linear stability analysis of an inviscid column of
cylindrical liquid of radius $R_c$ that the column will be unstable if the axisymmetric wavelength of any
disturbance $\lambda_d$ is longer than its circumference, i.e. the wave number
$k^{*}=2\pi R_c/\lambda_d$ should be less than one. Later, the theoretical analysis was extended to more
realistic conditions by including viscosity. In the last three decades, several experimental and numerical
investigations have also been performed. To evaluate the axisymmetric LBE model, we study the Rayleigh capillary
instability for different wavenumbers. Initial studies carried out with $k^{*}>1$ showed that the liquid does not
break-up. We will now present results of cases with break-up.
Consider a cylindrical liquid column of radius $R_c=45$
subject to an axisymmetric co-sinusoidal wavelength $\lambda_d=320$, i.e. $k^{*}=0.88$.
To simulate the dynamics of instability for this wavenumber, we consider a domain discretized by $321\times 151$ lattice
sites with $\rho_g=0.1$, $\rho_l=0.4$, $\nu_g=\nu_l=6.6667\times 10^{-2}$ and $\kappa=0.1$. Since $k^{*}<1$, it is
expected that the liquid column would eventually breakup. Figure \ref{fig:rayleigh1} shows the configurations of the liquid
column at different times. As time progresses, the imposed interfacial disturbances on the
\begin{figure}
\begin{center}
\includegraphics[height=6.50in,clip=]{fig7.EPS}
\caption{\label{fig:rayleigh1}Configurations of a cylindrical liquid column at different times undergoing Rayleigh
breakup and satellite droplet formation;
$k^{*}=0.88$, $\rho_g=0.1$, $\rho_l=0.4$, $\nu_l=\nu_g=6.6667\times 10^{-2}$. Quantities are in lattice units.}
\end{center}
\end{figure}
liquid column grow. At $t=28000, 46000$ $52000$, the cross-section of the column becomes
progressively thinner in the center, and by mass conservation, the ends becomes larger. At $t=60000$, notice
that a bead-type structure is formed at the ends and with a thin ligament between them. Such a structure has been
observed in experiments~\cite{eggers97} and in other numerical simulations~\cite{ashgriz95}. Eventually, the column
breaks up forming a thin ligament in the middle, which then becomes a satellite droplet.
Let us now increase the wavelength of the disturbance to $\lambda_d=600$, keeping the physical parameters the same as before.
We consider a domain represented by $601\times 151$ lattice sites. Since,
$R_c=45$ as before, the wavenumber is $0.47$. Figure \ref{fig:rayleigh2} shows the temporal evolution of the configurations of
the liquid column at this reduced wavenumber. The axisymmetric disturbance grows with time.
\begin{figure}
\begin{center}
\includegraphics{fig8.EPS}
\caption{\label{fig:rayleigh2}Configurations of a cylindrical liquid column at different times undergoing Rayleigh
breakup and satellite droplet formation;
$k^{*}=0.47$, $\rho_g=0.1$, $\rho_l=0.4$, $\nu_l=\nu_g=6.6667\times 10^{-2}$. Quantities are in lattice units.}
\end{center}
\end{figure}
Since the wavelength is longer, it can be noticed that the ligament that is formed during the Rayleigh instability
is also longer. As a result, after the column breaks up, a larger satellite droplet is formed.
To express the drop size distribution with wave numbers more quantitatively, we plot the non-dimensional
size of the main and satellite drops, $r^{*}=R/R_c$, as a function of wave number, $k^{*}$ in Fig. \ref{fig:rayleigh3}.
It may be noted that Rayleigh's
\begin{figure}
\begin{center}
\includegraphics{fig9.eps}
\caption{\label{fig:rayleigh3}Drop sizes resulting from Rayleigh breakup of liquid cylindrical column as a
function of wave number $k^{*}$. Quantities are dimensionless.}
\end{center}
\end{figure}
original analysis predicts only the onset of breakup and not the formation of satellite droplets. To predict analytically
satellite droplet formation, it has been shown that at least a third-order
perturbation analysis of the Navier-Stokes equations (NSE) is needed~\cite{lafrance75}. Computations based on
direct solutions of the NSE also predict the formation of the satellite droplets.
To evaluate the
drop size distribution computed using the axisymmetric LBE model, we consider the experimental data of
Rutland and Jameson (1971),~\nocite{rutland71} the experimental data and analytical solution based on a
third-order perturbation analysis of the NSE by Lafrance (1975),~\nocite{lafrance75}
a boundary integral solution of the NSE by Mansour and Lundgren (1990)~\nocite{mansour90} and a finite element solution of
the NSE by Ashgriz and Mashayek (1995).~\nocite{ashgriz95} It can be seen in the figure that as long as the wavenumber
is less than one, as expected there will be a satellite droplet formation. As the wavenumber is reduced, the sizes of
both the main drop and satellite droplet increases. The rate of increase of the size of the satellite droplet is greater
than that of the main drop. Notice that there is considerable scatter in the available data in the figure. The computed
results from the axisymmetric LBE model are presented for wavenumbers greater than or equal to $0.47$. Ignoring the
two experimental data points of Lafrance (1975) for the satellite drop sizes that deviate considerably from the others,
we find that the axisymmetric model is able to reproduce the drop size distribution quantitatively within $12\%$.
The axisymmetric model has been employed to study head-on collisions of drops of radii $R_1$ and $R_2$ approaching each other
with a relative velocity $U$. The dynamics and outcome of colliding drops is characterized mainly by the Weber number,
$We$ defined by $We=\rho_{l} (R_{1}+R_{2}) U^{2} / \sigma$~\cite{qian97}. Additional parameters that may have an influence
are the Ohnesorge number, $Oh$, defined by $Oh=16\mu_{l}/\sqrt{\rho_{l}R_{1}\sigma}$ and ratios of liquid and gas
densities($r$) and dynamic viscosities ($\lambda$). According to experiments~\cite{qian97}, it is expected that lower $We$
collisions
lead to coalescence while higher $We$ to separation by reflexive action. Figures \ref{fig:collWe20} and
\ref{fig:collWe100} present drop configurations
at $We=20$ and $We=100$ respectively. Notice that at $We=20$, the drops coalesce, while at $We=100$, they eventually separate
\begin{figure}
\begin{center}
\includegraphics{fig10.EPS}
\caption{\label{fig:collWe20} Colliding drops at different times $T$; $We=20$, $Oh=0.589$, $r=4$, $\lambda=1$.
Time is normalized by the relative velocity between the drops and their diameter. Axes are in lattice units.}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics{fig11.EPS}
\caption{\label{fig:collWe100} Colliding drops at different times $T$; $We=100$, $Oh=0.589$, $r=4$, $\lambda=1$.
Time is normalized by the relative velocity between the drops and their diameter. Axes are in lattice units.}
\end{center}
\end{figure}
with the formation of a satellite droplet, which are consistent with experimental observations.
Also notice that for the latter
case, the temporarily coalesced drop undergoes various stages of deformation which are consistent with a recent theoretical
analysis~\cite{roisman04}. Additional details of these and other studies of drop collisions are given in Ref.~\cite{premnath04a}.
\section{\label{sec:summary}Summary}
In this paper, a LB model for axisymmetric multiphase flows is developed.
The axisymmetric model is developed by adding source terms to the
standard Cartesian BGK LBE. The source terms, which are temporally and
spatially dependent, represent the axisymmetric contributions of the order parameter,
which distinguish the different phases, as well as
inertial, viscous and surface tension forces. Consistency of the model in achieving the
desired axisymmetric flow multiphase behavior is established through the Chapman-Enskog
multiscale analysis. The analysis shows that the axisymmetric macroscopic conservation
equations are recovered in the continuum limit.
An axisymmetric model with reduced compressibility effects is then
developed to improve its computational stability. In this version,
a transformation is introduced to the distribution function in the LBE such that it
reduces the compressibility effects.
Comparisons of computed axisymmetric equilibrium drop formation
and oscillations, Rayleigh capillary instability, breakup and formation of satellite
drops liquid cylindrical liquid columns and the outcomes of head-on drop collisions with
available data show satisfactory agreement.
The maximum error for the frequency of drop oscillations is less than $5.1\%$ and that for drop sizes as a
result of Rayleigh breakup is $12\%$.
\begin{acknowledgments}
The authors thank Dr.\ X. He for helpful discussions and the Purdue University Computing Center (PUCC)
and National Center for Supercomputing Applications (NCSA) for providing access to computing resources.
\end{acknowledgments}
|
{
"timestamp": "2005-03-18T21:14:18",
"yymm": "0503",
"arxiv_id": "physics/0503160",
"language": "en",
"url": "https://arxiv.org/abs/physics/0503160"
}
|
\section{Introduction}
\subsection{Metric stability for random walks} In the study of a dynamical system, some of the most
important questions concerns the stability of their dynamical
properties under (most of the) perturbations: how robust are
they?
Here we are mainly interested in the stability of metric
(measure-theoretical) properties of dynamical systems. A
well-known example is given by ($C^2$) Markov expanding maps on
the circle: this is a class stable under perturbations and all of
them have an absolutely continuous and ergodic invariant probability
satisfying certain decay of correlations estimatives. In
particular, in the measure theoretical sense, most of the orbits
are dense in the phase space.
Now let us study a slightly more complicated situation: consider a
$C^2$ Markov almost onto expanding map of the interval $f\colon I \rightarrow
I$ with bounded distortion and large images (see Section 2 for details) and let $\psi \colon I \rightarrow \mathbb{Z}$ be a function
which is constant in each interval of the Markov partition of $f$.
We can define $F\colon I \times \mathbb{Z} \rightarrow I \times
\mathbb{Z}$ as
$$F(x,n):= (f(x), \psi(x) + n).$$
The second entry of $(x,n)$ will be called its {\bf state}. We
also assume that
\begin{equation}\label{cond1}
\inf \psi > -\infty
\end{equation}
and that $F$ is topologically mixing.
\begin{figure}
\centering \psfrag{f}{$f$}
\psfrag{Fx}[][][0.8]{$F$}
\psfrag{Fy}[][][0.8]{$F$}
\psfrag{x}[][][0.8]{$x$}
\psfrag{y}[][][0.8]{$y$}
\psfrag{fx}[][][0.8]{$f(x)$}
\psfrag{fy}[][][0.8]{$f(y)$}
\psfrag{a}[][][0.8]{$i$}
\psfrag{b}[][][0.8]{$i+1$}
\psfrag{c}[][][0.8]{$i+2$}
\psfrag{d}[][][0.8]{$i-1$}
\psfrag{e}[][][0.8]{$i-2$}
\psfrag{psix}[][][0.8]{$\psi(x)=-1$}
\psfrag{psiy}[][][0.8]{$\psi(y)=1$}
\includegraphics[width=1.0\textwidth]{figure1.eps}
\caption{A deterministic random walk}
\end{figure}
The map $F$ is refereed to in literature in many ways: as a
"skew-product between $f$ and the translation on the group
$\mathbb{Z}$", a "group extension of $f$", or even a
"deterministic random walk generated by $f$", and its metric
behavior is very well studied: for instance, are most the orbits
recurrent? Everything depends on the {\bf mean drift }
$$M = \int \psi d\mu,$$
where $\mu$ is the absolutely continuous invariant probability of
$f$ (the function $\psi$ will be called {\bf drift function}).
Indeed, note that
$$F^n(x,i)= (\ f^n(x)\ ,\ i + \sum_{k=0}^{n-1}\psi(f^k(x))\ ).$$
By the Birkhoff Ergodic Theorem
$$
\lim_{n \rightarrow \infty}\ \frac{\pi_2(F^n(x,i))- \pi_2(x,i)}{n}
= \lim_{n \rightarrow \infty}\ \frac{1}{n}
\sum_{k=0}^{n-1}\psi(f^k(x))= M.$$ for almost every $x \in I$
(here $\pi_2(x,n):=n$). In particular if $M \neq 0$ then almost every point $(x,i)
\in I \times \mathbb{Z}$ is {\bf transient}: in other words we
have
$$\lim_{n\rightarrow \infty} |\pi_2(F^n(x,i))|= \infty.$$
So most of the points are not recurrent.
On the other hand, if $M=0$, most of points are
recurrent (see Guivarc'h \cite{guivarch}): by the Central Limit Theorem for expanding maps (here we need to assume that $\psi$ in no constant and $f \in aO$: see Section 2) of the
interval
$$sup_{\epsilon \in \mathbb{R}} \ | \mu( x \in I\colon \frac{\sum_{k=0}^{n-1}\psi(f^k(x))}{\sigma \sqrt{n}} \leq \
\epsilon) - \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\epsilon}
e^{-\frac{u^2}{2}} \ du | \leq \frac{C}{\sqrt{n}},$$
Given $\delta >0$ we can easily obtain, taking
$\epsilon = n^{-1/4}$ and applying Borel-Cantelli Lemma, that
$$\mu(A_{+}):=\mu( x \in I \colon \ \limsup_{n\rightarrow \infty}
\frac{\sum_{k=0}^{n-1}\psi(f^k(x))}{ \sqrt[2 + \delta]{n}}=\infty)\geq \frac{1}{2},$$
$$\mu(A_{-}):=\mu( x \in I \colon \ \liminf_{n\rightarrow \infty}
\frac{\sum_{k=0}^{n-1}\psi(f^k(x))}{ \sqrt[2 + \delta]{n}}=-\infty)\geq \frac{1}{2}.$$
Clearly $A_{+}$ and $A_{-}$ are invariant sets: the ergodicity of
$f$ implies that $$\mu(A_{+}~\cap~A_{-})~=~1.$$
By the conditions on $\psi$ in Eq. (\ref{cond1}), that $f$ is expanding with distortion control
and that $F$ is transitive, we can easily conclude that almost every point in
$I\times \mathbb{Z}$ is a $F$-recurrent point.
Note that the random walk $F$ is a dynamical system quite similar to
expanding circle maps: $F$ is an expanding map, with good
bounded distortion properties; but the lack of compactness of the
phase space allows the non-existence of an invatiant
probability absolutely continuous with respect to the Lebesgue measure on $I\times \mathbb{Z}$. Moreover, in general the random walk is not even
recurrent and the recurrence property lost its stability: given a
recurrent random walk $(f,\psi)$, it is possible to obtain a
transient random walk by changing a little bit $f$ and $\psi$.
Since the non compactness of the phase space seems to be the
origin of the lack of stability of recurrence and transience
properties, a natural question is to ask if such properties are
stable by compact perturbations. The answer is yes. Indeed, as we are going to see
in Theorems \ref{sttr}-\ref{strec}, the transience and recurrence
are preserved even by non-compact perturbations which decreases
fast away from state $0$. For instance,we can choose perturbations like
$$\tilde{F}(x,n)=(f_n(x),\psi(x)+n),$$
where, for some $\lambda \in [0, 1)$,
\begin{equation}\label{decay}|f_n-f|_{C^3} \leq
\lambda^{|n|}.\end{equation} The notations and conventions are more or less
obvious: we postponed the rigorous definitions to the next
section.
With respect to the stability of transience and recurrence, there
is a previous quite elegant result by R. L. Tweedie \cite{tw}: if
$p_{ij}$ are the transition probabilities of a Markov chain on
$\mathbb{Z}$, then any perturbation $\tilde{p}_{ij}$ so that
$$ (1+\epsilon_i)^{-1}p_{ij} \leq \tilde{p}_{ij} \leq p_{ij}
(1+\epsilon_i), \ j \neq i,$$ and
$$\prod_{i=0}^{\infty} (1+\epsilon_i) < \infty$$
preserves the recurrence or transience of the original Markov
chain. But Tweedie argument does not seem to work in our setting.
Our result coincides with Tweedie result in the very special case
where $f$ and $f_n$ are linear Markov maps and $\epsilon_i \sim
C\lambda^{|i|}$.
In the transient case we can tell a little more: there will be a
conjugacy between the original random walk $f$ and its
perturbation which is a martingale strongly quasisymmetric
map (for short, mSQS-map) with respect to certain dynamically defined set of partitions.
Unlike the usual class of one-dimensional quasisymmetric
functions, which does not share many of most interesting
properties of higher dimensional quasisymmetric maps, the
one-dimensional mSQS-maps are
much closer to their high-dimensional cousins, as quasiconformal maps in dimension $2$. For instance, they are absolutely
continuous.
We also study the behavior of the Hausdorff dimension of dynamically defined sets: Denote by $\Omega_+(F)$ the set of points which have non-negative states along the positive orbit by $F$. We prove that $\Omega_+(F)$ has Hausdorff dimension strictly smaller than one if and only if $\Omega_+(\tilde{F})$ has dimension less than one for all perturbation satisfying Eq. (\ref{decay}). Furthermore we give a variational characterization for the Hausdorff dimension $HD(\Omega_+(F))$ as the minimum of $HD(\Omega_+(\tilde{F}))$, where $\tilde{F}$ runs on the set of such perturbations. For these results we study of the stability of the multifractal spectrum of the random walk $F$ under those perturbations.
\subsection{Applications to (generalized) renormalization theory}An unimodal map is a map with an unique critical point. Under reasonable conditions (real-analytic maps with negative Schwarzian derivative and non-flat critical point) two non renormalizable unimodal maps with the same topological entropy are indeed topologically conjugated. A key question in one-dimensional dynamics is about the regularity of the conjugacy: is it H\"older? Is it absolutely continuous? Since Dennis Sullivan work in the 80's the quasisymmetry of the conjugacy became a very useful tool to obtain deep results in one-dimensional dynamics. Lyubich proved that under the reasonable condition above the conjugacy between two non renormalizable unimodal maps is quasisymmetric. Later on, the density of the hyperbolic maps in the real quadratic family was proved verifying the quasisymmetry of the conjugacies for all combinatorics, including infinitely renormalizable ones.
Note that quasisymmetric maps are not, in general, absolutely continuous: they do not even preserve (in general) sets of Hausdorff dimension one. Are the conjugacy between unimodal maps absolutely continuous? The answer is no: M.~Martens and W.~de~Melo \cite{mm} proved that under the reasonable conditions above an absolutely continuous conjugacy is actually $C^\infty$, provided the unimodal maps
\begin{itemize}
\item[ ] \
\item[{\it i.}] {\it do not have a periodic attractor,}\\
\item[{\it ii.}]{\it are not infinitely renormalizable, }\\
\item[{\it iii.}] {\it do not have a wild attractor (the topological and measure-theoretical attractor must coincide).} \\
\end{itemize}
Since we can change the eigenvalues of the periodic points of maps preserving its topological class, and the eigenvalues are preserved by $C^1$ conjugacies, we conclude that in general a conjugacy between unimodal maps is not absolutely continuous.
Condition i. is clearly necessary. This work (Theorem \ref{apl1}) shows that the Condition ii. is necessary proving that the conjugacy between two arbitrary Feigenbaum unimodal maps with same critical order is {\em always } absolutely continuous . Actually the conjugacy is martingale strongly quasisymmetric with respect to a set of dynamically defined partitions.
Condition iii. is never violated when the critical point is quadratic. But for certain topological classes of unimodal maps wild attractor appears when the order of the critical point increases: Fibonacci maps are the simplest kind of such maps. We are going to prove (Theorem \ref{apl2}) that a Fibonacci map with even order has a wild attractor if and only if all Fibonacci maps with the same even order are conjugated to each other by an absolutely continuous mapping (in particular all these Fibonacci maps have a wild attractor). So Condition iii. is necessary.
In both examples above the study about perturbations of transient and recurrent random walks are going to be crucial, as the (generalized) renormalization theory for unimodal maps: for these maps it is possible to construct an induced map which is essentially a perturbation of a deterministic random walk. In the Fibonacci case the transience of this random walk is equivalent to the existence of a wild attractor. The random walk associated to the Feigenbaum map will always be transient.
For both Feigenbaum and Fibonacci maps there are infinitely many periodic points (indeed in the Fibonacci case the periodic points are also dense in the maximal invariant set). It is well known that the conjugacy between critical circle maps with same irrational rotation number and satisfying certain Diophantine condition is absolutely continuous, but we think that these are the first interesting examples of a similar phenomena for maps with many periodic points.
\section{Expanding Markov maps, random walks and its perturbations}
In this article we will deal with maps
$$F\colon I \times \mathbb{Z} \rightarrow I \times \mathbb{Z}$$
which are piecewise $C^2$ diffeomorphisms, which means that there is a partition $\mathcal{P}^0$ of $I \times \mathbb{Z}$ so that each element $J \in \mathcal{P}^0$ is an open interval where $F|_{\overline{J}}$ is a $C^2$ diffeomorphism. Denote $I_n=I\times \{ n \}$. Denote by $m$ the Lebesgue measure in the in $I\times\mathbb{Z}$, that is, if $A\subset I\times\mathbb{Z}$ is a Borelian set then
$$m(A)=\sum_n m_I(\pi(A\cap I_n)),$$
where $m_I$ is the Lebesgue measure in the interval $I$ and $\pi(n,x)=x$.
If $A_J$ denotes the unique affine transformation which maps the interval $J$ to $[0,1]$ and preserves orientation, then define, for each $J \in \mathcal{P}^0$, $$\tau_J^F:= A_{J}\circ F^{-1} \circ A_{F(J)}^{-1}.$$
Throughout this article we will assume that $F$ satisfies some of the following properties:
\begin{itemize}
\item[\ ] \ \\
\item {\bf Markovian (Mk)}: For each $J \in \mathcal{P}^0$,
$F(J)$ is a connected union of elements in $\mathcal{P}^0$. In particular we can write $F(x,n)=(f_n(x), n + \psi(x,n))$, where $f_n\colon I \rightarrow I$ is a piecewise $C^2$ diffeomorphism relative to the partition $\mathcal{P}^0_n:=\{J \in \mathcal{P}^0\colon \ J \subset I_n \}$ and $\psi\colon I \times \mathbb{Z} \rightarrow \mathbb{Z}$, called the {\bf drift function}, is constant on each element of $\mathcal{P}^0$.\\
\item {\bf Lower Bounded Drift (LBD)} $F$ is Markovian and $\min \psi > -\infty$.\\
\item {\bf Large Image (LI)}: $F$ is Markovian and there exists $\delta > 0$ so that for each $J \in \mathcal{P}^0$ we have $|F(J)|\geq \delta$.\\
\item {\bf Onto (On)}: $F$ is Markovian and for each $J \in \mathcal{P}^0$ we have $F(J)=I^n$, for some $n \in \mathbb{Z}$.\\
\item {\bf Bounded Distortion (BD)}: There exists $C > 0$ so that every $J \in \mathcal{P}^0_n$ and map $\tau_J$ is a $C^2$ function satisfying
$$\sup_{J} \Big| \frac{D^2\tau_J}{(D\tau_J)^2} \Big| \leq C.$$ \\
\item {\bf Strong Bounded Distortion (sBD)}: There exists $C > 0$ so that every $J \in \mathcal{P}^0_n$ and map $\tau_J$ is a $C^2$ function satisfying
$$\sup_{J} \Big| \frac{D^2\tau_J}{(D\tau_J)^2} \Big| \leq C|J|.$$ \\
\item {\bf Expansivity (Ex)}: If $J \in \mathcal{P}^0_n:=\{ J \in \mathcal{P}^0\colon \ J \subset I_n\}$, denote $\phi_J:=f_n^{-1}|_{f_n(J)}$. Then either $\phi_J$ can be extended to a function in a $\delta$-neighborhood of $J$ so that $$ S\phi_J > 0,$$ where $S\phi_J$ denotes the Schwarzian derivative of $\phi_J$, or there exists $\theta \in (0,1)$ so that
$$|\phi'_J| < \theta$$
on $I$. \\
\item {\bf Regularity a (Ra)}: There exists $N \in \mathbb{N}$, $\delta > 0$ and $C > 0$ with the following properties: the intervals in $\mathcal{P}^0_n$ are positioned in $I_n$ in such way that the complement of $$\bigcup_{J \in \mathcal{P}^0_n}\ int \ J$$ contains at most $N$ accumulation points $$c_1^n < c_2^n < \dots < c^{n}_{i_n},$$ with $i_n\leq N$, which is in the interior of $I_n$. Furthermore $|c^n_{i+1}-c^n_i|\geq \delta$. Moreover, given $P$ and $Q \in \mathcal{P}^0_n$ so that $\overline{P} \cap \overline{Q} \neq \phi$ then
$$\frac{1}{C} \leq \frac{|P|}{|Q|} \leq C.$$
\item {\bf Regularity b (Rb)}: Assume $Ra$. There exists $C > 0$, $\lambda \in (0,1)$, $\delta > 0$ so that for each $1< i<i_n$ we can find a point $$d_i^n \in (c^n_i,c^n_{i+1}),$$
which does not belong to any $P \in \mathcal{P}^0_n$, and $$\min \{|c^n_{i+1}-d_i^n|, |d_i^n- c^n_{i}| \} \geq \delta$$ with the following property: If $J$ is a connected component of $$I_n \setminus \{d_i^n, c^n_j\}_{i,j}$$
then we can enumerate the set $$\{P\}_{P \in \mathcal{P}^0_n,\ P \subset J}=\{J_i\}_{i \in \mathbb{N}}$$ in such way that $\partial J_{i} \cap \partial J_{i+1} \neq \phi$ for each $i$ and $$\frac{|J_{i+j}|}{|J_{i}|}\leq C\lambda^j$$ for $i \geq 0$, $j > 0$.\\
\item {\bf Good Drift (GD)}: , if $\psi$ is the drift function of the random walk then there exists $\gamma \in (0,1)$ and $C > 0$ so that
$$m(\{(x,n) \ s.t. \ \psi(x,n)\geq k \}) \leq C\gamma^k.$$ \\
\item {\bf Transitive (T)}: $F$ has a dense orbit.
\end{itemize}
For convenience of the notation if for instance $F$ is Markovian and it has Bounded Distortion, we will write $F \in Mk+BD$.
A {\bf deterministic random walk} (or simply random walk) is a map $$F \in Mk+LBD+LI+Ex+BD+GD.$$ It is generated by the pair $(\{ f_n\},\psi)$ if
$$F(x,n):= (f_n(x),\psi(x,n)+n).$$
When $f_n=f \in Mk$ and $\psi(x,n)=\psi(x)$, we say that $F$ is the
{\bf spatially homogeneous deterministic random walk} generated by the pair $(f,\psi)$. There is a large
literature about such random walks. We will sometimes assume the following property:
\begin{itemize}
\item[\ ] \ \\
\item {\bf Almost Onto (aO)}: For every $i, j \in \Lambda$ there exists a finite sequence $i=i_0,i_1,i_2,\dots,i_{n-1},i_n=j \in \Lambda$ so that $$f(I_{i_k})\cap f(I_{i_{k+1}})\neq \emptyset$$ for each $k < j$. \\
\end{itemize}
Denote $\pi(x,n):= \pi_2(x,n):=n$. A random walk is called {\bf transient} if for almost every $(x,n) \in I \times \mathbb{Z}$
$$\lim_{k\rightarrow \infty} |\pi_2(F^k(x,n))| = \infty,$$
and it is {\bf recurrent} if for almost every $(x,n) \in I \times \mathbb{Z}$
$$\# \{ k \colon \pi_2(F^k(x,n))=n \} = \infty.$$
Making use of usual bounded distortion tricks it is easy to show that every $F \in Mk+LI+Ex+BD+T$ is either recurrent or transient.
A (topological) {\bf perturbation} of a random walk is a random walk $\tilde{F}$, generated by a pair $(\{\tilde{f}_i\},\tilde{\psi})$, so that $F\circ H = H \circ \tilde{F}$ for some homeomorphism $$H\colon I \times \mathbb{Z} \rightarrow I \times \mathbb{Z}$$
which preserves states: $\pi_2(H(x,i))=i$.
Define $\mathcal{P}^n(F) := \vee_{i=0}^{n-1} F^{-i}\mathcal{P}^0(F)$. If $F$ and $\tilde{F}$ are random walks and $h$ is a topological conjugacy that preserves states between $F$ and $\tilde{F}$, then for each interval $L$ such that $L\subset J \in \mathcal{P}^n(F)$, define
$$dist_n(L):= \sup_{x \in L} \Big|\ln \frac{DF^n(x)}{DF^n(y)} \Big|,$$
Similarly, if $x \in J \in \mathcal{P}^n(F)$ define
$$dist_n(x):= dist_n(J)$$
and
$$dist_\infty(x):= \sup_n dist_n(x).$$
Another kind of random walk which will have a central role in our
results are those which are {\bf asymptotically small} perturbations: these are perturbations $(\{\tilde{f}_i\},\tilde{\psi})$ of a homogeneous random walk $(\{f_i\},\psi)$ such that there exists $\lambda \in (0,1)$ and $C >0$ satisfying either
\begin{equation}\label{asymp} |\log \frac{DF(H(p))}{D\tilde{F}(p)}| \leq C\lambda^{|\pi_2(p)|},\end{equation}
if $\psi$ is bounded, or
\begin{equation}\label{asymp2} |\log \frac{DF(H(p))}{D\tilde{F}(p)}| \leq C\lambda^{\pi_2(p)},\end{equation}
for $\pi_2(p)\geq 0$ and $DF(H(p))=D\tilde{F}(p)$ otherwise, if $\psi$ has only a lower bound.
It is easy to see that properties $Ra$, $Rb$ and $GD$ are invariant by asymptotically small perturbations (if we allow to change the constants described in these properties).
Let $F=(\{f_i\}_i,\psi)$ be a random walk, where $\psi$ is Lebesgue integrable on compact subsets of $I \times \mathbb{Z}$. We say that $F$ is {\bf strongly transient} if $K > 0$ and
$$\mathbb{E}(\psi\circ F^n | \mathcal{P}^{n-1}(F)) > K$$
for every $n\geq 1$. We will also say that $F$ is $K$-strongly transient. Here we are considering conditional expectations relative to the Lebesgue measure. As the notation suggest, every strongly transient random walk is transient. Moreover we have the following large deviations result:
\begin{prop}\label{largedeviationsst}\label{bru} Every $K$-strongly transient random walk $F\in Ra+Rb$ is transient. Furthermore for every small $\epsilon > 0$ there exist $\lambda \in [0,1)$ and $C > 0$ so that for each $P \in \mathcal{P}^0$ we have
$$m( p \in P \colon \ \pi_2(F^n(p))-\pi_2(p) < (K - \epsilon) n )\leq C\lambda^n |P|.$$
\end{prop}
We will postpone the proof of this result to Section \ref{sectiontransience}.
\begin{rem}{\rm By the Birkhoff Ergodic Theorem it is easy to see that a sufficiently high iteration of a homogeneous random walk with positive mean drift is strongly transient (see the proof of Proposition \ref{homstr} for details). }
\end{rem}
\section{Statements of results}
\subsection{Stability of transience}
\begin{thm}[Stability of Transience I]\label{sttr} Assume that
the random walk $F$ defined by the pair $(\{f_i\}_i,\psi)$ is
strongly transient. Then every asymptotically small perturbation $G$ of $F$ is also transient.
Indeed there is a topological conjugacy between $F$ and $G$ which
is an absolutely continuous map and preserves the states.
\end{thm}
We have a similar theorem for all transient homogeneous random
walks:
\begin{thm}[Stability of Transience II]\label{sttrho}\label{abscont} Suppose that the homogeneous random walk $F$ defined by
the pair $(f,\psi)$ is transient. Then every asymptotically small perturbation of $F$ is topologically conjugated to $F$ by
an absolutely continuous map which preserves the states.
\end{thm}
We can be more precise regarding the regularity of the conjugacy
if the drift is non-negative:
Let $\mathcal{A}_0,\ \mathcal{A}_1, \ \cdots
, \mathcal{A}_n, \ \mathcal{A}_{n+1}, \cdots $ be a succession of partitions by intervals of $I \times \mathbb{Z}$, such that $\mathcal{A}_{n+1}$ refines $\mathcal{A}_{n}$ and whose union generates the Borelian
algebra of $\sqcup_n I_n$. We say that $h\colon \sqcup_n I_n \rightarrow \sqcup_n I_n$ is a {\bf martingale strongly quasisymmetric (mSQS)} map with respect to the {\bf stochastic basis } $\cup_n \mathcal{A}_n$ if there exist $C > 0$ and $\alpha \in (0,1]$ so that
$$ \frac{m(h(B))}{|h(J)|} \leq C \left( \frac{m(B)}{|J|} \right )^\alpha$$
for all Borelian $B \subset J \in \cup_n \mathcal{A}_n$, and the same inequality holds replacing $h$ by $h^{-1}$ and $\cup_n \mathcal{A}_n$ by $\cup_n h(\mathcal{A}_n)$.
\begin{thm}[Strongly quasisymmetric rigidity]\label{sqr} Let $F$ be either a strongly transient random walk or a
transient homogeneous random walk with positive mean drift. Moreover assume in both cases that $\psi\geq 0$. Then every asymptotically small perturbation $G$ of
$F$ is topologically conjugated to $F$ by
an absolutely continuous map $h$ which preserves the states.
Furthermore $h$ on $\cup_{i\geq 0} I_i$ is a martingale strongly quasisymmetric mapping with respect
to the stochastic basis $\cup_i \mathcal{P}^i.$
\end{thm}
\subsection{Stability of recurrence}
In the recurrent case, we are going to restrict ourselves to the
stability of the metric properties of homogeneous random walks under asymptotically
small perturbations: it is easy to see that the recurrence
is not stable by perturbations which are not asymptotically small. Nevertheless
\begin{thm}[Stability of Recurrence]\label{strec} Suppose that $F \in aO+T$ is a
recurrent homogeneous random walk generated by the pair
$(f,\psi)$. Then every asymptotically small perturbation of $F$ is also recurrent.
\end{thm}
If $p$ is a periodic point with prime period $n$ then $DF^n(p)$ is called the spectrum of the periodic point $p$.
Note that we can not expect, as in the transient case, an
absolutely continuous conjugacy which preserves states between $F$
and $G$, once asymptotic small perturbations do not
preserve (in general) the spectrum of the periodic points and:
\begin{prop}[Rigidity]\label{rigidity} Suppose that the random walk $F \in On$ generated by
a pair $(\{f_i \}_i,\psi)$ is recurrent.
If there is an absolutely continuous conjugacy which preserves
states $H$ between $F$ and a random walk $G$, then $H$ is $C^1$ in
each state. In particular the spectrum of the corresponding
periodic points of $F$ and $G$ are the same.
\end{prop}
The reader should compare this result with similar results by Shub and Sullivan \cite{ss} for expanding maps on the circle and de Melo and Martens \cite{mm} for unimodal maps.
\subsection{Stability of the multifractal spectrum}
Let $F$ be a random walk and denote
$$\Omega_+(F):=\{p \colon \ \pi_2(F^jp)\geq 0, \ for \ j\geq 0 \},$$
$$\Omega_+^k(F):= \{(x,k) \colon \ \pi_2(F^j(x,k))\geq 0, \ for \ j\geq 0 \}$$
and
$$\Omega_{+\beta}^k(F):= \{ (x,k) \in \Omega_+^k\ s.t\ \ \underline{\lim}_{\ n} \ \frac{\pi_2(F^n(x,k))}{n}\geq\beta\}$$
\begin{thm}\label{multi} Let $F\in Ra + Rb+ On$ be a random walk. Then, for all $k \in \mathbb{Z}$ and $\beta > 0$ the Hausdorff dimension $HD(\Omega_{+\beta}^k)$ is
invariant
by asymptotically small perturbations.
\end{thm}
Besides its inner interest, the previous result will be useful by
other reason:
\begin{prop}\label{ms}\label{inv} Let $F \in Ra+Rb+On$ be a homogeneous random walk. Then
$$HD(\Omega_+^k(F))= \lim_{\beta\rightarrow 0^+} HD(\Omega_{+\beta}^k(F)).$$
\end{prop}
and as a consequence of Theorem \ref{multi} and Proposition \ref{ms}:
\begin{thm}\label{omega} Let $F\in Ra+Rb+On$ be a homogeneous random walk. If $G$ is an
asymptotically small perturbation of $F$ then
\begin{equation}\label{ineq} HD(\Omega_{+}^k(G)) \geq HD(\Omega_{+}^k(F)).\end{equation}
\end{thm}
We can not replace the inequality in Eq. (\ref{ineq}) by an
equality. Indeed, even if $HD(\Omega_{+}^k (F)) < 1$, we have that
$sup \ HD(\Omega_{+}^k(G)) =1$, where the supremum is taken on all
asymptotically small perturbations $G$ of $F$. Nevertheless:
\begin{thm}\label{menor} Let $F \in Ra+Rb+On$ be the homogeneous random walk generated by the pair $(f,\psi)$.
Consider $M=\int \psi d\mu$, where $\mu$ is the unique
absolutely continuous invariant measure of $f$.
\begin{itemize}
\item[-] If $M >0$ then for all
asymptotically small perturbations $G$ of $F$ we have $m
(\Omega_{+}(G))> 0$.\\
\item[-] If $M =0$ then for all
asymptotically small perturbations $G$ of $F$ we have $HD
(\Omega_{+}(G))=1$ but $m
(\Omega_{+}(G))= 0$.\\
\item[-] If $M < 0$ then for all
asymptotically small perturbations $G$ of $F$ we have $HD
(\Omega_{+}(G))< 1$.
\end{itemize}
\end{thm}
\begin{rem}{\rm Since the authors are more familiar with deterministic rather than stochastic terminology, we stated and proved the results in this work for determinist random walks. However we believe that the above results could be easily translated to the theory of chains with complete connections (g-measures, chains of infinite order) and one-side shifts on an infinite alphabet. }
\end{rem}
\subsection{Applications to renormalization
theory of one-dimensional maps}
\begin{thm}\label{apl1} Let $f$ and $g$ be unimodal maps which are infinitely
renormalizable with the same bounded combinatorial type and even
critical order. Then the continuous conjugacy $h$ between $f$ and
$g$ is a strongly quasisymmetric mapping with respect to a certain
stochastic basis of intervals $\mathcal{P}$.\end{thm}
The set of intervals $\mathcal{P}$ is defined using a map induced
by $f$. See the details in Section \ref{apl}.
Let $\mathcal{F}_d$ be the class of
analytic maps with schwarzian negative derivative which are infinitely renormalizable in the Fibonacci sense with even critical order $d$ (see
Section \ref{aplf} for definitions). If $f$ is a Fibonacci map, denote by $J_{\mathbb{R}}(f)$ the maximal invariant set of $f$. Let $\mathcal{F}_d^{uni}$ be the class of Fibonacci {\it unimodal} maps with negative Schwarzian derivative.
\begin{thm}[Metric Universality]\label{juliathm} For each even critical order $d$, one of the
following statements holds:
\begin{itemize}
\item $HD(J_{\mathbb{R}}(f)) < 1$, for all $f \in \mathcal{F}_d$.
\item $HD(J_{\mathbb{R}}(f))= 1$ and $m(J_{\mathbb{R}})=0$ for all $f \in
\mathcal{F}_d$.
\item $HD(J_{\mathbb{R}}(f))= 1$ and $f$ has a wild attractor (in
particular, $m(J_{\mathbb{R}}(f))> 0$) for all $f \in \mathcal{F}_d$
\end{itemize}
\end{thm}
\begin{thm}[Measurable Deep Point]\label{deep} Let $f \in \mathcal{F}_d$, and assume that $0$ is its critical point. If $J_{\mathbb{R}}(f)$ has positive Lebesgue measure then there exists $\alpha > 0$ and $C > 0$ so that
$$m(x \in (-\delta,\delta)\colon \ x \not\in J_{\mathbb{R}}(f))\leq C\delta^{1+\alpha}.$$
\end{thm}
\begin{rem} {\rm Indeed $\alpha$ can be taken depending only on $d$. }\end{rem}
\begin{thm}\label{apl2}For each even critical order $d$, the following statements are
equivalent:
\begin{enumerate}
\item There exists $f \in \mathcal{F}_d$ such that $m(J_{\mathbb{R}}(F)) > 0$.
\item There exists $f \in \mathcal{F}_d$ with a wild attractor.
\item There exist maps $f,g \in \mathcal{F}_d^{uni}$ which are
conjugated by a continuous absolutely continuous maps $h$, but $f$ has a periodic point $p$ whose eigenvalue is different from the eigenvalue of the periodic point $h(p)$ of $g$. \item All maps in
$\mathcal{F}_d$ have wild attractors. \item All maps in
$\mathcal{F}_d^{uni}$ can be conjugated with each other by an
absolutely continuous conjugacy.
\end{enumerate}
\end{thm}
\section{Preliminaries}
\subsection{Probabilistic tools.}
We are going to collect here a handful of probabilistic tools which are going to be useful along the article. A good reference for these results is \cite{broise}.
Most of the probabilistic results in dynamical systems (large deviation, central limit theorem) assumes the observable $\psi$ is quite regular: usual regularity assumptions are either Holder continuity or bounded variation. Fix $f \in Mk+BD$. We are interested in $\mathcal{P}^0$-measurable observables with integer values which do not have such properties. Fortunally this is almost true: Denote by $ \mathcal{O}(f)$ the class of $\mathcal{P}^0$-measurable functions $\psi\colon I \rightarrow \mathbb{Z}$ so that
\begin{itemize}
\item[ ] \
\item[-] $\psi \in L^2(\mu)$,\\
\item[-] If $P$ denotes the Perron-Frobenius-Ruelle operator of $f$, then $P\psi$ has bounded variation.\\
\end{itemize}
For instance, if $(f,\psi) \in Mk+ sBD+Ra+Rb+GD$ then $\psi \in \mathcal{O}(f)$. Let $\mu$ be the absolutely continuous invariant measure of a Markov map $f$ and let $\psi\colon I \rightarrow \mathbb{R}$ be a measurable function.
\begin{prop}[Large Deviations Theorem \cite{broise}]\label{ldt} For every $\psi \in \mathcal{O}(f)$ and $\epsilon > 0$ there exists $\gamma \in (0,1)$ so that
$$\mu(\{x \in I \colon |\frac{1}{n} \sum_{i=0}^{n-1}\psi(f^i(x)) - \int \psi d\mu|\geq \epsilon \}) \leq \gamma^n$$
\end{prop}
Up to simple modifications in the proofs in \cite{broise}, we have
\begin{prop}[Proposition 6.1 of \cite{broise}] For every $\psi \in \mathcal{O}(f)$ the limit
$$\sigma^2 := \lim_{n\rightarrow \infty} \int \left( \frac{1}{\sqrt{n}} \sum_{k=0}^{n-1}\psi(f^k(x)) \right) ^2 d\mu$$ exists. Furthermore $\sigma^2=0$ if and only if there exists a function $\alpha \in L^2(\mu)$ so that $$\psi = \alpha\circ f - \alpha. $$
\end{prop}
and
\begin{prop}[Central Limit Theorem: Theorem 8.1 in \cite{broise}]\label{clt}For every $\psi \in \mathcal{O}(f)$ so that $\sigma^2\neq 0$ we have \begin{equation}\label{cltt} sup_{\epsilon \in \mathbb{R}} \ | \mu( x \in I\colon \frac{\sum_{k=0}^{n-1}\psi(f^k(x))}{\sigma \sqrt{n}} \leq \
\epsilon) - \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\epsilon}
e^{-\frac{u^2}{2}} \ du | \leq \frac{C}{\sqrt{n}},\end{equation}
\end{prop}
Indeed we are going to see that the assumption $\sigma^2 \neq 0$ is very weak: to this end we need the following result:
\begin{prop}[Theorem 3.1 in \cite{ad}]\label{cocycle} Let $f\colon \cup_i I_i \rightarrow I$ be a map in Mk + BD + Ex + Ra + Rb. Let $\psi \colon \cup_i I_i \rightarrow \mathbb{S}^1$ be a $\mathcal{P}_0$-measurable function. If $$\psi = \frac{\alpha\circ f}{\alpha},$$ where $\alpha$ is measurable, then $\alpha$ is $\mathcal{P}^\star$-measurable, where $\mathcal{P}^\star$ is the finest partition of $I$ so that $f(I_i)$ is included in an atom of $\mathcal{P}^\star$ for each $i \in \Lambda$.
\end{prop}
\begin{prop}\label{cco} Let $\psi \colon \cup_i I_i \rightarrow \mathbb{Z}$ be a $\mathcal{P}^0$-measurable function. If $\psi = \alpha\circ f - \alpha$, where $\alpha$ is measurable, then $\alpha$ is constant on $f(I_i)$, for each $i \in \Lambda$.
\end{prop}
\begin{proof} Note that we can assume that $\alpha(x) \in \mathbb{Z}$, for every $x$. Indeed, the relation $\psi = \alpha\circ f - \alpha$ implies that the function $\beta(x)=\alpha(x) \mod 1$ is $f$-invariant, so we can replace $\alpha$ by $\alpha -\beta$, if necessary. Fix an irrational number $\gamma$. Then
$$e^{2\pi \gamma \psi(x)i} = \frac{e^{2\pi \gamma \alpha(f(x))i}}{e^{2\pi \gamma \alpha(x)i}},$$
so by Proposition \ref{cocycle} we have that $e^{2\pi \gamma \alpha(x)i}$ is a $\mathcal{P}^\star$-measurable function. Since $j \in \mathbb{Z} \rightarrow e^{2\pi \gamma j i} \in \mathbb{S}^1$ is one-to-one, we get that $\alpha$ is $\mathcal{P}^\star$-measurable.
\end{proof}
A Markov map $f$ is almost onto if and only if $\mathcal{P}_0^\star= \{ I\}$, so
\begin{cor}On the conditions of Proposition \ref{cco}, if $f$ is almost onto then $\alpha$ is constant.\end{cor}
\begin{cor} For every nonconstant $\psi \in \mathcal{O}(f)$ we have that $\sigma^2 \neq 0$. In particular the Central Limit Theorem as given in Eq. (\ref{cltt}) holds for every non-constant $\psi$.\end{cor}
Let $\mathcal{A}_0 \subset \mathcal{A}_1 \subset \mathcal{A}_2 \subset \dots $ be an increasing sequence of $\sigma$-subalgebras of a probability space $(\Omega,\mathcal{A},\mu)$. A {\bf martingale difference sequence } is a sequence of functions $\psi_n\colon \Omega \rightarrow \mathbb{R}$, where $\psi_n$ is $\mathcal{A}_n$-measurable for $n\geq 1$, so that
$$\mathbb{E}(\psi_n | \mathcal{A}_{n-1}) =0$$
for every $n$. Here $\mathbb{E}(\psi | \mathcal{B})$ denotes de conditional expectation of $\psi$ relative to the sub-algebra $\mathcal{B}$. When $\mathcal{B}$ is generated by atoms $\{J_i \}_i$ then $\mathbb{E}(\psi | \mathcal{B})$ is the function defined as
$$\mathbb{E}(\psi | \mathcal{B})(x)= \frac{1}{\mu(J_i)} \int_{J_i} \psi \ d\mu$$
for every $x \in J_i$.
The following Proposition is the classic Azuma-Hoeffding inequality: see, for instance Exercise E14.2 in \cite{williams}:
\begin{prop}[Azuma-Hoeffding inequality]\label{azuma} Let $\psi_n$ as above and furthermore assume that
$$|| \psi_i ||_{\infty} = c_i < \infty.$$
Define
$$\psi:= \sum_{i=1}^{n} \psi_i.$$
Then
$$\mu(x \in \Omega\colon \ |\psi - \mathbb{E}(\psi)| > t ) \leq 2\exp ({-\frac{t^2}{2\sum_{i=1}^{n} c_i^2}}).$$
\end{prop}
\subsection{How to construct asymptotically small perturbations.}
As we will see in the next Proposition, it is easy to construct asymptotically small perturbations of a random walk:
\begin{prop}\label{identity} \label{how}Let $F$ and $G$ be random walks satisfying the properties $LI$, $Ex$, $sBD$, $Ra$ and $Rb$, where $G$ is a topological perturbation of $F$. Assume that there exist $C >0$ and $\lambda \in (0,1)$ with the following properties: if $I^n_j$ is as in properties $Ra$ and $Rb$, then
\begin{itemize}
\item[ ] \
\item[i.] For every $I^n_j \in \mathcal{P}^0_n$ we have
$$| log \frac{|I^n_{j+1}|}{|I^n_j|}\frac{|H(I^n_j)|}{|H(I^n_{j+1})|}| \leq C\lambda^{|n|+|j|}.$$\\
\item[ii.] For every $J \in \mathcal{P}^0_n$ we have
$$ |\tau_{J}^F - \tau_{H(J)}^G|_{C^2} \leq C\lambda^{|n|}.$$\\
\item[iii.] If $I_i^n=[a_i^n,b_i^n]$ then
$$\max_i \max \{|a_i^n - H(a_i^n)|, |b_i^n - H(b_i^n)| \} \leq C\lambda^{|n|}.$$\\
\item[iv.] Either $\psi$ is a bounded funtion or $\psi$ has a lower bound and $F = G$ on $\cup_{n<0}I_n$.
\end{itemize}
Then $G$ is an asymptotically small perturbation of $F$. Furthermore there exist $\beta \in [0,1)$ and $C > 0$ so that
$$|H(p)-p|\leq C\beta^{|\pi_2(p)|}.$$
\end{prop}
\begin{proof}
We will assume that $\psi$ is bounded: the other case is analogous. Consider $(x,n) \in I\times \mathbb{Z}$ and $(y,n)=H(x,n)$. Denote $(x_i,n_i):=F^i(x,n)$, $(y_i,n_i):=G^i(y,n)$.
Denote $\delta_i = |y_i-x_i|$ and $\tilde{\delta}_i=|A_{G(H(J_i))}(y_i)-A_{F(J_i)}(x_i)|$. Here $(x_i,n_i)~\in~J_i \in~\mathcal{P}^0$. It is easy to conclude, using iii. and property $LI$, that
\begin{equation}\label{til} \tilde{\delta}_i \leq \frac{\delta_i}{|F(J)|} + C\lambda^{|n_i|}\end{equation}
and making use of ii. to get
$$|\tau_{H(J)}^G(A_{G(H(J_i))}(y_i))-\tau_J^F(A_{F(J_i)}(x_i))| \leq D\tau^F_J(z_i) \frac{\delta_i}{|F(J)|} +C\lambda^{|n_i|}. $$ Here $z_i \in [0,1]$. Since $D\tau^F_J(z_i)|F(J)|/|J|\leq \lambda$ (property $Ex$), we get, using again $iii.$ \begin{equation} \label{rec} \delta_{i-1} \leq \lambda \delta_i + C\lambda^{|n_i|}.\end{equation}
Because $\psi$ is bounded, $|n_{i+1}-n_i|\leq B=\max |\psi|$. So if $i< n/2B$ then $|n_i| > |n_0|/2$. Since $\delta_{[\frac{n}{2B}]} \leq 1$, Eq. (\ref{rec}) implies
$$|H(x,n)-(x,n)|=|y_0-x_0|\leq C\lambda^{\frac{|n|}{2}}.$$
In particular, by Eq. (\ref{til}) and property ii., we have
\begin{equation} \label{normalizado} |D\tau_{H(J)}^G(A_{G(H(J_0))}(y_1))-D\tau_J^F(A_{F(J_0)}(x_1))|\leq C\lambda^{\frac{|n|}{2}}.\end{equation}
By $Ra+Rb$ there exists $\theta \in (0,1)$ so that
\begin{equation}\label{lowerb} \theta^{|i|}\leq |I^n_i|.\end{equation}
Let $i$ so that $J=I^n_i$.
{\it Case A.} $|i|\geq |n/2|(\log \lambda/\log \theta)$: Due i. and iii. and property $Ra$, there exists $C > 0$ so that
$$ |\log \frac{|H(I^n_i)|}{|I^n_i|}| \leq C \lambda^n.$$ Together with $sBD+LI$ and $iii.$, this implies that for every $p \in I^n_i$, with $|i|\geq |n/2|(\log \lambda/\log \theta)$, we have
$$|\log \frac{DG(H(p))}{DF(p)}| \leq C\lambda^{\frac{|n|}{2}\frac{\log \lambda}{\log \theta}}.$$
{\it Case B.} $|i| < |n/2|(\log \lambda/\log \theta)$: In this case, by iii. and Eq. (\ref{lowerb}) we have
$$\log \frac{|H(I^n_i)|}{|I^n_i|} \leq C\frac{|H(b^n_i)-b^n_i| + |H(a^n_i)-a^n_i|}{|b^n_i - a^n_i|} \leq C\lambda^{\frac{|n|}{2}}. $$
Now using Eq. (\ref{normalizado}) we can easilly obtain
$$|\log \frac{DG(H(p))}{DF(p)}|\leq C\lambda^{\frac{|n|}{2}}.$$
\end{proof}
\section{Stability of transience}\label{sectiontransience}
We will begin this section with the large deviations result to strongly transient random walks:
\begin{proof}[\bf Proof of Proposition \ref{largedeviationsst}] Fix $\epsilon > 0$ small. We intend to apply the Azuma-Hoeffding inequality, but since $\psi$ is not necessarily bounded, we need to make some adjustments first: Fix $P \in \mathcal{P}^0(F)$ and define $\mathcal{F}_0:=\{P\}$ and $\mathcal{F}_n:=\{Q\}_{Q\subset P,\ Q \in \mathcal{P}^n(F)}$. Since $F\in GD$, by the usual distortion control tricks for $F$, we can find $M > \min \psi$ such that $\alpha(x):= \min \{\psi(x),M \}$ satisfies
\begin{equation}\label{perturbacao} \mathbb{E}(\alpha\circ F^n | \mathcal{F}_{n-1}) \geq K-\epsilon/4 \end{equation}
for every $n\geq 1$. Here we are considering conditional expectations relative to the probability
$$\mu_P(A):= \frac{m(A)}{|P|},$$
where $m$ is the Lebesgue measure.
Define the martingale difference sequence
$$\Psi_n:= \alpha\circ F^n - \mathbb{E}(\alpha\circ F^n | \mathcal{F}_{n-1}).$$
Of course $||\Psi_n||_\infty \leq M$, if $M$ is large enough. By the Azuma-Hoeffding inequality we have
$$m( p \in P \colon \ |\sum_{i=1}^{n} \Psi_i(p) | > t ) \leq 2 \exp( -\frac{t^2}{2nM^2})|P|.$$
Taking $t=\epsilon n/4$ we obtain
\begin{equation}\label{dcorr} m( p \in P \colon \ |\sum_{i=1}^{n} \Psi_i(p) | > \frac{\epsilon}{4} \ n ) \leq 2 \exp( -\frac{\epsilon^2 n}{32M^2})|P|.\end{equation}
Since
$$\pi_2(F^{n+1}p)-\pi_2(F(p))= \sum_{i=1}^n \psi(F^i(p)) \geq \sum_{i=1}^{n} \alpha(F^i(p)) = \sum_{i=1}^n \Psi_i(p) + \sum_{i=1}^n \mathbb{E}(\alpha\circ F^i|\mathcal{F}_{i-1})(x)$$
$$\geq \sum_{i=1}^n \Psi_i(p) + (K-\epsilon/4) n.$$
Due Eq. (\ref{dcorr}), this implies that
$$m( p \in P \colon \ \pi_2(F^{n}p)-\pi_2(F(p))=\sum_{i=1}^{n-1} \psi(F^i(p)) < (K -\epsilon/2 ) \ (n -1)) \leq C_1 \exp( -\frac{\epsilon^2 n}{32M^2}) |P|.$$
Let $n_0$ be such that $\min \psi > -\epsilon (n_0-1)/2 -\epsilon + K$. Then for $n\geq n_0$ we have that $$\pi_2(F^{n}p)-\pi_2(p) < (K-\epsilon) n$$ implies $$\pi_2(F^{n}p)-\pi_2(F(p)) < (K -\epsilon/2 ) \ (n -1).$$ So
$$m( p \in P \colon \ \pi_2(F^{n}p)-\pi_2(p) < (K -\epsilon ) \ n) \leq C_2 \exp( -\frac{\epsilon^2 n}{32M^2}) |P|$$
for every $n$.
This completes the proof.
\end{proof}
\begin{prop}\label{prws}\label{homstr} Let $F$ be either strongly transient or a homogeneous random walk with positive mean drift. Then any asymptotically small perturbation $G$ of $F$ has the following property: there exists $\lambda \in [0,1)$, $C > 0$ and $\tilde{K} > 0$ so that for every $P \in \mathcal{P}^0(G)$
$$m(p \in P \colon \ \sum_{i=0}^{n-1}\psi(G^i(p)) < \tilde{K} n )\leq C\lambda^n |P|.$$
In particular $G$ is also transient.
\end{prop}
\begin{proof}
We will carry out the proof assuming the strongly transience: the homogeneous case is analogous:
Fix $\epsilon > 0$. Let $\tilde{\delta}_1 > 0$ be small enough such that
$$(1-\tilde{\delta_1})(K-\epsilon) + \tilde{\delta_1} \min \psi> K-2\epsilon.$$
Due the bounded distortion of $G$, there exists $\delta_1 > 0$ such that for every $n\geq 1$ and every $P \in \mathcal{P}^{n-1}(G)$, interval $Q \subset G^n(P)$, and set $A\subset Q$ satisfying
$$\frac{m(A)}{m(Q)} \geq 1 -\delta_1$$
we have
\begin{equation}\label{distg}\frac{m(P\cap G^{-n}A)}{m(P\cap G^{-n}Q)} \geq 1 -\tilde{\delta}_1.\end{equation}
By Proposition \ref{largedeviationsst} we have
\begin{equation}\label{unpum} m( p \in P \colon \ \sum_{i=0}^{n-1} \psi(F^i(p)) < (K -\epsilon) n\ for \ some \ n\geq n_0) \leq C_1
\exp( -C_2n_0)|P|.\end{equation}
Since $G$ is an asymptotically small perturbation, Eq. (\ref{asymp}) implies that
\begin{equation}\label{uv} m( p \in P \colon \ \sum_{i=0}^{n-1} \psi(G^i(p)) < (K -\epsilon) n\ for \ some \ n\geq n_0) \leq C_3
\exp( -C_4n_0) |P|\end{equation}
provided that $P \in \mathcal{P}^0_j$, $j \geq 2\ |\min \psi|\ n_0$. In particular there exists $n_0=n_0(\delta_1)$ such that for every $P \in \mathcal{P}^0_j$, $j \geq 2\ |\min \psi|\ n_0$, we have
\begin{equation} \label{rato} m(\tilde{\Omega}_P)\geq (1-\delta_1)|P|,\end{equation}
where
$$\tilde{\Omega}_P:= \{
p \in P \colon \ \pi_2(G^n(p))\geq |\min \psi|\ n_0 \ for \ all \ n\geq 0 \ and \ \pi_2(G^n(p))-\pi_2(p)\geq (K -\epsilon)n \ for \ n\geq n_0 \}.$$
By the $GD$ condition, there exists $n_1$ such that for $n \geq n_1$ we have
$$m(p \in P\colon \ there \ exists \ i\leq n \ s.t. \ \psi(F^i(p))\geq n)\leq \frac{\delta_1}{4} $$
By Eq (\ref{unpum}) there exists $n_2>n_1$ such that
\begin{equation}\label{unpdois} m( p \in P \colon \ \sum_{i=0}^{n_2-1} \psi(F^i(p)) > (K -\epsilon) n_2) \geq (1-\frac{\delta_1}{4})|P|.\end{equation}
So
\begin{equation}\label{unptres} m( p \in P \colon \ \sum_{i=0}^{n_2-1} \psi(F^i(p)) > (K -\epsilon) n_2 \ and \ \psi(F^i(p))< n_2 \ for \ every \ i\leq n_2) \geq (1-\frac{\delta_1}{2})|P|.\end{equation}
Note that for $p$ in the set in Eq (\ref{unptres}) we have $\pi_2(G^i(p))-\pi_2(p)\leq (n_2)^2$ for every $i\leq n_2$. Since $G$ is an asymptotically small perturbation of $F$, this observation and Eq. (\ref{unptres}) implies that there exists $n_3 >> (n_2)^2$ such that for $P \in \mathcal{P}^0_j$, with $j\leq - n_3$, we have
\begin{equation} m( p \in P \colon \ \sum_{i=0}^{n_2-1} \psi(G^i(p)) > (K -\epsilon) n_2 \ and \ \psi(G^i(p))< n_2 \ for \ every \ i\leq n_2) \geq (1-\delta_1)|P|.\end{equation}
So
\begin{equation}\label{pum} m( p \in P \colon \ \sum_{i=0}^{n_2-1} \psi(G^i(p)) > (K -\epsilon) n_2) \geq (1-\delta_1)|P|.\end{equation}
{\it Claim $A$:} Almost every point $x \in I\times \{j \}$, $j\leq -n_3$, visits at least once (and consequently infinitely many times) the set
\begin{equation}\label{tend} \bigcup_{j\geq-n_3} I\times \{j \}\end{equation}
Indeed, define a new random walk $\tilde{G}\colon I\times \mathbb{Z}\rightarrow I\times \mathbb{Z}$
$$\tilde{G}(x,n):=(\tilde{g}_n(x),n+\tilde{\psi}(x,n))$$
in the following way. Let $T$ be an integer larger than $n_2(K-\epsilon)$. If $n \geq -n_3$ then define $\tilde{g}_n\colon I \rightarrow I$ as an affine expanding map, onto on each element of $\mathcal{P}_n^{n_2}$, and $\tilde{\psi}(x,n)=T$.
For $(x,n)$, with $n< -n_3$, define $\tilde{G}(x,n)=G^{n_2}(x,n)$. In this case
$$\tilde{\psi}(x,n)=\sum_{i=0}^{n_2-1} \psi(G^i(x,n)).$$
It is not difficult to see that the $\tilde{G}$-orbit of a point $(x,n)$, with $n< -n_3$, visits the set in Eq. (\ref{tend}) at least once then the $G$-orbit of $(x,n)$ visits the same set at least once.
To prove the claim, it is enough to show that $\tilde{G}$ is strongly transient. Indeed, let $P$ be an element of the Markov partition $\mathcal{P}^{k-1}_j(\tilde{G})$. If $\pi_2(\tilde{G}^i(P))\geq -n_3$, for some $i\leq k$ then $\pi_2(\tilde{G}^{k}(P))\geq -n_3$, so
\begin{equation} \label{stest1}\frac{1}{|P|} \int_P \tilde{\psi}\circ \tilde{G}^k \ dm= \frac{1}{|P|} \int_P T \ dm \geq (K-\epsilon)n_2.\end{equation}
Otherwise $\pi_2(\tilde{G}^i(P))< -n_3$ for every $i\leq k$. In particular $\tilde{G}^i=G^{i n_2}$ on $P$, for every $i\leq k$. Note that $$\tilde{G}^{k}P= \bigcup_i Q_i,$$
where $Q_i \in \mathcal{P}^0_j(G)$ (this is a consequence of the Markovian property of $G$), with $j < -n_3$. By Eq. (\ref{pum}) we have
$$m(q \in Q_i\colon \tilde{\psi}(q) \geq (K-\epsilon)n_2)\geq(1-\delta_1)|Q_i|,$$
so by the distortion control in Eq. (\ref{distg}) we obtain
$$m(p \in P\cap \tilde{G}^{-k} Q_i\colon \tilde{\psi}(\tilde{G}^kp) \geq (K-\epsilon)n_2)\geq(1-\tilde{\delta}_1)|P\cap \tilde{G}^{-k} Q_i|,$$
consequently
\begin{equation}\label{stest2} \int_P \tilde{\psi}\circ \tilde{G}^k \ dm=\sum_i \int_{P\cap \tilde{G}^{-k} Q_i} \tilde{\psi}\circ \tilde{G}^k \ dm \end{equation}
$$ \geq \sum_i
((1-\tilde{\delta}_1)(K-\epsilon)n_2 + \tilde{\delta}_1 n_2 \min \psi) |P\cap \tilde{G}^{-k} Q_i| $$
$$\geq \sum_i (K -2\epsilon)n_2 |P\cap \tilde{G}^{-k} Q_i|= (K -2\epsilon)n_2 |P|$$
Eq. (\ref{stest1}) and (\ref{stest2}) imply that $\tilde{G}$ is strongly transient, so by Proposition \ref{largedeviationsst}, $\tilde{G}$ is transient. This concludes the proof of the claim.
{\it Claim $B$:} The $G$-orbit of almost every point of $I\times \mathbb{Z}$ eventually arrives at $\tilde{\Omega}_P$, for some $P \in \mathcal{P}^0_j$, with $j> 2|\min \psi|n_0$.
Since $F$ is transient and $G$ is topologically conjugate to $F$ the set
$$\Omega:= \{ p \colon \ -n_3\leq \pi_2(p) \leq 2|\min \psi| n_0 \ and \ \lim_n \pi_2(G^n(p))=+\infty\}$$
is dense on
$$\bigcup_{j=-n_3}^{2|\min \psi| n_0 } I\times \{j\}.$$
This implies that for every non-empty open set $O \subset I_j$, with $-n_3\leq j \leq 2|\min \psi| n_0$ we have
\begin{equation}\label{meio}
m((x,j) \in O \colon \exists \ k\geq 0 \ s.t. \ G^k(x,j) \in \tilde{\Omega}_P, \ with \ P \in \mathcal{P}^0_q(G), \ q > 2|\min \psi|n_0)> 0.
\end{equation}
Indeed, pick a point $p \in O\cap \Omega$. By property $Ex$ and the definition of $\Omega$, there exists $k$ and $Q \in \mathcal{P}_j^k(G)$ such that $Q \subset O$, $P=G^k(Q)\in \mathcal{P}^0_q$, with $q> 2|\min \psi|n_0$. By Eq. (\ref{rato}) we have $m(\tilde{\Omega}_P)> 0$, so
$$m(O\cap G^{-k}\tilde{\Omega}_P)\geq m(Q\cap G^{-k}\tilde{\Omega}_P) > 0.$$
By the property $LI$, Eq. (\ref{meio}) and (\ref{rato}), and the bounded distortion of the iterations of $G$, it follows that there exists $\delta_3 > 0$ such that for every $i$ and every $Q \in \mathcal{P}^{i-1}(G)$ such that $\pi_2(G^iQ)\geq -n_3$ we have that
\begin{equation}\label{ubc} m(p \in Q\colon \ \exists k\geq 0 \ s.t. \ G^kp \in \tilde{\Omega}_P, \ with \ P \in \mathcal{P}^0_q(G), \ q > 2|\min \psi|n_0)\geq \delta_3 |Q|
\end{equation}
We will show Claim $B$ by contradiction. Suppose that it does not hold. Then there is a set $W$ of positive measure whose $G$-orbit of its elements never hits $\tilde{\Omega}_P$ for any $P\in \mathcal{P}^0_j$, with $j\geq 2 |\min \psi| n_0$. Pick a Lebesgue density point $p$ of $W$ whose $G$-orbit visits
$$\bigcap_{j=-n_3}^{2|\min \psi|n_0} I\times\{j\}$$
infinitely many times, which is possible due Claim A. In particular there exists a sequence $Q_k \in \mathcal{P}^{n_k-1}(G)$ such that $|Q_k|\rightarrow_n 0$, $p \in Q_k$, $\pi_2(G^{n_k}Q_k)\geq -n_3$ and
$$\lim_k \frac{m(Q_k\cap W)}{|Q_k|}=1.$$
That contradicts Eq. (\ref{ubc}). This concludes the proof of Claim $B$.
Note that Claim $B$ implies the following: almost every point in $I \times \{j\}$ belongs to the set
$$\Lambda_j := \bigcup_{k\geq 0} \Lambda_j^k,$$
where $$\Lambda_j^k:= \{p \in I \times \{j\}\colon \pi_2(G^n(p))-\pi_2(G^k(p))\geq (K-\epsilon)(n-k), \ for \ every \ n\geq k +n_0 \}.$$
Let $k_0$ be large enough such that for every $-n_3\leq j\leq 2|\min \psi|n_0$ we have
$$m(A\cap \bigcup_{k\leq k_0} \Lambda_j^k) \geq (1-\delta_1) |A|$$
for every interval $A\subset I\times \{j\}$ satisfying $|A|\geq \delta$, where $\delta >0$ is as in the property $LI$. Pick $n_4$ satisfying $n_4\geq k_0+n_0$ and
$$n_4 > \frac{-k_0\min \psi}{\epsilon}-k_0.$$
It is easy to see that if $p \in \bigcup_{k\leq k_0} \Lambda_j^k$ then
$$\pi_2(G^{n_4}p)-\pi_2(p)=\sum_{i=0}^{n_4-1}\psi(G^ip)\geq (K-2\epsilon)n_4.$$
In a argument similar to the proof of Claim $A$, consider the random walk $\hat{G}$ defined in the following way: if $\pi_2(p)\leq -n_3$ define $\hat{G}(p)=G^{n_2}$. If $\pi_2(p)\geq 2|\min \psi|n_0$ define $\hat{G}(p)=G^{n_0}$. Finaly if $-n_3< \pi_2(p)< 2|\min \psi|n_0$ define $\hat{G}(p)=G^{n_4}$. The random walk $\hat{G}$ is $3\hat{K}$-strongly transient, for some $\hat{K} > 0$. The proof is quite similar to the proof of the strong transience of $\tilde{G}$, so we let it to the reader. So $\hat{G}$ is transient. It is easy to see that this implies that $G$ is transient. Finally Proposition \ref{bru} implies that
$$m( p \in P \colon \ \pi_2(\hat{G}^n(p))-\pi_2(p) < 2\hat{K} n )\leq C\hat{\lambda}^n |P|,$$
for some $\hat{\lambda} \in (0,1)$, which implies
$$m(Y^n_P)\leq C\hat{\lambda}^n |P|,$$
where
$$Y^n_P:=\{p \in P \colon \ \exists \ m \geq n \ s.t. \ \pi_2(\hat{G}^m(p))-\pi_2(p) < 2\hat{K} m\}.$$
Let $n_5=\max\{n_0,n_4,n_2\}$. Let $p \in P$ be such that
$$ \pi_2(G^i(p))-\pi_2(p) < \frac{\hat{K}}{n_5} i.$$
There exists $m$ such that $\hat{G}^m(p)=G^j(p)$, with $i \geq j$, $|i-j|\leq n_5$. Note that $$m\leq i\leq j + n_5\leq (m+1)n_5,$$
so we can find $i_0$ such that for every $i\geq i_0$ we have
$$ \frac{-n_5 \min \psi}{m} + \hat{K}\frac{m+1}{m}< 2\hat{K}.$$
So
$$\pi_2(\hat{G}^m(p))-\pi_2(p)= \pi_2(G^j(p))- \pi_2(G^i(p))+ \pi_2(G^i(p))- \pi_2(p)$$
$$\leq - n_5 \min \psi + \frac{\hat{K}}{n_5} i\leq- n_5 \min \psi + \hat{K}(m+1)< 2\hat{K} m, $$
where $$m\geq \frac{i}{n_5}-1.$$
This implies
$$\{ p \in P\colon \pi_2(G^i(p))-\pi_2(p) < \frac{\hat{K}}{n_5} i \}\subset Y^{ \frac{i}{n_5}-1}_P,$$
so
$$m( p \in P\colon \pi_2(G^i(p))-\pi_2(p) < \frac{\hat{K}}{n_5} i )\leq C\hat{\lambda}^{i/n_5}|P|$$
This completes the proof.
\end{proof}
Let $n > 0$ and $j$ be integers and $F$ be a deterministic random walk. Then any connected component $C$ of $F^{-n}
\ int \ I_j$ is called a {\bf cylinder}. The {\bf lenght} $\ell(C)$ of the cylinder $C$ is $n$. If $C$ is a cylinder of lenght $n$ so that $F^i(C) \subset I_{j_i}$, for $i<n$, we will denote $C=C(j_0,j_1,\dots,j_n)$.
\begin{prop}\label{comeco} Let $F$ be a random walk induced by the pair
$(\{f_i\},\psi)$. Assume that there exists $\epsilon > 0$ so that
for $K
> 0$, we have
$$m(\{ p \in I_n \colon \psi(p) < -K\}) \leq \frac{1}{K^{2+\epsilon}},$$
provided $n \geq n_0$. Then
$$\lim_k m( \{p \in I_{n_k}\colon \text{ there exists } i \leq k^2
\text{ so that } \psi(F^i(p)) < -k\})=0,$$ uniformly for all
sequence satisfying $n_k
> k^3 + n_0$.
\end{prop}
\begin{rem} For a homogeneous random walk, the condition on $\psi$
is equivalent to $1_{I_0}\cdot\psi \in
L^{2+\epsilon}(m)$.\end{rem}
Let $F$ and $G$ be random walks which are topologically
conjugated by a homeomorphism $h$ that preserves states. For any $p
\in I \times \mathbb{Z}$ define
$$dist_i(p):= \big|\log \frac{DG^i(h(p))}{ DF^i(p)}\big|$$
and
$$C_p:= \sup_{i\geq 0} dist_i(p).$$
For each $n_0 \in \mathbb{Z}\cup\{-\infty\}$ define
$$\Omega_{n_0+}(F):= \{p \colon \pi_2(F^n(p))\geq n_0, \text{ for
all } n \geq n_0\}.$$
In particular $\Omega_{-\infty+}(F)=I\times \mathbb{Z}$.
\begin{prop}\label{abs} Let $F$ and $G$ be random walks which are conjugated
by a homeomorphism $h$ which preserves states. Suppose that there
exists a $F$-forward invariant set $\Lambda$ so that
\begin{itemize}
\item[ ] \
\item[\it -H1:] $C_p:=\sup_{i\geq 0} \ dist_i(p) < \infty$, for each
$p \in \Lambda$.\\
\end{itemize}
then $h$ is absolutely continuous on $\cup_i F^{-i}\Lambda$ and
$h^{-1}$ is absolutely continuous on $\cup_i G^{-i}h(\Lambda)$.
Furthermore, if
\begin{itemize}
\item[ ] \
\item[\it -H2:] There exists $C > 0$, $M > 0$ and $n_0 \in \mathbb{Z}\cup\{-\infty\}$ so that for every $n\geq n_0$ with $n \in \mathbb{Z}$ and $P \in \mathcal{P}^0_n$,
$$m(p \in P \cap \Lambda \colon \ C_p \leq C ) \geq M |P|.$$
\end{itemize}
then $h$ is absolutely continuous on $\cup_i
F^{-i}(\Omega_{n_0+}(F))$ and $h^{-1}$ is absolutely continuous on
$\cup_i G^{-i}(\Omega_{n_0+}(G))$. In particular when $n_0=-\infty$ we have that $h$ and $h^{-1}$ are absolutely continuous on $I\times \mathbb{Z}$.
\end{prop}
\begin{proof}For each $j \in \mathbb{N}$ denote
$$\Lambda_j:= \{ p \in \Lambda \colon \sup_i \ dist_i(p) \leq
j\}.$$ Note that $\Lambda_i$ is forward invariant.
We claim that $h$ is absolutely continuous on $\Lambda_j$ and
$h^{-1}$ is absolutely continuous on $h(\Lambda_j)$. Indeed, for
each $p \in \Lambda_j$ and $k \in \mathbb{N}$, denote
$F^ip=(x_i,n_i)$. Denote by $J_k(x) \in \mathcal{P}^k$ the unique interval which
contains $x$ so that $F^k$ maps $J_k(x)$ diffeomorphically onto
$Q_k \subset I_{n_k}$. There is some ambiguity here if $x$ is in the boundary
of $J_k(x)$, but these points are countable, so they are
irrelevant for us.
If we use the analogous notation to $h(x)$ and $G$, we have
$h(J_k(x))=J_k(h(x))$ and, due the bounded distortion property of
the random walks $F$ and $G$, there exist $C_1, C_2
> 0$ such that
$$C_1 e^{-dist_k(p)} \leq \frac{|h(J_k(x))|}{|J_k(x)|} \leq C_2
e^{dist_k(p)}.$$ So, if $p \in \Lambda_j$ then
\begin{equation}\label{dis} C_1e^{-j} \leq \frac{|h(J_k(x))|}{|J_k(x)|} \leq
C_2 e^j, \ \text{ for all } k \in \mathbb{N}.\end{equation}
Let $A \subset \Lambda_j$ be a set with positive Lebesgue measure.
We claim that $h(A)$ also has positive Lebesgue measure. Indeed,
choose a compact set $K \subset A$ with positive Lebesgue measure.
Denote $U_k:= \cup_{x \in K} J_k(x)$. Since $|J_k(x)|\leq
\lambda^k$, we have that $\lim_k m(U_k)=m(K)$ and $\lim_k
m(h(U_k))=m(h(K))$. Since $U_k$ is a countable disjoint union of
intervals of the type $J_k(x)$, by Eq. (\ref{dis})
\begin{equation}\label{dist2}
C_1e^{-j} \leq \frac{m(h(U_k))}{m(U_k)} \leq C_2 e^j, \ so \
C_1e^{-j}\leq \frac{m(h(K))}{m(K)} \leq C_2
e^j,\end{equation} and we conclude that $h(K)$ also has positive
Lebesgue measure. An identical argument shows that, if $A \in
\Lambda_j$ has positive Lebesgue measure, then $h^{-1}A$ also has
positive Lebesgue measure. The proof of the claim is finished and
so $h$ and $h^{-1}$ are absolutely continuous on $\Lambda=\cup_j
\Lambda_j$ and $h(\Lambda)=\cup_j h(\Lambda_j)$.
Now it is easy to conclude that $h$ and $h^{-1}$ are absolutely continuous on
$\cup_i F^{-i}\Lambda$ and $\cup_i G^{-i}h(\Lambda)$.
Now assume $H2$. We claim that $\cup_i F^{-i}\Lambda$ has full
Lebesgue measure on $\Omega_{n_0+}(F)$. Indeed, Assume that $m(
\Omega_{n_0+}(F)\setminus \cup_i F^{-i}\Lambda) > 0$ and choose
a Lebesgue density point $p$ of this set. Then
$$ \lim_k \frac{m(J_k(p)\cap \Omega_{n_0+}(F)\setminus \cup_i F^{-i}\Lambda )}{|J_k(x)|} =1.$$
Due the bounded distortion of $F$, if $F^k(p)=(x_k,n_k)$ and $F^k(J_k(x))=Q_k \subset I_{n_k}$, with $n_k\geq n_0$, where $Q_k$ is a union of intervals in $\mathcal{P}^0_{n_k}$, then
$$ \limsup_k \frac{m(Q_k\cap \Lambda)}{|Q_k|}
\leq C(1- \liminf_k \frac{m(J_k(x)\cap \Omega_{n_0+}(F)\setminus
\cup_i F^{-i}\Lambda)}{|J_k(x)|}) =0,$$ which contradicts H2.
Since $dist_k(p)$ is uniformly bounded with respect to $k$ and $p$ on the set $\{p \in P\cap \Lambda \colon \ C_p \leq C \}$, we
can use an argument identical to the proof of Eq. (\ref{dist2}) to
conclude that
$$\frac{m(p \in P\cap \Lambda \colon \ C_p \leq C )}
{m(h(p) \in h(P)\cap h(\Lambda) \colon \ C_p \leq C )}\leq
C_1,$$ so $m(h(P\cap \Lambda\colon C_p\leq C)) \geq \tilde{C}M|h(P)|$, for all $P \in \mathcal{P}^0_n$, $n\geq n_o$ and
using an argument as above, we conclude that $\cup_i
G^{-i}h(\Lambda)$ has full Lebesgue measure on $\Omega_{n_0+}(G)$.
Since $h$ ($h^{-1}$) is absolutely continuous on $\cup_i
F^{-i}\Lambda$ ($\cup_i G^{-i}h(\Lambda)$) and
$$m(\Omega_{n_0+}(F)\setminus \cup_i
F^{-i}\Lambda)=m(h(\Omega_{n_0+}(F)\setminus \cup_i
F^{-i}\Lambda))=m(\Omega_{n_0+}(G)\setminus \cup_i
G^{-i}h(\Lambda))=0,$$
we have that $h$ and $h^{-1}$ are absolutely continuous on
$\Omega_{n_0+}(F)$ and $\Omega_{n_0+}(G)$. Now it is easy to prove that $h$ is absolutely continuous on $\cup_i F^{-i}\Omega_{n_0+}(F)$ and $h^{-1}$ is absolutely continuous on $\cup_i G^{-i}\Omega_{n_0+}(G)$.
\end{proof}
\begin{proof}[{\bf Proof of Theorem \ref{sttr}}] By Proposition \ref{homstr}, $G$ is transient. In particular for all $n_0 \in \mathbb{Z}$ the sets
$$\cup_i F^{-i}\Omega_{n_0+}(F) \text{ and } \cup_i G^{-i}\Omega_{n_0+}(G)$$ have full Lebesgue measure. So by Proposition
\ref{abs}, to prove that $h$ and $h^{-1}$ are absolutelly continuous, it is enough to find a forward invariant set satisfying
the assumptions H1 and H2 for some $n_0 \in \mathbb{Z}$. Indeed, fix $\delta > 0$ (we will choose $\delta$ latter).
Consider the $F$-forward invariant set
$$\Lambda =\Lambda_\delta := \{ p \ \colon \
\liminf_k \frac{\pi_2(F^k(p))-\pi_2(p)}{k} \geq \frac{\delta}{3} \}.$$
We claim that $\Lambda$ satisfies H1. Indeed take $x \in \Lambda$.
Then, for $k \geq k_0(x)$ we have $n_k:=\pi_2(F^k(p)) \geq k
\delta/4$. So
\begin{equation} \label{distcont} dist_k(x)\leq \sum_{i=0}^{k-1} |\log \frac{DF(F^{i+1}(p))}{ DG(h(F^{i+1}(p)))}| \end{equation}
$$\leq \sum_{i=0}^{k_0-1} |\log \frac{ DF(F^{i+1}(p))}{ DG(h(F^{i+1}(p)))}| +\sum_{i=k_0}^{k-1} |\log \frac{ DF(F^{i+1}(p))}{
DG(h(F^{i+1}(p)))}|$$
$$\leq \sum_{i=0}^{k_0-1} |\log \frac{ DF(F^{i+1}(p))}{ DG(h(F^{i+1}(p)))}| +
\sum_{i=k_0}^{k-1} \lambda^{n_i}$$
$$\leq \sum_{i=0}^{k_0-1} |\log \frac{ DF(F^{i+1}(p))}{ DG(h(F^{i+1}(p)))}| +
\sum_{i=k_0}^{\infty} \lambda^{i\delta/4 }$$
$$ \leq K_p + C(\delta).$$
To prove that $\Lambda$ satisfies H2, By Proposition \ref{largedeviationsst} for each $P \in \mathcal{P}^0_i$ we have
\begin{equation}\label{estexp} m( p \in P \colon \ \pi_2(F^k(p))-\pi_2(p) < \delta k )\leq C\lambda^k |P|,\end{equation}
provided $\delta$ is small enough. From Eq. (\ref{estexp}) we obtain
\begin{equation}\label{estr2} \mu( p \in P \colon \ \pi_2(F^n(p))-\pi_2(p) \geq \delta n \text{ for all } n\geq n_0 )\geq (1-C\lambda^{n_0}) |P|.\end{equation}
In particular, we have that, for every $n$, \begin{equation}\label{estest} \pi_2(F^n(p))\geq \delta (n-n_0) + \pi_2(p)+ n_0 \min \psi.\end{equation} in the set in Eq. (\ref{estr2}). Using the same argument as in Eq. (\ref{distcont}) we can easily obtain $H2$ from Eq. (\ref{estest}) and Eq. (\ref{estr2}), choosing $n_0$ large enough.
\end{proof}
\begin{proof}[{\bf Proof of Theorem \ref{sttrho}}] Observe that using the argument in the proof of Proposition \ref{homstr}, an induced map of a homogeneous random walk with positive drift is strongly transient. From this the proof of Theorem \ref{sttrho} goes exactly as the Theorem \ref{sttr}. \end{proof}
\begin{proof}[{\bf Proof of Theorem \ref{sqr}}] By Proposition \ref{prws},
for every $i$ we have
$$m( p \in I_i \ \colon \ \frac{\pi_2(F^k(p))-\pi_2(p)}{k} \leq \delta
) \leq C\theta^{k}.$$
and furthermore $\theta:=\theta(\delta)$ tends to $0$ when $\delta$ tends to
zero. Using an argument as in the proof of Theorem \ref{sttr} we can conclude that
\begin{equation} \label{equsei} m( p \in I_i \ \colon \ \frac{\pi_2(F^k(p))-\pi_2(p)}{k} \geq \delta
\ for \ k \geq k_0) \geq 1- C\theta^{k_0}\end{equation}
In particular we can use the argument in the proof of Theorem \ref{sttrho} to conclude that the conjugacy $h$ is absolutely continuous. Indeed, Eq. (\ref{equsei}) implies
\begin{equation} \label{bdinf} m( p \in I_i \ \colon \ dist_{k}(x) \geq \delta n+C \ for \ some \ k)\leq C\theta^n.\end{equation}
where $\delta= \sup_p dist_1(p)$. Firstly we will prove Theorem \ref{sqr} when $\delta$ is small.
Denote $\Lambda_1:= \{ p \in I_i \ \colon h'(x) \leq 1\}$ and, for $n \geq 1$
$$\Lambda_n:= \{ p \in I_i \ \colon \ {e}^{\delta(n-1)} < h'(x) \leq {e}^{\delta n}\}.$$ By Eq. (\ref{bdinf}) we have $m(\Lambda_n)\leq C\theta^n$.
Let $B \subset I_i$ be an arbitrary Lebesgue measurable set. Let $k_1$ be so that
$$ \theta^{k_1+1}< |B|\leq \theta^{k_1}.$$
Since $h$ is absolutely continuous we have
$$|h(B)|= \int_B h'\ dm$$
$$ = \sum_{n=0}^{k_1} \int_{B\cap \Lambda_n} h' \ dm + \sum_{n=k_1+1}^{\infty} \int_{B\cap \Lambda_n} h' \ dm$$
$$ \leq \sum_{n=0}^{k_1} C\theta^{k_1}e^{\delta n} + \sum_{n=k_1+1}^{\infty} C(e^{\delta}\theta)^n$$
$$ \leq C (e^{\delta}\theta)^{k_1}\leq C |B|^{1+ \frac{\delta}{\ln \theta}}.$$
Now if $B \subset J \in \mathcal{P}^n$ and $F^n(J)=Q \subset I_i$, with $|Q|\geq C$ (due Property LI), then due the bounded distortion of $F$
$$\frac{|h(B)|}{|h(J)|}\leq C \frac{|h(F^n(B)|}{|h(Q)|} \leq C \Big( \frac{ |F^n(B)|}{|Q|} \Big)^{1+ \frac{\delta}{\ln \theta}}\leq C\Big( \frac{ |B|}{|J|} \Big)^{1+ \frac{\delta}{\ln \theta}}.$$
To prove a similar inequality to $h^{-1}$, define
$$\tilde{\Lambda}_n:= \{ p \in I_i \ \colon \ {e}^{\delta(n-1)} < (h^{-1})'(x) \leq {e}^{\delta n}\}.$$
of course
$$h^{-1}\tilde{\Lambda}_n = \{ p \in I_i \ \colon \ {e}^{-\delta(n)} < h'(x) \leq {e}^{-\delta (n-1)}\},$$
so by Eq. (\ref{bdinf}) we obtain
$$m(h^{-1}\tilde{\Lambda}_n)\leq \theta^n.$$
In particular
$$m(\tilde{\Lambda}_n) = \int_{h^{-1}\tilde{\Lambda}_n} h'(x) \ dm \leq (e^{-\delta}\theta)^n$$
Note that this argument gives us an exponential upper bound even if $\delta$ is large.
Now we can switch the roles of $F$ and $G$ to obtain the inequality to $h^{-1}$,
which shows that $h$ is a mSQS-homeomorphism relative to the stochastic basis $\cup_n \mathcal{P}^n$.
To complete the proof when $\delta$ is not small do the following: find a continuous path of random walks $F_t$ so that $F_0=F$ and $F_1=G$, so that for every $t \in [0,1]$ we have that $F_t$ is a asymptotically small perturbation of $F$. By the argument above for every $t \in [0,1]$ there exists $\epsilon_t$ so that $F_{\tilde{t}}$ is mSQS-conjugated to $F_t$, provided $|\tilde{t}-t|\leq \epsilon_t$. Using the compactness of $[0,1]$ we can find a finite sequence of random walks $F_{t_0}=F, F_{t_1}, F_{t_2}, \dots F_{t_n}=G$ so that $F_{t_i}$ and $F_{t_{i+1}}$ are conjugated by a map $h_i$ which is mSQS with respect some dynamically defined stochastic basis. Composing these conjugacies we find a mSQS-conjugacy between $F$ and $G$.
\end{proof}
\section{Stability of recurrence}
To avoid a cumbersome notation, in this section we make the convention that all inequalities holds only for large $n$.
Moreover in this section we assume that $\psi$ is unbounded. Recall that in this case we assume that asymptotically small perturbations $G$ coincides with $F$ on negative states. The case where $\psi$ is bounded is similar.
The following is a easy consequence of the Central Limit Theorem for Birkhoff
sums (Proposition \ref{clt})
\begin{cor}\label{co}Let $a_n$ be a positive increasing sequence. Then
$$\mu( \frac{|S_n|}{\sqrt{n}} > a_n) \leq Ce^{-\frac{a_n^2}{2}} +
C\frac{1}{\sqrt{n}}.$$
\end{cor}
\begin{proof} Use Proposition \ref{clt} and and note that the estimative $$\int_{-\infty}^v
e^{-\frac{u^2}{2}} \ du \leq C e^{-\frac{v^2}{2} }$$
holds for $v << 0$.
\end{proof}
Given $n \in \mathbb{N}$, split $[0,2n]\cap \mathbb{N}$ in $\sqrt{\log n}$ blocks
(called main blocks) , denoted $B_j$, with length
$$\frac{n}{\log^{8j}n }, \ j=1,\dots, \sqrt{\log n},$$
and between the main blocks we put little blocks $H_j$, called
holes, of length $\log^4n$. These holes will warranty the
independence between the events in distinct main blocks. Put these
blocks in the following order:
$$\dots < B_{j+1} < H_{j+1} <
B_j < H_{j}< \dots,$$ with $min \ B_{\sqrt{\log n}} = 0.$
Note that we let most of the second half of the interval $[0,2n]\cap \mathbb{N}$
uncovered.
Define
$$S(j) = \sum_{i \in B_j} \psi\circ f^i$$
$$H(j) = \sum_{i \in H_j} \psi\circ f^i$$
Denote $|B_j|:= \max B_j - \min B_j$.
\begin{lem}\label{muitogrande} We have
$$\mu(\sum_{i=0}^{|B_j|} \psi\circ f^i \geq \frac{\sqrt{n}}{\log^{4j} n} \log^3n) \leq C \frac{\log^{4j} n}{\sqrt{n}}.$$
\end{lem}
\begin{proof} This follows from Corollary \ref{co}.
\end{proof}
\begin{prop}\label{aux1} For every $\epsilon > 0$ we have
$$\mu(S(j) > \frac{\sqrt{n}}{\log^{4j} n} \log^3n, \ for \ some \
j \leq \sqrt{\log n}) \leq C\frac{1}{\sqrt[2+\epsilon]{n}},$$
provided $n$ is large enough.
\end{prop}
\begin{proof} For $j \leq \sqrt{\log n}$ define
$$ \Lambda_j:=\{x \in I\colon \ S(j)(x) > \frac{\sqrt{n}}{\log^{4j} n} \log^3n \}$$
$$ = \{x \in I\colon \sum_{i < |B_j|} \psi\circ f^{i + \min B_j }(x) > \frac{\sqrt{n}}{\log^{4j} n} \log^3n \} $$and for each $P \in \mathcal{P}^{\min B_j}$ denote $\Lambda_j(P):= \Lambda_j\cap P$.
Due Lemma \ref{muitogrande} and the bounded distortion of $f^{\min B_j}$ on $P$ we have
$$m(\Lambda_j(P)) \leq C \frac{\log^{4j} n}{\sqrt{n}}|P|.$$
Summing on $j$ and $P$
$$m(\bigcup_{{ j }}\bigcup_{{ P }} \Lambda_j(P)) \leq \sqrt{\log n} \ \frac{\log^{4j} n}{\sqrt{n}} << C\frac{1}{\sqrt[2+\epsilon]{n}}.$$
\end{proof}
\begin{prop}\label{aux2} For every $\epsilon > 0$ and $d >0$ we have
\begin{equation}\label{mais} \mu(|\sum_{i \in H_j} \psi(f^i(x))| > \log^8 n, \ for \ some \
j \leq \sqrt{\log n}) \leq C\frac{1}{n^d},\end{equation} provided $n$
is large enough.
\end{prop}
\begin{proof} For $i \in H_j-1$, with $j \leq \sqrt{\log n}$, define
$$\Lambda_{i}:=\{x \in I\colon |\psi(f^i(x))| > \log^4 n. \}.$$
By expanding and bounded distortion properties of $f$ and condition $GD$ we have
that
$$\mu(\Lambda_i)\leq C\lambda^{\log^4 n}.$$
Since $|H_j|= \log^4 n$, if $x$ belongs to the set in Eq. (\ref{mais}) then $x \in \Lambda_{i}$, for some $i \in H_j-1$, with $j \leq \sqrt{\log n}$. So
$$\mu(|\sum_{i \in H_j} \psi(f^i(x))| > \log^8 n, \ for \ some \
j \leq \sqrt{\log n})$$
$$ \leq \mu(\bigcup_{{ j \leq \sqrt{\log n}}} \ \bigcup_{{ i \in H_j-1}} \Lambda_j)$$
$$\leq \sqrt{\log n}\ \log^4 n \ n^{\log \lambda \log^3n}$$
$$<< \frac{1}{n^d}, $$
where the least inequality holds for $n$ large enough.
\end{proof}
\begin{prop}[Independence between distant events] \label{inde}There exists $\lambda < 1$
so that the following holds: For all cylinders $C_1$ and $C_2$, we
have
$$\mu(C_1\cap f^{-(n+d)}C_2) = \mu(C_1
)\mu(C_2)(1+ O(\lambda^{d})).$$ Here $n=|C_1|$.
\end{prop}
\begin{proof}
Let $J$ be an interval in $C_1$ so that $f^n(J)=I$. Define the
measure $\rho(A):= \mu(f^{-n}A\cap J)/\mu(J)$. Note that by the
bounded distortion property of $f$, we have that $\log d\rho/dm$
is $\alpha$-Holder, where $\alpha$ does not depend on $n$.
Furthermore it is bounded by above by a constant which does not
depend on $n$. By the well-know theory of Perron-Frobenius-Ruelle
operators for Markov expanding maps, if $P$ is the
Perron-Frobenius-Ruelle operator of $f$, then there exists $\lambda
< 1$ so that
$$ P^d\frac{d\rho}{dm} = (1+ O(\lambda^d))\frac{d\mu}{dm}.$$
So
$$\frac{\mu(J\cap f^{-(n+d)}C_2)}{\mu(J)}$$
$$=\rho(f^{-d}C_2)= \int 1_{C_2}\circ f^d \ \frac{d\rho}{dm} dm $$
$$=\int 1_{C_2} \ P^d \frac{d\rho}{dm} dm$$
$$=(1 + O(\lambda^d))\int 1_{C_2} \ \frac{d\mu}{dm} dm$$
$$=(1 + O(\lambda^d))\mu(C_2).$$
Since $C_1$ is a disjoint union of intervals $J$ so that
$f^{n}J=I$, we finished the proof.
\end{proof}
\begin{cor}\label{aux3} There exists $M > 0$ so that
$$\mu( S_j < \frac{\sqrt{n}}{\log^{4j} n}\ M \text{ for all } j \leq
\sqrt{\log n}) \leq C\big( \frac{2}{3} \big)^{\sqrt{\log n}}$$
\end{cor}
\begin{proof}Choose $M >0 $ so that
$$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{M}
e^{-\frac{u^2}{2}} \ du < \frac{2}{3}$$
Consider the disjoint union of cylinders
$$C_j:= \{ x \ s.t. \ \sum_0^{|B_j|} \psi\circ f^i(x) < \frac{\sqrt{n}}{\log^{4j}
n}\ M\}.$$
The Central Limit Theorem tells us that if $n$ is large enough then
$$\mu(C_j) < \frac{2}{3}$$
for every $j \leq \sqrt{\log n}$.
Recall that between $B_j$ and $B_{j+1}$ there is a hole
with length $\log^4 n$. Applying $\sqrt{\log n}$ times Proposition
\ref{inde} , we obtain
$$\mu( S_j < \frac{\sqrt{n}}{\log^{4j} n}\ M \text{ for all } j \leq
\sqrt{\log n}) \leq \big( \frac{2}{3} \big)^{\sqrt{\log n}} (1 +
O(\lambda^{\log^4 n}))^{\sqrt{\log n}} \leq C\big( \frac{2}{3}
\big)^{\sqrt{\log n}}$$
\end{proof}
\begin{prop}\label{sei} There exists $C > 0$ so that for every $k$,
$$\mu( x \in I_k \colon \text{ there exists } i < \ell^3 \text{ so that }
\sum_{k=0}^{i} \psi\circ f^k(x) > \frac{\ell}{2}) \geq 1-
C\big(
\frac{2}{3} \big)^{\sqrt{3\log \ell}}$$
\end{prop}
\begin{proof} Let $M$ be as in Proposition . Denote $n = \ell^3$ and define
$$A_\ell:= \{x \colon \text{ there exists } i < \ell^3 \text{ so
that } \sum_{k=0}^{i} \psi\circ f^k(x) > \frac{\ell}{2}\},$$
$$B_\ell:= \{ x \colon \ |S_j(x)| < \frac{\sqrt{n}}{\log^{4j} n}
\log^3n, \text{ for all } j \leq \sqrt{\log n}\},$$
$$C_\ell:= \{x \colon S_j(x) \geq \frac{\sqrt{n}}{\log^{4j} n} \ M, \text{ for some } j \leq
\sqrt{\log n} \},$$
$$D_\ell:= \{x \colon |H_j(x)| \leq \log^8 n, \text{ for all }
j \leq \sqrt{\log n} \}.$$ We claim that if $\ell$ is large then
$B_\ell\cap C_\ell \cap D_\ell \subset A_\ell$. Indeed, let $x \in
B_\ell\cap C_\ell \cap D_\ell$. Then for some $j_0 \leq \sqrt{\log
n}$,
$$S(j_0) \geq \frac{\sqrt{n}}{\log^{4j_0} n} \ M.$$
We claim that, if $m = max \ B_{j_0}$, then
$$\sum_{0}^m \psi\circ f^i(x) > \frac{\ell}{2}.$$
Indeed, since $x \in
D_\ell$,
$$| \sum_{i\in H_{j}, \ j > j_0} \psi\circ f^i(x)| \leq \sqrt{\log n}
\log^8 n = o(\ell).$$ Moreover, since $ x \in B_\ell$,
$$| \sum_{i \in B_{j}, \ j > j_0} \psi\circ f^i(x)| \leq \sum_{j > j_0}
\frac{\sqrt{n}}{\log^{4j} n } \log^3 n \leq C \frac{\sqrt{n}}{\log^{4j_0 +4} n}. $$
So
$$\sum_{0}^m \psi\circ f^i(x) = \sum_{i \in B_{j_0}} \psi\circ
f^i(x) + \sum_{i \in B_{j}, \ j > j_0} \psi\circ f^i(x) + \sum_{i
\in H_{j}, \ j > j_0} \psi\circ f^i(x)$$
$$ \geq \big(M- \frac{C}{\log^4 n}\big) \frac{\sqrt{n}}{\log^{4j_0} n} + o(\ell) > C\ell - o(\ell)> \frac{\ell}{2},$$
and we finished the proof of the claim. To finish the proof, note that by Proposition \ref{aux1}, Corollary \ref{aux3} and Proposition \ref{aux2} $$\mu(A_\ell) \geq \mu(B_\ell\cap C_\ell \cap D_\ell) \geq 1-
C\frac{1}{\sqrt[2+\epsilon]{n}}-C\big(
\frac{2}{3} \big)^{\sqrt{\log n}} - C\frac{1}{n^d} \geq 1- C\big(
\frac{2}{3} \big)^{\sqrt{\log n}}.$$
\end{proof}
\begin{prop} There exist $\epsilon$ and $D$ so that for every $\ell \geq 0$,
$$\mu (\{ x \in I_\ell \colon \text{ there exists } i
\text{ so that } F^i(p) \in \bigcup_{t \in [\min \psi, -\min \psi]} I_t \text { and } dist_i(p) \leq D \}) \geq
\epsilon$$
\end{prop}
\begin{proof} Define, for $p \in C(i_0,i_1,\dots,i_{n-1})$,
$$Dist_n(p):= sup_{q \in C(i_0,i_1,\dots,i_{n-1})} \ dist_n(q).$$
We are going to prove by induction on $k$ that there is $C > 0$ so
that, if we define
$$B_k^{\ell}:= \{ p \in I_\ell \colon \text{ there exists } j \leq \sum_{i=0}^{k-1}
\frac{\ell^3}{2^{3i}}\text{ such that } \pi_2(F^j(p)) \leq
\frac{\ell}{2^k} \text{ and } Dist_j(p) \leq \sum_{i=0}^{k-1}
\frac{\ell^3}{2^{3i}}\theta^{\frac{\ell}{2^i}}\},$$ then
\begin{equation} \label{mest} \mu(B_k^{\ell}) \geq \prod_{i=0}^{k-1} \left( 1-
C\big(
\frac{2}{3} \big)^{\sqrt{\log \frac{\ell}{2^i}}} \right). \end{equation}
Indeed, take $p \in B_k^\ell$. Let $p \in
L=C(i_0,i_1,\dots,i_{j-1})$, where $j$ is as in the definition of
$B_k^\ell$. Note that $L \subset B_k^\ell$ and $F^n(L)=I_r$, for
some $r < \ell/2^k$. By Proposition \ref{sei},
\begin{equation}\label{oest} \mu( x \in I_r \colon \text{ there exists } i <
\frac{\ell^3}{2^{3k}} \text{ so that } \sum_{k=0}^{i} \psi\circ
f^k(x) > \frac{\ell}{2^{k+1}})\end{equation} $$ \geq 1-
C\big(
\frac{2}{3} \big)^{\sqrt{\log \frac{\ell}{2^k}}}.$$ Denote
$$D_L: = \{ x \in I_\ell\cap L \colon \text{ there exists } i <
\frac{\ell^3}{2^{3k}} \text{ so that } \sum_{k=0}^{i} \psi\circ
f^k(f^j(x)) > \frac{\ell}{2^{k+1}} \}$$ Due the bounded
distortion property for $F$, the estimative in Eq. (\ref{oest})
implies
\begin{equation} \label{estimative} \frac{\mu(D_L)}{|L|} \geq 1-
C\big(
\frac{2}{3} \big)^{\sqrt{\log \frac{\ell}{2^k}}}.\end{equation} For $x \in
D_L$ take the smallest $i$ so that $$\sum_{k=0}^{i}
\psi\circ f^k(f^j(x)) > \frac{\ell}{2^{k+1}}.$$
Then $\pi_2(F^{j+h}(p)) \geq \frac{\ell}{2^{k+1}}$, for every $0
\leq h < i$, so
$$Dist_i(F^{j}(p)) \leq \sum_{h=0}^{i} \theta^{\pi(F^{j+h}(p))}
\leq \frac{\ell^3}{2^{3k}} \theta^{\frac{\ell}{2^{k+1}}}.$$
So $D_L \subset B^{\ell}_{k+1}$. Since $B^{\ell}_{k}$ is a
disjoint union of cylinders $L$, the estimative in Eq.
(\ref{estimative}) implies Eq. (\ref{mest}).
Define
$$D:= \sum_{i=0}^{\infty}
\frac{\ell^3}{2^{3i}}\theta^{\frac{\ell}{2^i}} < \infty.$$
Let $k$
be so that $ 2^k \leq \ell \leq 2^{k+1}$. Now it is easy to check
that
$$\mu (\{ x \in I_\ell \colon \text{ there exists } i
\text{ so that } F^i(p) \in I_0 \text { and } dist_i(p) \leq D \})$$
$$ \geq C\mu(B^{\ell}_k) \geq \prod_{i=0}^{k-1} \left( 1-
C\big(\frac{2}{3} \big)^{\sqrt{\log \frac{\ell}{2^i}}} \right) \geq C
\prod_{i=0}^{k-1} \left( 1- C\big(\frac{2}{3} \big)^{\sqrt{\log \frac{2^k}{2^i}}}
\right)$$ $$ \geq \exp( -C\sum_{i=1}^{\infty} \big(\frac{2}{3} \big)^{\sqrt{i\log 2}}) >
\tilde{C}
> 0,$$
which finishes the proof.
\end{proof}
\begin{proof}{\bf Proof of the Stability of Recurrence (Theorem \ref{strec})} Because $G$ coincides with $F$ on negative states, and $F$ is recurrent, of course the orbit by $G$ of almost every point $p$ so that $\pi_2(p) < 0$ will entry
$$\cup_{i\geq 0} I^i.$$
So it is enough to prove that the orbit by $G$ of almost every point $p \in \cup_{i\geq 0} I^i$ hit $I^0$.
Let $\ell \geq 0$.
By the previous Proposition, there exist $D~>~0$ and $\epsilon >
0$ so that
$$A_\ell:= \{ p \in I_\ell \colon \text{ there exists } i \text{ so that } F^i(p)
\in \bigcup_{t =\min \psi}^{ -\min \psi} I_t \text{ and } Dist_i(p) < D \}$$ satisfies $\mu(A_\ell) >
\epsilon$, for all $\ell$.
Consider a cylinder $C_F=C_F(\ell,k_1,\dots,k_{i-1},0) \subset
A_\ell$, satisfying $k_j \neq 0$ for $0< j < i$ and $Dist_i(x) <
D$, for every $x \in C_F$. We claim that that corresponding
cylinder $C_G=C_G(\ell,k_1,\dots,k_{i-1},0)$ for the perturbed
random walk $G$ satisfies
$$\frac{1}{C} \leq \frac{|C_G|}{|C_F|} \leq C,$$
where $C$ depends only on $D$. Since $A_\ell$ is a disjoint union
of cylinders of this type, we obtain that $B_\ell = H(A_\ell)$
satisfies $m(B_\ell) > C\epsilon > 0$, for all $\ell$.
To prove that the set of points whose orbits returns infinitely
many times to $$\bigcup_{t = \min \psi}^{ -\min \psi} I_t$$ has full Lebesgue measure, it is enough to
prove that $\Lambda:=\cup_{j>0, \ell } G^{-j}B_\ell$ has full
Lebesgue measure.
Indeed, assume by contradiction that $\Lambda$ is not full. Choose
a Lebesgue density point $p$ of the complement of $\Lambda$. Then
there exist a sequence of cylinders
$C_k~=~C_G(\ell_0,\ell_1,\dots,\ell_k)$ so that $p \in C_k$ and
\begin{equation} \label{conv} \frac{m(C_k\setminus \Lambda)}{|C_k|} \rightarrow_k
1. \end{equation}
But $G^k(C_k)=I_{\ell_k}$, and $m(I_{\ell_k} \cap B_{\ell_k}) \geq
C\epsilon |I_{\ell_k}|$. By the bounded distortion property
$$\frac{m(\Lambda \cap C_k)}{|C_k|} > \frac{m(G^{-k}B_{\ell_k} \cap C_k)}{|C_k|}
> \tilde{C} \epsilon,$$
which contradicts Eq. (\ref{conv}). Now we can use that $G$ is transitive and has bounded distortion to prove that $G$ is recurrent.
\end{proof}
\begin{proof}[\bf Proof of Proposition \ref{rigidity}] Since $F$ is recurrent, almost every point of $I^0$ returns to $I^0$ at least once. So the first return map $R_F\colon I^0 \rightarrow I^0$ is defined almost everywhere is $I^0$ and the same can be said about $R_G$. Of course, the absolutely continuous conjucagy $H$ also cojugates the expanding Markovian maps $R_F$ and $R_G$. Using the same argument used in Shub and Sullivan \cite{ss} and Martens and de Melo \cite{mm}, we can prove that $H$ is actually $C^1$ on $I^0$. Using the dynamics, it is easy to prove that $H$ is $C^1$ everywhere.
\end{proof}
\section{Stability of the multifractal spectrum}
\subsection{Dynamical defined intervals and root cylinders} When we are dealing with Markov expanding maps with {\em finite} Markov partitions, for each arbitrary interval $J$ we can find an element of $\cup_j \mathcal{P}^j$ which covers $J$ and has more or less the same size that $J$. Note that this is no longer true when the Markov partitions is infinite. Since coverings by intervals are crucial in the study of the Hausdorff dimension of an one-dimensional set, this trick is very useful to estimate the dimension of dynamically defined sets, once we can replace an arbitrary covering by intervals by another one with essentially the same metric properties but whose elements are themselves {\em dynamically defined} sets (cylinders).
Consider $j\geq 0$ and let $\{C_i\}_i \subset \mathcal{P}^j$ be a finite or countable family of cylinders $\{C_i\}_{i \in \Theta} \subset \mathcal{P}^j$ such that $W:=\bigcup_i \overline{C_i}$ is connected and $int \ W$ does not contain any point $d^n_i$ (as defined in property $Rb$). Then $W$ is called a dynamically defined interval (dd-interval, for short). Define the root cylinder of $W$ as the unique cylinder $C_{i_0}$ with the following property: if $\sharp \Theta =\infty$ then $W$ is a semi-open interval and $C_{i_0}$ will be the cylinder so that $\partial C_{i_0}\cap \partial W \neq \emptyset$. Otherwise $W$ is closed and let $C_{i_0}$ be the unique cylinder such that $F=\partial C_{i_0}\cap \partial W$ is the boundary of a semi-open dd-interval which contains $W$. The following Lemmas are an easy consequence of the regularity properties $Ra+Rb$ and it will be useful to recover the trick described above for (certain) infinite Markov partitions. The proof is very simple.
\begin{lem}\label{rcia} For every $d \in (0,1)$ there exists $K > 1$ so that for every dd-interval $W:=\cup_i \overline{C_i}$ with root cylinder $C_{i_0}$ we have
$$\frac{1}{K}\leq \frac{|W|^\alpha}{\sum_{i}|C_i|^\alpha} \leq K$$
$$\frac{1}{K}\leq \frac{|C_{i_0}|^\alpha}{\sum_{i}|C_i|^\alpha} \leq K$$
for every $\alpha \geq d$.
Indeed the constant $K$ depends only on $d$ and constants in the properties $Ra+Rb+Ex+BD$.
\end{lem}
\begin{lem}\label{rci} Let $N$ be as in Properties $Ra+Rb$. For every $d \in (0,1)$ there exists $K > 1$ so that the following holds: For every interval $J \subset I\times \mathbb{Z}$ there exists $m$ dd-intervals $W_j$, all of same level, with $m \leq 2N$, satisfying the following properties:
\begin{itemize}
\item[-] The interior of these dd-intervals are pairwise disjoint.
\item[-] The closure of the union of $W_j$ covers $J$:
$$J \subset \overline{\bigcup_j W_j}. $$
\item[-] We have
$$ \frac{1}{K} \leq \frac{\sum_{i=1}^m |W_i|^{\alpha}}{|J|^\alpha} \leq K$$
for every $\alpha > d$.
\end{itemize}
Indeed the constant $K$ depends only on $d$ and constants in the properties $Ra+Rb+Ex+BD$.\end{lem}
\subsection{Dimension of dynamically defined sets}
Let $f \in Mk+ BD + Ex$ and denote by $\mathcal{P}^0$ its Markov partition.
Let $$\mathcal{I}:= \{ C_i\}_i \subset \cup_i \mathcal{P}^n$$ be a finite or countable family of
disjoint cylinders. Define the induced Markov map $f_{\mathcal{I}}\colon
\cup_i C_i \rightarrow I$ by
$$f_{\mathcal{I}}(x) = f^{\ell(C_i)-1}(x), \ if \ x \in C_i.$$
We can also define an induced drift function
$\Psi\colon \cup_{i} C_i \rightarrow \mathbb{Z}$
in the following way: Define, for $x \in C \in \mathcal{P}^n_0$,
$$\Psi_{\mathcal{I}}(x):= \sum_{i=0}^{n-1} \psi(f^i(x)).$$
On the same conditions on $x$, define $N_{\mathcal{I}}(x)=n$.
The maximal invariant set of $f_{\mathcal{I}}$ is
$$\Lambda(\mathcal{I}):= \{x \in I \colon \ f^j(x) \in \bigcup_i C_i, \ for \ all \ j\geq 0 \}.$$
Denote by $HD(\mathcal{I})$ the Hausdorff dimension of the maximal
invariant set of $f_{\mathcal{I}}$.
We are going to use the following result
\begin{prop}[Theorem 1.1 in \cite{mu2}] We have
$$HD(\mathcal{J})= \sup \{HD(\mathcal{I})\colon \mathcal{I} \subset \mathcal{J}, \ \mathcal{I} \ finite \}.$$
\end{prop}
The following result was proved to Markov maps with finite Markov
partition, however the proof can be adapted to our case.
Before to give the proof of Proposition \ref{ms} we need to introduce some tools which
are useful to estimate the Hausdorff dimension.
Let $\mathcal{J}$ as above. If there exists $\beta$ such that
$$\sum_{C \in \mathcal{J}} |C|^\beta =1,$$ we will call $\beta$ the {\bf virtual
Hausdorff dimension } of $f_{\mathcal{I}}$, denoted
$VHD(\mathcal{I})$. The virtual Hausdorff dimension is a nice way
to estimate $HD(\mathcal{I})$: indeed if $f_{\mathcal{I}}$ is linear on each interval of the Markov partition then these values coincide. When the distortion is positive, these values remain related, as expressed in the following result (which is
included, for instance, in the proof of Theorem 3, Section 4.2 of
\cite{pt}):
\begin{prop}\label{vhd} Let $\mathcal{I}$ be a finite family of disjoint
cylinders. Then
$$|HD(\mathcal{I})-VHD(\mathcal{I})| \leq \frac{d}{\log \lambda -
d},$$ where $$d := \sup_{C \in \mathcal{I}} \sup_{x, y \in C} \log
\frac{Df_{\mathcal{I}}(y)}{Df_{\mathcal{I}}(x)} \text{ and }
\lambda := \inf_{C \in \mathcal{I}} \inf_{x \in C} |Df_{\mathcal{I}}|.$$
\end{prop}
Recall that if $\mathcal{I}$ is finite then $f_{\mathcal{I}}$ has an invariant probability measure $\mu_{\mathcal{I}}$ supported on its maximal invariant set $\Lambda(\mathcal{I})$ such that for any subset $S \subset \Lambda(\mathcal{I})$ satisfying $\mu_{\mathcal{I}}(S)=1$ we have $HD(S)=HD(\mathcal{I})$.
Note that for a homogeneous random walk $F$
$$\Omega_+^k(F)= \{k\}\times\{x \in I\ s.t. \sum_{i=0}^{j} \psi(f^j(x))+ k\geq 0, \ for \ j\geq 0 \}$$
and
$$\Omega_{+\beta}^k(F)=$$
$$\{k\}\times \{ x \in I\ s.t. \
\sum_{j=0}^{n-1}\psi(f^j(x)) + k \geq 0\, \ for \ all \ n\geq 0 \ and \ \underline{\lim}_{\ n} \frac{1}{n}\sum_{j=0}^{n-1}\psi(f^j(x)) \geq\beta\}.$$
Define $\pi_1(x,n):=x$. The following is an easy consequence of this observation:
\begin{lem}\label{aux} If $F$ is a homogeneous random walk then
$\pi_1(\Omega_{+}^0(F)) \subset \pi_1(\Omega_{+}^k(F))$ and $\pi_1(\Omega_{+\beta}^0(F)) \subset \pi_1(\Omega_{+\beta}^k(F))$, for all $k\geq 0$. Furthermore $$HD(\Omega_{+}^0(F)) =HD(\Omega_{+}^k(F))$$ and
$$HD(\Omega_{+\beta}^0(F)) =HD(\Omega_{+\beta}^k(F)).$$
\end{lem}
\begin{prop}\label{sim} Let $F$ be a homogeneous random walk.
Then there exists a sequence of finite families of cylinders $$\mathcal{\mathcal{F}}_s \subset \cup_i \mathcal{P}^i_0$$ so that
\begin{itemize}
\item[ ] \
\item[-]$\Lambda(\mathcal{F}_s) \subset \Omega_+^0(F),$\\
\item[-] Denote $\beta_n:= \int \Psi_{\mathcal{F}_s}\ d\mu_{\mathcal{F}_s }$. Then $\beta_n > 0$. \\
\item[-] $\lim_{s\rightarrow \infty} HD(\mathcal{F}_s)=HD(\Omega_+^0(F)).$ \\
\end{itemize}
\end{prop}
\begin{proof}
Denote $d=HD \ \Omega_{+}^0(F)$. Given any $s \in \mathbb{N}^\star$, $m_{d_s}
\ \Omega_{+}(F)=\infty$, where $d_s:=d(1-1/s)$. Here $m_{D}$ denotes the $D$-dimensional Haussdorf measure. By Theorem 5.4 in \cite{f}, for each positive number $M$ we can find a
compact subset $\Lambda_s \subset \Omega_{+}^0(F)$ satisfying $m_{d_s} \
\Lambda_s = M$. We may assume that $\Lambda_s$ does not have isolated points. We will specify $M$ later.
In particular, for each $\epsilon$ small enough the following holds:
\begin{itemize}
\item[]\ \\
\item[i.]{\em For every} family of intervals $\{ J_i \}_i$ which covers $\Lambda_s$, with $|J_i| < \epsilon$ we have
$$\frac{M}{2}\leq \sum_i |J_i|^{d_s}.$$
\item[ii.] {\em There exists } a family of intervals $\{ J_i \}_i$, with $|J_i|\leq
\epsilon$, which covers $\Lambda_s$ and
$$\sum_i |J_i|^{d_s} \leq 2M.$$
Furthermore we can assume that $\partial J_i \subset \Lambda_s$.\\
\end{itemize}
Assume that $d_s\geq d/2$. By Lemma \ref{rci}, there exists some $K$ such that we can replace the special covering $\{ J_i\}$ in ii. by a
new covering by dd-intervals $\{W_i^\ell\}_{i, \ \ell}$, with root cylinders $R_i^\ell$, where
\begin{equation}\label{p1} J_i\cap \Lambda_s \subset \overline{ \bigcup_\ell W_i^\ell},\end{equation}
\begin{equation}\label{p2} W_i^\ell:=\bigcup_k \overline{C^{i\ell}_k}, \ for \ each \ \ell \leq m_{i\ell} \leq 2N,\end{equation}
\begin{equation}\label{p3} \frac{1}{K} \leq \frac{\sum_\ell |R^{\ell}_i|^{d_s}}{|J_i|^{d_s}}\leq K,\end{equation}
\begin{equation}\label{p4} \frac{1}{K} \leq \frac{\sum_k |C^{i\ell}_k|^{d_s}}{|R^{\ell}_i|^{d_s}}\leq K,\end{equation}
Indeed we can replace $W_i^\ell$ by a dd-subinterval of it, if necessary, in such way that $R_i^\ell \cap \Lambda_s \neq \phi$ and Eq. (\ref{p1}), Eq. (\ref{p2}), Eq. (\ref{p3}) and Eq. (\ref{p4}) hold, except perhaps the lower bound in Eq. (\ref{p3}). The above estimates, together to the fact that $\{ W^\ell_i \}$ covers $\Lambda_s$ (up to a countable set) gives
\begin{equation} \label{eqsb}\frac{M}{2K^2}\leq \sum_{i,\ell,k} |C^{i\ell}_k|^{d_s} \leq 2K^2M.\end{equation}
Since these intervals are cylinders, if necessary we can replace this family of cylinders by a subfamily of disjoint cylinders which covers $\Lambda_s$ up to a countable number of points and such that each cylinder intersects $\Lambda_s$. Indeed we can choose a finite subfamily $\mathcal{F}_s:=\{C_r\}_r$ satisfying
\begin{equation} \label{eqsb1}\frac{M}{3K^2}\leq \sum_r |C_r|^{d_s} \leq 2K^2M.\end{equation}
Let's call this finite subfamily $\mathcal{F}_s$. Note that, since $C_r\cap \Lambda_s\neq \emptyset$ we have that
$$\sum_{t=0}^{\ell} \psi(f^t(x)) \geq 0$$ for every $x \in C_r$ and $\ell \leq \ell(C_r)$. Choose a very small cylinder $\tilde{C}$ such that
$$\sum_{t=0}^{\ell} \psi(f^t(x)) \geq 0$$ for every $x \in \tilde{C}$ and $\ell < \ell(\tilde{C})$, and moreover satisfying
$$\sum_{t=0}^{\ell(\tilde{C})} \psi(f^t(x)) > 0$$
on $\tilde{C}$, and
\begin{equation} \label{neqsb}\frac{M}{3K^2}\leq |\tilde{C}|^{d_s} + \sum_r |C_r|^{d_s} \leq 3K^2M.\end{equation}
Add $\tilde{C}$ to the family $\mathcal{F}_s$. Then, if $\mu_s$ is the geometric invariant measure of $f_{\mathcal{F}_s}$, we have
$$\int \Psi_{\mathcal{F}_s} \ d\mu_s > 0.$$
And by Lemma \ref{vhd} and Eq. (\ref{neqsb})
$$| HD(\Lambda(f_{\mathcal{F}_s}))- d_s|\leq -\frac{C}{\log \epsilon}.$$
Since $\epsilon$ can be taken arbitrary, we can choose $\mathcal{F}_s$ such that $$HD(\Lambda(f_{\mathcal{F}_s}))\rightarrow_s d.$$
\end{proof}
\begin{cor} If $F$ is a homogeneous random walk we have that
$$HD(\Omega_+(F))= \lim_{\beta\rightarrow 0^+} HD(\Omega_{+\beta}(F))=\sup_{\beta > 0} HD(\Omega_{+\beta}(F)).$$
\end{cor}
\begin{proof} Due Lemma \ref{aux}, it is enough to prove the Corollary for $k=0$. Of course $\Omega_{+\beta}^0(F)\subset \Omega_{+}^0(F)$ and $\beta_0 \leq \beta_1$ implies $\Omega_{+\beta_1}^0(F) \subset \Omega_{+\beta_0}^0(F)$, so
$$\lim_{\beta\rightarrow 0^+} HD(\Omega_{+\beta}^0(F))=\sup_{\beta > 0} HD(\Omega_{+\beta}^0(F)) \leq HD(\Omega_{+}^0(F)).$$
To obtain the opposite inequality, let $\mathcal{F}_s$ be as in Proposition \ref{sim}. Denote
$$\gamma_s := \int \Psi_{\mathcal{F}_s} \ d\mu_{\mathcal{F}_s}, \text{ and } W_n := \int N_{\mathcal{F}_s} \ d\mu_{\mathcal{I}}$$
and $\beta_s:= \gamma_s/W_s$. Then by the Birkhoff Ergodic Theorem there is subset $T_s \subset \Lambda(\mathcal{I}_n)$ such that $\mu_{\mathcal{F}_s}(T_s)=1$ and
$$\lim_k \frac{1}{k} \sum_{i=0}^{k-1} \psi(f^i(x)) = \lim_k \frac{ \sum_{j=0}^{k-1} \Psi_{\mathcal{I}_n}(f^j_{\mathcal{F}_s}(x))}{ \sum_{j=0}^{k-1} N_{\mathcal{I}_n}(f^j_{\mathcal{F}_s}(x))}=\frac{\gamma_s}{W_s}=\beta_s > 0.$$
for every $x \in T_s$. Since the Hausdorff dimension of $\mu_{\mathcal{F}_s}$ is equal to $HD(\mathcal{F}_s)$, we have that $HD(T_s)=HD(\mathcal{F}_s)$. Note also that
$$T_s \subset \Omega_{+\beta_s}^0,$$
which implies $HD(\mathcal{F}_s)\leq HD(\Omega_{+\beta_s}^0)$, so by the choice of $\mathcal{F}_s$, we conclude that
$$ HD(\Omega_{+}^0)= \lim_s \ HD(\mathcal{F}_s) \leq \overline{\lim}_s \ HD(\Omega_{+\beta_s}^0)\leq \sup_{\beta > 0} \ HD(\Omega_{+\beta}^0).$$
\end{proof}
\begin{proof}[{\bf Proof of Theorem \ref{multi}.}] Define
$$\Gamma_n(F) := \{ x \in \Omega_{+\beta}^k(F) \ s.t. \ \pi_2(F^i(x,k))\geq \frac{\beta}{2} i, \ for \ all \ i\geq n\}.$$
Of course $$\Omega_{+\beta}^k(F) = \bigcup_n \Gamma_n(F).$$
To prove the Theorem, it is enough to verify that $HD (\Gamma_n(F))=HD (\Gamma_n(G))$. Indeed, for every $\epsilon > 0$ and $\alpha \in (HD(\Gamma_n(F)),1)$ there exists a covering of $\Gamma_n(F)$ by intervals $A_i$ so that
$$\sum_j |A_j|^\alpha \leq \epsilon.$$
Note that we can assume that $\partial A_j \subset \Gamma_n(F)$. Since $G$ is an asymptotically small perturbation of $F$, it is easy to see that $G$ also satisfies the properties $Ra+Rb$, replacing the points $c^n_i$ and $d^n_i$ by $h(c^n_i)$ and $h(d^n_i)$, and modifying the constant . Indeed can choose constants in the definitions of the properties $Ex+BD+Ra+Rb$ which works for both random walks, so we can take $K > 0$ in the statements of Lemma \ref{rci} and Lemma \ref{rcia} in such way that it works for both random walks.
In particular (as in the proof of Proposition \ref{sim}) for each $A_j$ we can find at most $2N$ dd-intervals $$W_j^\ell:=\bigcup_k \overline{C^{j\ell}_k}, \ with \ \ell \leq m_j \leq 2N$$
which satisfy
$$A_i\cap \Gamma_n(F) \subset \overline{\bigcup_{\ell} W^\ell_i},$$
and
$$\sum_{k, \ell} |C^{j \ell}_k|^\alpha\leq K |A_j|^\alpha.$$
Furthermore we can assume that the root $R_{j}^\ell$ of $W_j^\ell$ satisfies
\begin{equation}\label{abc}\frac{1}{K} \leq \frac{|R_{j}^\ell|^\alpha}{\sum_{k} |C^{j \ell}_k|^\alpha}\leq K\end{equation}
and $R^{j}_\ell \cap \Gamma_n(F)\neq \emptyset$.
The constant $K$ does not depend on $\alpha$, $j$ or $\ell$. In particular the union of all cylinders $C^{j\ell}_k$ covers $\Gamma_n(F)$ up to a countable set and
\begin{equation}\label{hdq} \sum_{j,k,\ell} |C^{j\ell}_k|^\alpha \leq K\epsilon.\end{equation}
Note that if $x \in \Gamma_n(F)$ then
$$dist_i(x)\leq r_n:=Cn + C\lambda^n$$
for every $i \in \mathbb{N}$. So
$$e^{-r_n} \leq \frac{|\mathcal{P}^i_F(x)|}{|\mathcal{P}^i_G(h(x))|} \leq e^{r_n}.$$
There is a point in the cylinder $R_j^\ell$ which belongs to $\Gamma_n(F)$, so
\begin{equation}\label{hdd} e^{-\alpha r_n}\leq \frac{|R_j^\ell|^\alpha}{|h(R_j^\ell)|^\alpha} \leq e^{\alpha r_n}.\end{equation}
Note that $h(W_j^\ell)=\bigcup_k \overline{h(C^{j\ell}_k)}$ is a dd-interval for $G$ and $h(R_j^\ell)$ is its root cylinder. So
\begin{equation}\label{fgh} \frac{1}{K}\leq \frac{|h(R_j^\ell)|^\alpha}{\sum_{i}|h(C^{j\ell}_i)|^\alpha} \leq K\end{equation}
But the union of the cylinders $h(C^{j\ell}_k)$ covers $\Gamma_n(G)$ up to a countable set and Eq. (\ref{abc}), Eq. (\ref{hdq}), Eq. (\ref{hdd}) and Eq. (\ref{fgh}) gives
$$\sum_{j,k,\ell} |h(C^{j\ell}_k)|^\alpha \leq K^3 e^{\alpha r_n}\epsilon.$$
Since $\alpha > HD(\Gamma_n(F))$ and $\epsilon$ is arbitrary we obtain that $HD(\Gamma_n(G)) \leq HD(\Gamma_n(F))$. Switching the roles of $F$ and $G$ in the above argument gives the opposite inequality.
\end{proof}
\begin{lem}\label{ol} Let $G \in On+Ra+Rb$ be a random walk. For every $\alpha > 0$ there exist $\epsilon$ and $C$
so that
\begin{equation}\label{best} \sum_{P \in \mathcal{P}^n, \ P \subset I_k} |P|^{1-\epsilon} \leq C (1+
\alpha)^n,\end{equation} for all $n$ and $k$.
\end{lem}
\begin{proof}For a random walk $G$, denote by $\mathcal{P}^n:=\{P^n_i\}_i$ the
Markov partition of $G^n$. Since $G \in BD+On+Ex$, for each $\delta > 0$, we can choose $n_0$ large
enough so that for every inverse branch $\phi$ of an iteration of
$G$ and an element $P \in \mathcal{P}^{n_0}$, we have
\begin{equation}\label{peq} 1-\delta \leq \frac{|D\phi(x)|}{|D\phi(y)|} \leq
1+ \delta.\end{equation}for every $x,y \in P$. F
Moreover note that for every $\epsilon <
1$ there exists a constant $K=K_ {\epsilon}
>1$ so that
\begin{equation}\label{bound} \sum_i |P^n_i|^{1-\epsilon} \leq K^n\end{equation}
for every $n$.
Denote $\mathcal{P}^{n_0}=\{ Q^j \}_j$ and $\mathcal{P}^{n_0+1}=\{ Q^j_k \}_{j,k}$, in such way that $Q_k^j \subset Q_j$. Indeed, since $G \in BD+Rb$, it is possible to order $Q_k^j$ so that there exists $C$ satisfying
$$\frac{|Q^j_k|}{|Q^j|}\leq C\lambda^k,$$
for every $j,k$. As a consequence the set of functions
$$h_j(\epsilon)= \sum_k
\frac{|Q^j_k|^{1-\epsilon}}{|Q^j|^{1-\epsilon}}$$ is a
equicontinuous set of functions in a small neighborhood of
$0$. In particular, since $h_j(0)=1$, there exists $\epsilon$ so
that, for every $j$,
\begin{equation} \label{uni} \sum_k \frac{|Q^j_k|^{1-\epsilon}}{|Q^j|^{1-\epsilon}} \leq
1+\delta.\end{equation}
For $n \leq n_0$, it follows from Eq. (\ref{bound}) that there
exists $C$ so that, for $n\leq n_0$, we have $\sum_{P \in
\mathcal{P}^n} |P|^{1-\epsilon} \leq C$. Assume by induction that
we have proved Eq. (\ref{best}) until some $n\geq n_0$. Denote by
$\{ \phi_j \}$ the inverse branches of $G^{n-n_0}$, with $Im \
\phi_i = P^{n-n_0}_i$. Then $\mathcal{P}^{n+1}= \{\phi_i(Q^j_k)
\}_{i,j,k}$ and $\mathcal{P}^n =\{\phi_i(Q^j) \}_{i,j}$. By the
distortion control in Eq. (\ref{peq}) and the estimative in Eq.
(\ref{uni}), for each $i, j$ we have
$$\frac{\sum_k |\phi_i(Q^j_k)|^{1-\epsilon}}{|\phi_i(Q^j)|^{1-\epsilon} } \leq
\frac{1+\delta}{1-\delta} \frac{\sum_k |Q^j_k|^{1-\epsilon}}{|Q^j|^{1-\epsilon} } \leq \frac{(1+\delta)^2}{1-\delta}. $$
So
$$\sum_{P \in \mathcal{P}^{n+1}} |P|^{1-\epsilon}= \sum_{i,j,k}
|\phi_i(Q^j_k)|^{1-\epsilon} \leq \sum_{i,j}
|\phi_i(Q^j)|^{1-\epsilon} \sum_k
\frac{|\phi_i(Q^j_k)|^{1-\epsilon}}{|\phi_i(Q^j)|^{1-\epsilon}}$$
$$ \leq \frac{(1+\delta)^2}{1-\delta}\sum_{i,j}
|\phi_i(Q^j)|^{1-\epsilon} = \frac{(1+\delta)^2}{1-\delta} \sum_{p
\in \mathcal{P}^n} |P|^{1-\epsilon}.$$ We finish the proof
choosing $\delta$ so that $(1+\delta)^2/(1-\delta) \leq
(1+\alpha)$.\end{proof}
From now on we are going to assume that the mean drift is
negative: $\int \psi \ d\mu < 0$.
\begin{lem}\label{bom} Let $G \in On+Ra+Rb$ be a random walk with negative mean drift. For every $\alpha > \int \psi \ d\mu$, there exists $\sigma < 1$
so that for any $n_1 \geq n_0$, with $n_0$ large
enough, \begin{equation}\label{nzero} m\{p \in I_ {n_1}\colon \
\pi_2(G^k(p))\geq n_0, \text{ for } k\leq n, \text{ and } \pi_2(G^n(p))
- n_1 \geq \alpha n \}\leq \sigma^{n}.\end{equation}
\end{lem}
\begin{proof} Denote $$\Lambda_{n_0,n_1}^n(G):=\{p \in I_ {n_1}\colon \
\pi_2(G^k(p))\geq n_0 \text{ for all } k \leq n \text{ and }
\pi_2(G^n(p)) - n_1 \geq \alpha n \}.$$ The statement for $F$ is
consequence of the large deviations estimative (see, for instance
\cite{broise})
$$m\{ p \in I\colon | \frac{\sum_{k=0}^{n-1} \psi(f^k(p))}{n} - \int \psi \ d\mu | \geq
K \}\leq C_K\sigma^n,$$ which holds for every $K > 0$. In
particular choosing $K= \alpha-\int \psi \ d\mu$ we get, for any
$n_0$, and $n_1\geq n_0$,
$$m\{p \in I_ {n_1}\colon
\pi_2(F^n(p)) -n_1 \geq \alpha n \}\leq \sigma^{n},
$$
which implies (of course)
\begin{equation}\label{unp} m(\Lambda_{n_0,n_1}^n(F))\leq \sigma^{n}.
\end{equation}
We are going to use this estimative to obtain Eq. (\ref{nzero})
for the perturbation of $F$.
Indeed, for every $\delta > 0$, there is $n_0$ so that
if $\pi_2(x)\geq n_0$ then
\begin{equation}\label{distor} 1-\delta \leq \frac{|DF(x)|}{|DG(H(x))|} \leq 1+
\delta,\end{equation}
Here $H$ is the topological conjugacy between $F$
and $G$ which preserves states. Note that $\Lambda_{n_0,n_1}^n(F)$
is a disjoint union of elements $Q_i \in \mathcal{P}^n(F)$, so
$\Lambda_{n_0,n_1}^n(G)$ is a disjoint union of the intervals
$H(Q_i)$. Due Eq. (\ref{unp}) and Eq. (\ref{distor}), we have
\begin{equation} \sum_i |H(Q_i)|\leq \sum_i (1+\delta)^n |Q_i| \leq (1+\delta)^n \sigma^n.\end{equation}
Take $n_0$ large enough so that $(1+\delta)\sigma < 1$.
\end{proof}
We would like to replace $n_0$ by an arbitrary state in Eq.
(\ref{nzero}). The following Lemma will be useful for this task:
\begin{lem}\label{absa} Let $p_n$ and $q_n$ sequences of non-negative real
numbers such that
\begin{enumerate}
\item $p_0+ q_0 \leq 1$,
\item There exists $\epsilon > 0$ and
$\ell \in \mathbb{N}$ such that $s_n:= p_n + q_n \leq
(1-\epsilon)^{\ell} p_{n-\ell} + q_{n-\ell}$ and $q_n \leq
C\sigma^n + \sum_{k=1}^{n} (1-\epsilon)^kp_{n-k}$ , for every
$n\geq 1$.
\end{enumerate}
Then there exists $\delta > 0$ such that $s_n \leq (1-\delta)^n$,
for every $n \in \mathbb{N}$.
\end{lem}
\begin{proof} If $n \geq \ell$, we have $s_n \leq (1-\epsilon)p_{n-\ell}+ q_{n-\ell}=
(1-\epsilon)s_{n-\ell} + \epsilon q_{n-\ell}$. It follows by
induction that if $n= i\ell + r$, with $r < \ell$, then $$s_n
\leq (1-\epsilon)^i s_r +
\sum_{k=0}^{i-1}\epsilon(1-\epsilon)^{k\ell} q_{n-(k+1)\ell}$$
$$\leq C(1-\epsilon)^{n/\ell} s_0 +
\sum_{k=0}^{n-1}\epsilon(1-\epsilon)^{k} q_{n-\ell -k}$$ Since
$q_{n-\ell} \leq C(1-\epsilon)^n + \sum_{k=1}^{n-1}
(1-\epsilon)^kp_{n-\ell-k}$, we obtain
$$s_n\leq (1-\epsilon)^{n/\ell} s_0 + \epsilon (1-\epsilon)^{n/\ell} + \sum_{k=1}^{n-1} \epsilon(1-\epsilon)^{k}(p_{n-\ell-k}
+ q_{n-\ell-k})$$
$$\leq (1-\epsilon)^{n/\ell}C(s_0 + \epsilon)+ \sum_{k=1}^{n-1} \epsilon(1-\epsilon)^{k}
s_{n-\ell-k},$$ for every $n\geq \ell$.
We claim that there exists $\delta < 1$ and $K$ so that $s_n \leq
K(1-\delta)^n$, for every $n$. Indeed, fix $\delta < 1$, For each
$n$, define $K_n := s_n/(1-\delta)^n$. Note that
\begin{equation} \label{final} s_n \leq (1-\epsilon)^{n/\ell}C(s_0 + \epsilon)+
\sum_{k=1}^{n-1} \epsilon(1-\epsilon)^{k} s_{n-\ell-k}$$ $$ \leq
(1-\epsilon)^{n/\ell}C(s_0 + \epsilon)+ \sum_{k=1}^{n-1}
\epsilon(1-\epsilon)^{k}
K_{n-\ell-k}(1-\delta)^{n-\ell-k}\end{equation}
$$\leq \big[ \big(
\frac{(1-\epsilon)^{1/\ell}}{1-\delta} \big)^n C(s_0 + \epsilon)+
\max_{i< \ n-\ell}K_i \ \frac{\epsilon}{(1-\delta)^\ell}
\sum_{k=1}^{n-1} \big(\frac{1-\epsilon}{1-\delta}\big)^k
\big](1-\delta)^n$$
Choose $\delta$ close enough to $1$ so that
$$\sigma_1:= \frac{(1-\epsilon)^{1/\ell}}{1-\delta} < 1, \ and $$
$$\sigma_2:= \frac{\epsilon}{(1-\delta)^\ell}
\sum_{k=1}^{\infty} \big(\frac{1-\epsilon}{1-\delta}\big)^k < 1.$$
Then by Eq. (\ref{final}) we have $K_{n} \leq \sigma_2\max_{i< \
n-\ell}K_i + C\sigma_1^n$, for every $n > \ell$, which easily
implies that $\max_i K_i < \infty$.
\end{proof}
Define $$\Omega_{+}^{n_1,n}:= \{p \in I_{n_1}\colon \pi_2(G^{k}(p)) \geq 0,
\text{ for } 0 \leq k \leq n \}.$$
\begin{lem}\label{estimativa} There exists $\delta < 1$ so that for every $n_1\geq
0$ there exists $C=C(n_1)$ satisfying
$$m(\Omega_{+}^{n_1,n}(G))\leq C(1-\delta)^n.$$
\end{lem}
\begin{proof}Take $n_0$ as in Lemma \ref{bom} and fix $n_1 \geq 0$. Define the sets and sequences
$$s_n:= m(\Omega_{+}^{n_1,n}) $$
$$p_n:= m(B^n), \text{ where } B^n:=\{ p \in \Omega_{+}^{n_1,n}\colon \ \pi_2(G^{n}(p))\in [0,n_0] \}, \ and $$
$$q_n:= m(C^n), \text{ where } C^n:=\{p \in \Omega_{+}^{n_1,n}\colon \ \pi_2(G^{n}(p)) > n_0 \}.$$
To prove Lemma \ref{estimativa}, it is enough to verify that
these sequences satisfy the assumptions of Lemma \ref{absa}.
Indeed, of course $p_0 + q_0 \leq 1$. To prove the other
assumptions, take $i \in [0,n_0]$. Since $G$ is topologically
transitive, there are $\ell_i \in \mathbb{N}$ and intervals $J_i
\subset I_i$ so that $\pi_2(G^{\ell_i}(J_i)) < 0$. Denote
$\ell=max_{\ 0\leq i\leq n_0} \ell_i$ and $r= min_{\ 0\leq i\leq
n_0} |J_i|/|I_i|$.
Clearly $\Omega^{n_1,n}_{+}(G)=B^n\cup C^n \subset B^{n-\ell}\cup
C^{n-\ell} $. Let $J \subset B^{n-\ell}$ be an interval so that
$G^{n-\ell}(J)=I_i$, with $0\leq i\leq n_0$. Note that
$B^{n-\ell}$ is a disjoint union of such intervals. By the bounded
distortion control for $G$,
\begin{equation}\label{ind} \frac{m(J\cap \Omega_{+}^{n_1,n}(G))}{m(J)} \leq
1- \frac{m(J\cap G^{-(n-\ell)}J_i)}{m(J)}\leq (1-\frac{r}{c})\end{equation}
Choose $\epsilon_0$ satisfying $(1-r/c)\leq (1-\epsilon_0)^\ell$.
Then Eq. (\ref{ind}) implies $$m(B^{n-\ell}\cap \Omega_{+}^{n_1,n}(G))\leq
(1-\epsilon_0)^\ell m(B^{n-\ell})$$ and we obtain
$$s_n = m(B^{n-\ell}\cap \Omega_{+}^{n_1,n}(G)) + m(C^{n-\ell}\cap
\Omega_{+}^{n_1,n}(G)) \leq (1-\epsilon_0)^\ell p_{n-\ell} +
q_{n-\ell}.$$
It remains to prove that $q_n \leq \sum_{k=1}^{n}
(1-\epsilon)^kp_{n-k}$. There are two kind of points $p$ in $C^n$:
{\em Type 1.} For every $j\leq n$ we have $\pi_2(G^j(p))\geq n_0$
(in particular $n_1\geq n_0$). We are going to estimate the
measure of the set of these points, denoted $\Theta_1^n$. It
follows from Lemma \ref{bom}, choosing, for instance, $\alpha=\int
\psi \ d\mu/2$, that
\begin{equation}\label{acima} m(\{p \in I_{n_1}\colon \ \pi_2(G^k(p))\geq n_0, \ for \ k\leq n
\text{ and } \pi_2(G^n(p)) \geq n_1 + \alpha n \}) \leq
C\sigma^n.\end{equation}
But the set in the r.h.s. of Eq. (\ref{acima}) coincides with
$\Theta^{n}_{1}$ provided $n\geq~(n_0-n_1)/\alpha$. So
$$m(\Theta_1^n)\leq C_{n_1}\sigma^n,$$
for some $\sigma < 1$ which does not depend on $n_1$.
{\em Type 2.} For some $j < n$ we have $\pi_2(G^j(p)) \leq n_0$.
Denote the set of these points by $\Theta^n_2$. Denote by
$\Theta_{2,k}^n$ the set of points $p$ so that $k\geq 1$ is the
smallest natural satisfying $\pi_2(G^{n-k}p)\leq n_0$. Clearly
$\Theta^n_2$ is a disjoint union of these sets. We are going to
estimate their measure. Note that $\Theta_{2,k}^n \subset
B^{n-k}$. The set $B^{n-k}$ is a disjoint union of intervals $L$
so that $\pi_2(G^{n-k}L)=I_i$, for some $i \leq n_0$. To estimate
$$\frac{m(\Theta_{2,k}^n\cap L)}{|L|} $$ note that $L \subset B^{n-k},$
and $\Theta_{2,k}^n\cap L$ is the set of points $p \in L$ so that
$\pi_2(G^{n-k+j}p) > n_0$, for every $0< j \leq k$. Define
$$L_y := \{ p \in L\colon \psi(G^{n-k}p)=y\}.$$
Firstly note that for $y \leq n_0 -i$ we have \begin{equation}
\label{estum} |L_y\cap \Theta_{2,k}^n|=0,\end{equation} since $p
\in L_y\cap \Theta_{2,k}^n$ satisfies $\pi_2(G^{n-k+1}p)= i+
\psi(G^{n-k}p)= i + y
> n_0$. In particular for $y < 0$ we have $|L_y\cap \Theta_{2,k}^n|=0$, which implies, due the bounded
distortion control
$$\frac{m(L\cap \Theta_{2,k}^n)}{|L|}\leq \frac{\sum_{y \geq 0}
|L_y|}{|L|} \leq (1-\delta),$$ for some $\delta < 1$ which does
not depends on $k$, $L$ or $n_1$, which implies
\begin{equation}\label{estquatro} m(\Theta^n_{2,k}) \leq
(1-\delta)m(B^{n-k})= (1-\delta)p_{n-k}.\end{equation}
Furthermore, using again the distortion control and the regularity
condition $GD$(big jumps are rare) we have
\begin{equation}\label{estdois} \frac{\sum_{y > -\alpha (k-1)} |L_y \cap \Theta_{2,k}^n|}{|L|}\leq
\frac{\sum_{y > -\alpha (k-1)} |L_y|}{|L|}\leq
C\gamma^{k},\end{equation} for some $C\geq 0$ and $\gamma < 1$.
To estimate $|L_y\cap \Theta_{2,k}^n|/|L_y|$, in the case $n_0-i
\leq y \leq -\alpha (k-1)$, recall that $G^{n-k+1}L_y=I_{i+y}$,
with $i+y
> n_0$. By Lemma \ref{bom}, we have
$$ m\{p \in I_ {i+y}\colon \ \pi_2(G^m(p))\geq n_0, \text{ for } m\leq
k-1, \text{ and } \pi_2(G^{k-1}(p))
\geq i+y + \alpha (k-1) \}\leq C\sigma^{k}.$$
Since $i+y+ \alpha (k-1) \leq n_0$, this implies that
$$ m\{p \in I_ {i+y}\colon \ \pi_2(G^m(p))\geq n_0, \text{ for every } m\leq
k-1 \}\leq C\sigma^{k}.$$
The points in $L_y\cap\Theta^n_{2,k}$ are exactly the points whose
$(n-k+1)$th-iteration belongs to the set in the estimative above.
Using the bound distortion control we have
$$ \frac{|L_y\cap \Theta^n_{2,k}| }{|L_y|} \leq C\sigma^k,$$
so
\begin{equation}\label{esttres}
\frac{|\sum_{n_0-i \leq y \leq -\alpha (k-1)}L_y\cap
\Theta^n_{2,k}| }{|L|} \leq C \frac{|\sum_{n_0-i \leq y \leq
-\alpha (k-1)}L_y\cap \Theta^n_{2,k}| }{\sum_{n_0-i \leq y \leq
-\alpha (k-1)} |L_y|} \leq C\sigma^k.
\end{equation}
Choose $\epsilon < \epsilon_0$ so that $ min\{ max\{ C\sigma^k,
C\gamma^k\}, 1-\delta\} \leq (1-\epsilon)^k$, for every $k \geq
0$, and put together Eq. (\ref{estum}), Eq. (\ref{estquatro}), Eq.
(\ref{estdois}) and Eq. (\ref{esttres}), to get $m(L\cap
\Theta_{2,k}^n)\leq (1-\epsilon)^k |L|$. Since $B^{n-k}$ is a
disjoint union of such intervals $L$, we obtain
$$m(\Theta_{2,k}^n)\leq (1-\epsilon)^k m(B^{n-k})=
(1-\epsilon)^kp_{n-k}$$ and now we can conclude with
$$q_n = m(\Theta^n_{1}) + \sum_{k} m(\Theta^n_{2,k}) \leq C_{n_1}\sigma^n + \sum_k (1-\epsilon)^kp_{n-k}.$$
\end{proof}
Now we are ready to prove Theorem \ref{menor}:
\begin{proof}[{\bf Proof of Theorem \ref{menor}.}] There are three cases:
{\bf $F$ is transient with $ M > 0$.} If $M > 0$ then the random walk $F$ is transient and it is easy to see that $m(\Omega_+(F)) > 0$. Since the conjugacy with an asymptotically small perturbation $G$ is absolutely continuous (Theorem \ref{abscont}), we conclude that $m(\Omega_+(G)) > 0$.
{\bf $F$ is recurrent ($ M = 0$).} if $M=0$ then $F$ and its asymptotically small perturbations are recurrent by Theorem \ref{strec}. In particular almost every point visits negative states infinitely many times, so $m(\Omega_+(G)) = 0$. It remains to prove that $HD \ \Omega_+(G)=1$. By Theorem \ref{omega} it is enough to verify that $HD \ \Omega_+(F)=1$. Indeed, it is easy to show using the Central Limit Theorem that if
$$\int \psi \ d\mu=0$$
then there exist $C > 0$ and and for each $n$, subsets $\mathcal{A}_n \subset \mathcal{P}^n$ so that
$$\sum_{i=0}^{n-1}\psi(f^i(x)) > 0$$
for all $x \in J \in \mathcal{A}_n$ and
$$m(\bigcup_{J \in \mathcal{A}_n} J) \geq C >0.$$
here $C$ does not depend on $n$. Replacing $\mathcal{A}_n$ by a finite subfamily, if necessary, we can apply Proposition \ref{vhd} to obtain
$$HD \ \Lambda(\mathcal{A}_n) = 1 - O(\frac{1}{n}).$$
If $\mu_{\mathcal{A}_n }$ is the geometric invariant measure of $f_{\mathcal{A}_n}$ then
$$\int \psi_{\mathcal{A}_n} \ d\mu_{\mathcal{A}_n} > 0$$
So by the Birkhoff Ergodic Theorem
\begin{equation} \label{conv1} \lim_{n\rightarrow \infty} \sum_{i=0}^{n-1}\psi(f^i(x)) = + \infty \end{equation}
in a set $S_n \subset \Lambda(\mathcal{A}_n)$ satisfying $\mu_{\mathcal{A}_n}(S_n)=1$, so $HD \ S_n = 1 - O(1/n)$. In particular the set $S$ of points satisfying Eq.(\ref{conv1}) has Hausdorff dimension $1$. We can decompose $S$ in subsets $B_j$ defined by
$$B_j :=\{ x \in S\colon {\min}_{n} \sum_{i=0}^{n-1} \psi(f^i(x)) \geq -j \}.$$
Clearly $\sup_j HD \ B_j=1$.
For each $j$ choose $k_j$ and $J_j \in \mathcal{P}^{k_j}$ so that for all $x \in J_j$ we have
$$\sum_{i=0}^{\ell-1}\psi(f^i(x)) \geq 0$$
for every $\ell\leq k_j$ and
$$\sum_{i=0}^{k_j} \psi(f^i(x)) \geq j.$$
Then $$(J_j \cap f^{-k_j}B_j)\times \{0\}$$
belongs to $\Omega_+(F)$, for every $j$. This implies $HD \ \Omega_+(F) \geq HD \ B_j$ so
$$ HD \ \Omega_+(F) \geq \sup_j HD \ B_j =1.$$
{\bf $F$ is transient with $ M < 0$.} By Lemma
\ref{estimativa}, there is some $\delta \in (0,1)$, which does not depend on
$n_1$, so that
\begin{equation} \label{useum} m(\Omega^{n_1,n})\leq
C(1-\delta)^n.\end{equation}
By Lemma \ref{ol}, there exists $\epsilon$ so that
\begin{equation}\label{usedois} \sum_{P \in \mathcal{P}^n, \ P \subset I_k} |P|^{1-\epsilon}
\leq C(1-\delta)^{-n/2}.\end{equation}
Denote by $\{J_i^n\}_i \subset \mathcal{P}^n$ the family of
disjoint intervals so that $\Omega^{n_1,n} = \cup_i J_i^n$. We
claim that there exists $C > 0$ satisfying
\begin{equation} \sum_i |J_i^n|^{1-\epsilon/4} \leq C(1-\delta)^n.\end{equation}
Since $sup_i \ |J^n_i|\rightarrow_n 0$, this proves that $HD \
\Omega_+^{n_1,\infty} \leq 1-\epsilon/4$.
Indeed,
$$\sum_i |J_i^n|^{1-\epsilon/4} = \sum_{ |J_i|> (1-\delta)^{2n/\epsilon}} |J_i^n|^{1-\epsilon/4} +
\sum_{ |J_i|\leq (1-\delta)^{2n/\epsilon}}
|J_i^n|^{1-\epsilon/4}$$
$$\leq (1-\delta)^{-n/2} \sum_{i} |J_i^n| +
(1-\delta)^{3n/2} \sum_{i} |J_i^n|^{1-\epsilon}$$
$$\leq C(1-\delta)^{n/2},$$
where in the last line we made use of Eq. (\ref{useum}) and Eq.
(\ref{usedois}). The proof is complete.
\end{proof}
\section{Applications to one-dimensional renormalization theory}
\subsection{(Classic) infinitely renormalizable maps}\label{apl} Consider a real analytic unimodal
maps $f\colon I \rightarrow I$, with negative Schwarzian
derivative and even order critical point. The map $f$ is called
infinitely renormalizable if there exists an sequence of natural
numbers $n_0 < n_1 < n_2 < \dots$ and a nested sequence of
intervals
$$I=I_0 \supset I_1 \supset I_2 \supset \cdots $$
so that
\begin{itemize}
\item$f^{n_k}\partial I_k \subset \partial I_k$,
\item $f^{n_k}I_k
\subset I_k$,
\item $f^{n_k}\colon I_k \rightarrow I_k$ is a unimodal map.
\end{itemize}
We say that $f$ has bounded combinatorics if there exists $C
>0$ so that $n_{k+1}/n_k \leq C$, for all $k$. Two infinitely renormalizable
maps $f$ and $g$ have the same
combinatorics if there exists a homeomorphism $h\colon I
\rightarrow I$ such that $f\circ h = h \circ g$.
The following result is a deep result in renormalization theory:
\begin{prop}[\cite{mc2}]\label{convren} Let $f$ and $g$ be two infinitely
renormalizable unimodal maps with the same bounded combinatorics
and same even order. Then for every $r > 0$ there exists $C > 0$
and $\lambda < 1$ so that
$$||\frac{1}{|I_k^f|}\ f^{n_k}(|I_k^f| \cdot ) - \frac{1}{|I_k^g|}\ g^{n_k}(|I_k^g| \cdot )||_{C^r}
\leq C\lambda^k.
$$
\end{prop}
Here $|I^f_k|$ denotes the length of $I_k^f$.
\begin{figure}
\centering
\psfrag{f}{$f$}
\psfrag{g}{$f^2$}
\psfrag{h}{$f^4$}
\psfrag{p1}[][]{$p_1$}
\psfrag{p1l}[][]{$p_1'$}
\psfrag{p2}[][]{$p_{2}$}
\psfrag{p2l}[][]{$p_{2}'$}
\psfrag{p3}[][]{$p_{3}$}
\psfrag{p3l}[][]{$p_{3}'$}
\includegraphics[width=0.70\textwidth]{figure2.eps}
\caption{The "Bat" map: the induced map $F$ for a Feigenbaum unimodal map}
\end{figure}
\begin{proof}[{\bf Proof of Theorem \ref{apl1}.}] Let $f$ be an infinitely renormalizable map with bounded
combinatorics. We are going to define an induced map $F\colon
I\rightarrow I$, following Y. Jiang (see \cite{jianga},
\cite{jiangb}): Let $p_k$ be the periodic point in $\partial
I_k$. Define $E$ as the set
$$\{1,-1,-p_k, p_k,f(p_k), -f(p_k),\dots, f^{n_k-1}(p_k),-f^{n_k-1}(p_k)\}-\{f(p_k),-f(p_k) \}.$$
The set $E$ cuts $I_{k-1}\setminus I_k$ in $m_k$ intervals.
Denote these intervals $M_{k-1,i}$, with $i=1,\dots,m_k$. For each $x \in M_{k-1,i}$, define $n(x)\geq 1$ as the minimal positive integer so that $$I_k\subset f^{n(x)n_{k-1}} M_{k-1,1}.$$ Note that $f^{n(x)n_{k-1}}$ does not have critical points on $ M_{k-1,i}$. Define
the induced map $F$, which is defined everywhere in $I$, except
for a countable set of points:
$$F(x):=f^{n(x)}(x), \ for \ x \in I_k\setminus I_{k+1}.$$
See in Fig. 2 the induced map for an infinitely renormalizable
maps satisfying $n_{i+1}=2n_i$ for all $i$ (the so called
Feigenbaum maps). The map $F$ is Markovian with respect to the
partition
$$\mathcal{P}:=\{ M_{k,i} \}_{k \in \mathbb{N},i \leq m_k}.$$
Furthermore, if $f$ and $g$ have the same bounded combinatorics
and even order, then by Proposition \ref{convren}, the
corresponding induced maps $F$ and $G$ satisfies
$$||\ \frac{1}{|I_k^f|}\ F(|M_{k,i}^f| \cdot + |I_k^f|-|M_{k,i}^f|)-
\frac{1}{|I_k^g|}\ G(|M_{k,i}^g| \cdot + |I_k^g|-|M_{k,i}^g|)\
||_{C^r([0,1])} \leq C\lambda^k.$$
Define $L_k$ as, say, the right component of $I_k\setminus
I_{k+1}$ and $\gamma_k \colon I \rightarrow L_k$ as the unique
bijective order preserving affine map between this two intervals.
We are going to define a random walk $\mathcal{F} \colon I \times \mathbb{N} \rightarrow I \times \mathbb{N}$ from the map $F$ in the following
way:
$$
\mathcal{F}(x,k):=
\begin{cases}
(\gamma^{-1}_i \circ F\circ \gamma_k (x),i) & \text{if $F\circ \gamma_k (x) \in L_i$;}\\
(\gamma^{-1}_i \circ (-F)\circ \gamma_k (x),i) & \text{if $F\circ \gamma_k (x) \in -L_i$.}\\
\end{cases}
$$
It is easy to see that we can extend $\mathcal{F}\colon I \times \mathbb{Z} \rightarrow I \times \mathbb{Z}$ to a strongly transient deterministic random walk with non-negative drift. Furthermore if $g$ is another infinitely
renormalizable map with the same bounded combinatorics that $f$
then by Proposition \ref{convren} and Proposition \ref{how} the corresponding random walk $\mathcal{G}$ is an asymptotically small
perturbation of $\mathcal{F}$. So we can apply Theorem \ref{sqr} to conclude
that there is a conjugacy between $F$ and $G$ which is strongly
quasisymmetric with respect to the nested sequence of partitions
defined by the random walk $\mathcal{F}$. We can now easily
translate this result in terms of the original unimodal maps $f$
and $g$ saying that the continuous conjugacy $h$ between $f$ and
$g$ is a strongly quasisymmetric mapping with respect to
$\mathcal{P}$.
\end{proof}
\begin{rem}\label{quotient}{\rm An interesting case is when the unimodal map $f$ is a periodic point to the renormalization operator: there exists $n_0$ and $\lambda$, with $|\lambda|< 1$ so that
$$\frac{1}{\lambda}f^{n_0}(\lambda x)=f(x).$$
In this case, if we take $n_k=kn_0$, then the induced map $F$ will satisfy the functional equation
\begin{equation}\label{fe} F(\lambda x)=\lambda F(x).\end{equation}
Define the relation $\sim$ in the following way: $$x\sim y \text{ iff there exists $i \in \mathbb{Z}$ so that } x=\pm \lambda^i y.$$
By Eq. (\ref{fe}), $F$ preserves this relation, so we can take the quotient of $F$ by the relation $\sim$. Note that
$$L_0 = \mathbb{R}^\star/\sim.$$
It is easy to see that if $q=F/\sim\colon L_0\rightarrow L_0$ is a Markov expanding map. Now define $\psi\colon L_0 \rightarrow \mathbb{Z}$ as $\psi(x)= k$, if $f(x) \in I_{k}\setminus I_{k+1}$. Then $\mathcal{F}$ is exactly the homogeneous random walk defined by the pair $(q,\psi)$. }
\end{rem}
\subsection{Fibonacci maps} \label{aplf}
The Fibonacci renormalization is the simplest way to generalize
the concept of classical renormalization as described in Section
\ref{apl}. Actually we could prove all the results stated for
Fibonacci maps to a wider class of maps: maps which are infinitely
renormalizable in the generalized sense and with periodic
combinatorics and bounded geometry, but we will keep ourselves in the simplest case to
avoid more technical definitions and auxiliary results with its
long proofs.
Consider the class of real analytic maps $f$ with $Sf < 0$ and
defined in a disjoint union of intervals $I^0_1 \sqcup I^1_1$,
where $-I_1^0=I_1^0$, so that
\begin{itemize}
\item[ ]\
\item[{\it -}] The map $f\colon I^1_1 \rightarrow I^0_0:=f(I^1_1)$ is a
diffeomorphism. Furthermore $I^1_1$ is compactly contained in
$I^0_0$.\\
\item[{\it -}] The map $f\colon I^0_1 \rightarrow I^0_0$ is an even map
which has as $0$ as its unique critical point of even order.\\
\end{itemize}
We say that $f$ is {\bf Fibonacci renormalizable} if
$$f(0) \in I_1^1, \ f^2(0) \in I^1_0 \ and \ f^3(0) \in I^1_0.$$
In this case, the Fibonacci renormalization of $f$ is defined as
the first return map to the interval $I_1^0$ restricted to the
connected components of its domain which contain the points $f(0)$
and $f^2(0)$. This new map is denoted $\mathcal{R}f$: it could be
Fibonacci renormalizable again and so on, obtaining an infinite
sequence of renormalizations $\mathcal{R}f$, $\mathcal{R}^2f$,
$\mathcal{R}^3f$, $\dots$.
We will denote the set of infinitely renormalizable maps in
the Fibonacci sense with a critical point of order $d$ by
$\mathcal{F}_d$. A map $f \in \mathcal{F}_d$ will be called a {\bf Fibonacci map}.
As in the original map $f$, the $n$-th renormalization $f_n:=
\mathcal{R}^nf$ of $f$ is a map defined in two disjoint intervals,
denoted $I^n_0$ and $I_n^1$, where $-I^n_0=I^n_0$. Indeed $f_n$
on $I_n^0$ is a unimodal restriction of the $S_n$-th iteration of
$f$, where $\{ S_n \}$ is the Fibonacci sequence
$$S_0 = 1, \ S_1=2, \ S_2=3, \ S_3= 5, \ \dots \ , S_{k+2} = S_{k+1}
+ S_k, \dots$$ and $f_n$ on $I_n^1$ is the restriction of the
$S_{n-1}$-th iteration of $f$.
\begin{figure}
\centering \psfrag{f}{$f$}
\psfrag{un}[][][0.8]{$u_n$}
\psfrag{unl}[][][0.8]{$u_n'$}
\psfrag{un1}[][][0.8]{$u_{n+1}$}
\psfrag{un1l}[][][0.8]{$u_{n+1}'$}
\psfrag{unm1}[][][0.8]{$u_{n-1}$}
\psfrag{unm1l}[][][0.8]{$u_{n-1}'$}
\psfrag{unm2}[][][0.8]{$u_{n-2}$}
\psfrag{pn}[][][0.8]{$p_n$}
\psfrag{pnl}[][][0.8]{$p_n'$}
\psfrag{pn1}[][][0.8]{$p_{n+1}$}
\psfrag{pn1l}[][][0.8]{$p_{n+1}'$}
\includegraphics[width=\textwidth]{figure3.eps}
\caption{On the left figure the (green) solid curves represents the part of the $f^{S_n}$ used in the definition of the induced map. On the right figure the (red) solid curve is the part of $f^{S_n}$ which coincides with the $n$-th Fibonacci renormalization on its central domain. }
\end{figure}
\begin{figure}
\centering
\psfrag{f}{$f$}
\psfrag{un}[][][0.8]{$u_n$}
\psfrag{unl}[][][0.8]{$u_n'$}
\psfrag{un1}[][][0.8]{$u_{n+1}$}
\psfrag{un1l}[][][0.8]{$u_{n+1}'$}
\psfrag{unm1}[][][0.8]{$u_{n-1}$}
\psfrag{unm1l}[][][0.8]{$u_{n-1}'$}
\psfrag{unm2}[][][0.8]{$u_{n-2}$}
\psfrag{pn}[][][0.8]{$p_n$}
\psfrag{pnl}[][][0.8]{$p_n'$}
\psfrag{pn1}[][][0.8]{$p_{n+1}$}
\psfrag{pn1l}[][][0.8]{$p_{n+1}'$}
\psfrag{pnm1}[][][0.8]{$p_{n-1}$}
\includegraphics[width=\textwidth]{figure4.eps}
\caption{The (red) curves inside the medium square is the graph of the $n$-th Fibonacci renormalization $f_n$. The (red and blue) curves inside the largest square is the graph of an extension of $f_n$ which has the same maximal invariant set. }
\label{figure:extension}
\end{figure}
Denote by $p_k$ the sequence of points $p_k \in \partial I^k_0$ so that
$$f_k(p_{k+1})=p_k$$
and denote $I^k_0=[p_k, p_k']$.
It is possible to define a sequence $u_k$ of points satisfying
\begin{itemize}
\item[ ] \
\item[{\it 1.}] $\dots < \ p_{k+1} < u_k < p_k < \dots < p_0, $ \\
\item[{\it 2.}] $f^{S_k}$ is monotone on $[0,u_k]$, \\
\item[{\it 3.}] $f^{S_k}(u_{k+1})=u_k$, \\
\item[{\it 4.}] $f^{S_k}(u_k)=u_{k-2}$. \\
\end{itemize}
We are going to define an induced map for an infinitely
renormalizable map in the Fibonacci sense in the following way:
Firstly, define $f_{-1}\colon I^0_0 \setminus I^1_0$ as an
$C^3$ monotone extension of $f_0$ on $I^1_1$ which has negative
Schwarzian derivative and bounded distortion. Define $F\colon I^0_0
\rightarrow \mathbb{R}$ as
$$F(x) := f^{S_i}(x) \ \ if \ \ x \in [u_i, -u_i]\setminus [u_{i+1},-u_{i+1}]$$
for each $i\geq 0$.
Define $L_i$ as, say, the right component of $[u_i, -u_i]\setminus [u_{i+1},-u_{i+1}]$ and $\gamma_i \colon I \rightarrow L_i$ as the unique
bijective order preserving affine map between these two intervals.
We are ready to define the random walk $\mathcal{F}\colon I \times \mathbb{Z} \rightarrow I \times \mathbb{Z}$ as
$$
\mathcal{F}(x,k):=
\begin{cases}
(\gamma^{-1}_i \circ F\circ \gamma_k (x),i) & \text{if $F\circ \gamma_k (x) \in L_i$,}\\
(\gamma^{-1}_i \circ (-F)\circ \gamma_k (x),i) & \text{if $F\circ \gamma_k (x) \in -L_i$.}\\
\end{cases}
$$
There is a very special Fibonacci map $f^\star$, called the Fibonacci fixed point (see, for instance \cite{smfib}), whose induced map $F^\star$ satisfies (choosing a good $u_0$)
$$F^\star(\lambda x) = \pm \lambda F^\star(x)$$
for some $\lambda \in (0,1)$. In this case we can use the argument in Remark \ref{quotient} to conclude that $\mathcal{F}^\star$ is a homogeneous random walk. For an arbitrary Fibonacci map $f$, $\mathcal{F}$ is not homogeneous, however due Proposition \ref{how} and the following result $\mathcal{F}$ is an asymptotically small perturbation of $\mathcal{F}^\star$:
\begin{prop}[see \cite{smfib}] For each even integer larger than two the following holds: for every Fibonacci map $f$, denote
$$g_i = \alpha_{i}^{-1} \circ f^{S_i}\circ \alpha_{i+1}\colon I \rightarrow I,$$
where $\alpha_i\colon I \rightarrow [u_i^{f},-u_i^{f}]$ is an bijective affine map so that $\alpha_i^{-1}(f_{i+1}(0)) > 0$ and consider the correspondent maps $g_i^\star$ for $f^\star$. Then
$$|| g_i - g_i^{\star}||_{C^r} \leq K_r\rho^{i}$$
for some $\rho < 1$ and every $r \in \mathbb{N}$.
\end{prop}
The {\bf real Julia set} of $f$, denoted $J_{\mathbb{R}}(f)$, is
the maximal invariant of the map $$f\colon I_0^1 \sqcup I_1^1
\rightarrow I_0^0,$$ in other words,
$$J_{\mathbb{R}}(f_j):= \cap_i f^{-i}_j I^j_0. $$
Denote
$$\Omega_{+}^j(F):= \{ (x,i) \ s.t. \
\pi_2(F^n(x,i)) \geq j\, \ for \ all \ n\geq 0\}.$$
\begin{prop}\label{ji} There exists some $k_0$ so that
$$ \Omega_{+}^{j+1}(F) \subset J_{\mathbb{R}}(f_j) \subset \Omega_{+}^{j-1}(F).$$ In particular
\begin{equation}\label{esthaus}
HD \ \Omega_{+}^{j+1}(F) \leq HD \ J_{\mathbb{R}}(f_j) \leq HD \ \Omega_{+}^{j-1}(F),
\end{equation}
and, for the Fibonacci fixed point, since $\Omega_{+}^{j+1}(F)$ is an affine copy of $\Omega_{+}^{j-1}(F)$ we have
\begin{equation}
HD \ \Omega_{+}^{j}(F) = HD \ J_{\mathbb{R}}(f).
\end{equation}
for all $j\geq 0$.
\end{prop}
\begin{proof} Denote by $F_\ell$ the restriction of $F$ to $\cup_{i\geq \ell}L_i$. Then the maximal invariant set of $F_\ell$
$$\Lambda(F_\ell):= \cap_{i \in \mathbb{N}} F^{-i}\mathbb{R}$$
is $\Omega_{+}^{\ell}(F)$. Consider the extension of $f_j$ described in Fig. (\ref{figure:extension}). Let's call this extension $\tilde{f}_j$. An easy analysis of its graph shows that $f_j$ and $\tilde{f}_j$ have the same maximal invariant set. We claim that $\tilde{f}_{j+1}$ is just a map induced by $\tilde{f}_{j}$. Indeed, the restriction of $\tilde{f}_{j+1}$ to $[u_{j+1},u_{j+1}']$ coincides with $\tilde{f}_j^2$ on the same interval. On the rest of $\tilde{f}_{j+1}$-domain $\tilde{f}_{j+1}$ coincides with $\tilde{f}_{j}$.
\begin{figure}
\centering \psfrag{f}{$f$}
\psfrag{f2}[][][0.6]{$f_n$}
\psfrag{f1}[][][0.6]{$f_{n+1}$}
\psfrag{f0}[][][0.6]{$f_{n+2}$}
\psfrag{f1p}[][][0.6]{$f_{n+3}$}
\psfrag{f2p}[][][0.6]{$f_{n+4}$}
\psfrag{2}[][][0.6]{$u_n$}
\psfrag{1}[][][0.6]{$u_{n+1}$}
\psfrag{0}[][][0.6]{$u_{n+2}$}
\psfrag{1p}[][][0.6]{$u_{n+3}$}
\psfrag{2p}[][][0.6]{$u_{n+4}$}
\psfrag{2t}[][][0.6]{$u_n'$}
\psfrag{1t}[][][0.6]{$u_{n+1}'$}
\psfrag{0t}[][][0.6]{$u_{n+2}'$}
\psfrag{1pt}[][][0.6]{$u_{n+3}'$}
\psfrag{2pt}[][][0.6]{$u_{n+4}'$}
\includegraphics[width=0.730\textwidth]{figure5.eps}
\caption{Induced map $F$ for a Fibonacci map}
\end{figure}
By consequence, for $i \geq j$ the map $\tilde{f}_i$ is induced by $\tilde{f}_j$ and, since $F_{j+1}$ restricted to $L_i$ is equal to $\tilde{f}_i$, we obtain that $F_{j+1}$ is a map induced by $\tilde{f}_{j}$. In particular
$$\Lambda(F_{j+1}) \subset \Lambda(\tilde{f}_j)=J_{\mathbb{R}}(f_j).$$
To prove that $\Lambda(\tilde{f}_j) \subset \Lambda(F_{j-1})$, we are going to prove that
\begin{equation}\label{contido} x \in \Lambda(\tilde{f}_j) \ implies \ F_{j-1}(x) \in \Lambda(\tilde{f}_j).\end{equation}
If $x$ belongs to the interval $I^j_1 \subset L_{j-1}$, where $\tilde{f}_j$ coincides with $F_{j-1}$, then $F_{j-1}(x) \in \Lambda(\tilde{f}_j)$. Otherwise $x \in I^j_0 \subset \cup_{i\geq j}L_i$, so $x \in \Lambda(\tilde{f}_j)\cap \ L_i$, for some $i\geq j$, then $F_{j-1}$ is an iteration of $\tilde{f}_j$ on $L_i$, so $F_{j-1}(x) \in \Lambda(\tilde{f}_j)$. This finishes the proof of Eq. (\ref{contido}). Since $\Lambda(\tilde{f}_j)$ is invariant by the action of $F_{j-1}$ we have $\Lambda(\tilde{f}_j) \subset \Lambda(F_{j-1})$.\end{proof}
\begin{proof}[{\bf Proof of Theorem \ref{juliathm}}] Consider the homogeneous random walk $F^\star= (g,\psi)$ induced by $f^\star$. Denote
$$M = \int \psi \ d\mu,$$
where $\mu$ is the absolutely continuous invariant measure of $g$. Using Thorem \ref{menor}, there are three cases:
\vspace{4mm}
{\bf 1. ${ \mathbf M < 0}$.} In this case $\mathbf F^\star$ is transient and we have that $HD \ \Omega_+(F) < 1$ for every asymptotically small perturbation of $F^\star$, in particular when $F$ is a random walk induced by a Fibonacci map $f$. By Proposition \ref{ji}, $HD \ J_{\mathbb{R}}(f) < 1$.
\vspace{4mm}
{\bf 2. $\mathbf M =0$.} In this case every asymptotically small perturbation $G$ of $F^\star$ is recurrent and $m(\Omega_+(G))=0$ but $HD \ \Omega_+(G)=1$. By Proposition \ref{ji} we obtain $m(J_\mathbb{R}(f)) =0$ and $HD \ J_\mathbb{R}(f)=1$.
\vspace{4mm}
{\bf 3. $\mathbf M > 0$.} In this case $\mathbf F^\star$ is transient with $m(\Omega_+(F^\star)) > 0$ and the conjugacy between $F^\star$ and any asymptotically small perturbation of it is absolutely continuous on $\Omega_+^i(F^\star)$. In particular $m(\Omega_+(F)) > 0$ for every random walk $F$ induced by a Fibonacci map $f$ so $m(J_\mathbb{R}(f)) > 0$ by Proposition \ref{ji}.\end{proof}
A map $f\colon I \rightarrow I$ is called a unimodal map if $f$ has a unique critical point, with even order $d$, which is a maximum, and $f(\partial I) \subset \partial I$. We will assume that $f$ is real analytic, symmetric with respect the critical point and $Sf < 0$. If the critical value is high enough, then $f$ has a reversing fixed point $p$. Let $I_0^0:=[-p,p]$. Consider the map of first return $R$ to $f$: if $x \in I_0^0$ and $f^r(x) \in I_0^0$, but $f^n(x) \not\in I_0^0$ for $i < r$, define $$R(x):=f^r(x).$$
If there exists exactly two connected components $I_1^0$ and $I_1^1$ of the domain of $R$ containing points in the orbit of the critical point, and furthermore the map $$R\colon I_1^0 \cup I_1^1 \rightarrow I_0^0$$ is a Fibonacci map, then we will called $f$ an {\bf unimodal Fibonacci map}. The class of all unimodal Fibonacci maps will be denoted $\mathcal{F}^{uni}_d$.
\begin{proof}[{\bf Proof of Theorem \ref{deep}}]We will use the notation in the proof of Theorem \ref{juliathm}. Since $m(J_{\mathbb{R}}(f)) > 0$, we conclude that the mean drift $M$ is positive. by Proposition \ref{prws}
any asymptotically small perturbation $G$ of $\mathcal{F}^\star$ has the following property: there exists $\lambda \in [0,1)$, $C >0$ and $K > 0$ so that for every $P \in \mathcal{P}^0(G)$
$$m(p \in P \colon \ \sum_{i=0}^{n-1}\psi(G^i(p)) < K n )\leq C\lambda^n |P|.$$
This implies that
$$m(p \in I_j \colon \ \sum_{i=0}^{\ell}\psi(G^i(p)) \geq K \ell \text{ for every } \ell\geq n )\geq (1-C\lambda^n).$$
so if $j= n|min \psi|$ we obtain
$$m(I_j\cap \Omega_+^j(G))\geq 1-C\lambda^{C_1 j}.$$
here $c_1 > 0$. If $G$ is a random walk induced by a Fibonacci map $g$ then this implies that for $j$ large
$$m(L_j\setminus J_{\mathbb{R}}(g))= m((-L_j)\setminus J_{\mathbb{R}}(g)) \leq C\lambda^{C_1 j}|L_j|.$$
Since $$[-u_{j+1},u_{j+1}]=\bigcup_{i\geq j} L_i\cup(-L_i),$$
we conclude that
\begin{equation}\label{soalguns} m([u_{j+1},-u_{j+1}]\setminus J_{\mathbb{R}}(g)) \leq C\lambda^{c_1 j}|u_{j+1}|.\end{equation}
For every $\delta$, choose $j$ so that $|u_{j+2}|\leq \delta \leq |u_{j+1}|$. Because $|u_{j+2}|> \theta |u_{j+1}| $, where $\theta \in (0,1)$ does not depend on $j$, we have that $|u_{j}|\geq C\theta^j$. Together with Eq. (\ref{soalguns}) this implies
$$m([-\delta,\delta]\setminus J_{\mathbb{R}}(g)) \leq C\lambda^{C_1 j}|u_{j+1}|\leq C|u_{j+1}|^{1+ \alpha}\leq C|\delta|^{1+ \alpha}.$$\end{proof}
\begin{proof}[\bf Proof of Theorem \ref{apl2}]We will prove each one of the following implications:
{\bf (1) implies (2):} From the proof of Theorem \ref{juliathm}, if $m(J_{\mathbb{R}}(f))> 0$ for some $f \in \mathcal{F}_d$ the mean drift $M$ of the homogeneous random walk $\mathcal{F}^\star$ of $f^\star$ is positive. So $\mathcal{F}^\star$ (and all its asymptotically small perturbations) is transient (to $+\infty$). In terms of the original Fibonacci map $f$, this means that almost every orbit in $J_{\mathbb{R}}(f)$ accumulates in the post-critical set: So $f$ has a wild attractor.
{\bf (2) implies (3):} if there exists a wild attractor for $f$ then $m(J_{\mathbb{R}}(f))> 0$. From the proof of Theorem \ref{juliathm} we obtain that the mean drift $M$ of $\mathcal{F}^\star$ is positive. So there exists a absolutely continuous conjugacy between $\mathcal{F}^\star$ and any asymptotically small perturbation of $\mathcal{F}^\star$. This implies that any two maps $f_1, f_2 \in \mathcal{F}_d$ admits a continuous and absolutely continuous conjugacy $$h\colon J_{\mathbb{R}}(f_1) \rightarrow J_{\mathbb{R}}(f_2).$$
Now consider two arbitrary maps $g_1, g_2 \in \mathcal{F}_d^{uni}$. Then we already know that there exists an absolutely continuous conjugacy
$$h\colon J_{\mathbb{R}}(R_{g_1}) \rightarrow J_{\mathbb{R}}(R_{g_2})$$ between the induced Fibonacci maps $R_{g_1}$ and $R_{g_2}$ associated to $g_1$ and $g_2$. Of course $h$ is just the restriction of a topological conjugacy between $g_1$ and $g_2$. By a Block and Lyubich result (see, for instance, page 332 in \cite{ms}), every map of $\mathcal{F}_d^{uni}$ is ergodic with respect the Lebesgue measure. Since $g_1$ and $g_2$ have wild attractors, this implies that the orbit of almost every point $x \in I$ hits $J_{\mathbb{R}}(R_{g_1}$ at least once. Let $n(x)$ be a time when this happens.
So consider a arbitrary measurable set $B \subset I$ so that $m(B)>0$. Then for at least one $n_0 \in \mathbb{N}$ the set
$$B_{n_0}:=\{x \in B\colon \ n(x)=n_0 \}$$
has positive Lebesgue measure. This implies that $f^{n_0}B_{n_0}$ has positive Lebesgue measure, so $m(h(f^{n_0}B_{n_0})) > 0$. Now it is easy to conclude that $m(h(B_{n_0})$ and $h(B) > 0$. Switching the places of $g_1$ and $g_2$ in this argument we can conclude that $h$ is absolutely continuous on $I$.
Finally note that the eigenvalues of the periodic points are not constant on the class $\mathcal{F}_d^{uni}$.
{\bf (3) implies (4):} By the argument in Martens and de Melo \cite{mm}, if a Fibonacci map does not have a wild attractor then any continuous absolutely continuous conjugacy with other Fibonacci map is $C^1$: in particular the conjugacy preserves the eigenvalues of the periodic points. So if (3) holds then we can use the same argument in the proof of the previous implication to conclude that every Fibonacci map has a wild attractor.
{\bf (4) implies (5):} The proof goes exactly as the proof of (2)$\Rightarrow$ (3).
{\bf (5) implies (1):} The proof goes exactly as the proof of (3)$\Rightarrow$ (4).
\end{proof}
|
{
"timestamp": "2010-01-12T17:48:24",
"yymm": "0503",
"arxiv_id": "math/0503736",
"language": "en",
"url": "https://arxiv.org/abs/math/0503736"
}
|
\section{Introduction to Non-Commutative Worlds}
Aspects of gauge
theory, Hamiltonian mechanics and quantum mechanics arise naturally in the mathematics of a non-commutative
framework for calculus and differential geometry. This paper consists in two sections. This first section sketches
our results in this domain in general. The second section gives a derivation of a generalization of the Feynman-Dyson
derivation of electromagnetism using our non-commutative context and using diagrammatic techniques.
The first section is based on the paper \cite{NCW}. The second section is a new approach to issues in \cite{NCW}.
\bigbreak
Constructions are performed in a Lie algebra $\cal A.$
One may take $\cal A$ to be a specific matrix Lie algebra, or abstract Lie algebra.
If $\cal A$ is taken to be an abstract Lie algebra, then it is convenient to use the universal
enveloping algebra so that the Lie product can be expressed as a commutator. In making general constructions of
operators satisfying certain relations, it is understood that one can always begin with a free algebra and make
a quotient algebra where the relations are satisfied.
\bigbreak
On $\cal A,$ a variant of calculus is built by
defining derivations as commutators (or more generally as Lie products). For a fixed $N$ in $\cal A$ one defines
$$\nabla_N : \cal A \longrightarrow \cal A$$ by the formula
$$\nabla_{N} F = [F, N] = FN - NF.$$
$\nabla_N$ is a derivation satisfying the Leibniz rule.
$$\nabla_{N}(FG) = \nabla_{N}(F)G + F\nabla_{N}(G).$$
\bigbreak
There are many motivations for replacing derivatives by commutators. If $f(x)$ denotes (say) a function of a real variable $x,$
and $\tilde{f}(x) = f(x+h)$ for
a fixed increment $h,$ define the {\em discrete derivative} $Df$ by the formula $Df = (\tilde{f} - f)/h,$ and find that
the Leibniz rule is not satisfied. One has the basic formula for the discrete derivative
of a product: $$D(fg) = D(f)g + \tilde{f}D(g).$$
Correct this deviation from the Leibniz rule by introducing a new non-commutative operator $J$ with the property that
$$fJ = J\tilde{f}.$$ Define a new discrete derivative in an extended non-commutative algebra by the formula
$$\nabla(f) = JD(f).$$ It follows at once that
$$\nabla(fg) = JD(f)g + J\tilde{f}D(g) = JD(f)g + fJD(g) = \nabla(f)g + f\nabla(g).$$
Note that $$\nabla(f) = (J\tilde{f} - Jf)/h =
(fJ-Jf)/h = [f, J/h].$$ In the extended algebra, discrete derivatives are represented by commutators, and satisfy the Leibniz rule.
One can regard discrete calculus as a subset of non-commutative
calculus based on commutators.
\bigbreak
In $\cal A$ there are as many derivations as there are elements of the
algebra, and these derivations behave quite wildly with respect to one another. If
one takes the concept of {\em curvature} as the non-commutation of
derivations, then $\cal A$ is a highly curved world indeed. Within $\cal A$ one can build
a tame world of derivations that mimics the
behaviour of flat coordinates in Euclidean space. The description of the
structure of $\cal A$ with respect to these flat coordinates contains many of the
equations and patterns of mathematical physics.
\bigbreak
\noindent The
flat coordinates $X_i$ satisfy the equations below with the $P_j$ chosen to represent differentiation with
respect to $X_j.$:
$$[X_{i}, X_{j}] = 0$$
$$[P_{i},P_{j}]=0$$
$$[X_{i},P_{j}] = \delta_{ij}.$$
Derivatives are represented by commutators.
$$\partial_{i}F = \partial F/\partial X_{i} = [F, P_{i}],$$
$$\hat{\partial_{i}}F = \partial F/\partial P_{i} = [X_{i},F].$$
Temporal derivative is represented by commutation with a special (Hamiltonian) element $H$ of the algebra:
$$dF/dt = [F, H].$$
(For quantum mechanics, take $i\hbar dA/dt = [A, H].$)
These non-commutative coordinates are the simplest flat set of
coordinates for description of temporal phenomena in a non-commutative world.
Note:
\noindent {\bf Hamilton's Equations.} $$dP_{i}/dt = [P_{i}, H] = -[H, P_{i}] = -\partial H/\partial X_{i}$$
$$dX_{i}/dt = [X_{i}, H] = \partial H/\partial P_{i}.$$
These are exactly Hamilton's equations of motion. The pattern of
Hamilton's equations is built into the system.
\bigbreak
\noindent {\bf Discrete Measurement.} Consider a time series $\{X, X', X'', \cdots \}$ with commuting scalar values.
Let $$\dot{X} = \nabla X = JDX = J(X'-X)/\tau$$ where $\tau$ is an elementary time step (If $X$ denotes a times series value at time
$t$, then
$X'$ denotes the value of the series at time $t + \tau.$). The shift operator $J$ is defined by the equation
$XJ = JX'$
where this refers to any point in the time series so that $X^{(n)}J = JX^{(n+1)}$ for any non-negative integer $n.$
Moving $J$ across a variable from left to right, corresponds to one tick of the clock. This discrete,
non-commutative time derivative satisfies the Leibniz rule.
\bigbreak
This derivative $\nabla$ also fits a significant pattern of discrete observation. Consider the act of observing $X$ at a given time
and the act of observing (or obtaining) $DX$ at a given time.
Since $X$ and $X'$ are ingredients in computing $(X'-X)/\tau,$ the numerical value associated with $DX,$ it is necessary to let the
clock tick once, Thus, if one first observe
$X$ and then obtains $DX,$ the result is different (for the $X$ measurement) if one first obtains $DX,$ and then observes $X.$ In the
second case, one finds the value $X'$ instead of the value $X,$ due to the tick of the clock.
\bigbreak
\begin{enumerate}
\item Let $\dot{X}X$ denote the sequence: observe $X$, then obtain $\dot{X}.$
\item Let $X\dot{X}$ denote the sequence: obtain $\dot{X}$, then observe $X.$
\end{enumerate}
\bigbreak
The commutator $[X, \dot{X}]$ expresses the difference between these two orders of discrete measurement.
In the simplest case, where the elements of the time series are commuting scalars, one has
$$[X,\dot{X}] = X\dot{X} - \dot{X}X =J(X'-X)^{2}/\tau.$$
Thus one can interpret the equation $$[X,\dot{X}] = Jk$$ ($k$ a constant scalar) as $$(X'-X)^{2}/\tau = k.$$ This means
that the process is a walk with spatial step $$\Delta = \pm \sqrt{k\tau}$$ where $k$ is a constant. In other words, one
has the equation
$$k = \Delta^{2}/\tau.$$
This is the diffusion constant for a Brownian walk.
A walk with spatial step size $\Delta$ and time step $\tau$ will satisfy the commutator equation above
exactly when the square of the spatial step divided by the time step remains constant. This
shows that the diffusion constant of a Brownian process is a structural property of that process, independent of considerations of
probability and continuum limits.
\bigbreak
\noindent {\bf Heisenberg/Schr\"{o}dinger Equation.} Here is how the Heisenberg form of Schr\"{o}dinger's equation fits in
this context. Let the time shift operator be given by the equation
$J=(1 + H\Delta t/i \hbar).$ Then the non-commutative version of the discrete time derivative is expressed by the commutator
$$\nabla\psi = [\psi, J/\Delta t],$$ and we calculate
$$\nabla \psi = \psi[(1 + H \Delta t/i \hbar)/\Delta t] -
[(1 + H\Delta t/i \hbar)/\Delta t] \psi = [\psi, H]/i \hbar,$$
$$i \hbar \nabla \psi = [\psi, H].$$
This is exactly the Heisenberg version of the Schr\"{o}dinger equation.
\bigbreak
\noindent {\bf Dynamics and Gauge Theory.} One can take the general dynamical
equation in the form
$$dX_{i}/dt = {\cal G}_{i}$$ where $\{ {\cal G}_{1},\cdots, {\cal G}_{d} \}$
is a collection of elements of $\cal A.$ Write ${\cal G}_{i}$
relative to the flat coordinates via ${\cal G}_{i} = P_{i} - A_{i}.$
This is a definition of $A_{i}$ and $\partial F/\partial X_{i} = [F,P_{i}].$ The formalism of gauge theory appears
naturally. In particular, if $$\nabla_{i}(F) = [F, {\cal G}_{i}],$$ then one has
the curvature $$[\nabla_{i}, \nabla_{j}]F = [R_{ij}, F]$$
and
$$R_{ij} = \partial_{i} A_{j} - \partial_{j} A_{i} + [A_{i}, A_{j}].$$ This is the well-known formula for the curvature of a gauge
connection. Aspects of geometry arise naturally in this context, including the Levi-Civita
connection (which is seen as a consequence of the Jacobi identity in an appropriate non-commutative world).
\bigbreak
One can consider the consequences of the commutator $[X_{i}, \dot{X_{j}}] = g_{ij}$,
deriving that
$$\ddot{X_{r}} = G_{r} + F_{rs}\dot{X^{s}} + \Gamma_{rst}\dot{X^{s}}\dot{X^{t}},$$
where $G_{r}$ is the analogue of a scalar field, $F_{rs}$ is the analogue of a gauge field and $\Gamma_{rst}$ is the Levi-Civita
connection associated with $g_{ij}.$
This decompositon of the acceleration is uniquely determined by the given framework.
\bigbreak
One can use this context to revisit the Feynman-Dyson derivation of electromagnetism from commutator equations,
showing that most of the derivation is independent of any choice of commutators, but highly dependent upon the choice of definitions
of the derivatives involved. Without any assumptions about initial commutator equations, but taking the right (in some sense simplest)
definitions of the derivatives one obtains a significant generalization of the result of Feynman-Dyson.
\bigbreak
\noindent {\bf Electromagnetic Theorem.} (See Section 2.) With the appropriate [see below] definitions of the operators, and taking
$$\nabla^{2} = \partial_{1}^{2} + \partial_{2}^{2} + \partial_{3}^{2}, \,\,\, B = \dot{X} \times \dot{X} \,\,\, \mbox{and} \,\,\, E =
\partial_{t}\dot{X}, \,\,\, \mbox{one has}$$
\begin{enumerate}
\item $\ddot{X} = E + \dot{X} \times B$
\item $\nabla \bullet B = 0$
\item $\partial_{t}B + \nabla \times E = B \times B$
\item $\partial_{t}E - \nabla \times B = (\partial_{t}^{2} - \nabla^{2})\dot{X}$
\end{enumerate}
\bigbreak
The key to the proof of this Theorem is the definition of the time derivative. This definition is as follows
$$\partial_{t}F = \dot{F} - \Sigma_{i}\dot{X_{i}}\partial_{i}(F) = \dot{F} - \Sigma_{i} \dot{X_{i}}[F, \dot{X_{i}}]$$
for all elements or vectors of elements $F.$ The definition creates a
distinction between space and time in the non-commutative world. A calculation ( done diagrammatically in
Figure 3) reveals that
$$\ddot{X} = \partial_{t}\dot{X} + \dot{X} \times (\dot{X} \times \dot{X}).$$
This suggests taking $E = \partial_{t}\dot{X}$ as the electric field, and $B = \dot{X} \times \dot{X}$
as the magnetic field so that the Lorentz force law
$$\ddot{X} = E + \dot{X} \times B$$
is satisfied.
\bigbreak
\noindent
This result is applied to produce many discrete models of the Theorem. These models show that, just as the commutator $[X, \dot{X}] =
Jk$ describes Brownian motion in one dimension, a generalization of electromagnetism describes the interaction of triples of time
series in three dimensions.
\bigbreak
\noindent {\bf Remark.} While there is a large
literature on non-commutative geometry, emanating from the idea of replacing a space by its ring of functions, work discussed herein
is not written in that tradition. Non-commutative geometry does occur here, in the sense of geometry occuring in the context of
non-commutative algebra. Derivations are represented by commutators. There are relationships between the present work and the
traditional non-commutative geometry, but that is a subject for further exploration. In no way is this paper intended to be an
introduction to that subject. The present summary is based on
\cite{Kauff:KP,KN:QEM,KN:Dirac,KN:DG,Twist,NonCom,ST,Aspects,Boundaries,NCW} and the references cited therein.
\bigbreak
The following references in relation to non-commutative calculus are useful in
comparing with the present approach \cite{Connes, Dimakis, Forgy, MH}. Much of the present work is the fruit of a long
series of discussions with Pierre Noyes, influenced at critical points by Tom Etter and Keith Bowden.
Paper \cite{Mont} also works with minimal coupling for the Feynman-Dyson derivation. The first remark about
the minimal coupling occurs in the original paper by Dyson \cite{Dyson}, in the context of Poisson brackets.
The paper \cite{Hughes} is worth reading as a companion to Dyson. It is the purpose of this summary to indicate how
non-commutative calculus can be used in foundations.
\bigbreak
\section{Generalized Feynman Dyson Derivation}
In this section we assume that specific time-varying coordinate elements $X_{1},X_{2},X_{3}$ of the algebra $\cal{A}$ are given.
{\it We do not assume any commutation relations about $X_{1},X_{2},X_{3}.$}
\bigbreak
In this section we no longer avail ourselves of the commutation relations that are in back of the original
Feynman-Dyson derivation. We do take the definitions of
the derivations from that previous context. Surprisingly, the result is very similar to the one of Feynman and Dyson, as
we shall see.
\bigbreak
Here $A \times B$ is the non-commutative vector cross product:
$$(A \times B)_{k} = \Sigma_{i,j = 1}^{3} \epsilon_{ijk}A_{i}B_{j}.$$ (We will drop this summation sign
for vector cross products from now on.)
Then, with $B = \dot{X} \times \dot{X},$ we have $$B_{k} = \epsilon_{ijk}\dot{X_{i}}\dot{X_{j}} =
(1/2)\epsilon_{ijk}[\dot{X_{i}},\dot{X_{j}}].$$ The epsilon tensor $\epsilon_{ijk}$ is defined for the indices $\{ i,j,k \}$ ranging
from $1$ to $3,$ and is equal to
$0$ if there is a repeated index and is ortherwise equal to the sign of the permutation of $123$ given by $ijk.$
We represent dot products and cross products in diagrammatic tensor notation as indicated in Figure 1 and Figure 2.
In Figure 1 we indicate the epsilon tensor by a trivalent vertex. The indices of the tensor correspond to labels for the
three edges that impinge on the vertex. The diagram is drawn in the plane, and is well-defined since the epsilon tensor is
invariant under cyclic permutation of its indices.
\bigbreak
We will define the fields $E$ and $B$ by the equations
$$B = \dot{X} \times \dot{X} \,\,\, \mbox{and} \,\,\, E =
\partial_{t}\dot{X}.$$
We will see that $E$ and $B$ obey a
generalization of the Maxwell Equations, and that this generalization describes specific discrete models.
The reader should note that this means that a significant part of the {\it form} of electromagnetism is
the consequence of choosing three coordinates of space, and the definitions of spatial and temporal derivatives with respect to them.
The background process that is being described is otherwise aribitrary, and yet appears to obey physical laws once these
choices are made.
\bigbreak
In this section we will use diagrammatic matrix methods to carry out the mathematics.
In general, in a diagram for matrix or tensor composition, we sum over all indices labeling any edge in the diagram that has no free
ends. Thus matrix multiplication corresponds to the connecting of edges between diagrams, and to the summation over
common indices. With this interpretation of compositions, view the first identity in Figure 1. This is a fundmental identity about
the epsilon, and corresponds to the following lemma.
\bigbreak
\begin{center}
$$ \picill4inby4.2in(EpsilonIdentity) $$
{ \bf Figure 1 - Epsilon Identity}
\end{center}
\bigbreak
\noindent {\bf Lemma.} (View Figure 1) Let $\epsilon_{ijk}$ be the epsilon tensor taking values $0$, $1$ and $-1$ as follows: When
$ijk$ is a permuation of $123$, then $\epsilon_{ijk}$ is equal to the sign of the permutation. When $ijk$ contains a repetition from
$\{1,2,3 \},$ then the value of epsilon is zero.
Then $\epsilon$ satisfies the following identity in terms of the Kronecker delta.
\begin{center}
$$ \picill4inby1in(LabeledEpsilonIdentity) $$
\end{center}
\bigbreak
$$\Sigma_{i} \,\epsilon_{abi}\epsilon_{cdi} = -\delta_{ad}\delta_{bc} + \delta_{ac}\delta_{bd}.$$
\bigbreak
\noindent The proof of this identity is left to the reader. The identity itself will be referred to as the {\em epsilon identity}.
The epsilon identity is a key structure in the work of this section, and indeed in all formulas involving the vector cross product.
\bigbreak
The reader should compare the formula in this Lemma with the diagrams in Figure 1. The first two diagram are two versions of the
Lemma. In the third diagram the labels are capitalized and refer to vectors $A,B$ and $C.$ We then see that the epsilon identity
becomes the formula $$A \times (B \times C) = (A \bullet C)B - (A \bullet B)C$$ for vectors in three-dimensional space
(with commuting coordinates, and a generalization of this identity to our non-commutative context. Refer to Figure 2 for the
diagrammatic definitions of dot and cross product of vectors. We take these definitions (with implicit order of multiplication)
in the non-commutative context.
\bigbreak
\begin{center}
$$ \picill4inby5.2in(DefiningDiff) $$
{ \bf Figure 2 - Defining Derivatives}
\end{center}
\bigbreak
\noindent {\bf Remarks on the Derivatives.}
\begin{enumerate}
\item Since we do not assume that $[X_{i}, \dot{X_{j}}] = \delta_{ij},$ nor do we assume $[X_{i},X_{j}]=0,$ it will not follow that
$E$ and $B$ commute with the $X_{i}.$
\item We define $$\partial_{i}(F) = [F, \dot{X_{i}}],$$ and the reader should note
that, these spatial derivations are no longer flat in the sense of section 1 (nor were they in the original Feynman-Dyson derivation).
See Figure 2 for the diagrammatic version of this definition.
\item We define $\partial_{t} = \partial/\partial t$ by the equation
$$\partial_{t}F = \dot{F} - \Sigma_{i}\dot{X_{i}}\partial_{i}(F) = \dot{F} - \Sigma_{i} \dot{X_{i}}[F, \dot{X_{i}}]$$
for all elements or vectors of elements $F.$ We take this equation as the global definition
of the temporal partial derivative, even for elements that are not commuting with the $X_{i}.$ This notion of temporal partial
derivative
$\partial_{t}$ is a least relation that we can write to describe the temporal relationship of an arbitrary non-commutative vector
$F$ and the non-commutative coordinate vector $X.$ See Figure 2 for the diagrammatic version of this definition.
\item In defining $$\partial_{t}F = \dot{F} - \Sigma_{i}\dot{X_{i}}\partial_{i}(F),$$ we
are using the definition itself to obtain a notion of the variation of $F$ with respect to time. The definition itself creates a
distinction between space and time in the non-commutative world.
\item The reader will have no difficulty verifying the following formula:
$$\partial_{t}(FG) = \partial_{t}(F)G + F\partial_{t}(G) + \Sigma_{i}\partial_{i}(F)\partial_{i}(G).$$
This formula shows that $\partial_{t}$ does not satisfy the Leibniz rule in our non-commutative context.
This is true for the original Feynman-Dyson context, and for our generalization of it. All derivations in this theory that are defined
directly as commutators do satisfy the Leibniz rule. Thus $\partial_{t}$ is an operator in our theory that does not have a
representation as a commutator.
\item We define divergence and curl by the equations
$$\nabla \bullet B = \Sigma_{i=1}^{3} \partial_{i}(B_{i})$$ and
$$(\nabla \times E)_{k} = \epsilon_{ijk}\partial_{i}(E_{j}).$$
See Figure 2 and Figure 4 for the diagrammatic versions of curl and divergence.
\end{enumerate}
\bigbreak
Now view Figure 3. We see from this Figure that it follows directly from the definition of
the time derivatives (as discussed above) that
$$\ddot{X} = \partial_{t}\dot{X} + \dot{X} \times (\dot{X} \times \dot{X}).$$
This is our motivation for defining
$$E = \partial_{t}\dot{X}$$ and
$$B = \dot{X} \times \dot{X}.$$
With these definition in place we have
$$\ddot{X} = E + \dot{X} \times B,$$ giving an analog of the Lorentz force law for
this theory.
\bigbreak
Just for the record, look at the following algebraic calculation for this derivative:
$$ \dot{F} = \partial_{t}F + \Sigma_{i} \dot{X_{i}}[F, \dot{X_{i}}]$$
$$ = \partial_{t}F + \Sigma_{i} (\dot{X_{i}}F \dot{X_{i}} - \dot{X_{i}} \dot{X_{i}} F)$$
$$ = \partial_{t}F + \Sigma_{i} (\dot{X_{i}}F \dot{X_{i}} - \dot{X_{i}} F_{i} \dot{X}) + \dot{X_{i}} F_{i} \dot{X} - \dot{X_{i}}
\dot{X_{i}} F$$
Hence $$ \dot{F} = \partial_{t}F + \dot{X} \times F + (\dot{X} \bullet F) \dot{X} - (\dot{X} \bullet \dot{X}) F$$
(using the epsilon identity).
Thus we have
$$\ddot{X} = \partial_{t} \dot{X} + \dot{X} \times (\dot{X} \times \dot{X}) + (\dot{X} \bullet \dot{X}) \dot{X} - (\dot{X} \bullet
\dot{X})\dot{X},$$
whence
$$\ddot{X} = \partial_{t}\dot{X} + \dot{X} \times (\dot{X} \times \dot{X}).$$
\bigbreak
In Figure 4, we give the derivation that $B$ has zero divergence.
\begin{center}
$$ \picill4inby6in(Xdoubledot) $$
{ \bf Figure 3 - The Formula for Acceleration}
\end{center}
\bigbreak
\begin{center}
$$ \picill4inby5in(DivB) $$
{ \bf Figure 4 - Divergence of $B$ }
\end{center}
\bigbreak
Figures 5 and 6 compute derivatives of $B$ and the Curl of $E,$ culminating in the formula
$$\partial_{t}B + \nabla \times E = B \times B.$$
In classical electromagnetism, there is no term $B \times B.$ This term is an artifact of our non-commutative context.
In discrete models, as we shall see at the end of this section, there is no escaping the effects of this term.
\bigbreak
\begin{center}
$$ \picill4inby5in(Bdot) $$
{ \bf Figure 5 - Computing $\dot{B}$}
\end{center}
\bigbreak
\begin{center}
$$ \picill4inby6in(CurlE) $$
{ \bf Figure 6 - Curl of $E$}
\end{center}
\bigbreak
\begin{center}
$$ \picill4inby6in(CurlB) $$
{ \bf Figure 7 - Curl of $B$}
\end{center}
\bigbreak
Finally, Figure 7 gives the diagrammatic proof that
$$\partial_{t}E - \nabla \times B = (\partial_{t}^{2} - \nabla^{2})\dot{X}.$$
This completes the proof of the Theorem below.
\bigbreak
\noindent {\bf Electromagnetic Theorem} With the above definitions of the operators, and taking
$$\nabla^{2} = \partial_{1}^{2} + \partial_{2}^{2} + \partial_{3}^{2}, \,\,\, B = \dot{X} \times \dot{X} \,\,\, \mbox{and} \,\,\, E =
\partial_{t}\dot{X} \,\,\, \mbox{we have}$$
\begin{enumerate}
\item $\ddot{X} = E + \dot{X} \times B$
\item $\nabla \bullet B = 0$
\item $\partial_{t}B + \nabla \times E = B \times B$
\item $\partial_{t}E - \nabla \times B = (\partial_{t}^{2} - \nabla^{2})\dot{X}$
\end{enumerate}
\bigbreak
\noindent {\bf Remark.} Note that this Theorem is a non-trivial generalization of the Feynman-Dyson derivation of electromagnetic
equations. In the Feynman-Dyson case, one assumes that the commutation relations
$$[X_{i}, X_{j}] = 0$$ and
$$[X_{i}, \dot{X_{j}}] = \delta_{ij}$$ are given, {\em and} that the principle of commutativity is assumed, so that
if $A$ and $B$ commute with the $X_{i}$ then $A$ and $B$ commute with each other. One then can interpret $\partial_{i}$ as a
standard derivative with $\partial_{i}(X_{j}) = \delta_{ij}.$ Furthermore, one can verify that $E_{j}$ and $B_{j}$ both commute with
the $X_{i}.$ From this it follows that $\partial_{t}(E)$ and $\partial_{t}(B)$ have standard intepretations and that $B \times B = 0.$
The above formulation of the Theorem adds the description of $E$ as $\partial_{t}(\dot{X}),$ a non-standard use of
$\partial_{t}$ in the original context of Feyman-Dyson, where $\partial_{t}$ would only be defined for those $A$ that commute with
$X_{i}.$ In the same vein, the last formula $\partial_{t}E - \nabla \times B = (\partial_{t}^{2} - \nabla^{2})\dot{X}$ gives a way
to express the remaining Maxwell Equation in the Feynman-Dyson context.
\bigbreak
\noindent {\bf Remark.} Note the role played by the epsilon tensor $\epsilon_{ijk}$ throughout the construction of
generalized electromagnetism in this section. The epsilon tensor is the structure constant for the Lie algebra of the rotation
group $SO(3).$ If we replace the epsilon tensor by a structure constant
$f_{ijk}$ for a Lie algebra ${\cal G}$of dimension $d$ such that the tensor is invariant under cyclic permutation ($f_{ijk} =
f_{kij}$), then most of the work in this section will go over to that context. We would then have $d$ operator/variables $X_1,
\cdots X_d$ and a generalized cross product defined on vectors of length $d$ by the equation
$$(A \times B)_{k} = f_{ijk}A_{i}B_{j}.$$
The Jacobi identity for the Lie algebra ${\cal G}$ implies that this cross product will satisfy
$$A \times (B \times C) = (A \times B) \times C + [B \times (A ] \times C)$$
where $$([B \times (A ] \times C)_{r} = f_{klr}f_{ijk}A_{i}B_{k}C_{j}.$$ This extension of the Jacobi identity
holds as well for the case of non-commutative cross product defined by the epsilon tensor.
It is therefore of interest to explore the structure of generalized non-commutative electromagnetism over other Lie algebras
(in the above sense). This will be the subject of another paper.
\bigbreak
\subsection{Discrete Thoughts}
In the hypotheses of the Electromagnetic Theorem, we are free to take any non-commutative world, and
the Electromagnetic Theorem will
satisfied in that world. For example, we can take each $X_{i}$ to be an arbitary time series of real or complex numbers, or
bitstrings of zeroes and ones. The global time derivative is defined by $$\dot{F} = J(F' - F) = [F, J],$$ where $FJ =
JF'.$ This is the non-commutative discrete context discussed in sections 1. We will write
$$\dot{F} = J\Delta(F)$$ where $\Delta(F)$ denotes the classical discrete derivative
$$\Delta(F) = F' -F.$$
With this interpretation
$X$ is a vector with three real or complex coordinates at each time, and
$$B = \dot{X} \times \dot{X} = J^{2}\Delta(X') \times \Delta(X)$$ while
$$E = \ddot{X} - \dot{X} \times (\dot{X} \times \dot{X}) = J^{2}\Delta^{2}(X) - J^{3} \Delta(X'') \times ( \Delta(X') \times
\Delta(X)).$$ Note how the non-commutative vector cross products are composed through time shifts in this context of temporal
sequences of scalars. The advantage of the generalization now becomes apparent. We can create very simple models of generalized
electromagnetism with only the simplest of discrete materials. In the case of the model in terms of triples of time series, the
generalized electromagnetic theory is a theory of measurements of the time series whose key quantities are
$$\Delta(X') \times \Delta(X)$$ and
$$\Delta(X'') \times (\Delta(X') \times \Delta(X)).$$
\bigbreak
It is worth noting the forms of the basic derivations in this model. We have, assuming that $F$ is a commuting scalar (or vector of
scalars) and taking $\Delta_{i} = X_{i}' - X_{i},$
$$\partial_{i}(F) = [F, \dot{X_{i}}] =[F, J\Delta_{i}] = FJ\Delta_{i} - J\Delta_{i}F = J(F'\Delta_{i} - \Delta_{i}F)
= \dot{F}\Delta_{i}$$ and for the temporal derivative we have
$$\partial_{t}F = J[1 - J \Delta' \bullet \Delta]\Delta(F)$$ where
$\Delta = (\Delta_{1}, \Delta_{2}, \Delta_{3}).$
\bigbreak
\noindent {\bf Acknowledgement.} Most of this effort was sponsored by the Defense
Advanced Research Projects Agency (DARPA) and Air Force Research Laboratory, Air
Force Materiel Command, USAF, under agreement F30602-01-2-05022. The U.S. Government is authorized to reproduce and distribute
reprints for Government purposes notwithstanding any copyright annotations thereon. The
views and conclusions contained herein are those of the authors and should not be
interpreted as necessarily representing the official policies or endorsements,
either expressed or implied, of the Defense Advanced Research Projects Agency,
the Air Force Research Laboratory, or the U.S. Government. (Copyright 2005.)
It gives the author great pleasure to acknowledge support from NSF Grant DMS-0245588 and to thank
Pierre Noyes and Keith Bowden for continuing conversations related to the contents of this paper.
\bigbreak
|
{
"timestamp": "2005-04-03T20:31:16",
"yymm": "0503",
"arxiv_id": "quant-ph/0503198",
"language": "en",
"url": "https://arxiv.org/abs/quant-ph/0503198"
}
|
\section{Introduction}
Recently, numerous efforts are being made to describe the
large-scale evolution of the Earth ecosystem. Due to the
ecosystem's complexity, however, such a task is extremely
difficult~(\cyt{pimm}). That is why researchers in this field have
to turn to very simplified and abstract models that hopefully
still contain relevant factors. Such an approach proved to be
successful, e.g., in modelling of some aspects of extinction
dynamics (\cyt{NEWMAN}). Indeed, in very simple models of
ecosystems certain properties of extinctions as, e.g., the
distribution of sizes or durations of extinctions, seem to agree,
at least qualitatively, with palaeontological data
(\cyt{BAKSNEPP}; \cyt{SOLE}). In these models, the dynamics
spontaneously drives the ecosystem toward the scale-invariant
state with extinctions described by some power-law
characteristics. However, since the accuracy of fossil data is
rather limited, especially with respect to events on a large
timescale, the applicability of such models should be considered
with care.
The suggestion that the extinction dynamics is not scale invariant
but it has a characteristic timescale was made by Raup and
Sepkoski (\cyt{raupsep}). While analysing fossil data, they
noticed that during the last 250 My mass extinctions on Earth
appeared more or less cyclically with a period of approximately
26My. Although their analysis was initially questioned
(\cyt{patterson}), some other works confirmed Raup and Sepkoski's
hypothesis (\cyt{fox}; \cyt{prokoph}; \cyt{plotnick}). The
suggested large periodicity of mass extinctions turned out to be
very difficult to explain. Indeed, 26My does not seem to match any
of known Earth cycles and some researchers have been looking for
more exotic explanations involving astronomical effects
(\cyt{theories1}; \cyt{theories2}), increased volcanic activity
(\cyt{stot1}), or the Earth's magnetic field reversal
(\cyt{stot2}). So far, however, none of these proposals has been
confirmed. One should also note that the most recent analysis of
palaeontological data that span last 542My strongly supports the
periodicity of mass extinctions albeit with a larger cycle of
about 62My (\cyt{rohde}).
Lacking a firm evidence of an exogenous cause, one can ask whether
the periodicity of extinctions can be explained without referring
to such a factor. It is already well known that a periodic
behaviour of a system is not necessarily the result of periodic
driving. In particular, since the seminal works of Lotka and
Volterra, it is known that spontaneous oscillations of the
population size might appear in various prey-predator systems~
(\cyt{LV}). However, the period of oscillations in such systems is
determined by the growth and death rate coefficients of
interacting species and is of the order of a few years rather than
tens of millions. Consequently, if the periodicity of mass
extinctions is to be explained within a model of interacting
species, a different mechanism that generates long-period
oscillations must be at work.
Recently, a multi-species prey-predator model has been introduced,
where long-term oscillatory behaviour is observed
(\cyt{lipowski}). Only some preliminary studies of basic
properties of this model have been made, and the objective of the
present paper is to provide its more detailed analysis. In this
model the period of oscillations is determined by the inverse of
the mutation rate and as we argue, it should be several orders of
magnitude longer than in the Lotka-Volterra oscillations. The
mechanism that generates oscillations in our model can be briefly
described as follows: A coevolution of predator species induced by
the competition for food and space causes a gradual increase of
their size. However, such an increase leads to the overpopulation
of large predators and a shortage of preys. It is then followed by
a depletion of large species and a subsequent return to the
multi-species stage with mainly small species that again gradually
increase their size and the cycle repeats. Numerical calculations
for our model show that the longevity of a species depends on the
evolutionary stage at which the species is created. A similar
pattern has been observed in some palaeontological data
(\cyt{aimiller}) and, to our knowledge, the presented model is the
first one that reproduces such a dependence. Let us notice that
the oscillatory behaviour in a prey-predator system that was also
attributed to the coevolution has been already examined by
Dieckmann et al.~(\cyt{DIECKMANN}). In their model, however, the
number of species is kept constant and it cannot be applied to
study extinctions. Moreover, the idea that an internal ecosystem
dynamics might be partially responsible for the long-term
periodicity in the fossil records was suggested by
Stanley~(\cyt{STANLEY}) and later examined by Plotnick and
McKinney (\cyt{plotnick1993}). However, in his approach mass
extinctions are triggered by external impacts. Their approximately
equidistant separation is the result of a delayed recovery of the
ecosystem. In our approach no external factor is needed to trigger
such extinctions and sustain their approximate periodicity.
\section{Model}
Numerical simulations and models of various levels of description
have been frequently used to study extinctions of species
(\cyt{NEWMAN}). In the simplest cases, the dynamics of models was
formulated at the level of species and had to refer to the notion
of fitness that is not commonly accepted. In more recent
approaches, an individual-oriented dynamics has often been used
and although computationally more demanding, such models are
considered as more adequate (\cyt{higgs}; \cyt{stauffer};
\cyt{fdl}). Our model uses the individual-oriented dynamics but in
addition it is spatially extended. Such a feature increases the
computational complexity even more but it also takes into account,
e.g., dynamically generated spatial inhomogeneities that sometimes
are known to play an important role. Our model can be also
considered as a multi-species generalization of the already
studied spatially extended prey-predator model (\cyt{lip1999};
\cyt{lip2000}). Some other multi-species lattice models were also
studied in various contexts (\cyt{pekal}; \cyt{SATO};
\cyt{DIECKMANN2000}).
Our model describes a multi-species prey-predator system defined
on a square lattice of linear size $N$ (\cyt{lipowski}). At each
site of a lattice $i$ there is an operator $x_i$ that specifies
whether this site is occupied by a prey ($x_i=1$), by a predator
($x_i=2$), by both of them ($x_i=3$), or is empty ($x_i=0$). Each
predator is characterized by its size $m \ (0<m<1)$ that
determines its consumption rate and at the same time its strength
when it competes with other predators. Only approximately the size
$m$ can be considered as related with physical size. Predators and
preys evolve according to rules typical to such systems (e.g.,
predators must eat preys to survive, preys and predators can breed
provided that there is an empty site nearby, etc.). In addition,
the relative update rate for preys and predators is specified by
the parameter $r \ (0<r<1)$ and during breeding mutations are
taking place with the probability $p$. More detailed definition of
the model dynamics is given below:\\
(a) Choose a site at random (the chosen site is denoted by $i$).\\
(b) Provided that $i$ is occupied by a prey (i.e., if $x_i=1$ or
$x_i=3$) update the prey with the probability $r$. If at least one
neighbor (say $j$) of the chosen site is not occupied by a prey
(i.e., $x_j=0$ or $x_j=2$), the prey at the site $i$ produces an
offspring and places it on an empty neighboring site (if there are
more empty sites, one of them is chosen randomly). Otherwise
(i.e., if there are no empty sites) the prey does not breed.\\
(c) Provided that $i$ is occupied by a predator (i.e., $x_i=2$ or
$x_i=3$) update the predator with the probability $(1-r)m_i$,
where $m_i$ is the size of the predator at site $i$. If the chosen
site $i$ is occupied by a predator only ($x_i=2$), it dies, i.e.,
the site becomes empty ($x_i=0$). If there is also a prey there
($x_i=3$), the predator consumes the prey (i.e., $x_i$ is set to
2) and if possible, it places an offspring at an empty neighboring
site. For a predator of the size $m_i$ it is possible to place an
offspring at the site $j$ provided that $j$ is not occupied by a
predator ($x_j=0$ or $x_j=1$) or is occupied by a predator
($x_j=2$ or $x_j=3$) but of a smaller size than $m_i$ (in such a
case the smaller-size predator is replaced by an offspring of the
larger-size predator). The offspring inherits its parent's size
with the probability $1-p$ and
with the probability $p$ it gets a new size that is drawn from a uniform distribution.\\
At first sight one can think that such a model describes an
ecosystem with two trophic levels (preys and predators) and only
with predators being equipped with evolutionary abilities, which
would be of course highly unrealistic. Let us notice, however,
that expansion of predators sometimes proceeds at the expense of
smaller-size predators. Thus, predators themselves are involved
in prey-predator-like interactions. Perhaps it would be more
appropriate to consider unmutable preys as a renewable (at a
finite rate) source of, e.g., energy, and predators as actual
species involved in various prey-predator interactions and
equipped with evolutionary abilities.
\section{Results}
To examine the behaviour of this model we used numerical
simulations. Our results, shown in
Figs.\ref{okna}-\ref{spsize}, are obtained for $r=0.2$ but we
expect (\cyt{lipowski}) that the behaviour of the model should be
qualitatively the same for any $r<0.27$ (a brief discussion of the
behaviour of the model for $r>0.27$ is given at the end of this
section).
\subsection{Oscillatory behaviour}
In Fig.\ref{okna}B one can see that, indeed, the number of species
$s$ exhibits pronounced irregular oscillations. These oscillations
are coupled with more regular oscillations of the averaged (over
all predators) size $m_a$ (Fig.\ref{okna}A) and maxima of $s$
correspond approximately to minima of $m_a$ and vice versa. To
have a better understanding of the behaviour of the model we also
calculated the size $m_d$ of the dominant species (i.e., the
predator species with the largest number of individuals) and the
results are shown in Fig.\ref{okna}A.
\begin{figure}
\vspace{-0cm} \centerline{ \epsfxsize=9cm \epsfbox{okna_jtb.eps} }
\caption{The results of numerical simulations ($N=500$,\
p=0.00001). Data on both panels are obtained from the same run. In
our simulations a unit of time is defined as a single on average
update of each site. (A) The time dependence of the average size
$m_a$ (dashed line) and the size of the dominant species $m_d$
(short, horizontal intervals). (B) The number of species $s$
(continuous line) and the averaged lifetime of species (+). After
extinction the number of species drops 3-4 times. To reduce
stochastic noise in the calculation of the average lifetime data
are collected in time windows of the width $\Delta t = 3000$.}
\label{okna}
\end{figure}
These results indicate that the behaviour of our model can be
described as follows: In a species-rich interval the size $m_a$ is
typically quite low and there is an abundance of preys. In such a
case predators of a large size are in a more favorable position
(because a larger predator can replace a smaller predator) and as
a result $m_a$ and $m_d$ increase. The process of increasing the
size is gradual and involves a large number of species and is not
related to a creation of a single (very-efficient) species, as
suggested previously (\cyt{lipowski}). The increased size $m$
implies a higher consumption rate and due to a finite recovery
rate of preys the large-size species, that at this stage dominate
the system, are running out of food. At first sight one might
expect that further evolution will gradually reduce $m_a$ and
$m_d$. Numerical results show, however (Fig.\ref{okna}A), that
after reaching a local maximum, $m_d$ jumps to a very low value.
This indicates that abrupt changes take place in the model after
which large-size species are no longer dominant and vast majority
of them become extinct. At the same time, however, a lot of new,
mainly small-size species is created and that increases the
diversity $s$ (although we do not suggest that this was really the
cause, a succession of small mammals after large dinosaurs could
be a vivid example of such a change). In such a way the system
returns to the initial species-rich state. Such a cycle is also
illustrated in Fig.\ref{sizes} that shows the distribution of size
$m$ at various stages of the evolution\footnote{ Dynamics of the
model is also illustrated with a Java applet available at: http://
spin.amu.edu.pl/lipowski/prey\_pred.html}.
\begin{figure}
\vspace{0cm} \centerline{ \epsfxsize=9cm \epsfbox{sizes_jtb.eps} }
\caption{Four panels show distribution of sizes of species at
various stages of evolutionary cycle ($N=1000$,\ p=0.0001). The
upper panel shows the time dependence of the number of species
$s$.} \label{sizes}
\end{figure}
Gradual increase of size of species recalls the Cope's rule that
states that species tend to increase body size over geological
time. This rule is not commonly accepted among paleontologists and
evolutionists and was questioned on various grounds
(\cyt{STANLEY1973}). However, recent studies of fossil records of
mammal species are consistent with this rule (\cyt{ALROY};
\cyt{VAN}). Perhaps our model could suggests a way to obtain a
theoretical justification of this rule.
From the above description, it is expected that the periodicity of
such a cycle increases when the mutation rate $p$ decreases, and
such a behaviour is confirmed with more detailed calculations
(\cyt{lipowski}). In particular, already for $p=10^{-5}$ the
estimated (\cyt{lipowski}) periodicity of oscillations in our
model is approximately 1000 times larger than that of the
Lotka-Volterra oscillations in the corresponding single-predator
system. It shows that the oscillations in our model are indeed
long-period and, perhaps for smaller $p$, on a timescale close to
26My.
Although very complicated, in principle, it should be possible to
estimate the value of the mutation probability $p$ from the
mutational properties of living species. Let us notice that in our
model mutations produce an individual that might be substantially
different from its parent. In Nature, this is typically the result
of many cumulative mutations and thus we expect that $p$ is indeed
a very small quantity. Actually, $p$ should be considered rather
as a parameter related with the speed of morphological and
speciation processes that are known to be typically very slow
(\cyt{gingerich}). Perhaps a modification of the mutation
mechanism where a new species will be only a small modification of
its parental species could be more suitable for comparison with
living species, but it might require longer calculations.
Alternatively, one can try to estimate the parameters $p$, $N$,
and $r$ (or at least their ratios) by matching the behaviour of
our model with some characteristics of the ecosystem such as the
period of oscillations (26My), fraction of extinct species during
a mass extinction or the average lifetime of species as compared
with the periodicity of mass extinctions.
The oscillatory behaviour sets probably the largest timescale in our
model. However, on the shorter timescale some characteristics,
such as, e.g., the number of species, exhibit strong fluctuations
(Fig.~\ref{okna}B). On such a timescale some distributions might
be very broad and resemble power-law distributions. Indeed, such a
behaviour was demonstrated for the distribution of lifetimes of
species in our model (\cyt{lipowski}).
The increase of the size of species in our model resembles the
fitness-increasing evolution in the real ecosystem. It is tempting
to consider present-day large mammals as highly adapted dominant
species and, in the context of our model, located perhaps close to
the local maximum in the fitness space (as in Fig.\ref{okna}A). If
so, then according to our model, the next dominant species most
likely will be a small-size species that at the moment might not
even exist. Its dominance will be possible due to drastic and
inevitable changes of our ecosystem. Putting aside the validity of
our model, such a scenario does not seem unlikely.
\subsection{Longevity of species}
An analysis of palaeontological data (\cyt{aimiller}) shows that
the longevity is larger for species created after mass extinctions
than for other species. To compare such a result with the
predictions of our model, we calculated the average lifetime of
species. It turns out, however, that important contributions to
this quantity are coming from short-living species and their
lifetime is essentially independent on the evolutionary phase at
which they are born. To reduce this effect we took into account
only the species that lived longer than a given threshold, which
we set equal to 30. Fossils of species of short lifetime are
rather scarce and palaeontological data also reflect a similar
bias toward long-lifetime species. The obtained results are shown
in Fig.\ref{okna}B. Although still strongly fluctuating, they
clearly show that the lifetime is correlated with the global
evolution of the ecosystem and they qualitatively agree with
palaeontological data. In particular, the maximum lifetime appears
for species born shortly after a large and abrupt decrease of the
size of the dominant species (crash). Apparently, species created
at this time find most favourable conditions while the worst
conditions exist shortly before a crash. Again using the analogy
with the real ecosystem and humans, the model predicts (not
counter-intuitively) that species created during our dominance
will have a rather short lifetime.
\begin{figure}
\vspace{0cm} \centerline{ \epsfxsize=9cm\epsfbox{rate_tau_jtb.eps}
}
\caption{The average lifetime of species $\tau$ as a function of
the size $m$ ($N=500$).} \label{rate_tau.eps}
\end{figure}
For a species to have a very small size $m$ is usually a
disadvantage since such a species will loose in competition with
other species. On the other hand, a large size implies a high
consumption rate and such a species might suffer from lack of
food. It means that a lifetime of a species as a function of $m$
should have a maximum at a certain intermediate value and
numerical calculations confirm such a behaviour (see
Fig~\ref{rate_tau.eps}). Some data on distribution of sizes in
Pleistocene and Recent molluscan faunas do show some maximum
(\cyt{jablonsky}) but a more detailed comparison cannot be done
yet.
As our last result, we present the calculation of the average
population size of species of a given lifetime (Fig.\ref{spsize}).
Although all the curves look qualitatively similar, one can notice
a small difference between short- and long-lifetime species. This
difference is better seen on the rescaled plot
(Fig.\ref{spsize}B). This data suggest that population sizes for
species of a lifetime much shorter than the periodicity of
extinctions (which in this case (\cyt{lipowski}) is around 3000)
after rescaling fall into a single curve. For species of a
lifetime comparable or larger than the periodicity of extinctions
the data will deviate from such a universal curve. Although we are
not aware of any palaeontological data of this kind, a comparison
could provide an interesting test of our model.
\begin{figure}
\vspace{0cm} \centerline{ \epsfxsize=9cm \epsfbox{spsize_jtb.eps}}
\caption{The analysis of the time dependence of the population
size. (A) The average population size of species with a given
lifetime ($p=0.001$, $N=500$). (B) Some data from panel (A)
rescaled (i.e., multiplied by some factors in both directions) in
such a way that the lifetime and the maximal population size
overlap. For species with the lifetime equal to 100, 150, and 200,
the rescaled population sizes nearly overlap. Some deviations from
the overlapping data can be seen for the lifetime 1000 and 2000.}
\label{spsize}
\end{figure}
\subsection{Unique code and the emergence of a multi-species ecosystem}
All living cells use the same code that is responsible for the
transcription of information from DNA to proteins (\cyt{orgel};
\cyt{szathmary}). It suggests that at a certain point of evolution
of life on Earth a replicator that invented this apparently
effective mechanism was able to eliminate replicators of all other
species (if they existed) and establish, at least for a short
time, a single-species ecosystem. Although this process is still
to a large extent mysterious, one expects that subsequent
evolution of these successful replicators leads to their
differentiation and proliferation of species. In such a way the
ecosystem shifted from a single- to multi-species one
(\cyt{lipowski2000}).
It seems to us that the present model might provide some insight
into this problem. As we have already mentioned, the oscillatory
behaviour appears in our model only for the relative update rate
$r<0.27$. When preys reproduce faster ($r>0.27$), a different
behaviour can be seen (\cyt{lipowski}) and the model reaches a
steady state with almost all predators belonging to the same
species with the size $m$ close to 1. Only from time to time a new
species is created with even larger $m$ and a change of the
dominant species might take place. In our opinion, it is possible
that at the very early period of evolution of life on Earth, the ecosystem
resembled the case $r>0.27$. This is because at that time
substrates ('preys') were renewable faster than primitive
replicators ('predators') could use them. If so, every invention
of the increase of the efficiency ('size') could invade the entire
system. In particular, the invention of the coding mechanism could
spread over the entire system. A further evolution increased the
efficiency of predators and that effectively shifted the
(single-species) ecosystem toward
the $r<0.27$ (multi-species, oscillatory) regime.\\
\par
\section{Conclusions}
In the present paper we examined a spatially extended
multi-species prey-predator model. In a certain regime in this
model densities of preys and predators as well as the number of
species show long-term oscillations, even though the dynamics of
the model is not exposed to any external periodic forcing. It
suggests that the oscillatory behaviour of the Earth ecosystem
predicted by Raup and Sepkoski could be simply a natural feature
of its dynamics and not the result of an external factor. Some
predictions of our model such as the lifetime of species or the
time dependence of their population sizes might be testable
against palaeontological data. The prediction that a lifetime of
species depends on the evolutionary stage at which it was created,
that qualitatively agrees with fossil data ~(\cyt{aimiller}),
suggests that a further study of this model would be desirable.
Certainly, our model is based on some restrictive assumptions that
drastically simplifies the complexity of the real ecosystem. We
hope, however, that it includes some of its important
ingredients: replication, mutation, and competition for resources
(food and space). As an outcome, the model shows that typically
there is no equilibrium-like solution and the ecosystem remains in
an evolutionary cycle. The model does not include geographical
barriers but let us notice that palaeontological data that suggest
the periodicity of mass extinctions are based only on marine
fossils (\cyt{rohde}). More realistic versions should take into
account additional trophic levels, gradual mutations, or sexual
reproduction. One should also notice that the palaeontological
data are mainly at a genus, and not species level. It would be
desirable to check whether the behaviour of our model is in some
sense generic or it is merely a consequence of its specific
assumptions. An interesting possibility in this respect could be
to recast our model in terms of Lotka-Volterra like equations and
use the methodology of adaptive dynamics developed by Dieckmann et
al. (\cyt{DIECKMANN}).
Of course, the real ecosystem was and is exposed to a number of
external factors such impacts of astronomical objects, volcanism
or climate changes. Certainly, they affect the dynamics of an
ecosystem and contribute to the stochasticity of fossil data.
Filtering out these factors and checking whether the main
evolutionary rhythm is indeed set by the ecosystem itself, as
suggested in the present paper, is certainly a difficult task but
maybe worth an effort.
\vspace{5mm}
\noindent {\bf Acknowledgements}\\
The research grant 1 P03B 014 27 from KBN is gratefully
acknowledged. We thank Department of Physics of the University of
Aveiro (Portugal) for giving us access to computing facilities.\\
\vspace{5mm}
\begin{center}
REFERENCES
\end{center}
|
{
"timestamp": "2006-01-06T21:14:37",
"yymm": "0503",
"arxiv_id": "q-bio/0503020",
"language": "en",
"url": "https://arxiv.org/abs/q-bio/0503020"
}
|
\section{Quarkonia and heavy flavors: what is different at the LHC}
\label{widatl}
With a nucleus-nucleus center-of-mass energy nearly 30 times larger
than the one reached at RHIC, the LHC will open a new era for studying
the properties of strongly interacting matter at extreme energy
densities~\cite{Carminati:2004fp}. One of the most exciting aspects
of this new regime is the abundant production rate of hard probes
which can be used, for the first time, as high statistics probes of
the medium~\cite{Bedjidian:2003gd}. Futhermore, heavy flavor
measurements at the LHC should provide a comprehensive understanding
of open and hidden heavy flavor production at very low $x$ values,
where strong nuclear gluon shadowing is expected. The heavy flavor
sector at LHC energies is subject to other significant differences
with respect to SPS and RHIC energies. First, the large production
rate offers the possibility to use a large variety of observables.
Then, the magnitude of most of the in-medium effects is dramatically
enhanced. Some of these aspects are discussed hereafter.
\subsection{New observables}
The Table~\ref{qqbar} shows the number of $c\bar{c}$ and $b\bar{b}$
pairs produced in central A-A collisions at SPS, RHIC and LHC. From
RHIC to LHC, there are 10 times more $c\bar{c}$ pairs and 100 times
more $b\bar{b}$ pairs produced. Therefore, while at SPS only
charmonium states are experimentally accessible and at RHIC it remains
to be seen how much of the bottom sector can be explored, at the LHC both
charmonia and bottomonia can be used, thus providing powerful probes
for Quark Gluon Plasma (QGP) studies. In fact, since the
$\Upsilon(1S)$ state only dissolves significantly above the critical
temperature~\cite{Digal:2001ue}, at a value which might only be
reachable above that of RHIC, the spectroscopy of the $\Upsilon$
family at the LHC should reveal unique characteristics of the
QGP~\cite{Gunion:1996qc}. In addition to the centrality dependence of
the $\Upsilon$ yield, the study of the $\Upsilon^\prime/\Upsilon$
ratio versus transverse momentum ($p_{\rm T}$) is believed to be of
crucial interest~\cite{Gunion:1996qc} (see below).
\begin{table}[ht]
\centering
\caption{Number of $c\bar{c}$ and $b\bar{b}$ pairs produced in central
heavy-ion collisions ($b=0$) at SPS (Pb-Pb), RHIC (Au-Au), and LHC
(Pb-Pb) energies. $b\bar{b}$ production is negligible at the SPS.}
\label{qqbar}
\begin{tabular}{lccc}
\hline\noalign{\smallskip}
& SPS & RHIC & LHC \\
\noalign{\smallskip}\hline\noalign{\smallskip}
N($c\bar{c}$) & 0.2 & 10 & 130 \\
N($b\bar{b}$) & -- & 0.05 & 5 \\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}
On the other hand, studies with open heavy flavors also benefit from
high statistics measurements. In particular, as shown in the
following, the reconstruction of the $p_{\rm T}$ distribution of $D^0$
mesons in the hadronic channel should provide valuable information on
in-medium induced $c$ quark energy loss.
\subsection{Large quarkonium nuclear absorption}
Charmonium measurements at the SPS have shown that a detailed
understanding of the normal nuclear absorption is mandatory in order
to reveal any anomalous suppression behavior~\cite{louis}. According
to Ref.~\cite{Bedjidian:2003gd}, the following observations can be
made:
\begin{itemize}
\item the J/$\psi$ nuclear absorption in central Pb-Pb collisions is
two times larger at the LHC than at the SPS;
\item the J/$\psi$ nuclear absorption in central Ar-Ar collisions at
the LHC is similar to the one in central Pb-Pb collisions at the SPS;
\item the $\Upsilon$ nuclear absorption in central Pb-Pb collisions at
the LHC is similar to the J/$\psi$ nuclear absorption in central Pb-Pb
collisions at the SPS.
\end{itemize}
\subsection{Large resonance dissociation rate}
It has been realized that, in addition to the normal nuclear
absorption, the interactions with comoving hadrons and the melting by
color screening, quarkonia can also be significantly destroyed by
gluon ionization~\cite{Xu:1995eb}. Since this mechanism results from
the presence of quasi-free gluons, it starts being effective for
temperatures above the critical temperature but not necessarily above
the resonance dissociation temperature by color screening. Recent
estimates~\cite{Bedjidian:2003gd} (see Ref.~\cite{Blaschke:2004dv} for
an update) of the quarkonium dissociation cross-sections show that
none of the J/$\psi$ mesons survives the deconfined phase at the LHC
and that about 80\,\% of the $\Upsilon$ are destroyed. Significant
information about the initial temperature and lifetime of the QGP
should be extracted from the $\Upsilon$ suppression pattern.
\subsection{Large charmonium secondary production}
An important yield of secondary charmonia is expected from $B$ meson
decays~\cite{Eidelman:2004wy}, $D\overline{D}$
annihilation~\cite{Ko:1998fs}, statistical
hadronization~\cite{Braun-Munzinger:2000px} and kinetic
recombination~\cite{Thews:2000rj}. Contrary to the two first
processes, the two last ones explicitly assume the formation of a
deconfined medium. The underlying picture is that charmonium
resonances form by coalescence of free $c$ and $\bar{c}$ quarks in the
QGP~\cite{Thews:2000rj} or at the hadronization
stage~\cite{Braun-Munzinger:2000px}. According to these models, the
QGP should lead to an increase of the J/$\psi$ yield versus
centrality, roughly proportional to ${\rm N}^2(c\bar{c})$, instead of
a suppression. Due to the large number of $c\bar{c}$ pairs produced
in central heavy ion collisions at the LHC, these models predict a
spectacular enhancement of the J/$\psi$ yield; up to a factor 100
relative to the primary production
yield~\cite{Bedjidian:2003gd,Andronic:2003zv}. Although the
statistical accuracy of the present RHIC data cannot confirm or rule
out such mechanisms~\cite{robert}, it is interesting to extrapolate
from secondary charmonium production at RHIC to secondary bottomonium
production at the LHC. Indeed, the expected multiplicity of
$b\bar{b}$ pairs at the LHC is roughly equal to the expected
multiplicity of $c\bar{c}$ pairs at RHIC (Table~\ref{qqbar}).
Therefore, if secondary production of charmonia is observed at RHIC,
it is conceivable to expect the same formation mechanism for
bottomonium states at the LHC.
\subsection{Complex structure of dilepton yield}
The dilepton mass spectrum at the LHC exhibits new features,
illustrated in Fig.~\ref{smbat2}. It can be seen that, with a low
$p_{\rm T}$ threshold of around 2~GeV/$c$ on the decay leptons,
unlike-sign dileptons from bottom decay dominate the dilepton
correlated component over all the mass range. These dileptons have
two different origins. In the high invariant mass region, each lepton
comes from the direct decay of a $B$ meson (the so-called $BB$-diff
channel). In the low invariant mass region, both leptons come from
the decay of a single $B$ meson via a $D$ meson (the so-called
$B$-chain channel). Next to leading order processes, such as gluon
splitting, also populate significantly the low mass dilepton spectrum
due to their particular kinematics. Then, as discussed in more detail
below, a substantial fraction of the J/$\psi$ yield arises from bottom
decays. Finally, a sizeable yield of like-sign correlated dileptons
from bottom decays is present. This contribution arises from the
peculiar decay chain of $B$ mesons and from $B$ meson oscillations
(see below). Its yield could be even larger than the yield of
unlike-sign correlated dileptons from charm.
\begin{figure}
\begin{center}
\resizebox{0.35\textwidth}{!}{\includegraphics{smbat2.epsi}}
\end{center}
\caption{Invariant mass spectra of dimuons produced in central
($b<3$~fm) Pb-Pb collisions in the ALICE forward muon
spectrometer~\cite{smbat}, with a $p_{\rm T}$ cut of 2~GeV/$c$
applied to each single muon. The lines correspond to: like-sign
correlated dimuons from bottom (dotted); unlike-sign correlated
dimuons from charm (dash-dotted) and from bottom (dashed);
unlike-sign correlated and unlike-sign non-correlated pairs (solid).}
\label{smbat2}
\end{figure}
\section{The LHC heavy ion program}
The LHC will be operated several months per year in pp mode and
several weeks in heavy-ion mode. The corresponding effective time for
rate estimates is $10^7$~s for pp and $10^6$~s for heavy-ion
operation. As described in Ref.~\cite{Carminati:2004fp}, the
``heavy-ion runs'' include, during the first five years of operation,
one Pb-Pb run at low luminosity, two Pb-Pb runs at high luminosity,
one p-A run and one light-ion run. In the following years different
options will be considered, depending on the first results. Three of
the four LHC experiments are expected to take heavy-ion data.
\subsection{ALICE}
ALICE (A Large Ion Collider Experiment) is the only LHC experiment
dedicated to the study of nucleus-nucleus collisions~\cite{ALICEWEB}.
The detector is designed to cope with large charged particle
multiplicities which, in central Pb-Pb collisions, are expected to be
between 2000 and 8000 per unit rapidity at mid rapidity. The detector
consists of a central barrel ($|\eta|<0.9$), a forward muon
spectrometer $(2.5<\eta<4$) and several forward/backward and central
small acceptance detectors. Heavy flavors will be measured in ALICE
through the electron channel and the hadron channel in the central
barrel as well as through the muon channel in the forward region.
Note that, contrary to the other LHC experiments, ALICE will be able
to access most of the signals down to very low $p_{\rm T}$.
\subsection{CMS}
CMS (Compact Muon Solenoid)~\cite{CMSWEB} is designed for high $p_{\rm
T}$ physics in pp collisions but has a strong heavy ion
program~\cite{cms}. This program includes jet reconstruction,
quarkonia measurements (in the dimuon channel) and high mass dimuon
measurements. The detector acceptance, for quarkonia measurements,
ranges from $-2.5$ to 2.5 in $\eta$, with a $p_{\rm T}$ threshold of
3~GeV/$c$ on single muons. Such a $p_{\rm T}$ cut still allows the
reconstruction of $\Upsilon$ states down to $p_{\rm T} = 0$ but limits
J/$\psi$ measurements to high $p_{\rm T}$.
\subsection{ATLAS}
Like CMS, ATLAS (A Toroidal LHC ApparatuS)~\cite{ATLASWEB} is designed
for pp physics. The detector capabilities for heavy ion physics have
been investigated recently~\cite{atlas}. As far as heavy flavors are
concerned, the physics program will focus on measurements of $b$-jets
and $\Upsilon$. The detector acceptance for muon measurements is
large in $\eta$ ($|\eta|<2.4$) but, like CMS, is limited to high
$p_{\rm T}$.
\section{Selected physics channels}
\subsection{Quarkonia}
\subsubsection{Centrality dependence of resonance yields}
The centrality dependence of the quarkonium yield, in the $\mu\mu$
channel, has been simulated in the ALICE detector. From the results,
displayed in Table~\ref{smbatTable}, the following comments can be
made. The statistics of J/$\psi$ events is large and should allow for
narrower centrality bins. The $\psi^\prime$ measurement is rather
uncertain, because of the small signal to background ratio (S/B). The
$\Upsilon$ and $\Upsilon^\prime$ statistics and significance are quite
good and the corresponding S/B ratios are almost always greater
than~1. On the other hand, the $\Upsilon^{\prime\prime}$ suffers from
limited statistics. The resonances will also be measured in the
dielectron channel in ALICE~\cite{TRDTP}, and in the dimuon channel in
CMS~\cite{cms} and ATLAS~\cite{atlas}, providing consistency
cross-checks and a nice complementarity in acceptance. A recent
study~\cite{sudhir} demonstrated the capabilities of ALICE to
measure the resonance azimuthal emission angle with respect to the
reaction plane. Such measurements are of particular importance given
the latest RHIC results on open charm elliptic
flow~\cite{Kelly:2004qw}.
\begin{table}[ht]
\caption{Preliminary yield (S), signal over background (S/B) and
significance (${\rm S}/\sqrt{\rm S+B}$) for quarkonium resonances
measured versus centrality in the ALICE forward muon
spectrometer~\cite{smbat}. The input cross-sections are taken from
Ref.~\cite{Bedjidian:2003gd}. Shadowing is taken into account. Any
other suppression or enhancement effects are not included. The
numbers correspond to one month of Pb-Pb data taking and are extracted
with a 2$\sigma$ mass cut.}
\label{smbatTable}
\begin{tabular}{lllllll}
\hline\noalign{\smallskip}
& $b$ (fm) & 0-3 & 3-6 & 6-9 & 9-12 & 12-16 \\
\noalign{\smallskip}\hline\noalign{\smallskip}
& S $(\times 10^3)$ & 86.48 & 184.6 & 153.3 & 67.68 & 10.46 \\
J/$\psi$ & S/B & 0.167 & 0.214 & 0.425 & 1.237 & 6.243 \\
& ${\rm S}/\sqrt{\rm S+B}$ & 111.3 & 180.4 & 213.8 & 193.4 & 94.95 \\
\noalign{\smallskip}\hline\noalign{\smallskip}
& S $(\times 10^3)$ & 1.989 & 4.229 & 3.547 & 1.565 & 0.24 \\
$\psi^\prime$ & S/B & 0.009 & 0.011 & 0.021 & 0.063 & 0.273 \\
& ${\rm S}/\sqrt{\rm S+B}$ & 4.185 & 6.902 & 8.604 & 9.641 & 7.171 \\
\noalign{\smallskip}\hline\noalign{\smallskip}
& S $(\times 10^3)$ & 1.11 & 2.376 & 1.974 & 0.83 & 0.118 \\
$\Upsilon$ & S/B & 2.084 & 2.732 & 4.31 & 7.977 & 12.01 \\
& ${\rm S}/\sqrt{\rm S+B}$ & 27.39 & 41.71 & 40.03 & 27.16 & 10.42\\
\noalign{\smallskip}\hline\noalign{\smallskip}
& S $(\times 10^3)$ & 0.305 & 0.653 & 0.547 & 0.229 & 0.032 \\
$\Upsilon^\prime$ & S/B & 0.807 & 1.043 & 1.661 & 2.871 & 4.319 \\
& ${\rm S}/\sqrt{\rm S+B}$ & 11.68 & 18.26 & 18.48 & 13.02 & 5.077 \\
\noalign{\smallskip}\hline\noalign{\smallskip}
& S $(\times 10^3)$ & 0.175 & 0.376 & 0.312 & 0.13 & 0.019 \\
$\Upsilon^{\prime\prime}$ & S/B & 0.566 & 0.722 & 1.18 & 1.936 & 3.024 \\
& ${\rm S}/\sqrt{\rm S+B}$ & 7.951 & 12.55 & 13 & 9.274 & 3.73 \\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}
\subsubsection{$\Upsilon^\prime/\Upsilon$ ratio versus $p_{\rm T}$}
The $p_{\rm T}$ dependence of resonance suppression was recognized
very early as a relevant observable to probe the characteristics of
the deconfined medium~\cite{Blaizot:1987ha}. Indeed, the $p_{\rm T}$
suppression pattern of a resonance is a consequence of the competition
between the resonance formation time and the QGP temperature, lifetime
and spatial extent. However, quarkonium suppression is known to
result not only from deconfinement but also from nuclear effects like
shadowing and absorption. In order to isolate pure QGP effects, it
has been proposed to study the $p_{\rm T}$ dependence of quarkonium
ratios instead of single quarkonium $p_{\rm T}$ distributions. By
doing so, nuclear effects are washed out, at least in the $p_{\rm T}$
variation of the ratio\footnote{Using ratios has the additional
advantage that systematical detection inefficiencies cancel out to
some extent.}. Following the arguments of Ref.~\cite{Gunion:1996qc},
the capabilities of the ALICE muon spectrometer to measure the $p_{\rm
T}$ dependence of the $\Upsilon^\prime/\Upsilon$ ratio in central
(10\,\%) Pb-Pb collisions have been recently
investigated~\cite{ericTHESIS}. Two different QGP models with
different system sizes were considered. The results of the
simulations (Fig.~\ref{ericFIG}) show that, with the statistics
collected in one month of data taking, the measured
$\Upsilon^\prime/\Upsilon$ ratio exhibit a strong sensitivity to the
characteristics of the QGP. Note that in the scenario of the upper
right panel of Fig.~\ref{ericFIG} the expected suppression is too
large for any measurement beyond the $p_{\rm T}$ integrated one.
\begin{figure*}
\begin{center}
\resizebox{0.74\textwidth}{!}{\includegraphics{results.eps}}
\end{center}
\caption{$\Upsilon^\prime/\Upsilon$ ratio versus $p_{\rm T}$ for two
different QGP models with different system
sizes~\cite{ericTHESIS}. The solid curves correspond to the
``theoretical ratios''. The triangles show the expected
measurements with the ALICE forward muon spectrometer in one month
of central (10\,\%) Pb-Pb data taking (the open triangles
correspond to the $p_{\rm T}$ integrated ratios). Error bars are
of statistical origin only. The horizontal solid lines show the
expected value of the ratio in pp collisions. More details on the
ingredients used in the different scenarios can be found in
Ref~\cite{Gunion:1996qc}.}
\label{ericFIG}
\end{figure*}
\subsubsection{Secondary J/$\psi$ from bottom decay}
A large fraction of the J/$\psi$ yield arises from the decay of $B$
mesons. The ratio ${\rm N}(b\bar{b}\rightarrow J/\psi)/{\rm N}({\rm
direct}~J/\psi)$ can be determined as follows. The number of directly
produced J/$\psi$ in central (5\,\%) Pb-Pb collisions is
0.31~\cite{Bedjidian:2003gd}\footnote{Including shadowing and no
feed-down from higher states.}. The corresponding number of
$b\bar{b}$ pairs (with shadowing) amounts to
4.56~\cite{Bedjidian:2003gd}. The $b\rightarrow {\rm J}/\psi$
branching ratio is $1.16\pm0.10\,\%$~\cite{Eidelman:2004wy}. Therefore
${\rm N}(b\bar{b}\rightarrow {\rm J}/\psi)/{\rm N}({\rm direct}~{\rm
J}/\psi) = 34\,\%$ in $4\pi$. These secondary J/$\psi$ mesons from
$b$ decays, which are not QGP suppressed, must be subtracted from the
measured J/$\psi$ yield prior to J/$\psi$ suppression
studies\footnote{In addition, 1.5\,\% of $B$ mesons decay into
$\chi_{c1}(1P)$ which subsequently decay into $\gamma$J$/\psi$ with a
31\,\% branching ratio~\cite{Eidelman:2004wy}.}. They can further be
used in order to measure the $b$ cross-section in pp
collisions~\cite{Acosta:2004yw}, to estimate shadowing in p-A
collisions and to probe the medium induced $b$ quark energy loss in
A-A collisions. Indeed, it has been shown~\cite{Lokhtin:2001nh} that
the $p_{\rm T}$ and $\eta$ distributions of those J/$\psi$ exhibit
pronounced sensitivity to $b$ quark energy loss. In addition, a
comparison between high mass dileptons and secondary J/$\psi$
distributions could clarify the nature of the energy
loss~\cite{Lokhtin:2001nh}.
Due to the large life-time of $B$ mesons, J/$\psi$ from bottom decay
is the only source of J/$\psi$ not coming from the primary
vertex\footnote{J/$\psi$ from statistical hadronization, kinetic
recombination and $D\overline{D}$ annihilation are usually quoted as
secondary J/$\psi$ but they originate from the primary vertex.}. The
best way to identify them is, therefore, to reconstruct the invariant
mass of dileptons with displaced vertices i.e.\ with impact parameter,
$d0$, above some threshold. Simulations have shown that such
measurements can successfully be performed with dielectrons measured
in the central part of ALICE using the ITS, the TPC and the
TRD~\cite{TRDTP} and with dimuons in CMS~\cite{Lokhtin:2001nh}, thanks
to the excellent spatial resolution of the inner tracking devices of
these experiments. It should also be possible to disentangle the two
sources of J/$\psi$ from the slopes of the overall measured J/$\psi$
$p_{\rm T}$ distributions since primary J/$\psi$ have a harder
spectrum~\cite{TRDTP}. Finally, a recent study~\cite{andreas} has
demonstrated the possibility to isolate J/$\psi$ from bottom decay in
pp collisions, without secondary vertex reconstruction, by triggering
on three muon events in the ALICE forward muon spectrometer. Indeed,
in standard (dimuon) pp events, the J/$\psi$ peak contains 85\,\% of
primary J/$\psi$ and 15\,\% of J/$\psi$ from $B$ meson decays. The
situation is totally inverted in tri-muon events because a
$B\overline{B}$ pair can easily produce many decay leptons. In such
events the J/$\psi$ peak contains 85\,\% of secondary J/$\psi$ from
bottom decay and 15\,\% of direct J/$\psi$~\cite{andreas}. It is
obvious that this analysis technique becomes less and less efficient
as the track multiplicity increases. Nevertheless, it could still be
performed for light-ion systems.
\subsection{Open heavy flavors}
\subsubsection{Open bottom from single leptons with displaced vertices}
As mentionned above, the $d0$ distributions of leptons from heavy
meson decays exhibit a significantly large tail because heavy mesons
have a larger life-time than other particles decaying into leptons.
Therefore, inclusive measurements of open heavy flavors can be
achieved from the identification of the semi-leptonic decay of heavy
mesons~\cite{TRDTP}. Recent simulation studies~\cite{rosario}
performed with the ALICE central detectors show that with
$d0>180~\mu{\rm m}$ and $p_{\rm T}>2$~GeV/$c$, the monthly expected
statistics of electrons from $B$ decays in central Pb-Pb collisions is
$5\cdot 10^4$ with a contamination of only 10\,\%, mainly coming from
charm decays. The deconvolution of $d0$ distributions by imposing
different $p_{\rm T}$ cuts should allow charm measurements as well.
Furthermore, such analyses should give access to the $p_{\rm T}$
distribution of $D$ and $B$ mesons by exploiting the correlation
between the $p_{\rm T}$ of the decay lepton and that of its
parent~\cite{TRDTP}.
\subsubsection{Open bottom from single muons and unlike-sign dimuons}
The possibility to measure the differential $B$ hadron inclusive
production cross-section in central Pb-Pb collisions at the LHC has
recently been investigated by means of analyses similar to the ones
performed in p$\bar{\rm p}$ collisions at the Tevatron. This study is
based on unlike-sign dimuon mass and single muon $p_{\rm T}$
distributions measured with the ALICE forward muon
spectrometer~\cite{rachid}. The principle is first to apply a low
$p_{\rm T}$ threshold on single muons in order to reject background
muons (mainly coming from charm decays) and therefore to maximize the
$b$ signal significance. Then, fits are performed to the total
(di)muon yield with fixed shapes for the different contributing
sources and the bottom amplitude as the only free parameter. The $B$
hadron production cross-section is then obtained after corrections for
decay kinematics and branching ratios and muon detection acceptance
and efficiencies. This allows to extract the signal over a broad
range in $p_{\rm T}$ (Fig.~\ref{rachidFIG}). A large statistics is
expected~\cite{rachid} thus allowing detailed investigations on $b$
quark production mechanisms and in-medium energy loss. On the other
hand, such a measurement, which can be performed for different
centrality classes, provides the most natural normalization for
$\Upsilon$ suppression studies.
\begin{figure}[ht]
\begin{center}
\resizebox{0.40\textwidth}{!}{\includegraphics{rachid4.eps}}
\end{center}
\caption{Differential $B$ hadron inclusive production cross-section
in the most central (5\,\%) Pb-Pb collisions~\cite{rachid}.
Measurements from unlike-sign dimuons at low and high mass and from
single muons (symbols) are compared to the input distribution
(curve). Statistical errors (not shown) are negligible.}
\label{rachidFIG}
\end{figure}
\subsubsection{Open bottom from like-sign dileptons}
As shown in Fig.~\ref{smbat2}, a sizable fraction of like-sign
correlated dileptons arise from the decay of $B$ mesons. These
dileptons have two different origins:
\begin{itemize}
\item{The first decay generation of $B$ mesons contains $\sim 10\,\%$ of
primary leptons and a large fraction of $D$ mesons which decay
semi-leptonically with a branching ratio of $\sim 12\,\%$. Therefore a
$B\overline{B}$ pair is a source of like-sign correlated dileptons via
channels like:\\
\hspace*{0.5cm}$B^+$ $\rightarrow$ $\overline{D}^0$ $e^+$ $\nu_e$,
$\overline{D}^0$ $\rightarrow$ $e^-$ anything\\
\hspace*{0.5cm}$B^-$ $\rightarrow$ $D^0$ $\pi^-$, $D^0$ $\rightarrow$
$e^+$ anything\\ where the $B^+B^-$ pair produces a correlated
$e^+e^+$ pair in addition to the two correlated $e^+e^-$ pairs;}
\item{The two neutral $B^0\overline{B}^0$ meson systems
$B^0_d\overline{B}^0_d$ and $B^0_s\overline{B}^0_s$ undergo the
phenomenon of particle-antiparticle mixing (or oscillation). The
mixing parameters\footnote{Time-integrated probability that a produced
$B^0_d$ ($B^0_s$) decays as a $\overline{B}^0_d$ ($\overline{B}^0_s$)
and vice versa.} are $\chi_d~= 0.17$ and $\chi_s\ge
0.49$~\cite{Eidelman:2004wy}. Therefore, a $B^0_{d}\overline{B}^0_d$
($B^0_s\overline{B}^0_s$) pair produces, in the first generation of
decay leptons, $70\,\%$ ($50\,\%$) of unlike-sign correlated lepton pairs
and $30\,\%$ ($50\,\%$) of like-sign correlated lepton pairs.}
\end{itemize}
This component is accessible experimentally from the subtraction of
so-called event-mixing spectrum from the like-sign
spectrum~\cite{Crochet:2001qd}. The corresponding signal is a
reliable measurement of the bottom cross-section since i) $D$ mesons
do not oscillate~\cite{Eidelman:2004wy} and ii) most (if not all)
leptons from the second generation of $D$ meson decay can be removed
by a low $p_{\rm T}$ threshold of about 2~GeV/$c$.
\subsubsection{Hadronic charm}
In the central part of ALICE, heavy mesons can be fully reconstructed
from their charged particle decay products in the ITS, TPC and
TOF~\cite{Dainese:2003zu}. Not only the integrated yield, but also
the $p_{\rm T}$ distribution can be measured. The most promising
decay channel for open charm detection is the $D^0 \rightarrow
K^-\pi^+$ decay (and its charge conjugate) which has a branching ratio
of 3.8\,\% and $c\tau=124~\mu{\rm m}$. The expected rates (per unit
of rapidity at mid rapidity) for $D^0$ (and $\overline{D}^0$) mesons,
decaying in a $K^\mp\pi^\pm$ pair, in central (5\,\%) Pb-Pb at
$\sqrt{s}=5.5~{\rm TeV}$ and in pp collisions at $\sqrt{s}=14~{\rm
TeV}$, are $5.3\cdot 10^{-1}$ and $7.5\cdot 10^{-4}$ per event,
respectively. The selection of this decay channel allows the direct
identification of the $D^0$ particles by computing the invariant mass
of fully-reconstructed topologies originating from displaced secondary
vertices. The expected statistics are $\sim 13\,000$ reconstructed
$D^0$ in $10^7$ central Pb-Pb collisions and $\sim 20\,000$ in $10^9$
pp collisions. The significance is larger than 10 for up to about
$p_{\rm T}=10$~GeV/$c$ both in Pb-Pb and in pp collisions. The
cross section can be measured down to $p_{\rm T} = 1$~GeV/$c$ in Pb-Pb
collisions and down to almost $p_{\rm T} = 0$ in pp collisions.
\begin{figure}[ht]
\begin{center}
\resizebox{0.46\textwidth}{!}{\includegraphics{andrea2.epsi}}
\end{center}
\caption{Ratio of the nuclear modification factors for $D^0$ mesons
and for charged (non-charm) hadrons with and without energy loss
and dead cone effect~\cite{Dainese:2003wq}. Errors corresponding
to the case ``no energy loss'' are reported. Vertical bars and
shaded areas correspond to statistical and systematic errors,
respectively.}
\label{andrea2}
\end{figure}
The reconstructed $D^0$ $p_{\rm T}$ distributions can be used to
investigate the energy loss of $c$ quarks by means of the nuclear
modification factor $R_{\rm
AA}^{D^0}$~\cite{Dainese:2003zu,Dainese:2003wq}. Even more
interesting is the ratio of the nuclear modification factors of $D^0$
mesons and of charged (non-charm) hadrons ($R_{D/h}$) as a function of
$p_{\rm T}$. Apart from the fact that many systematic uncertainties
on $R_{\rm AA}^{D^0}$ cancel out with the double ratio, $R_{D/h}$
offers a powerful tool to investigate and quantify the so-called dead
cone effect (Fig.~\ref{andrea2}).
\subsubsection{Electron-muon coincidences}
The semi-leptonic decay of heavy mesons involves either a muon or an
electron. Therefore, the correlated $c\bar{c}$ and $b\bar{b}$
cross-sections can be measured in ALICE from unlike-sign electron-muon
pairs where the electron is identified in the central part and the
muon is detected in the forward muon spectrometer. The $e\mu$ channel
is the only leptonic channel which gives a direct access to the
correlated component of the $c\bar{c}$ and $b\bar{b}$ pairs. Indeed,
in contrast to $e^+e^-$ and $\mu^+\mu^-$ channels, neither a
resonance, nor direct dilepton production, nor thermal production can
produce correlated $e\mu$ pairs. Within ALICE, the $e\mu$ channel has
the additional advantage that the rapidity distribution of the
corresponding signal extends from $\sim 1$ to $\sim 3$, therefore
bridging the acceptances of the central and the forward parts of the
detector~\cite{Lin:1998bd}. Electron-muon coincidences have already
been successfully measured in pp collisions at $\sqrt{s}=60~{\rm
GeV}$~\cite{Chilingarov:1979ur} and in p-nucleus collisions at
$\sqrt{s}=29~{\rm GeV}$~\cite{Akesson:1996wf}. Preliminary
simulations have shown the possibility, with ALICE, to measure the
correlated $e\mu$ signal after appropriate background
subtraction~\cite{MUONTDR}.
\section{Summary}
The heavy flavor sector will bring fantastic opportunities for
systematic explorations of the dense partonic system formed in heavy
ion collisions at the LHC through a wide variety of physics channels.
In addition to the channels discussed here, further exciting
possibilities should be opened with, for example, charmed baryons,
high mass dileptons, quarkonia polarization and dilepton correlations.
\section*{Acknowledgments}
I am grateful to A.~Andronic, M.~Bedjidian, A.~Dainese, S.~Grigoryan,
R.~Guernane, G.~Martinez and A.~Morsch for their help in preparing
this paper.
|
{
"timestamp": "2005-04-05T09:10:37",
"yymm": "0503",
"arxiv_id": "nucl-ex/0503008",
"language": "en",
"url": "https://arxiv.org/abs/nucl-ex/0503008"
}
|
\section{Introduction}
It is interesting for theoretical and practical reasons to study
coherent and squeezed states associated to the quantum Hopf
algebras \cite{kn:Dri86,kn:Reshe,kn:Ma}. The Hopf algebra structure
of a quantum algebra provides us with useful technical elements such
as the coproduct, for exemple. In the case of boson quantum
algebras, the special coproduct properties are useful to
characterize multi-particle Hamiltonians \cite{kn:TsoPaJa}. For
example, in the case of the Poincar\'{e} quantum algebra, the
coproduct have been brought to bear to the study the fusion of
phonons \cite{kn:Cele2}. In general, the concept of deformed quantum
Lie algebras found various applications in quantum optics, quantum
field theory, quantum statistical mechanics, supersymmetric quantum
mechanics and some purely mathematical problems. For instance, in
the case of the $su_q (2)$ algebra, it has been found that the $su_q
(2)$ effective Hamiltonians reproduce accurately the physical
properties of the $su(2) \oplus h(2)$ models \cite{kn:BaCiHeRe}. On
the other side, there are some works showing that quantum algebras
are connected with paragrassmann algebras \cite{kn:Spiri,kn:Fi}.
Paragrassmann algebras are relevant in the studies of theories that
show the necessity of unusual statistic \cite{kn:Ru}, for instance,
the studies of anyons and topological field theories
\cite{kn:MaWi,kn:AnBa}.
Now, to associate coherent and squeezed states to a quantum
deformed Lie algebra one can use the algebra eigentates (AES)
technique. The AES associated to a real Lie algebra have been
defined as the set of eigenstates of an arbitrary complex linear
combination of generators of the considered algebra
\cite{kn:Brif}. The AES associated to a quantum real deformed Lie
algebra can be defined in a similar way. Indeed, if $A_k (q), \,
k=1,2, \ldots, n$ denote the generators of this deformed algebra
in a given representation, parametrized by the set of deformation
parameters $q,$ then the AES associated to this deformed algebra
will be given by the set of solutions of the eigenvalue equation
\begin{equation} \sum_{k=1}^n \alpha_k A_k (q) |\psi \rangle = \lambda |\psi
\rangle, \qquad \alpha_k, \lambda \in {\mathbb C}. \end{equation}
The purpose of this work is to compute the AES of the deformed
quantum Heisenberg Lie algebras \cite{kn:HL94}, obtained by
applying the R-matrix methods \cite{kn:Dri86}, and find new
classes of deformed harmonic oscillator coherent and squeezed
states. We will see that these states will be new deformations of
the standard coherent and squeezed states of the harmonic
oscillator system and we will recover them in the limit when the
deformation parameters go to zero. The approach of AES also gives
us the possibility to construct, starting from a deformed algebra,
some Hamiltonians, of physical systems to which these deformed
coherent and squeezed states are associated, similarly as for
algebras and superalgebras \cite{kn:NaVh1,kn:NaVh3}.
It is important to mention that the deformed coherent states
obtained by this method differ from the $q$--deformed coherent
states associated to a $q$-deformed oscillator algebra, which is
not a Hopf algebra, constructed by considering either deformed
exponential functions, eigenstates of a given deformed
annihilation operator, a generalization of the usual form of the
standard coherent states, a resolution of the identity technique
or a generalized group theoretical techniques
\cite{kn:Dellinas,kn:Cquesne,kn:BjuPSt}.
The paper is organized as follows. In section \ref{sec-two}, a
Fock space representation of deformed quantum algebras associated
to the Heisenberg algebra $h(2)$ is given. In section
\ref{sec-three}, we compute the AES associated to these algebras
and obtain new classes of deformed coherent and squeezed states
that are true deformations of the standard coherent and squeezed
states associated to the harmonic oscillator system. These states
are parametrized by the deformation parameters which will be
considered as real numbers and also as real paragrassmann numbers.
In section \ref{sec-cuatro}, we compute the product of the
dispersions of the position and linear momentum operators of a
particle in these states when the parameters of deformation are
small. We compare them with the corresponding results obtained in
the minimum uncertainty states \cite{kn:NaVh1}. Some details of
calculations are presented in the Appendices \ref{sec-appa} and
\ref{sec-appb}. We also give general expressions of these
dispersions, in the case where a non trivial one parameter algebra
deformation family is concerned, for all values of the
deformation parameter. Finally, we construct a class of
$\eta$--pseudo Hermitian Hamiltonians \cite{kn:AMostafazadeh} to
which a subset of these deformed states are the associated
coherent states.
\section{Deformed quantum Heisenberg algebras in the Fock representation space}
\label{sec-two} We are considering in this work, the deformed
Heisenberg quantum algebras obtained by V. Hussin and A. Lauzon
\cite{kn:HL94}. They have been obtained using the well-known
$R$--matrix method \cite{kn:Dri86} and are mainly of two types.
The first one is formed by the generators $A,B,C$ which satisfy
\begin{equation} \left[A,B\right] = 0, \qquad \left[B,C\right] = - { 2 z \over
p^2 } ( \cosh( pB ) - 1 ), \qquad \left[A,C\right] = {1 \over p}
\sinh (pB) . \label{com-he1} \end{equation} It is denoted by
$ {\cal U}_{z,p} \, (h(2)),$ where $p$ and $z$ are different from
zero.
Let us mention that the invertible change of basis \begin{eqnarray} {\tilde A} = A, \qquad {\tilde B} = { 2
\over p} \sinh \left( {p B \over 2} \right), \qquad {\tilde C} = {
1 \over \cosh \left( {p B \over 2} \right) } \ C , \label{optilde}
\end{eqnarray} leads to the new deformed algebra $ \ {\tilde {\cal
U}}_{z,0} \, (h(2)):$ \begin{equation} [{\tilde A} , {\tilde B}] =0, \qquad
[{\tilde B} , {\tilde C}] = - z {\tilde B}^2, \qquad [{\tilde A} ,
{\tilde C}] = {\tilde B}. \label{com-he1tilde} \end{equation} This means that
we get the same commutation relations as in \eqref{com-he1}
when $p$ goes to zero. As it has been pointed out by Ballesteros et al. \cite{kn:BaHePr}, here the
$p$ parameter is superfluous and the families of bialgebras $
{\cal U}_{z,p} \, (h(2))$ and $ {\cal U}_{z,0} \, (h(2))$ are
isomorphic (these families are identified there as of type $I_+ $)
on the condition that the coproduct form stands invariant
\cite{kn:BaCeOl}.
The second quantum deformation of $h(2)$ is given by \begin{equation}
\label{altype2} \left[A,B\right] = \left[B, C\right] =0, \qquad
\left[A,C\right] = {e^{pB} - e^{- qB} \over p+q} .
\end{equation} and is denoted by ${\cal U}_{p,q} \, (h(2)),$ where
$p,q \ne 0.$ It corresponds to so-called so called type $II$
bialgebras in \cite{kn:BaHePr}. When $p=q,$ we find the quantum
Heisenberg algebra obtained in Celeghini et al. \cite{kn:Cele}
(see also \cite{kn:BaCeOl}), i.e., \begin{equation} \left[A,B\right] =
\left[B,C\right] =0, \qquad \left[A,C\right] = {1 \over p} \sinh
(p B). \label{cel-et-al} \end{equation}
Let us now give a boson realization of these
deformed Lie algebras, in terms of the usual creation operator,
$a^\dagger, $ and annihilation operator, $a,$ associated to the
standard quantum harmonic oscillator system. For ${\tilde {\cal
U}}_{z,0} \, (h(2))$ given in \eqref{com-he1tilde}, it is given by
\begin{equation} \label{cas1} \tilde A = - a^\dagger, \qquad \tilde B = e^{z
a^\dagger}, \qquad \tilde C = e^{z a^\dagger} \ a . \end{equation}
From \eqref{optilde} and \eqref{cas1}, we thus get a
realization of ${\cal U}_{z,p} \, (h(2))$ as \begin{equation}
\label{op-def-one-zp} A = - a^\dagger, \qquad B = {2\over p}
\sinh^{-1} \left( {p \over 2} e^{z a^\dagger} \right), \qquad
C = e^{ z a^\dagger} \sqrt{1+ {\left({p \over 2} e^{z
a^\dagger}\right)}^2 } a. \end{equation}
Another realization of ${\tilde {\cal U}}_{z,0} \, (h(2))$ is \begin{equation}
\label{cas2} \tilde A = a, \qquad \tilde B = e^{- z a}, \qquad
\tilde C = a^\dagger \ e^{-za}. \end{equation} We thus get another
realization of ${\cal U}_{z,p} \, (h(2))$ as \begin{equation}
\label{op-def-two-zp} A = a, \qquad \qquad B = {2\over p}
\sinh^{-1} \left( {p \over 2} e^{- z a} \right), \qquad C =
a^\dagger e^{- z a} \sqrt{1+ {\left({p \over 2} e^{-z a}\right)}^2
}. \end{equation}
When $z$ goes to zero, the operators \eqref{op-def-one-zp} become
\begin{equation} \label{def1} A=- a^\dagger , \qquad B= {2 \over p} \sinh^{-1}
\left({p\over 2} \right) I, \qquad C = \sqrt{1+ {p^2 \over 4}} a,
\end{equation} while the operators \eqref{op-def-two-zp} become \begin{equation}
\label{def2} A = a, \qquad B= {2 \over p} \sinh^{-1} \left({p\over
2} \right) I, \qquad C = \sqrt{1+ {p^2 \over 4}} a^\dagger. \end{equation}
The operators \eqref{def1} or \eqref{def2} thus constitute a
realization of deformed Heisenberg algebra \eqref{cel-et-al}. When
$p$ goes to zero, we regain $h(2).$
The algebra \eqref{altype2} is clearly isomorphic to $h(2)$ if
we introduce \begin{equation} \tilde A =A, \qquad \tilde C = C, \qquad \tilde B
= {e^{pB} - e^{-qB} \over p+q}. \end{equation}
So to obtain new class of deformed coherent and squeezed states
using the AES method we will deal in the following with ${\tilde
{\cal U}}_{z,0} \, (h(2))$ and ${\cal U}_{z,p} \, (h(2)).$
\section{AES and deformed coherent and squeezed states}
\label{sec-three} In this section, we compute the AES associated
to ${\tilde {\cal U}}_{z,0} \, (h(2))$ and ${\cal U}_{z,p} \,
(h(2)), $ using the representations obtained in the preceding
section. We thus get new classes of deformed coherent and squeezed
states associated to the harmonic oscillator system.
\subsection{Deformed algebra eigenstates for \mathversion{bold} ${\tilde {\cal U}}_{z,0} \,(
h(2))$} \label{sec-aes-He} We start with ${\tilde {\cal U}}_{z,0}
\,( h(2))$ as given by \eqref{com-he1tilde} using the realizations
\eqref{cas1} and \eqref{cas2}. The AES are thus defined as the set
of solutions of the eigenvalue equation \begin{equation} \label{aes-10}
[\alpha_+ {\tilde A} + \alpha_0 {\tilde B} + \alpha_- {\tilde C}
] |\psi\rangle = \alpha |\psi\rangle, \qquad \alpha_- , \alpha_0 ,
\alpha_+ , \alpha \in {\mathbb C}. \end{equation}
\subsubsection{Deformed harmonic oscillator coherent and squeezed
states}\label{sub-sec-coh-squee} Let us take first the
realization \eqref{cas1}. Thus, if $\alpha_- \ne 0, $ equation
\eqref{aes-10} can be written in the form \begin{equation} \label{eigen-10}[
e^{z a^\dagger} a + \mu a^\dagger + \nu e^{z a^\dagger}] |\psi
\rangle = \lambda |\psi \rangle, \qquad \mu,\nu, \lambda \, \in
{\mathbb C}. \end{equation} By defining \begin{equation} \label{psi-varphi} |\psi\rangle =
e^{- \nu a^\dagger} |\varphi\rangle \end{equation} and using $ e^{-\nu
a^\dagger} \, a \,e^{\nu a^\dagger}= a + \nu, $ equation
\eqref{eigen-10} can be reduced to \begin{equation} \label{reduce-001}[ e^{z
a^\dagger} a + \mu a^\dagger ] |\varphi \rangle = \lambda |\varphi
\rangle, \qquad \mu, \lambda \, \in {\mathbb C}. \end{equation} To solve this
eigenvalue equation, let us consider the Bargmann space ${\cal F}$
of analytic functions $f(\xi)$ ($ \xi \, \in {\mathbb C}),$
provided with the scalar product \begin{equation} (f_1, f_2) = \int_{{\mathbb
C}} \overline{f_1 (\xi) } f_2 (\xi) e^{-{\bar \xi} \xi} {d {\bar
\xi} d\xi \over 2 \pi i}, \qquad \forall \, f_1,f_2 {\in \cal F}.
\end{equation} It is well-know that any function $f \in {\cal F}$ can be
expressed as a linear combination of orthonormalized functions
$u_n (\xi)= {\xi^n \over \sqrt{n!}}, \, n=0,1,2, \ldots,$
verifying \begin{equation} \label{ortho-umn} (u_m, u_n) = \int_{{\mathbb C}}
\overline{u_m (\xi)} u_n (\xi) e^{-{\bar \xi} \xi} {d {\bar \xi}
d\xi \over 2 \pi i} = \delta_{mn}, \end{equation} that is \begin{equation} f(\xi) =
\sum_{n=0}^\infty c_n u_{n} (\xi), \end{equation} with \begin{equation} c_n =
\int_{{\mathbb C}} \overline{u_n (\xi)} f (\xi) e^{-{\bar \xi}
\xi} {d {\bar \xi} d\xi \over 2 \pi i}. \end{equation}
Let us assume a solution of \eqref{reduce-001} of the type \begin{equation}
\label{type-sol} | \varphi \rangle = \sum_{n=0}^{\infty} c_n
|n\rangle, \end{equation} where the set of states $\{ |n\rangle
\}_{n=0}^{\infty}$ form the basis of the standard Fock oscillator
space, verifying the orthogonality relation \begin{equation} \langle m
|n\rangle =\delta_{mn}. \label{st-mn-ortho} \end{equation} As usually, the
action of the operators $a$ and $a^\dagger $ on these states is
given by \begin{equation} a | n \rangle = \sqrt{n} |n-1\rangle, \qquad
a^\dagger | n \rangle = \sqrt{n+1} |n+1\rangle. \end{equation} Let us take
$| {\bar \xi} \rangle $ to be the standard coherent states
associated to the harmonic oscillator system, that is \begin{equation} | {\bar
\xi} \rangle = e^{{\bar \xi} a^\dagger} |0 \rangle =
\sum_{n=0}^{\infty} {{(\bar \xi )}^n \over \sqrt{n!}} |n\rangle.
\end{equation} Then, according to the orthogonality property
\eqref{st-mn-ortho}, the projection of $|\varphi \rangle$ on the
coherent state $| {\bar \xi} \rangle $ is given by the analytic
function \begin{equation} \label{var-xi} \varphi (\xi) = \langle {\bar \xi} |
\varphi \rangle =\sum_{n=0}^{\infty} c_n u_{n} (\xi). \end{equation} The
action of the operators $a^\dagger $ and $a$ in this
representation corresponds to \begin{equation} \label{act-xi}\langle {\bar
\xi} | a^\dagger |\varphi \rangle = \xi \varphi (\xi), \qquad \;
\langle {\bar \xi} | a |\varphi \rangle= {d \varphi \over d\xi}
(\xi). \end{equation} respectively. Thus, by projecting both sides of the
eigenvalue equation \eqref{reduce-001} on the coherent states $|
{\bar \xi} \rangle$ and then using \eqref{act-xi}, we can write it
as \begin{equation} \label{eigen-fock} \left( e^{z\xi} {d\over d\xi} + \mu
\xi\right) \varphi (\xi) = \lambda \varphi (\xi). \end{equation} The general
solution of this differential equation is given by \begin{equation} \varphi
(\xi) = C_0 (\lambda,\mu, z ) \, \exp\left(\sum_{k=0}^{\infty}
{{(-z\xi)}^k \over (k+1)!} \left( \lambda \xi - {k+1 \over k+2}\mu
\xi^2 \right) \right), \label{solgen-varphi} \end{equation} where $ C_0 $ is
an arbitrary constant which can be fixed from the normalization
condition \begin{equation} \label{con-nor-var}( \varphi , \varphi) =
\int_{{\mathbb C}} \overline{\varphi (\xi )} \varphi (\xi)
e^{-{\bar \xi} \xi} {d {\bar \xi} d\xi \over 2 \pi i} =1.\end{equation} Let
us notice that in the particular limit when $z$ goes to zero, the
solution \eqref{solgen-varphi}, becomes the symbol for the
squeezed states \cite{kn:Dodo} associated to the standard harmonic
oscillator, that is \begin{equation} \label{sym-squee} \varphi (\xi) = C_0
(\lambda,\mu,0) \, \exp\left( \lambda \xi - {\mu \over 2} \xi^2
\right) . \end{equation} This quantity is normalizable only if $|\mu| <1 $
\cite{kn:NORu}.
When $z\ne 0,$ the solution \eqref{solgen-varphi} can be written
in the form \begin{equation} \varphi (\xi) = C_0 (\lambda, \mu, z ) \,
\exp\left( {\lambda \over z} - {\mu \over z^2} \right) \,
\exp\left( e^{-z \xi} {(\mu - \lambda z + \mu z \xi) \over z^2}
\right). \label{solgen-varphi2} \end{equation}
Going back to the expression \eqref{var-xi}, we get the
coefficients $c_n, \, n=0,1,\ldots, $ as \begin{eqnarray} c_n =
\int_{{\mathbb C}} \overline{u_n (\xi )} \varphi (\xi)
e^{-{\bar \xi} \xi} {d {\bar \xi} d\xi \over
2 \pi i} &=& C_0 (\lambda, \mu, z ) \, \exp\left( {\lambda \over
z} - {\mu \over z^2} \right) \nonumber \\ & & \int_{{\mathbb C}}
{{\bar \xi}^n \over \sqrt{n!}} \exp\left( e^{-z \xi} {(\mu -
\lambda z + \mu z \xi) \over z^2} \right)
e^{-{\bar \xi} \xi} {d {\bar \xi} d\xi \over
2 \pi i} . \end{eqnarray} By using the polar change of variables $\xi =
\rho e^{i \vartheta},$ this last equation can be written in the
form \begin{eqnarray} c_n &=& C_0 (\lambda, \mu, z ) \, \exp\left( {\lambda
\over z} - {\mu \over z^2} \right) \nonumber \\ & &
\int_{0}^\infty \int_{0}^{2 \pi} { \rho^{n+1} e^{- \rho^2 } \over
\sqrt{n!}} e^{-i n \vartheta} \exp\left( {e^{-z \rho e^{i
\vartheta}} \over z^2} ( \mu - \lambda z + \mu z \rho e^{i
\vartheta})\right) { d\rho d\vartheta \over \pi} . \end{eqnarray} Let us
write the exponential factor in the form \begin{eqnarray} && \nonumber
\exp\left( {e^{-z \rho e^{i \vartheta}} \over z^2} ( \mu -
\lambda z + \mu z \rho e^{i \vartheta})\right) \\ \nonumber &=&
\sum_{k=0}^{\infty} { \exp\left(-z k \rho e^{i \vartheta}\right)
\over k! } {\left( \mu - \lambda z + u z \rho e^{i \vartheta}
\over z^2\right) }^{k} \nonumber \\ &=& \sum_{k,l=0}^{\infty}
\sum_{m=0}^{k} {k \choose m} \rho^{l+m} e^{i (l+m) \vartheta} {{(
- z k )}^l {(\mu z)}^m {(\mu - \lambda z)}^{k-m}\over k! \, l! \,
z^{2k}} \end{eqnarray} to get \begin{eqnarray} c_n &=&
C_0 (\lambda, \mu, z ) \, \exp\left( {\lambda
\over z} - {\mu \over z^2} \right) \, \sum_{k,l=0}^{\infty}
\sum_{m=0}^{k} {k \choose m} {{( - z k )}^l {(\mu z)}^m {(\mu -
\lambda z)}^{k-m}\over \sqrt{n!} \; k! \, l! \, z^{2k}} \nonumber
\\ && \left(\int_{0}^{\infty} \rho^{m+l+n+1} e^{- \rho^2}
d\rho\right) \left(\int_{0}^{2 \pi} e^{i (l+m-n)\vartheta}
{d\vartheta \over \pi} \right). \end{eqnarray} Using the known results \begin{equation}
\int_{0}^{2 \pi} e^{i (l+m-n)\vartheta} {d\vartheta \over \pi} = 2
\delta_{l+m-n,0}, \end{equation} \begin{equation}\int_{0}^{\infty} \rho^{m+l+n+1} e^{-
\rho^2} d\rho = {1\over 2} \Gamma\left({m+l+n \over 2} +1
\right),\end{equation} and performing the sum over the index $l,$ the
expression for the coefficients $c_n$ reduces to \begin{equation}
\label{coeff-cn} c_n = C_0 (\lambda, \mu, z ) \, \exp\left(
{\lambda \over z} - {\mu \over z^2} \right) {z^n \over \sqrt{n!}}
\, \sum_{k=0}^{\infty} \sum_{m=0}^{k_{<}} {n \choose m} {{( -k
)}^{n-m} \over (k-m)!} {\left({\mu \over z^2}\right)}^m
{\left({\mu\over z^2} - {\lambda \over z} \right)}^{k-m} ,\end{equation}
where $k_{<}$ denotes the minimum between $k$ and $n.$ This last
expression can be written in the form \begin{equation} c_n = C_0 (\lambda,
\mu, z ) \, {z^n \over \sqrt{n!}} \, \sum_{m=0}^{n}
\sum_{j=0}^{n-m} {n \choose m} {(-1)}^{n-m} \upsilon_{mj}
{\left({\mu \over z^2}\right)}^m {\left({\mu\over z^2} -
{\lambda \over z} \right)}^j , \label{mejorcn} \end{equation} where the
coefficients $\upsilon_{mj}$ are obtained from \begin{equation} {k^{n-m} \over
(k-m)! } = \sum_{j=0}^{n-m} {\upsilon_{mj} \over (k-m-j)!}. \end{equation}
Thus the coefficients $c_n, $ $n=1,2,\ldots, $ represent
polynomials of degree $n-1$ in the z variable. For example, $ c_1
= \lambda C_0, $ \begin{equation} c_2 = C_0 \sqrt{2!} \left[ \left({\lambda^2
\over 2!} - {\mu \over 2}\right) - {\lambda \over 2 } z \right],
\quad \nonumber c_3 = C_0 \sqrt{3!} \left[ \left( {\lambda^3 \over
3!} - {\mu \lambda \over 2} \right) + \left({\mu \over 3} -
{\lambda^2 \over 2}\right) z + {\lambda \over 6} z^2 \right]. \end{equation}
The normalization constant $C_0$ can be now computed. Indeed,
inserting \eqref{mejorcn} into \eqref{var-xi} and the resulting
expression into the normalization condition \eqref{con-nor-var},
using the orthogonality relation \eqref{ortho-umn}, we get \begin{eqnarray}
\nonumber C_0 \, ( \lambda, \mu, z ) &=& \Biggl[
\sum_{n=0}^{\infty} \, {z^{2n} \over n!} \, \sum_{m=0}^{n}
\sum_{r=0}^{n} \sum_{j=0}^{n-m} \sum_{l=0}^{n-r} {n \choose m}{n
\choose r} {(-1)}^{m+r} \upsilon_{mj} \upsilon_{rl} \\ & & {\left(
{\mu \over z^2 } \right)}^m {\left( {{\bar \mu} \over z^2 }
\right)}^r {\left({\mu\over z^2} - {\lambda \over z} \right)}^j
{\left({{\bar \mu} \over z^2} - {{\bar \lambda} \over z}
\right)}^l \Biggr]^{-{1 \over 2 } } ,
\label{nor-fac}\end{eqnarray} which has been chosen real. The
convergence of these series it not easy to determine.
In the case where $z=0,$ as we have already mentioned, the series $\sum_{n=0}^\infty {|c_n|}^2$ converges for all
$\lambda$ provided that $|\mu| < 1.$ In the case $\mu=0,$ this series becomes
\begin{equation}
\sum_{n=0}^\infty {|c_n|}^2 = {|C_0 (\lambda,z)|}^2
\exp\left({\lambda \over z}\right) \sum_{n=0}^\infty {{\biggl(- {
\lambda \over z}\biggr)}^{n} \over
n!} \exp\left( -{{\bar \lambda} \over z} \sum_{k=1}^\infty {{(z^2
n)}^k \over k!}\right).
\end{equation} It converges for all $z > 0$ provided that the phase $\theta$ in
$\lambda= \beta e^{i \theta}$ satisfies $ - {\pi \over 2} \le
\theta \le {\pi \over 2}, $ whereas for all $z<0, $ it converges
if $ {\pi \over 2} \le \theta \le {3 \pi \over 2}. $
Finally, we can show that the normalized algebra eigenstates
$|\varphi \rangle,$ solving \eqref{reduce-001} , can be expressed
in terms of a deformed squeezed operator acting on the ground
state of the standard harmonic oscillator, that is \begin{equation} |\varphi
\rangle = C_0 \, ( \lambda, \mu, z )\exp \left(
\sum_{k=0}^{\infty} {{(-z a^\dagger )}^k \over (k+1)!} \left(
\lambda a^\dagger - { k + 1 \over k+2 } \mu {(a^\dagger)}^2
\right) \right) |0 \rangle. \label{re-norm-squee} \end{equation} Also,
combining this last equation with equation \eqref{psi-varphi}, we
get the algebra eigenstates solving \eqref{eigen-10} to be the
deformed coherent states \begin{equation} |\psi \rangle = N_0 ( \lambda, \mu,
\nu, z ) \,\exp \left( \sum_{k=0}^{\infty} {{(-z a^\dagger )}^k
\over (k+1)!} \left( \lambda a^\dagger - { k + 1 \over k+2 } \mu
{(a^\dagger)}^2 \right) \right) e^{-\nu a^\dagger}|0 \rangle,
\label{eigen-10-aes} \end{equation} where $N_0 \, ( \lambda, \mu, \nu, z )$
is a normalization constant which can computed in the same way as
$ C_0 \, ( \lambda, \mu, z ).$
\subsubsection{Perturbed squeezed states}
\label{sec-perturba-z-real} Let us now assume that $z$ is a
small perturbation parameter of order $k_0 -1$, where $k_0$ is an
integer greater or equal to $2$. From \eqref{re-norm-squee},
neglecting the terms containing the power of $ z $ greater than
$k_0 -1$, we can write \begin{eqnarray} \nonumber |\varphi \rangle &
\thickapprox & C_0 (\lambda, \mu, z, k_0) \Biggl[ 1 +
\sum_{k=1}^{k_0 - 1} {{(-z a^\dagger )}^k \over (k+1)!} \left(
\lambda a^\dagger - { k + 1 \over k+2 } \mu {(a^\dagger)}^2
\right) \\ &+& \cdots + { 1 \over (k_0 -1)!} {\Biggl( {-z
a^\dagger \over 2!} \left(\lambda a^\dagger - {2\over 3} \mu
{(a^\dagger)}^2 \right) \Biggr)}^{k_0 -1} \Biggr] \exp\left(
\lambda a^\dagger - {\mu \over 2} {(a^\dagger)}^2 \right) |0
\rangle. \label{def-squee-st}\end{eqnarray}
These states can be normalized in the standard form. For
instance, when $k_0 = 2, $ $\mu = \delta e^{i \phi},$ $\lambda =
\beta e^{i \theta}, $ where $\phi$ and $\theta $ are real phases,
$ 0 \le \delta < 1, $ and $\beta \ge 0,$ a normalized version of
the deformed squeezed states \eqref{def-squee-st}, is given by
\begin{eqnarray} \nonumber |\varphi \rangle &\thickapprox& \Omega (\delta,
\phi, \beta, \theta) \left[ 1 + z \left( {\delta e^{i \phi} \over
3} {(a^\dagger)}^3 - {\beta e^{i \theta} \over 2} {(a^\dagger)}^2
\right) \right] \\ & & S \left( - \arctan (\delta )e^{i \phi}
\right) D \left( {\beta e^{i\theta} \over \sqrt{1 - \delta^2}}
\right) |0 \rangle, \label{nor-z-def} \end{eqnarray} where \begin{eqnarray} \Omega
(\delta, \phi, \beta, \theta) &=& 1 + {z \beta \over 2
{(1-\delta^2)}^2} \Biggl[ \left( 2 \delta^2 +
\beta^2 \left( {1 + \delta^2 \over 1-\delta^2} \right) \right) \cos\theta \nonumber \\
&-& \delta \left( 1 + \delta^2 +
{ 2 \beta^2 \over 1-\delta^2 } \right) \cos(\phi - \theta)
\nonumber \\ &+& \delta^2 \beta^2 \left( 1 + {2 \delta^2 \over
3(1-\delta^2)} \right) \cos(2 \phi - 3 \theta) - {2\delta \beta^2
\over 3(1-\delta^2)} \cos(\phi - 3 \theta) \Biggr]. \end{eqnarray} Here $
S(\chi) = \exp\left[ - \left( \chi {{(a^\dagger)}^2 \over 2} -
{\bar \chi} {a^2 \over 2} \right)\right] $ is the standard
unitary squeezed operator \cite{kn:ruso} and $ D (\lambda) =
\exp\left( \lambda a^\dagger - {\bar \lambda} a \right) $ the
standard displacement operator \cite{kn:pere}.
\subsubsection{Deformed squeezed and coherent states parametrized by paragrassmann numbers}
\label{sec-paragra} Let us now use the realization \eqref{cas2}
of ${\tilde {\cal U}}_{z,0} \, (h(2))$. In the case $\alpha_+ \ne
0, $ equation \eqref{aes-10} can be now written in the form \begin{equation}
\label{eigen-100}[a + \mu a^\dagger e^{- z a} + \nu e^{-z a}]
|\psi \rangle = \lambda |\psi \rangle, \qquad \mu,\nu, \lambda \,
\in {\mathbb C}. \end{equation} There are two types of equations to solve.
The first type is obtained when $\mu \ne 0 $ and $\nu \ne 0.$ We
can take \begin{equation} |\psi \rangle = \exp\left({\nu \over \mu} a \right)
|\varphi\rangle \end{equation} and use the relation, $ \exp\left(- {\nu \over
\mu} a \right) a^\dagger \exp\left({\nu \over \mu} a \right) =
a^\dagger - {\nu \over \mu}, $ to reduce \eqref{eigen-100} to the
form \begin{equation} \label{eigen-200} [a + \mu a^\dagger e^{- z a} ] |\varphi
\rangle = \lambda |\varphi \rangle, \qquad \mu, \lambda \, \in
{\mathbb C}. \end{equation} If $\nu =0$ and $\mu \ne 0,$ we see from
\eqref{eigen-100} that the same type of eigenvalue equation must
be solved. The second type is obtained when $\mu =0. $ The
eigenvalue equation is \begin{equation} \label{eigen-30} [a + \nu e^{- z a} ]
|\psi \rangle = \lambda |\psi \rangle, \qquad \nu, \lambda \, \in
{\mathbb C}.\end{equation}
We begin with the resolution of Equation \eqref{eigen-200}. Let us
assume $| \varphi \rangle $ to be again a solution of the type
\eqref{type-sol}. Thus, proceeding as in the preceding section,
the eigenvalue equation satisfied by the symbol $\varphi (\xi),$
in the Bargmann representation, is given by \begin{equation} \label{fock-para}
\left({d \over d \xi} + \mu \, \xi \, e^{- z {d \over d \xi}} \right) \varphi (\xi)
= \lambda \varphi (\xi ), \qquad \mu, \lambda \, \in {\mathbb
C}. \end{equation} To solve this equation, let us assume that $z$ is a
real paragrassmann number \cite{kn:Fi,kn:Ru}, that is $z^{k_0}=0,$
for some integer $k_0 \ge 1.$ A detailed procedure of resolution
of this equation is given in the Appendix \ref{sec-appa}. Let us
notice that the case $k_0=1,$ i.e., $z=0,$ is somewhat trivial
since the eigenfunctions $\varphi (\xi)$ solving
\eqref{fock-para}, are given by the standard squeezed symbol
\eqref{sym-squee}. When $k_0=2,$ or $z^2=0,$ i.e., when $z$ is a
odd Grassmann number \cite{kn:Dewit,kn:Corn}, the eigenvalue
equation \eqref{fock-para} becomes \begin{equation} \left( (1- \mu z \xi) {d
\over d \xi} + \mu \, \xi \right) \varphi (\xi)
= \lambda \varphi (\xi ), \qquad \mu, \lambda \in {\mathbb
C}. \end{equation} There are two independent solutions (see Appendix
\ref{sec-appa}). The normalizable solution of this eigenvalue
equation, is given by the deformed squeezed symbol \begin{equation} \varphi
(\lambda, \mu, z) (\xi) = C_0 (\lambda,\mu,z) \left[1 + z \mu
\left( \lambda {\xi^2 \over 2} - \mu {\xi^3 \over 3}
\right)\right] \exp\left(\lambda \xi - {\mu \over 2} \xi^2
\right). \label{nor-sol-uno}\end{equation} A normalized version of these
states, in the Fock space representation, is given by \begin{eqnarray}
\nonumber |\varphi \rangle &=& {\tilde \Omega} (\delta, \phi,
\beta, \theta) \left[1 + z \delta \left( {\delta e^{2i \phi} \over
3} {(a^\dagger)}^3 - {\beta e^{i ( \theta + \phi )} \over 2}
{(a^\dagger)}^2 \right)
\right] \\ & & S \left( - \arctan (\delta )e^{i \phi} \right) D \left(
{\beta e^{i\theta} \over \sqrt{1 - \delta^2}} \right) |0 \rangle,
\end{eqnarray} where $\lambda$ and $\mu$ have been chosen as in the
preceding subsection and \begin{eqnarray} {\tilde \Omega} (\delta, \phi,
\beta, \theta) &=& 1 - {z \delta \beta \over 2 {(1-\delta^2)}^2}
\Biggl[ \left( 2 \delta^2 +
\beta^2 \left( {1 + \delta^2 \over 1-\delta^2} \right) \right) \cos(\theta- \phi) \nonumber \\
&-& \delta \left( 1 + \delta^2 +
{ 2 \beta^2 \over 1-\delta^2 } \right) \cos \theta
\nonumber \\ &+& \delta^2 \beta^2 \left( 1 + {2 \delta^2 \over
3(1-\delta^2)} \right) \cos(\phi - 3 \theta) - {2\delta \beta^2
\over 3(1-\delta^2)} \cos(2 \phi - 3 \theta) \Biggr].\end{eqnarray} When
$k_0 = 3, $ or $z^3=0,$ the eigenvalue equation \eqref{fock-para}
becomes the second order differential equation \begin{equation}
\label{sec-order-eq}\left( {1\over 2} \mu z^2 \xi {d^2 \over
d\xi^2} + (1- \mu z \xi) {d \over d \xi} \right) \varphi (\xi)
= (\lambda - \mu \, \xi ) \varphi (\xi ), \qquad \mu, \lambda, \, \in {\mathbb
C}. \end{equation} According to the results obtained in Appendix
\ref{sec-appa}, the general solution of this equation can be
expanded in the form \begin{equation} \label{sol-type-3} \varphi (\xi) =
\varphi_0 (\xi) + z \varphi_1 (\xi) + z^2 \varphi_2 (\xi), \end{equation}
with \begin{eqnarray} \varphi_0 (\xi) &=& C_0 \exp\left(\lambda \xi - \mu
{\xi^2 \over
2}\right) , \\
\varphi_1 (\xi) &=& \left[ \mu \left( \lambda {\xi^2 \over 2} -
\mu {\xi^3 \over 3} \right) C_0 + C_1 \right] \exp\left(\lambda
\xi - \mu {\xi^2 \over 2}\right) , \\ \nonumber \varphi_2 (\xi)
&=& \Biggl[ \left( \mu (\mu - \lambda^2 ) {\xi^2 \over 4} +
{2\over 3} \mu^2 \lambda \xi^3 + \mu^2 (\lambda^2 - 3 \mu )
{\xi^4 \over 8} - \lambda \mu^3 {\xi^5 \over 6} +
\mu^4 {\xi^6 \over 18} \right) C_0 \\
&+& \mu \left( \lambda {\xi^2 \over 2} - \mu {\xi^3 \over 3} \right)
C_1 + C_2 \Biggr] \exp\left(\lambda \xi - \mu {\xi^2 \over
2}\right), \end{eqnarray} where $C_0, $ $C_1$ and $C_2$ are arbitrary
integration constants. Three independent solutions may thus be
obtained. The first one is obtained by taking $C_1 = C_2 =0.$ We get
\begin{eqnarray} \nonumber \varphi (\xi) &=& C_0 \Biggl[ 1 + z \mu \left(
\lambda {\xi^2 \over 2} - \mu {\xi^3 \over 3} \right) + z^2 \Biggl(
\mu (\mu -
\lambda^2 ) {\xi^2 \over 4} + {2\over 3} \mu^2 \lambda \xi^3 \\
\nonumber &+& \mu^2 (\lambda^2 - 3 \mu ) {\xi^4 \over 8} -
\lambda \mu^3 {\xi^5 \over 6} + \mu^4 {\xi^6 \over 18} \Biggr)
\Biggr] \exp\left(\lambda \xi - \mu {\xi^2 \over 2}\right)\\ &=&
C_0 \exp\left[ z \mu \left( \lambda {\xi^2 \over 2} - \mu {\xi^3
\over 3} \right) + z^2 f(\xi)\right]\exp\left(\lambda \xi - \mu
{\xi^2 \over 2}\right),
\end{eqnarray}
where \begin{equation} f(\xi) = \Biggl( \mu (\mu - \lambda^2 ) {\xi^2 \over 4}
+ {2\over 3} \mu^2 \lambda \xi^3 - 3 \mu^3 {\xi^4 \over 8}
\Biggr). \end{equation} This solution can be normalized and represents a
second order paragrassmann deformation of squeezed states
associated to the standard harmonic oscillator.
The other independent solutions are given respectively by \begin{equation}
\varphi (\xi) = C_1 \, z \Biggl[ 1 + z \mu \left( \lambda {\xi^2
\over 2} - \mu {\xi^3 \over 3} \right) \Biggr] \exp\left(\lambda
\xi - \mu {\xi^2 \over 2}\right) \end{equation} and \begin{equation} \varphi (\xi) = C_2
\, z^2
\exp\left(\lambda \xi - \mu {\xi^2 \over 2}\right). \end{equation} These
solutions can not be normalized since $z^k, \, k=1,2, $ are not invertible paragrassmann numbers and $ z^k = 0, \,
k= 3,4, \ldots. $
The higher order paragrassmann deformations of the squeezed
states associated to the standard harmonic oscillator can be
obtained following a similar procedure (see Appendix
\ref{sec-appa}).
In the case of eigenvalue equation \eqref{eigen-30}, the
differential equation to solve is given by \begin{equation}
\label{eigen-30-diff} \left( {d\over d\xi} + \nu e^{- z {d\over
d\xi}} \right)
\varphi (\xi) = \lambda \varphi (\xi), \qquad \nu, \lambda, \, \in
{\mathbb C}.\end{equation} Proceedings as before and considering the results
of Appendix \ref{sec-appa}, the normalizable solutions of this
last equation, when $k_0= 1,2,3,$ are given respectively by the
deformed coherent symbols \begin{equation} \varphi^{(1)}(\xi) = C_0 \exp\biggl(
(\lambda - \nu) \xi \biggr), \end{equation} \begin{equation} \varphi^{(2)}(\xi)= C_0
\left[ 1 + z (\lambda - \nu) \nu \xi \right] \exp\biggl( (\lambda
-\nu) \xi\biggr)\end{equation} and \begin{eqnarray} \varphi^{(3)} (\xi)&=& C_0
\biggl\{1 + z (\lambda - \nu) \nu \xi
+ z^2 \biggl[\left( \frac{{\lambda }^2\,\nu
}{2} \nonumber +
2\,\lambda \,{\nu }^2 -
\frac{3\,{\nu }^3}{2} \right) \xi \\ &+&
\left(
\frac{{\lambda }^2\,{\nu }^2 }{2} -
\lambda \,{\nu }^3 +
\frac{{\nu }^4}{2}\right) \, {\xi }^2 \biggr] \biggr\} \exp\biggl( (\lambda -\nu) \xi
\biggr).
\end{eqnarray}
Theses solutions can be normalized and represent zero, first and
second order paragrassmann deformations, respectively, of coherent
states associated to the standard harmonic oscillator. For higher
values of $ k_0,$ we must proceed as in Appendix \ref{sec-appa}.
\subsection{Deformed algebra eigenstates for \mathversion{bold} ${\cal U}_{z,p}
(h(2))$} It is interesting to compute the AES associated to ${\cal
U}_{z,p} (h(2)), \ z,p \ne 0, $ and compare it with the ones
associated to ${\cal U}_{z,0} (h(2)).$ As we have noticed in
section \ref{sec-two}, these quantum algebras are isomorphic in
the sense that there is a nonlinear change of basis transforming
one to the other. In general, the existence of this isomorphism
does not imply the existence of an internal homomorphism at the
AES level. Indeed, by definition, the eigenvalue equation
determining the set of AES deals with an arbitrary linear
combination of the deformed algebra generators, then from the
inverses of transformations \eqref{optilde} and the solvable
structure of the commutation relations \eqref{com-he1tilde}, it is
impossible to find an internal homomorphism, at the AES level,
transforming the eigenvalue equation with $z,p \ \ne 0 $ to the
eigenvalue equation with $z \ne 0, p=0.$
To see that, in this section, we consider the two parameters
deformed algebra ${\cal U}_{z,p} (h(2))$ as given by
\eqref{com-he1}, and compute the AES using the particular
realization \eqref{op-def-one-zp}. More precisely, we have to
solve the eigenvalue equation \begin{equation} \left[ e^{ z a^\dagger}
\sqrt{1+ {\left({p \over 2} e^{z a^\dagger}\right)}^2 } a + \mu
a^\dagger + {2 \nu \over p} \sinh^{-1} \left( {p\over 2} e^{z
a^\dagger} \right) \right] |\psi\rangle = \lambda |\psi\rangle,
\qquad \mu,\nu,\lambda \in {\mathbb C}. \end{equation} In the Bargmann
representation, this equation becomes the first order differential
equation \begin{equation}\left[ e^{ z \xi} \sqrt{1+ {\left({p \over 2} e^{z
\xi}\right)}^2 } {d\over d\xi} + \mu \xi + {2 \nu \over p}
\sinh^{-1} \left( {p\over 2} e^{z \xi} \right) \right] \psi(\xi) =
\lambda \psi (\xi), \qquad \mu,\nu,\lambda \in {\mathbb C}. \end{equation}
When $z=0,$ we easily get the standard squeezed symbols \begin{equation}
\psi_{o,p} (\xi) = C_0 (p,\lambda , \mu ,\nu)
\exp\left[\left(\lambda - {2\nu \over p} \sinh^{-1} (p/2)\right)
\xi - \mu {\xi^2 \over 2} \right]. \end{equation} These symbols correspond to
the Bargmann representation of the AES associated to
the deformed quantum Heisenberg algebra realization
\eqref{def1}. Moreover, when $p$ goes to zero, these symbols
becomes the standard squeezed symbols associated to $h(2).$
When $z \ne 0,$ making the change of variable $\zeta = e^{z\xi}, $
rearranging the terms and using the method of characteristics
curves to separate the differentials, we get \begin{equation} {d\psi \over
\psi} (\zeta) = {\left[ \lambda - {\mu \over z} \ln \zeta - {2\nu
\over p} \sinh^{-1} {\left( p \zeta \over 2\right)} \right] \over
z \, \zeta^2\, \sqrt{1 + {p^2 \zeta^2 \over 4} }} d\zeta.\end{equation}
Integrating both sides of this equation and then exponentiating,
we get \begin{eqnarray} \psi_{z,p} (\zeta) &=& C_0 (\lambda, \mu,\nu ; z, p )
\, \exp\Biggl[ {\sqrt{1 + {p^2 \zeta^2 \over 4}} \over z^2 \zeta}
\biggl( (1+ \ln \zeta) \mu - \lambda z + {2 \nu z \over p}
\sinh^{-1}(\frac{p \,\zeta}{2}) \biggr) \nonumber \\ &-& {\mu p
\over 2 z^2} \sinh^{-1}(\frac{p \,\zeta}{2}) - {\nu \over z} \ln
\zeta \Biggr]. \label{so--gen-zp}\end{eqnarray} This result includes the
ones obtained for \eqref{eigen-10} when $p$ goes to zero.
Moreover, when we set also $\nu=0,$ we regain
\eqref{solgen-varphi2}.
\subsubsection{Perturbed two parameters deformation coherent and
squeezed states} Up to first order of approximation in $z$ and
$p^2,$ the deformed symbol \eqref{so--gen-zp} writes \begin{eqnarray}
\psi_{z,p} (\xi) &\approx& \tilde C_0 (\lambda, \mu,\nu ; z, p )
\biggl[1 + z \left( {\mu \xi^3 \over 3} - {\lambda \xi^2 \over 2}
\right) \nonumber \\&+& { p^2 \over 4} \left( {\mu \xi^2 \over 4}
- \left({\lambda \over 2} - {\nu \over 3} \right)\xi \right)
\biggr] \, \exp\left( (\lambda - \nu) \xi - {1\over 2} \mu \xi^2
\right). \end{eqnarray} In the case $\mu = \delta e^{i \phi},$ $\lambda =
\beta e^{i \theta}$ and $\nu = - \gamma e^{i \eta},$ where $\gamma
\ge 0,$ a normalized version of these states, in the Fock
representation, is given by \begin{eqnarray} \nonumber |\psi \rangle &\approx
& {\tilde \Omega} (\delta, \phi, \beta, \theta, \gamma, \eta)
\biggl\{1 + \left[ z \left( {\delta e^{i \phi} \over 3}
{(a^\dagger)}^3 - {\beta e^{i \theta} \over 2} {(a^\dagger)}^2
\right) \right] \\ \nonumber &+& {p^2 \over 4} \left[{\delta e^{i
\phi} \over 4} {(a^\dagger)}^2 - \left({\beta e^{i \theta} \over
2} + {\gamma e^{i \eta} \over 3} \right) a^\dagger
\right]\biggr\}
\\ & & S \left( - \arctan (\delta )e^{i \phi} \right) D \left( {{\tilde
\beta}
e^{i{\tilde \theta}} \over \sqrt{1 - \delta^2}} \right) |0
\rangle, \label{def-squee-sta}\end{eqnarray} where \begin{eqnarray} {\tilde \Omega}
(\delta, \phi, \beta, \theta, \gamma, \eta ) &=& 1 + {z \over 2
{(1-\delta^2)}^2} \Biggl\{ {\tilde \beta} \Biggl[ \left( 2
\delta^2 +
{\tilde \beta}^2 \left( {1 + \delta^2 \over 1-\delta^2} \right) \right) \cos{\tilde \theta} \nonumber \\
&-& \delta \left( 1 + \delta^2 +
{ 2 {\tilde \beta}^2 \over 1-\delta^2 } \right) \cos(\phi - {\tilde \theta})
\nonumber \\ &+& \delta^2 {\tilde \beta}^2 \left( 1 + {2
\delta^2 \over 3(1-\delta^2)} \right) \cos(2 \phi - 3 {\tilde
\theta}) - {2\delta {\tilde \beta}^2 \over 3(1-\delta^2)}
\cos(\phi - 3 {\tilde \theta}) \Biggr] \nonumber \\\nonumber &-&
\gamma \Biggl[ {\tilde \beta}^2 \cos(\eta - 2 {\tilde \theta}) -
\delta (2{\tilde \beta}^2 + 1 - \delta^2 ) \cos(\eta - {\tilde
\theta}) \\ \nonumber &+& \delta^2 {\tilde \beta}^2 \cos(2 \phi-
\eta - 2 {\tilde \theta}) \Biggr]\Biggr\} - {p^2 \over 16
{(1-\delta^2)}^2} \Biggl\{ \delta {\tilde \beta}^2 (3
\cos(\phi - 2 {\tilde \theta}) \\ &+& {2\gamma\over 3}
{\tilde \beta} (1 -\delta^2) \biggl( \cos(\eta - {\tilde \theta}) + \delta \cos
(\phi -\eta - {\tilde \theta}\biggr) -
2 {\tilde \beta}^2 - \delta^2 +
\delta^4 \Biggr\}, \nonumber \\
\end{eqnarray}
where \begin{equation} {\tilde \beta} = \sqrt{\beta^2 + \gamma^2 + 2 \beta
\gamma \cos(\eta-\theta)}, \qquad {\tilde \theta}= \tan^{-1}
\left( { \beta \sin \theta + \gamma \sin \eta \over \beta \cos
\theta + \gamma \cos \eta } \right). \end{equation} We notice that, in the
case $\gamma =0$ and $p=0,$ these normalized states become the
normalized states given in equation \eqref{nor-z-def}.
\section{Some properties of the deformed states}
\label{sec-cuatro} In this section, we will give some properties
of the deformed states found in preceding section. From Fock space
representation, we will deduce the physical quantities $X$ and
$P,$ representing the position and linear momentum of a particle,
respectively, and compute the corresponding dispersions in both
the perturbed deformed states associated to ${\cal U}_{z,p}
(h(2))$ and the deformed states associated to $ {\tilde {\cal
U}}_{z,0} (h(2)).$ We will also connect the last states with an
$\eta$-pseudo Hermitian Halmiltonian \cite{kn:AMostafazadeh}.
\subsection{Squeezing properties} \label{sec-xp-squeezed} First,
let us consider the squeezing properties of $X$ and $P.$ In the
Fock space representation, these quantities are given by the
hermitian operators (we have assumed that the mass, angular
frequency and Planck's constant are all equal to 1) \begin{equation} X = {(a +
a^\dagger) \over \sqrt{2}}, \qquad P= i {(a^\dagger - a )\over
\sqrt{2}}. \label{xp-def} \end{equation} They verify the canonical
commutation relation \begin{equation} [X,P] = i I . \end{equation} The dispersion of these
quantities, computed on a specific normalized particle state
$|\psi\rangle,$ is defined as \begin{equation} {(\Delta X )}^2 = \langle \psi |
X^2 | \psi \rangle - {(\langle \psi | X | \psi \rangle)}^2
\label{x-disper} \end{equation} and \begin{equation} {(\Delta P )}^2 = \langle \psi | P^2
| \psi \rangle - {(\langle \psi | P | \psi \rangle)}^2.
\label{p-disper} \end{equation} The product of these dispersions satisfies
the Schr\"{o}dinger-Robertson uncertainty relation (SRUR)
\cite{kn:SchRo,kn:Mer} \begin{equation} {(\Delta X )}^2 \, {(\Delta P )}^2 \ge
{1 \over 4} \biggl( \langle I \rangle^2 + \langle F \rangle^2
\biggr) = {1 \over 4} \biggl( 1 + \langle F \rangle^2 \biggr) ,
\label{SchRo-prin} \end{equation} where $F$ is the anti-commutator $ F = \{ X
- \langle X \rangle I, P- \langle P \rangle I\}.$ The mean value
of $F$ is a correlation measure between $X$ and $P.$ When $\langle
F \rangle = 0 ,$ we regain the standard Heisenberg uncertainty
principle.
The minimum uncertainty states (MUS) are states that satisfy the
equality in \eqref{SchRo-prin}. They are called coherent states
when the dispersions of both $X$ and $P$ are the same and
squeezed states when these dispersions are different to each
other. The states for which the dispersion of $X$ is greater than
the one of $P$ are called $X$-squeezed whereas the states for
which the dispersion of $P$ is greater than the one of $X$ are
called $P$-squeezed.
We are interested to compute the dispersions of $X$ and $P,$ in
the deformed squeezed states \eqref{def-squee-sta}, when $\nu=0,$
or $\gamma=0.$ More precisely, we want to study the effect of the
deformation parameters on the squeezed properties of these
quantities. As we have seen, when $z$ and $p$ go to zero, the
states \eqref{def-squee-sta} becomes the standard harmonic
oscillator squeezed states. In such a case, we know that the
dispersions of $X$ and $P$ are independent of $\lambda = \beta
e^{i \theta },$ and given by \cite{kn:NaVh1} \begin{equation} {{(\Delta X )}_0
}^2 = {1 - 2 \delta \cos\phi + \delta^2 \over 2 (1-\delta^2)}
\qquad {\rm and} \qquad {{(\Delta P )}_0 }^2 = {1 + 2 \delta
\cos\phi + \delta^2 \over 2 (1-\delta^2)}. \end{equation} All these states
are MUS, that is, they satisfy the equality in \eqref{SchRo-prin}.
When $\gamma=0,$ the square of the mean value of $X,$ in the
states \eqref{def-squee-sta}, to first order of approximation in
$z$ and $p^2,$ is given by \begin{eqnarray} \langle \psi |X |\psi \rangle^2
& \approx & 2 \biggl(\mathrm{Re\,} \Gamma_{01}\biggr) \, \mathrm{Re\,} \Biggl\{
\biggl( 1+ 4 \epsilon (z,p) \biggr) \Gamma_{01} \nonumber
\\\nonumber &+& 2 z \, \Biggl( {\delta e^{-i \phi} \over 3}
\Gamma_{04} - {\beta e^{-i \theta} \over 2} \Gamma_{03} + {\delta
e^{i \phi} \over 3} \Lambda_{13} - {\beta e^{i \theta} \over 2 }
\Lambda_{12} \Biggr)
\\ &+& {p^2 \over 2} \, \Biggl( {\delta e^{-i \phi} \over
4} \Gamma_{03} - {\beta e^{-i \theta} \over 2} \Gamma_{02} +
{\delta e^{i \phi} \over 4} \Lambda_{12} - {\beta e^{i \theta}
\over 2} \Lambda_{11} \Biggr) \Biggr\}, \label{moyx2} \end{eqnarray} where
$\epsilon (z,p)= {\tilde \Omega}(\delta,\phi,\beta,\theta,0,0) - 1
$ and $\Gamma_{kl}$ and $\Lambda_{kl},$ $k,l=1,2,\ldots,$ are
matrix elements defined in Appendix \ref{sec-appb}. According to
\eqref{xp-def}, we have the same expression for the square of the
mean value of $P,$ but taking the imaginary part in place of the
real part.
On the other hand, the mean value of $X^2$ in the states
\eqref{def-squee-sta}, to first order of approximation in $z$ and
$p^2,$ is given by \begin{eqnarray} \nonumber \langle \psi |X^2 |\psi
\rangle & \approx & { 1 \over 2}+ \biggl( 1 + 2 \epsilon (z,p)
\biggr) (\Gamma_{11} + \mathrm{Re\,} \Gamma_{02}) \\ \nonumber &+& z
\, \mathrm{Re\,} \Biggl( {\delta e^{-i \phi} \over 3} \Gamma_{05} -
{\beta e^{-i \theta} \over 2} \Gamma_{04} + {\delta e^{i \phi}
\over 3} \Lambda_{23} - {\beta e^{i \theta} \over 2 } \Lambda_{22}
\Biggr)
\\\nonumber &+& {p^2 \over 4} \, \mathrm{Re\,} \Biggl( {\delta e^{-i
\phi} \over 4} \Gamma_{04} - {\beta e^{-i \theta} \over 2}
\Gamma_{03} + {\delta e^{i \phi} \over 4} \Lambda_{22} - {\beta
e^{i \theta} \over 2} \Lambda_{21} \Biggr) \nonumber \\ & + & z
\Biggl( {\delta e^{-i \phi} \over 3} (\Lambda_{41} - \Gamma_{03})
- {\beta e^{-i \theta} \over 2} (\Lambda_{31} - \Gamma_{02}) +
{\delta e^{i \phi} \over 3}(\Lambda_{14} - \Lambda_{03})\nonumber
\\ \nonumber &-& {\beta e^{i \theta} \over 2 } (\Lambda_{13} -
\Lambda_{02})\Biggr) + {p^2 \over 4} \Biggl( {\delta e^{-i \phi}
\over 4} ( \Lambda_{31} - \Gamma_{02}) - {\beta e^{-i \theta}
\over 2} ( \Lambda_{21} - \Gamma_{01}) \nonumber \\&+& {\delta
e^{i \phi} \over 4} (\Lambda_{13} - \Lambda_{02}) - {\beta e^{i
\theta} \over 2} (\Lambda_{12} - \Lambda_{01}) \Biggr).
\label{x2moy} \end{eqnarray} Again, according to \eqref{xp-def}, we have
the same expression for the mean value of $P^2,$ but taking the
negative of the real part in place of the real part.
Combining \eqref{moyx2} with \eqref{x2moy}, according to equation
\eqref{x-disper}, we get the dispersion of $X$. In the same way,
we can obtain the dispersion of $P.$ Inserting the matrix elements
$\Gamma_{ij}$ and $\Lambda_{ij},$ as given in the Appendix
\ref{sec-appb}, we can compute these dispersions explicitly.
Figure \ref{fig:varXPz0.0-0.020p=0} show the dispersions of $X$
and $P$ in the minimum uncertainty squeezed states in dashed
lines, and in the deformed squeezed states in solid lines, as a
function of $\phi$ for fixed valued of the parameters $\delta,
\beta, \theta$ and $p$ ($\delta=0.5, \, \beta=2.0, \, \theta =0.8
\, \pi, p=0.00 $) and for special values of $z=0.0010, 0.0015,
0.0020$ (from the smaller to the greater gray level).
\begin{figure}[h] \centering
\begin{picture}(31.5,21)
\put(0,0){\framebox(31.5,21){}}
\put(1,2.1){\includegraphics[width=70mm]{nalvar1.eps}}
\put(28.9,3){\scriptsize{$\phi$}}
\put(27.6,5){\scriptsize{${(\Delta P)}^2$}}
\put(26.9,5){\vector(-1,0){3}}
\put(25.6,15.4){\scriptsize{${(\Delta X)}^2$}}
\put(25.4,15.4){\vector(-1,0){3}}
\end{picture}
\caption{Graphs of the dispersions of $X$ and $P$ as functions of
$\phi$ for $p=0$ and $z=0.000,0.0010,0.0015,0.0020.$ }
\label{fig:varXPz0.0-0.020p=0}
\end{figure} We observe
that, as a consequence of the small deformations in the parameters
$z$ the squeezing properties of $X$ and $P$ have not been
essentially changed. Thus, in all the cases, we have $P$--squeezed
states when $- {\pi \over 2} < \phi < {\pi \over 2}, $ and
$X$--squeezed states when ${\pi \over 2} < \phi < {3 \pi \over 2}.
$ Also we observe that the product of the dispersions of $X$ and
$P$ in the deformed squeezed states, for a given value of $\phi,$
is always greater than the product of the dispersions in the
minimum uncertainty states, as required by the SRUR. These
difference is more remarkable for values of $\phi$ in the range
${\pi \over 2} \le \phi < {3\pi \over 2}. $ Let us notice that
when $\phi = \pm {\pi \over 2},$ the MUS are coherent states, in
the sense of the SRUR, i,e., the dispersion of $X$ and $P,$ are
the same. Indeed, in all these cases, ${{(\Delta X)}_0}^2 =
{{(\Delta P)}_0}^2 = 0.83. $ This value is conserved by the
product of the dispersions of $X$ and $P$ in the deformed squeezed
states when $\phi = - {\pi \over 2},$ but when $ \phi = {\pi
\over 2},$ it grows quickly as $z$ increases.
Figure \ref{fig:varXPz-3p0-0.11} show the dispersions of $X$ and
$P$ in the minimum uncertainty squeezed states in dashed lines,
and in the deformed squeezed states in solid lines, as a function
of $\phi$ for fixed valued of the parameters $\delta, \beta,
\theta$ and $z$ ($\delta=0.5, \, \beta=2.0, \, \theta =0.8 \, \pi,
z=0.0030 $) and for special values of $p=0.00, 0.06, 0.11$ (from
the greater to the smaller gray level). \begin{figure}[h]
\centering
\begin{picture}(31.5,21)
\put(0,0){\framebox(31.5,21){}}
\put(1,2.1){\includegraphics[width=70mm]{nalvar2.eps}}
\put(28.9,3){\scriptsize{$\phi$}}
\put(27.8,5.5){\scriptsize{${(\Delta P)}^2$}}
\put(27.5,5.5){\vector(-1,0){2.5}}
\put(25.3,15.4){\scriptsize{${(\Delta X)}^2$}}
\put(25,15.4){\vector(-1,0){2.5}}
\end{picture}
\caption{Graphs of the dispersions of $X$ and $P$ as functions of
$\phi$ for $z= 0.0030, p=0.00,0.06,0.11, \beta=2.0, \theta =0.8
\pi $ and $ \delta= {0.5}.$} \label{fig:varXPz-3p0-0.11}
\end{figure} We observe that the
product of dispersions of $X$ and $P$ decreases when $p$
increases. Thus the influence of the $p$ parameter on the first
order in $z$ deformed states is to reduce the uncertainty product
of $X$ and $P$ and to bring closer this quantity to the minimum
uncertainty values.
\begin{figure}[h]
\centering
\begin{picture}(31.5,21)
\put(0,0){\framebox(31.5,21){}}
\put(1,2.1){\includegraphics[width=70mm]{nalvar43.eps}}
\put(28.9,3){\scriptsize{$\delta$}}
\put(27.6,7){\scriptsize{${(\Delta P)}^2$}}
\put(26.9,7){\vector(-1,-1){2}}
\put(14.9,13.4){\scriptsize{${(\Delta X)}^2$}}
\put(16.8,12.9){\vector(1,-1){2}}
\end{picture}
\caption{Graphs of the dispersions of $X$ and $P$ as functions of
$\delta$ for $z= 0.0025, p=0.01, \beta=2.0, \theta =0.8 \pi $ and
$ \phi= {\pi\over 6} .$} \label{fig:artvap01}
\end{figure}
Figure \ref{fig:artvap01} shows the typical behavior of the
dispersions of $X$ and $P$ in the minimum uncertainty squeezed
states in dashed lines, and in the deformed squeezed states in
solid lines, as a function of $\delta $ for $\phi=0.5, \,
\beta=2.0, \, \theta =0.8 \, \pi, z=0.0025 $ and $p=0.001.$ We
observe again that, as a consequence of the small deformations in
$z$ and $p,$ the squeezing properties of $X$ and $P$ have not been
essentially changed. Thus, the figure shows the behavior of
$P$--squeezed and $P$-deformed squeezed states. When $ 0< \delta
\lesssim 0.75,$ the product of the dispersions of $X$ and $P,$ in
the deformed squeezed states is always greater than the
corresponding product in the minimum uncertainty squeezed states,
as required by the SRUR. For higher values of $\delta,$ only the
dashed lines represent the true behavior of the dispersions of $X$
and $P.$ Indeed, the approximation for the deformed squeezed
states, in this region, is not valid. These states are no longer
normalizable.
\subsection{General formulas for the dispersions of $X$ and $P$ in
the \boldmath $z$ deformed states} The mean values of $X^k,
k=1,2, \ldots,$ in the states \eqref{re-norm-squee} can be
expressed in the forme \begin{equation} \biggl. \langle \varphi | X^{k} |
\varphi \rangle ={ {\partial^k \over
\partial \tau^k} \widetilde{\langle \varphi} | e^{\tau X} |
\widetilde{\varphi \rangle} \biggr|_{\tau=0} \over
\widetilde{\langle \varphi } | \widetilde{ \varphi \rangle}} ={
{\partial^k \over
\partial \tau^k}\biggl\{ e^{- {\tau^2 \over 4}} \ \widetilde{\langle \varphi} | e^{{\tau\over \sqrt2} a}
e^{{\tau\over \sqrt2} a^\dagger}| \widetilde{\varphi \rangle} \biggr\}\biggr|_{\tau=0} \over
\widetilde{\langle \varphi } | \widetilde{ \varphi \rangle}}, \end{equation}
where \begin{equation} \widetilde{|\varphi \rangle} = \exp\left( e^{-z
a^\dagger} {(\mu - \lambda z + \mu z a^\dagger) \over z^2} \right)
|0\rangle. \end{equation} Inserting these results into \eqref{x-disper} and
evaluating we get \begin{equation} \label{x-disper-tau} {(\Delta X )}^2 = - { 1
\over 2} + { {\partial^2 \over
\partial \tau^2} \widetilde{\langle \varphi} | e^{{\tau\over \sqrt2} a}
e^{{\tau\over \sqrt2} a^\dagger}| \widetilde{\varphi \rangle} \biggr|_{\tau=0} \over
\widetilde{\langle \varphi } | \widetilde{ \varphi \rangle}} -
{\Biggl( { {\partial \over
\partial \tau } \widetilde{\langle \varphi} | e^{{\tau\over \sqrt2} a}
e^{{\tau\over \sqrt2} a^\dagger}| \widetilde{\varphi \rangle} \biggr|_{\tau=0} \over
\widetilde{\langle \varphi } | \widetilde{ \varphi \rangle}}
\Biggr)}^2 . \end{equation} To compute the matrix element
$\widetilde{\langle \varphi} | e^{{\tau\over \sqrt2} a}
e^{{\tau\over \sqrt2} a^\dagger} | \widetilde{\varphi \rangle}, $
we can firstly write
\begin{equation}
e^{{\tau\over \sqrt2} a^\dagger}| \widetilde{\varphi \rangle}
= \sum_{n=0}^{\infty} C_n (\tau) |n\rangle
\end{equation}
and then to compute the coefficients $C_n (\tau), \ n=0,1,2,\ldots
, $ in the Bargman representation, in the same way as we have do
it in section \eqref{sub-sec-coh-squee}. That is \begin{equation}
\label{varphi-tilde-modif} \widetilde{\langle \varphi} |
e^{{\tau\over \sqrt2} a}
e^{{\tau\over \sqrt2} a^\dagger} | \widetilde{\varphi \rangle} =
\sum_{n=0}^{\infty} {\bar C}_{n} (\tau) C_{n} (\tau),
\end{equation}
where \begin{equation} \label{cn-tau} C_n (\tau) = {1\over \sqrt{n!}}
\sum_{r=0}^{n}
{n \choose r} {\left({\tau \over \sqrt2}\right)}^r \ z^{n-r} \
\sum_{k=0}^{\infty} \sum_{m=0}^{{\tilde k}_< } {n-r \choose m}
{{(- k)}^{n-r-m} \over (k-m)!} {\left({\mu \over z^2 }\right)}^m \
{\left({\mu \over z^2} - {\lambda \over z}\right)}^{k-m}, \end{equation} with
${\tilde k}_< $ the minimum between $k$ and $n-r .$
Inserting \eqref{varphi-tilde-modif} into \eqref{x-disper-tau}
and evaluating again we get \begin{eqnarray} \nonumber {(\Delta X )}^2 &=& -
{ 1 \over 2} + {\sum_{n=0}^\infty \Biggl. \biggl[ {\bar C}_n
(\tau) C_n^{\prime \prime} (\tau) + {\bar C}_n^{\prime \prime} C_n
(\tau) + 2 {\bar C}_n^{\prime} C_n^{\prime} (\tau) \biggr]
\Biggr|_{\tau=0} \over \sum_{n=0}^\infty {\bar C}_n (0) C_n(0) }
\\&-& {\Biggl( {\sum_{n=0}^\infty \Biggl. \biggl[ {\bar C}_n
(\tau) C_n^{\prime} (\tau) + {\bar C}_n^{\prime} (\tau) C_n (\tau)
\biggr] \Biggl|_{\tau=0}\over \sum_{n=0}^\infty {\bar C}_n (0)
C_n(0) }\Biggr)}^2 ,\label{gen-for-x} \end{eqnarray} where, for instance,
$C_{n}^{\prime} (\tau) = {d C_n \over d\tau} (\tau) . $ From
\eqref{cn-tau}, we obtain \begin{equation} C_{n}^{\prime} (0) ={1\over
\sqrt{n!}}
{n \choose 1} {1 \over \sqrt2} \ z^{n-1} \
\sum_{k=0}^{\infty} \sum_{m=0}^{{\tilde k}_1 } {n-1 \choose m}
{{(- k)}^{n-1-m} \over (k-m)!} {\left({\mu \over z^2 }\right)}^m \
{\left({\mu \over z^2} - {\lambda \over z}\right)}^{k-m}, \end{equation} when
$n= 1, 2, \ldots, $ with ${\tilde k}_1 $ the minimum between $k$
and $n-1,$ \begin{equation} C_{n}^{\prime \prime} (0) ={1\over \sqrt{n!}}
{n \choose 2} \ z^{n-2} \
\sum_{k=0}^{\infty} \sum_{m=0}^{{\tilde k}_2 } {n-2 \choose m}
{{(- k)}^{n-2-m} \over (k-m)!} {\left({\mu \over z^2 }\right)}^m \
{\left({\mu \over z^2} - {\lambda \over z}\right)}^{k-m}, \end{equation} when
$n= 2, 3, \ldots, $ with ${\tilde k}_2 $ the minimum between $k$
and $n-2,$ and \begin{equation} C_0^\prime (0) = C_0^{\prime \prime} (0) =
C_1^{\prime \prime} (0)=0.\end{equation} The formula to the dispersion of $P$
can be obtained from \eqref{gen-for-x} changing the $\tau $
argument of $C_n (\tau) $ by $ i \tau $ and then deriving and
evaluating to $\tau = 0.$ Thus, dispersions formulas of $X$ and
$P$ at all order in $z$ can be obtained. The first order
perturbation formulas of these dispersions must correspond to the
dispersions obtained in the preceding subsection, in the limit
when $p$ goes to zero.
\subsection{\boldmath $\eta$-pseudo Hermitian and Hermitian Hamiltonians} In
this section we show that the subset of deformed coherent states
\eqref{eigen-10-aes}, corresponding to the eigenvalue $\lambda=0,$
are the coherent states associated to an $\eta$-pseudo Hermitian
Hamiltonian \cite{kn:AMostafazadeh} but also, up to a similarity
transformation, the coherent states associated to a Hermitian
Hamiltonian, both isospectral to the harmonic oscillator
Hamiltonian. Indeed, when $\lambda=0,$ the eigenstates
\eqref{eigen-10-aes} correspond to the solutions of the eigenvalue
equation \begin{equation} \label{eigen-10-reduc} {\cal A} |\psi \rangle = -
\nu |\psi \rangle, \qquad \nu \in {\mathbb C}, \end{equation} where ${\cal
A} = a + \mu a^\dagger e^{- z a^\dagger}. $ These solutions can
be written in the form \begin{equation} \label{eigenatilde} |\psi ; - \nu
\rangle = {\tilde N}_0 (\mu, - \nu, z ) \ G (\mu , z ) \ e^{-
\nu a^\dagger} |0\rangle, \end{equation} where \begin{equation} G (\mu, z ) = \exp \left(
- \mu \ \sum_{k=0}^{\infty} {{(-z a^\dagger )}^k \over k!}
{{(a^\dagger)}^2 \over (k+2)} \right) , \label{eigen-aes-reduc} \end{equation} and $ {\tilde N}_0 \, (\mu, - \nu, z
)$ is a normalization constant.
Let us now to define the operator \begin{equation} \label{H-pseudo} {\cal H} =
G \ a^\dagger a \ G^{-1}, \end{equation} which satisfies \begin{equation} {\cal
H}^\dagger = \eta {\cal H} \eta^{-1},\end{equation} where $\eta$ is the
hermitian operator \begin{equation} \eta (\mu , z) = {(G^{-1})}^\dagger G^{-1}.
\end{equation} Thus ${\cal H}$ is an $\eta$-pseudo Hermitian Hamiltonian
\cite{kn:AMostafazadeh}. Moreover, as \begin{equation} G a^\dagger G^{-1} =
a^\dagger, \qquad G a G^{-1} = {\cal A}, \end{equation} we get \begin{equation}
\label{exp-H-pseudo} {\cal H} = a^\dagger {\cal A} = a^\dagger
\left( a + \mu a^\dagger e^{- z a^\dagger} \right) = a^\dagger a +
\mu e^{- z a^\dagger} {(a^\dagger)}^2 .\end{equation} On the other hand, by
construction, it is easy to verify that \begin{equation}\label{com-h2-hab} [
{\cal H}, {\cal A}]= - {\cal A}, \qquad [{\cal H}, a^\dagger]=
a^\dagger, \qquad [{\cal A}, a^\dagger]=1 \end{equation} and \begin{equation}
\label{h-on-ezero}{\cal H} |E_0 \rangle = 0, \end{equation} where \begin{equation} |E_0
\rangle ={\tilde N}_0 \, (\mu, 0, z ) G (\mu, z) |0\rangle. \end{equation}
This state is thus an eigenstate of ${\cal A}$ corresponding to
the eigenvalue $\nu = 0.$ Thus, according to \eqref{com-h2-hab}
and \eqref{h-on-ezero}, the hamiltonian ${\cal H}$ is isospectral
to the harmonic oscillator Hamiltonian. ${\cal A}$ represents an
annihilation operator for this system and their eigenstates
\eqref{eigenatilde} are the associated coherent states of ${\cal
H}.$
Let us mention that $\cal H$ verifies all the useful properties of
pseudo-Hermitian operators \cite{kn:Mostafazadeh-2004}. For
instance, $\cal H$ is Hermitian on the physical Hilbert space
$\mathfrak{H}_{{\rm phys}}$ spanned by their corresponding
eigenstates $ | \psi_n \rangle \propto {(a^\dagger)}^n G
|0\rangle, \ n=0,1,2,\ldots,$ endowed with the positive-definite
inner product $\langle \cdot | \eta \ \cdot \rangle.$ Also, $\cal
H$ may be mapped to a Hermitian Hamiltonian ${\tilde {\cal H}}$ by
a similarity transformation ${\tilde {\cal H}}= {\hat \rho} {\cal
H} {\hat \rho}^{-1}, $ where ${\hat \rho}(\mu,z)=
\sqrt{\eta(\mu,z)}= \sqrt{{G^{-1}}^\dagger G^{-1}},$ is a
Hermitian operator on a Hilbert space $\mathfrak{H}$ formed of
same vectorial space $\mathfrak{H}_{{\rm phys}} $ but endowed with
the original inner product $\langle \cdot | \cdot \rangle.$ Thus,
in our case, according to \eqref{H-pseudo}, the Hermitian
Hamiltonian ${\tilde {\cal H}},$ is unitarily equivalent to the
standard harmonic oscillator Hamiltonian and is given by \begin{equation}
\label{hami-tilde} {\tilde {\cal H}} = {\hat \rho} \ G \
a^\dagger a \ G^{-1} \ {\hat \rho}^{-1}. \end{equation} Indeed, \begin{equation} {\hat
\rho} G {({\hat \rho} G)}^{\dagger} = {\hat \rho} G G^{\dagger}
{\hat \rho}^{\dagger} = {\hat \rho} \eta^{-1} {\hat \rho} = {\hat
\rho} {({\hat \rho}^2)}^{-1} {\hat \rho} = I \end{equation} and \begin{equation} {({\hat
\rho} G)}^{\dagger} {\hat \rho} G = G^{\dagger} {\hat
\rho}^{\dagger} {\hat \rho} G = G^{\dagger} {\hat \rho}^2 G =
G^{\dagger} {(G^{-1})}^{\dagger} G^{-1} G = I, \end{equation} that is
${({\hat \rho} G)}^{\dagger}={({\hat \rho} G)}^{-1}, $ i.e., $
{\hat \rho} G $ is an unitary operator.
Let us notice that in absence of deformation ($z=0$) the operator
$\hat \rho $ is given by \begin{equation} \label{ro-u-0} \hat \rho (\mu,0) =
\sqrt{\exp\left({\bar \mu} {a^2 \over 2 } \right)\exp\left( \mu {
{a^\dagger}^2 \over 2} \right)} = {\biggl[\exp\left( \int_{0}^{1}
[{\bar \mu} K_- + \mu K_+ + \varsigma (s) K_3 ] d s
\right)\biggr]}^{1\over2}, \end{equation} where $K_- = {a^2 \over 2},$ $K_+ =
{{(a^\dagger)}^2 \over 2} $ and $K_3 = {1\over 4}(a a^\dagger +
a^\dagger a) $ are the standard bosonic realizations of the
$su(1,1)$ Lie algebra generators verifying the commutation
relations \begin{equation} [K_-, K_+] = 2 K_3, \qquad [K_3, K_{\pm}] = \pm
K_{\pm}\end{equation} and \begin{equation} \varsigma (s) = - 2 {d \over ds} \ln q(s) , \end{equation}
where \begin{equation} q(s)= \cosh(|\mu|(1-s)) + |\mu| \sinh(|\mu|(1-s)). \end{equation}
In this case, the Hamiltonian \eqref{hami-tilde}
becomes \begin{equation} \label{htilde-ro-u-0} {\tilde {\cal H}} = {\hat \rho}
(\mu,0) \ [ a^\dagger a + \mu {(a^\dagger)}^2 ]\ {\hat \rho}^{-1}
(\mu,0), \end{equation} and represents a Hermitian Hamiltonian describing two
photon processes in a single mode. To know the explicit form of
this Hamiltonian we must firstly factorize the operator
\eqref{ro-u-0} in the form of a product of exponential operators
of each $su(1,1)$ generators and then insert it into
\eqref{htilde-ro-u-0}. This process requires to solve some
Ricatti type differential equations.
For small values of $z,$ the Hamiltonian \eqref{hami-tilde}
describes corrections to the energy of this system as a
consequence of the deformation. In general, when $z\ne 0,$ the
Hamiltonian \eqref{hami-tilde} represents multi-photon processes
in a single mode.
The generalized coherent states associated to the system described
by \eqref{hami-tilde}, can be easily obtained from the coherent
states associated to the standard harmonic oscillator. Indeed,
they are given by \begin{equation} \label{gen-zmunu-ch} | \nu, z, \mu \rangle =
{\hat \rho} (\mu, z) \ G(\mu, z) D(\nu) |0\rangle, \end{equation} where
$D(\nu)$ is the standard unitary displacement operator defined at
the end of subsection \ref{sec-perturba-z-real}. These coherent
states correspond to the coherent states associated to the
pseudo-Hermitian Hamiltonian \eqref{exp-H-pseudo}, up to the
transformation $ {\hat \rho} (\mu, z),$ and are eigenstates of the
annihilation operator ${\tilde {\cal A}} = {\hat \rho} (\mu, z)\
{\cal A} \ {\hat \rho}^{-1} (\mu, z)$ corresponding to the
eigenvalue $\nu.$
\section{Conclusions}
In this paper, we have found some realizations of the deformed
quantum Heisenberg Lie algebra ${\cal U}_{z,p} (h(2)),$ in terms of
the usual creation and annihilation operators associated with Fock
space representation of the standard harmonic oscillator. The method
used to get these realizations can be easily applied to find the
realizations of other quantum Hopf algebras and super-algebras, such
as the bosonic and fermionic oscillators Hopf algebras\cite{HLR96}
or the quantum super-Heisenberg algebra, that can also be obtained
by using the $R$-matrix approach.
We have computed the AES associated to ${\cal U}_{z,p} (h(2)).$ We
have seen that the set of AES contains the set of coherent and
squeezed states associated to the standard harmonic oscillator
system but also a new class of deformed coherent and squeezed
states, parametrized by the deformation parameters. We have
studied the behavior of the dispersions of the position and linear
momentum operators of a particle in a class of perturbed squeezed
states and we have compared them with the behavior of these
dispersions in the minimum uncertainty squeezed states. Also we
have computed these dispersions on the deformed states associated
to ${\cal U}_{z,0} (h(2)),$ for all values of the $z$ parameter.
To first order in $z,$ these last dispersions reduce to the
perturbed ones obtained to ${\cal U}_{z,p} (h(2)),$ when p goes to
zero. Besides, we have constructed a $\eta$-pseudo Hermitian
Hamiltonian \cite{kn:AMostafazadeh} to which a subset of the set
of algebra eigenstates associated to ${\cal U}_{z,0} (h(2)),$ are
the coherent states. From this point of view, our deformed
states are linked to Hamiltonians presenting important physical
aspects \cite{kn:Mostafazadeh-2004}. Indeed, our pseudo-Hermitian
Hamiltonian verifies naturally all the properties of
pseudo-Hermitian Hamiltonians such as the existence of associated
biorthonormal basis, resolution of the identity, positive-definite
inner product, physical Hilbert space, unitary and invertible
operators mapping the pseudo-Hermitian operators to the Hermitian
ones, etc. Thus, with the help of pseudo-Hermitian quantum
mechanics techniques we are allowed to compute, for instance, the
spectrum, the eigenstates and the associated coherent states of
complicated deformed Hermitian Hamiltonians describing
multi-photon processes in a single mode. Also, we can compute more
easily quantities such as mean values of physical observables and
transition amplitudes. Moreover, it could be interesting to know,
at least for small values of the deformation parameter $z,$ the
explicit form of the resolution of the identity verified by the
generalized coherent states \eqref{gen-zmunu-ch}. Indeed, this
fact could have important consequences, for instance, in the study
of corrections to the time evolution of the quantum fluctuations
associated to the quadratures of the position and linear momentum
of a system characterized by a Hamiltonian describing one and two
photon processes in a single mode \cite{kn:Wei-Min}. This is a no
trivial problem and it could be developed elsewhere.
On the other hand, we have found new classes of deformed
squeezed states, parametrized by a real paragrassmann number,
i.e., a number $z$ such that $z^{k_0}=0, $ for some $k_0 \, \in
{\mathbb N}. $ These states can be normalized, even if $z$ is
considered as a complex paragrassmann number. In this last case,
when $k_0 =2,$ we can should interpret $z$ as an odd complex
Grassmann number and compare this new classes of deformed
squeezed states with the ones associated to the
$\eta$-super-pseudo-Hermitian Hamiltonians \cite{kn:NaVh3}.
\section*{Acknowledgments} The author would like to thank V. Hussin for
valuable discussions and suggestions. He also thanks the referees
for valuable suggestions about this article. The author's research
was partially supported by research grants from NSERC of Canada.
\renewcommand{\theequation}{\thesection.\arabic{equation}}
|
{
"timestamp": "2005-03-23T17:59:48",
"yymm": "0503",
"arxiv_id": "math-ph/0503055",
"language": "en",
"url": "https://arxiv.org/abs/math-ph/0503055"
}
|
\section{Introduction}
Freeze out (FO) is a term referring to the stage of expanding or exploding matter when
its constituents (particles) loose contact, collisions cease and the local dynamical
equilibrium no longer can be maintained.
When the local equilibrium is significantly perturbed, the microscopic length and time scales
become comparable to the characteristic macroscopic ones and the hydrodynamical approach used
to describe the evolution of matter breaks down.
In the absence of collisions the momentum distribution of the particles "freezes out", hence
the name kinetic freeze out.
\\ \indent
The final break-up corresponds to a "phase-transition" from an interacting fluid to a non-interacting
gas of particles, where the interactions between the constituents ceases suddenly when reaching
the "critical" FO temperature of the order of the pion mass
$T_{FO}\approx 140$ MeV, as first assumed by Landau \cite{Landau_1}.
Consequently, FO is a discontinuity in space-time represented by a space-time boundary or
FO hypersurface, taken at the critical temperature [i.e., FO isotherm].
Across such FO hypersurface the properties of matter change suddenly.
We denote the two sides of the FO hypersurface as Pre FO and Post FO sides.
Originally, the Post FO distribution function was assumed to be an equilibrated J\"uttner distribution
function boosted with the local flow velocity on the actual side of the FO hypersurface.
This approximation and method corresponds to the so-called "sudden" FO model
by Cooper and Frye \cite{Cooper-Frye}.
\\ \indent
The Cooper-Frye type of FO process is the zero thickness limit of a more realistic, so-called
"gradual" FO process, where the FO description applies over a finite space-time domain, i.e. FO layer.
Inside the finite FO layer the properties of the matter change gradually trough interactions, while the
frozen out particles are formed and emitted at different "temperatures" which correspond to the
actual temperature of the interacting matter, gradually during the whole evolution of the matter.
\\ \indent
The basic philosophy of this paper is similar to the recent work \cite{article_1}, which introduces
and analyzes in detail the gradual FO description for space-like FO situations.
Through the paper we are going to use the notation from Ref. \cite{article_1}, recall
the governing equations and the major results for comparison.
We have made this paper sole and complete and widely understandable without the need to consult
our previous work where the original ideas were first introduced.
\\ \indent
Time-like discontinuities represent the overall sudden change in a finite volume where the events happen
simultaneously at causally disconnected points of the hypersurface with a time-like normal vector.
For example the assumption of instantaneous or isochronous FO [i.e., happening at a constant time in
the center of mass system], belongs to this category.
\\ \indent
The aim of this paper is to present an analyze a simple gradual kinetic FO process with time-like normal vectors.
In the first part of this study we introduce and generalize the gradual kinetic FO treatment for a finite
time-like FO layer in a fully covariant footing, while in the second part we analyze its outcome.
\section{Freeze-out from a finite time-like layer}\label{Fid}
The basis of the gradual FO method is to separate the "full" $f=f(x,p)$ distribution function into still
interacting and already frozen out parts, $f=f^i + f^f$, and describe the evolution of both components
in a self consistent way \cite{grassi_1, cikk_1, cikk_2, cikk_3, hama_1}.
This can be achieved by introducing the so-called escape rate, $\mathcal{P}_{esc}(s,p)$, used
to drain particles, which no longer collide from the interacting component, $f^i$, and to
gradually build up the free component, $f^f$.
For the better understanding of the model we use Fig. \ref{figure_1t}, and assume that the FO
of particles starts from the inside boundary of the FO layer, $S_1$ (thick line).
Within the FO layer of finite thickness, $L$, the density of interacting particles
decreases and disappears once we reach the outside boundary, $S_2$ (thin line),
of the FO layer.
\\ \indent
In general, the kinetic description of freeze out leads to a complex multidimensional problem.
To clarify the basic properties of the FO process through a finite layer we may essentially
reduce the number of variables, assuming that the dominant change in the distribution function
happens in the direction of the FO normal vector, while it is negligible along the directions
perpendicular to it, (e.g. in a spherically symmetric system the change happens in radial direction,
and it is negligible in the perpendicular directions).
Thus, the FO process can be effectively described as a one-dimensional process and the space-time domain where such a process takes place can be viewed as a FO layer, where the FO normal vector
is tied to the direction of the density decrease arising from velocity divergence at a curved FO surface.
\\ \indent
If we have a space-like normal vector, $d\sigma_{\mu}=(0,1,0,0)$, the resulting equations can be transformed
into a frame where the process is stationary, while in the case of a time-like normal vector,
$d\sigma_{\mu}=(1,0,0,0)$, the equations can be transformed into a frame where the process is
uniform and time-dependent.
In this paper we only discuss FO processes inside a finite time-like layer.
\\ \indent
Here we recall the governing equations [i.e., Eqs. (13-14)] from Ref. \cite{article_1},
which can be used in both time-like and space-like FO cases.
The equations depend on the projection, $s = x^{\mu} d\sigma_{\mu}$, in the direction of the FO normal
vector, $d\sigma_{\mu}$, where the four vector $x^{\mu}$ denotes the particle coordinate,
having its origin at the inner surface of the FO layer.
Thus,
\begin{eqnarray}\label{first} \nonumber
\partial_s f^{i} (s,p) \! &=& \! - \mathcal{P}_{esc}(s,p) \, f^{i}(s,p) + \frac{f^{i}_{eq}(s,p) - f^{i}(s,p)}{\lambda_{th}} \, , \\
\partial_s f^{f} (s,p) \! &=& \! + \mathcal{P}_{esc}(s,p) \, f^{i}(s,p) \, ,
\end{eqnarray}
where using the relaxation time approximation we ensure that the interacting component approaches
the equilibrated J\"uttner distribution, $f^i_{eq}$, with $\lambda_{th}$ relaxation time
(or relaxation length in the space-like case).
\\ \indent
The escape rate, $\mathcal{P}_{esc}(s,p)$, describes the escape of particles from the interacting component
into the free component and it is defined as:
\begin{equation}\label{esc2}
\mathcal{P}_{esc}(s,p) = \frac{1}{\lambda(s)}
\left[ \frac{L}{L - s} \, \frac {p^\mu d\sigma_\mu}{p^\mu u_\mu}\right] \Theta(p^{\mu}d\sigma_{\mu}) \, ,
\end{equation}
where the parameter, $L$, is the "proper" thickness of the FO layer and it is an invariant scalar.
The proper thickness is analogous to the proper time, that is the time measured in the rest
frame of the particle.
In our case the local rest frame is the rest frame attached to the FO front (RFF), (see \ref{frames}),
and thus the proper "thickness" of the FO layer is the invariant proper time interval between the start
of the process and its end, (see Fig. \ref{figure_1t} at point A).
Furthermore, $p^{\mu}$ is the four-momentum of particles, $d\sigma_{\mu}$ is the normal in the FO direction,
while $u^{\mu}$ is the flow velocity normalized to unity.
The initial characteristic time is denoted by $\tau_0$, and the $\Theta(p^{\mu}d\sigma_{\mu})$ function,
was first introduced by Bugaev \cite{Bugaev_1} to ensure that all particles leave to the outside.
In case of time-like FO this condition is always satisfied and does not lead to any additional constraint.
\begin{figure}[t!]
\centering
\includegraphics[width=8.5cm, height = 8.2cm]{figure_1t.eps}
\caption{(Color online) The figure shows a finite FO layer with varying thickness.
The normal to a surface element is $d\sigma_{\mu}$, thus between A and B the surface is
time-like, [i.e., $d\sigma^{\mu} d\sigma_{\mu} = + 1$],
from B down to C it is space-like, [i.e., $d\sigma^{\mu} d\sigma_{\mu} = -1$].
The change from time-like to space-like surface happens where the normal of the surface is light-like,
but not necessarily has its origin at the center of the system.
The momentum of particles is $p^{\mu}$ and the four vector $x^{\mu}$ denotes the particle coordinate,
having its origin at the inner surface of the FO layer.
On the time-like FO region all particles emerging from a point on the Pre FO side will propagate
to the Post FO side, while the particles originating from the space-like part of the FO surface
are divided between Pre FO and Post FO parts.
Only those particles cross the surface which have their momentum enclosed by the light cone and
the Post FO surface, [i.e., if $p^{\mu}d\sigma_{\mu}>0$].}
\label{figure_1t}
\end{figure}
\\ \indent
A qualitative expression of the escape rate for both time-like and space-like FO situations is based
on the following simple assumptions.
Particles with higher momentum in the FO direction will freeze out first.
The particles closer to the outside boundary of the FO layer have a greater chance
to freeze out since the probability to find another particle to collide with is smaller
as the system became sparser, (given by the $L/(L-s)$ factor).
In our special case the FO direction is parallel to the time-axis, thus all particles will freeze out
irrespective of their momenta.
However, particles emitted at later times will freeze out "faster" since they have less chance
to collide with other particles in the diluted system.
Please note that, although $\tau_0$ is assumed to be a constant for simplicity, the characteristic
FO length is increasing with time or distance such as, $\tau_0(L - s)/L$.
The detailed treatment and analysis of the escape rate can be found in \cite{article_1, ModifiedBTE_1, QM05_1}.
\\ \indent
Now, if we describe the time evolution of the particle FO, then $d\sigma_{\mu} = (1,0,0,0)$,
$x^{\mu}d\sigma_{\mu} = t$ and $p^{\mu}d\sigma_{\mu} = p^0$, thus eq. (\ref{first}) leads to
\begin{eqnarray}\label{first-rethermalized} \nonumber
\partial_t f^{i} &=& - \frac{1}{\tau_0} \left( \frac{L}{L-t} \right)\!
\left( \frac{p^0}{p^{\mu} u_{\mu}} \right) f^{i}
+ \frac{ f^{i}_{eq} - f^{i}}{\tau_{th}} \, , \\
\partial_t f^{f} &=& + \frac{1}{\tau_0} \left(\frac{L}{L-t} \right)\!
\left(\frac{p^0}{p^{\mu} u_{\mu}}\right) f^{i} \, ,
\end{eqnarray}
where the interacting component approaches the equilibrated J\"uttner distribution, $f_{eq}$,
with $\tau_{th}$ relaxation time.
This is a common simplification and the practical reason to use it is to calculate the quantities
depending on the equilibrium distribution function.
Although the present solution mathematically is achieved taking an infinitely short relaxation time, in reality,
one can show, see Ref. \cite{article_1}, that the complete thermalization of the interacting component can
be achieved with good accuracy if $\tau_0$ is smaller than $\tau$ by a factor of 2 or more.
For a thorough analysis of this approach, see Refs. \cite{article_1, sven} and the Appendix.
Here we mention that in our calculations we use only one type of particles,
namely massless pions, therefore the chemical composition of our system remains unchanged during
the kinetic FO.
Furthermore, for simplicity we assume simultaneous chemical and thermal equilibration, thus the
chemical potential of the massless pion gas is $\mu=0$ during the FO process.
\\ \indent
Earlier in Ref. \cite{cikk_5}, the gradual time-like FO description was modeled with equations
having a similar form to eqs. (\ref{first-rethermalized}), but it was only treating the simplest
case when $u^{\mu}=d\sigma_{\mu}$, while the FO was lasting infinitely long.
The model was based on the idea of the boost invariant Bjorken hydrodynamical model \cite{Bjorken},
where the evolution of matter is a function of the proper time, $\tau$, only, while the flow
of matter is parallel to the normal vector of the proper time hyperbolas in every point.
The covariant equations, eqs. (\ref{first-rethermalized}), return those equations if we change
the time variable, $dt= d\tau$, and neglect the $L/(L-t)$ factor.
However, the new equations allow $u^{\mu}\neq d\sigma_{\mu}$ and also a possibility to finish the FO process
within a given duration.
\subsection{Reference frames}
\label{frames}
Before proceeding further, first we define the reference frames in which our calculations
will be handled.
\begin{figure}[!hbt]
\centering
\includegraphics[width = 8.4cm, height = 7.2cm]{figure_2t.eps}
\caption{(Color online) A simple FO hypersurface in RFG with coordinates [t,x],
where $u^{\mu} = (1,0,0,0)_{RFG}$.
The normal vectors of the FO front, $d\sigma_{\mu}$, are time-like at points, A, B, C,
while the normal vectors are space-like at points, D, E, F.
At point B in RFF with coordinates [t',x'], where $d\sigma_{\mu}=(1,0,0,0)_{RFF}$ and $u^{\mu} = \gamma_{\sigma}(1,-v_{\sigma},0,0)_{RFF}$.
Note that RFF moves together with the FO front, while on the figure the origin of RFF is shifted to match
the origin of RFG.
}
\label{figure_2t}
\end{figure}
\\ \indent
On the Pre FO side the matter is parameterized by an equilibrium distribution function,
as required by the hydrodynamical description of evolution.
The frame where the matter is at rest is the local Rest Frame of the Gas (RFG), where $u^{\mu} = (1,0,0,0)_{RFG}$.
On the Post FO side we can use the frame which is attached to the FO front, that is the
Rest Frame of the Front (RFF). In RFF the normal vector to the FO hypersurface for the time-like part is always
$d\sigma_{\mu} = (1,0,0,0)_{RFF}$, while on the space-like part is always $d\sigma_{\mu} = (0,1,0,0)_{RFF}$.
\\ \indent
If we are in the RFG then the four-flow is always $u^{\mu} = (1,0,0,0)_{RFG}$.
If we take different characteristic points, for example points, A, B and C, on the FO hypersurface
then the normal vector is different at different points of the hypersurface in RFG, see Fig. \ref{figure_2t}.
To calculate the parameters of the normal vector, $d\sigma_{\mu}$, for different cases in the RFF
we make use of the Lorentz transformation.
\\ \indent
The normal vector of the time-like part of the FO hypersurface may be defined as the local $t'$-axis,
while the normal vector of the space-like part may be defined as the local $x'$-axis.
This defines the axes of RFF.
\subsection{Conservation laws}\label{conservation}
The change of conserved quantities caused by the particle transfer from the interacting matter
into the free matter can be obtained in terms of distribution function of the interacting matter
calculated from eqs. (\ref{first-rethermalized}) as:
\begin{eqnarray}
dN^{\mu}_{i} (t) \! &=& \! dt \! \! \int \frac{d^3 p}{p^0} \, p^{\mu} \, \partial_{t} f^{i} \\ \nonumber
\! &=& \! - \frac{dt}{\tau_0} \frac{L}{L-t} \int \frac{d^3 p}{p^0}\, p^{\mu}
\, \frac{p^{\rho} d\sigma_{\rho}}{p^{\rho}u_{\rho}} f^i_{eq}(t,p) \, ,
\end{eqnarray}
while the change in the energy-momentum as:
\begin{eqnarray}
dT^{\mu\nu}_{i} (t) \! &=& \! dt \! \! \int \frac{d^3 p}{p^0} \, p^{\mu} p^{\nu} \, \partial_{t} f^{i} \\ \nonumber
\! &=& \! - \frac{dt}{\tau_0} \frac{L}{L-t} \int \frac{d^3 p}{p^0}\, p^{\mu} p^{\nu}
\, \frac{p^{\rho} d\sigma_{\rho}}{p^{\rho}u_{\rho}} f^i_{eq}(t,p)\, .
\end{eqnarray}
The equilibrium distribution function for massless baryonfree particles is:
\begin{equation}
f^i_{eq}(t,p) = \frac{g}{(2\pi \hbar)^3} \,
\exp{\left[-\frac{{\gamma(p^0 - jup \cos{\theta}_{\vec{p}})}}{T}\right]} \, ,
\end{equation}
where the four-momentum of particles is $p^{\mu} = (p^0,\vec{p})$, $p=|\vec{p}|$, $p^x = p \cos \theta_{\vec{p}}$,
the flow velocity of the interacting matter in RFF is $u^{\mu} = \gamma(1,v,0,0)_{RFF}$,
$\gamma = 1/\sqrt{1 - v^2}$, $u = |v|$, $j = \textrm{sign} (v)$,
and $g$ is the degeneracy of particles.
The results of the calculations in the RFF can be found in Appendix B.
\\ \indent
The change in energy density after a step $dt$ is
\begin{equation}\label{energy_density}
d e_{i}(t) = u_{\mu,i}(t) \, dT^{\mu\nu}_{i}(t) \, u_{\nu,i}(t) \, ,
\end{equation}
from which by using a simple EoS, $e = \sigma_{SB} T^4$, for a baryonfree massless gas
with $\sigma_{SB} = \frac{\pi^2}{10}$, we can calculate the change
in the temperature and Landau's flow velocity similarly to \cite{article_1, cikk_1, cikk_3}.
Thus,
\begin{eqnarray}\label{landau}
d \ln T &=& \frac{\gamma^{2}}{4\sigma_{SB} T^{4}} \bigg[ dT^{00}_i -
2vdT^{0x}_i + v^{2} dT^{xx}_i \bigg] \, ,\\ \nonumber
d v &=& \frac{3}{4 \sigma_{SB} T^{4}}\bigg[ -v dT^{00}_i + (1+v^{2}) dT^{0x}_i -v dT^{xx}_i \bigg]\,.
\end{eqnarray}
while in the massless limit the above equations lead to:
\begin{eqnarray} \label{massless_landau}
d \ln T &=& - \frac{dt}{\tau_0} \left(\frac{L}{L-t} \right)\frac{3n\gamma}{4\sigma_{SB} T^{3}} \, ,\\ \nonumber
d v &=& - \frac{dt}{\tau_0} \left(\frac{L}{L-t} \right)\frac{3 n v}{4 \gamma \sigma_{SB} T^{3}} \, .
\end{eqnarray}
\section{Results and discussions}
In this section we will present the results for the Post FO distribution and the relevant quantities
calculated form this model.
We will present our results for two different cases, for infinite FO (I) and finite FO (F).
\begin{itemize}
\item [ I) \,]
The system is characterized by an infinitely long FO duration, [i.e., $L/(L-t) \rightarrow 1$ and $(t = 100 \tau_0)$,
where most of the matter is frozen out].
The results are shown on
Figs. \ref{figure_3t}, \ref{figure_5t}, \ref{figure_7t}, \ref{figure_8t}.
\item [ F) \,] A finite FO process happening in a finite FO layer, where
$(L=10 \tau_0)$. The results are shown on
Figs. \ref{figure_4t}, \ref{figure_6t}, \ref{figure_7t}, \ref{figure_9t}, \ref{figure_10t}.
\end{itemize}
\subsection{The evolution of temperature of the interacting component}
The first set of figures, Fig. \ref{figure_3t} and Fig. \ref{figure_4t}, shows the gradual decrease in
temperature of the interacting component calculated in RFF.
\\ \indent
Comparing Fig. \ref{figure_3t} with Fig. \ref{figure_4t}, we see the difference between the
finite and infinite FO.
The FO in a finite layer is faster than in an infinite layer, "per se".
The temperature curves belonging to different initial flow velocities but with opposite
sign [i.e., $v_0 = -0.5$ and $v_0=0.5$] are the same.
This is so since for time-like FO we do not have any constraint on the momenta,
such as the cut-off factor $\Theta(p^{\mu}d\sigma_{\mu})$, and the initial momentum distribution
is symmetric over the time axis.
In the case of time-like FO the gradual cooling of the matter is faster and "smoother" compared
to space-like FO, since the most energetic particles freeze out unrestricted in direction
thus the remaining interacting component cools down faster.
\\ \indent
The matter with higher initial flow velocity, $v_0$, cools faster, but for small differences
between the initial flow velocities, the resulting difference in the temperature is negligible.
If the interacting gas has a higher flow velocity in RFF, the escape rate is bigger for
higher values of the flow velocity.
This expresses the fact that if the matter flows we remove energy faster form the
interacting component.
\\
\begin{figure}[t!]
\centering
\includegraphics[width=8.5cm, height = 5.5cm]{figure_3t.eps}
\caption{(Color online) The temperature of the interacting component in RFF, calculated
for an infinitely lasting FO.
The initial temperature is \mbox{$T_0 = 170\,$ MeV}, the parameter, $v_0$, is the initial
flow velocity.
This corresponds to case I.}
\label{figure_3t}
\end{figure}
\\
\begin{figure}[hbt!]
\centering
\includegraphics[width=8.5cm, height = 5.5cm]{figure_4t.eps}
\caption{(Color online) The temperature of the interacting component in RFF, calculated
for a finite $(L=10 \tau_0)$ FO time, where the initial temperature is \mbox{$T_0 = 170\,$ MeV},
and $v_0$ is the initial flow velocity.
This corresponds to case F.}
\label{figure_4t}
\end{figure}
\subsection{The evolution of common flow velocity of the interacting component in RFF and RFG}
The second set of figures, Figs. \ref{figure_5t}, \ref{figure_6t}, shows the evolution of
the flow velocity of the interacting component calculated for a baryonfree massless gas in RFF.
\\ \indent
Again, comparing Fig. \ref{figure_5t} with Fig. \ref{figure_6t}, we can see the difference between
finite and infinite FO.
In the case of finite FO the velocity decrease is much faster than in the case of infinite FO.
\\ \indent
Furthermore, we notice that the flow velocity of the interacting component tends to
zero, while here we recall to compare, that in case of space-like FO it tends to $-1$.
Again, this is due to the cut-off factor which retains particles propagating with negative momenta
in space-like directions.
In RFF the quantities change discontinuously at the light cone and that is why we
have different results for the final flow velocity comparing space-like and time-like cases.
However, in RFG all quantities are continuous when crossing the light cone, see Ref. \cite{article_1,QM05_1}.
\\
\begin{figure}[!t]
\centering
\includegraphics[width=8.5cm, height = 5.5cm]{figure_5t.eps}
\caption{(Color online) The evolution of the flow velocity of the interacting component calculated for an
infinitely lasting FO, corresponding to case I.
The initial temperature is $T_0 = 170\,$ MeV, and $v_0$ is the initial flow velocity of the gas.}
\label{figure_5t}
\end{figure}
\begin{figure}[!hbt]
\centering
\includegraphics[width=8.5cm, height = 5.5cm]{figure_6t.eps}
\caption{(Color online) The evolution of the flow velocity of the interacting component calculated
for a finite $(L=10 \tau_0)$ FO, corresponding to case F.
The initial temperature is $T_0 = 170\, $ MeV, and $v_0$ is the initial
flow velocity of the gas.}
\label{figure_6t}
\end{figure}
\subsection{The transverse momentum and the contour plots of the Post FO distribution}
The third set of figures, Figs. \ref{figure_7t}, and \ref{figure_8t}, shows the evolution of the local
transverse momentum distribution and the corresponding contour plots of the Post FO momentum distribution.
\\ \indent
We have presented a one-dimensional model here, but we assume that it is applicable for the direction
transverse to the beam in heavy ion experiments.
The plots presented should be related to the transverse momentum distribution of measured particles.
\\
\begin{figure}[!hbt]
\centering
\includegraphics[width=8.5cm, height = 5.5cm]{figure_7t.eps}
\caption{(Color online) The local transverse momentum (here $p_x$) distribution for a baryonfree and massless
gas at $(p_y = 0)$.
The calculations were done for an infinite FO with thin lines and for
a finite FO $(L=10\tau_0)$ with marker lines.
The initial flow velocity and temperature are, $u^{\mu}=(1,0,0,0)_{RFG}$ and $T_0 = 170\,$MeV.
The transverse momentum spectrum at the end is obviously curved due to the FO process
for low momenta.}
\label{figure_7t}
\end{figure}
\\ \indent
From Fig. \ref{figure_7t} we see that the Post FO momentum distributions for the
infinite and finite FO cases are qualitatively identical.
At the early stages of the FO process (for values of $t \simeq \tau_0$) the distribution
of particles in the two cases match.
This property persists until the end of the FO process, and can be seen on Fig. \ref{figure_7t}, where
the local transverse momentum distribution calculated for an infinite and finite FO are identical.
The maximum is increasing with $t$ as indicated in Figs. \ref{figure_7t} and \ref{figure_8t}.
Thus, the final Post FO distributions do not differ if we switch from an infinitely long to a finite
layer FO description for any initial flow velocity.
This means that our finite layer FO description was done correctly.
These important features of the model were already discussed in detail in Refs. \cite{article_1, QM05_2}.
\\ \indent
The overall conclusion is that the resulting Post FO distributions are non-thermal distributions even
for time-like FO processes.
The distributions strongly deviate from thermal distributions in the low momentum region [i.e., $p_x < 300$ MeV].
If one decreases the duration of total FO time, the gradual FO process would still produce similar
particle spectra until the duration is not less than $2\tau_0$.
Below that value the spectra becomes less curved and in the limit when the duration approaches zero
the final momentum spectra corresponds to a constant temperature equilibrium distribution function.
For $L < 2\tau_0$ the FO process does not have enough time to significantly change the shape of the
final spectrum.
\\ \indent
Here one may go further and intuitively say that the FO process has a maximal lifespan,
even though the parameter, L, was not defined in this work in such way that it would
allow us to exactly calculate its limits from first principles.
Of course for such a statement to hold one would need a realistic full scale fluid dynamical
simulation including chemistry, secondaries and the expansion of the system.
However, our results concluded from this simple model could still hold valuable in the realistic
FO modeling in complex fluid dynamical simulations.
\begin{figure}[!t]
\centering
\includegraphics[width=8.5cm, height=3.8cm]{figure_8t.eps}
\caption{(Color online) The Post FO distribution, $f_{free}(x,\vec{p})$, at point A of Fig. \ref{figure_2t}, for an infinitely
long FO length.
The figures correspond to different time points, {\bf $t = 1 \tau_0,\ 10 \tau_0,\ 100 \tau_0$}
respectively.
Contour lines are given at values represented on the figure.
The initial flow velocity and temperature are; $u^{\mu}=(1,0,0,0)_{RFG}$ and $T_0 = 170\,$ MeV.
The maximum is increasing with $t$ as indicated on Fig. \ref{figure_7t}.
}
\label{figure_8t}
\end{figure}
\subsection{The boosted Post FO distributions}
The fourth set of figures, Fig. \ref{figure_9t} and Fig. \ref{figure_10t}, shows the
final Post FO distributions calculated for different flow velocities in RFF and the
boosted post FO (J\"uttner) distributions in RFG.
The FO distributions corresponding to different initial flow velocities,
$u^{\mu}=\gamma_{\sigma}(1,-v_{\sigma},0,0)_{RFF}$, generally lead to
non-equilibrated and anisotropic Post FO distributions.
However, boosting the distribution from point A to points B or C leads to a more elongated FO distribution
in the direction of the boost than the calculated Post FO distributions at those points,
(compare the contours with the same values given on Figs. \ref{figure_9t} and \ref{figure_10t}).
\begin{figure}[!t]
\centering
\includegraphics[width=8.5cm, height =3.8cm]{figure_9t.eps}
\caption{(Color online) The final Post FO distribution, $f_{free}(x,\vec{p})$, at points A, B, C, of Fig. \ref{figure_2t},
calculated for a finite FO time, $L = 10\tau_0$.
The contour plots correspond to different initial flow velocities,
$u^{\mu}=\gamma_{\sigma}(1,-v_{\sigma},0,0)_{RFF}$, with $v_{\sigma} = 0, 0.5c, 0.9c$, respectively
where the initial temperature is $T_0 = 170\, $ MeV.
Note that, the Post FO distributions at points B and C are not the boosted distributions of point A.
The difference is that the Post FO distribution is much less elongated than the boosted J\"uttner, because
it is a superposition of sources with decreasing speed in RFF as indicated in Fig. \ref{figure_6t}.}
\label{figure_9t}
\end{figure}
\\ \indent
This is also an important outcome of our analysis leading to the conclusion that assuming an
isotropic equilibrated J\"uttner distribution at time-like parts the FO hypersurface, other
than at point A on Fig. \ref{figure_2t} where $u^{\mu} = d\sigma_{\mu}$, in general cannot hold
\cite{gorenstein}.
More importantly, the common practice of boosting the J\"uttner distribution or any
post FO distribution function instead of calculating it from the conservation laws,
similarly as it was done here, leads to a noticeable difference in the final particle spectra.
\subsection{The non-equilibrated post FO distributions revised}
Here we present another important result of our study, following the approach from
Ref. \cite{cikk_5}, where an infinitely long FO was studied with momentum independent escape rate.
We can reproduce that earlier result by taking $u^\mu(t_0)=d\sigma^\mu=(1,0,0,0)$, and can calculate the temperature decrease using eqs. (\ref{massless_landau}), in the case of a massless baryonfree matter for an
infinitely lasting FO:
\begin{equation}
T(t) = T(t_0) \, \exp \left(-\frac{k}{\tau_0} (t-t_0) \right)\, ,
\end{equation}
where $k = 3 /(4 \pi^2\sigma_{SB})$. In the general case, the flow velocity is not zero, hence
one has to solve the system of equations from eqs. (\ref{massless_landau}).
The distribution function of interacting particles (in the fast rethermalization limit) at any time $t$ is:
\begin{equation}
f^i(t,p) = \frac{1}{(2\pi)^3} \, \exp \left(-\frac{p^0}{T(t_0)} \, e^{k (t-t_0)/\tau_0}\right) \, ,
\end{equation}
Now, we can solve the equation for the free component from eqs. (\ref{first-rethermalized}), therefore the distribution of free particles at time $t$ is:
\begin{eqnarray}\label{exp_int_t}
f^f(t,p) &=& \frac{1}{\tau_0}\int_{t_0}^{t} f^i(t',p) \\ \nonumber
&=&\frac{k^{-1}}{(2\pi^3)} \, \left[ \textrm{Ei} \left( -\frac{p^0}{T(t)} \right)
- \textrm{Ei} \left( -\frac{p^0}{T(t_0)}\right) \right]\, ,
\end{eqnarray}
which for $t\rightarrow \infty$ leads to:
\begin{equation}\label{exp_int}
f^f = \frac{k^{-1}}{(2\pi^3)} \, \textrm{Ei} \left( -\frac{p^{\mu} u_{\mu}(t_0)}{T(t_0)}\right)
\end{equation}
where $\textrm{Ei}$ is the exponential integral function defined in Appendix B eq. (\ref{exponential_integral}).
Thus, we got a simple formula, similarly to the one in Ref. \cite{cikk_5}, which correctly parameterizes
the non-equilibrated post FO distribution function when $u^{\mu} = (1,0,0,0)$.
It was actually shown in Ref. \cite{article_1} that the post FO distribution is not
sensitive to momentum dependence of the escape rate, so we assume that this simple formula
is valid for any initial flow velocity $u^\mu(t_0)$.
\\ \indent
Here we will use the above formula to plot the post FO distribution, for finite layers, with $L > 2\tau_0$,
and extend this approximation to the general case when $u^{\mu}(t_0)\neq d\sigma_{\mu}$.
On Fig. \ref{figure_12t}, we have plotted the final FO distribution functions calculated
using eq. (\ref{exp_int}) with lines, and the finite FO $(L=3\tau_0)$ calculation with marker lines.
The results are matching, which is a remarkable result, thus we conclude that this simple approximation
is applicable for the description of gradual FO thorough finite time-like layers and correctly
approximates its post FO distribution functions.
\begin{figure}[!t]
\centering
\includegraphics[width=8.5cm, height =3.8cm]{figure_10t.eps}
\caption{(Color online) The post FO distribution at point A and boosted to the frames at points B and C as
depicted in Fig. \ref{figure_2t}.
The different figures correspond to different initial normal vectors,
$d\sigma^{\mu}=\gamma_{\sigma}(1,v_{\sigma},0,0)_{RFG}$,
where {\bf $v_{\sigma} = 0, 0.5c, 0.9c$}, and the initial temperature is $T_0 = 170\,$ MeV.}
\label{figure_10t}
\end{figure}
\begin{figure}[!hbt]
\centering
\includegraphics[width=8.5cm, height = 5.5cm]{figure_12t.eps}
\caption{(Color online) The local transverse momentum (here $p_x$) distribution for a baryonfree and massless
gas at $(p_y = 0)$.
The calculations were done for an infinite FO with thin lines using eq. (\ref{exp_int})
and for a finite FO $(L=3\tau_0)$ with marker lines.
The initial temperature was $T_0 = 170\,$MeV.
The different initial flow velocities are given on the figure legend.}
\label{figure_12t}
\end{figure}
\section{Conclusions}
In this work we have presented a simple kinetic freeze out model for a finite time-like layer.
We have demonstrated that FO across time-like surfaces leads to non-equilibrated and anisotropic distributions.
These distributions in general cannot be Lorentz transformed to a frame where the distribution is isotropic.
The only exception is when the normal to the FO hypersurface is parallel to the local flow velocity.
Our analysis shows that the usual practice of assuming a J\"uttner distribution as a Post FO distribution is
in general not valid!
\\ \indent
We can also see that while the boosted J\"uttner distribution is elongated in the boost direction,
i.e. in the direction of $d\sigma_{\mu}$, the Post FO distribution is close to a spherical and isotropic
distribution at low momenta, and becomes elongated only at higher momenta, see Fig. \ref{figure_9t}.
This special Post FO distribution leads to a curved "$p_{t}$ - spectrum".
Here, we can also demonstrate (as in earlier works \cite{article_1, cikk_1, cikk_2, cikk_3}) that non-equilibrium
processes in kinetic FO lead to observable effects.
\\ \indent
We observe that the J\"uttner distribution is not a good approximation for the Post FO distribution,
just like in the case of a space-like FO.
While in the case of space-like FO the Cancelling-J\"uttner distribution introduced in Ref. \cite{karolis}
is satisfactory, in the case of time-like FO, we have found a simple formula to use.
\\ \indent
Now, one may ask the question whether we observe this additional low $p_t$ effect in the experimental data.
This effect has several possible explanations: products of low momentum resonance decays,
the transverse expansion of the system, and possibly due to the long gradual FO with rethermalization.
As already discussed, during such scenario the particles are freezing out at different
gradually decreasing temperatures, thus correspondingly the final FO spectrum is a
superposition of thermal distributions with different temperatures.
Although the low $p_t$ enhanced non-thermal spectrum of massless pions is a necessary outcome of long gradual
FO with rethermalization, for the heavy particles (if these are in the mixture with pions) it is almost
unobservable, see Ref. \cite{cikk_5}.
\\ \indent
In our simplistic study we have found that FO in layer of finite thickness below $L < 2\tau_0$ will
not show a sharp peak at low momentum in the transverse momentum spectrum.
Thus, naively one can conclude from a simple fit that the FO in heavy-ion collisions happens
in a narrow or wide FO layer.
However, such conclusion would be premature without including the expansion of the system and
calculate two particle correlations.
At the moment FO in a long finite layer, $L > 2\tau_0$, cannot be excluded.
If one assumes gradual FO with non-thermal post FO spectra, then one may also fit the data but with
different flow velocity and slope parameter, where the curvature of the pion spectra at low $p_t$ will
be partly due to FO and partly due to the expansion of the system.
\section{Outlook}
We do not aim directly to apply the results presented here to experimental heavy ion collision data,
instead our purpose was to study qualitatively the basic features of the freeze out process, and
to demonstrate the applicability of this covariant formulation for FO in a finite layer.
\\ \indent
Here we note that our model may be applicable in CFD calculations, where one has both time-like
and space-like parts of the full FO hypersurface, thus the gradual FO calculation must be done over
the full FO hypersurface, with varying flow velocities and normal vectors.
The method should be applied after reaching the $T_{FO}$ critical temperature and calculate the Post FO
momentum distribution function starting form the inner FO hypersurface.
Such a calculation should be compared to the Cooper-Frye ansatz in the first place and then to
experimental results, similarly to Refs. \cite{grassi_1, hama_2, grassi_2}.
\\ \indent
A successful application of this model was already used to study the impact of nucleon mass shift
on the freeze out process \cite{sven}.
This analysis will be carried forward to calculate the impact of mass shift on other particles,
such as pions and kaons, which will help us study the effect of mass shift on two-particle correlations.
An even more interesting study based on our analysis will estimate the effect of expansion on the
time-like freeze out process using the Bjorken model \cite{new_article}.
Therefore, we believe that our model may give a better description and understanding of the final
observables which are calculated using the single (and two) particle distribution functions.
\section*{ACKNOWLEDGMENTS}
The authors, L. P. Csernai, E. Moln\'ar, A. Ny\'iri and K. Tamosiunas thank
the hospitality of the University of Cape Town, where parts of this work were done.
E. Moln\'ar, also thanks the hospitality of the Babe\c s-Bolyai University of Cluj.
\\ \indent
Enlightening discussions with Cs. Anderlik,
T. S. Bir\'o, J. Cleymans, A. Dumitru and S. Zschocke are gratefully acknowledged.
\section*{APPENDIX A}
Here we discuss the properties of the rethermalization term from eq. (\ref{first-rethermalized})
and its consequences on finishing the FO process in a finite layer.
\begin{figure}[t!]
\centering
\includegraphics[width=8.6cm, height = 2.4cm]{figure_11t.eps}
\caption{(Color online) A schematic view of the FO process for a linear density profile.
Initially the mean free path is $\lambda_{mfp} = a_0$, the relaxation length is $\lambda_{th} = 2 \lambda_{mfp}$,
the initial characteristic length is $\lambda_0$, while the length of the FO layer is $L = 4\lambda_0$.}
\label{figure_11t}
\end{figure}
\\ \indent
From kinetic theory we know that if the following conditions:
\begin{equation}
\tau_{mfp} < \tau_{th} < \tau_0 \quad \text{or} \quad
\lambda_{mfp} < \lambda_{th} < \lambda_{0}\, ,
\end{equation}
between the average length between the collisions, the relaxation length, and the characteristic length
are satisfied, then we can use the Boltzmann Transport Equation (BTE) for the evolution of the single
particle distribution function, $f(x,p)$.
\\ \indent
For better understanding we first assume a linear decrease of the interacting particle density during
freeze out, such as, $n(x) = (L - x)/L$, where the mean free path of interacting particles is,
$\lambda_{mfp}(x) \approx 1/n(x)$.
Using the relaxation time approximation for the FO process, for example at $x>2\lambda_0$ the density
of the interacting particles already decreased to $n(x) < n_0/2$, while the mean free path increased to
$\lambda_{mfp}(x) > a =2a_0 $, see Fig. \ref{figure_11t}.
Consequently, by the end of the FO process, the thermalization length becomes longer than the initial characteristic
length of the system.
Although, $\tau_0$ is constant (scale parameter), the characteristic FO length is actually not constant during the evolution.
Thus, using the rethermalization approximation the error we introduce within is of the order of $\tau_{th}/\tau_0$.
\\ \indent
In our model the change in the density is generally given as:
\begin{equation}\label{density}
d n_{i}(x) = u_{i,\mu}(x) \, dN^{\mu}_{i}(x) \, .
\end{equation}
This leads to an exponentially fast decrease of particle density, therefore more than $95\%$ of the interacting matter is frozen out before $\tau_{th}\simeq \tau_0$, thus we can safely use the
relaxation time approximation in our calculations.
\section*{APPENDIX B}
The changes in the particle four current and energy momentum tensor are:
\begin{eqnarray} \nonumber
d N^{0}_i (t) &=& - \frac{dt}{\tau_0} \frac{L}{L-t} \frac{n}{4ju\gamma}
\Bigg\{- G_1^+(m) + \, G_1^-(m) \Bigg\} \\ \nonumber
&\buildrel m=0 \over \longrightarrow & - \frac{dt}{\tau_0} \frac{L}{L-t} n
\Bigg[\frac{(3+v^2)}{3}\,\gamma^2 \Bigg] \, ,
\end{eqnarray}
\begin{eqnarray} \nonumber
d N^{x}_i (t) &=& \frac{dN^{0}_i (t)}{ju} - \\ \nonumber
&&\frac{dt}{\tau_0} \frac{L}{L-t} \frac{n}{4ju\gamma}
\Bigg\{- 2b\Big[2 K_1(a) + aK_0(a) \Big]\Bigg\} \\
&\buildrel m=0 \over \longrightarrow & - \frac{dt}{\tau_0} \frac{L}{L-t} n
\Bigg[\frac{(3+v^2)}{3v}\,\gamma^2 - \frac{1}{v}\Bigg] \, ,
\end{eqnarray}
\begin{eqnarray} \nonumber
d T^{00}_i (t) &=& - \frac{dt}{\tau_0} \frac{L}{L-t} \frac{nT}{4ju\gamma}
\Bigg\{ - G_2^+(m) + \, G_2^-(m) \Bigg\} \\ \nonumber
&\buildrel m=0 \over \longrightarrow & - \frac{dt}{\tau_0} \frac{L}{L-t} nT
\Bigg[ 3(1+v^2)\,\gamma^3 \Bigg] \, ,
\end{eqnarray}
\begin{eqnarray} \nonumber
d T^{0x}_i (t) &=& \frac{dT^{00}_i(t) }{ju} -
\frac{dt}{\tau_0} \frac{L}{L-t} \frac{nT}{4ju\gamma} \\ \nonumber
&&\Bigg\{- 2b^2(3 + u^2) K_2(a) - 2ab^2 K_1(a) \Bigg\} \\ \nonumber
&\buildrel m=0 \over \longrightarrow & - \frac{dt}{\tau_0} \frac{L}{L-t} nT
\Bigg[ \frac{3(1+v^2)\,\gamma^3}{v} - \frac{\gamma (3+v^2)}{v}\Bigg] \, ,
\end{eqnarray}
\begin{eqnarray} \nonumber
d T^{xx}_i (t) &=& \frac{dT^{0x}_i(t)}{ju} - \frac{T}{\gamma ju} \Bigg[ dN^{x}_i(t) -
\frac{dN^{0}_i(t)}{ju} \Bigg] \\ \nonumber
&-& \frac{dt}{\tau_0} \frac{L}{L-t} \frac{nT}{4ju\gamma}
\Bigg\{-\frac{2b^2}{ju} (1 + 3u^2) K_2(a) \\ \nonumber
&-& 2juab^2 K_1(a) \Bigg\} \\ \nonumber
&\buildrel m=0 \over \longrightarrow & - \frac{dt}{\tau_0} \frac{L}{L-t} nT
\Bigg[ \frac{3(1+v^2)\,\gamma^3}{v^2}
- \frac{\gamma (3+v^2)}{v^2} \\ \nonumber
&+& \frac{1}{\gamma v^2} - \frac{(1 + 3v^2)\gamma}{v^2}\Bigg] \, ,
\end{eqnarray}
\begin{eqnarray} \nonumber
d T^{yy}_i (t)&=& - \, \frac{dT^{xx}_i(t)}{2}
- \frac{dt}{\tau_0} \frac{L}{L-t} \frac{nT}{8ju\gamma} \\ \nonumber
&& \Bigg\{ - G^+_{3}(m) + G^+_{3}(m) \Bigg\} \\ \nonumber
&\buildrel m=0 \over \longrightarrow & - \frac{dt}{\tau_0} \frac{1}{2}
\Bigg[ dT^{xx}(t) + dT^{00}(t)\Bigg] \, ,
\end{eqnarray}
and
\begin{eqnarray}
d T^{zz}_i (t) = d T^{yy}_i (t)\, ,
\end{eqnarray}
where $a = \frac{m}{T}$, $b=a\gamma$ and
$n = 4 \pi T^3 a^2 K_2(a)\, g \frac{e^{\mu/T}}{(2 \pi \hbar)^3}$ is the particle
density, while $g$ is the degeneracy factor.
Furthermore, the definition of the modified Bessel function of the second kind $K_{n}(z)$ for $n>-1$, is
\begin{equation}
K_{n}(z) = \frac{2^n \, n!}{(2n)!} \, z^{-n}
\int_{z}^{\infty} dx \,e^{-x} \,(x^2 - z^2)^{n-\frac{1}{2}} \, .
\end{equation}
The analytically not integrable functions $G_n^- (m) $ and
$G_n^+ (m) $, where $n>-2$, depend on other quantities such as $u$ and $T$.
However, in the case when, $m \rightarrow 0$, the dependence on other quantities persists and hence
we only denote the mass dependence of the functions defined as:
\begin{eqnarray} G_n^{\pm}(m)
&=&\frac{1}{T^{n+2}}\int_{0}^{\infty} d p \, p \, \Big( \sqrt{p^2 + m^2} \Big)^n \, \\ \nonumber
&\times&\Gamma \Big(0,\frac{\gamma}{T}
\sqrt{p^2 + m^2} \pm \frac{\gamma jup}{T}\Big) \, .
\end{eqnarray}
The exponential integral function is defined as:
\begin{equation}\label{exponential_integral}
\textrm{Ei} (z) = \int_{z}^{\infty} dx \, \frac{e^{-x}}{x} \, .
\end{equation}
|
{
"timestamp": "2007-06-16T21:27:36",
"yymm": "0503",
"arxiv_id": "nucl-th/0503048",
"language": "en",
"url": "https://arxiv.org/abs/nucl-th/0503048"
}
|
\section{The spectral sequence}
By a Grothendieck spectral sequence we mean a composite functor spectral sequence as described by Cartan and Eilenberg (\cite{cartaneilenberg}, Chapter XVII \S7). The Lyndon--Hochschild--Serre spectral sequence (\cite{cartaneilenberg}, Chapter XVI \S6 (6)) can be viewed this way: see for example Rotman's account, Theorem 11.45 of \cite{rotman}.
Consider a group extension $K\rightarrowtail G\twoheadrightarrow Q$. Write $\operatorname{\Mo\HYPHEN}{\mathbb Z} G$ for the category of right ${\mathbb Z} G$-modules.
The zeroth cohomology functor $H^0(G,{\phantom M})$ is the fixed point functor $({\phantom M})^G$ and the LHS spectral sequence arises from the factorization of this functor through the category of ${\mathbb Z} Q$-modules as illustrated below:
\[
\xymatrix{
\operatorname{\Mo\HYPHEN}{\mathbb Z} G\ar[rr]^{({\phantom M})^G}\ar[dr]_{H^0(K,{\phantom M})=({\phantom M})^K}&&\operatorname{\Mo\HYPHEN}{\mathbb Z}.\\
&\operatorname{\Mo\HYPHEN}{\mathbb Z} Q\ar[ur]_{H^0(Q,{\phantom M})=({\phantom M})^Q}\\
}
\]
One needs to know that the module categories are abelian categories with enough injectives and that the $K$-fixed point functor carries injective $G$-modules to injective $Q$-modules.
Now suppose that $\mathcal S$ is an admissible family of subgroups of $G$.
\begin{definition}\label{new cat}
We write $$\operatorname{\Mo\HYPHEN}{\mathbb Z} G/\mathcal S$$ for the full subcategory of $\operatorname{\Mo\HYPHEN}{\mathbb Z} G$ whose objects are those ${\mathbb Z} G$-modules $M$ for which
$M=\bigcup_{H\in\mathcal S}M^H.$
\end{definition}
\begin{remark}\label{rem} Although this definition makes perfect sense for any family $\mathcal{S}$ of subgroups, it behaves particularly well when $\mathcal{S}$ is admissible and we shall always assume that this is so. Given this assumption and an arbitrary ${\mathbb Z} G$-module $M$ then $\bigcup_{H\in\mathcal S}M^H$ is a ${\mathbb Z} G$-submodule of $M$ which belongs to the new category. We write $H^0(\mathcal{S},M)$ for this submodule. Notice that the definition of admissible family is designed exactly so that this works: $H^0(\mathcal{S},M)$ inherits an action of $G$ because $\mathcal{S}$ is closed under conjugation and $H^0(\mathcal{S},M)$ is an additive subgroup of $M$ because $\mathcal{S}$ is downwardly directed. Thus $H^0(\mathcal{S},M)$ is an object of the new category $\operatorname{\Mo\HYPHEN}{\mathbb Z} G/\mathcal S$ and {\em by definition} all objects of $\operatorname{\Mo\HYPHEN}{\mathbb Z} G/\mathcal S$ arise this way.
\end{remark}
It is easy to see that $\operatorname{\Mo\HYPHEN}{\mathbb Z} G/\mathcal S$ is an abelian category:
it will be the intermediary for our spectral sequence, replacing $\operatorname{\Mo\HYPHEN}{\mathbb Z} Q$.
Note that in case $\mathcal S$ consists of a single (necessarily normal) subgroup $K$ then $\operatorname{\Mo\HYPHEN}{\mathbb Z} G/\mathcal S$ is naturally equivalent to the category of right modules for the quotient group $Q=G/K$.
We now have two functors. The first is mentioned already in Remark \ref{rem}:
the assignment
$$H^0(\mathcal S,{\phantom M}):\operatorname{\Mo\HYPHEN}{\mathbb Z} G\to\operatorname{\Mo\HYPHEN}{\mathbb Z} G/\mathcal S$$
defined by
$$H^0(\mathcal S,M)=\bigcup_{H\in\mathcal S}M^H$$
is functorial in $M$. Secondly, we can restrict the $G$-fixed point functor to the new category so we have a functor
$$H^0(G/\mathcal S,{\phantom M}):\operatorname{\Mo\HYPHEN}{\mathbb Z} G/\mathcal S\to\operatorname{\Mo\HYPHEN}{\mathbb Z}$$
defined by
$$H^0(G/\mathcal S,M)=M^G.$$
\begin{remark}\label{rem2}
The $G$-fixed point functor $({\phantom M})^G:\operatorname{\Mo\HYPHEN}{\mathbb Z} G\to\operatorname{\Mo\HYPHEN}{\mathbb Z}$ now factors through
$\operatorname{\Mo\HYPHEN}{\mathbb Z} G/\mathcal S$ as the composite: $H^0(\mathcal S,{\phantom M})$ followed by
$H^0(G/\mathcal S,{\phantom M})$
as illustrated in the diagram below.
\[
\xymatrix{
\operatorname{\Mo\HYPHEN}{\mathbb Z} G\ar[rr]^{({\phantom M})^G}\ar[dr]_{H^0(\mathcal S,{\phantom M})}&&\operatorname{\Mo\HYPHEN}{\mathbb Z}.\\
&\operatorname{\Mo\HYPHEN}{\mathbb Z} G/\mathcal S\ar[ur]_{H^0(G/\mathcal S,{\phantom M})}\\
}
\]
This will provide the basis for a Grothendieck spectral sequence.
\end{remark}
Beware that the notation $H^0(G/\mathcal S,{\phantom M})$ is not intended to imply the construction of any kind of object $G/\mathcal S$, but is simply notation for the functor. This functor necessarily has to be distinguished from $H^0(G,{\phantom M})$ which has a different domain. The notation {\em is} intended to suggest an analogy with the classical situation when $\mathcal S$ consists of a single normal subgroup $K$.
The analogy works well, and raises the interesting question whether there is any kind of natural object which deserves to be named $G/\mathcal S$. In \S6 we show that a certain completion of $G$ appears to be the object one should expect.
\begin{lemma}\label{injectives}\
\begin{enumerate}
\item
If $I$ is an injective ${\mathbb Z} G$-module then $H^0(\mathcal S,I)$ is injective in $\operatorname{\Mo\HYPHEN}{\mathbb Z} G/\mathcal S$.
\item
$\operatorname{\Mo\HYPHEN}{\mathbb Z} G/\mathcal S$ has enough injectives.
\item
The functor $H^0(\mathcal S,{\phantom M})$ has right derived functors
$H^n(\mathcal S,{\phantom M})$ and there are natural isomorphisms
$$H^n(\mathcal S,{\phantom M})\cong\colimf H^n(H,M).$$
\end{enumerate}
\end{lemma}
\begin{proof} (i) Let $I$ be an injective ${\mathbb Z} G$-module. To show that $H^0(\mathcal S,I)$ is injective we need to address the extension problem as illustrated below:
\[
\xymatrix{
0\ar[r]&M\ar[d]_\phi\ar[r]&N\ar@{-->}[dl]^{\widehat\phi{\text?}}\\
&H^0(\mathcal S,I)&\\
}
\]
Using the injectivity of $I$ we can find $\widehat\phi$ to make a commutative diagram
\[
\xymatrix{
0\ar[r]&M\ar[d]_\phi\ar[r]&N\ar[ddl]^{\widehat\phi}\\
&H^0(\mathcal S,I)\ar[d]&\\
&I&
}
\]
and since $N$ belongs to the subcategory, $\widehat\phi$ has image in $H^0(\mathcal S,I)$. This proves (i). For part (ii), observe that we can embed any ${\mathbb Z} G/\mathcal S$-module $M$ into an injective ${\mathbb Z} G$-module and then we have $M=H^0(\mathcal S,M)\hookrightarrow H^0(\mathcal S,I)$: by part (i), we have now embedded $M$ into an injective object of $\operatorname{\Mo\HYPHEN}{\mathbb Z} G/\mathcal S$.
(iii) Derived functors are defined in the standard way using injective resolutions
over ${\mathbb Z} G$, applying the functor $H^0(\mathcal S,{\phantom M})$ and passing to the cohomology of the resulting cochain complex.
The isomorphism is easily established.
\end{proof}
\begin{lemma}\label{lem:der quot}
The functor $H^0(G/\mathcal S,{\phantom M})$ has right derived functors
$H^n(G/\mathcal S,{\phantom M})$.
\end{lemma}
\begin{proof}
This time we work with injective resolutions in the category $\operatorname{\Mo\HYPHEN}{\mathbb Z} G/\mathcal S$,
apply the $G$-fixed point functor and pass to cohomology. In general, there is no simple interpretation of the derived functors.
\end{proof}
Remark \ref{rem2}, Lemma \ref{injectives} and Lemma \ref{lem:der quot} together provide the ingredients necessary for a Grothendieck spectral sequence and our main tool is established:
\begin{theoremA}\label{A}
Let $G$ be a group and let $\mathcal S$ be an admissible family of subgroups.
There is a Grothendieck spectral sequence
$$H^p(G/\mathcal S,H^q(\mathcal S,M))\implies H^{p+q}(G,M)$$
which is natural in the $G$-module $M$.
\end{theoremA}
As a routine feature of any first/third quadrant spectral sequence we have the following: (see for example \cite{rotman}, Theorem 11.43).
\begin{corollaryA}\label{corollaryA}
With $G$ and $\mathcal S$ as above,
\begin{enumerate}
\item there is a five term exact sequence analogous to the standard inflation-restriction sequence:
{\small{$$0\to H^1(G/\mathcal S,H^0(\mathcal S,M))\to H^1(G,M)\to
H^1(\mathcal S,M)^G\to H^2(G/\mathcal S,H^0(\mathcal S,M))\to H^2(G,M);$$}}
\item the inflation map $$H^n(G/\mathcal S,M)\to H^n(G,M)$$ (which is defined for $M$ in the subcategory $\operatorname{\Mo\HYPHEN}{\mathbb Z} G/\mathcal S$) is an isomorphism when $n=0$ and is injective when $n=1$.
\end{enumerate}
\end{corollaryA}
\section{Continuity of functors}
We need to consider continuity issues for the new functors.
\begin{definition}\label{def:cont}
Let $\mathcal C$ and $\mathcal D$ be abelian categories with all small filtered colimits.
Let $F:\mathcal C\to\mathcal D$ be a functor. We say that $F$ is {\em continuous at zero} if and only if
$${\displaystyle\lim_{\buildrel\longrightarrow\over\lambda}\ } F(M_\lambda)=0$$ whenever $(M_\lambda)$ is a small filtered colimit system in $\mathcal C$ which is {\em vanishing}: i.e.
$${\displaystyle\lim_{\buildrel\longrightarrow\over\lambda}\ } M_\lambda=0.$$
\end{definition}
First we recall a basic result from Bieri's notes, essentially the content of (\cite{bieri-qmw}, Theorem 1.3 (i)$\iff$(iiib)):
\begin{lemma}\label{bieri1}
A group $G$ is of type $\operatorname{FP}_n$ if and only if the cohomology functors $H^i(G,{\phantom M})$ are continuous at zero for all $i\le n$.
\end{lemma}
\begin{lemma}\label{lem:fpinfty}
Let $\mathcal S$ be an admissible family in $G$. If all members of $\mathcal S$ have type $\operatorname{FP}_\infty$ then the functors $H^n(\mathcal S,{\phantom M})$ are continuous at zero for all $n$.
\end{lemma}
\begin{proof} By Lemma \ref{bieri1}, the $\operatorname{FP}_\infty$ condition guarantees that the functors $H^{m}(H,{\phantom M})$ are continuous at zero for all $H\in\mathcal S$ and all $m$.
The result now follows from the natural isomorphism of Lemma \ref{injectives}(iii).
\end{proof}
The next lemma is a version of Strebel's criterion \cite{strebel}.
\begin{lemma}\label{strebel}
Let $G$ be a group of finite cohomological dimension. Suppose that for all vanishing filtered colimit systems $(P_\lambda)$ of projective modules $P_\lambda$, and all $m\in{\mathbb Z}$,
$${\displaystyle\lim_{\buildrel\longrightarrow\over\lambda}\ } H^m(G,P_\lambda)=0.$$ Then $G$ is of type $\operatorname{FP}$.
\end{lemma}
\begin{proof}
We shall use the following notation: write $FM$ for the free module on the underlying set of non-zero elements of a module $M$. Then $F$ is functorial in $M$ and the inclusion $M\setminus\{0\}\hookrightarrow M$ induces a natural surjection $FM\twoheadrightarrow M$ whose kernel, $\Omega M$, is also functorial. Moreover both $F$ and $\Omega$ take vanishing filtered colimit systems to vanishing filtered colimit systems.
By Lemma \ref{bieri1} it suffices to prove that
$H^m(G,{\phantom M})$ is continuous at zero for all $m$. This is proved by downward induction on $m$: it is trivial for all $m$ greater than the cohomological dimension of $G$, so we fix $m$ and assume as inductive hypothesis that $H^{m+1}(G,{\phantom M})$ is continuous at zero. Let $(M_\lambda)$ be a vanishing filtered colimit system of modules.
Then we have short exact sequences
$$\Omega M_\lambda\rightarrowtail FM_\lambda\twoheadrightarrow M_\lambda.$$ On passing to cohomology and taking colimits we have the exact sequence
$${\displaystyle\lim_{\buildrel\longrightarrow\over\lambda}\ } H^m(G,FM_\lambda)\to {\displaystyle\lim_{\buildrel\longrightarrow\over\lambda}\ } H^m(G,M_\lambda)\to {\displaystyle\lim_{\buildrel\longrightarrow\over\lambda}\ } H^{m+1}(G,\Omega M_\lambda).$$
We need to prove that the central group here is zero. The right hand group vanishes by induction and the left hand group vanishes by hypothesis. The result follows from exactness.
\end{proof}
\begin{lemma}\label{lem:van1}
Let $L$ be a near-normal subgroup of type $\operatorname{FP}_\infty$ and infinite index in a group $G$.
Let $M$ be a ${\mathbb Z} L$-module. Let $\mathcal S$ be the set of subgroups commensurable with $L$. Then
$$\left(H^m(\mathcal S,M\otimes_{{\mathbb Z} L}{\mathbb Z} G)\right)^G=0$$ for all integers $m$.
\end{lemma}
\begin{proof}
We proceed in steps.
\begin{enumerate}
\item[Step 1.] The case $m=0$.
\end{enumerate}
We use only on the fact that all members of $\mathcal S$ have infinite index in $G$.
Note that the calculation simplifies because
$$\left(H^0(\mathcal S,M\otimes_{{\mathbb Z} L}{\mathbb Z} G)\right)^G=(M\otimes_{{\mathbb Z} L}{\mathbb Z} G)^G,$$
the set of $G$-fixed points.
Let $T$ be a right transversal to $L$ in $G$, i.e. $G$ is the disjoint union of the cosets $Lt$, for $t\in T$. Any non-zero element of $M\otimes_{{\mathbb Z} L}{\mathbb Z} G$ has a unique expression as a finite sum
$$m_1\otimes t_1+\dots+m_s\otimes t_s$$
where the $m_i\in M$ are non-zero and the $t_i$ are distinct elements of $T$.
Let $X=Lt_1\cup\dots\cup Lt_s$. We can choose $g\in G$ such that $Xg\cap X=\emptyset$. To see this, suppose for a contradiction that $Xg\cap X=\emptyset$ for all $g$. Then $G$ is the union of the sets $t_i^{-1}Lt_j$ over all $i,j$: this expresses $G$ as a finite union of cosets of subgroups and implies that at least one of the subgroups has finite index in $G$ which is contrary to our assumption. For a $g$ such that $Xg\cap X=\emptyset$ we have
$$(m_1\otimes t_1+\dots+m_s\otimes t_s)g\ne m_1\otimes t_1+\dots+m_s\otimes t_s$$
so there are no non-zero fixed points.
\begin{enumerate}
\item[Step 2.] In case $M$ is injective as a ${\mathbb Z} L$-module and $m\ge1$.
\end{enumerate}
We prove the stronger statement that for any $H$ in $\mathcal S$,
$$H^m(H,M\otimes_{{\mathbb Z} L}{\mathbb Z} G)=0.$$
This argument uses both the commensurability and the type $\operatorname{FP}_\infty$.
Fix any $H$. Mackey decomposition yields
$$M\otimes_{{\mathbb Z} L}{\mathbb Z} G\cong\bigoplus_tMt\otimes_{{\mathbb Z}[L^t\cap H]}{\mathbb Z} H$$
as ${\mathbb Z} H$-modules, where $t$ runs over a set of $(L,H)$ double coset representatives in $G$. Each summand $Mt\otimes_{{\mathbb Z}[L^t\cap H]}{\mathbb Z} H$ is injective as an $H$-module because $M$ is injective over $L$ and, using commensurability,
$L^t\cap H$ has finite index in $H$. In positive dimensions, the cohomology of any group vanishes on injective modules. Here we can take advantage of the fact that $H$ has type $\operatorname{FP}_\infty$ to see that the cohomology also vanishes on arbitrary direct sums of injective modules.
\begin{enumerate}
\item[Step 3.] The general case.
\end{enumerate}
The general case can be deduced by dimension shifting. We use induction on $m$, the case $m=0$ being covered by step (i). Choose any short exact sequence
$$M\rightarrowtail I \twoheadrightarrow M'$$ with $I$ injective over $L$. Then we have a short exact sequence of induced modules:
$$M\otimes_{{\mathbb Z} L}{\mathbb Z} G\rightarrowtail I\otimes_{{\mathbb Z} L}{\mathbb Z} G\twoheadrightarrow M'\otimes_{{\mathbb Z} L}{\mathbb Z} G.$$ Using the long exact sequence of cohomology together with step (ii) reduces the $m$-dimensional matter for $M$ to the $(m-1)$-dimensional matter for $M'$ and the result follows by induction. Of course, in case $m=1$, Step 2 does not cover everything but then Step 1 can be used as well.
\end{proof}
\begin{proposition}\label{bieri2}
Let $n$ be natural number.
Let $G$ be a finitely generated group of cohomological dimension $\le n+1$. Let $K$ be near-normal subgroup of infinite index in $G$ such that:
\begin{enumerate}
\item
$K$ is of type $\operatorname{FP}$;
\item
$H^m(K,P)=0$ whenever $P$ is a projective module and $m\ne n$.
\end{enumerate}
Then $G$ is of type $\operatorname{FP}$.
\end{proposition}
\begin{proof} Let $\mathcal S$ be the family of all subgroups commensurable with $K$. Let $P$ be a projective ${\mathbb Z} G$-module. All the members of $\mathcal S$ inherit properties (i) and (ii) and it follows from Lemma \ref{injectives}(iii) and Lemma \ref{bieri1} that $\mathcal S$ itself inherits the properties as well:
\begin{enumerate}
\item
$H^m(\mathcal S,{\phantom M})$ is continuous at zero for all $m$;
\item
$H^m(\mathcal S,P)=0$ whenever $P$ is a projective ${\mathbb Z} G$-module and $m\ne n$.
\end{enumerate}
The spectral sequence of Theorem A therefore collapses to a single column and we find that
$$H^{n+1}(G,P)\cong H^1(G/\mathcal S,H^n(\mathcal S,P)),$$
$$H^{n}(G,P)\cong H^0(G/\mathcal S,H^n(\mathcal S,P)),$$
and
$$H^{m}(G,P)=0$$ when $m\notin\{n,n+1\}$ because of (ii) for $m<n$ and the constraint on dimension of $G$ for $m>n+1$.
By definition, $H^0(G/\mathcal S,H^n(\mathcal S,{\phantom M}))=(H^n(\mathcal S,{\phantom M}))^G$ and this functor vanishes on all induced modules (i.e. modules of the form $A\otimes{\mathbb Z} G$ where $A$ is an abelian group) by Lemma \ref{lem:van1}. Since projective modules are direct summands of free modules which are in turn examples of induced modules, we have
$$H^{n}(G,P)=0.$$
Now we can apply Strebel's criterion Lemma \ref{strebel}. If $(P_\lambda)$ is a vanishing filtered colimit system of projective modules then we only have to check that
$${\displaystyle\lim_{\buildrel\longrightarrow\over\lambda}\ } H^{n+1}(G,P_\lambda)=0$$
and this will follow from the isomorphism
$${\displaystyle\lim_{\buildrel\longrightarrow\over\lambda}\ } H^{n+1}(G,P_\lambda)={\displaystyle\lim_{\buildrel\longrightarrow\over\lambda}\ } H^1(G/\mathcal S,H^n(\mathcal S,P_\lambda))$$
together with the observation that both the functors $H^1(G/\mathcal S,{\phantom M})$ and $H^n(\mathcal S,{\phantom M})$ are continuous at zero. The second observation is part of (i) above. The first observation follows from the injectivity of the inflation maps
$$H^1(G/\mathcal S,H^n(\mathcal S,P_\lambda))\to
H^1(G,H^n(\mathcal S,P_\lambda))$$
at the start of the five term exact sequence, Corollary A: note that $H^1(G,{\phantom M})$ is continuous at zero because $G$ is finitely generated.
\end{proof}
\section{A Decomposition Theorem}
The goal of this section is to prove a decomposition theorem for certain groups. We take this in two stages. The first stage, Theorem B, is easy to state. The second stage Theorem C requires a little more introduction although it is easy to prove by combining Theorem B with results \cite{bf} of Bestvina and Feighn.
\begin{theoremB}\label{B} Let $n$ be a fixed natural number. Suppose that $G$ is a group with the following properties:
\begin{itemize}
\item[$(\alpha)$]
$G$ is finitely generated;
\item[$(\beta)$]
$G$ has cohomological dimension $\le n+1$;
\item[$(\gamma)$]
$G$ has a near-normal $\operatorname{PD}^n$-subgroup $K$.
\end{itemize}
Then either $K$ has finite index in $G$ or $G$ splits over a subgroup commensurable with $K$.
\end{theoremB}
We recall some definitions. An {\em $n$-dimensional duality group} (over ${\mathbb Z}$) is a group $G$ which affords a dualizing module $D$ so that there are natural isomorphisms
$$H^i(G,M)\cong \operatorname{Tor}^{{\mathbb Z} G}_{n-i}(M,D)$$ for all $i$.
{\em Poincar\'e duality groups} ($\operatorname{PD}^n$-groups) are duality groups for which $D$ has underlying additive group ${\mathbb Z}$. We say that a group $G$ {\em splits} over a subgroup $H$ if and only if $G$ is isomorphic to a free product with amalgamation $K*_HL$ with $K\ne H\ne L$ or and HNN-extension $K*_H$. According to the standard theory of group actions on trees which is described by Serre \cite{serre} and Dicks--Dunwoody \cite{dicks}, $G$ splits over $H$ if and only if there is a $G$-tree with no fixed vertex, one orbit of edges and in which $H$ is the stabilizer of one of the edges.
The proof of Theorem B uses Dunwoody's graph cutting methods, \cite{dunwoody}.
\begin{proof}[Proof of Theorem B]\
We may as well assume that $K$ has infinite index in $G$.
Let $\mathcal S$ be the family of all subgroups commensurable with $K$. Then $\mathcal S$ is an admissible family of $\operatorname{PD}^n$-subgroups of $G$ all of which have infinite index in $G$.
The hypotheses of Proposition \ref{bieri2} are satisfied and therefore $G$ has type $\operatorname{FP}$.
The next step is to prove
\begin{itemize}
\item[{\bf Claim 1.}] $G$ is an $(n+1)$-dimensional duality group over any field $k$.
\end{itemize}
To be an $(n+1)$-dimensional duality group over a (non-zero) commutative ring $k$ it is necessary and sufficient that the following three conditions hold.
\begin{itemize}
\item
$G$ is of type $\operatorname{FP}$ over $k$ and of cohomological dimension $\le n+1$.
\item
The cohomology groups $H^i(G,kG)$ are zero for $i\le n$.
\item $H^{n+1}(G,kG)$ is flat as a $k$-module.
\end{itemize}
When these conditions hold, $G$ has dualizing module $H^{n+1}(G,kG)$: this is a left $kG$-module via the left action of $G$ on $kG$.
Since $kG$ is an instance of an induced ${\mathbb Z} G$-module, the first two conditions are already established. The third condition is automatically satisfied if $k$ is a field. We now work over the field ${\mathbb F}$ of two elements and show how to identify $H^n(\mathcal S,{\mathbb F} G)$ with a certain set of subsets of $G$.
Let $\mathcal P$ denote the powerset of $G$. This is viewed as an $({\mathbb F} G,{\mathbb F} G)$-bimodule with symmetric difference of subsets providing the additive structure and left/right multiplication by elements of $G$ for the action.
Let $\mathcal B$ be the set of subsets $B$ of $G$ which satisfy the following condition
\begin{itemize}
\item
There is a subgroup $H\in\mathcal S$ and a finite subset $F$ of $G$ such that $B=HFH$. I.e. $B$ is a finite union of double cosets of some $\mathcal S$-subgroup.
\end{itemize}
The set $\mathcal B$ is an $({\mathbb F} G,{\mathbb F} G)$-sub-bimodule.
\begin{itemize}
\item[{\bf Claim 2.}]
$H^n(\mathcal S,{\mathbb F} G)$ is isomorphic to $\mathcal B$ as a $({\mathbb F} G,{\mathbb F} G)$-bimodule.
\end{itemize}
We need to understand the connecting maps which are involved in the colimit formulation for $H^*(\mathcal S,{\mathbb F} G)$. When $H\subseteq L$ are $\operatorname{PD}^n$-groups then $H$ has finite index in $L$ and there are commutative diagrams
\[
\xymatrix{
H^i(L,M)\ar[r]\ar[d]^{\text{Res}}&\operatorname{Tor}_{n-i}^{{\mathbb F} L}(M,{\mathbb F})\ar[d]^{\text{Tr}}\\
H^i(H,M)\ar[r]&\operatorname{Tor}_{n-i}^{{\mathbb F} H}(M,{\mathbb F})\\
}
\]
where the horizontal maps are the duality isomorphisms, the left hand vertical map is the ordinary restriction map in cohomology and the right hand map is {\em transfer}. When $i=n$ this specializes to
\[
\xymatrix{
H^n(L,M)\ar[r]\ar[d]^{\text{Res}}&M\otimes_{{\mathbb F} L}{\mathbb F}\ar[d]^{\text{Tr}}\\
H^n(H,M)\ar[r]&M\otimes_{{\mathbb F} H}{\mathbb F}\\
}
\]
and here the transfer map is easy to describe: it is given by
$$m\otimes1\mapsto\sum_{t\in T}mt\otimes1$$ where $T$ is a left transversal to $H$ in $L$ (i.e. $L$ is the disjoint union of the left cosets $tH$, $t\in T$). We are interested in taking $M:={\mathbb F} G$. Now ${\mathbb F} G\otimes_{{\mathbb F} L}{\mathbb F}$ is isomorphic as left ${\mathbb F} G$-module to the submodule of $\mathcal P$ comprising subsets which are finite unions of right cosets of $L$. Similarly $H$ is isomorphic to the left module of finite unions of cosets of $H$. From this viewpoint, the transfer map is simply induced by inclusion of sets. On passing to the colimit over all members of $\mathcal S$ we obtain Claim 2.
We know that $H^{n+1}(G,{\mathbb F} G)$ is non-zero and we shall take advantage of this to construct a graph with more than one end on which $G$ acts in a useful way. We have
$$0\ne H^{n+1}(G,{\mathbb F} G)\cong H^1(G/\mathcal S,H^n(\mathcal S,{\mathbb F} G)).$$
Substituting into the five term exact sequence we obtain an exact sequence
$$0\to H^{n+1}(G,{\mathbb F} G)\to H^1(G,H^n(\mathcal S,{\mathbb F} G))\to
H^1(\mathcal S,H^n(\mathcal S,{\mathbb F} G))^G.$$
Thus we have an exact sequence
$$0\to H^{n+1}(G,{\mathbb F} G)\to H^1(G,\mathcal B)\to
H^1(\mathcal S,\mathcal B).$$
This identifies the dualizing module of $G$ over ${\mathbb F}$ with the kernel of a restriction map in first cohomology. To compute the first cohomology of $G$ observe that the power set $\mathcal P$ of $G$ is a coinduced ${\mathbb F} G$-module on which cohomology vanishes and we can view $\mathcal B$ as a submodule. The short exact sequence $\mathcal B\to\mathcal P\to \mathcal P/\mathcal B$ yields the exact sequence
$$0\to\mathcal B^G\to\mathcal P^G\to(\mathcal P/\mathcal B)^G\to H^1(G,\mathcal B)\to0$$ in cohomology. Note that $\mathcal P^G=\{\emptyset,G\}={\mathbb F}$ and the assumption that all members of $\mathcal S$ have infinite index in $G$ implies that $\mathcal B^G=0$. So the exact sequence simplifies to
$$0\to{\mathbb F}\to(\mathcal P/\mathcal B)^G\to H^1(G,\mathcal B)\to0$$
The first cohomology group is most easily viewed as the quotient derivations modulo inner derivations. Writing $\mathcal P_{\mathcal S}$ for the preimage of $(\mathcal P/\mathcal B)^G$ under the natural map
$\mathcal P\to\mathcal P/\mathcal B$ we have the commutative diagram
\[
\xymatrix{
&&0\ar[d]&0\ar[d]\\
&&\mathcal B\ar@{=}[r]\ar[d]&\operatorname{Ider}(G,\mathcal B)\ar[d]\\
0\ar[r]&{\mathbb F}\ar[r]\ar@{=}[d]&\mathcal P_{\mathcal S}\ar[r]\ar[d]&\operatorname{Der}(G,\mathcal B)\ar[r]\ar[d]&0\\
0\ar[r]&{\mathbb F}\ar[r]&(\mathcal P/\mathcal B)^G\ar[r]\ar[d]&H^1(G,\mathcal B)\ar[r]\ar[d]&0\\
&&0&0\\
}
\]
with exact rows and columns. The set $\mathcal P_{\mathcal S}$ consists of those subsets $B$ of $G$ such that for all $g\in G$, the symmetric difference $B+Bg$ belongs to $\mathcal B$. Such a set gives rise to a derivation defined by
$$g\mapsto B+Bg.$$ We have identified $H^{n+1}(G,{\mathbb F} G)$ with the kernel of the restriction map $H^1(G,\mathcal B)\to H^1(\mathcal S,\mathcal B)$. Let $\xi$ be a non-zero element of this kernel. Then $\xi$ is represented by a derivation $\delta:G\to\mathcal B$ and this restricts to an inner derivation on some subgroup $H$ in $\mathcal S$. Our derivation arises from a choice of $B\in\mathcal P$:
$$\delta g=B+Bg$$
for all $g$. The restriction condition says that there is a set $A\in\mathcal B$ such that
$$B+Bh=A+Ah$$ for all $h\in H$. We can choose a subgroup $L$ contained in $H$ which is also a member of $\mathcal S$ such that
$$A=LAL.$$
On restriction to $L$ we find that
$$B+B\ell=\emptyset$$ for all $\ell\in L$. This says that $B=BL$. It follows that the number of ends of the pair $G,L$ is at least $2$:
$$e(G,L)\ge2.$$
In the terminology of Dunwoody and Swenson, $L$ has codimension one in $G$ and a splitting of $G$ can be found using methods closely related to theirs, \cite{ds}.
Here we shall give an argument based on the earlier result \cite{dunwoody} of Dunwoody.
Let $X$ be a finite set of generators for $G$.
We now construct a graph $\Gamma$ with a left action of $G$. The vertex set $V$ of $\Gamma$ is the set of cosets $gL$ of $L$.
The edge set $E$ of $\Gamma$ is defined to be a subset of $V\times V$:
$$E=\{(gL,gxL):\ g\in G, x\in X\}.$$
A typical edge $(gL,gxL)$ has initial vertex $gL$ and terminal vertex $gxL$.
Clearly $\Gamma$ admits a left action of $G$. Also $\Gamma$ is connected because $X$ generates $G$.
For each vertex $gL$ in $\Gamma$ either $gL\subseteq B$ or $gL\subseteq B^*$.
\begin{itemize}
\item[{\bf Claim 3.}]
There are only finitely many edges $e$ of $\Gamma$ having one vertex in $B$ and one vertex in $B^*$
\end{itemize}
To establish the claim, consider an edge $(gL,gxL)$ with $gL\subset B$ and $gxL\subset B^*$. Then $g\in B\setminus Bx^{-1}$. Similarly, if $gL\subset B^*$ and $gxL\subset B$ then $g\in Bx^{-1}\setminus B$. Thus, if the edge $(gL,gxL)$ has exactly one of its vertices in $B$ then this reasoning shows that
$$g\in \bigcup_{x\in X}B+Bx^{-1},$$ and also
$$gx\in \bigcup_{x\in X}B+Bx.$$
Set $$Y:=\left(\bigcup_{x\in X}(B+Bx^{-1})\cup (B+Bx)\right)L.$$ Then $Y$ is a union of finitely many left cosets of $L$ and every edge with exactly one vertex in $B$ has both its vertices in $Y$. This proves the Claim 3. It now follows from Dunwoody's result \cite{dunwoody} that $G$ splits over a subgroup commensurable with $L$.
\end{proof}
For Theorem C shall need to appeal to a result about group actions on trees. The following is the main theorem of \cite{bf}.
\begin{theorem}[Bestvina and Feighn (1991)]\label{bf}
Let $G$ be a group of type $\operatorname{FP}_2$ over ${\mathbb F}$. Then there exists an integer $\gamma(G)$ such that the following holds.
If $T$ is a reduced $G$-tree with small edge stabilizers, then the number of vertices in $T/G$ is bounded by $\gamma(G)$.
\end{theorem}
The precise statement in \cite{bf} assumes that $G$ is finitely presented. However, in the subsequent remark (8), the authors state that the result holds for {\em almost finitely presented groups}, i.e. groups of type $\operatorname{FP}_2$ over ${\mathbb F}$, and that their proof requires absolutely no change. The reason for this is that finite presentation is used only to manufacture a connected $2$-dimensional CW-complex $X$ on which $G$ acts freely and cocompactly in which every {\em track} separates: this last condition is guaranteed when $H^1(X,{\mathbb F})=0$ and the construction of $X$ is therefore possible for any almost finitely presented $G$.
The following definitions are supplied in \cite{bf} and we restate them so that the reader can see exactly how Theorem \ref{bf} applies to our situation.
\begin{definition}\label{def:min red hyp}
Let $G$ be a group and let $T$ be a $G$-tree.
\begin{enumerate}
\item
The action of $G$ on $T$ is said to be {\em minimal} if and only if there are no proper invariant subtrees.
\item The $G$-tree $T$ is called {\em reduced} if and only if it is minimal and in addition every vertex of valency $2$ properly contains the stabilizers of the two incident edges.
\item When $T$ is minimal, it is called {\em hyperbolic} if and only if there exist two hyperbolic elements in $G$ (these being elements which have no fixed points but do have an invariant line, called the axis) whose axes intersect in a compact set. In this case, $G$ contains a free group on $2$ generators: in fact, sufficiently high powers of the two group elements freely generate a free group.
\end{enumerate}
\end{definition}
\begin{definition}\label{def:small}
A group $G$ is called {\em small} if and only if it does not admit a hyperbolic action on any minimal $G$-tree. In particular, if $G$ has no non-cyclic free subgroups then $G$ is small. Polycyclic-by-finite groups are small.
\end{definition}
The following result is a considerable generalization of the main theorem of \cite{phk-commentari}. Note that the hypothesis {\em all subgroups commensurable with $K$ are small} is clearly satisfied if $K$ is polycyclic-by-finite. The main theorem of \cite{phk-commentari} deals with the very special case when $K$ is infinite cyclic and $G$ has cohomological dimension $\le2$.
\begin{theoremC}\label{C}
Let $G$ be a group satisfying the conditions $(\alpha),(\beta),(\gamma)$ of Theorem B. Suppose that all subgroups commensurable with $K$ are small. Then there is a $G$-tree $T$ such that all vertices and edges have stabilizers commensurable with $K$.
\end{theoremC}
\begin{proof} We give the general argument below, but first, for motivation, we consider a special case which indicates how the general argument must proceed.
By Theorem B, $G$ splits over a subgroup commensurable with $K$.
Suppose for the sake of argument that $G=U*_HV$ is a free product with amalgamation where $H$ is commensurable with $K$.
Since $G$ and $H$ are both finitely generated it is necessarily the case that $U$ and $V$ are also finitely generated. Both $U$ and $V$ satisfy all the hypotheses of Theorem $B$ and we deduce that either $|U:H|$ is finite or $U$ splits over a subgroup $L$ commensurable with $H$. If the latter holds, let $T$ be the corresponding $U$-tree. The subgroup $H$ acts on $T$ and has finite orbits on the edges of $T$. Therefore there must be at least one vertex fixed by $H$. This means that the splitting of $U$ is compatible with the original splitting of $G$ and we can find a $G$-tree combining both splittings of $G$ in which there are two orbits of edges corresponding to the subgroups $H$ and $L$. The process can be continued but we can appeal to Theorem \ref{bf} to be sure that it breaks off. When it breaks off we reach a situation where all the vertex groups as well as all the edge groups are commensurable with $K$.
More precisely, choose a reduced $G$-tree $T$ with finitely many orbits of edges, stabilizers all commensurable with $K$, and subject to these conditions, with the maximum possible number of orbits of vertices. The existence of such is guaranteed by Theorem \ref{bf}.
This $G$-tree has finitely many orbits of edges and vertices and the fact that $G$ itself and all the edge stabilizers are finitely generated forces the vertex stabilizers to be finitely generated.
We claim that every vertex stabilizer here is also commensurable with $K$. Suppose not. Then there is a vertex $v$ whose stabilizer $G_v$ is not commensurable with $K$. Let $e$ be an edge incident with $v$. Since $G_e$ and $K$ {\em are} commensurable we have that $G_v$ and $G_e$ are not commensurable. However, $G_e\subseteq G_v$ and therefore $G_e$ has infinite index in $G_v$. We may now apply Theorem B to split $G_v$ over a subgroup $H$ commensurable with $G_e$.
Let $T'$ be the corresponding $G_v$-tree and let $e_*$ be an edge of $T'$ with stabilizer $H$. Let $E_0$ be the set of edges which are incident with the vertex $v$ in the original tree $T$. For each $e_0\in E_0$, let $G_{e_0}$ denote the stabilizer of $e_0$ for the action of $G$ on $T$. Since $G_{e_0}\subseteq G_v$ we have an action of $G_{e_0}$ on $T'$. Moreover, $|G_{e_0}:G_{e_0}\cap H|$ is finite and so the $G_{e_0}$-orbit of $e_*$ in $T'$ is finite. It follows that $G_{e_0}$ fixes a vertex $v(e_0)$ of $T'$. We can now build a new $G$-tree
by blowing up each of the vertices in the $G$-orbit of $v$ using the tree $T'$.
We remove the vertices $v\cdot G$ of the orbit of $v$ and replace them with the forest $T'\times_{G_v}G$. The loose edge $e_0$ can be joined to the vertex $v(e_0)$ in the primary copy $T'\times 1$ of $T'$ and we can repeat this for the other edges incident with $v$. The process can be carried equivariantly over the orbits of loose edges. In this way we obtain a $G$-tree with a greater number of orbits of vertices. This contradicts the assumption and Theorem C follows.
\end{proof}
Other approaches to constructing the simplicial actions on trees for Theorem B and to deducing Theorem C from Theorem B can be found in the work of Dunwoody and Swenson \cite{ds} and of Mosher, Sageev and Whyte \cite{msw1,msw2}. However the construction of almost invariant sets through the use of our new spectral sequence is novel and essential for our arguments: we do not know of any alternative to this line of reasoning beyond the two dimensional case considered in \cite{phk-commentari}.
\section{An application to Poincar\'e duality groups}
We illustrate the potential of Theorems B and C by using them to establish the following result which tidies up and extends some of the considerations in \cite{kr}.
\begin{theorem}\label{pd}
Let $G$ be a $\operatorname{PD}^{n+1}$-group and let $H$ be a $\operatorname{PD}^n$-subgroup. Then either $|\operatorname{Comm}_G(H):H|$ is finite or $|G:\operatorname{Comm}_G(H)|$ is finite. In the latter case there is a subgroup $K$ commensurable with $H$, normal in $\operatorname{Comm}_G(H)$ such that $\operatorname{Comm}_G(H)/H$ is either infinite cyclic or infinite dihedral.
\end{theorem}
This result should not be regarded as an advance in itself. Indeed it is clear that this and further results can and have been proved by other methods in the work of Scott and Swarup \cite{ss}. However, it gives an indication of how our results may prove helpful in the study of Poincar\'e duality groups.
The following will be needed in our proof of \ref{pd}.
\begin{lemma}\label{asc}
Let $G$ be a group which is the union of a strictly ascending sequence
$$B_0<B_1<B_2<\dots$$ of $\operatorname{PD}^n$-groups $B_i$. Then $G$ has cohomological dimension $n+1$.
\end{lemma}
\begin{proof}
$H^{n+1}(G,M)$ is isomorphic to ${\displaystyle\lim_{\longleftarrow}}^1 H^n(B_i,M)$ for any $G$-module $M$. Taking $M={\mathbb F} G$ one can calculate that this ${\displaystyle\lim_{\longleftarrow}}^1 $ does not vanish.
$H^n(B_i,{\mathbb F} G)$ is isomorphic to ${\mathbb F} G/B_i$ and the connecting maps in the limit system
$$\dots\to {\mathbb F} G/B_i\to \dots \to {\mathbb F} G/B_1\to {\mathbb F} G/B_0$$
all strict inclusions. (See the argument for Claim 2 in the proof of Theorem B.)
We can view the inverse limit system as sitting inside the constant system
$({\mathbb F} G/B_0)$ in which the connecting maps are the identity maps. We thus have a short exact sequence of limit systems
$$({\mathbb F} G/B_i)\rightarrowtail({\mathbb F} G/B_0)\twoheadrightarrow (U_i)$$
where the connecting maps in the quotient system $(U_i)$ are surjections with non-zero kernels. Applying the ${\displaystyle\lim_{\longleftarrow}}$-${\displaystyle\lim_{\longleftarrow}}^1 $ exact sequence we obtain the exact sequence
$${\mathbb F} G/B_0\to{\displaystyle\lim_{\longleftarrow}} U_i\to{\displaystyle\lim_{\longleftarrow}}^1 {\mathbb F} G/B_i\to0.$$
The left hand map here cannot be surjective because ${\mathbb F} G/B_0$ is countable whereas ${\displaystyle\lim_{\longleftarrow}} U_i$ has cardinality $2^{\aleph_0}$. Therefore the right hand group is non trivial, completing the proof.
\end{proof}
We shall also need Strebel's fundamental dimension theorem \cite{strebdim} which says that all subgroups of infinite index in a $\operatorname{PD}^{n+1}$-group have cohomological dimension $\le n$.
\begin{proof}[Proof of \ref{pd}]
As a first step we prove the
\begin{itemize}
\item[{\bf Claim.}]
Every finitely generated $S\le\operatorname{Comm}_G(H)$ such that $|H:H\cap S|<\infty$ is either commensurable with $H$ or of finite index in $G$.
\end{itemize}
Let $S$ be such a subgroup. We can apply Theorem B to the group $S$ with subgroup $H\cap S$. This shows that either $|S:H\cap S|$ is finite or $S$ splits over a subgroup commensurable with $H$. In the latter case, the proof of Theorem B shows that $S$ is a duality group of dimension $n+1$ over any field and so by Strebel's theorem it must have finite index in $G$.
We consider two cases which together cover all eventualities.
\begin{itemize}
\item[Case 1.]
There is a finitely generated subgroup $S$ of $\operatorname{Comm}_G(H)$ with
$$|S:H|=\infty.$$
We show that in this case, $\operatorname{Comm}_G(H)$ has finite index in $G$ and the existence of $K$ can be established.
\end{itemize}
Applying Theorem B, we know that $S$ splits over a subgroup commensurable with $H$.
Since the edge group in the splitting of $S$ is commensurable with $H$, it is finitely generated. Since $S$ is also finitely generated it follows that the vertex groups in the splitting of $S$ are finitely generated. The vertex groups have infinite index in $S$ and the above shows that they are commensurable with $H$. Therefore either $S=J*_KL$, a free product with amalgamation in which $J\ne K\ne L$ are all commensurable with $H$, or $S=B*_K,t$ is an HNN-extension in which the base and associated subgroups are commensurable with $H$.
If $S$ is a non-ascending HNN-extension then one can use the Kurosh subgroup theorem to exhibit finitely generated subgroups of $S$ which have infinite index and which contain an infinite index subgroup commensurable with $H$. This contradicts the claim.
Similarly, if $G$ is an amalgamation in which one of the indices
$|J:K|$, $|L:K|$ is $\ge3$ then we can again find intermediate finitely generated subgroups which contradict the claim.
If $S=B*_Bt$ is a strictly ascending HNN-extension then the chain of subgroups
$$B\subset B^t\subset B^{t^2}\subset\dots$$ has union of cohomological dimension $n+1$ by Lemma \ref{asc}. But this contradicts Strebel's theorem and so cannot happen.
Therefore either $S$ is an amalgamation in which $|J:K|=|L:K|=2$ in which case $K\lhd S$ and $S/K\cong D_\infty$, or $G$ is a stationary HNN-extension meaning $B=K\lhd S$ and $S/K\cong C_\infty$.
Now $S$ has finite index in $G$ and normalizes $K$. Therefore $K$ has only finitely many distinct conjugates. Thus the subgroup $K_0$
defined by
$$K_0:=\bigcap_{g\in\operatorname{Comm}_G(H)}K^g$$ is a finite intersection of subgroups commensurable with $H$ and is therefore itself commensurable with $H$. Also $K_0$ is normal in $\operatorname{Comm}_G(H)$ and the corresponding quotient is virtually cyclic. If we set $K_1/K_0$ equal to the largest finite normal subgroup of $\operatorname{Comm}_G(H)$ then the quotient $\operatorname{Comm}_G(H)/K_1$ is either infinite cyclic or infinite dihedral.
\begin{itemize}
\item[Case 2.]
For every finite subset $F$ of $\operatorname{Comm}_G(H)$,
$$|\langle H\cup F\rangle:H|<\infty.$$
We show that in this case, $H$ has finite index in $\operatorname{Comm}_G(H)$.
\end{itemize}
In this case, if $|\operatorname{Comm}_G(H):H|$ is infinite then we can choose a strictly ascending chain of finite extensions of $H$ by successively increasing the size of finite set $F$. The union of such a chain has cohomological dimension $n+1$ by Lemma \ref{asc} and therefore finite index in $G$. This is a contradiction because $G$ is finitely generated while the union of the chain is not. Thus
$|\operatorname{Comm}_G(H):H|<\infty$ as claimed.
\end{proof}
\section{An easy application in which the spectral sequence collapses}
We conclude by mentioning a very simple application of the theory motivated by the notion of complete cohomology as described in \cite{bensoncarlson,goichot,mislin}. This again is a significant generalization of one of the results in \cite{phk-commentari}.
\begin{definition}\label{def:already complete}
We shall say that a group $G$ {\em already has complete cohomology} if and only if the cohomology functors $H^n(G,{\phantom M})$ vanish on projective modules for all $n$. This is equivalent to asserting that the natural map $H^n(G,{\phantom M})\to\widehat H^n(G,{\phantom M})$ is always an isomorphism. For example, free abelian groups of infinite rank already have complete cohomology whereas finite groups never enjoy the property.
\end{definition}
\begin{theorem}\label{star}
Let $G$ be a group and let $\mathcal S$ be an admissible family of subgroups which already have complete cohomology. Then $G$ already has complete cohomology.
\end{theorem}
\begin{proof}
The new spectral sequence collapses because the hypotheses along with Lemma \ref{injectives}(iii) ensure that
$$H^*(\mathcal S,P)=0$$ for all projective modules $P$.
\end{proof}
This provides a transparent argument for proving one of the results of \cite{browngeoghegan} that
$$H^*(F,{\mathbb Z} F)=0$$ where $F$ denotes Thompson's group given by the presentation
$$F=\langle x_0,x_1,x_2,\dots:\ x_i^{-1}x^{{\phantom 1}}_jx^{{\phantom 1}}_i=x^{{\phantom 1}}_{j+1}\ (i<j)\rangle.$$
Let $A$ be the subgroup of $F$ generated by $\{x^{{\phantom 1}}_{2n+1}x_{2n}^{-1}:\ n\ge0\}$ and let $F_0$ be the subgroup of index $2$ in $F$ comprising elements which can be expressed as even weight words in the $x_i$. Thus
$$A\subset F_0\subset F.$$
Clearly it suffices to prove that
$$H^*(F_0,{\mathbb Z} F)=0,$$
and for this we only need to observe that the admissible family $\mathcal S$ of subgroups generated by $A$ consists entirely of free abelian groups of infinite rank. We shall write $A_m$ for the subgroup of $A$ generated by
$\{x^{{\phantom 1}}_{2n+1}x_{2n}^{-1}:\ n\ge m\}$. The admissible family $\mathcal S$ is the set of subgroups which are finite intersections of conjugates $A^g$ with $g\in F_0$. Here are the precise details of the argument.
\begin{lemma}\label{lem:Thompson}
\begin{enumerate}
\item $x^{{\phantom 1}}_1x_0^{-1}$, $x^{{\phantom 1}}_3x_2^{-1}$, $x^{{\phantom 1}}_5x_4^{-1}$, $x^{{\phantom 1}}_7x_6^{-1}$, $\dots$ is a sequence of distinct elements of $F$ which freely generate the abelian group $A$.
\item For any finite subset $X$ of $F_0$, there is an $m\ge0$ such that
$$A_m\subseteq\bigcap_{g\in X}A^g.$$
\item Every member of $\mathcal S$ is free abelian of infinite rank.
\item $H^*(F,{\mathbb Z} F)=0$.
\end{enumerate}
\end{lemma}
\begin{proof}
\begin{enumerate}
\item Notice that if $0\le m<n$ then the relations in our given presentation of $F$ immediately yield
$$x_{2m}^{-1}\left(x^{{\phantom 1}}_{2n+1}x_{2n}^{-1}\right)x_{2m}
=x^{{\phantom 1}}_{2n+2}x_{2n+1}^{-1}$$
and
$$x_{2m+1}^{-1}\left(x^{{\phantom 1}}_{2n+1}x_{2n}^{-1}\right)x_{2m+1}=
x^{{\phantom 1}}_{2n+2}x_{2n+1}^{-1}.$$
Thus $x^{{\phantom 1}}_{2m+1}x_{2m}^{-1}$ commutes with $x^{{\phantom 1}}_{2n+1}x_{2n}^{-1}$. This shows that $A$ is abelian. One can see quite easily that $A$ is free abelian on the stated generators. One way is to use the known representation of $F$ as a group of piecewise linear maps of the unit interval $[0,1]$. Alternatively, let $F_m$ be the subgroup of $F$ generated by the $x_i$ with $i\ge m$. Then each $F_m$ is isomorphic to $F=F_0$ and $F_m$ is an ascending HNN-extension over $F_{m+1}$. One can now check inductively that for each $m$, the subgroup
$$x^{{\phantom 1}}_1x_0^{-1},\dots,x^{{\phantom 1}}_{2m+1}x_{2m}^{-1}$$ is free abelian of rank $m+1$ and lies in the centralizer of $F_{2m+2}$.
\item Let $g$ be a word in the alphabet $x_i^{\pm1}$, $i\ge0$ with exponent sum $j$. The relations defining $F$ show that for all sufficiently large $n$,
$$g^{-1}x_ng=x_{n+j}.$$ If $g$ is an element of $F_0$ then it can be expressed as a word with even exponent sum and therefore
$$A_m\subset A\cap A^g$$ for sufficiently large $m$. The result for an intersection of finitely many conjugates now follows from this.
\item This follows at once.
\item Theorem \ref{star} shows that $H^*(F_0,{\mathbb Z} F)=0$ and then we can apply the ordinary LHS spectral sequence to the group extension $F_0\rightarrowtail F\twoheadrightarrow{\mathbb Z}/2{\mathbb Z}$.
\end{enumerate}
\end{proof}
\section{Connection with Galois Cohomology}
As a concluding remark we mention that although the derived functors
$$H^n(G/\mathcal S,{\phantom M})$$
are not at first sight easily related to any familiar functors, there are nevertheless close connections with Galois cohomology of profinite groups.
As an example we shall see that if $G$ is a residually finite group and $\mathcal S$ is the family of all subgroups of finite index then $\operatorname{\Mo\HYPHEN}{\mathbb Z} G/\mathcal S$ is the category of discrete modules for the profinite completion $\widehat G$ of $G$ and $H^n(G/\mathcal S,{\phantom M})$ is isomorphic to the continuous (Galois) cohomology functor $H^n_{\operatorname{Gal}}(\widehat G,{\phantom M})$.
We refer the reader to \cite{serrecohom} for an introduction to the theory.
In general, given a group $G$ and admissible family $\mathcal S$,
it is not immediately clear that one can form a completion analogous to the profinite completion because it may happen that $\mathcal S$ contains few, or possibly no, normal subgroups. For example if $G$ is the Baumslag--Solitar group with presentation
$$\langle x,y:\ y^{-1}x^2y=x^3\rangle$$ and $\mathcal S$ is the set of subgroups commensurable with the infinite cyclic subgroup $\langle x\rangle$ then no member of $\mathcal S$ is normal.
Nevertheless,
one can always form a completion $\widehat G_{\mathcal S}$ for this example and any other. One can endow the completion with a product which makes it into a monoid. We do not know whether this monoid is necessarily a group in all cases, but we can show that it is a group in all the applications and examples considered in this paper. Define $\widehat G_{\mathcal S}$ to be the set of all functions $f:\mathcal S\to \mathcal P(G)$ which satisfy the conditions
\begin{itemize}
\item $f(H)\in H\backslash G$ for all $H\in\mathcal S$ and
\item $f(K)\subseteq f(H)$ whenever $K\subseteq H$ are members of $\mathcal S$.
\end{itemize}
In effect, we associate the coset space $H\backslash G$ to $H$ and observe that an inclusion $K\subset H$ induces a natural surjection $K\backslash G\twoheadrightarrow H\backslash G$. Then $\widehat G_{\mathcal S}$ is the inverse limit
${\displaystyle\lim_{\longleftarrow}}\ H\backslash G$. For any $f\in\widehat G_{\mathcal S}$ and any $H\in\mathcal S$ there exists $x\in G$ such that $f(H)=Hx$. It is useful to introduce the notation $H^f$ for the conjugate $H^x$ because it depends only on $f$ and $H$; not on the particular coset representative $x$.
To make $\widehat G_{\mathcal S}$ into a monoid we define the product by setting:
$$f\cdot f'(H)=f(H)f'(H^f)$$
for each $H\in \mathcal S$. On the right, the product is carried through using the standard convention $AB=\{ab:a\in A,b\in B\}$ for sets $A,B\subseteq G$.
It is straightforward to check that $H^{f\cdot f'}=(H^f)^{f'}$.
To check the associative law:
\begin{eqnarray*}
(f\cdot f')\cdot f''(H)&=&f\cdot f'(H)f''\big(H^{f\cdot f'}\big)\\
&=&f(H)f'(H^f)f''\big(H^{f\cdot f'}\big),\\
\end{eqnarray*}
and
\begin{eqnarray*}
f\cdot(f'\cdot f'')(H)&=&f(H)(f'\cdot f'')(H^f)\\
&=&f(H)f'(H^f)f''\big((H^f)^{f'}\big).\\
\end{eqnarray*}
The function $e$ defined by $e(H)=H$ for all $H$ is the identity element.
There is a natural map $$\widehat{\phantom g}:G\to \widehat G_{\mathcal S}$$ defined by sending each $g\in G$ to the function $\widehat g$ defined by $\widehat g(H)=Hg$. This is a homomorphism which carries $1\in G$ to
$e\in\widehat G_{\mathcal S}$ because
$$\widehat g\cdot\widehat{g'}(H)=%
\widehat g(H)\widehat{g'}(H^g)=Hgg'.$$
The question of whether or not $\widehat G_\mathcal S$ is a group appears to be subtle.
If $f\in\widehat G_{\mathcal S}$ has an inverse $f^{-1}$ then expanding the definition of $f\cdot f^{-1}(H)$ shows that
$$f^{-1}(H^f)=f(H)^{-1}.$$
This shows that $f^{-1}$ is uniquely determined on the subset $\{H^f:H\in\mathcal S\}$ of $\mathcal S$. In general, for fixed $f$, the map
$H\mapsto H^f$ is injective and preserves the poset structure of $\mathcal S$. To see this observe that for any finite subset $\mathcal F$ of $\mathcal S$ there is a group element $x$ such that $f(H)=Hx$ and $H^f=H^x$ for all $H\in\mathcal F$. Thus the map $H\mapsto H^f$ is given locally as conjugation by a single group element.
\begin{definition}
We shall say that the admissible family $\mathcal S$ is {\em stable} if and only if for all $K\le H$ in $\mathcal S$ there exists $L\in\mathcal S$ such that $L\le K$ and $L$ is normal in $H$.
\end{definition}
\begin{lemma}
If $\mathcal S$ is a stable admissible family of subgroups of $G$ then $\widehat G_{\mathcal S}$ is a group.
\end{lemma}
\begin{proof}
Fix $f\in\widehat G_{\mathcal S}$. Fix $H\in\mathcal S$. Choose $x\in f(H)$. Choose $K$ so that $\mathcal S\ni K\le H\cap H^f$ and $K\lhd H^f$.
Choose $t\in f(K^{x^{-1}})$. Define
$$f^{-1}(H)=Ht^{-1}.$$
\begin{itemize}
\item[{\bf Claim 1.}] $f^{-1}$ is well defined.
\end{itemize}
To define $f^{-1}(H)$ we have made three choices, namely $x,K,t$. Note that $f(H)=Hx$, $H^f=H^x$ and $K^{x^{-1}}\le (H^f)^{x^{-1}}=H$ so necessarily $t\in f(H)$, $f(H)=Ht$ and $xt^{-1}\in H$. Thus $t^{-1}x\in H^x=H^f$ normalizes $K$ and $K^{x^{-1}}=K^{t^{-1}}$ so that $f(K^{t^{-1}})=tK$.
Consider now a different sequence $y,L,u$ of the three choices made in the same way as $x,K,t$. Then $f(H)=Hy=Hu$ and $f(L^{u^{-1}})=uL$. Downward directedness guarantees that $tK\cap uL\ne\emptyset$. Choose $v\in tK\cap uL$. Then $tK=vK$ and $uL=vL$. Since $K$ and $L$ are contained in $H$ it follows that $Ht^{-1}=Hv^{-1}=Hu^{-1}$ as required.
\begin{itemize}
\item[{\bf Claim 2.}] $f^{-1}$ belongs to $\widehat G_{\mathcal S}$.
\end{itemize}
Given subgroups $H'\le H$ in $\mathcal S$ we can make the three choices $x,K,t$ so that $x\in f(H')$, $K\subseteq H'$ and $K\lhd H$. The choices then simultaneously supply the definitions of $f^{-1}(H')$ and $f^{-1}(H)$ so that
$f^{-1}(H')=H't^{-1}$ and $f^{-1}(H)=Ht^{-1}$ have the required compatibility for the inverse system.
\begin{itemize}
\item[{\bf Claim 3.}] $f^{-1}$ is inverse to $f$.
\end{itemize}
This follows: for any $H$ we can choose $t\in G$ such that $f(H)=Ht$,
$f(H^{t^{-1}})=tH$, $f^{-1}(H)=Ht^{-1}$ and so
$$f^{-1}\cdot f(H)=f^{-1}(H)f(H^{f^{-1}})=Ht^{-1}tH=H=e(H).$$
Thus $f^{-1}\cdot f=e$ and since $e$ is a two-sided identity element it follows from elementary group theory that $f^{-1}$ is a two-sided inverse as required.
\end{proof}
Every object of $\operatorname{\Mo\HYPHEN}{\mathbb Z} G/\mathcal S$ can be endowed with a natural action of $\widehat G_{\mathcal S}$ as follows.
Let $M$ be a ${\mathbb Z} G/\mathcal S$-module, let $f\in \widehat G_{\mathcal S}$ and let $m\in M$. Then we set
$$m\cdot f:=m.f(H)$$ where $H$ is any choice of member of $\mathcal S$ which fixes $m$. Although the subgroups of $\mathcal S$ may not be normal, it is still true that the underlying set of $\widehat G_{\mathcal S}$ can be viewed as an inverse limit:
$$\widehat G_{\mathcal S}=\limf H\backslash G$$
taken over the discrete coset spaces
$$H\backslash G=\{Hg:\ g\in G\}.$$
Then we may endow $\widehat G_{\mathcal S}$ with the inverse limit topology and it becomes a topological group. One could then go on to consider the category of discrete $\widehat G_{\mathcal S}$-modules: this is equivalent to the category of $\operatorname{\Mo\HYPHEN} G/\mathcal S$-modules. The cohomology functors $H^n(G/\mathcal S,{\phantom M})$ can be identified with the continuous cohomology functors of the topological group defined for example by using continuous cocycles in the standard bar resolution construction.
Of course, if $\mathcal S$ consists of normal subgroups, or more generally if
$$\bigcap_{g\in G}H^g$$ belongs to $\mathcal S$ for all $H\in\mathcal S$, then $\widehat G_{\mathcal S}$ is easier to define: one can think straightforwardly in terms of the inverse limit of quotient groups.
|
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"timestamp": "2005-03-24T08:06:47",
"yymm": "0503",
"arxiv_id": "math/0503514",
"language": "en",
"url": "https://arxiv.org/abs/math/0503514"
}
|
\section{Figure captions}
\noindent
Fig.1. a)The complex plane of frequence.
The cuts of $\Pi^{0R}$ Eq.(\ref{3}) are shown. Numbers stand
for the cuts (\ref{12}). b)The complex plane of $z_1$. The cut
corresponds to the cut $I$ in Fig.a. c)The complex plane of
frequence. The shading marks the unphysical sheets described in
the text.
\noindent
Fig.2. Solutions $\omega_{sd}(k)$ to Eq.(\ref{5}) at $F_0<0$.
The variable $\gamma$ is
$\gamma = \frac{m}{kp_F}Im~\omega_{sd}$.
The shading with the
right slope marks the unphysical sheet $I$.
The horizontal shading marks the unphysical sheet $\tilde I$.
\noindent
Fig.3. The comparision of the solutions in the kinetic theory
and in the RPA. The solid (dashed) lines correspond to solutions
obtained at $F_0=-1.2$ ($F_0=-1.1$).
\noindent
Fig.4. The branches of solution $\omega_{sd}(k)$ obtained in
RPA at different values of $F_0$. The curve (1) corresponds to
$F_0=-0.4$;
(2) $F_0=-0.9$; (3) $F_0=-1.02$; (4) $F_0=-1.1$; (5) $F_0=-1.2$.
The shading marks the sheet $I$.
\noindent
Fig.5. The solutions to Eq. (\ref{5}) $\omega_s$ at $F_0>0$.
Two symmetric solutions are presented.
\noindent
Fig.6. The complex plane of frequence. The solutions
$\omega_s(k)$ to Eq.(\ref{1}) obtained in RPA are presented. The
curve $1$ is calculated at $F_0=1$ and the curve $2$ at
$F_0=2$ . The wave vector $k_d$ marks the point when the Landau
damping starts at this $F_0$:
$\frac{k_d}{p_F}=0.13$ when $F_0=1$,
and $\frac{k_d}{p_F}=0.52$ at $F_0=2$.
\noindent
Fig.7. The branch $\omega_s(k)$ at $F_0=2$ is shown for the
different models. The solid line is for the solutions of
Eq.(\ref{5}). The real $\omega_r$ and the
imaginary $\omega_i$ parts of the RPA
solutions are presented by the dashed curves. The dotted line
stands for the $\omega_i$ calculated by Eq.(\ref{14}).
\newpage
\begin{figure
\centering{\epsfig{figure=f1a.eps,width=7cm}
\epsfig{figure=f1b.eps,width=7cm}}
\vspace{1cm}
\centering{\epsfig{figure=f1c.eps,width=7cm}}
\caption{}
\end{figure}
\begin{figure
\centering{\epsfig{figure=f2.eps,width=9cm}}
\caption{}
\end{figure}
\begin{figure
\centering{\epsfig{figure=f3.eps,width=9cm}}
\caption{}
\end{figure}
\begin{figure
\centering{\epsfig{figure=f4a.eps,width=9cm}}
\caption{}
\end{figure}
\begin{figure
\centering{\epsfig{figure=f5.eps,width=9cm}}
\caption{}
\end{figure}
\begin{figure
\centering{\epsfig{figure=f5r.eps,width=9cm}}
\caption{}
\end{figure}
\begin{figure
\centering{\epsfig{figure=kdep1.eps,width=9cm}}
\caption{}
\end{figure}
\end{document}
|
{
"timestamp": "2005-10-25T08:28:26",
"yymm": "0503",
"arxiv_id": "nucl-th/0503085",
"language": "en",
"url": "https://arxiv.org/abs/nucl-th/0503085"
}
|
\section{Introduction}
M.-H. Schwartz in \cite {Sch1, Sch2} introduced the technique of radial extension of stratified vector fields and frames on singular varieties, and used this to construct cocycles representing classes in the cohomology
$H^*(M, M\setminus V)$, where $V$ is a singular variety embedded in a complex manifold $M$; these
are now called {\it the Schwartz classes} of $V$. A basic property of radial extension is that the index of the vector fields (or frames) constructed in this way is the same when measured in the strata or in the ambient space; this is called the Schwartz index of the vector field (or frame).
MacPherson in \cite{MP} introduced the notion of the local Euler obstruction, an invariant defined at each point of a singular variety using an index of an appropriate radial 1-form, and used this (among other things) to construct the homology Chern classes of singular varieties. Brasselet and Schwartz in \cite {BS} proved that the Alexander isomorphism
$H^*(M, M\setminus V) \cong H_*(V)$ carries the Schwartz classes into the MacPherson classes; a key ingredient for this proof is their {\it proportionality theorem} relating the Schwartz index and the local Euler obstruction.
These were the first indices of vector fields and 1-forms in the literature.
Later in \cite {GSV} was introduced another index for vector fields on isolated hypersurface singularities, and this definition was extended in \cite{SS} to vector fields on complete intersection germs. This is known as the GSV-index and one of its main properties is that it is invariant under perturbations of both, the vector field and the functions that define the singular variety. The definition of this index was recently extended in \cite {BSS1} for vector fields with isolated singularities on hypersurface germs with non-isolated singularities, and it was proved that this index satisfies a proportionality property analogous to the one proved in
\cite{BS} for the Schwartz index and the local Euler obstruction, the proportionality factor being now the Euler-Poincar\'e characteristic of a local Milnor fiber.
In \cite{EG1} Ebeling and Gusein-Zade observed that when dealing with singular varieties, 1-forms have certain advantages over vector fields, as for instance the fact that for a vector field on the ambient space the condition of being tangent to a (stratified) singular variety is very stringent, while every 1-form on the ambient space defines, by restriction, one on the singular variety.
They adapted the definition of the GSV-index to 1-forms on complete intersection germs with isolated singularities, and proved a very nice formula for it in the case when the form is holomorphic, generalizing the well-known formula of L\^e-Greuel for the Milnor number of a function.
This article is about 1-forms on complex analytic varieties and it is particularly relevant when the variety has non-isolated singularities. We show in section 2 how the radial extension technique of M.-H. Schwartz can be adapted to 1-forms, allowing us to define {\it the Schwartz index} of 1-forms with isolated singularities
on singular varieties. Then we see (section 3) how MacPherson's local Euler obstruction, adapted to 1-forms in general, relates to the Schwartz index, thus obtaining a
proportionality theorem for these indices
analogous to the one in \cite{BS} for vector fields.
We then extend (in section 4) the definition of the GSV-index to 1-forms with isolated singularities on (local) complete intersections with non-isolated singularities that satisfy the Thom $a_f$-condition (which is always satisfied if the variety is a hypersurface), and we prove the corresponding proportionality theorem for this index. When the form is the differential of a holomorphic function $h$, this index measures the number of critical points of a generic perturbation of $h$ on a local Milnor fiber, so it is analogous to invariants studied by a number of authors (see for instance \cite {Go, IS,STV}). Section 1 is a review of well-known facts about real and complex valued 1-forms.
The radial extension of 1-forms can be made global on compact varieties, and it can also be made
for frames of differential 1-forms. One gets in this way
the dual Schwartz classes of singular varieties, which equal the usual ones
up to sign. One also has the dual Chern-Mather classes of $V$, already envisaged in \cite{Sa}, and the proportionality formula 3.3 can be used as in \cite{BS} to express the dual Chern-Mather classes as ``weighted" dual Schwartz classes, the weights been given by the local Euler obstruction. Similarly,
in analogy with Theorem 1.1 in \cite {BSS1},
the corresponding GSV-index and the proportionality Theorem 4.4 extend to frames and can be used to express the dual Fulton-Johnson classes of singular hypersurfaces embedded with trivial normal bundle in compact complex manifolds,
as ``weighted" dual Schwartz classes, the weights been now given by the Euler-Poincar\'e characteristic of the local Milnor fiber.
This work was done while the second and third named authors were visiting the
``Institut de Math\'ematiques de Luminy", France; they acknowledge the support of the CNRS,
France and the ``Universit\'e de la M\'editerran\'ee".
The authors thank J. Sch\"urmann for his comments and suggestions on the first version of the paper. In particular, he gave us an
alternative proof of Theorem 3.3 in the case of the differential form associated to a Morse function,
using stratified Morse theory and the micro-local index formula in \cite {Schu2}.
\section{Some basic facts about 1-forms}
In this section we study some basic facts about the geometry of 1-forms and the interplay between real and complex valued 1-forms on (almost) complex manifolds, which plays an important role in the sequel.
The material here is all contained in the literature; we include it for completness
and to set up our notation with no possible ambiguities.
We give precise references when appropriate.
Let $M$ be an almost complex manifold of real dimension $2m>0$. Let $TM$ be its complex tangent bundle. We denote by $T^*M$ the cotangent bundle of $M$, dual of $TM$; each fiber $(T^*M)_x$ consists of the $\mathcal C$-linear maps $TM_x \to \mathcal C$. Similarly, we denote by $T_\mathbbm{R} M$ the underlying real tangent bundle of $M$; it is a real vector bundle of fiber dimension $2m$, endowed with a canonical orientation. Its dual $T_\mathbbm{R}^*M$ has as fiber the $\mathbbm{R}$-linear maps $(T_{\mathbbm{R}}M)_x \to \mathbbm{R}$.
\vskip 0.3cm \noindent
{\bf 1.1 Definition.} Let $A$ be a subset of $M$. By a real (valued) 1-form $\eta$ on $A$ we mean the restriction to $A$ of a continuous section of the bundle $T_\mathbbm{R}^*M$, i.e., for each $x \in A$,
$\eta_x$ is an $\mathbbm{R}$- linear map $(T_{\mathbbm{R}}M)_x \to \mathbbm{R}$. We usually drop the word ``valued" here and speak only of real 1-forms on $A$.
Similarly, a complex 1-form $\omega$ on $A$ means the restriction to $A$ of a continuous section of the bundle $T^*M$, i.e., for each $x \in A$,
$\omega_x$ is a $\mathcal C$-linear map $(TM)_x \to \mathcal C$.
\vskip.1cm
Notice that the kernel of a real form $\eta$ at a point $x$ is either the whole fiber $(T_\mathbbm{R} M)_x$ or a real
hyperplane in it. In the first case we say that $x$ is a singular point (or zero) of $\eta$. In the second case
the kernel $ker\, \eta_x$ splits $(T_\mathbbm{R} M)_x$ in two half spaces $(T_\mathbbm{R} M^{\pm})_x$; in one of these the form takes positive values, in the other $\eta(v)$ is negative.
We recall that a vector field $v$ in $\mathbbm{R}^N$ is radial at a point $x_o$ if it is transversal to every sufficiently
small sphere around $x_o$ in $\mathbbm{R}^N$. The duality between real 1-forms and vector fields assigns to each
tangent vector $\partial / \partial x_i$ the form $dx_i$ (extending it by linearity to all tangent vectors).
This refines the classical duality that assigns to each hyperplane in
$\mathbbm{R}^{N}$ the line orthogonal to it and motivates the following definition (c. f. \cite {EG1, EG2}):
\vskip 0.3cm \noindent
{\bf 1.2 Definition.} A real 1-form $\eta$ on $M$ is {\bf radial} (outwards-pointing) at a point $x_o \in M$ if, locally, it is dual over
$\mathbbm{R}$ to a radial outwards-pointing vector field at $x_o$. Inwards-pointing radial vector fields are defined similarly.
In other words, $\eta$ is {\bf radial} at a point $x_o$ if it is everywhere positive when evaluated in some radial vector field at $x_o$.
\vskip.1cm
Thus, for instance, if for a fixed $x_o \in M$ we let $\rho_{x_o}(x)$ be the function $\Vert x - x_o \Vert^2$ (for some Riemmanian metric), then its differential is a radial form.
\vskip 0.3cm \noindent
{\bf 1.3 Remark.}
The concept of radial forms was introduced in \cite{EG1}. In \cite {EG2}
radial forms are defined using more relaxed conditions than we do here.
However this is a concept "imported" from the corresponding notion of radial vector fields, so
we use definition 1.2.
\vskip.1cm
A complex 1-form $\omega$ on $A \subset M$ can be written in terms of its real and imaginary parts:
$$\omega \,=\, Re\,(\omega) \,+\, i\, Im \,(\omega)\,.$$
Both $ Re\,(\omega)$ and $ Im \,(\omega)$ are real 1-forms, and the linearity of $\omega$ implies that for each tangent vector one has:
$$ Im \,(\omega)(v) \,=\, - Re \, (\omega)(iv) \,,$$
thus
\[\omega (v) \,=\, Re\,(\omega) (v) \,-\, i\, Re \,(\omega) (iv)\,. \]
In other words the form $\omega$ is determined by its real part and one has a 1-to-1 correspondence between real and complex forms, assigning to each complex form its real part, and conversely, to a real 1-form $\eta$ corresponds the complex form $\omega$ defined by:
\[\omega(v) \,=\, \eta(v) - i \eta(iv)\,. \]
This statement (noted in \cite {EG2},\cite{GMP})
refines the obvious fact that a complex hyperplane $P$ in $\mathcal C^m$, say defined by a linear form $H$, is the intersection of
the real hyperplanes $\widehat H := \{Re\, H = 0\}$ and $\,i\, \widehat H$. This justifies the following definition:
\vskip 0.3cm \noindent
{\bf 1.4 Definition.} A complex 1-form $\omega$ is {\bf radial} at a point $x \in M$ if its real part is radial at $x$.
\vskip.1cm
Recall that the Euler class of an oriented vector bundle
is the primary obstruction for constructing a non-zero section \cite {St}.
In the case of the bundle $T^*_\mathbbm{R} M$,
this class equals the Euler class $\hbox{Eu}(M)$ of the underlying real tangent bundle $T_\mathbbm{R} M$, since they are isomorphic. Thus, if $M$ is compact then its Euler class evaluated on the orientation cycle of
$M$ gives the Euler-Poincar\'e characteristic $\chi(M)$. We can say this in different words: let
$\eta$ be a real 1-form on $M$ with isolated (hence finitely many) singularities $x_1, \cdots,x_r$. At
each $x_i$ this 1-form defines a map,
$\mathbbm{S}_\varepsilon \buildrel{\eta/\Vert \eta \Vert}\over \longrightarrow \mathbbm{S}^{2m-1}$, from a small sphere in $M$ around
$x_i$ into the unit sphere in the fiber $(T^*_\mathbbm{R} M)_x$. The degree of this map is the
{\bf Poincar\'e-Hopf} local index of $\eta$ at $x_i$, that we may denote by
$\hbox{Ind}_{PH}(\eta, x_i)$.
Then the total index of $\eta$ in $M$ is by definition the sum of its local indices at the $x_i$ and it equals $\chi(M)$. Its Poincar\'e dual class in
$H^{2m}(M)$ is the Euler class of $T^*_\mathbbm{R} M \cong T_\mathbbm{R} M$.
\vskip.1cm
More generally, if $M$ is a compact, $C^\infty$ manifold of real dimension $2m$ with
non-empty boundary $\partial M$ and a complex structure in its tangent bundle, one can speak of real and complex valued 1-forms as above.
Elementary obstruction theory (see \cite {St}) implies that one can always find real and complex
1-forms on $M$ with isolated singularities, all contained in the interior of $M$. In fact, if a
real 1-form
$\eta$ is defined in a neighborhood of $\partial M$ in $M$ and it is non-singular there, then we can always extend it to the interior of $M$ with finitely many singularities, and its total
index in $M$ does not depend on the choice of the extension.
\vskip 0.3cm \noindent
{\bf 1.5 Definition.} Let $M$ be an almost complex manifold with boundary $\partial M$ and let
$\omega$ be a (real or complex) 1-form on $M$, non-singular on a neighborhood of $\partial M$;
let $Re \, \omega$ be its real part if $\omega$ is a complex form, otherwise $Re \, \omega = \omega$ for real forms.
The form $\omega$ is {\bf radial} at the boundary if for each vector $v(x) \in TM$, $x \in \partial M$,
which is normal to the boundary (for some metric),
pointing outwards of $M$, one has $Re \, \omega(v(x)) > 0$.
\vskip.2cm
By the
theorem of Poincar\'e-Hopf for manifolds with boundary, if a real 1-form $\eta$ is radial at the boundary and $M$ is compact, then the total index of $\eta$ is $\chi(M)$.
\vskip.1cm
We now make similar considerations for complex 1-forms. We let $M$ be a compact, $C^\infty$ manifold of real dimension $2m$ (with or without boundary $\partial M$), with
a complex structure in its tangent bundle $TM$. Let $T^*M$ be as before, the cotangent bundle of $M$, i.e., the bundle of complex valued continuous 1-forms. The top Chern class $c^m(T^*M)$ is the
primary obstruction for constructing a section of this bundle, i.e., if $M$ has empty boundary, then
$c^m(T^*M)$ is the number of points, counted with their local indices, of the zeroes of a section
$\omega$ of $T^*M$ (i.e., a complex 1-form) with isolated singularities (i.e., points where it vanishes).
It is well known (see for instance \cite {Mi}) that one has:
$$c^m(T^*M) \,=\, (-1)^m \,c^m(TM)\,.$$
This corresponds to the fact that at each isolated singularity $x_i$ of $\omega$ one has two local indices: one of them is the index of its real part defined as above, $\hbox{Ind}_{PH}(Re \, \omega, x_i)$; the other is the degree of the map
$\mathbbm{S}_\varepsilon \buildrel{\omega/\Vert \omega \Vert}\over \longrightarrow \mathbbm{S}^{2m-1}$, that we denote by
$\hbox{Ind}_{PH}(\omega, x_i)$. These two indices are related by the equality:
\[\hbox{Ind}_{PH}(\omega, x_i) \,=\, (-1)^m \, \hbox{Ind}_{PH}(Re \, \omega, x_i)\,, \]
and the index on the right corresponds to the local Poincar\'e-Hopf index of the vector field defined
by duality near $x_i$. For instance, the form $\omega = \sum z_i dz_i$ in $\mathcal C^m$ has index $1$ at
$0$, while its real part $\sum(x_i dx_i - y_i dy_i)$ has index $(-1)^m$.
If we take $M$ as above, compact and with possibly non-empty boundary, and $\omega$ is a complex 1-form
with isolated singularities in the interior of $M$ and radial on the boundary, then (by the previous considerations) the total index of $\omega$ in $M$ is $(-1)^m \, \chi (M)$. We summarize some of the previous discussion in the following theorem (c.f. \cite {EG1, EG2}):
\vskip 0.3cm \noindent
\proclaim{1.6 Theorem}{Let $M$ be a compact, $C^\infty$ manifold of real dimension $2m$ (with or without boundary $\partial M$), with a complex structure in its tangent bundle $TM$. Let $T^*_\mathbbm{R} M$ and $T^*M$ be as before, the bundles of real and complex valued continuous 1-forms on $M$, respectively. Then:
\vskip.1cm \noindent {\bf i) } Every real 1-form $\eta$ on $M$ determines a complex 1-form $\omega$ by the formula
$$ \omega(v) \,=\, \eta(v) - i \eta(iv) \,;$$
so the real part of $\omega$ is $\eta$.
\vskip.1cm \noindent {\bf ii) } The local Poincar\'e-Hopf indices at an isolated singularity of a complex 1-form and its real part are related by:
\[
{\rm{Ind}}_{PH}(\omega, x_i) \,=\, (-1)^m \, {\rm{Ind}}_{PH}(Re \, \omega, x_i)\,.
\]
\vskip.1cm \noindent {\bf iii) } If a real 1-form on $M$ is radial at the boundary $\partial M$, then its total Poincar\'e-Hopf
index in $M$ is $\chi(M)$. In particular, a radial real 1-form has local index 1.
\vskip.1cm \noindent {\bf iv) } If a complex 1-form on $M$ is radial at the boundary $\partial M$, then its total Poincar\'e-Hopf index in $M$ is $(-1)^m \chi(M)$.
}
\vskip 0.3cm \noindent
{\bf 1.7 Remark.} One may consider frames of complex 1-forms on $M$ instead of a single 1-form. This means considering sets of $k$ complex 1-forms, whose singularities are the points where these forms become linearly dependent over $\mathcal C$. By definition (see \cite {St}) the primary obstruction for constructing such a frame is the Chern class $c^{m-k+1}(T^*M)$, so these classes also have an expression similar to 1.6 but using indices of frames of 1-forms. One always has
$c^i(T^*M) = (-1)^i c^i(TM)$. Thus the Chern classes, and all the Chern numbers of $M$, can be computed using indices of either vector fields or 1-forms.
\section{Radial extension and the Schwartz index}
In the sequel we will be interested in considering forms defined on singular varieties in a complex manifold, so we introduce some standard notation.
Let $V$ be a reduced, equidimensional complex analytic space of dimension $n$ in a complex manifold $M$ of dimension $m$,
endowed with a Whitney stratification $\{V_\alpha\}$ adapted to $V$, i.e., $V$ is union of strata.
The following definition is an immediate extension for 1-forms of the corresponding (standard)
definition for functions on stratified spaces
in terms of its differential (c.f. \cite {EG2, GMP, Le1}).
\vskip 0.3cm \noindent
{\bf 2.1 Definition. } Let $\omega$ be a (real or complex) 1-form on $V$,
i.e., a continuous section of either $T^*_\mathbbm{R} M \vert_V$ or $T^*M \vert_V$.
A singularity of $\omega$ with respect to the Whitney stratification $\{V_\alpha\}$ means a point $x$ where
the kernel of $\omega$ contains the tangent space of the corresponding stratum.
This means that the pull back of the form to $V_\alpha$ vanishes at $x$.
In section 1 we introduced the notion of radial forms, which is dual to
the "radiality" for vector fields. We now extend this notion relaxing the condition of radiality in the
directions tangent to the strata. From now on, unless it is otherwise stated explicitely,
by a singularity of a 1-form on $V$ we mean a singularity in the stratified sense, i.e., in the sense of 2.1.
\vskip 0.3cm \noindent
{\bf 2.2 Definition. } Let $\omega$ be a (real or complex) 1-form on $V$. The form is {\bf normally radial}
at a point $x_o \in V_\alpha \subset V$ if it is radial when restricted to vectors which are not tangent to the stratum $V_\alpha$ that contains $x_o$. In other words, for every vector $v(x)$ tangent to $M$ at a point $x \notin V_\alpha$, $x$ sufficiently close to $x_o$ and $v(x)$ pointing outwards a tubular neighborhood of the stratum $V_\alpha$, one has $Re \, \omega(v) > 0$ (or $Re \, \omega(v) < 0$ for all such vectors;
if $\omega$ is real then it equals $Re \, \omega$).
\vskip.2cm
Obviously a radial 1-form is also normally radial, since it is radial in all directions.
For each point $x$ in a stratum $V_\alpha$, one has a neighborhood $U_x$ of $x$ in $M$
which is diffeomorphic to the product
$U_\alpha \times \mathbbm{D}_\alpha,$
where $U_\alpha = U_x \cap V_\alpha$ and $\mathbbm{D}_\alpha$ is a small disc in $M$ transversal to $V_\alpha$.
Let $\pi$ be the projection $\pi : U_x \to U_\alpha$ and $p$ the projection $p: U_x \to \mathbbm{D}_\alpha$.
One has an isomorphism:
\[\;T^* U_x \,\cong
\pi^* T^*{U_\alpha} \oplus p^* T^*\mathbbm{D}_\alpha\,.\]
That a (real or complex) 1-form $\omega$ be normally radial at $x$ means that up to a local change of coordinates in $M$, $\omega$ is the direct sum of the pull back of a (real or complex) form on $U_\alpha$, i.e., a section
of the (real or complex) cotangent bundle $T^*U_\alpha$, and a section
of the (real or complex) cotangent bundle $T^*\mathbbm{D}_\alpha$ which is a radial form in the disc.
\vskip.1cm
It is possible to make for 1-forms
the classical construction of {\bf radial extension} introduced by M.-H. Schwartz in \cite{Sch1, Sch2} for stratified vector fields and frames. Locally, the construction can be described as follows.
We consider first real 1-forms.
Let $\eta$ be a 1-form on $U_\alpha$, denote by $\widehat \eta$ its pull back to a section of $\pi^* T^*_{\mathbbm{R}}{U_\alpha}$. This corresponds to the {\bf parallel extension} of stratified vector fields
done by Schwartz. Now look at the function $\rho$
given by the square of the distance to the origin in $\mathbbm{D}_\alpha$. The form $p^* d\rho$ on $U_x$
vanishes on $ U_\alpha $ and away from $U_\alpha $ its
kernel is transversal to the strata of $V$ by Whitney conditions.
The sum $\eta' = \widehat \eta + p^*d\rho$ defines a normally radial 1-form on $U_x$ which coincides with $\eta$ on $U_\alpha $; away from $U_\alpha $ its kernel is transversal to the strata of $V$. Thus,
if $\eta$ is non-singular at $x$, then $\eta'$ is non-singular everywhere on
$U_x$. If $\eta$ has an isolated singularity at $x \in V_\alpha$, then $\eta'$ also has an isolated singularity there. In particular, if the dimension of the stratum $V_\alpha$ is zero then $\eta'$ is
a radial form in the sense of section 1.
Following the terminology of \cite{Sch1, Sch2} we say that the form $\eta'$ is obtained from
$\eta$ by {\bf radial extension}.
Since the index
in $M$ of a normally radial form is its index in the stratum times the index of a radial form in
the disc $\mathbbm{D}_\alpha$, we obtain the following important property of forms constructed by radial extension.
\vskip 0.3cm \noindent
\proclaim{2.3 Proposition}{Let $\eta$ be a real 1-form on the stratum $V_\alpha$ with an isolated singularity
at a point $x$ with local Poincar\'e-Hopf index ${\rm{Ind}}_{PH}(\eta, V_\alpha; x)$. Let $\eta'$ the 1-form on
a neighborhood of $x$ in $M$ obtained by radial extension. Then the
index of $\eta$ in the stratum equals the index of $\eta'$ in $M$:
$${\rm{Ind}}_{PH}(\eta, V_\alpha; x)\,=\, {\rm{Ind}}_{PH}(\eta', M; x)\,. $$}
\noindent{\bf 2.4 Definition.} The {\bf Schwartz index} of the continuous real 1-form $\eta$ at an isolated
singularity $x \in V_\alpha \subset V$, denoted $\hbox{Ind}_{Sch}(\eta, V; x)$,
is the Poincar\'e-Hopf index of the 1-form $\eta'$ obtained from $\eta$ by radial extension; or equivalently, if the stratum of $x$ has dimension more than 0,
$\hbox{Ind}_{Sch}(\eta, V; x)$ is the Poincar\'e-Hopf index at $x$ of
$\eta$ in the stratum $V_\alpha$.
\vskip.2cm
If $x$ is an isolated singularity of $V$ then every 1-form on $V$ must be singular at $x$ since its kernel
contains the ``tangent space" of the stratum. In this case the
index of the form in the stratum is defined to be 1, and this is consistent with the previous definition since in this case the radial extension of $\eta$ is actually radial at $x$, so it has index 1 in the ambient space.
The previous process is easily adapted to give radial extension for
complex 1-forms. Let $\omega$ be such a form on $V_\alpha$; let $\eta$ be its real part. We extend $\eta$ as above, by radial extension, to obtain a real 1-form $\eta'$ which is normally radial at $x$. Then we use statement i) in Theorem 1.6 above to obtain a complex 1-form $\omega'$ on $U_x $
that extends $\omega$ and is also normally radial at $x$. If we prefer, we can make this process in a different but equivalent way: first make a parallel extension of
$\omega$ to $U_x $ as above, using the projection $\pi$; denote
by $\widehat \omega$ this complex 1-form. Now use 1.6.i) to define a complex 1-form
$\widehat { d\rho}$ on $ U_x $ whose real part is $d\rho$, and take the direct sum of
$\widehat \omega$ and
$\widehat {d\rho}$ at each point to obtain the extension $\omega'$. We say that
$\omega'$ is obtained from $\omega$ by {\bf radial extension}.
We have the equivalent of Proposition 2.3 for complex forms, modified with the appropriate signs:
$$(-1)^s \, \hbox{Ind}_{PH}(\omega, V_\alpha; x)\,=\, (-1)^m \, \hbox{Ind}_{PH}(\omega', M; x)\,, $$
where $2s$ is the real dimension of $V_\alpha$ and $2m$ that of $M$.
\vskip 0.3cm \noindent {\bf 2.5 Definition.} The {\bf Schwartz index} of the continuous complex 1-form $\omega$ at an isolated
singularity $x \in V_\alpha \subset V$, denoted $\hbox{Ind}_{Sch}(\omega, V; x)$, is $(-1)^n$-times the index of its real part:
$$\hbox{Ind}_{Sch}(\omega, V; x) \,=\, (-1)^n \hbox{Ind}_{Sch}(Re\, \omega, V; x)\;.$$
\section[Local Euler obstruction]
{Local Euler obstruction and the Proportionality Theorem}
We are now concerned only with a local situation, so we take $V$ to be embedded in an
open ball $\mathbbm{B} \subset \mathcal C^m$ centered at the origin $0$. On the regular part of $V$ one has the map
$\sigma : V_{reg} \to G_{n,m}$
into the Grassmannian of $n (= \text{dim}\,V)$-planes in $\mathcal C^m$, that assigns to each point the corresponding tangent space of $V_{reg}$. The
Nash blow up $\widetilde V \buildrel {\nu}\over \to V$ of $V$
is by definition the closure in $\mathbbm{B} \times G_{n,m}$ of the
graph of the map $\sigma$.
One also has the Nash bundle $\widetilde T \buildrel {p}\over \to \widetilde V $,
restriction to $\widetilde V $ of the tautological bundle over $\mathbbm{B} \times G_{n,m}$.
The corresponding dual bundles of complex and real 1-forms are denoted by
$\widetilde T^* \buildrel {p}\over \to \widetilde V $ and
$\widetilde T^*_\mathbbm{R} \buildrel {p}\over \to \widetilde V $, respectively. Observe that a point in ${\widetilde T^*}$ is a triple $(x,P,\omega)$ where $x$ is in $V$, $P$ is an $n$-plane in the tangent space $T_x \mathbbm{B}$ which is limit of a sequence $\{(T V_{reg})_{x_i}\}$, where the $x_i$ are points in the regular part of $V$ converging to $x$, and $\omega$ is a $\mathcal C$-linear map $P \to \mathcal C$. (Similarly for $\widetilde T^*_\mathbbm{R}$.)
Let us denote by $\rho$ the function given by the square of the distance to $0$.
We recall that MacPherson in \cite {MP} observed that the Whitney condition (a) implies that the pull-back of the differential $d\rho$ defines a never-zero section $\widetilde {d\rho}$ of
$\widetilde T^*_\mathbbm{R}$ over $\nu^{-1}(\mathbbm{S}_\varepsilon \cap V) \subset \widetilde V$, where $\mathbbm{S}_\varepsilon$ is the boundary of a small ball $\mathbbm{B}_\varepsilon$ in $\mathbbm{B}$ centered at $0$. The obstruction for extending $\widetilde {d\rho}$ as a
never-zero section of $\widetilde T^*_\mathbbm{R}$ over $\nu^{-1}(\mathbbm{B}_\varepsilon \cap V) \subset \widetilde V$ is a
cohomology class in
$H^{2n}(\nu^{-1}(\mathbbm{B}_\varepsilon \cap V), \nu^{-1}(\mathbbm{S}_\varepsilon \cap V); \mathbbm{Z})$, and
MacPherson defined {\bf the local Euler obstruction} $\hbox{Eu}_V(0)$ of $V$ at
$0$ to be the integer obtained by evaluating this class
on the orientation cycle
$[\nu^{-1}(\mathbbm{B}_\varepsilon \cap V), \nu^{-1}(\mathbbm{S}_\varepsilon \cap V)]$.
More generally, given a section $\eta$ of $T^*_{\Bbb R}\Bbb B|_A$, $A\subset V$, there is a canonical way of constructing a section $\tilde\eta$ of $\widetilde T^*_\mathbbm{R}|_{\tilde A}$,
$\tilde A=\nu^{-1}A$, which is described in the following. The same construction
works for complex forms. First, taking the pull-back $\nu^*\eta$, we get a section of
$\nu^*T^*_{\Bbb R}\Bbb B|_V$. Then $\tilde\eta$ is obtained by projecting $\nu^*\eta$
to a section of $\tilde T^*_\mathbbm{R}$ by the canonical bundle homomorphism
$$
\nu^*T^*_{\Bbb R}\Bbb B|_V\longrightarrow\tilde T^*_\mathbbm{R}.
$$
Thus the value of $\tilde\eta$ at a point $(x,P)$ is simply the restriction of the linear
map $\eta(x): (T_{\Bbb R}\Bbb B)_x\to\Bbb R$ to $P$.
We call $\tilde\eta$ the {\bf canonical lifting} of $\eta$.
By the Whitney condition (a), if
$a \in V_\alpha$ is the limit point of the sequence $\{ x_i \}\in V_{\rm reg}$ such that $P =
\lim (TV_{\rm reg})_{x_i} $ and if
the kernel of $\eta$ is transversal to $V_\alpha$, then the linear form $\widetilde \eta$ will be non-vanishing on $P$. Thus, if $\eta$
has an isolated singularity
at the point $0 \in V$ (in the stratified sense), then we have a never-zero section $\widetilde \eta$
of the dual Nash bundle $\widetilde T^*_\mathbbm{R}$ over $\nu^{-1}(\mathbbm{S}_\varepsilon \cap V) \subset \widetilde V$. Let
$o(\eta) \in H^{2n}(\nu^{-1}(\mathbbm{B}_\varepsilon \cap V), \nu^{-1}(\mathbbm{S}_\varepsilon \cap V); \mathbbm{Z})$ be the
cohomology class of the obstruction cycle
to extend this to a section of $\widetilde T^*_\mathbbm{R}$ over $\nu^{-1}(\mathbbm{B}_\varepsilon \cap V)$. Then define (c.f. \cite {BMPS, EG2}):
\vskip 0.3cm \noindent
{\bf 3.1 Definition.} The {\bf local Euler obstruction} of the real differential form $\eta$ at an isolated singularity is the integer
$\hbox{Eu}_{V}(\eta,0)$
obtained by evaluating the obstruction cohomology class $o(\eta)$ on the orientation cycle
$[\nu^{-1}(\mathbbm{B}_\varepsilon \cap V), \nu^{-1}(\mathbbm{S}_\varepsilon \cap V)]$.
\vskip 0.3cm \noindent
The local Euler obstruction $\hbox{Eu}_{V}(0)$
of MacPherson corresponds to taking the differential of the squared function distance to $0$.
In the complex case, one can perform the same construction, using the
corresponding complex bundles. If $\omega$ is a complex differential form,
section of $T^*\Bbb B|_A$ with an isolated singularity, one can define the local Euler obstruction
$\hbox{Eu}_{V}(\omega,0)$.
Notice that it is equal to that of its real part up to sign:
\[\hbox{Eu}_{V}(\omega,0) \,=\, (-1)^n \hbox{Eu}_{V}(Re\, \omega, 0) \,. \tag{3.2}\]
This is an immediate consequence of the relation between the Chern classes of a complex vector bundle and those of its dual (see for instance \cite {Mi}).
We note that the idea to consider the (complex) dual Nash bundle was already present in
\cite{Sa}, where Sabbah introduces a local Euler obstruction ${\rm E\check u}_V (0)$ that satisfies
${\rm E\check u}_V (0) = (-1)^{n}{\rm Eu}_V(0)$. See also Sch\"urmann \cite{Schu1}, sec. 5.2.
\vskip.1cm
Just as for vector fields (see \cite {BS}), one has in this situation the following:
\vskip 0.3cm \noindent
\proclaim{3.3 Theorem} {Let $ V_\alpha \subset V$ be the stratum containing $0$,
${\rm{Eu}}_V(0)$ the local Euler obstruction of $V$ at $0$ and
$\omega$ a (real or complex) 1-form on $V_\alpha$ with an isolated singularity at $0$.
Then the local Euler obstruction of the
radial extension $\omega'$ of $\omega$ and the Schwartz index of $\omega$ at $0$ are related by the
following proportionality formula:
\[{\rm{Eu}}_{V}(\omega',0) \,=\, {\rm{Eu}}_{V} (0)\cdot {\rm{Ind}}_{Sch}(\omega, V; 0) \,.\]}
\vskip 0.3cm \noindent
{\bf Proof}
By 3.2 and Theorem 1.6 above, it is enough to prove 3.3 for either real or complex 1-forms, each case implying the other. We prove it for real forms.
Let $\eta$ and $\eta'$ be as above. Also, let $\eta_{rad}$
denote a radial form at $0$.
By construction and definition, we have
\[
\hbox{Ind}_{PH}(\eta,V_\alpha; 0)=\hbox{Ind}_{PH}(\eta',\Bbb B;0)=\hbox{Ind}_{Sch}(\eta,V;0).\tag{3.4}
\]
By definition of $\hbox{Ind}_{PH}(\eta',\Bbb B;0)$, there is a homotopy
$$
\Psi:[0,1]\times\Bbb S_\varepsilon\longrightarrow T^*_\mathbbm{R}\Bbb B|_{\Bbb S_\varepsilon}
$$
such that its image satisfies:
\[
\partial\hbox{Im}\Psi=\eta'(\Bbb S_\varepsilon)-\hbox{Ind}_{PH}(\eta',\Bbb B;0)
\cdot \eta_{rad}(\Bbb S_\varepsilon)\tag{3.5}
\]
as chains in $T^*_{\Bbb R}\Bbb B|_{\Bbb S_\varepsilon}$. The restriction of $\Psi$ gives a homotopy
$$
\psi:[0,1]\times(\Bbb S_\varepsilon\cap V)\longrightarrow T^*_{\Bbb R}\Bbb B|_{\Bbb S_\varepsilon\cap V}
$$
such that (c.f. (3.4))
$$
\partial\hbox{Im}\psi=\eta'(\Bbb S_\varepsilon\cap V)-\hbox{Ind}_{Sch}(\eta,V;0)
\cdot\eta_{rad}(\Bbb S_\varepsilon\cap V).
$$
Now we can lift $\psi$, $\eta'$ and $\eta_{rad}$ to sections
$\nu^*\psi$, $\nu^*\eta'$ and $\nu^*\eta_{rad}$ of $\nu^*T^*_{\Bbb R}\Bbb B$ to get
a homotopy
$$
\nu^*\psi:[0,1]\times\nu^{-1}(\Bbb S_\varepsilon\cap V)
\longrightarrow\nu^*T^*_{\Bbb R}\Bbb B|_{\nu^{-1}(\Bbb S_\varepsilon\cap V)}
$$
and, since $\nu$ is an isomorphism away from the singularity of $V$, we still have
\[
\partial\hbox{Im}\nu^*\psi=\nu^*\eta'(\nu^{-1}(\Bbb S_\varepsilon\cap V))-\hbox{Ind}_{Sch}(\eta,V;0)
\cdot\nu^*\eta_{rad}(\nu^{-1}(\Bbb S_\varepsilon\cap V))
\]
as chains in $\nu^*T^*_{\Bbb R}\Bbb B|_{\nu^{-1}(\Bbb S_\varepsilon\cap V)}$. Recall that we get the canonical liftings
$\tilde\psi$, $\tilde\eta'$ and $\tilde\eta_{rad}$ of $\psi$, $\eta'$ and $\eta_{rad}$
by taking the images of $\nu^*\psi$, $\nu^*\eta'$ and $\nu^*\eta_{rad}$ by the canonical
bundle homomorphism
$\nu^*T^*_{\Bbb R}\Bbb B \longrightarrow \tilde T^*_{\Bbb R}$.
Thus we have
\[
\partial\hbox{Iml}\tilde\psi=\tilde\eta'(\nu^{-1}(\Bbb S_\varepsilon\cap V))-\hbox{Ind}_{Sch}(\eta,V;0)
\cdot\tilde\eta_{rad}(\nu^{-1}(\Bbb S_\varepsilon\cap V))
\]
as chains in
$\tilde T^*_{\Bbb R}|_{\nu^{-1}(\Bbb S_\varepsilon\cap V)}$.
The forms $\tilde\eta'$ and $\tilde\eta_{rad}$ are non-vanishing on
$\nu^{-1}(\Bbb S_\varepsilon\cap V)$, by the Whitney condition, and by definition of the Euler obstructions, we have
the theorem.
\ensuremath{\Box}
\section{The GSV-index}
We recall (\cite {GSV, SS}) that the GSV-index of a vector field $v$ on an isolated complete intersection germ $V$ can be defined to be the Poincar\'e-Hopf index of an extension of $v$ to a Milnor fiber $F$. Similarly, the GSV-index of a 1-form $\omega$ on $V$ can be defined to be the Poincar\'e-Hopf index of the form on $F$, i.e., the number of singularities of $\omega$ in $F$ counted with multiplicities \cite {EG1}. When $V$ has
non-isolated singularities one may not have a Milnor fibration in general, but one does if $V$ has
a Whitney stratification with Thom's $a_f$-condition, $f =(f_1,\cdots,f_k)$ being the functions that define $V$ (c.f. \cite {Le2, LT, BSS1}).
Let $(V,0)$ be a complete intersection of complex dimension $n$ defined in a ball $\mathbbm{B}$ in
$\mathcal C^{n+k}$ by functions
$f =(f_1,\cdots,f_k)$, and assume $0$ is a singular point of $V$ (not necessarily an isolated singularity). As before, we endow $\mathbbm{B}$ with a Whitney stratification adapted to $V$, and we assume that we can choose
$\{V_\alpha\}$ so that it satisfies the $a_f$-condition of Thom (see for instance \cite{LT}). In particular one always has such a stratification if $k = 1$, by \cite{Hi}.
Let $\omega$ be as before, a (real or complex) 1-form on $\mathbbm{B}$, and assume its restriction
to $V$ has an isolated singularity at $0$. This means that the kernel of $\omega(0)$ contains the tangent space of the stratum $V_\alpha$ containing $0$,
but everywhere else it is transversal to each stratum
$V_\alpha \subset V$. Now let $F = F_t$ be a Milnor fiber of $V$, i.e., $F = f^{-1}(t) \cap \mathbbm{B}_\varepsilon$, where
$\mathbbm{B}_\varepsilon$ is a sufficiently small ball in $\mathbbm{B}$ around $0$ and $t \in \mathcal C^k$ is a regular value of $f$ with
$\Vert t \Vert$ sufficiently small with respect to $\varepsilon$. Notice that the $a_f$-condition implies that for every sequence $t_n$ of regular values converging to $0$, and for every sequence $\{x_n\}$ of points in the corresponding Milnor fibers converging to a point $x_o \in V$ so that the sequence of tangent spaces
$\{(TF)_{x_n}\}$ has a limit $T$, one has that $T$ contains the space
$(TV_\alpha)_{x_o}$, tangent to the stratum that contains $x_o$. By transversality this implies that
choosing the regular value $t$ sufficiently close to $0$ we can assure that the kernel of
$\omega$ is transversal to the Milnor fiber at every point in its boundary $\partial F$. Thus its pull-back
to $F$ is a 1-form on this smooth manifold, and it is never-zero on its boundary, thus
$\omega$ has a well defined Poincar\'e-Hopf index in $F$ as in section 1. This index is well-defined and depends only on the restriction of
$\omega$ to $V$ and the topology of the Milnor fiber $F$, which is well-defined once we fix the defining function $f$ (which is assumed to satisfy the $a_f$-condition for some Whitney stratification).
\vskip 0.3cm \noindent
{\bf 4.1 Definition} The GSV-index of $\omega$ at $0 \in V$ relative to $f$, $\hbox{Ind}_{GSV}(\omega,0)$, is the
Poincar\'e-Hopf index of $\omega$ in $F$.
\vskip.2cm
In other words this index measures the number of points (counted with signs) in which a generic perturbation of $\omega$ is tangent to $F$.
In fact the inclusion $F \buildrel{i}\over \to M$ pulls the form $\omega$ to a section of the
(real or complex, as the case may be) cotangent bundle of $F$, which is never-zero near the
boundary because $\omega$ has an isolated singularity at $0$ and, by hypothesis, the map $f$ satisfies the $a_f$-condition of Thom. If the form $\omega$ is real then
\[ \hbox{Ind}_{GSV}(\omega,0) \,=\, \hbox{Eu}(F; \omega)[F] \;, \tag{4.2}\]
where $\hbox{Eu}(F; \omega) \in H^{2n}(F, \partial F)$ is the Euler class of the real cotangent bundle
$T^*_\mathbbm{R} F$ relative to the section defined by $\omega$ on the boundary, and $[F]$ is the orientation cycle of the pair $(F,\partial F)$.
If $\omega$ is a complex form, then one has:
\[ \hbox{Ind}_{GSV}(\omega,0) \,=\, c^n(T^*F; \omega)[F] \;, \tag{4.3.i}\]
where $c^n(T^*F; \omega)$ is the top Chern class of the cotangent bundle of $F$ relative to the form $\omega$ on its boundary. In this case one can, alternatively, express this index as the relative Chern class:
\[ \hbox{Ind}_{GSV}(\omega,0) \,=\, c^n(T^*M \vert_F; \Omega)[F] \;, \tag{4.3.ii}\]
where $\Omega$ is the frame of $k+1$ complex 1-forms on the boundary of $F$ given by
\[\Omega \,=\, (\omega, df_1, df_2, \cdots, df_k)\,,\]
since the forms
$(df_1,\cdots,df_k)$ are linearly independent everywhere on $F$.
Notice that if the form $\omega$ is holomorphic, then this index
is necessarily non-negative because it can be regarded as an intersection number of complex submanifolds. For every complex 1-form one has:
\[ \hbox{Ind}_{GSV}(\omega,0) \,=\, (-1)^n \hbox{Ind}_{GSV}(Re\, \omega,0) \,.\]
We remark that if $V$ has an isolated singularity at $0$, this is the index envisaged in \cite{EG1}, i.e.,
the degree of the map from the link $K$ of $V$ into the Stiefel manifold of complex (k+1)-frames
in the dual $\mathcal C^{n+k}$ given by the map $(\omega, df_1,\cdots,df_k)$. Also notice that this index is somehow dual to the index defined in \cite {BSS1} for vector fields, which is related to the top Fulton-Johnson class of singular hypersurfaces.
So, given the (non-isolated) complete intersection singularity $(V,0)$ and a (real or complex) 1-form
$\omega$ on $V$ with an isolated singularity at $0$, one has three different indices: the Euler obstruction (section 2), the GSV-index just defined and
the index of its pull back to a 1-form on the stratum of $0$. One also has the index of the form in the ambient manifold $M$.
For forms obtained by radial extension, the index in the stratum equals its index in $M$, and this is by definition the Schwartz index.
The following proportionality theorem is analogous to the one in \cite {BSS1} for vector fields.
\vskip 0.3cm \noindent
\proclaim{4.4 Theorem} {Let $\omega$ be a (real or complex) 1-form on the stratum $V_\alpha$ of $0$ with an isolated singularity at $0$. Then the GSV index of its radial extension $\omega'$
is proportional to the Schwartz index, the proportionality factor being the Euler-Poincar\'e characteristic of the Milnor fiber $F$:
\[{\rm{Ind}}_{GSV}(\omega',0)\,=\, {\chi(F) \cdot \rm{Ind}}_{Sch}(\omega, V; 0) \,.\]
}
\vskip.1cm \noindent
{\bf Proof.}
It is enough to prove 4.4 either for complex forms or for real forms, each one implying the other. The proof is similar to that of 3.3.
Let $\omega'$ and $\omega_{rad}$ be as in the proof of Theorem 3.3. Then 4.4 is proved by
taking the retriction to $F$ of each section in (3.5) as a differential form, noting that $\hbox{Ind}_{GSV}(\omega_{rad},0)\,=\, \chi(F)$. \ensuremath{\Box}
\vskip 0.3cm \noindent
{\bf 4.5 Remark.} We notice that 4.2 and 3.3 can also be proved using the stability of the index under
perturbations; this works for vector fields too. More precisely, one can easily show that the Euler obstruction $\hbox{Eu}_V(\omega,x)$ and the GSV-index are stable when we perturb the 1-form (or the vector field) in the stratum and then extend it radially; then the sum of the indices at the singularities of the new 1-form (vector field) give the corresponding index for the original singularity. This implies the proportionality of the indices.
|
{
"timestamp": "2005-05-11T10:07:43",
"yymm": "0503",
"arxiv_id": "math/0503428",
"language": "en",
"url": "https://arxiv.org/abs/math/0503428"
}
|
\section{Introduction}
This paper is an attempt to understand topological properties of
Lie group actions. Its starting point was the following theorem
of McDuff--Slimowitz~\cite{MSlim} concerning circle subgroups of
${\rm Symp}(M,\omega)$,
the group of symplectomorphisms of a symplectic manifold $(M,{\omega})$.
Recall that an effective circle action
is {\bf semifree} if
the stabilizer subgroup of each point in $M$ is either
the circle itself or the trivial group.
Also, we say that a circle subgroup ${\Lambda}$ of a topological group ${\mathcal H}$ is {\bf
essential in} ${\boldsymbol {\mathcal H}}$ if it represents a nonzero element in $\pi_1({\mathcal H})$ and
{\bf inessential in} ${\boldsymbol {\mathcal H}}$ otherwise.
\begin{theorem}\labell{thm:sfr}
Any semifree circle action on a closed symplectic manifold $(M,\omega)$
is essential in ${\rm Symp}(M,{\omega})$.
\end{theorem}
This is obvious if the
action is not Hamiltonian since in this case the flux homomorphism
$$
{\rm Flux}: \pi_1({\rm Symp}(M,{\omega})) \longrightarrow H^1(M,{\mathbb R})
$$
does not vanish on ${\Lambda}$. However, if the action is Hamiltonian with
generating Hamiltonian $K:M\longrightarrow {\mathbb R}$ then the result
is not so easy: the proof in~\cite{MSlim} involved studying
the Hofer length of the corresponding paths $\phi_t^K, t\in [0,T],$
in ${\rm Ham}(M,{\omega})$.
The first result in this paper uses the theorem above to answer
a question posed by Alan Weinstein in~\cite{Wei}.
Let $G$ be a semisimple Lie group with Lie algebra $\fg$.
Let $M \subset \fg^*$ be a coadjoint orbit,
together with the Kostant--Kirillov symplectic form $\omega$.
If the coadjoint action of $G$ on $M$ is effective,
then $G$ is naturally a subgroup of ${\rm Ham}(M,\omega)$,
the group of Hamiltonian symplectomorphisms of $(M,\omega)$.
This inclusion induces a natural map
from the fundamental group of $G$ to the fundamental group
of ${\rm Ham}(M,\omega)$.
Weinstein
asks when this map is injective.
We prove that this map is injective
for all compact semisimple Lie groups.
In~\cite{Vina} Vina established a
special case of this result by quite different methods.
\begin{theorem}\label{cor:coadj}
Let a compact semisimple Lie group $G$
act effectively on a coadjoint orbit $(M,\omega)$.
Then the inclusion $G\longrightarrow {\rm Ham}(M, {\omega})$ induces an
injection from $\pi_1(G)$ to $\pi_1({\rm Ham}(M,{\omega})).$
\end{theorem}
In view of Theorem~\ref{thm:sfr}, this is an
immediate consequence of the following result,
which we prove in Section \ref{ss:coadj}.
\begin{proposition}\label{WQ}
Let a compact semisimple Lie group $G$
act effectively on a coadjoint orbit $(M,\omega)$.
Then every nontrivial element in $\pi_1(G)$ may be represented
by a circle that acts semifreely on $M$.
\end{proposition}
Theorem~\ref{thm:sfr} immediately implies that if a compact Lie group
$G$ acts effectively on a closed symplectic manifold $(M,\omega)$,
then any semifree circle
subgroup $\Lambda \subset G$ is essential in $G$.
The other results in the paper generalize this claim.
Observe that Theorem~\ref{thm:sfr}
does not immediately extend to
the smooth (non-symplectic) category. For example, Claude
LeBrun pointed out to us that the circle action on $S^4$
induced by the diagonal
action of $S^1$ on ${\mathbb C}^2 = {\mathbb R}^4 \subset {\mathbb R}^5$ is semifree but gives a
nullhomotopic loop since $\pi_1(SO(5)) = {\mathbb Z}/2{\mathbb Z}$.
Nevertheless, the semifree condition does have consequences
in the smooth category, even
if the action is only semifree on
a neighborhood of a component of the fixed point set;
we shall say that such components are
{\bf semifree}. Further, given a circle subgroup
$\Lambda \subset G$ we say that $g \in G$ {\bf reverses}
${\boldsymbol \Lambda}$ {\bf in} ${\boldsymbol G}$
if $g t g^{-1} = t^{-1}$ for all $t\in {\Lambda}$.
Finally, a component $F$ of the fixed point set $M^{\Lambda}$ of $\Lambda$
is {\bf symmetric in} ${\boldsymbol G}$
if there is an element $g \in G$ whose action on $M$ fixes $F$
pointwise and which reverses $\Lambda$.
\begin{thm}\label{thm:main0}
Let $\Lambda$ be a circle subgroup
of a compact Lie group $G$ which acts effectively on a connected manifold $M$.
If there is a semifree component of the fixed point set $M^\Lambda$
which is not symmetric in $G$, then ${\Lambda}$ is essential in $G$.
\end{thm}
\begin{example}\label{ex}\rm
First, let $G = SU(2)$ act on ${\mathbb{CP }}^2$ by the defining representation on
the first two copies of ${\mathbb C}$,
and let $\Lambda \subset G$ be the circle subgroup
given by $\lambda \cdot [z_0:z_1: z_2]\mapsto
[\lambda z_0: \lambda^{-1} z_1: z_2]$.
This action has a semifree fixed point, namely $[0,0,1]$.
Moreover, this circle subgroup
is inessential in $G$.
Therefore, by the theorem above, there exists $g \in G$
which reverses the circle action and
fixes $[0,0,1]$. In fact, we can take any $g$ which
lies in the normalizer $N(\Lambda)$ but not in $\Lambda$ itself.
Note that $g^2 = -{{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l$ for any such $g$.
In contrast, consider the natural action
of $G= PU(3)$ on ${\mathbb{CP }}^2$, and
let $\Lambda \subset G$ be the circle subgroup
given by $\lambda \cdot [z_0:z_1: z_2]\mapsto
[\lambda^2 z_0: z_1: z_2]$.
This action is semifree and essential,
but is not reversed by any $g \in G$.
To see this, note that the circle has order $3$ in $\pi_1(G)$,
whereas every circle that can be reversed has order $1$ or $2$.
\end{example}
One can weaken the semifree hypothesis in the above theorem,
at the cost of adding
a global
isotropy assumption and working once more in the symplectic category.
We say that a circle action
has {\bf at most twofold isotropy} if every point which is not either fixed
or free has stabilizer ${\mathbb Z}/(2).$
Recall, also, that a symplectic action of $G$ on $(M,{\omega})$ is
Hamiltonian if it is given by an equivariant moment map $\Phi:M\longrightarrow \fg^*$.
\begin{thm} \labell{thmtwo}
Let $\Lambda$ be a circle subgroup
of a compact Lie group $G$ which acts effectively on a connected
symplectic manifold $(M,\omega)$.
If $\Lambda$ has
at most twofold isotropy and if there is no $g \in G$
which reverses $\Lambda$, then
${\Lambda}$ is essential in $G$.
\end{thm}
\begin{example}\rm
This theorem does not extend to circle actions
which have at most threefold isotropy.
For example, the action of $S^1$ on ${\mathbb{CP }}^3$
given by $\lambda \cdot [x,y,z,w] = [\lambda^2 x, \lambda^{-1}y,
\lambda^{-1}z,w]$ is inessential in $PU(4)$.
However, since $F_{\rm max}$ and $F_{\rm min}$ are not diffeomorphic,
this action has no reversor.
We also need the symplectic hypothesis.
To see this, consider the obvious action of $SU(3)$ on $S^6: =
{\mathbb C}^3\cup\{\infty\}$. The subgroup ${\Lambda}: = {\rm diag\,}(\lambda^2,
\lambda^{-1},\lambda^{-1})$ acts with at most twofold isotropy but
has no reversor.
\end{example}
\begin{remark} \rm
If $G$ is a simple group, we do not
need to assume that $(M,\omega)$ is symplectic in Theorem~\ref{thmtwo};
we only need to assume that there exists a point $p$ which
is fixed by a maximal torus containing $\Lambda$ but
is not fixed by all of $G$.
Note that, in contrast, in the example above,
the only points fixed by ${\Lambda}$ are
fixed by all of $G$.
\end{remark}
\begin{remark}\rm
If $\Lambda$ is {\em any} circle subgroup of
$SO(3)$ -- or indeed a subgroup of any simple group
of type $B_n$, $C_n$, or $F_4$ --
then there exists a $g \in G$ which reverses $\Lambda$. In this case,
Theorem~\ref{thmtwo} is trivial and the force of
Theorem~\ref{thm:main0}
is that we can choose $g$ so that it also fixes $p$.
\end{remark}
\begin{remark}\rm
In the proof of the above theorems,
we pick a maximal torus $T$ which contains $\Lambda$.
The reversor $g$ that we construct lies in the normalizer $N(T)$ and
has the property that $g^2$ lies in $T$. However,
as we saw in Example~\ref{ex}, $g^2$ may not be equal to the identity.
\end{remark}
Theorems \ref{thm:main0} and \ref{thmtwo}
have the following easy corollaries:
\begin{corollary}
Consider a Hamiltonian
circle action $\Lambda$ on a closed symplectic manifold
$(M,\omega)$ with moment map $K: M \longrightarrow {\mathbb R}$,
normalized so that $\int_M K \omega^n = 0$.
If $F$ is a semifree fixed component, then $\Lambda$
is essential in every compact subgroup
$G \subset {\rm Symp}(M,\omega)$ that contains it, unless there is
a symplectomorphism $g$ of $M$ that fixes $F$ and reverses ${\Lambda}$.
In this case, all the following hold:
\begin{enumerate}
\item $K(g(p)) = - K(p)$ for all $p \in M^{{\Lambda}}$.
\item There is a one-to-one correspondence between the
positive weights at $p$ and the negative weights at $g(p)$, and vice versa.
\item $g$ induces an isomorphism on the image of the restriction
map in equivariant cohomology $H^*_{S^1}(M) \longrightarrow H^*_{S^1}(M^{\Lambda})$.
\item $g(F) = F$.
In particular,
\begin{enumerate}
\item $K(F) = 0$.
\item The sum of the weights at $F$ is zero.
\end{enumerate}
\end{enumerate}
\end{corollary}
\begin{corollary}
Consider a Hamiltonian
circle action $\Lambda$ on a closed symplectic manifold
$(M,\omega)$ with moment map $K: M \longrightarrow {\mathbb R}$,
normalized so that $\int_M K \omega^n = 0$.
If the action has at most twofold isotropy, then $\Lambda$
is essential in every compact subgroup
$G \subset {\rm Symp}(M,\omega)$ that contains it, unless there is
a symplectomorphism $g$ of $M$ that reverses ${\Lambda}$. In this case,
all the following hold:
\begin{enumerate}
\item $K(g(p)) = - K(p)$ for all $p \in M^{{\Lambda}}$.
\item There is a one-to-one correspondence between the
positive weights at $p$ and the negative weights at $g(p)$, and vice versa.
\item $g$ induces an isomorphism on the image of the restriction
map in equivariant cohomology $H^*_{S^1}(M) \longrightarrow H^*_{S^1}(M^{\Lambda})$.
\end{enumerate}
\end{corollary}
It is unknown whether the existence of such $g$ is necessary for ${\Lambda}$ to be inessential in ${\rm Symp}(M,{\omega})$.
We make partial progress towards answering this question in
\cite{MT}.
All the results in this paper are proved by
a case by case study of the structure of semisimple Lie algebras.
\section{Coadjoint orbits}\label{ss:coadj}
In this section, we prove Proposition~\ref{WQ}.
We begin with a brief review of a few facts about Lie groups.
Each simply connected compact semisimple Lie group is a product of simple
factors, and its center is the product of the centers of its
simple factors. Moreover,
since its Lie algebra splits into a corresponding sum, the
coadjoint orbits also are products of coadjoint orbits of simple groups.
Therefore, we may assume that $G$ is simple.
Let $G$ be a compact simple Lie group.
Let $\widetilde{G}$ denote the universal cover of $G$, and
${\widehat{G}}$ denote the quotient of $G$ by its center.
Let $\fg$ denote the Lie algebra of $G$, and
let $\ft$ denote the Lie algebra of a maximal torus
$T \subset G$.
Let $\ell \subset \ft$, $\Tilde{\ell} \subset \ft$, and $\widehat{\ell} \subset \ft$ be the lattices consisting of
vectors $\xi \in \ft$ whose exponential is the identity
in $G$, $\widetilde{G}$, and ${\widehat{G}}$, respectively.
There is a one-to-one correspondence between $\ell$ and circle subgroup
of $G$, $\Tilde{\ell}$ and circle subgroups
of $\widetilde{G}$, and $\widehat{\ell}$ and circle subgroups of ${\widehat{G}}$,
given by sending $\lambda$ to $t \rightarrow \exp(t\lambda)$.
Note that $\Tilde{\ell} \subseteq \ell \subseteq \widehat{\ell}$.
Because $\widetilde{G}$ is simply
connected,
$$
\pi_1(G) \;\cong \;\ell/\Tilde{\ell} \;\subseteq \;\widehat{\ell}/\Tilde{\ell} \;\cong \; \pi_1({\widehat{G}}).
$$
Let $\ft^*$ denote the dual to $\ft$, and
let $\Delta \subset \ft^*$ denote the set of {\bf roots} of $G$, i.e. the
nonzero weights of the adjoint action $T$ on $\fg_{\mathbb C}$,
where $\fg_{\mathbb C}$ is the complexification of $\fg$.
The lattice $\widehat{\ell}$ is dual to the lattice in $\ft^*$ generated by the
roots, i.e. ${\lambda}\in \widehat{\ell}$ precisely when $\eta({\lambda})\in {\mathbb Z}$ for all
$\eta\in \Delta$.
If we use the Killing form $(\cdot, \cdot)$ to identify
$\ft$ and $\ft^*$, then $\Tilde{\ell}$ is generated by the set
$$
\left\{ \left. \frac{2 \eta}{(\eta,\eta)}\ \right| \eta \in \Delta \right\}.
$$
Further the set of weights at any fixed point $p$
for the action of $T$ on $M$ is a nonempty subset of the set of roots.
Therefore
the result will follow if we find a representative $\lambda$
for each nontrivial class in $\widehat{\ell}/\Tilde{\ell}$ such that
$|\eta(\lambda)| \leq 1$ for every $\eta \in \Delta$.
We will check this on a case by case basis;
in each case we will use the Killing
form to identify $\ft$ and $\ft^*$.
Let $( \cdot, \cdot)$ be the standard metric on ${\mathbb R}^k$
with the standard basis $e_1,\dots,e_k$, and define
$$
\epsilon_i = e_i - \frac{1}{k} \sum_{j=1}^k e_j.
$$
{\medskip}
{\noindent}{\bf (I)}\,\,
For the group $A_n$, where $n \geq 1$,
$\ft = \ft^* = \left\{ \lambda \in {\mathbb R}^{n+1} \left| \ \sum \lambda_i = 0 \right. \right\}$
and the roots are $\epsilon_i - \epsilon_j = e_i - e_j$ for $i \neq j$.
Hence
$$
\widehat{\ell} = \{ \lambda \in \ft \mid \lambda_i - \lambda_j \in {\mathbb Z} \ \forall \ i,j \},
\ \mbox{and} \quad
\Tilde{\ell} = \ft \cap {\mathbb Z}^{n+1}.
$$
As representatives for the quotient $\widehat{\ell}/\Tilde{\ell} \cong {\mathbb Z}/(n+1)$, we take
$\lambda = \sum_{i=1}^k \epsilon_i$ for $0 \leq k \leq n$.
{\medskip}
{\noindent}{\bf (II)}\,\,
For the group $B_n$, where $n \geq 2$, $\ft^* = {\mathbb R}^n$ and
the roots are $\pm e_i$ and $\pm e_i \pm e_j$ for $i \neq j$.
Hence
$$
\widehat{\ell} = {\mathbb Z}^n,\ \mbox{and} \qquad
\Tilde{\ell} = \left\{ \lambda \in {\mathbb Z}^n
\left| \ \sum \lambda_i \in 2{\mathbb Z} \right. \right\}.
$$
As representatives of the quotient $\widehat{\ell}/\Tilde{\ell} \cong {\mathbb Z}/(2) $,
we take $0$ and $e_1$.
{\medskip}
{\noindent}{\bf (III)}\,\,
For the group $C_n$, where $n \geq 3$, $\ft^* = {\mathbb R}^n$
and the roots are $\pm 2 e_i$ and $\pm e_i \pm e_j$ for $i \neq j$.
Hence
$$
\widehat{\ell} = \{ \lambda \in {\mathbb R}^n \mid \lambda_i \pm \lambda_j \in {\mathbb Z},
\ \forall \ i,j \},
\ \mbox{and} \qquad
\Tilde{\ell} = {\mathbb Z}^n.
$$
As representatives of the quotient $\widehat{\ell}/\Tilde{\ell} \cong {\mathbb Z}/(2)$ ,
we take $0$ and $\frac{1}{2} \sum_{i=1}^n e_i$.
{\medskip}
{\noindent}{\bf (IV)}\,\,
For the group $D_n$, where $n \geq 4$, $\ft^* = {\mathbb R}^n$ and the
roots are $\pm e_i \pm e_j$ for $i \neq j$. Hence
$$
\widehat{\ell} =
\{ \lambda \in {\mathbb R}^n \mid \lambda_i \pm \lambda_j \in {\mathbb Z}, \ \forall \ i,j \},
\ \mbox{and} \qquad
\Tilde{\ell} = \left\{ \lambda \in {\mathbb Z}^n
\left| \ \sum \lambda_i \in 2{\mathbb Z} \right. \right\}.
$$
The quotient $\widehat{\ell}/\Tilde{\ell}$ is isomorphic to
${\mathbb Z}/(2) \oplus {\mathbb Z}/(2)$ if $n$ is even, and
to ${\mathbb Z}/(4)$ if $n$ is odd.
Either way, as representatives of $\widehat{\ell}/\Tilde{\ell}$, we take
$0$, $e_1$, $\frac{1}{2}\sum_{i=1}^n e_i$
and $\frac{1}{2}\sum_{i=1}^n e_i - e_n$.
{\medskip}
{\noindent}{\bf (V, a)}\,\,
For the group $E_6$, $\ft^* = {\mathbb R}^6$
and the roots are $2 \epsilon$, $\epsilon_i - \epsilon_j$,
and $\epsilon_i + \epsilon_j + \epsilon_k \pm \epsilon$
for $i, j,$ and $k$ distinct,
where $\epsilon = \frac{1}{2 \sqrt{3}}(1,1,1,1,1,1)$.
Hence,
$$
\widehat{\ell} =
\Big\{ n \epsilon + (\xi_1,\ldots,\xi_6) \in {\mathbb R}^6 \Big|
\sum_{i=1}^6 \xi_i = 0, n \in {\mathbb Z}, \
\frac{n}{2} + 3 \xi_i \in {\mathbb Z} \ \mbox{and} \
\xi_i - \xi_j \in {\mathbb Z} \ \forall \ i,j \Big\},
\ \mbox{and} $$
$$
\Tilde{\ell} =
\Big\{ n \epsilon + (\xi_1,\ldots,\xi_6) \in {\mathbb R}^6 \Big|
\sum_{i=1}^6 \xi_i = 0, n \in {\mathbb Z} \ \mbox{and} \
\frac{n}{2} + \xi_i \in {\mathbb Z} \ \forall \ i \Big\}.
$$
As representatives of the quotient $\widehat{\ell}/\Tilde{\ell} \cong {\mathbb Z}/(3)$ ,
we take $0$, $\epsilon_1 + \epsilon_2$, and $-\epsilon_1 - \epsilon_2$.
{\medskip}
{\noindent}{\bf (V, b)}\,\,
For the group $E_7$,
$\ft = \ft^* = \left\{ \lambda \in {\mathbb R}^{8} \mid \sum \lambda_i = 0 \right\}$,
and the roots are $\epsilon_i - \epsilon_j$, and
$\epsilon_i + \epsilon_j + \epsilon_k + \epsilon_l$ for
$i, j, k,$ and $l$ distinct. Hence
$$\widehat{\ell} =
\{ \lambda \in \ft \mid
4 \lambda_i \in {\mathbb Z} \ \mbox{and} \ \lambda_i - \lambda_j \in {\mathbb Z}\ \forall\ i, j \},
\ \mbox{and} \quad \Tilde{\ell}
= \{ \lambda \in \ft \mid
\lambda_i \pm \lambda_j \in {\mathbb Z}\ \forall \ i, j
\}.
$$
As representatives for the quotient $\widehat{\ell}/\Tilde{\ell} \cong {\mathbb Z}/2{\mathbb Z}$, we take
$0$ and $\epsilon_1 + \epsilon_2$.{\medskip}
Every group of type $E_8$, $F_4$, and $G_2$ is simply connected,
so no further argument is necessary.
\hfill$\Box$\medskip
\section{Lie Group Actions}\label{sec:lie}
This section contains proofs of Theorems \ref{thm:main0} and \ref{thmtwo}. We begin by stating a lemma about root systems, that is proved at the end. We shall
always assume that the positive Weyl chamber is closed.
\begin{lemma}\labell{claims}
Let $G$ be a simply connected compact simple Lie group.
Let $\ft$ be the Lie algebra of a maximal torus $T \subset G$.
Let $\Tilde{\ell}$ be the integral lattice, let $\Delta$ denote the set of roots,
and let $W$ denote the Weyl group. Use the Killing form to identify $\ft$ and
$\ft^*$.
Fix $\lambda \in \Tilde{\ell}$.
Choose a positive Weyl chamber which contains $\lambda$.
Let $\delta \in \Delta$ denote the highest root.
Then the following claims hold:
\smallskip
{\noindent}{\rm (a)}
If $(\lambda,\delta) \leq 2$,
then there exist orthogonal roots $\eta_1,\ldots,\eta_k \in \Delta$
so that $\lambda = \sum a_i \eta_i$ and so that
$(\lambda,\eta_i) = a_i (\eta_i,\eta_i) = 2$ for all $i$.
\smallskip
{\noindent}{\rm (b)}
Let $L \subset \Delta$ be a set of roots
which contains every root $\eta \in \Delta$
such that $\delta + \eta$ or $\delta - \eta$ is also a root. Assume
also that $L$ is closed under addition, that is, it
contains every root which can be written as the sum of
roots in $L$. Then $L$ contains all roots.
\smallskip
{\noindent}{\rm (c)}
If $(\lambda,\delta) > 2$ and $-{\rm id}: \ft \longrightarrow \ft$ is not an
element of the Weyl group,
then for every nonzero weight $\alpha \in \Tilde{\ell}^*$
there exists $\sigma \in W$ so that $|(\sigma \cdot \alpha,\lambda)| > 1$.
\smallskip
{\noindent}{\rm (d)}
If $-{\rm id}: \ft \longrightarrow \ft$ is not an
element of the Weyl group,
then $\delta$ is the only root which lies in the positive Weyl chamber.
\end{lemma}
Using this result, we can find find elements which reverse certain
circle subgroups of simply connected compact simple Lie groups.
Note that because $G$
is simply connected, every circle subgroup of $G$ is inessential in $G$.
\begin{lemma} \labell{le:simple}
Let $\Lambda $ be a circle subgroup of a simply connected compact
simple Lie group $G$.
\smallskip
{\noindent}{\rm (i)}
Let $\rho : G \longrightarrow {\rm GL}(V)$ be a nontrivial representation of $G$.
If $\Lambda$ acts semifreely on $V$ then there exists
$g \in G$ that reverses $\Lambda$.
\smallskip
{\noindent}{\rm (ii)}
Let $H \varsubsetneq G$ be a proper subgroup containing $\Lambda$.
If the adjoint action of $\Lambda$ on $\fg/{\mathfrak h}$ is semifree,
then there exists $h \in H$ that reverses $\Lambda$.
\smallskip
{\noindent}{\rm (iii)}
Let $H \varsubsetneq G$ be a proper subgroup containing a maximal
torus which contains $\Lambda$.
If the natural action of $\Lambda$ on $G/H$ has at most twofold isotropy,
then there exists $g \in G$ that reverses $\Lambda$.
\end{lemma}
The assumption in (ii) above is a special case of (i) since
the representation $V$ is restricted; however,
the conclusion is stronger since it asserts that the reversor lies
in $H$. Statement (ii) and (iii) are also related: the former makes a strong
assumption about
the action induced by ${\Lambda}$ on the tangent space to $G/H$
at the fixed point $eH$, the latter makes a weaker assumption about the
action at all the fixed points on $G/H$.
We will now use the claims in Lemma \ref{claims} to prove Lemma \ref{le:simple}.
Let $T$ be a maximal torus which contains $\Lambda$.
Let $\Tilde{\ell} \subset \ft$ denote the integral lattice.
Let ${\lambda} \in\Tilde{\ell}$ be the vector corresponding to $\Lambda$.
Choose a positive Weyl chamber which contains $\lambda$.
Let $\delta \in \Delta$ denote the highest root.
Recall that the Weyl group $W$ is the quotient $N(T)/T$, where $N(T)$ is the
the normalizer of $T$ in $G$.
Every root $\eta$ gives rise to an element $w_\eta \in W$
whose action on $\ft^*$ is given by
$w_\eta(\beta) = \beta - \frac{2(\eta,\beta)}{(\eta,\eta)} \eta$.
{\medskip}
{\noindent}
{\bf Proof of Lemma~\ref{le:simple} (i).}
Let $\rho : G \longrightarrow {\rm GL}(V)$ be a nontrivial representation of $G$.
Assume that $\Lambda$ acts semifreely on $V$.
Suppose first that $(\lambda,{\delta}) \leq 2$.
By claim (a), there exist
orthogonal roots $\eta_1,\ldots,\eta_k \in \Delta$ so that
$\lambda = \sum a_i \eta_i$.
Since the roots are orthogonal,
for each $\eta_i$ the associated element of the Weyl group $w_{\eta_i}$
takes $\eta_i$ to $-\eta_i$ and leaves $\eta_j$ fixed for
all $j \neq i$. Hence, their product
$w = w_{\eta_1} \cdots w_{\eta_n}$ takes
$\lambda$ to $-\lambda$, and so reverses ${\Lambda}$.
So assume instead that
$(\lambda,{\delta}) > 2$.
If $-{\rm id}$ is in the Weyl group, then statement (i) is trivial.
So we assume that it is not.
Let $T$ act on $V$ via restriction,
and pick any nonzero weight $\alpha \in \Tilde{\ell}^*$ in the weight
decomposition.
By claim (c), we can find some $\sigma \in W$
such that $|(\sigma \cdot \alpha, \lambda) | >1.$
Since $\sigma \cdot \alpha$ also
appears in the weight decomposition,
this contradicts the assumption that the action of $\Lambda$ on $V$
is semifree.\hfill$\Box$\medskip
{\noindent}
{\bf Proof of Lemma~\ref{le:simple} (ii).}
Let $H \subseteq G$ be a proper subgroup which contains $\Lambda$.
Assume that the adjoint action of $\Lambda$ on $\fg/{\mathfrak h}$ is semifree.
Let $L$ be the set of roots $\eta \in \Delta$ so that the
associated weight space $E_\eta \subset \fg_{\mathbb C}$ lies in ${\mathfrak h}_{\mathbb C}$.
Clearly, if $|(\eta, \lambda)| > 1$, then $\eta \in L$.
Suppose first that $(\lambda,{\delta}) \leq 2$.
By claim (a), there exist orthogonal
roots $\eta_1,\ldots,\eta_k \in \Delta$
so that $\lambda = \sum a_i \eta_i$ and so
that $(\lambda,\eta_i) = 2$ for every $i$.
Since $(\eta_i,\lambda) = 2$,
$\eta_i$ lies in $L$ for all $i$.
Hence, the associated element of the Weyl group
$w_{\eta_i}$ lies in $H$ for all $i$.
Thus $w = w_{\eta_1} \cdots w_{\eta_k}$ must lie in $H$.
So assume instead that $(\lambda,\delta) > 2$.
We see immediately that $\delta$ and $-\delta$ lie in $L$.
If $\eta$, $\eta'$ and $\eta + \eta'$
are all roots, then $[E_\eta,E_\eta'] = E_{\eta + \eta'}$.
Hence, since ${\mathfrak h}_{\mathbb C}$ is closed under Lie bracket,
if $\eta$ and $\eta'$ are in $L$ then $\eta + \eta' \in L$ also,
that is, $L$ is closed under addition.
Additionally, if $\eta$ and $\eta'$ are
roots such that $\delta = \eta + \eta'$, then either
$(\lambda,\eta) > 1$ or $(\lambda,\eta') > 1$.
If the former holds, then $\eta$ and $-\eta$ lie in $L$.
Since $L$ is closed under addition, so do $\eta' $ and $-\eta'$.
The other case is identical.
Thus, claim (b) implies that every root lies in $L$.
This contradicts the claim that $H$ is a proper subgroup.\hfill$\Box$\medskip
{\noindent}
{\bf Proof of Lemma~\ref{le:simple} (iii).}
Let $H \subsetneq G$ be a proper subgroup which contains the
maximal torus $T$, and assume that the
natural action of
${\Lambda}\subset T$
on $G/H$ has at most twofold isotropy.
If $(\lambda,\delta) \leq 2$, then part (iii)
follows by the argument used to prove part (i).
So assume that $(\lambda, \delta) > 2$.
We may also assume that
$-{\rm id}$ does not lie in the Weyl group,
because otherwise the claim is trivial.
Since $H \subset G$ is proper, there exists at least one root
$\eta$ so that the associated weight space $E_\eta$ is not contained in ${\mathfrak h}_{\mathbb C}$.
Then there is $\sigma \in W$ so that the root
$\sigma \cdot \eta$ lies in the
positive Weyl chamber. Hence by (d) $\sigma \cdot \eta = {\delta}$, and so
$|(\sigma \cdot \eta,\lambda)| > 2.$
Choose $\tilde{\sigma} \in N(T)$
which descends to $\sigma$. Then $\tilde{\sigma} H$ is
a fixed point for $T$, and $\sigma \cdot \eta$ is one of
the weights for ${\Lambda}$
at this fixed point.
This contradicts the fact that the action has at most twofold isotropy.\hfill$\Box$\medskip
We are now ready to deduce Theorems~\ref{thm:main0} and~\ref{thmtwo}.
In both cases, we will do this by proving the contrapositive,
that is, we will assume that $\Lambda$ is an inessential circle subgroup
and use this to construct a reversor.
Let $\widetilde{G}$ denote the universal cover of $G$. Then
$\widetilde{G}$ is the direct product of a compact simply connected
semisimple Lie group and a vector space.
Since $\Lambda$ is inessential, it
lifts to a circle subgroup of $\widetilde{G}$.
Since this lift must lie in the compact part of $\widetilde{G}$, we
may assume without loss of generality that
$\widetilde{G}$ is a compact simply connected semisimple Lie group.
In fact, it is enough to prove these claims for the universal
cover of $G$, as long as we no longer insist on an effective action
but instead allow a finite number of elements of the group to act trivially
on $M$.
Thus we may assume that $G$ is the product of compact simple and
simply connected groups
$G_1 \times \cdots \times G_n$.
Let $\Lambda_i$ be the projection of $\Lambda$ to $G_i$.
Without loss of generality, we may assume that ${\Lambda}_i\ne \{{\rm id}\}$ for
all $i$.
{\medskip}
{\noindent}
{\bf Proof of Theorem~\ref{thm:main0}.}
Let $G = G_1\times \cdots\times G_n$ as above.
Choose $p\in F$ and let
$H \subset G$ be the stabilizer of $p$. Then ${\Lambda}\subset H$.
There exists a representation $V$ of $H$, called the {\bf isotropy
representation}, so that a neighborhood of the $G$-orbit
through $p$ is equivariantly diffeomorphic to
a neighborhood of the zero section of $G \times_H V$.
Fix some simple factor $G_i$, and let $H_i = H \cap G_i$.
Assume first that $H_i$ is a proper subgroup.
Note that $\fg_i$ is invariant under the action of $\Lambda$.
Thus, since $\Lambda$ acts semifreely on $\fg/{\mathfrak h}$ via the adjoint action,
$\Lambda_i$ acts semifreely on $\fg_i/{\mathfrak h}_i$.
Thus, by Lemma~\ref{le:simple} (ii)
there exists an element $h_i \in H_i$ that reverses $\Lambda_i$.
So assume on the contrary that $H_i = G_i$.
Let $\Lambda'$ be the projection of $\Lambda$ onto
the product of all the simple factors except $G_i$.
Since ${\Lambda}_i\subset G_i\subset H$ and ${\Lambda}\subset H$,
we must have ${\Lambda}'\subset H$. Hence ${\Lambda}'$ acts on $V$.
For any integer $k$, let $V_k$ denote the subspace of
$V$ on which $\Lambda'$ acts with weight $k$.
Since $\Lambda'$ commutes with $G_i$, $V_k$
is a representation of $G_i$.
Since only a finite number of elements of $G$ act trivially
on $M$, $G_i$ must act nontrivially on
$G \times_H V$, and hence also on $V$.
Therefore, there is some $k$ so that the
representation of $G_i$ on $V_k$ is nontrivial.
Because $G_i$ is simple,
$\Lambda_i$ must act with both positive
and negative weights on $V_k$.
But the weights for the action of $\Lambda$ on
$V_k$ are the weights for the action of $\Lambda_i$ shifted by $k$.
Hence,
because $F$ is a semifree fixed point component,
$k = 0$ and the action of $\Lambda_i$
on $V_k$ is itself semifree.
Therefore by Lemma~\ref{le:simple} (i) there exists
$h_i \in G_i = H_i$ that reverses $\Lambda_i$.
Since $h_i$ reverses $\Lambda_i$ for each
$i$, $g = (h_1,\ldots,h_n)$ reverses $\Lambda$, as required.
Moreover, since
$H_1 \times \cdots \times H_n\subset H$
(in general they are not equal), $g$
lies in $H$, and hence fixes $p$.
\hfill$\Box$\medskip
{\noindent}
{\bf Proof of Theorem~\ref{thmtwo}.}
Fix some simple factor $G_i$.
Let $W$ be the Weyl group of $G_i$.
Let $T \subset G_i$ be a maximal torus of $G_i$ containing $\Lambda_i$.
Let $\Phi: M \longrightarrow \ft^*$ be the moment map for the $T$-action.
Pick any $\xi \in \ft$ so that the one parameter subgroup
generated by $\xi$ is dense in $T$.
Let $p$ be any point which maps to
the minimum value of $\Phi^\xi$, the
component of $\Phi$ in the direction $\xi$.
By construction, $p$ is a fixed point for $T$.
Assume first that $\Phi^\xi(p) = 0$, that is,
the function $\Phi^\xi$ is nonnegative on
$M$. Since the moment polytope $\Phi(M)$ is
invariant under the Weyl group $W$,
this implies that
$\Phi^{{\sigma} \cdot \xi}$ is also nonnegative on $M$
for all ${\sigma}\in W$.
Because $G_i$ is simple and $\xi$ is a generic point of $\ft$,
for any nonzero $x\in \ft^*$ there exists an
element ${\sigma}\in W$ such that $({\sigma}\cdot\xi,x) < 0$.
Applying this to $x\in \Phi(M){\smallsetminus} \{0\}$, we see that
$\Phi(M)$ must be the single point $\{0\}$,
which is impossible, because the action is effective.
Therefore, $\Phi(p) \neq 0$.
Now let us reconsider the action of $G$ on $M$.
Let $H$ be the stabilizer of $p$ in $G$, and let $H_i = H \cap G_i$.
Since $\Lambda$ acts with
at most twofold isotropy on
$G/H\subset M$, $\Lambda_i$
acts with at most twofold isotropy on $G_i/H_i$.
Since $\Phi(p)$ is not zero, the stabilizer of $\Phi(p)$ in $G_i$.
is a proper subgroup of $G_i$. Since $\Phi$ is equivariant,
this implies that $H_i$ is a proper subgroup of $G_i$.
By Lemma \ref{le:simple} (iii), this implies that
there exists $g_i \in G_i$ which reverses $\Lambda_i$.
Then $(g_1,\ldots,g_n)$ reverses ${\Lambda}$.
\hfill$\Box$\medskip
{\noindent}
{\bf Proof of Lemma \ref{claims}.}
We now prove claims (a)-(d) on a case by case basis,
using the classification of compact simple Lie groups.
We will use the notation of
\S\ref{ss:coadj}.
Note, however, that here $G =\widetilde{G} $ since $G$ is simply connected.
{\medskip}
{\noindent}{\bf (I)}\,\,
Recall that for the group $A_n$, where $n \geq 1$, $\ft = \ft^*
= \{ \xi \in {\mathbb R}^n \mid \sum \xi_i = 0 \}$, the
roots are $\epsilon_i - \epsilon_j
= e_i-e_j$
for $i \neq j$, and
the integral lattice is $\Tilde{\ell} = {\mathbb Z}^{n+1} \cap \ft$.
The positive Weyl chamber is\footnote
{For uniformity, we shall always
use the lexigraphical order to choose the positive Weyl chamber.}
$$
\{\xi \in \ft \mid
\xi_1 \geq \cdots \geq \xi_{n+1}\}.
$$
The highest root is $\delta = e_1-e_{n+1}$.
If $(\lambda,\delta) = \lambda_1 - \lambda_{n+1} \leq 2$,
then $|\lambda_i| \leq 1$
for all $i$. Since $\sum_i \lambda_i = 0$ and $\lambda_i \in {\mathbb Z}$ for all $i$,
there are an equal number of $+1$'s and $-1$'s, and the rest are $0$'s.
Hence, $\lambda$ is the sum of orthogonal roots of the form
$\eta = e_i - e_j$. Since $(\eta,\eta) = 2$,
this proves claim (a).
Since $\delta = (e_1 - e_k) + (e_k - e_{n+1})$,
the roots $\pm (e_1 - e_k)$ and $\pm (e_k - e_{n+1})$ lie in $L$ for
all $1 < k < n+1$.
If neither $i$ nor $j$ is equal equal to $1$, then
$e_i - e_j = -(e_1 - e_i) + (e_1 - e_j)$ is also in $L$.
This proves claim (b).
We now prove (c).
The weight lattice is
$\Tilde{\ell}^* = \{ \alpha \in \ft \mid \alpha_i - \alpha_j \in {\mathbb Z} \ \;
\forall \ i, j \}$.
By permuting the coordinates of $\alpha$,
we may assume $\alpha_1 \geq \cdots \ge \alpha_n$.
Since $\alpha \neq 0$, there exists $k \in (1,\ldots,n)$
such that $\alpha_{k} - \alpha_{k+1} >0$; since
this difference lies in ${\mathbb Z}$, it must be at least $1$.
Since $\lambda_1 - \lambda_{n+1} = \lambda_1 + \sum_{i=1}^n \lambda_i > 2$ and
$\lambda_i \geq \lambda_{i + n - k}$,
$\sum_{i=1}^k \lambda_i + \sum_{i = 1}^{n+1 - k} \lambda_i > 2$.
Therefore, either
$\sum_{i=1}^k \lambda_i >1$ or $ \sum_{i =1}^{n+1-k} \lambda_i > 1$.
In the former case,
$$
(\alpha,\lambda) = \sum_{j=1}^n \left( (\alpha_j - \alpha_{j+1}) \sum_{i=1}^j
\lambda_i \right) \geq (\alpha_{k} - \alpha_{k+1}) \sum_{i=1}^k \lambda_i > 1.
$$
In the latter case, let $\alpha'$ be obtained from $\alpha$ by
the permutation which
reverses the coordinates, so that $\alpha'_i = \alpha_{n + 2 - i}$.
Then
$$
(\alpha',\lambda) = \sum_{j=1}^n \left( (\alpha_{n + 2 - j} - \alpha_{n + 1 - j})
\sum_{i=1}^j
\lambda_i \right) \leq (\alpha_{k + 1} - \alpha_{k}) \sum_{i=1}^{n + 1 - k}
\lambda_i < -1.
$$
The only facts we have used are that
$\ft = \{ \xi \in {\mathbb R}^n \mid \sum_i\xi_i = 0 \}$,
that the Weyl group contains the permutation group $S_n$, and that
$\alpha_i - \alpha_j \in {\mathbb Z}$ for any $\alpha \in \Tilde{\ell}^*$.
Finally, $\delta$ is the only root in the positive Weyl chamber.
{\medskip}
{\noindent}{\bf (II)}\,\,
Recall that for the group $B_n$, where $n \geq 2$,
$\ft = \ft^* = {\mathbb R}^n$, the roots are $\pm e_i$ and $\pm e_i \pm e_j$ for $i \neq j$,
and the integral lattice is $\Tilde{\ell} = \{ \xi \in {\mathbb Z}^n \mid
\sum_i \xi_i \in 2 {\mathbb Z} \}.$
The positive Weyl chamber is
$\{\xi \in \ft \mid \xi_1 \geq \cdots \geq \xi_n \geq 0 \}$.
The highest root is $\delta = e_1 + e_2$.
If $(\lambda,\delta) =
\lambda_1 + \lambda_2 \leq 2$, then either
$\lambda_1 = 2$ and $\lambda_i = 0$ for all $i
\neq 1$, or $\lambda_i \leq 1$ for all $i$.
Either way, since $\sum_i \lambda_i \in 2 {\mathbb Z}$,
we can write $\lambda$ as the sum
of orthogonal roots $\eta_i$ such that $(\eta_i,\eta_i) = 2$.
Since $\delta = (e_1 - e_k) + (e_2 + e_k) = (e_1 + e_k) + (e_2 - e_k)$,
the roots $\pm e_1 \pm e_k$ and $\pm e_2 \pm e_k$
lie in $L$ for $k \neq 1 $ or $2$.
Since $\delta = (e_1) + (e_2)$, the roots $\pm e_1$ and $\pm e_2$ lie in $L$.
Every root can be written as a sum of these roots.
Since $-{\rm id}$ lies in the Weyl group, we are done.
{\medskip}
{\noindent}{\bf (III)}\,\,
Recall that for the group $C_n$, where $n \geq 3$, $\ft = \ft^* = {\mathbb R}^n$,
the roots are $\pm 2 e_i$ and $\pm e_i \pm e_j$ for $i \neq j$, and the
integral lattice is $\Tilde{\ell} = {\mathbb Z}^n$.
The positive Weyl chamber is
$\{\xi \in \ft \mid \xi_1 \geq \cdots \geq \xi_n \geq 0\ \}$.
The highest root is $\delta = 2 e_1$.
If $(\lambda,\delta) = 2 \lambda_1 \leq 2$, then
$\lambda_i \leq 1$ for all $i$.
Since $\lambda \in {\mathbb Z}^n$, we can write $\lambda$
as half the sum of orthogonal roots of the form $ 2 e_i$.
Note that $(\lambda, 2 e_i) = 2$.
Since $\delta = (e_1 - e_k) + (e_1 + e_k)$,
the roots $\pm e_1 \pm e_k$
lie in $L$ for $k \neq 1 $.
Every root can be written as a sum of these roots.
Since $-{\rm id}$ lies in the Weyl group, we are done.
{\medskip}
{\noindent}{\bf (IV)}\,\,
Recall that for the group $D_n$, where $n \geq 4$,
$\ft = \ft^* = {\mathbb R}^n$, the roots are $\pm e_i \pm e_j$ for $i \neq j$,
and the integral lattice is
$\Tilde{\ell} =
\{ \xi \in {\mathbb Z}^n \mid \sum \xi_i \in 2{\mathbb Z} \}.$
The positive Weyl chamber is
$\{ \xi \in \ft \mid \xi_1 \geq \cdots \geq \xi_{n-1} \geq |\xi_n| \}$.
The highest root is $\delta = e_1 + e_2$.
If $(\lambda,\delta) =
\lambda_1 + \lambda_2 \leq 2$, then either
$\lambda_1 = 2$ and $\lambda_i = 0$ for all $i
\neq 1$, or $|\lambda_i| \leq 1$ for all $i$.
Either way, since $\sum_i \lambda_i \in 2 {\mathbb Z}$,
we can write $\lambda$ as the sum
of orthogonal roots $\eta_i$ such that $(\eta_i,\eta_i) = 2$.
Since $\delta = (e_1 - e_k) + (e_2 + e_k) = (e_1 + e_k) + (e_2 - e_k)$,
the roots $\pm e_1 \pm e_k$ and $\pm e_2 \pm e_k$
lie in $L$ for $k \neq 1 $ or $2$.
Every root can be written as a sum of these roots.
Now assume that $(\delta,\lambda) = \lambda_1 + \lambda_2 > 2$.
Consider a nonzero weight $\alpha \in \Tilde{\ell}^* = \{ \alpha \in {\mathbb R}^n \mid
\alpha_i \pm \alpha_j \in {\mathbb Z} \ \forall\
i, j \}.$
By applying the Weyl group, we may assume $\alpha$ lies in the positive Weyl chamber.
Since $\lambda$ also lies in the positive Weyl chamber,
$\alpha_i \lambda_i \geq 0$ for all $i \neq n$.
Moreover, since $\alpha_{n-1} \geq |\alpha_n|$, and
$\lambda_{n-1} \geq |\lambda_n|$, $\alpha_{n-1} \lambda_{n-1} +
\alpha_n \lambda_n \geq 0$.
Therefore, $\alpha_3 \lambda_3 + \cdots + \alpha_n \lambda_n \geq 0.$
(Here, we have used that $n \geq 4$.)
Since $\alpha$ is nonzero, either $\alpha_1 \geq 1$,
or $\alpha_1 = \alpha_2 = \frac{1}{2}$.
In either case, $\alpha_1 \lambda_1 + \alpha_2 \lambda_2 > 1$ .
(In the first case, we use the fact that $\lambda_1 + \lambda_2 > 2$ and
$\lambda_1 \geq \lambda_2$ implies that $\lambda_1 > 1$.)
Therefore,
$(\alpha,\lambda) \geq \alpha_1 \lambda_1 + \alpha_2 \lambda_2 > 1$,
This proves claim (c).
Finally, $\delta$ is the only root in the positive Weyl chamber.
{\medskip}
{\noindent}{\bf (V, a)}\,\,
Recall that for the group $E_6$, $\ft = \ft^* = {\mathbb R}^6$
and the roots are $2 \epsilon$, $\epsilon_i - \epsilon_j$,
and $\epsilon_i + \epsilon_j + \epsilon_k \pm \epsilon$
for $i, j,$ and $k$ distinct,
where $\epsilon = \frac{1}{2 \sqrt{3}}(1,1,1,1,1,1)$. Therefore
$$
\Tilde{\ell} = \Big\{ n\epsilon + (\xi_1,\ldots,\xi_6)\; \Big|\; \sum_{i=1}^6 \xi_i = 0, n \in {\mathbb Z},
\mbox{and} \ \frac{n}{2}+ \xi_i \in {\mathbb Z} \ \forall \ i \Big\}.
$$
The positive Weyl chamber is
$$
\Bigl\{ n \epsilon + (\xi_1,\ldots,\xi_6) \in \ft\; \Big| \;
\sum_{i=1}^6 \xi_i = 0, \xi_2 \geq \cdots \geq \xi_6,\; \xi_1 +
\xi_5 + \xi_6 \geq n/2 \geq 0\Bigr\}.
$$
(Note that these conditions imply $\xi_1 \geq \xi_2$.)
The highest root is $\delta = \epsilon_1 - \epsilon_6$.
Write
$\lambda = n\epsilon + (\xi_1,\ldots,\xi_6)$,
where $\sum_i \xi_i = 0$.
Assume that $(\lambda,\delta) = \xi_1 - \xi_6 \leq 2$.
Combining the inequalities
$\xi_1 - \xi_6 \leq 2$,
$\xi_4 \geq \xi_5$, $\xi_4 \geq \xi_6$, and
$\xi_1 + \xi_5 + \xi_6 \geq \frac{n}{2}$,
we see that $\xi_4 \geq \frac{n-4}{6}$.
Since also $\xi_2 + \xi_3 + \xi_4 \leq 0$,
$\xi_2 \geq \xi_4$, and $\xi_3 \geq \xi_4$, we have
$\xi_4 \leq 0$.
Moreover, in both cases, if the final inequality in the
sentence is an equality, so are all the preceding ones.
Since $n \geq 0$, $0 \geq \xi_4 \geq -\frac{4}{6}$.
Since $\lambda \in \Tilde{\ell}$, $\xi_4 = 0$ or $\xi_4 = -\frac{1}{2}$.
In the former case, $\xi_2 = \xi_3 = \xi_4 = 0$, so
$\xi_1 + \xi_5 + \xi_6 = 0$, so $n = 0$.
Hence, $\lambda = (\epsilon_1 - \epsilon_6).$
In the latter case, $n$ is odd, so $\xi_4 = -\frac{1}{2} \geq \frac{n-4}{6}$
implies that $n = 1$.
In this case, $\lambda = (\epsilon_1 - \epsilon_6)
+ (\epsilon + \epsilon_1 + \epsilon_2 + \epsilon_6)$.
This proves claim (a).
Since $\delta = (\epsilon_1 -\epsilon_i) + (\epsilon_i - \epsilon_6),$
the roots $\pm (\epsilon_1 - \epsilon_i)$ and $\pm(\epsilon_i -
\epsilon_6)$
lie in $L$ for all $1 < i < 6$.
Moreover, $\delta = (\epsilon + \epsilon_1 + \epsilon_i + \epsilon_j)
- (\epsilon + \epsilon_i + \epsilon_j +\epsilon_6)$,
so the roots $\pm(\epsilon + \epsilon_1 + \epsilon_i + \epsilon_j)$
and
$\pm(\epsilon + \epsilon_i + \epsilon_j + \epsilon_6)$
lie in $L$ for all $1 < i < j < 6$.
Since, for example, $\epsilon + \epsilon_1 + \epsilon_2 + \epsilon_3 =
\epsilon - \epsilon_4 - \epsilon_5 - \epsilon_6$,
it follows easily that $L$ contains all roots.
Let $\alpha \in \Tilde{\ell}^*$ be a nonzero weight.
Write $\lambda = n{\varepsilon} + \xi$ as before.
By applying the Weyl group, we may assume that
$\alpha = m\epsilon + (\zeta_1,\ldots,\zeta_6)$ is in the positive Weyl
chamber.
Since $(\alpha,\lambda) \geq (\zeta,\xi)$, it is enough
to show that $(\zeta,\xi) > 1$.
This fact now follows
from the argument from $A_5$,
since $(\delta,{\lambda}) = ({\delta},\xi)> 2$,
since the
Weyl group contains the permutation group $S_5$,
and since $\zeta$ must
satisfy $\zeta_i - \zeta_j \in {\mathbb Z}$.
Finally, $\delta$ is the only root in the positive Weyl chamber.
{\medskip}
{\noindent}{\bf (V, b)}\,\,
Recall that for the group $E_7$, $\ft =
\ft^* = \{ \xi \in {\mathbb R}^8 \mid \sum \xi_i = 0 \}$,
the roots are $\epsilon_i - \epsilon_j$
and $\epsilon_i + \epsilon_j + \epsilon_k + \epsilon_l$ for $i,j,k,$ and $l$
distinct, and the integral lattice is
$\Tilde{\ell} = \{\xi \in \ft \mid \xi_i \pm \xi_j \in {\mathbb Z} \ \forall \ i,j \}.$
The positive Weyl chamber is
$$
\{\xi \in \ft \mid \xi_2 \geq \cdots \geq \xi_8 \ \mbox{and}\ \xi_1 + \xi_6
+ \xi_7 + \xi_8 \geq 0 \}.
$$
(Note that this automatically implies that $\xi_1 \geq \xi_2$.)
The highest
root is $\delta = \epsilon_1 - \epsilon_8$.
Assume that $(\delta,\lambda) = \lambda_1 - \lambda_8 \leq 2$.
Combining the inequalities $\lambda_1 - \lambda_8 \leq 2$,
$\lambda_1 + \lambda_6 + \lambda_7 + \lambda_8 \geq 0$,
and $\lambda_5 \geq \lambda_i$ for $i = 6,7$ and $8$, we see
that $\lambda_5 \geq -\frac{1}{2}$.
Since also $\lambda_2 + \lambda_3 + \lambda_4 + \lambda_5 \geq 0$
and $\lambda_i \geq \lambda_5$ for $i = 2,3$ and $4$, $\lambda_5 \leq 0$.
Moreover, in both cases, if the last inequality in the
sentence is an equality, all the inequalities are equalities.
Since $\lambda \in \ell$,
the only possibilities are $\lambda_5 = 0$ or $\lambda_5 = -\frac{1}{2}.$
In the former case, we must have $\lambda = \epsilon_1 - \epsilon_8$.
In the latter case, the only possibilities
are
$\lambda = (\epsilon_1 + \epsilon_2 + \epsilon_3 + \epsilon_4) +
(\epsilon_1 - \epsilon_4)$, or
$\lambda = (\epsilon_1 + \epsilon_2 + \epsilon_3 + \epsilon_4) +
(\epsilon_1 - \epsilon_4) + (\epsilon_2 - \epsilon_3)$.
The proves claim (a).
Since $\delta = (\epsilon_1 - \epsilon_i) + (\epsilon_i - \epsilon_8)$,
the roots $\pm(\epsilon_1 - \epsilon_i)$ and $\pm(\epsilon_i - \epsilon_8)$
lie in $L$ for all $1<i<8.$
Since $\delta = (\epsilon_1 + \epsilon_i + \epsilon_j + \epsilon_k)
- (\epsilon_i + \epsilon_j + \epsilon_k + \epsilon_8)$,
the roots
$\pm (\epsilon_1 + \epsilon_i + \epsilon_j + \epsilon_k)$
and
$\pm (\epsilon_i + \epsilon_j + \epsilon_k + \epsilon_8)$
also lie in $L$ for all $1<i<j<k<8$.
All roots can be written as a sum of these roots. This proves claim (b).
Since $\Tilde{\ell} \subset {\mathbb Z}^8 \cap \ft$, $\alpha_i - \alpha_j \in {\mathbb Z}$ for
every $\alpha \in \Tilde{\ell}^* \subset \ft^*$.
Hence, the argument for claim (c) follows from the argument for $A_7$.
Finally, $\delta$ is the only root in the positive Weyl chamber.
{\medskip}
{\noindent}{\bf (V, c)}\,\,
For the group $E_8$,
$\ft = \ft^* = \{\xi \in {\mathbb R}^9 \mid \sum \xi_i = 0$\}
and the roots are $\epsilon_i - \epsilon_j$, and
$\pm( \epsilon_i + \epsilon_j + \epsilon_k)$
for $i, j$ and $k$ distinct.
Hence the integral lattice is
$$\Tilde{\ell} = \{\xi \in \ft \mid
3 \xi_i \in {\mathbb Z} \ \mbox{and}\ \xi_i - \xi_j \in {\mathbb Z} \ \forall \ i,j \}.$$
The positive Weyl chamber is
$$
\{\xi \in \ft \mid \xi_2 \geq \cdots \geq \xi_9 \ \mbox{and} \ \xi_2
+ \xi_3 + \xi_4 \leq 0 \}.
$$
(Note that these conditions imply that $\xi_1 \geq \xi_2$.)
The highest root is $\delta = \epsilon_1 - \epsilon_9$.
Assume that $(\delta,\lambda) = \lambda_1 - \lambda_9 \leq 2$.
Combining the inequalities
$$
\lambda_1 - \lambda_9 \leq 2,\quad
\lambda_1 + \lambda_5 + \lambda_6 + \lambda_7 + \lambda_8 + \lambda_9 \geq 0,
\quad
\lambda_i \leq \lambda_4,i > 4,
$$
we see that
$\lambda_4 \geq - \frac{1}{3}$.
Since $\lambda_2 + \lambda_3 + \lambda_4 \leq 0$
and $\lambda_2 \geq {\lambda}_3\geq \lambda_4$,
$\lambda_4 \leq 0$.
Moreover, in both cases, if the last inequality in the
sentence is an equality, all the inequalities
are equalities.
Since $\lambda \in \ell$, the only possibilities are
$\lambda_4 = 0$ or $\lambda_4 = -\frac{1}{3}$.
In the former case,
$\lambda = \epsilon_1 - \epsilon_9$.
In the latter case,
$\lambda = (\epsilon_1 - \epsilon_9) + (\epsilon_1 + \epsilon_2 + \epsilon_9)$.
Claim (a) follows.
We now notice that
$\delta = \epsilon_1 - \epsilon_9 = (\epsilon_1 - \epsilon_k) +
(\epsilon_k - \epsilon_9) = (\epsilon_1 + \epsilon_i + \epsilon_j)
- (\epsilon_i + \epsilon_j + \epsilon_9)$ for all $1 < k < 9$ and $1 < i < j < 9$.
Therefore, the corresponding roots
$\pm(\epsilon_1 - \epsilon_k)$, $\pm(\epsilon_k - \epsilon_9)$,
$\pm(\epsilon_1 + \epsilon_i + \epsilon_j)$, and $\pm(\epsilon_i + \epsilon_j +
\epsilon_9)$ all lie in $L$.
Since every root can be written as a sum of these roots, claim (b) follows.
Since $\Tilde{\ell} \subset {\mathbb Z}^9 \cap \ft$, $\alpha_i - \alpha_j \in {\mathbb Z}$ for
every $\alpha \in \Tilde{\ell}^* \subset \ft^*$.
Hence, the argument for claim (c) carries over
from the argument for the group $A_8$.
Finally, $\delta$ is the only root in the positive Weyl chamber.
{\medskip}
{\noindent}{\bf (VI)}\,\,
For the group $F_4$, $\ft = \ft^* = {\mathbb R}^4$.
The roots are $ \pm e_i$, $e_i \pm e_j$ for $i \neq j$, and
$\frac{1}{2}(\pm e_1 \pm e_2 \pm e_3 \pm e_4)$.
Hence the integral lattice is
$\Tilde{\ell} = \{ \xi \in {\mathbb Z}^4 \mid \sum \xi_i \in 2 {\mathbb Z} \}$.
The positive Weyl chamber is
$$
\{\xi \in \ft \mid \xi_2 \geq \xi_3 \geq \xi_4 \geq 0 \ \mbox{and}
\ \xi_1 \geq \xi_2 + \xi_3 + \xi_4 \}.
$$
(Note that automatically $\xi_1 \geq \xi_2$.)
The highest root is $\delta = e_1 + e_2$.
The argument for claim (a) carries over
word for word from the argument for $B_4$.
Notice that
if $k=3$ or $4$
\begin{eqnarray*}
\delta & = & e_1 + e_2 = (e_1) + (e_2)\; =\; (e_1 - e_k) + (e_2 + e_k)
\;=\; (e_1 + e_k) + (e_2 - e_k) \\
&=&
\frac{1}{2}(e_1 + e_2 + e_3 + e_4) + \frac{1}{2}(e_1 + e_2 - e_3 -
e_4)\\
& = & \frac{1}{2}(e_1 + e_2 - e_3 + e_4) + \frac{1}{2}(e_1 + e_2 + e_3 - e_4).
\end{eqnarray*}
Hence, the corresponding roots all lie in $L$.
Since every root can be written as the sum of these roots, this
proves claim (b).
Since $-{\rm id}$ lies in the Weyl group, we are done.
{\medskip}
{\noindent}{\bf (VII)}\,\,
For the group $G_2$, $\ft = \ft^* = \{\xi \in {\mathbb R}^3 \mid \sum \xi_i = 0$\}.
The roots are $\pm \epsilon_i$ and $\epsilon_i - \epsilon_j$
for $i$ and $j$ distinct.
The positive Weyl chamber is
$\{ \xi \in \ft^* \mid 0 \geq \xi_2 \geq \xi_3 \}$.
(Note that automatically $\xi_1 \geq \xi_2$.)
The integral lattice is $\Tilde{\ell} ={\mathbb Z}^3 \cap \ft$.
The highest root is $\delta = \epsilon_1 - \epsilon_3$.
The argument for claim (a) follows
the argument for $A_3$ word for word.
Since $\delta = (\epsilon_1 - \epsilon_2) + (\epsilon_2 - \epsilon_3)
= (\epsilon_1) + (-\epsilon_3)$,
the roots $\pm (\epsilon_1 - \epsilon_2)$, $\pm (\epsilon_2 - \epsilon_3)$,
$\pm \epsilon_1$ and $\pm \epsilon_3$ all lie in $L$.
Since every root can be written as a sum of these roots, claim (b) follows.
Since $-{\rm id}$ lies in the Weyl group, we are done.
{\medskip}
|
{
"timestamp": "2005-03-22T18:36:20",
"yymm": "0503",
"arxiv_id": "math/0503467",
"language": "en",
"url": "https://arxiv.org/abs/math/0503467"
}
|
\section{Introduction}
In heavy ion collisions an extended hot and dense fireball medium
is created. The properties (mass, width, momentum distribution,
yield) of the produced resonances depend on the fireball
conditions of temperature and pressure. During the fireball
expansion the short lived resonances and their hadronic decay
daughters may interact with the medium. Two freeze-out surfaces
can be defined, chemical and thermal, representing the conditions
when inelastic and elastic interactions cease respectively. In a
dynamical evolving system produced resonances decay and may get
regenerated. Hadronic decay daughters of resonances which decay
inside the medium may also scatter with other particles from the
medium. For SPS and RHIC energies these are mostly pions. This
results in a signal loss, because the reconstructed invariant mass
of the decay daughters no longer matches that of the parent.
Leptonic decay daughters on the other hand are unaffected by the
nuclear medium due to their small interaction cross section. The
rescattering and regeneration (pseudo-elastic) processes for
resonances and their decay particles depend on the individual
cross sections and are dominant after chemical but before the
kinetic freeze-out. These interactions can result in changes of
the reconstructed resonance yields, momentum spectra, widths and
mass positions. Rescattering will decrease the measured resonance
yields while regeneration will increase them.
Microscopic model calculations attempt to include every step in a
heavy ion interaction in terms of elastic and inelastic
interactions of hadrons and strings. They are therefore better
able to describe the rescattering and regeneration of the
resonances from fireball interactions. The prediction of a
specific model (UrQMD) is a signal loss for some of the resonances
due to more rescattering than regeneration in the low momentum
region p$_{\rm T}<1$~GeV for the hadronic decay channels
\cite{ble02,ble02b}. Comparisons between the yield and momentum
spectra of the hadronic and leptonic decay channels can indicate
the magnitude of the rescattering and regeneration contribution
between chemical and thermal freeze-out. In order to try to
understand the medium effect during the evolution and expansion of
the hot and dense fireball, we compare resonance yields and
spectra (width and mass) from elementary p+p and heavy ion
collisions and the results from the leptonic and hadronic decay
channels. An observed difference may give an indication of
in-medium modification of resonance properties.
\section{Resonance Reconstruction}
The signal loss due to rescattering is caused by the method of
measurement, the invariant mass is not properly reconstructed if
one of the decay daughters rescatters with another particle of the
surrounding medium. All the resonances are reconstructed by the
invariant mass of the decay daughters. The decay candidates are
identified by different techniques, their energy loss (dE/dx),
energy or displaced vertex (V0-reconstruction). The resonance
signal is obtained by the invariant mass reconstruction of each
daughter combination and subtraction of the combinatorial
background calculated by mixed event or like-signed techniques.
The resonance ratios, spectra and yields are measured at
mid-rapidity for RHIC at $\sqrt{s_{\rm NN}} = $ 200 GeV and over
4$\pi$ for SPS at $\sqrt{s_{\rm NN}} = $ 17 GeV. The central
trigger selection for Au+Au collisions at RHIC takes the 5\% or
10\% and for Pb+Pb collisions at SPS the 5\% of the most central
inelastic interactions. The setup for the p+p interaction is a
minimum bias trigger.
\section{Resonance Yields}
The resonance multiplicities at mid-rapidity for p+p and
peripheral to central Au+Au collisions at RHIC energies are
obtained for $\phi$(1020) \cite{ma04}, $\Delta(1232)^{++}$
\cite{mar04qm}, K(892) \cite{zha04} $\Lambda$(1520)
\cite{gau04,mar03} and $\Sigma$(1385) \cite{sal04}. In order to
compare different collision systems we normalize the yield to the
yield of the corresponding measured ground state particle. Under
the assumption that the Au+Au collision system is only a
superposition of p+p collisions we would expect the same
resonance/non-resonance ratio. Fig.\ref{part} shows the
resonance/non-resonance ratios normalized to the K(892)/K
measurement in p+p. The $\Lambda$(1520)/$\Lambda$ and the K(892)/K
ratio decreases from p+p to peripheral and central Au+Au
collisions.
\begin{figure}[htb]
\centering
\includegraphics[width=0.8\textwidth]{resonances_ncharge_sqm2004.eps}
\caption{Resonance/non-resonance ratios of $\phi$/K$^{-}$
\cite{ma04}, $\Delta^{++}$/p \cite{mar04qm}, $\rho/\pi$
\cite{fac04}, K(892)/K$^{-}$ \cite{zha04} and
$\Lambda$(1520)/$\Lambda$ \cite{gau04,mar03} for p+p and Au+Au
collisions at $\sqrt{s_{\rm NN}} = $ 200 GeV at mid-rapidity. The
ratios are normalized to the K(892)/K$^{-}$ ratio measurement in
p+p. Statistical and systematic errors are included.}
\label{part}
\end{figure}
The observed ratios can not be described by thermal model
predictions \cite{pbm01}, most likely because rescattering of the
decay daughters in the medium and regeneration are contributing to
the yield. If only rescattering occurs then the shorter lifetime
of the K(892) (4 fm/c) compared to the $\Lambda$(1520) (13 fm/c)
would result in a larger suppression for K(892)/K than for the
$\Lambda$(1520)/$\Lambda$ ratio. This implies that the
regeneration cross section is larger for the K+$\pi$ channel than
for the K+p channel. The $\phi$(1020)/K ratio is constant in all
collision systems within errors and can be described with the
thermal model, which is expected because only a small fraction of
the $\phi$(1020) are decaying inside the fireball due to the long
lifetime of the $\phi$(1020) (46 fm/c). The expected contribution
of rescattering for the short lived $\Delta$(1232) (1.7 fm/c) is
larger than that for the K(892) and the $\Lambda$(1520). However
the $\Delta$(1232)/p ratio does not decrease from p+p to Au+Au
collisions and is on the order of 41\% $\pm$ 22\% higher than the
thermal model prediction. This indicates a large cross section for
the regeneration of $\Delta$(1232) resonance in the p+$\pi$
channel. the $\Delta$(1232) can be re-created until T = 80-90 MeV
close to the kinetic freeze-out \cite{ble04}. The
$\Sigma$(1385)/$\Lambda$ ratio appears to follow the same trend as
the $\Delta$(1232)/p \cite{sal04}. This implies that the
$\Lambda$+$\pi$ regeneration cross section is nearly as high as
the p+$\pi$ regeneration cross section. From this observation we
can conclude that there is a ranking order of the cross section
for the different
regeneration processes: \\
$\sigma_{p+\pi}$ $\geq$ $\sigma_{\Lambda+\pi}$ $>$
$\sigma_{K+\pi}$ $>$ $\sigma_{K+p}$. The microscopic model
calculations (UrQMD) are able to reproduce the
resonance/non-resonance ratios in Au+Au collisions for most
resonances \cite{ble02,ble02b}. However the UrQMD prediction for
the $\Sigma$(1385)/$\Lambda$ ratio is in the order of 40\% $\pm$
20\% too high. In this calculation the assumption was made that
the $\Lambda$+$\pi$ regeneration cross section is the same than
for p+$\pi$. The trend of data would suggest that the
$\Lambda$+$\pi$ regeneration cross section is smaller than the
p+$\pi$ cross section.
\begin{figure}[htb]
\vspace{0.5cm}
\centering
\includegraphics[width=0.5\textwidth,angle=-90]{pt_thermal_2.eps.eps}
\caption{Transverse momentum distribution of $\Delta^{++}$,
$\rho$, K(892) and $\phi$(1020) in central ($\rho$ peripheral)
Au+Au collisions from the STAR experiment at RHIC compared to
thermal model predictions \cite{flo04}.}
\label{ptmodel}
\end{figure}
\section{Momentum Distribution}
Fig.~\ref{ptmodel} shows the momentum distribution of
$\Delta^{++}$, $\rho$, K(892) and $\phi$(1020) from central Au+Au
collisions ($\rho$ peripheral) from the STAR experiment at RHIC
compared to thermal model predictions from W. Florkowski. The
measured K(892) distribution deviates from the model predictions
in the low momentum region. This observation is consistent with
the UrQMD prediction of a signal loss due to rescattering in the
low momentum region. Based on the similarity in the trends between
the $\Lambda$(1520)/$\Lambda$ and the K(892)/K in Fig.~\ref{part},
one would also expect a signal loss in the low momentum region for
the $\Lambda$(1520) compared to the thermal model predictions. The
good agreement of the $\Delta^{++}$ momentum distribution with the
model indicats that the regeneration also takes place
predominantly in the low momentum region.
This low momentum signal loss of resonances due to rescattering in
results in a higher inverse slope parameter and a higher
$\langle$p$_{\rm T}$$\rangle$. The STAR data from p+p and Au+Au
collisions at $\sqrt{s_{\rm NN}} = $ 200 GeV confirm this trend. A
strong increase $\langle$p$_{\rm T}$$\rangle$ for resonances is
observed from p+p to the most peripheral Au+Au measurement. The
same trend is not present for the ground state particles (see
Fig~\ref{resopt}) \cite{ma04,mar04qm,zha04}.
\begin{figure}[h]
\centering
\vspace{0.5cm}
\includegraphics[width=0.55\textwidth]{resonancept.eps}
\caption{The $\langle$p$_{\rm T}$$\rangle$ for resonances
and ground state particles in p+p and Au+Au collisions
versus number of charged particles \cite{ma04,mar04qm,zha04,sal04}.}
\label{resopt}
\end{figure}
\section{Time Scale}
Depending on the length of the time interval between chemical and
kinetic freeze-out, $\Delta \tau$, the magnitude of the
suppression factor of the measured resonance will change due to
contributions from rescattering and regeneration. A model using
thermally produced particle yields at chemical freeze-out and an
additional rescattering phase, including the lifetime of the
resonances and decay product interactions within the expanding
fireball, can yield an estimated $\Delta \tau$
\cite{tor01,tor01a,mar02}. This model does not include
regeneration and therefore predicts a lower limit of the lifetime
between the two freeze-out surfaces. The two ratios K(892)/K and
$\Lambda$(1520)/$\Lambda$ are expected to have a larger
rescattering contribution. A $\Delta\tau$ $>$ 4~fm/c results if
chemical freeze-out occurs at 160 MeV.
\section{Leptonic and Hadronic Decay Channels}
In heavy ion collisions direct comparisons of the spectra and
yields obtained from leptonic and hadronic decay channels of a
single resonance may show the influence of the hadronic
interaction phase after chemical freeze-out. The $\phi$(1020) is
one of the resonances where we have measurements of the leptonic
and hadronic decay channel. At SPS energies the reconstruction of
the $\phi$(1020) in the different decay channels seemingly leads
to differing $\phi$(1020) kinematics and yields ($\phi$ puzzle).
\begin{figure}[htb]
\vspace{0.8cm}
\centering
\includegraphics[width=0.6\textwidth]{kolo_phi_eps_rot_single2.eps}
\caption{Transverse momentum distribution of the hadronic decay
$\phi$(1020) $\rightarrow$ K$^{+}$ + K$^{-}$ from NA49
\cite{fri97} and the leptonic decay $\phi$(1020) $\rightarrow$
$\mu^{+}$ + $\mu^{-}$ from NA50 \cite{wil99}.}
\label{phi}
\end{figure}
Fig.~\ref{phi} shows the transverse momentum distribution from the
hadronic decay $\phi$~$\rightarrow$~K$^{+}$~+~K$^{-}$ (NA49) and
the leptonic decay $\phi$~$\rightarrow$~$\mu^{+}$~+$\mu^{-}$
(NA50) \cite{fri97,wil99}. The inverse slope parameter from fits
to the momentum spectra, indicated as lines, are
T~=~305~$\pm$~15~MeV for hadronic decay and T~=~218~$\pm$~10~MeV
for leptonic decay. The extracted yield from the extrapolation of
the momentum spectrum of the leptonic decay is a factor of
4~$\pm$~2 higher than the one for the hadronic decay. Measurements
of the $\phi$(1020) reconstructed via the hadronic and leptonic
decay from CERES presented by A. Marin \cite{mari04} at this
conference confirm the NA49 results ($\phi$ $\rightarrow$ K$^{+}$
+ K$^{-}$) in terms yield and momentum distribution and the NA50
yield for the $\phi$ $\rightarrow$ $e^{+}$ + $e^{-}$ decay. First
results from NA60 experiment show an improved invariant mass
signal (significance $>$ 20) for the $\phi$ $\rightarrow$
$\mu^{+}$ + $\mu^{-}$ channel \cite{dam04}, which should result in
a conclusive contribution to the $\phi$ puzzle at SPS.
Microscopic calculations (UrQMD) estimate a suppression of 20-30\%
of the $\phi$(1020) yield in the hadronic decay channel due to
rescattering of the kaon decay daughters in the low momentum
region p$_{\rm T}$~$<$~1~GeV \cite{ble02,ble02b}. The rescattering
is negligible for the leptonic decay due to the very low cross
section of interaction with the hadronic phase. Therefore the
lower signal in the low momentum region of the hadronic decay
(NA49) compared to the leptonic decay (NA50) is in agreement with
the model. However the signal loss of 20-30\% from the model
calculation is not sufficient to explain the factor of 4~$\pm$~2
in the measured yield of the data.
This allows for possible medium effects on the resonance
production which are likely to occur at an earlier stage, before
chemical freeze-out. Alternative calculations to describe in
medium modification of the $\phi$(1020) resonance were published
recently by K. Haglin and E. Kolomeitsev
\cite{hag04,hag04a,kol99}. Here the lifetime of the $\phi$(1020)
resonance is modified towards smaller lifetimes due to
modification of the spectral functions in the hot and dense
fireball and therefore more of the $\phi$(1020) resonances decay
inside the medium. This will introduce a larger signal loss due to
rescattering of the hadronic decay daughters.
\section{Feeddown from Resonances}
Finally I would like to conclude with a small remark. If we
interpret particle spectra of ground state particles we have to
take into account that a large fraction of the particles are
coming from resonance feeddown, as already pointed out by E.
Schnedermann et al. \cite{sch93}. For the proton we have 42\% from
$\Lambda$'s, 21\% from $\Delta$'s, and 11\% from $\Sigma^{0}$'s
(statistical model \cite{raf}). Therefore only 26\% of the protons
are primary produced protons. 35\% of the $\Lambda$'s are from
$\Sigma$(1385) and 20\% from $\Sigma^{0}$'s (statistical model)
decays. If we take the contribution of multiple rescattering and
regeneration processes during the expansion of the fireball source
into account, the number of primary particles will be further
reduced, because the regeneration does not necessarily involve the
actual resonance decay particles. Since the lifetimes of the
$\rho$ and $\Delta$(1232) are very short compared to the lifetime
of the fireball, we would expect a larger number of $\pi$'s and
protons coming from a $\Delta$(1232) decay than from higher mass
baryons. Therefore many $\pi$'s and protons are coming from a
later stage of the evolution of the fireball source and their
momentum distribution might be different from the primary produced
particles. Conclusions based on the momentum distributions of
particle spectra in terms of flow and freeze-out temperatures have
to take the contribution from resonance decays into account.
\section*{REFERENCES}
|
{
"timestamp": "2005-03-25T22:04:18",
"yymm": "0503",
"arxiv_id": "nucl-ex/0503011",
"language": "en",
"url": "https://arxiv.org/abs/nucl-ex/0503011"
}
|
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\head Introduction\endhead
\subhead 0.1\endsubhead
Let $\kk$ be an algebraically closed field of characteristic exponent $p\ge1$. Let $G$
be a reductive connected algebraic group over $\kk$. Let $\cu$ be the variety of
unipotent elements of $G$. The unipotent classes of $G$ are the orbits of the
conjugation action of $G$ on $\cu$. The theory of Dynkin and Kostant \cite{\KO}
provides a classification of unipotent classes of $G$ assuming that $p=1$. It is known
that this classification remains valid when $p\ge2$ is assumed to be a good prime for
$G$. But the analogous classification problem in the case where $p$ is a bad prime for
$G$ is more complicated. In every case a classification of unipotent classes is known:
see \cite{\WA} for classical groups and \cite{\EN,\SH,\MI} for exceptional groups; but
from these works it is difficult to see the general features of the classification.
One of the aims of this paper is to present a picture of the unipotent elements which
should apply for arbitrary $p$ and is as close as possible to the picture for $p=1$.
In 1.4 we observe that the set of unipotent classes in $G$ can be parametrized by a set
$\cs^p(\WW)$ of irreducible representations of the Weyl group $\WW$ which can be
described apriori purely in terms of the root system. This explains clearly why the
classification is different for small $p$.
In 1.1 we restate in a more precise form an observation of \cite{\LN} according to
which $\cu$ is naturally partitioned into finitely many "unipotent pieces" which are
locally closed subvarieties stable under conjugation by $G$; the classification of
unipotent pieces is independent of $p$. For $p=1$ or a good prime, each unipotent piece
is a single conjugacy class. When $p$ is a bad prime a unipotent piece is in general a
union of several conjugacy classes. Also each unipotent piece has
some topological properties which are independent of $p$ (for example, over a finite
field, the number of points of a unipotent piece is given by a formula independent of
the characteristic).
Another aim of this paper is the study of $\cb_u$, the variety of Borel subgroups of
$G$ containing a unipotent element $u$. It is known \cite{\SP} that when $p$ is a good
prime, the $l$-adic cohomology spaces of $\cb_u$ are pure. We would like to prove a
similar result in the case where $p$ is a bad prime. We present a method by which this
can be achieved in a number of cases. Our strategy is to extend a technique from
\cite{\DLP} in which (assuming that $p=1$), $\cb_u$ is analyzed by first partitioning
it into finitely many smooth locally closed subvarieties using relative position of a
point in $\cb_u$ with a canonical parabolic attached to $u$. Much of our effort is
concerned with trying to eliminate reference to the linearization procedure of
Bass-Haboush (available only for $p=1$) which was used in an essential way in
\cite{\DLP}. Our approach is based on a list of properties $\fP_1-\fP_8$ of unipotent
elements of which the first five (resp. last three) are expected to hold in general
(resp. in many cases). All these properties are verified for general linear and
symplectic groups (any $p$) in \S2, \S3. In writing \S3 (on symplectic groups mostly
with $p=2$) I found that the treatment in \cite{\WA} is not sufficient for this paper's
purposes; I therefore included a treatment which does not rely on \cite{\WA}.
{\it Notation.} When $p>1$ we denote by $\kk_p$ an algebraic closure of the field with
$p$ elements. Let $\cb$ the variety of Borel subgroups of $G$. If $\G'$ is a subgroup
of a group $\G$ and $x\in\G$ let $Z_{G'}(x)=\{z\in\G';zx=xz\}$. For a finite set $Z$
let $|Z|$ be the cardinal of $Z$. Let $l$ be a prime number invertible in $\kk$. For
$a,b\in\ZZ$ let $[a,b]=\{z\in\ZZ;a\le z\le b\}$.
\head Contents\endhead
1. Some properties of unipotent elements.
2. General linear groups.
3. Symplectic groups.
4. The group $A^1(u)$.
5. Study of the varieties $\cb_u$.
\head 1. Some properties of unipotent elements\endhead
\subhead 1.1\endsubhead
$G$ acts naturally, by conjugation, on $\Hom(\kk^*,G)$ (homomorphisms of algebraic
groups). The set of orbits $\Hom(\kk^*,G)/G$ is naturally in bijection with the
analogous set $\Hom(\CC^*,G')/G'$ where $G'$ is a connected reductive group over $\CC$
of the same type as $G$. (Both sets may be identified with the set of Weyl group orbits
on the group of $1$-parameter subgroups of some maximal torus.) Let $\tD_{G'}$ be the
set of all $\o\in\Hom(\CC^*,G')$ such that there exists a homomorphism of algebraic
groups $\ti\o:SL_2(\CC)@>>>G'$ with $\ti\o\left(\sm t&0\\0&t\i\esm\right)=\o(t)$ for
all $t\in\CC^*$. Now $\tD_{G'}$ is $G'$-stable; it has been described explicitly by
Dynkin. Let $\tD_G$ be the unique $G$-stable subset of $\Hom(\kk^*,G)$ whose image in
$\Hom(\kk^*,G)/G$ corresponds under the bijection
$\Hom(\kk^*,G)/G\lra\Hom(\CC^*,G')/G'$ (as above) to the image of $\tD_{G'}$ in
$\Hom(\CC^*,G')/G'$. Let $D_G$ be the set of sequences
$\l=(G^\l_0\sps G^\l_1\sps G^\l_2\sps\do)$ of closed connected subgroups of $G$ such
that for some $\o\in\tD_G$ we have (for $n\ge0$):
$\Lie G^\l_n=\{x\in\Lie G;\lim_{t\in\kk^*;t\to 0}t^{1-n}\Ad\o(t)x=0\}$.
\nl
Now $G$ acts on $D_G$ by conjugation and the obvious map $\tD_G@>>>D_G$ induces a
bijection $\tD_G/G@>\si>>D_G/G$ on the set of orbits. If $\l\in D_G$ and $g\in G$ then
$G^{g\l g\i}_n=gG^\l_ng\i$ for $n\ge0$. Moreover, $G^\l_0$ is a parabolic subgroup of
$G$ with unipotent radical $G^\l_1$ and $G^\l_n$ is normalized by $G^\l_0$ for any $n$.
Moreover,
(a) $G^\l_2/G^\l_3$ is a commutative connected unipotent group;
(b) the conjugation action of $G^\l_0$ on $G^\l_2/G^\l_3$ factors through an action of
$\bG^\l_0:=G^\l_0/G^\l_1$ on $G^\l_2/G^\l_3$.
\nl
Note also that $G^\l_n$ for $n\ne0,2$ are uniquely determined by $G^\l_0,G^\l_2$.
Let $\bla$ be a $G$-orbit in $D_G$. Then $\tH^\bla:=\cup_{\l\in\bla}G^\l_2$ is a closed
irreducible subset of $\cu$ (since for $\l\in\bla$, $G^\l_2$ is a closed irreducible
subset of $\cu$ stable under conjugation by $G^\l_0$ and $G/G^\l_0$ is projective). Let
$$H^\bla=\tH^\bla-\cup_{\bla'\in D_G/G;\tH^{\bla'}\subsetneqq\tH^\bla}\tH^{\bla'}.$$
For $\l\in D_G$ let $X^\l=G^\l_2\cap H^\bla$ where $\bla$ is the $G$-orbit of $\l$.
Then $H^\bla$ is an open dense subset of $\tH^\bla$ stable under conjugation by $G$ and
$X^\l$ is an open dense subset of $G^\l_2$ stable under conjugation by $G^\l_0$. (We
use that $D_G/G$ is finite.) Hence $H^\bla$ is locally closed in $\cu$. The subsets
$H^\bla(\bla\in D_G/G)$ are called the {\it unipotent pieces} of $G$.
We state the following properties $\fP_1-\fP_5$.
$\fP_1$. {\it The sets $X^\l(\l\in D_G)$ form a partition of $\cu$.}
$\fP_2$. {\it Let $\bla\in D_G/G$. The sets $X^\l(\l\in\bla)$ form a partition of
$H^\bla$. More precisely, $H^\bla$ is a fibration over $\bla$ with smooth fibres
isomorphic to $X^\l$ ($\l\in\bla$); in particular, $H^\bla$ is smooth.}
$\fP_3$. {\it The locally closed subets $H^\bla(\bla\in D_G/G)$ form a (finite)
partition of $\cu$.}
$\fP_4$. {\it Let $\l\in D_G$. We have $G^\l_3X^\l=X^\l G^\l_3=X^\l$.}
$\fP_5$. {\it Assume that $\kk=\kk_p$. Let $F:G@>>>G$ be the Frobenius map
corresponding to a split $\FF_q$-rational structure with $q-1$ sufficiently divisible.
Let $\l\in D_G$ be such that $F(G^\l_n)=G^\l_n$ for all $n\ge0$ and let $\bla$ be the
$G$-orbit of $\l$. Then $|H^\bla(\FF_q)|,|X^\l(\FF_q)|$ are polynomials in $q$ with
integer coefficients independent of $p$.}
Assume first that $p=1$ or $p\gg0$. By the theory of Dynkin-Kostant, for $\l\in D_G$
there is a unique open $G^\l_0$-orbit $X'{}^\l$ in $G^\l_2$; we then have a bijection
of $D_G/G$ with the set of unipotent classes on $G$ which to the $G$-orbit $\bla$ of
$\l\in D_G$ associates the unique unipotent class $H'{}^\bla$ of $G$ that contains
$X'{}^\l$; moreover, if $g\in X'{}^\l$ then $Z_{G^\l_0}(g)=Z_G(g)$. As stated by
Kawanaka \cite{\KA}, the same holds when $p$ is a good prime of $G$ (but his argument
is rather sketchy). To show that $\fP_1-\fP_3$ holds when $p$ is a good prime it then
suffices to show that $X^\l=X'{}^\l$ for any $\l$. It also suffices to show that
$X'{}^\l=G^\l_2\cap H'{}^\bla$ for $\l\in\bla$ as above. (Assume that
$g\in G^\l_2\cap H'{}^\bla,g\n X'{}^\l$. Let $g'\in X'{}^\l$. By the definition of
$X'{}^\l$ and the irreducibility of $G^\l_2$, the dimension of the $G^\l_0$-orbit of
$g$ is strictly smaller than the dimension of the $G^\l_0$-orbit of $g'$. Hence
$\dim Z_{G^\l_0}(g)>\dim Z_{G^\l_0}(g')$. We have
$\dim Z_G(g)\ge\dim Z_{G^\l_0}(g)$, $\dim Z_{G^\l_0}(g')=\dim Z_G(g')$ hence
$\dim Z_G(g)>\dim Z_G(g')$. This contradicts the fact that $g,g'$ are $G$-conjugate.)
In this case we have $H^\bla=H'{}^\bla$ and $\tH^\bla$ is the closure of $H'{}^\bla$.
We expect that $\fP_1-\fP_5$ hold in general. In the case where $G=GL_n(\kk)$ (any $p$)
the validity of $\fP_1-\fP_5$ follows from 2.9. In the case where $G$ is a symplectic
group (any $p$) the validity of $\fP_1-\fP_5$ follows from 3.13, 3.14. If $G$ is of
type $E_n$ (any $p$) then one can deduce $\fP_1-\fP_5$ from the various lemmas in
\cite{\MI}, or rather from the extensive computations (largely omitted) on which those
lemmas are based; it would therefore be desirable to have an independent verification
of these properties. The case of special orthogonal groups will be considered
elsewhere.
\mpb
We note the following consequence of $\fP_1$.
(c) {\it If $\l\in D_G$ and $u\in X^\l$ then $Z_G(u)\sub G^\l_0$.}
\nl
Let $g\in G$. Then $gug\i\in X^{g\l}$. Hence if $g\in Z_G(u)$ we have $u\in X^{g\l}$.
Thus, $X^{g\l}\cap X^\l\ne\em$. From $\fP_1$ we see that $g\!\l=\l$. In particular
$gG^\l_0g\i=G^\l_0$ and $g\in G^\l_0$, as required.
\subhead 1.2\endsubhead
Let $\l\in D_G$. We assume that $\fP_1-\fP_4$ hold for $\l$. Let
$\p^\l:G^\l_2@>>>G^\l_2/G^\l_3$ be the obvious homomorphism. By $\fP_4$ we have
$X^\l=(\p^\l)\i(\bX^\l)$ where $\bX^\l$ is a well defined open dense subset of
$G^\l_2/G^\l_3$ stable under the action of $\bG^\l_0$. We wish to consider some
properties of the sets $\bX^\l$ which may or may not hold for $G$.
$\fP_6$. {\it If $u\in X^\l$ then $uG^\l_3=G^\l_3u$ is contained in the
$G^\l_0$-conjugacy class of $u$. Hence $\g\m(\p^\l)\i(\g)$ is a bijection between the
set of $\bG^\l_0$-orbits in $\bX^\l$ and the set of $G^\l_0$-conjugacy classes in
$X^\l$.}
$\fP_7$. {\it Let $\g$ be a $\bG^\l_0$-orbit in $\bX^\l$. Let $\hag$ be the union of
all $\bG^\l_0$-orbits in $\bX^\l$ whose closure contains $\g$. Thus, $\hag$ is an open
subset of $\bX^\l$ and $\g$ is a closed subset of $\hag$. There exists a variety $\g_1$
and a morphism $\r:\hag@>>>\g_1$ such that the restriction of $\r$ to $\g$ is a finite
bijective morphism $\s:\g@>>>\g_1$ and the map of sets $\s\i\r:\hag@>>>\g$ is
compatible with the actions of $\bG^\l_0$.}
$\fP_8$. {\it There exists a finite set $I$ and a bijection $J\m\Ph_J$ between the set
of subsets of $I$ and the set of $G^\l_0$-orbits in $X^\l$ such that for any $J\sub I$,
the closure of $\Ph_J$ in $X^\l$ is $\cup_{J';J\sub J'}\Ph_{J'}$. Moreover, if $\kk,q$
are as in $\fP_5$ then there exists a function $I@>>>\{2,4,6,\do\},i\m c_i$ such that
$|\Ph_J(\FF_q)|=\prod_{i\in J}(q^{c_i}-1)|\Ph_\em(\FF_q)|$ for any $J\sub I$.}
When $p=1$ or $p\gg0$ property $\fP_6$ can be deduced from the theory of
Dynkin-Kostant; properties $\fP_7,\fP_8$ are trivial. In the case where $G=GL_n(\kk)$
(any $p$) the validity of $\fP_6$ follows from 2.9; properties $\fP_7,\fP_8$ are
trivial. In the case where $G$ is a symplectic group (any $p$) the validity of
$\fP_6-\fP_8$ follows from 3.14. $\fP_6$ is false for $G$ of type $G_2$, $p=3$.
\subhead 1.3\endsubhead
Let $\VV$ be a finite dimensional $\QQ$-vector space. Let $R\sub\VV^*=\Hom(\VV,\QQ)$ be
a (reduced) root system, let $\che R\sub\VV$ be the corresponding set of coroots and
let $\WW\sub GL(\VV)$ be the Weyl group of $R$. Let $\b\lra\che\b$ be the canonical
bijection $R\lra\che R$. Let $\Pi$ be a set of simple roots for $R$ and let
$\che\Pi=\{\che\a;\a\in\Pi\}$. Let
$\Th=\{\b\in R;\b-\a\n R\qua\frl\a\in\Pi\}$,
$\ti\Th=\{\b\in R;\che\b-\che\a\n\che R\qua\frl\a\in\Pi\}$,
$\ti\ca=\{J\sub\Pi\cup\ti\Th;J\text{ linearly independent in }\VV^*\}$.
\nl
For any prime number $r$ let $\ca_r$ be the set of all $J\sub\Pi\cup\Th$ such that $J$
is linearly independent in $\VV^*$ and the torsion subgroup of
$\sum_{\a\in\Pi}\ZZ\a/\sum_{\b\in J}\ZZ\b$ has order $r^k$ for some $k\in\NN$.
For any $J\in\ca_r$ or $J\in\ti\ca$ let $\WW_J$ be the subgroup of $\WW$ generated by
the reflections with respects to roots in $J$. For $W'=\WW$ or $\WW_J$ let $\Ir(W')$ be
the set of (isomorphism classes) of irreducible representations of $W'$ over $\QQ$. For
$E\in\Ir(W')$ let $b_E$ be the smallest integer $\ge0$ such that $E$ appears with
non-zero multiplicity in the $b_E$-th symmetric power of $\VV$ regarded as a
$W'$-module; if this multiplicity is $1$ we say that $E$ is good. If $J$ is as above
and $E\in\Ir(W')$ is good then there is a unique $\tE\in\Ir(\WW)$ such that $\tE$
appears in $\Ind_{\WW_J}^\WW E$ and $b_{\tE}=b_E$; moreover, $\tE$ is good. We set
$\tE=j_{\WW_J}^\WW E$.
Let $\cs_\WW\sub\Ir(\WW)$ be the set of special representations of $\WW$ (see
\cite{\LC}). Now any $E\in\cs_\WW$ is good. Following \cite{\LC}, let $\cs^1_\WW$ be
the set of all $E\in\Ir(\WW)$ such that $E=j_{\WW_J}^\WW E_1$ for some $J\in\ti\ca$ and
some $E_1\in\cs_{\WW_J}$. (Note that $\WW_J$ is like $\WW$ with the same $\VV$ and with
$R$ replaced by the root system with $J$ as set of simple roots; hence $\cs_{\WW_J}$ is
defined.) Now any $E\in\cs^1_\WW$ is good.
For any prime number $r$ let $\cs^r_\WW$ be the set of all $E\in\Ir(\WW)$ such that
$E=j_{\WW_K}^\WW E_1$ for some $K\in\ca_r$ and some $E_1\in\cs^1_{\WW_K}$. (Note that
$\WW_K$ is like $\WW$ with the same $\VV$ and with $R$ replaced by the root system with
$K$ as set of simple roots; hence $\cs^1_{\WW_K}$ is defined.) Now any $E\in\cs^1_\WW$
is good.
We have $\cs^1(\WW)\sub\cs^r(\WW)$. We have $\cs^1(\WW)=\cs^r(\WW)$ if $r$ is a good
prime for $\WW$ and also in the following cases $\WW$ of type $G_2,r=2$; $\WW$ of type
$F_4$, $r=3$; $\WW$ of type $E_6$; $\WW$ of type $E_7$, $r=3$; $\WW$ of type $E_8$,
$r=5$. If $\WW$ is of type $G_2$ and $r=3$ then $\cs^r(\WW)-\cs^1(\WW)$ consists of a
single representation of dimension $1$ coming under $j_{\WW_J}^\WW$ from a $\WW_J$ of
type $A_2$. If $\WW$ is of type $F_4$ and $r=2$ then $\cs^r(\WW)-\cs^1(\WW)$ consists
of four representations of dimensions $9/4/4/2$ coming under $j_{\WW_J}^\WW$ from a
$\WW_J$ of type $C_3A_1/C_3A_1/B_4/B_4$. If $\WW$ is of type $E_7$ and $r=2$ then
$\cs^r(\WW)-\cs^1(\WW)$ consists of a single representation of dimensions $84$ coming
under $j_{\WW_J}^\WW$ from a $\WW_J$ of type $D_6A_1$. If $\WW$ is of type $E_8$ and
$r=2$ then $\cs^r(\WW)-\cs^1(\WW)$ consists of four representations of dimensions
$1050/840/168/972$ coming under $j_{\WW_J}^\WW$ from a $\WW_J$ of type
$E_7A_1/D_5A_3/D_8/E_7A_1$. If $\WW$ is of type $E_8$ and $r=3$ then
$\cs^r(\WW)-\cs^1(\WW)$ consists of a single representation of dimensions $175$ coming
under $j_{\WW_J}^\WW$ from a $\WW_J$ of type $E_6A_2$.
\subhead 1.4\endsubhead
Let $\WW$ be the Weyl group of $G$. Let $u$ be a unipotent element in $G$. Springer's
correspondence (generalized to arbitrary characteristic) associates to $u$ and the
trivial representation of $Z_G(u)/Z_G(u)^0$ a representation $\r_u\in\Ir(\WW)$.
Moreover $u\m\r_u$ defines an injective map from the set of unipotent classes in $G$ to
$\Ir(\WW)$. Let $\cx^p(\WW)$ be the image of this map ($p$ as in 0.1). We state:
(a) {\it If $p=1$ we have $\cx^1(\WW)=\cs^1(\WW)$.} (See \cite{\LC}).
(b) {\it If $p>1$ we have $\cx^p(\WW)=\cs^p(\WW)$.}
\nl
The proof of (b) follows from the explicit description of the Springer correspondence
for small $p$ given in \cite{\LS},\cite{\SPAII}.
\head 2. General linear groups\endhead
\subhead 2.1\endsubhead
Let $\bcc$ be the category whose objects are $\ZZ$-graded $\kk$-vector spaces
$\bV=\op_{a\in\ZZ}\bV_a$ such that $\dim\bV<\iy$; the morphisms are linear maps
respecting the grading. Let $\bV\in\bcc$. For $j\in\ZZ$ let
$\End_j(\bV)=\{T\in\Hom(\bV,\bV);T(\bV_a)\sub\bV_{a+j}\qua\frl a\}$. Let
$\End_2^0(\bV)$ be the set of all $\nu\in\End_2(\bV)$ that satisfy the {\it Lefschetz
condition}: $\nu^n:\bV_{-n}@>>>\bV_n$ is an isomorphism for any $n\ge0$. Let
$\nu\in\End_2^0(\bV)$. Define a graded subspace $P^\nu=\bV^{\prim}$ of $\bV$ by
$P^\nu_a=\{x\in\bV_a;\nu^{1-a}x=0\}$ for $a\le0$, $P^\nu_a=0$ for $a>0$. A standard
argument shows that $N^{(a-c)/2}:P^\nu_c@>>>\bV_a$ is injective if
$c\in a+2\ZZ,c\le a\le-c$ and we have
(a) $\op_{c\in a+2\ZZ;c\le a\le-c}P^\nu_c@>\si>>\bV_a,
(z_c)\m\sum_{c\in a+2\ZZ;c\le a\le-c}N^{(a-c)/2}z_c$.
\nl
We show:
(b) {\it Let $j\in\NN$, $R\in\End_{j+2}(\bV)$. Then $R=T\nu-\nu T$ for some
$T\in\End_j(\bV)$.}
\nl
Let $c\le0$. Since $\nu^{1-c}:\bV_{j-c}@>>>\bV_{j+c+2}$ is surjective, the induced map
$\Hom(P^\nu_{-c},\bV_{j-c})@>>>\Hom(P^\nu_{-c},\bV_{j+c+2})$
\nl
is surjective. Hence there exists $\t_c\in\Hom(P^\nu_{-c},\bV_{j-c})$ such that
$\nu^{1-c}\t_c=-\sum_{i+i'=-c}\nu^iR\nu^{i'}$.
\nl
For $k\in[0,-c]$ we define $\t_{c,k}\in\Hom(P^\nu_c,\bV_{c+2k+j})$ by $\t_{c,0}=\t_c$
and $\t_{c,k}=\nu\t_{c,k-1}+R\nu^{k-1}$ for $k\in[1,-c]$. Then
$\nu\t_{c,-c}+R\nu^{-c}=0$. Let $T:\bV@>>>\bV$ be the unique linear map such that
$T(\nu^kx)=\t_{c,k}(x)$ for $x\in P^\nu_c,c\le0,k\in[0,-c]$. This $T$ has the required
property.
\subhead 2.2\endsubhead
Let $\cc$ be the category whose objects are $\kk$-vector spaces of finite dimension;
morphisms are linear maps. Let $V\in\cc$. A collection of subspaces
$V_*=(V_{\ge a})_{a\in\ZZ}$ of $V$ is said to be a {\it filtration} of $V$ if
$V_{\ge a+1}\sub V_{\ge a}$ for all $a$, $V_{\ge a}=0$ for some $a$, $V_{\ge a}=V$ for
some $a$. We say that $V$ is {\it filtered} if a filtration $V_*$ of $V$ is given.
Assume that this is the case. We set $\gr V_*=\op_{a\in\ZZ}\gr_aV_*\in\bcc$ where
$\gr_aV_*=V_{\ge a}/V_{\ge a+1}$. For any $j\in\ZZ$ let
$E_{\ge j}V_*=\{T\in\End(V);T(V_{\ge a})\sub V_{\ge a+j}\qua\frl a\}$. Any such $T$
induces a linear map $\bT\in\End_j(\gr V_*)$.
\subhead 2.3\endsubhead
Let $V\in\cc$. Let $\Nil(V)=\{T\in\End(V);T\text{ nilpotent }\}$. Let $N\in\Nil(V)$.
When $p=1$, the Dynkin-Kostant theory associates to $1+N$ a canonical filtration
$V^N_*$ of $V$; in terms of a basis of $V$ of the form
(a) $\{N^kv_r;r\in[1,t],k\in[0,e_r-1]\}$ with $v_r\in V,e_r\ge1,N^{e_r}v_r=0$ for
$r\in[1,t]$,
\nl
$V^N_{\ge a}$ is the subspace spanned by
$\{N^kv_r;r\in[1,t],k\in[0,e_r-1],2k+1\ge e_r+a\}$. This subspace makes sense for any
$p$ and we denote it in general by $V^N_{\ge a}$; it is independent of the choice of
basis: we have
$V^N_{\ge a}=\sum_{j\ge\max(0,a)}N^j(\ker N^{2j-a+1})$.
\nl
The subspaces $V^N_{\ge a}$ form a filtration $V^N_*$ of $V$; thus, $V$ becomes a
filtered vector space. From the definitions we see that
(b) {\it$N\in E_{\ge2}V^N_*$ and $\bN\in\End_2(\gr V_*^N)$ belongs to
$\End_2^0(\gr V_*^N)$.}
\nl
Note that for any $j\ge1$,
(c) {\it$\dim P^{\bN}_{1-j}$ is the number of Jordan blocks of size $j$ of $N:V@>>>V$.}
\nl
From 2.1(a) we deduce that for any $n\ge0$:
(d) $\dim P^{\bN}_{-n}=\dim\gr_{-n}V^N_*-\dim\gr_{-n-2}V^N_*$.
\subhead 2.4\endsubhead
According to \cite{\DEII, 1.6.1},
(a) {\it if $V_*$ is a filtration of $V$ and $N\in E_{\ge2}V_*$ induces an element
$\nu\in\End_2^0(\gr V_*)$ then $V_*=V_*^N$.}
\nl
We show that $V_{\ge a}=V^N_{\ge a}$ for all $a$. Let $e$ be the smallest integer
$\ge0$ such that $N^e=0$. We argue by induction on $e$. If $a\ge e$ then
$\nu^a:\gr_{-a}V_*@>>>\gr_aV_*$ is both $0$ and an isomorphism hence
$V_{\ge-a}=V_{\ge1-a}$ and $V_{\ge a}=V_{\ge a+1}$. Thus,
$V_{\ge e}=V_{\ge e+1}=\do=0$ and $V_{\ge1-e}=V_{\ge-e}=\do=V$. Similarly,
$V^N_{\ge e}=V^N_{\ge e+1}=\do=0$ and $V^N_{\ge1-e}=V^N_{\ge-e}=\do=V$. Hence
$V_{\ge a}=V^N_{\ge a}$ if $a\ge e$ or if $a\le1-e$. This already suffices in the case
where $e\le1$. Thus we may assume that $e\ge2$. Now
$\nu^{e-1}:\gr_{1-e}V_*@>>>\gr_{e-1}V_*$ is an isomorphism that is,
$N^{e-1}:V/V_{\ge2-e}@>>>V_{\ge e-1}$ is an isomorphism. We see that
$V_{\ge e-1}=N^{e-1}V$ and $V_{\ge2-e}=\ker(N^{e-1})$. Hence if $2-e\le a\le e-1$ we
have $N^{e-1}V\sub V_{\ge a}\sub\ker(N^{e-1})$; let $V'_{\ge a}$ be the image of
$V_{\ge a}$ under the obvious map $\r:\ker(N^{e-1})@>>>V':=\ker(N^{e-1})/N^{e-1}V$. For
$a\le1-e$ we set $V'_{\ge a}=V'$ and for $a\ge e$ we set $V'_{\ge a}=0$. Now
$(V'_{\ge a})_{a\in\ZZ}$ is a filtration of $V'$ satisfying a property like (a) (with
$N$ replaced by the map $N':V'@>>>V'$ induced by $N$). Since $N'{}^{e-1}=0$, the
induction hypothesis applies to $N'$; it shows that $V'_{\ge a}=V'{}^{N'}_{\ge a}$ for
all $a$. Since for $2-e\le a\le e-1$, $V_{\ge a}=\r\i(V'_{\ge a})$, it follows that
$V_{\ge a}=\r\i(V'{}^{N'}_{\ge a})$; similarly, $V^N_{\ge a}=\r\i(V'{}^{N'}_{\ge a})$
hence $V_{\ge a}=V^N_{\ge a}$. This completes the proof.
With notation in the proof above we have:
$V^N_{\ge a}=0$ for $a\ge e$,
$V^N_{\ge a}=V$ for $a\le1-e$,
$V^N_{\ge a}=\r\i(V'{}^{N'}_{\ge a}),V'{}^{N'}_{\ge a}=\r(V^N_{\ge a})$ for $e\ge2$ and
$2-e\le a\le e-1$,
$V^N_{\ge e-1}=N^{e-1}V$ if $e\ge1$,
$V^N_{\ge2-e}=\ker(N^{e-1})$ if $e\ge1$.
We have $\gr_aV^N_*=0$ for $a\ge e$ and for $a\le-e$.
Note also that the proof above provides an alternative (inductive) definition of
$V^N_{\ge a}$ which does not use a choice of basis.
\subhead 2.5\endsubhead
Let $V,N$ be as in 2.3. Let $V_*=V^N_*$. Let $\nu=\bN\in\End_2(\gr V_*)$. We can find a
grading $V=\op_{a\in\ZZ}V_a$ of $V$ such that
(a) $NV_a\sub V_{a+2}$ and $V_{\ge a}=V_a\op V_{a+1}\op\do$ for all $a$.
\nl
For example, in terms of a basis of $V$ as in 2.3(a), we can take $V_a$ to be the
subspace spanned by $\{N^kv_r;r\in[1,t],k\in[0,e_r-1],2k+1=e_r+a\}$. Taking direct sum
of the obvious isomorphisms $V_a@>\si>>\gr_aV_*$ we obtain an isomorphism of graded
vector spaces $V@>\si>>\gr V_*$ under which $N$ corresponds to $\nu$. It follows that
(b) $N\in\End_2^0(V)$ (defined in terms of the grading $\op_aV_a$).
\nl
We note the following result.
(c) {\it Let $n\ge0$ and let $x\in P^\nu_{-n}$. There exists a representative $\dx$ of
$x$ in $V_{\ge-n}$ such that $N^{n+1}\dx=0$.}
\nl
Let $V_a$ be as above. There is a unique representative $\dx$ of $x$ in $V_{\ge-n}$
such that $\dx\in V_{-n}$. We have $N^{n+1}\dx\in V_n$ and the image of $N^{n+1}\dx$
under the canonical isomorphism $V_n@>\si>>\gr_nV_*^N$ is $0$; hence $N^{n+1}\dx=0$.
Let $E_{\ge1}^NV_*=\{S\in E_{\ge1}V_*;SN=NS\}$,
$\End_1^\nu(\gr V_*)=\{\s\in\End_1(\gr V_*),\s\nu=\nu\s\}$. We show:
(d) {\it The obvious map $E_{\ge1}^NV_*@>>>\End_1^\nu(\gr V_*),S\m\bS$ is surjective.}
\nl
Let $\s\in\End_1^\nu(\gr V_*)$. Let $V_a$ be as above. In terms of these $V_a$ we
define $V@>\si>>\gr V_*$ as above. Under this isomorphism, $\s$ corresponds to a linear
map $S:V@>>>V$. Clearly, $S\in E_{\ge1}^NV_*$ and $\bS=\s$.
\subhead 2.6\endsubhead
Let $V,N$ be as in 2.3. Let $V_*=V_*^N$. Now $1+E_{\ge1}V_*$ is a subgroup of $GL(V)$
acting on $N+E_{\ge3}V_*$ by conjugation. We show that
(a) {\it the conjugation action of $1+E_{\ge1}V_*$ on $N+E_{\ge3}V_*$ is transitive.}
\nl
We must show: if $S\in E_{\ge3}V_*$ then there exists $T\in E_{\ge1}V_*$ such that
$(1+T)N=(N+S)(1+T)$ that is, $TN-NT=S+ST$. We fix subspaces $V_a$ as in 2.5. We have
$S=\sum_{j\ge3}S_j$ where $S_j\in\End(V)$ satisfy $S_jV_a\sub V_{a+j}$ for all $a$. We
seek a linear map $T=\sum_{j\ge1}T_j$ where $T_j\in\End(V)$ satisfy
$T_jV_a\sub V_{a+j}$ for all $a$ and
$\sum_{j\ge1}(T_jN-NT_j)=\sum_{j\ge3}S_j+\sum_{j'\ge3,j''\ge1}S_{j'}T_{j''}$ that is,
$(*)$ $T_jN-NT_j=S_{j+2}+\sum_{j'\in[1,j-1]}S_{j+2-j'}T_{j'}$ for $j=1,2,\do$.
\nl
We show that this system of equations in $T_j$ has a solution. We take $T_1=0$. Assume
that $T_j$ has been found for $j<j_0$ for some $j_0\ge2$ so that $(*)$ holds for
$j<j_0$. We set $R=S_{j_0+2}+\sum_{j'\in[1,j_0-1]}S_{j+2-j'}T_{j'}$. Then
$R(V_a)\sub V_{a+j_0+2}$ for any $a$. The equation $T_{j_0}N-NT_{j_0}=R$ can be solved
by 2.1(b) (see 2.5(b)). This shows by induction that the system $(*)$ has a solution.
(a) is proved.
We now show:
(b) {\it if $\tN\in N+E_{\ge3}V_*$ then $V^{\tN}_*=V_*$.}
\nl
Indeed by (a) we can find $u\in1+E_{\ge1}V_*$ such that $\tN=uNu\i$. Since $V_*^N$ is
canonically attached to $N$, we have $V^{uNu\i}_{\ge a}=u(V^N_{\ge a})=V^N_{\ge a}$ and
(b) follows. For example,
(c) {\it if $\tN=c_1N+c_2N^2+\do+c_kN^k$ where $c_i\in\kk,c_1\ne0$ then
$V^{\tN}_*=V^N_*$.}
\nl
We may assume that $c_1=1$. Since $c_2N^2+\do+c_kN^k\in E_{\ge4}V_*\sub E_{\ge3}V_*$,
(b) is applicable and (c) follows.
\subhead 2.7\endsubhead
Let $V,N$ be as in 2.3. Let $V_*=V_*^N$. Let $\nu=\bN\in\End_2(\gr V_*)$. Let $r\ge2$
be such that $N^r=0$ on $V$. Let $W$ be an $N$-stable subspace of $V$ such that there
exists an $N$-stable complement of $W$ in $V$, $N:W@>>>W$ has no Jordan block of size
$\ne r,r-1$ and $N^{r-2}=0$ on $V/W$. Then $W_*=W_*^N$ is defined. Define a linear map
$\mu:\gr V_*@>>>\gr W_*$ as follows. Let $x\in\gr_aV_*$. We have uniquely
$x=\sum_{c\in a+2\ZZ;c\le a\le-c}\nu^{(a-c)/2}x_c$ where $x_c\in P^\nu_c$; we set
$$\mu(x)=\sum_{c\in a+2\ZZ;c\le a\le-c,c=1-r\text{ or }2-r}\nu^{(a-c)/2}x_c.$$
Let $\cx$ be the set of $N$-stable complements of $W$ in $V$. Then $\cx\ne\em$. For
$Z\in\cx$ define $\Pi_Z:V@>>>W$ by $\Pi_Z(w+z)=w$ where $w\in W,z\in Z$. Let
$\baP_Z:\gr V_*@>>>\gr W_*$ be the map induced by $\Pi_Z$. We show that
(a) $\Pi_Z(V_{\ge a})\sub W_{\ge a}$ for all $a$ and $\baP_Z=\mu$.
\nl
We have $V_{\ge a}=W_{\ge a}\op Z_{\ge a}$. If $x\in V_{\ge a},x=w+z,w\in W_{\ge a},
z\in Z_{\ge a}$, then $\Pi_Z(x)=w$. Thus $\Pi_Z(V_{\ge a})\sub W_{\ge a}$. We can find
direct sum decompositions $W=\op_mW_m,Z=\op_mZ_m$ such that $NW_m\sub W_{m+2}$,
$NZ_m\sub Z_{m+2}$ and $N^m:W_{-m}@>\si>>W_m$, $N^m:Z_{-m}@>\si>>Z_m$ for $m\ge0$ (see
2.5). Let $V_a=W_a\op Z_a$. Define $V_a^{\prim},W_a^{\prim},Z_a^{\prim}$ as in 2.1 in
terms of $N$. We have $V_a^{\prim}=W_a^{\prim}\op Z_a^{\prim}$. We must show that
$\baP_Z(x)=\mu(x)$ for $x\in\gr_aV_*$. It suffices to show: if $w\in W_a,z\in Z_a$ and
$w+z=\sum_{c\in a+2\ZZ;c\le a\le-c}\nu^{(a-c)/2}x_c$ where $x_c\in V_c^{\prim}$ then
$w=\sum_{c\in a+2\ZZ;c\le a\le-c,1-r\le c\le2-r}\nu^{(a-c)/2}x_c$. We have
$x_c=w_c+z_c$ where $w_c\in W_c^{\prim},z_c\in Z_c^{\prim}$ and
$w=\sum_{c\in a+2\ZZ;c\le a\le-c}\nu^{(a-c)/2}w_c$. Now if $W_c^{\prim}\ne0$ then
$1-r\le c\le2-r$. Hence
$w=\sum_{c\in a+2\ZZ;c\le a\le-c,1-r\le c\le2-r}\nu^{(a-c)/2}w_c$.
Also, $Z_{1-r}^{\prim}=Z_{2-r}^{\prim}=0$ since $N:Z@>>>Z$ has no Jordan blocks of size
$\ge r-1$. Thus if $c\in a+2\ZZ;c\le a\le-c,1-r\le c\le2-r$ then $z_c=0$ and
$x_c=w_c$. Thus $w=\sum_{c\in a+2\ZZ;c\le a\le-c,1-r\le c\le2-r}\nu^{(a-c)/2}x_c$, as
required.
Let $Z,Z'\in\cx$. By the previous argument, $\Pi_Z,\Pi_{Z'}:V@>>>W$ both map
$V_{\ge a}$ into $W_{\ge a}$ and induce the same map $\gr V_*@>>>\gr W_*$. It follows
that $\Pi_Z-\Pi_{Z'}:V@>>>W$ maps $V_{\ge a}$ into $W_{\ge a+1}$. In other words,
(b) {\it if $x\in V_{\ge a}$ and $x=w+z=w'+z'$ where $w,w'\in W,z\in Z,z'\in Z'$, then
$w-w'\in W_{\ge a+1}$.}
\nl
Define $\Ph\in GL(V)$ by $\Ph(x)=x$ for $x\in W$, $\Ph(x)=x'$ for $x\in Z$ where
$x'\in Z'$ is given by $x-x'\in W$. We show:
(c) $(1-\Ph)V_{\ge a}\sub V_{\ge a+1}$ {\it for any $a$.}
\nl
Let $x\in V_{\ge a}$. We have $x=w+z=w'+z'$ where $w,w'\in W,z\in Z,z'\in Z'$. We have
$\Ph(x)=w+z'$ hence $(1-\Ph)(x)=(w+z)-(w+z')=z-z'=w'-w$ and this belongs to
$W_{\ge a+1}$ by (b).
We show:
(d) $\Ph N=N\Ph$.
\nl
Indeed, for $x=x_1+x_2,x_1\in W,x_2\in Z$ we have $Nx=Nx_1+Nx_2$ with $Nx_1\in W$,
$Nx_2\in Z$ and $x_2-x'_2\in W$ with $x'_2\in Z'$. We have $Nx_2-Nx'_2\in W$ with
$Nx_2\in Z$, $Nx'_2\in Z'$. Hence $\Ph(Nx)=Nx_1+Nx'_2=N(x_1+x'_2)=N\Ph(x)$, as
required.
\subhead 2.8\endsubhead
Let $V,N$ be as in 2.3. Let $r\ge1$ be such that $N^r=0$. A subspace $W$ of $V$ is said
to be {\it $r$-special} if $NW\sub W$, $N:W@>>>W$ has no Jordan blocks of size $\ne r$
and $N^{r-1}=0$ on $N/W$. We show:
(a) {\it If $W,W'$ are $r$-special subspaces then there exists a subspace $X$ of $V$
such that $NX\sub X,W\op X=V,W'\op X=V$.}
\nl
We argue by induction on $r$. If $r=1$ the result is obvious; we have $W=W'=V$. Assume
that $r\ge2$. Let $V'=\ker N^{r-1},V''=\ker N^{r-2}$. Let $E\sub W,E'\sub W'$ be such
that $W=E\op NE\op\do N^{r-1}E$, $W'=E'\op NE'\op\do N^{r-1}E'$. Clearly,
$E\cap V'=0$, $E'\cap V'=0$, $NE\sub V'$, $NE\cap V''=0$, $NE'\sub V'$,
$NE'\cap V''=0$. Let $E''$ be a subspace of $V'$ such that $E''$ is a complement of
$NE\op V''$ in $V'$ and a complement of $NE'\op V''$ in $V'$. (Such $E''$ exists since
$\dim(NE\op V'')=\dim(NE'\op V')=\dim E+\dim V''=\dim E'+\dim V''$.) Then
$W_1=(E''\op NE)+N(E''\op NE)+\do+N^{r-2}(E''\op NE)$,
$W'_1=(E''\op NE')+N(E''\op NE')+\do+N^{r-2}(E''\op NE')$
\nl
are $(r-1)$-special subspaces of $V'$. By the induction hypothesis we can find an
$N$-stable subspace $X_1$ of $V'$ such that $V_1\op X_1=V',V'_1\op X_1=V'$. Then
$X=(E''+N(E'')+\do+N^{r-2}(E''))+X_1$ has the required properties.
(b) {\it If $W,W'$ are $r$-special subspaces then there exists $g\in1+E_{\ge1}V_*$ such
that $g(W)=W'$, $gN=Ng$.}
\nl
Let $X$ be as in (a). Define $g\in GL(V)$ by $g(x)=x$ for $x\in X$ and $g(w)=w'$ for
$w\in W$ where $w'\in W'$ is given by $w-w'\in X$. Then $g(W)=W'$, $(g-1)X=0$ and
$(g-1)W\sub X$. Clearly, $gN=Ng$. We have $V_{\ge a}=W_{\ge a}\op X_{\ge a}$. It
suffices to show that $(g-1)(W_{\ge a})\sub X_{\ge a+1}$. Now $X=X_{\ge2-r}$. We have
$W=W_{\ge1-r},W_{\ge2-r}=W_{\ge3-r}=NW,W_{\ge4-r}=W_{\ge5-r}=N^2W,\do$. Now if
$a\le1-r$ then $(g-1)W_{\ge a}=(g-1)W\sub X=X_{\ge a+1}$. If $a=2-r$ or $a=3-r$ then
$(g-1)W_{\ge a}=(g-1)NW=N(g-1)W\sub NX=NX_{\ge2-r}\sub X_{\ge4-r}\sub X_{\ge a+1}$.
\nl
If $a=4-r$ or $a=5-r$ then
$(g-1)W_{\ge a}=(g-1)N^2W=N^2(g-1)W\sub N^2X=N^2X_{\ge2-r}\sub X_{\ge6-r}\sub
X_{\ge a+1}$.
\nl
Continuing in this way, the result follows.
\subhead 2.9\endsubhead
Let $V\in\bcc$. Let $G=GL(V)$. For any filtration $V_*$ of $V$ let
$$\x(V_*)=\{N\in\Nil(V);V^N_*=V_*\}=\{N\in E_{\ge2}V_*;\bN\in\End_2^0(\gr V_*)\}$$
(see 2.3(b), 2.4(a)). The following three conditions are equivalent:
(i) $\x(V_*)\ne\em$;
(ii) $\End_2^0(\gr V_*)\ne\em$;
(iii) $\dim\gr_nV_*=\dim\gr_{-n}V_*\ge\dim\gr_{-n-2}V_*$ for all $n\ge0$.
\nl
We have (i)$\imp$(ii) by the definition of $\x(V_*)$; we have (ii)$\imp$(iii) by
2.3(d). The fact that (iii)$\imp$(ii) is easily checked. If (ii) holds we pick for any
$a$ a subspace $V_a$ of $V_{\ge a}$ complementary to $V_{\ge a+1}$ and an element in
$\End_2^0(V)$ (defined in terms of the grading $\op_aV_a$). This element is in
$\x(V_*)$ and (i) holds.
Let $\fF_V$ be the set of all filtrations $V_*$ of $V$ that satisfy (i)-(iii). From the
definitions we have a bijection
(a) $\fF_V@>\si>>D_G,V_*\m\l$
\nl
($D_G$ as in 1.1) where $\l=(G^\l_0\sps G^\l_1\sps G^\l_2\sps\do)$ is defined in terms
of $V_*$ by $G^\l_0=E_{\ge0}V_*\cap G$ and $G^\l_n=1+E_{\ge n}V_*$ for $n\ge1$.
The sets $\x(V_*)$ (with $V_*\in\fF_V)$ form a partition of $\Nil(V)$. (If
$N\in\Nil(V)$ we have $N\in\x(V_*)$ where $V_*=V^N_*$).
Let $V_*\in\fF_V$. Let $\Pi=E_{\ge0}V_*\cap G$. We show that $\x(V_*)$ is a single
$\Pi$-conjugacy class. Let $N,N'\in\x(V_*)$. Since $V^N_*=V^{N'}_*$ we see from 2.3(d)
that $\dim P^{\bN}_{1-j}=\dim P^{\bN'}_{1-j}$ for any $j\ge0$. Using 2.3(c) we see that
for any $j\ge0$, $N,N'$ have the same number of Jordan blocks of size $j$. Hence there
exists $g\in G$ such that $N'=gNg\i$. For any $a$,
$gV_{\ge a}^N=V_{\ge a}^{N'}=V_{\ge a}^N$ hence $gV_{\ge a}=V_{\ge a}$. We see that
$g\in E_{\ge0}$ hence $g\in\Pi$, as required. Taking in the previous argument $N'=N$,
we see that, if $N\in\x(V_*)$ and $g\in G$ satisfies $gNg\i=N$ then $g\in\Pi$. Now any
element in $E_{\ge2}V_*-\x(V_*)$ is in the closure of $\x(V_*)$ (since $E_{\ge2}V_*$ is
irreducible and $\x(V_*)$ is open in it (and non-empty) hence it is in the closure of
the $G$-conjugacy class containing $\x(V_*)$. We show that it is not contained in that
$G$-conjugacy class. (Assume that it is. Then we can find $N\in\x(V_*)$ and
$N'\in E_{\ge2}V_*-\x(V_*)$ that are $G$-conjugate. Then the $\Pi$-orbit $\Pi(N)$ of
$N$ in $E_{\ge2}V_*$ is $\x(V_*)$ hence is dense in $E_{\ge2}V_*$ while the $\Pi$-orbit
$\Pi(N')$ of $N'$ is contained in the proper closed subset $E_{\ge2}V_*-\x(V_*)$ of
$E_{\ge2}V_*$; hence $\dim\Pi(N)=\dim(E_{\ge2}V_*)>\dim\Pi(N')$. It follows that $a<a'$
where $a$ (resp. $a'$) is the dimension of the centralizer of $N$ (resp. $N'$) in
$\Pi$. Let $\ta$ (resp. $\ta'$) be the dimension of the centralizer of $N$ (resp. $N'$)
in $G$. By an earlier argument we have $a=\ta$. Obviously, $a'\le\ta'$. Since $N,N'$
are $G$-conjugate, we have $\ta=\ta'$. Thus, $\ta=a<a'\le\ta'=\ta$, contradiction.) We
see that $1+\x(V^*)=X^\l$ where $V_*\m\l$ as in (a) and $X^\l$ is as in 1.1. Thus
$\fP_1$ holds for $G$. From this $\fP_2,\fP_3$ follow; $H^\bla$ in $\fP_2$ is a single
conjugacy class in this case. Also, $\fP_8$ is trivial since $G^\l_0$ acts transitively
on $X^\l$. Now $\fP_5$ is easily verified. $\fP_6$ (hence $\fP_4$) follows from 2.6(a);
$\fP_7$ is trivial in this case.
\head 3. Symplectic groups\endhead
\subhead 3.1\endsubhead
In this section, any text marked as $\sp\do\sp$ applies only in the case $p=2$.
For $V,V'\in\cc$ let $\Bil(V,V')$ be the space of all bilinear forms $V\T V'@>>>\kk$.
For $b\in\Bil(V,V')$ define $b^*\in\Bil(V',V)$ by $b^*(x,y)=b(y,x)$. We write $\Bil(V)$
instead of $\Bil(V,V)$. Let $\Symp(V)$ be the set of non-degenerate symplectic forms on
$V$.
Let $\bV\in\bcc$. We say that $\lar_0\in\Symp(\bV)$ is {\it admissible} if
$\la x,y\ra_0=0$ for $x\in\bV_a,y\in\bV_{a'},a+a'\ne0$. Assume that
$\lar_0\in\Symp(\bV)$ is admissible and that $\nu\in\End_2^0(\bV)$ is {\it
skew-adjoint} that is, $\la\nu(x),y\ra_0+\la x,\nu(y)\ra_0=0$ for $x,y\in\bV$. For
$n\ge0$ we define a bilinear form $b_n:P^\nu_{-n}\T P^\nu_{-n}@>>>\kk$ by
$b_n(x,y)=\la x,\nu^ny\ra_0$. We show:
(a) $b_n(x,y)=(-1)^{n+1}b_n(y,x)$ for $x,y\in P^\nu_{-n}$.
\nl
Indeed,
$$b_n(x,y)=\la x,\nu^ny\ra_0=(-1)^n\la\nu^nx,y\ra_0=(-1)^{n+1}\la y,\nu^nx\ra_0
=(-1)^{n+1}b_n(y,x),$$
as required. We show:
(b) {\it$b_n$ is non-degenerate.}
\nl
Let $y\in P^\nu_{-n}$ be such that $\la x,\nu^n y\ra_0=0$ for all $x\in P^\nu_{-n}$. If
$x'\in P^\nu_{-n-2k},k>0$, we have
$\la\nu^kx',\nu^ny\ra_0=\pm\la x,\nu^{n+k}y\ra_0=\pm\la x,0\ra_0=0$. Since
$\bV_{-n}=\sum_{k\ge0}\nu^kP^\nu_{-n-2k}$, we see that $\la x,\nu^ny\ra_0=0$ for all
$x\in\bV_{-n}$. Since $\la\bV_m,\nu^ny\ra_0=0$ for $m\ne-n$ we see that
$\la\bV,\nu^ny\ra_0=0$. By the non-degeneracy of $\lar_0$, it follows that $\nu^ny=0$.
Since $\nu^n:\bV_{-n}@>\si>>\bV_n$, it follows that $y=0$, as required. We show:
(c) {\it if $n\ge0$ is even then $b_n$ is a symplectic form. Hence $\dim P^\nu_{-n}$ is
even.}
\nl
Indeed, for $x\in P^\nu_{-n}$ we have
$\la x,\nu^nx\ra_0=\pm\la\nu^{n/2}x,\nu^{n/2}x\ra_0=0$.
\subhead 3.2\endsubhead
Let $V\in\cc$ and let $\lar\in\Symp(V)$. Let
$Sp(\lar)=\{T\in GL(V);T\text{ preserves }\lar\}$.
\nl
For any subspace $W$ of $V$ we set $W^\pe=\{x\in V;\la x,W\ra=0\}$. A filtration $V_*$
of $V$ is said to be {\it self-dual} if $(V_{\ge a})^\pe=V_{\ge1-a}$ for any $a$. It
follows that
(a) $\la V_{\ge a},V_{\ge a'}\ra=0$ if $a+a'\ge1$.
\nl
It also follows that there is a unique admissible $\lar_0\in\Symp(\gr V_*)$ such that
for $x\in\gr_aV_*,y\in\gr_{-a}V_*$ we have $\la x,y\ra_0=\la\dx,\dy\ra$ where
$\dx\in V_{\ge a},\dy\in V_{\ge-a}$ represent $x,y$. Moreover,
(b) {\it there exists a direct sum decomposition $\op_{a\in\ZZ}V_a$ of $V$ such that
$V_{\ge a}=V_a\op V_{a+1}\op\do$ for all $a$ and $\la V_a,V_{a'}\ra=0$ for all $a,a'$
such that $a+a'\ne0$.}
Let $\cm_{\lar}$ be the set of $N\in\Nil(V)$ such that
$\la Nx,y\ra+\la x,Ny\ra+\la Nx,Ny\ra=0$ for all $x,y$ or equivalently
$1+N\in Sp(\lar)$. Define an involution $N\m N^\da$ of $\cm_{\lar}$ by
$\la x,Ny\ra=\la N^\da x,y\ra$ for all $x,y\in V$ or equivalently by
$N^\da=(1+N)\i-1=-N+N^2-N^3+\do$.
Let $N\in\cm_{\lar}$. We set $V_*=V^N_*$. By 2.6(c) we have $V^{N^\da}_*=V_*$. We
show:
(c) {\it the filtration $V_*$ is self-dual.}
\nl
We argue by induction on $e$ as in 2.4. If $a\ge e$ then $V_{\ge a}=0,V_{\ge1-a}=V$ and
(c) holds. If $a\le1-e$ then $V_{\ge a}=V,V_{\ge1-a}=0$ and (c) holds. If $e\le1$ this
already suffices. Hence we may assume that $e\ge2$ and $2-e\le a\le e-1$ hence
$2-e\le1-a\le e-1$. Let $V'=\ker(N^{e-1})/\Im(N^{e-1})$. Let $\r:\ker(N^{e-1})@>>>V'$
be the canonical map. We have $N^{e-1}V=\ker((N^\da)^{e-1})^\pe=\ker(N^{e-1})^\pe$
since $(N^\da)^{e-1}=(-N)^{e-1}$. Hence $\lar$ induces $\lar'\in\Symp(V')$. Also $N$
induces a linear map $N':V'@>>>V'$ such that $N'\in\cm_{\lar'}$. By the induction
hypothesis, $V'{}^{N'}_{\ge1-a}$ is the perpendicular in $V'$ of $V'{}^{N'}_{\ge a}$.
Hence $V_{\ge a}=\r\i(V'{}^{N'}_{\ge a})$ is the perpendicular in
$V$ of $V_{\ge1-a}=\r\i(V'{}^{N'}_{\ge1-a})$. This completes the proof.
Let $\nu\in\End_2^0(\gr V_*)$ be the endomorphism induced by $N$. We show that
(d) {\it$\nu$ is skew-adjoint (with respect to $\lar_0$ on $\gr V^N_*$).}
\nl
It suffices to show that, if $a+a'+2=0$ and $x\in V_{\ge a'}$, $y\in V_{\ge a}$ then
$\la Nx,y\ra+\la x,Ny\ra=0$. It suffices to show that $\la Nx,Ny\ra=0$. From (a),(b) we
see that $\la V_{\ge-1-a},Ny\ra=0$ hence it suffices to show that $Nx\in V_{\ge-1-a}$.
We have $Nx\in V_{\ge a'+2}\sub V_{\ge-1-a}$ since $a'+2>-1-a$. This proves (c).
\subhead 3.3\endsubhead
$\sp$ In this subsection we assume that $p=2$. Let $V,\lar,N,\nu,\lar_0$ be as in 3.2.
Let $V_*=V^N_*$. Then $b_n\in\Bil(P_{-n}^\nu)$ is defined for $n\ge0$, see 3.1. Let
$\cl$ be the set of all even integers $n\ge2$ such that $b_{n-1},b_{n+1}$ are
symplectic forms. Let $\cl'$ be the set of all even integers $n\ge2$ such that
$b_{n-1},b_{n+1},b_{n+3},\do$ are symplectic forms or equivalently, if
$\la z,\nu^{n-1}(z)\ra_0=0$ for all $z\in\gr_{1-n}V_*$. (Assume first that
$b_{n-1},b_{n+1},b_{n+3},\do$ are symplectic forms. By 2.1(a), any $z\in\gr_{1-n}V_*$
is of the form $\sum_{k\ge0}\nu^kz_k$ where $z_k\in P^\nu_{1-n-2k}$. For $k\ge0$ we
have $\la\nu^kz_k,\nu^{n-1}(\nu^kz_k)\ra_0=0$ since $b_{n+2k-1}$ is symplectic. Since
$z'\m\la z',\nu^{n-1}(z')\ra_0$ is additive in $z'$ it follows that
$\la z,\nu^{n-1}(z)\ra_0=0$. Conversely, assume that $\la z,\nu^{n-1}(z)\ra_0=0$ for
any $z\in\gr_{1-n}V_*$. In particular, for $k\ge0$ and $z_k\in P^\nu_{1-n-2k}$ we have
$\la \nu^kz_k,\nu^{n-1}(\nu^kz_k)\ra_0=0$ that is, $\la z_k,\nu^{n-1+2k}z_k)\ra_0=0$.
We see that $b_{n+2k-1}$ is symplectic.)
Clearly, $\cl'\sub\cl$.
For $n\in\cl$, we define $q_n:P^\nu_{-n}@>>>\kk$ by $q_n(x)=\la\dx,N^{n-1}\dx\ra$ where
$\dx\in V_{\ge-n}$ is a representative for $x\in P^\nu_{-n}$ such that $N^{n+1}\dx=0$
(see 2.5(c)). We show that $q_n(x)$ is well defined. It suffices to show that if
$y\in V_{\ge1-n},N^{n+1}y=0$ then $\la\dx+y,N^{n-1}(\dx+y)\ra=\la\dx,N^{n-1}\dx\ra$
that is, $\la y,N^{n-1}(y)\ra+\la\dx,N^{n-1}(y)\ra+\la y,N^{n-1}(\dx)\ra=0$. Since
$N^{n+1}(\dx)=0$, we have
$$\la\dx,N^{n-1}(y)\ra+\la y,N^{n-1}(\dx)\ra=\la y,(N^{n-1}+(N^\da)^{n-1})(\dx)\ra
=\la y,N^n(\dx)\ra.$$
This is zero, since $y\in V_{\ge1-n},N^n(\dx)\in V_{\ge n}$ and $1-n+n=1$. It remains
to show that $\la y,N^{n-1}(y)\ra=0$. It suffices to show that
$\la z,\nu^{n-1}(z)\ra_0=0$ for all $z\in\gr_{1-n}V_*$ such that $N^{n+1}z=0$. By
2.1(a) any such $z$ is of the form $z_0+\nu z_1$ where
$z_0\in P^\nu_{1-n},z_1\in P^\nu_{-1-n}$. Now $z'\m\la z',\nu^{n-1}(z')\ra_0$ is
additive in $z'$ hence it suffices to show that $\la z_0,\nu^{n-1}(z_0)\ra_0=0$ and
$\la\nu(z_1),\nu^{n-1}(\nu(z_1))\ra_0=0$ for $z_0,z_1$ as above. This follows from our
assumption that $b_{n-1}$ and $b_{n+1}$ are symplectic.
We show:
(a) {\it For $x,y\in P^\nu_{-n}$ we have $q_n(x+y)=q_n(x)+q_n(y)+b_n(x,y)$.}
\nl
Let $\dx,\dy\in V_{\ge-n}$ be representatives for $x,y$ such that $N^{n+1}\dx=0$,
$N^{n+1}\dy=0$. We must show that
$\la\dx+\dy,N^{n-1}(\dx+\dy)\ra=\la\dx,N^{n-1}(\dx)\ra+\la\dy,N^{n-1}(\dy)\ra
+\la\dx,N^n(\dy)\ra$,
\nl
or that
$\la\dx,N^{n-1}(\dy)\ra+\la\dy,N^{n-1}(\dx)\ra+\la\dx,N^n(\dy)\ra=0$,
\nl
or that $\la\dx,((N^\da)^{n-1}+N^{n-1}+N^n)\dy\ra=0$. Since $n$ is even,
$(N^\da)^{n-1}+N^{n-1}+N^n$ is a linear combination of $N^{n+1},N^{n+2},\do$ and it
remains to use that $N^{n+1}(\dy)=0$.
For $n\in\cl'$, we define $Q_n:\gr_{-n}V_*@>>>\kk$ by $Q_n(x)=\la\dx,N^{n-1}\dx\ra$
where $\dx\in V_{\ge-n}$ is a representative for $x$. We show that $Q_n(x)$ is well
defined. It suffices to show that if $y\in V_{\ge1-n}$ then
$\la\dx+y,N^{n-1}(\dx+y)\ra=\la\dx,N^{n-1}\dx\ra$ that is,
$\la y,N^{n-1}(y)\ra+\la\dx,N^{n-1}(y)\ra+\la y,N^{n-1}(\dx)\ra=0$. We have
$\la\dx,N^{n-1}(y)\ra+\la y,N^{n-1}(\dx)\ra=\la y,(N^{n-1}+(N^\da)^{n-1})(\dx)\ra$
\nl
and this is a linear combination of terms $\la y,N^{n'}(\dx)\ra$ with $n'\ge n$. Each
of these terms is $0$ since $y\in V_{\ge1-n},N^{n'}(\dx)\in V_{\ge2n'-n}$ and
$1-n+2n'-n\ge1$. It remains to show that $\la y,N^{n-1}(y)\ra=0$. This follows from the
fact that $\la z,\nu^{n-1}(z)\ra_0=0$ for all $z\in\gr_{1-n}V_*$.
For $n\in\cl'$ we show:
(b) {\it if $x,y\in\gr_{-n}V_*$ then $Q_n(x+y)=Q_n(x)+Q_n(y)+\la x,\nu^ny\ra$.}
\nl
Let $\dx,\dy\in V_{\ge-n}$ be representatives for $x,y$. We must show that
$\la\dx+\dy,N^{n-1}(\dx+\dy)\ra=\la\dx,N^{n-1}(\dx)\ra+\la\dy,N^{n-1}(\dy)\ra
+\la\dx,N^n(\dy)\ra$,
\nl
or that
$\la\dx,N^{n-1}(\dy)\ra+\la\dy,N^{n-1}(\dx)\ra+\la\dx,N^n(\dy)\ra=0$,
\nl
or that $\la\dx,((N^\da)^{n-1}+N^{n-1}+N^n)\dy\ra$ is $0$. Since $n$ is even, this is a
linear combination of terms $\la \dx,N^{n'}(\dy)\ra$ with $n'>n$. Each of these terms
is $0$ since $N^{n'}(\dy)\in V_{\ge2n'-n},\dx\in V_{\ge-n}$ and $2n'-n-n\ge1$.
Now let $n\in\cl'$ and let $x\in\gr_{-n}V_*$. We can write $x=\sum_{k\ge0}\nu^kx_k$
where $x_k\in P^\nu_{-n-2k}$. We show that
(c) $Q_n(x)=\sum_{k\ge0}q_{n+2k}(x_k)$.
\nl
Let $\dx_k$ be a representative of $x_k$ in $V_{\ge-n-2k}$ such that
$N^{n+2k+1}\dx_k=0$. Then $\sum_{k\ge0}N^k\dx_k$ is a representative of $x$ in
$V_{\ge-n}$ and we must show:
$$\la\sum_{k\ge0}N^k\dx_k,N^{n-1}\sum_{k'\ge0}N^{k'}\dx_{k'}\ra=
\sum_{k\ge0}\la\dx_k,N^{n+2k-1}\dx_k\ra.$$
The left hand side is $\sum_{k,k'\ge0}\la N^k\dx_k,N^{n-1+k'}\dx_{k'}\ra$. If
$k\ge k'+2$ we have
$\la N^k\dx_k,N^{n-1+k'}\dx_{k'}\ra=\la\dx_k,(N^\da)^kN^{n-1+k'}\dx_{k'}\ra$
\nl
and this is zero since $N^{n+2k'+1}\dx_{k'}=0$. If $k'\ge k+2$ we have
$\la N^k\dx_k,N^{n-1+k'}\dx_{k'}\ra=\la (N^\da)^{n-1+k'}N^k\dx_k,\dx_{k'}\ra$
\nl
and this is zero since $N^{n+2k+1}\dx_k=0$. It suffices to show that
$$\align&\sum_{k\ge0}(\la N^k\dx_k,N^{n-1+k}\dx_k\ra
+\la N^{k+1}\dx_{k+1},N^{n-1+k}\dx_k\ra+\la N^k\dx_k,N^{n+k}\dx_{k+1}\ra)\\&
=\sum_{k\ge0}\la \dx_k,N^{n+2k-1}\dx_k\ra.\endalign$$
We have
$$\align&\la N^{k+1}\dx_{k+1},N^{n-1+k}\dx_k\ra+\la N^k\dx_k,N^{n+k}\dx_{k+1}\ra\\&
=\la N^{k+1}\dx_{k+1},(N^{n-1+k}+(N^\da)^{n-1}N^k)\dx_k\ra\\&=\la N^{k+1}\dx_{k+1},
(c_1N^{n+k}+c_2N^{n+k+1}+\do)\dx_k\ra\\&=c_1\la \nu^{k+1}x_{k+1},
\nu^{n+k}x_k\ra_0=c_1\la x_{k+1},\nu^{n+2k+1}x_k\ra_0=0.\endalign$$
(Here $c_1,c_2,\do\in\kk$.) It suffices to show that
$\la N^k\dx_k,N^{n-1+k}\dx_k\ra+\la\dx_k,N^{n+2k-1}\dx_k\ra$ is $0$. This equals
$$\align&\la \dx_k,(N^{n+2k-1}+(N^\da)^kN^{n-1+k})\dx_k\ra=\la \dx_k,(N^{n+2k}
+c'_1N^{n+2k+1}+\do)\dx_k\ra\\&
=\la x_k,\nu^{n+2k}x_k\ra_0=\la \nu^{k+n/2}x_k,\nu^{k+n/2}x_k\ra_0=0.\endalign$$
(Here $c'_1,c'_2,\do\in\kk$.) This completes the proof of (c).
We say that $(q_n)_{n\in\cl}$ are {\it the quadratic forms attached to $(N,\lar)$.} We
say that $(Q_n)_{n\in\cl'}$ are {\it the Quadratic forms attached to $(N,\lar)$}. $\sp$
\subhead 3.4\endsubhead
Let $V\in\cc$ and let $V_*$ be a filtration of $V$. We fix
$\lar_0\in\Symp(\gr V_*)$ which is admissible and $\nu\in\End_2^0(\gr V_*)$ which is
skew-adjoint with respect to $\lar_0$ (see 3.1). Then $P^\nu_{-n}$ are defined in terms
of $\gr V_*,\nu$ and $b_n\in\Bil(P^\nu_{-n})$ are defined as in 3.1 for any $n\ge0$.
Let $\cv=1+E_{\ge1}V_*$, a subgroup of $GL(V)$.
$\sp$ If $p=2$, let $\nn$ be the smallest even integer $\ge2$ such that
$b_{\nn-1},b_{\nn+1},b_{\nn+3},\do$ are symplectic or, equivalently, such that
$\la z,\nu^{\nn-1}(z)\ra_0=0$ for all $z\in\gr_{1-\nn}V_*$. Let
$Q:\gr_{-\nn}V_*@>>>\kk$ be a quadratic form such that
$Q(x+y)=Q(x)+Q(y)+\la x,\nu^\nn y\ra$ for all $x,y\in\gr_{-\nn}V_*$. $\sp$.
Let $\cz$ be the set of all pairs $(N,\lar)$ where $N\in\Nil(V)$, $\lar\in\Symp(V)$ are
such that $V^N_*=V_*$, $\la Nx,y\ra+\la x,Ny\ra+\la Nx,Ny\ra=0$ for $x,y\in V$, $N$
induces $\nu$ on $\gr V_*$, $\lar$ induces $\lar_0$ on $\gr V_*$; $\sp$ in the case
$p=2$ we require in addition that $Q_\nn$ defined in terms of $(N,\lar)$ as in 3.3 is
equal to $Q$. $\sp$
The proofs of Propositions 3.5, 3.6, 3.7 below are intertwined (see 3.11).
\proclaim{Proposition 3.5}In the setup of 3.4 let $\lar\in\Symp(V)$ be such that $V_*$
is self-dual with respect to $\lar$ and $\lar$ induces $\lar_0$ on $\gr V_*$. Let
$Y=Y_{\lar}=\{N;(N,\lar)\in\cz\}$. Let $U'=\cv\cap Sp(\lar)$, a subgroup of $Sp(\lar)$.
Then
(a) $Y\ne\em$;
(b) if $N\in Y$ and $z\in U'$ then $zNz\i\in Y$ (thus $U'$ acts an $Y$ by conjugation);
(c) the action (b) of $U'$ on $Y$ is transitive.
\endproclaim
The proof of (a) is given in 3.8. Now (b) follows immediately from 3.7(a).
We show that (c) is a consequence of 3.7(c). Assume that 3.7(c) holds. Let $N,N'\in Y$.
We have $(N,\lar)\in\cz,(N',\lar)\in\cz$ and by 3.7(c) there exists $g\in\cv$ such that
$N'=gNg\i$, $\la g\i x,g\i y\ra=\la x,y\ra$ for $x,y\in V$. Then $g\in U'$ and (c) is
proved (assuming 3.7(c)).
\proclaim{Proposition 3.6}In the setup of 3.4 let $N\in\Nil(V)$ be such that
$V^N_*=V_*$ and $N$ induces $\nu$ on $\gr V_*$. Let
$X=X_N=\{\lar;(N,\lar)\in\cz\}$. Let $U=U_N=\{T\in\cv;TN=NT\}$, a subgroup of
$GL(V)$. Then:
(a) $X\ne\em$;
(b) if $\lar\in X$ and $u\in U$ then the symplectic form $\lar'$ on $V$ given by
$\la x,y\ra'=\la u\i x,u\i y\ra$ belongs to $X$ (thus $U$ acts naturally an $X$);
(c) the action (b) of $U$ on $X$ is transitive.
\endproclaim
We show that (a) is a consequence of 3.7(a). By 3.7(a) there exists
$(N',\lar')\in\cz$. By 2.6(a) there exists $g\in\cv$ such that $N=gN'g\i$. Define
$\lar\in\Symp(V)$ by $\lar=\la g\i x,g\i y\ra'$. From 3.7(a) we see that
$(N,\lar)\in\cz$ hence $\lar\in X_N$. Thus $X_N\ne\em$, as required.
Now (b) follows immediately from 3.7(b). The proof of (c) is given in 3.9, 3.10.
\proclaim{Proposition 3.7}In the setup of 3.4,
(a) $\cz\ne\em$;
(b) if $(N,\lar)\in\cz$, $g\in\cv$ and $(N',\lar')$ is defined by $N'=gNg\i$,
$\la x,y\ra'=\la g\i x,g\i y\ra$ then $(N',\lar')\in\cz$ (thus $\cv$ acts naturally on
$\cz$);
(c) the action (b) of $\cv$ on $\cz$ is transitive.
\endproclaim
Clearly, (a) is a consequence of 3.5(a).
We prove (b). We have $V^{N'}_{\ge a}=gV^N_{\ge a}=V^N_{\ge a}=V_{\ge a}$. Next we must
show that $\la gNg\i x,y\ra'+\la x,gNg\i y\ra'+\la gNg\i x,gNg\i y\ra'=0$ for
$x,y\in V$ that is, $\la Ng\i x,g\i y\ra+\la g\i x,Ng\i y\ra+\la Ng\i x,Ng\i y\ra=0$
for $x,y\in V$. This follows from $\la Nx',y'\ra+\la x',Ny'\ra+\la Nx',Ny'\ra=0$ for
$x',y'\in V$. Next we must show that $gNg\i,N$ induce the same map
$\gr V_*@>>>\gr V_*$. (We must show: if $x\in V_{\ge a}$ then
$gNg\i(x)-Nx\in V_{\ge a+3}$; this follows from $g\in\cv$.) Next we must show that for
$x\in V_{\ge-a},y\in V_{\ge a}$ we have $\la x,y\ra'=\la x,y\ra$ that is
$\la g\i x,g\i y\ra=\la x,y\ra$. Set $g\i=1+S$ where $S\in E_{\ge1}V_*$. We must show
that $\la Sx,y\ra+\la x,Sy\ra+\la Sx,Sy\ra=0$. But $Sx\in V_{\ge1-a},y\in V_{\ge a}$
implies $\la Sx,y\ra=0$. Similarly $\la x,Sy\ra=0,\la Sx,Sy\ra=0$.
$\sp$ In the case where $p=2$ we see that the number $\nn$ defined in terms of $N,\lar$
is the same as that defined in terms of $N',\lar'$ and we must check that for
$x\in V_{\ge-\nn}$ we have $\la x,(gNg\i)^{\nn-1}x\ra'=\la x,N^{\nn-1}x\ra$
that is, $\la g\i x,N^{\nn-1}g\i x\ra=\la x,N^{\nn-1}x\ra$ that is,
$\la Sx,N^{\nn-1}x\ra+\la x,N^{\nn-1}Sx\ra+\la Sx,N^{\nn-1}Sx\ra=0$. We have
$\la Sx,N^{\nn-1}x\ra+\la x,N^{\nn-1}Sx\ra=\la x,(N^{\nn-1}+(N^\da)^{\nn-1}Sx\ra$. This
is a linear combination of terms $\la x,N^{n'}Sx\ra$ where $n'\ge\nn$; each of these
terms is zero since $x\in V_{\ge-\nn},N^{n'}Sx\in V_{\ge2n'-\nn+1}$ and
$-\nn+2n'-\nn+1\ge1$. Next we have $\la Sx,N^{\nn-1}Sx\ra=0$ since
$\la y,N^{\nn-1}y\ra=0$ for all $y\in V_{\ge1-\nn}$ by the definition of $\nn$. $\sp$
This completes the proof of (b).
We show that (c) is a consequence of 3.6(c). Let
$(N,\lar)\in\cz,(N',\lar')\in\cz$. By 2.6(a), since $V_*^N=V_*^{N'}$ and $N,N'$ induce
the same $\nu$, we can find $S\in E_{\ge1}V_*$ such that $R=1+S$ satisfies $N'R=RN$.
Define $\lar''\in\Symp(V)$ by $\la x,y\ra''=\la Rx,Ry\ra'$. From (b) we see that
$(R\i N'R,\lar'')\in\cz$ that is $(N,\lar'')\in\cz$. Thus $\lar\in X_N,\lar''\in X_N$.
By 3.6(c) we can find $S'\in E_{\ge1}V_*$ such that $R'=1+S'$ satisfies $R'N=NR'$ and
$\la x,y\ra=\la R'x,R'y\ra''$ for all $x,y$ that is, $\la x,y\ra=\la RR'x,RR'y\ra'$.
Then $RR'\in U'$ and $RR'N=RNR'=N'RR'$. Thus under the action (b), $RR'$ carries
$(N,\lar)$ to $(N',\lar')$. This proves (c) (assuming 3.6(c)).
\subhead 3.8. Proof of 3.5(a)\endsubhead
We choose a direct sum decomposition $\op_{a\in\ZZ}V_a$ of $V$ as in 3.2(b). Define
$N_2\in\End_2(V)$ by the requirement that $N_2:V_a@>>>V_{a+2}$ corresponds to
$\nu:\gr_aV_*@>>>\gr_{a+2}V_*$ under the obvious isomorphisms $V_a@>\si>>\gr_aV_*$,
$V_{a+2}@>\si>>\gr_{a+2}V_*$.
$\sp$. If $p=2$ we regard $Q$ as a quadratic form on $V_{-\nn}$ via the obvious
isomorphism $V_{-\nn}@>\si>>\gr_{-\nn}V_*$. $\sp$
We will construct a linear map $N=\sum_{j\ge1}N_{2j}$ where $N_2$ is as above and for
$j\ge2$, $N_{2j}\in\End(V)$ satisfy $N_{2j}V_a\sub V_{a+2j}$ for all $a$ and
$$\la\sum_{j\ge1}N_{2j}x,y\ra+\la x,\sum_{j\ge1}N_{2j}y\ra+
\la\sum_{j'\ge1}N_{2j'}x,\sum_{j''\ge1}N_{2j''}y\ra=0$$
for any $a,c$ and any $x\in V_a,y\in V_c$ that is,
$$\la N_{2j}x,y\ra+\la x,N_{2j}y\ra
+\sum_{j',j''\ge1;j'+j''=j}\la N_{2j'}x,N_{2j''}y\ra=0\tag a$$
for any $j\ge1$, any $a,c$ such that $a+c+2j=0$ and any $x\in V_a,y\in V_c$.
$\sp$ If $p=2$, we require in addition that $\la x,N^{\nn-1}x\ra=Q(x)$ for all
$x\in V_{-\nn}$ that is, $\sum_{i+i'=\nn-2}\la x,N_2^iN_4N_2^{i'}x\ra=Q(x)$ for all
$x\in V_{-\nn}$. $\sp$
We shall determine $N_j$ by induction on $j$. For $j=1$ the equation (a) is just
$\la N_2x,y\ra+\la x,N_2y\ra=0$ for any $a,c$ such that $a+c+2=0$ and any
$x\in V_a,y\in V_c$; this holds automatically by our choice of $N_2$. For $x\in V_a$
with $a<-2$ we set $N_4(x)=0$. Then the equation (a) for $j=2$ becomes
(b) $\la N_4x,y\ra+\la x,N_4y\ra=-\la N_2x,N_2y\ra$ for any $x\in V_{-2},y\in V_{-2}$,
$\la N_4x,y\ra=-\la N_2x,N_2y\ra$ for any $a>-2,x\in V_a,y\in V_{-a-4}$.
\nl
The second equation in (b) determines uniquely $N_4(x)$ for $x\in V_a,a>-2$. Since
$\la N_2x,N_2y\ra$ is a symplectic form on $V_{-2}$ we can find $[,]\in\Bil(V_{-2})$
such that $[x,y]-[y,x]=-\la N_2x,N_2y\ra$ for any $x,y\in V_{-2}$. There is a unique
linear map $N_4:V_{-2}@>>>V_2$ such that $\la N_4x,y\ra=[x,y]$ for any $x,y\in V_{-2}$.
Then equation (a) for $j=2$ is satisfied.
$\sp$ If $p=2$ the $N_4$ just determined satisfies
$\sum_{i+i'=\nn-2}\la x,N_2^iN_4N_2^{i'}x\ra=Q'(x)$ for all $x\in V_{-\nn}$, for some
quadratic form $Q':V_{-\nn}@>>>\kk$ not necessarily equal to $Q$. For $x,y\in V_{-\nn}$
we have (by the choice of $N_4$):
$$\align&Q'(x+y)-Q'(x)-Q'(y)=\sum_{i+i'=\nn-2}\la x,N_2^iN_4N_2^{i'}y\ra+
\sum_{i+i'=\nn-2}\la y,N_2^iN_4N_2^{i'}x\ra\\&=\sum_{i+i'=\nn-2}\la N_2^ix,
N_4N_2^{i'}y\ra+\sum_{i+i'=\nn-2}\la N_4N_2^ix,N_2^{i'}y\ra
=\sum_{i+i'=\nn-2}\la N_2N_2^ix,N_2N_2^{i'}y\ra\\&=
\sum_{i+i'=\nn-2}\la x,N_2^\nn y\ra=\la x,N_2^\nn y\ra=Q(x+y)-Q(x)-Q(y).\endalign$$
It follows that $Q'(x)=Q(x)+\th(x)^2$ where $\th\in\Hom(V_{-\nn},\kk)$. We try to find
$\z\in\End(V)$ with $\z(V_a)\sub V_{a+4}$ for all $a$ in such a way that (a) (for
$j=2$) remains true when $N_4$ is replaced by $N_4+\z$ and
$\sum_{i+i'=\nn-2}\la x,N_2^i(N_4+\z)N_2^{i'}x\ra=Q(x)$ for $x\in V_{-\nn}$. (Then
$N_4+\z$ will be our new $N_4$.) Thus we are seeking $\z$ such that
$\la\z(x),y\ra+\la x,\z(y)\ra=0$ for any $a,c$ with $a+c+4=0$ and $x\in V_a,y\in V_c$,
$\sum_{i+i'=\nn-2}\la x,N_2^i\z N_2^{i'}x\ra=\th(x)^2$ for $x\in V_{-\nn}$.
\nl
The first of these two equations can be satisfied for $(a,c)\ne(-2,-2)$ by defining
$\z(x)=0$ for $x\in V_a,a\ne-2$. Then in the second equation the terms corresponding to
$i'$ such that $2i'-\nn\ne-2$ are $0$. Thus it remains to find a linear map
$\z:V_{-2}@>>>V_2$ such that
$\la \z(x),y\ra+\la x,\z(y)\ra=0$ for any $x,y\in V_{-2}$,
$\la N_2^tx,\z N_2^tx\ra=\th(x)^2$ for $x\in V_{-\nn}$ where $t=(\nn-2)/2$.
\nl
Since $N_2^t:V_{-\nn}@>>>V_{-2}$ is injective (by the Lefschetz condition), there
exists $\th_1\in\Hom(V_{-2},\kk)$ such that $\th_1(N_2^tx)=\th(x)$ for all
$x\in V_{-\nn}$. We see that it suffices to find $\z\in\Hom(V_{-2},V_2)$ such that
$\la \z(x),y\ra+\la x,\z(y)\ra=0$ for any $x,y\in V_{-2}$,
$\la x',\z x'\ra=\th_1(x')^2$ for $x'\in V_{-2}$.
\nl
It also suffices to find $b_0\in\Bil(V_{-2})$ such that $b_0=b_0^*$ and
$b_0(x,x)=\th_1(x)^2$ for $x\in V_{-2}$. Such $b_0$ clearly exists. $\sp$.
This completes the determination of $N_4$.
Now assume that $j\ge3$ and that $N_{2j'}$ is already determined for $j'<j$. For
$x\in V_a$ with $a<-j$ we set $N_{2j}(x)=0$. Then equation (a) for our $j$ determines
uniquely $N_{2j}(x)$ for $x\in V_a$ with $a>-j$. Next, we can find $[,]\in\Bil(V_{-j})$
such that
$[x,y]-[y,x]=-\sum_{j',j''\ge1;j'+j''=j}\la N_{2j'}x,N_{2j''}y\ra$.
\nl
To see this we observe that the right hand side is a symplectic form that is,
$\sum_{j',j''\ge1;j'+j''=j}\la N_{2j'}x,N_{2j''}x\ra=0$. There is a unique
$N_{2j}\in\Hom(V_{-j},V_j)$ such that $\la N_{2j}x,y\ra=[x,y]$ for any $x,y\in V_{-j}$.
Then equation (a) for our $j$ is satisfied. This completes the inductive construction
of $N$. We have $V_*^N=V_*$ by 2.4(a). We see that $N\in Y$. This completes the proof.
\subhead 3.9\endsubhead
In this subsection we prove 3.6(c) in a special case. Let $n\in\ZZ_{>0}$. We have
$[-n,n]=I_0\sqc I_1$ where $I_\e=\{i\in[-n,n];i=\e\mod 2\}$ for $\e\in\{0,1\}$. For
$i\in[-n,n]$ define $|i|\in\{0,1\}$ by $i=|i|\mod2$ that is by $i\in I_{|i|}$. Let
$F_0,F_1\in\cc$. Let $V=\op_{i\in[-n,n]}F_i$ where $F_i=F_{|i|}$. A typical element of
$V$ is of the form $(x_i)_{i\in[-n,n]}$ where $x_i\in F_{|i|}$. Define $N:V@>>>V$ by
$(x_i)\m(x'_i)$ where $x'_i=x_{i-2}$ for $i\in[2-n,n]$, $x'_{-n}=0,x'_{1-n}=0$. We fix
$\lar_0\in\Symp(V)$ such that
$\la (x_i),(y_i)\ra_0=\sum_{i\in[-n,n]}(-1)^{(i-|i|)/2}b^{|i|}(x_i,y_{-i})$ where
$b^\e\in\Bil(F_\e)(\e\in\{0,1\}$ satisfy $b^{\e*}=(-1)^{1-\e}b^\e$, $b^\e$ is
non-degenerate, $b^0\in\Symp(F_0)$. Note that $\la Nx,y\ra_0+\la x,Ny\ra_0=0$ for
$x,y\in V$.
We assume: if $p\ne2$ then either $F_0=0$ or $F_1=0$; $\sp$ if $p=2,b^1$ is symplectic
and $n\ge2$, then we are given a quadratic form $Q:F_0@>>>\kk$ such that
$Q(x+y)=Q(x)+Q(y)+b^0(x,y)$ for $x,y\in F_0$. $\sp$
Let $X$ be the set of all $\lar\in\Symp(V)$ such that
$\la Nx,y\ra+\la x,Ny\ra+\la Nx,Ny\ra=0$ for $x,y\in V$ and $\la x,y\ra=\la x,y\ra_0$
if there exists $i$ such that $x_j=0$ for $j\ne i$ and $y_j=0$ for $j\ne-i$; $\sp$ if
$p=2,b^1$ is symplectic and $n\ge2$, we require also that $\la x,Nx\ra=Q(x_{-2})$ if
$x\in V$ is such that $x_j=0$ for $j\ne-2$. $\sp$
Setting $\la (x_i),(y_i)\ra=\sum_{i,j}b_{ij}(x_i,y_j)$ identifies $X$ with the set of
all families $(b_{ij})_{i,j\in[-n,n]}$ where $b_{ij}\in\Bil(F_{|i|},F_{|j|})$ are such
that
$b_{i-2,j}+b_{i,j-2}+b_{ij}=0$ if $i,j\in[2-n,n]$,
$b_{i,-i}=(-1)^{(i-|i|)/2}b^{|i|}$ for all $i\in[-n,n]$,
$b_{ii}\in\Symp(F_{|i|})$ for all $i\in[-n,n]$,
$b_{ij}^*=-b_{ji}$ for all $i,j\in[-n,n]$,
$b_{-2,0}(x,x)=Q(x)$ for $x\in F_0$ if $p=2,F_1=0$ and $n$ is even, $\ge2$.
\nl
(We have automatically $b_{ij}=0$ if $i+j\ge1$.)
Let $\D=\{T\in GL(V);TN=NT\}$, a subgroup of $GL(V)$; equivalently $\D$ is the set of
linear maps $T:V@>>>V$ of the form
(a) $T:(x_i)\m(x'_i),x'_i=\sum_{j\in[-n,i]}T_{i-j}^{|i|,|j|}x_j$
\nl
where $T_r^{\e,\d}\in\Hom(F_\d,F_\e)$ $(r\in[0,2n],\e,\d\in\{0,1\},r+\d=\e\mod 2)$ are
such that $T_0^{00},T_0^{11}$ are invertible and $T_{2n}^{1-|n|,1-|n|}=0$. Now $\D$
acts on $X$ by $T:\lar\m\lar'$ where $\la Tx,Ty\ra'=\la x,y\ra$, or equivalently by
$T:(b_{ij})\m(b'_{ij})$ where
$$b_{ij}(x,y)=\sum_{i'\in[i,n],j'\in[j,n]}b'_{i'j'}(T_{i'-i}^{|i'|,|i|}(x),
T_{j'-j}^{|j'|,|j|}(y)).$$
Let $\D_u=\{T\in\D;T_0^{00}=1,T_0^{11}=1\}$, a subgroup of $\D$. We show:
(b) {\it Let $k\in[1-n,0]$ and let $(\tb_{ij}),(b_{ij})$ be two points of $X$ such that
$b_{ij}=\tb_{ij}$ for $i+j\ge2k$. Then there exists $T\in\D_u$ such that
$T(b_{ij})=(b'_{ij})$ and $b'_{ij}=\tb_{ij}$ for $i+j\ge2k-2$.}
\nl
For $\e\in\{0,1\}$ we set $a^\e=\tb_{ij}$ for $i,j\in[-n,n],i+j=-1,i=\e\mod2$. Then
$a^\e$ are independent of choices; they are $0$ unless $p=2$. We have
$a^{\e*}=a^{1-\e}$. For $h\in\{2k-2,2k-1\}$ we set
$c^\e_h=(-1)^{(i-\e)/2}(b_{ij}-\tb_{ij})$ where $i,j\in[-n,n],i+j=h,i=\e\mod2$. Then
$c^\e_h$ is independent of $i,j$. We have $c^\e_{2k-1}=0$ unless $p=2$. We have
$c^{\e*}_{2k-2}=(-1)^{k-\e}c_{2k-2}^\e$, $c_{2k-1}^{\e*}=c_{2k-1}^{1-\e}$. Since
$b_{k-1,k-1}-\tb_{k-1,k-1}$ is symplectic, $c_{2k-2}^\e$ is symplectic where
$\e=k-1\mod2$.
{\it Case 1: $p\ne2$.} Let $\e\in\{0,1\}$ be such that $F_{1-\e}=0$. Since
$c^{\e*}_{2k-2}=(-1)^{k-\e}c_{2k-2}^\e$, we can find $\tc\in\Bil(F_\e)$ such that
$c_{2k-2}^\e=\tc+(-1)^{k-\e}\tc^*$. Since $b^\e$ is non-degenerate we can find
$\t\in\End(F_\e)$ such that $\tc(x,y)=b^\e(x,\t(y))$ for $x,y\in F_\e$. For
$i,j\in[-n,n],i+j=2k-2,i=\e\mod2$ and $x,y\in F_\e$ we have
$$\align&b_{ij}(x,y)-\tb_{ij}(x,y)=(-1)^{(i-\e)/2}(\tc(x,y)+(-1)^{k-\e}\tc(y,x))\\&=
\tb_{i,j+2-2k}(x,\t(y))-\tb_{j,i+2-2k}(y,\t(x))=\tb_{i,j+2-2k}(x,\t(y))
+\tb_{i+2-2k,j}(\t(x),y).\endalign$$
Let $T$ be as in (a) with $T_0^{00}=1,T_0^{11}=1$, $T_{2-2k}^{\e,\e}=\t$ and the other
components $0$. Define $(b'_{ij})$ by $T(b_{ij})=(b'_{ij})$. Then $(b'_{ij})$ has the
required properties.
$\sp$ {\it Case 2: $p=2,k=0$.} Since $b^0$ is non-degenerate we can find
$T_1^{0,1}\in\Hom(F_1,F_0)$ such that $c_{-1}^0(x,y)=\tb^0(x,T_1^{0,1}(y))$ for all
$x\in F_0,y\in F_1$. Then $c_{-1}^1(x,y)=\tb^0(T_1^{0,1}(x),y)$ for all
$x\in F_1,y\in F_0$. Thus, for $i\in I_0,j\in I_1,i+j=-1$ and $x\in F_0,y\in F_1$ we
have $b_{ij}(x,y)+\tb_{ij}(x,y)=\tb_{i,-i}(x,T_1^{0,1}(y))$; for
$i\in I_1,j\in I_0,i+j=-1$ and $x\in F_1,y\in F_0$ we have
$$b_{ij}(x,y)+\tb_{ij}(x,y)=\tb_{-j,j}(T_1^{0,1}(x),y).$$
Since $c_{-2}^{0*}=c_{-2}^0,b^{1*}=b^1$, we have $c_{-2}^0(y,y)=\th(y)^2$ for
$y\in F_0$, $b^1(x,x)=\th_1(x)^2$ for $x\in F_1$ where $\th\in\Hom(F_0,\kk)$,
$\th_1\in\Hom(F_1,\kk)$. If $b^1$ is not symplectic, we have $\th_1\ne0$. Hence there
exists $T_1^{1,0}\in\Hom(F_0,F_1)$ such that $\th(y)=\th_1(T_1^{1,0}(y))$ for all
$y\in F_0$. Then $c_{-2}^0(y,y)+b^1(T_1^{1,0}(y),T_1^{1,0}(y))=0$ for all $y\in F_0$.
Thus $c_{-2}^0+b^1(T_1^{1,0}\ot T_1^{1,0})$ is symplectic. This also holds if $b^1$ is
symplectic (in that case we have
$c_{-2}^0(y,y)=b_{-2,0}(y,y)-\tb_{-2,0}(y,y)=Q(y)-Q(y)=0$ for $y\in F_0$) and we take
$T_1^{1,0}=0$. Now $c_{-2}^1$ is also symplectic.
Since $a^{0*}=a^1$, $a^1(T_1^{1,0}\ot1)+a^0(1\ot T_1^{1,0})$ is symplectic. Similarly
$a^0(T_1^{0,1}\ot1)+a^1(1\ot T_1^{0,1})$ is symplectic. Hence
$c_{-2}^0+b^1(T_1^{1,0}\ot T_1^{1,0})+a^1(T_1^{1,0}\ot1)+a^0(1\ot T_1^{1,0})$ is
symplectic and $c_{-2}^1+a^0(T_1^{0,1}\ot1)+a^1(1\ot T_1^{0,1})$ is symplectic. Hence
we can find $\tc^0\in\Bil(F_0),\tc^1\in\Bil(F_1)$ such that
$$c_{-2}^0+b^1(T_1^{1,0}\ot T_1^{1,0})+a^1(T_1^{1,0}\ot1)+a^0(1\ot T_1^{1,0})=
\tc^0+\tc^{0*},$$
$$c_{-2}^1+a^0(T_1^{0,1}\ot1)+a^1(1\ot T_1^{0,1})=\tc^1+\tc^{1*}.$$
Since $b^0,b^1$ are non-degenerate we can find $T_2^{0,0}\in\End(F_0)$,
$T_2^{1,1}\in\End(F_1)$ such that $\tc^0(x,y)=b^0(x,T_2^{0,0}(y))$ for $x,y\in F_0$,
$\tc^1(x,y)=b^1(x,T_2^{1,1}(y))$ for $x,y\in F_1$. For $x,y\in F_0$ we have
$$\align&c_{-2}^0(x,y)+b^1(T_1^{1,0}(x)\ot T_1^{1,0}(x))+a^1(T_1^{1,0}(x),y)
+a^0(x,T_1^{1,0}(y))\\&=b^0(x,T_2^{0,0}(y))+b^0(T_2^{0,0}(x),y).\endalign$$
For $x,y\in F_1$ we have
$$c_{-2}^1(x,y)+a^0(T_1^{0,1}(x),y)+a^1(x,T_1^{0,1}(y))=b^1(x,T_2^{1,1}(y))
+b^1(T_2^{1,1}(x),y).$$
Thus, for $i,j\in I_0,i+j=-2$ and $x,y\in F_0$ we have
$$\align&b_{ij}(x,y)=\tb_{ij}(x,y)+\tb_{i+1,j}(T_1^{1,0}(x),y)+
\tb_{i,j+1}(x,T_1^{1,0}(y))\\&+\tb_{i+1,j+1}(T_1^{1,0}(x),T_1^{1,0}(y))+
\tb_{i,-i}(x,T_2^{0,0}(y))+\tb_{-j,j}(T_2^{0,0}(x),y);\endalign$$
for $i,j\in I_1,i+j=-2$ and $x,y\in F_1$ we have
$$\align&b_{ij}(x,y)=\tb_{ij}(x,y)+\tb_{i+1,j}(T_1^{0,1}(x),y)+\tb_{i,j+1}(x,T_1^{0,1}
(y))\\&+\tb_{i,-i}(x,T_2^{1,1}(y))+\tb_{-j,j}(T_2^{1,1}(x),y).\endalign$$
Let $T$ be as in (a) with $T_0^{00}=1,T_0^{11}=1,T_1^{1,0},T_1^{0,1},T_2^{1,1}$,
$T_2^{0,0}$ as above and the other components $0$. Define
$(b'_{ij})$ by $T(b_{ij})=(b'_{ij})$. Then $(b'_{ij})$ has the required properties.
{\it Case 3: $p=2,k=-1$.} In this case we have $n\ge2$. We first show that there
exists $\s\in\End(F_1)$ such that
$$b^1(x,\s(y))=b^1(\s(x),y),\quad c^1_{-4}(x,x)=b^1(x,\s(x))+b^1(\s(x),\s(x))\tag *$$
for $x,y\in F_1$. The functions $F_1@>>>\kk,x\m b^1(x,x),x\m c^1_{-4}(x,x)$ are
additive and homogeneous of degree $2$, hence are of the form
$x\m\th(x)^2,x\m\th_1(x)^2$ where $\th,\th_1\in\Hom(F_1,\kk)$. We can find a direct sum
decomposition $F_1=F'\op F''$ where $b^1(x',x'')=0$ for all $x'\in F',x''\in F''$,
$\th|_{F'}=0$, $F'=F_1$ if $\th=0$, $\dim F''\in\{1,2\}$ if $\th\ne0$. Define
$\s'\in\End(F')$ by $\th_1(x)\th_1(y)=b^1(x,\s'(y))$ for $x,y\in F'$. Then
$b^1(x,\s'(y))=b^1(\s'(x),y)$ for $x,y\in F'$, $\th_1(x)^2=b^1(x,\s'(x))+\th(\s'(x))^2$
for $x\in F'$.
If $\dim F''=1$ we have $\th|_{F''}\ne0$ and there is a unique $v\in F''$ such that
$\th(v)=1$. Let $\s'':F''@>>>F''$ be multiplication by $a$ where $a\in\kk$ satisfies
$a^2+a=\th_1(v)^2$. Then $\th_1(x)^2=b^1(x,\s''(x))+\th(\s''(x))^2$ for $x\in F''$ and
$b(x,\s''(y))=b(\s''(x),y)$ for $x,y\in F''$.
If $\dim F''=2$ we can find a basis $\{v,v'\}$ of $F''$ such that
$\th(v')=0,\th(v'')=1$. We set $b(v',v'')=f$. We have $f\ne0$. Define
$\s''\in\End(F'')$ by $\s''(v')=\ta f\i v'+\ta v''$, $\s''(v'')=\th_1(v'')^2f\i v'$
where $\ta\in\kk$ satisfies $\ta^2+\ta=\th_1(v')^2$. Then $b(x,\s''(y))=b(\s''(x),y)$
for $x,y\in F''$, $\th_1(x)^2=b(x,\s''(x))+\th(\s''(x))^2$ for $x\in F''$.
If $F''=0$ let $\s'':F''@>>>F''$ be the $0$ map.
Define $\s\in\End(F_1)$ by $\s(x)=\s'(x)$ if $x\in F'$, $\s(x)=\s''(x)$ if $x\in F''$.
Then $\s$ satisfies $(*)$. Since $b^0$ is non-degenerate we can find
$T_3^{0,1}\in\Hom(F_1,F_0)$ such that
$$c^0_{-3}(x,y)+a^0(x,\s(y))=b^0(x,T_3^{0,1}(y))$$
for $x\in F_0,y\in F_1$. For any $i\in I_0,j\in I_1,i+j=-3$ and $x\in F_{|i|}$,
$y\in F_{|j|}$ we have
$$b_{ij}(x,y)=\tb_{ij}(x,y)+\tb_{i,j+2}(x,\s(y))+\tb_{i,j+3}(x,T_3^{0,1}(y)).$$
It follows that for any $i\in I_1,j\in I_0,i+j=-3$ and $x\in F_{|i|},y\in F_{|j|}$ we
have
$$b_{ij}(x,y)=\tb_{ij}(x,y)+\tb_{i+2,j}(\s(x),y)+\tb_{i+3,j}(T_3^{0,1}(x),y).$$
Define $d_1\in\Bil(F_1)$ by $d_1(x,y)=\tb_{i,j+2}(x,\s(y))+\tb_{i+2,j}(\s(x),y)$ where
$i,j\in I_1,i+j=-4$. Using the first equality in $(*)$ we see that $d_1$ is independent
of the choice of $i,j$. Define $d\in\Bil(F_1)$ by
$$d(x,y)=c^1_{-4}(x,y)+d_1(x,y)+b^1(\s(x),\s(y))+a^0(T_3^{0,1}(x),y)
+a^1(x,T_3^{0,1}(y)).$$
We have $d(x,x)=0$ for $x\in F_1$. (We use $(*)$ and the identity
$\tb_{i,j+2}+\tb_{j,i+2}=b^1$ for $i,j\in I_1,i+j=-4$.) Thus, $d$ is symplectic hence
we can find $d'\in\Bil(F_1)$ such that $d=d'+d'{}^*$. Since $b^1$ is non-degenerate we
can find $T_4^{1,1}\in\End(F_1)$ such that $d'(x,y)=b^1(x,T_4^{1,1}(y))$ for
$x,y\in F_1$. We have
$$d(x,y)=b^1(x,T_4^{1,1}(y))+b^1(y,T_4^{1,1}(x))=b^1(x,T_4^{1,1}(y))
+b^1(T_4^{1,1}(x),y).$$
Hence for $i,j\in I_1,i+j=-4$ and $x,y\in F_1$ we have
$$\align&b_{ij}(x,y)=\tb_{ij}(x,y)+\tb_{i,j+2}(x,\s(y))+\tb_{i+2,j}(\s(x),y)\\&+
\tb_{i+2,j+2}(\s(x),\s(y))+\tb_{i+3,j}(T_3^{0,1}(x),y)+\tb_{i,j+3}(x,T_3^{0,1}(y))\\&
+b_{i,j+4}(x,T_4^{1,1}(y))+b_{i+4,j}(T_4^{1,1}(x),y).\endalign$$
For $i,j\in I_1,i+j=-2$ and $x,y\in F_1$ we have
$$b_{ij}(x,y)=\tb_{ij}(x,y)+\tb_{i,j+2}(x,\s(y))+\tb_{i+2,j}(\s(x),y)=\tb_{ij}(x,y).$$
Define $f\in\Bil(F_0)$ by $f(x,y)=c^0_{-4}(x,y)$. Then $f$ is symplectic. (We use that
$c_{-4}^0$ is symplectic.) Hence we can find $f'\in\Bil(F_0)$ such that $f=f'+f'{}^*$.
Since $b^0$ is non-degenerate we can find $T_4^{0,0}\in\End(F_0)$ such that
$f'(x,y)=b^0(x,T_4^{0,0}(y))$ for $x,y\in F_0$. We have
$$f(x,y)=b^0(x,T_4^{0,0}(y))+b^0(y,T_4^{0,0}(x))=b^0(x,T_4^{0,0}(y))
+b^0(T_4^{0,0}(x),y)$$
hence for $i,j\in I_0,i+j=-4$ and $x,y\in F_0$ we have
$$b_{ij}(x,y)=\tb_{ij}(x,y)+b_{i,j+4}(x,T_4^{0,0}(y))+b_{i+4,j}(T_4^{0,0}(x),y).$$
For $i,j\in I_0,i+j=-2$ and $x,y\in F_0$ we have
$$b_{ij}(x,y)=\tb_{ij}(x,y).$$
Let $T$ be as in (a) with $T_0^{00}=1,T_0^{11}=1$, $T_3^{0,1},T_4^{1,1},T_4^{0,0}$,
$T_2^{1,1}=\s$ as above and the other components $0$. Define $(b'_{ij})$ by
$T(b_{ij})=(b'_{ij})$. Then $(b'_{ij})$ has the required properties.
{\it Case 4: $p=2,k<-1$.} In this case we have $n\ge3$. Define $\e,\d\in\{0,1\}$ by
$\e=k-1\mod 2,\d=1-\e$. Since $b^\d$ is non-degenerate, we have
$c_{2k-2}^\d(x,y)=b^\d(x,\s(y))$ for $x,y\in F_\d$ where $\s\in\End(F_\d)$ is well
defined. Since $b^{\d*}=b^\d,c_{2k-2}^{\d*}=c_{2k-2}^\d$, we have
$b^\d(x,\s(y))=b^\d(\s(x),y)$. Since $b^\e$ is non-degenerate we can find
$T_{1-2k}^{\e,\d}\in\Hom(F_\d,F_\e)$ such that
$$c^\e_{2k-1}(x,y)+a^\e(x,\s(y))=b^\e(x,T_{1-2k}^{\e,\d}(y))$$
for $x\in F_\e,y\in F_\d$. For any $i\in I_\e,j\in I_\d,i+j=2k-1$ and $x\in F_\e$,
$y\in F_\d$ we have
$$b_{ij}(x,y)=\tb_{ij}(x,y)+\tb_{i,j-2k}(x,\s(y))
+\tb_{i,j+1-2k}(x,T_{1-2k}^{\e,\d}(y)).$$
It follows that for any $i\in I_\d,j\in I_\e,i+j=2k-1$ and $x\in F_\d,y\in F_\e$ we
have
$$b_{ij}(x,y)=\tb_{ij}(x,y)+\tb_{i-2k,j}(\s(x),y)
+\tb_{i+1-2k,j}(T_{1-2k}^{\e,\d}(x),y).$$
Define $d_1\in\Bil(F_\d)$ by $d_1(x,y)=\tb_{i,j-2k}(x,\s(y))+\tb_{i-2k,j}(\s(x),y)$
where $i,j\in I_\d,i+j=2k-2$. Using $b^\d(1\ot\s)=b^\d(\s\ot 1)$ we see that $d_1$ is
independent of the choice of $i,j$. Define $d\in\Bil(F_\d)$ by
$$d(x,y)=c^\d_{2k-2}(x,y)+d_1(x,y)+a^\e(T_{1-2k}^{\e,\d}(x),y)
+a^\d(x,T_{1-2k}^{\e,\d}(y)).$$
We have $d(x,x)=0$ for $x\in F_\d$. (This follows from the choice of $\s$ and the
identity $\tb_{i,j-2k}+\tb_{j,i-2k}=b^\d$ for $i,j\in I_\d,i+j=2k-2$.) Thus, $d$ is
symplectic hence we can find $d'\in\Bil(F_\d)$ such that $d=d'+d'{}^*$. Since $b^\d$ is
non-degenerate we can find $T_{2-2k}^{\d,\d}\in\End(F_\d)$ such that
$d'(x,y)=b^\d(x,T_{2-2k}^{\d,\d}(y))$ for $x,y\in F_\d$. We have
$$d(x,y)=b^\d(x,T_{2-2k}^{\d,\d}(y))+b^\d(y,T_{2-2k}^{\d,\d}(x))=
b^\d(x,T_{2-2k}^{\d,\d}(y))+b^\d(T_{2-2k}^{\d,\d}(x),y).$$
Hence for $i,j\in I_\d,i+j=2k-2$ and $x,y\in F_\d$ we have
$$\align&b_{ij}(x,y)=\tb_{ij}(x,y)+\tb_{i,j-2k}(x,\s(y))+\tb_{i-2k,j}(\s(x),y)+
\tb_{i+1-2k,j}(T_{1-2k}^{\e,\d}(x),y)\\&+\tb_{i,j+1-2k}(x,T_{1-2k}^{\e,\d}(y))
+b_{i,j+2-2k}(x,T_{2-2k}^{\d,\d}(y))+b_{i+2-2k,j}(T_{2-2k}^{\d,\d}(x),y).\endalign$$
For $i,j\in I_\d,i+j=2k$ and $x,y\in F_\d$ we have
$$b_{ij}(x,y)=\tb_{ij}(x,y)+\tb_{i,j-2k}(x,\s(y))+\tb_{i-2k,j}(\s(x),y)=\tb_{ij}(x,y).
$$
Define $f\in\Bil(F_\e)$ by $f(x,y)=c^\e_{2k-2}(x,y)$. Then $f$ is symplectic. (We use
that $c^\e_{2k-2}$ is symplectic.) Hence we can find $f'\in\Bil(F_\e)$ such that
$f=f'+f'{}^*$. Since $b^\e$ is non-degenerate we can find
$T_{2-2k}^{\e,\e}\in\End(F_\e)$ such that $f'(x,y)=b^\e(x,T_{2-2k}^{\e,\e}(y))$ for
$x,y\in F_\e$. We have
$$f(x,y)=b^\e(x,T_{2-2k}^{\e,\e}(y))+b^\e(y,T_{2-2k}^{\e,\e}(x))
=b^\e(x,T_{2-2k}^{\e,\e}(y))+b^\e(T_{2-2k}^{\e,\e}(x),y)$$
hence for $i,j\in I_\e,i+j=2k-2$ and $x,y\in F_\e$ we have
$$b_{ij}(x,y)=\tb_{ij}(x,y)+b_{i,j+2-2k}(x,T_{2-2k}^{\e,\e}(y))
+b_{i+2-2k,j}(T_{2-2k}^{\d,\d}(x),y).$$
For $i,j\in I_\d,i+j=2k$ and $x,y\in F_\e$ we have $b_{ij}(x,y)=\tb_{ij}(x,y)$.
Let $T$ be as in (a) with $T_0^{00}=1,T_0^{11}=1$, $T_{1-2k}^{\e,\d},T_{2-2k}^{\e,\e},
T_{2-2k}^{\d,\d}$, $T_{-2k}^{\d,\d}=\s$ as above and the other components $0$. Define
$(b'_{ij})$ by $T(b_{ij})=(b'_{ij})$. Then $(b'_{ij})$ has the required properties.
$\sp$
This completes the proof of (b).
We now verify the following special case of 3.6(c).
(c) {\it Let $(\tb_{ij}),(b_{ij})$ be two points of $X$. Then there exists $T\in\D_u$
such that $T(b_{ij})=(\tb_{ij})$.}
\nl
We first prove the following statement by induction on $k\in[-n,0]$.
($P_k$) {\it Assume in addition that $b_{ij}=\tb_{ij}$ for any $i,j$ with $i+j\ge2k$.
Then there exists $T\in\D_u$ such that $T(b_{ij})=(\tb_{ij})$.}
\nl
If $k=-n$ the result is obvious. Assume now that $k\in[1-n,0]$. By (b) we can find
$T'\in\D_u$ such that $T'(b_{ij})=(b'_{ij})$ and $b'_{ij}=\tb_{ij}$ for $i+j\ge2k-2$.
By the induction hypothesis we can find $T''\in\D_u$ such that
$T''(b'_{ij})=(\tb_{ij})$. Let $T=T''T'\in\D_u$. Then $T(b_{ij})=(\tb_{ij})$. This
completes the proof of $(P_k)$ for $k\in[-n,0]$. In particular $(P_0)$ holds and (c) is
proved.
\subhead 3.10. Proof of 3.6(c)\endsubhead
Let $\lar,\lar'$ be two elements of the set $X$ in 3.6. We must show that $\lar,\lar'$
are in the same $U$-orbit. We argue by induction on $e$, the smallest integer $\ge0$
such that $N^e=0$. If $e=0$ we have $V=0$ and the result is obvious. If $e=1$ we have
$N=0$. Then $V=\gr V_*$ canonically, $U=\{1\}$ and both $\lar,\lar'$ are the same as
$\lar_0$ hence the result is clear. We now assume that $e\ge2$.
$\sp$. Assume first that $p=2$. For $n\in\cl$ let $q_n:P^\nu_{-n}@>>>\kk$ be the
quadratic forms attached to $(N,\lar)$ in 3.3 and let $q'_n:P^\nu_{-n}@>>>\kk$ be the
analogous quadratic forms defined in terms of $(N,\lar')$. We show:
(a) {\it there exists $T\in U$ such that if $\lar''\in\Symp(V)$ is given by
$\la x,y\ra''=\la Tx,Ty\ra$ then for $n\in\cl$ the quadratic form $q''_n$ defined as in
3.3 in terms of $(N,\lar'')$ satisfies $q''_n=q'_n$.}
\nl
We are seeking an $S\in E_{\ge1}V_*$ such that $SN=NS$ and
$\la(1+S)\dx,(1+S)N^{n-1}\dx\ra=\la\dx,N^{n-1}\dx\ra'$ that is,
$\la(1+S)\dx,N^{n-1}(1+S)\dx\ra=\la\dx,N^{n-1}\dx\ra'$ that is,
$\la S\dx,N^{n-1}\dx\ra+\la \dx,N^{n-1}S\dx\ra+\la S\dx,N^{n-1}S\dx\ra
=\la\dx,N^{n-1}\dx\ra'+\la\dx,N^{n-1}\dx\ra$
\nl
for any $n\in\cl$ and any $\dx\in V_{\ge-n}$ such that $N^{n+1}\dx=0$. Now
$\la S\dx,N^{n-1}\dx\ra+\la\dx,N^{n-1}S\dx\ra=\la S\dx,(N^{n-1}+(N^\da)^{n-1})\dx\ra$
\nl
is a linear combination of terms $\la S\dx,N^{n'}\dx\ra$ with $n'\ge n$; each of these
terms is $0$ since $S\dx\in V_{\ge1-n},N^{n'}\dx\in V_{\ge2n'-n}$ and $1-n+2n'-n\ge1$.
Moreover, $\la S\dx,N^{n-1}S\dx\ra=\la \bS x,\nu^{n-1}\bS x\ra_0$ where
$x\in P^\nu_{-n}$ is the image of $\dx$ and
$\la\dx,N^{n-1}\dx\ra'+\la\dx,N^{n-1}\dx\ra=q'_n(x)+q_n(x)$. By the surjectivity of the
map $S\m\bS$ in 2.5(d), we see that it suffices to show that there exists
$\s\in\End_1^\nu(\gr V_*)$ (that is $\s\in\End_1(\gr V_*)$ such that $\s\nu=\nu\s$)
with $\la\s x,\nu^{n-1}\s x\ra_0=q'_n(x)+q_n(x)$ for any $n\in\cl$ and any
$x\in P^\nu_{-n}$.
For $n\in\cl'$ the last equation is automatically satisfied for any $\s$. (The left
hand side is zero by the definition of $\cl'$. The right hand side is equal by 3.3(c)
to $Q'_\nn(\nu^{(n-\nn)/2}x)+Q_\nn(\nu^{(n-\nn)/2}x)$ where $Q_\nn$ is the quadratic
form attached as in 3.3 to $(N,\lar)$ and $Q'_\nn$ is the analogous quadratic form
defined in terms of $(N,\lar')$. The last sum is zero since $Q_\nn=Q'_\nn=Q$.)
We see that it suffices to show that there exists $\s\in\End_1^\nu(\gr V_*)$ such that
$\la\s x,\nu^{n-1}\s x\ra_0=q'_n(x)+q_n(x)$ for any $n\in\cl-\cl'$ and any
$x\in P^\nu_{-n}$.
For $n\in\cl-\cl'$, the quadratic forms $q'_n,q_n$ have the same associated symplectic
form (see 3.3(a)); hence there exists $\th_n\in\Hom(P^\nu_{-n},\kk)$ such that
$q'_n(x)+q_n(x)=\th_n(x)^2$ for all $x\in P^\nu_{-n}$. Hence it suffices to show that
the linear map
$\r:\End_1^\nu(\gr V_*)@>>>\op_{n\in\cl-\cl'}\Hom(P^\nu_{-n},\kk)$
\nl
given by $\s\m(\th_n)$ where $\th_n(x)=\sqrt{\la \s x,\nu^{n-1}\s x\ra_0}$ for
$x\in P^\nu_{-n}$ is surjective. Let $\ce=\op_{n\ge0}\Hom(P^\nu_{-n},\gr_{1-n}V_*)$. We
have an isomorphism $\p:\End_1^\nu(\gr V_*)@>\si>>\ce$ given by $\s\m(\s_n)$ where
$\s_n\in\Hom(P^\nu_{-n},\gr_{1-n}V_*)$ is the restriction of $\s$. Define a linear map
$\r':\ce@>>>\op_{n\in\cl-\cl'}\Hom(P^\nu_{-n},\kk)$
\nl
by $(\s_n)\m(\th_n)$ where $\th_n(x)=\sqrt{\la \s_nx,\nu^{n-1}\s_nx\ra_0}$ for
$x\in P^\nu_{-n}$. We have $\r'\p=\r$. Hence it suffices to show that $\r'$ is
surjective. It also suffices to show that for any $n\in\cl-\cl'$ the linear map
$\r'_n:\Hom(P^\nu_{-n},\gr_{1-n}V_*)@>>>\Hom(P^\nu_{-n},\kk)$
\nl
given by $f\m\th$, where $\th(x)=\sqrt{\la fx,\nu^{n-1}fx\ra_0}$ for $x\in P^\nu_{-n}$,
is surjective. Define $g\in\Hom(\gr_{1-n}V_*@>>>\kk)$ by
$h\m\sqrt{\la h,\nu^{n-1}h\ra_0}$. Then $\r'_n(f)=g\circ f$ for
$f\in\Hom(P^\nu_{-n},\gr_{1-n}V_*)$. Hence to show that $\r'_n$ is surjective it
suffices to show that $g\ne0$. Since $n\in\cl-\cl'$, there exists $m$ odd such that
$m\ge n+3$ and $b_m$ is not symplectic. Hence there exists $u'\in P^\nu_{-m}$ such that
$\la u',\nu^mu'\ra_0\ne0$. We have $m=(n-1)+2k$ where $k$ is an integer $\ge2$. Let
$u=N^ku'\in\gr_{1-n}V_*$ and
$\la u,\nu^{n-1}u\ra_0=\la \nu^ku',\nu^{n-1+k}u'\ra_0=\la u',\nu^mu'\ra_0\ne0$.
\nl
Thus $g(u)\ne0$. We see that $g\ne0$, as required. This proves (a).
Note that $\lar''$ in (a) is in $X$ (in fact in the $U$-orbit of $\lar$). Replacing if
necessary $\lar$ by $\lar''$ we see that
(b) {\it we may assume that $\lar,\lar'$ are such that $q_n=q'_n$ for all $n\in\cl$.}
$\sp$
We now return to a general $p$. Let $r\ge e$. Let $F$ be a complement of
$V_{\ge2-r}=\ker N^{r-1}$ in $V_{\ge1-r}=V$ and let $F'$ be a complement of
$V_{\ge3-r}=\ker N^{r-2}+NV$ in $V_{\ge2-r}=\ker N^{r-1}$. Consider the linear map $\a$
of $F\op F'\op F\op\do\op F'\op F$ ($2r-1$ summands) into $V$ given by
$(x_{1-r},x_{2-r},\do,x_{r-2},x_{r-1})\m$
$x_{1-r}+Nx_{3-r}+\do+N^{r-1}x_{r-1}+x_{2-r}+Nx_{4-r}+\do+N^{r-2}x_{r-2}$.
\nl
(Here $x_i\in F$ if $i=r+1\mod2$ and $x_i\in F'$ if $i=r\mod2$.) Let $W$ be the image
of $\a$. We show that
(c) $\lar$ and $\lar'$ are non-degenerate on $W$.
\nl
We prove this only for $\lar$; the proof for $\lar'$ is the same. Assume that
$w=x_{1-r}+Nx_{3-r}+\do+N^{r-1}x_{r-1}+x_{2-r}+Nx_{4-r}+\do+N^{r-2}x_{r-2}$ with $x_i$
as above satisfies $\la w,W\ra=0$. We show that each $x_i$ is $0$. We have
$0=\la w,N^{r-1}F\ra=\la x_{1-r},N^{r-1}F\ra=0$. Using the non-degeneracy of $b_{r-1}$
we see that $x_{1-r}=0$ and
$w=Nx_{3-r}+\do+N^{r-1}x_{r-1}+x_{2-r}+Nx_{4-r}+\do+N^{r-2}x_{r-2}$. We have
$0=\la w,N^{r-2}F'\ra=\la x_{2-r},N^{r-2}F'\ra$. Using the non-degeneracy of $b_{r-2}$
we see that $x_{2-r}=0$ and
$w=Nx_{3-r}+\do+N^{r-1}x_{r-1}+Nx_{4-r}+\do+N^{r-2}x_{r-2}$. We have
$0=\la w,N^{r-2}F\ra=\la Nx_{3-r},N^{r-2}F\ra=-\la x_{3-r},N^{r-1}F\ra$. Using the
non-degeneracy of $b_{r-1}$ we see that $x_{3-r}=0$. Continuing in this way we see that
each $x_i$ is $0$. This proves (c).
The proof shows also that $\a$ is injective.
Let $Z=\{x\in V;\la x,W\ra=0\},Z'=\{x\in V;\la x,W\ra'=0\}$. From (c) we see that
$V=W\op Z=W\op Z'$.
Clearly, $W$ is $N$-stable hence $(1+N)$-stable. Since $1+N$ is an isometry of $\lar$
it follows that $Z$ is $(1+N)$-stable hence $N$-stable. Similarly, $Z'$ is $N$-stable.
Define $\Ph\in GL(V)$ by $\Ph(x)=x$ for $x\in W$, $\Ph(x)=x'$ for $x\in Z$ where
$x'\in Z'$ is given by $x-x'\in W$. We have $\Ph\in U$ (see 2.7(c),(d)). Define
${}'\lar\in\Symp(V)$ by ${}'\la x,y\ra=\la \Ph(x),\Ph(y)\ra'$. By 3.6(a), we have
${}'\lar\in X$.
Let ${}'Z=\{x\in V;{}'\la x,W\ra=0\}$. We show that ${}'Z=Z$. Let $x=x_1+x_2$ where
$x_1\in W,x_2\in Z$. We have $x_2=w+x'_2$, $w\in W,x'_2\in Z'$. For $w'\in W$ we have
$\la\Ph(x),w'\ra'=\la x_1+x'_2,w'\ra'=\la x_1,w'\ra'$. The condition that
$\la\Ph(x),W\ra'=0$ is that $\la x_1,W\ra'=0$ or that $x_1=0$ (using (c)) or that
$x\in Z$. Thus,
${}'Z=\{x\in V;\la \Ph(x),\Ph(W)\ra'=0\}=\{x\in V;\la \Ph(x),W\ra'=0\}=Z$ as required.
$\sp$ In the case where $p=2$ we show that for any $n\in\cl$ the quadratic form $q'_n$
attached to $(N,\lar')$ as in 3.3 is equal to the analogous quadratic form attached to
$(N,{}'\lar)$. We must show that, if $x\in V_{\ge-n}$, $N^{n+1}x=0$ then
$\la\Ph x,\Ph N^{n-1}x\ra'=\la x,N^{n-1}x\ra'$ that is,
$\la\Ph x,N^{n-1}\Ph x\ra'=\la x,N^{n-1}x\ra'$. Both sides are additive in $x$. We can
write $x=x_1+x_2$ where $x_1\in W,x_2\in Z$ satisfy $x_1,x_2\in V_{\ge-n}$,
$N^{n+1}x_1=0,N^{n+1}x_2=0$. We may assume that $x=x_1$ or $x=x_2$. When $x=x_1$ the
desired equality is obvious. Hence we may assume that $x\in Z$. Write $x=x'+w$,
$x'\in Z',w\in W$. We must show that $\la x+w,N^{n-1}x+N^{n-1}w\ra'=\la x,N^{n-1}x\ra'$
that is, $\la x,N^{n-1}w\ra'+\la w,N^{n-1}x\ra'+\la w,N^{n-1}w\ra'=0$ that is,
$\la x,(N^{n-1}+(N^\da)^{n-1})w\ra'+\la w,N^{n-1}w\ra'=0$ that is,
$\la x,N^nw\ra'+\la w,N^{n-1}w\ra'=0$ (we use $N^{n+1}w=0$) that is,
$\la x'+w,N^nw\ra'+\la w,N^{n-1}w\ra'=0$ that is,
$\la w,N^nw\ra'+\la w,N^{n-1}w\ra'=0$. Now $w\in W_{\ge1-n}$ (see 2.7(b)),
$N^nw\in W_{\ge n+1}$ hence $\la w,N^nw\ra'=0$. It remains to show
$\la w,N^{n-1}w\ra'=0$. Since $w\in W_{\ge1-n}$, $N^{n+1}w=0$, it suffices to show
$\la y,\nu^{n-1}y\ra_0=0$ for any $y\in\gr_{1-n}V_*$ such that $\nu^{n+1}y=0$. This has
already been seen in the proof in 3.3 that $q_n$ is well defined. $\sp$
Replacing $\lar'$ by ${}'\lar$ (which is in the same $U$-orbit) we see that condition
(b) is preserved (for $p=2$).
Thus, we may assume that $\lar,\lar'$ satisfy $Z=Z'$ and that for $p=2$ condition (b)
holds. Thus $V=W\op Z$ is an othogonal decomposition with respect to either $\lar$ or
$\lar'$. Let $\lar_W,\lar_Z$ be the restrictions of $\lar$ to $W,Z$. Let
$\lar'_W,\lar'_Z$ be the restrictions of $\lar'$ to $W,Z$. Let $U_1$ (resp. $U_2$) be
the analogue of $U$ for $W$ (resp. $Z$) defined in terms of $N$ and $W^N_*$ (resp.
$Z^N_*$). We have naturally $U_1\T U_2\sub U$.
We consider $5$ cases.
{\it Case 1: $p\ne2$.} Take $r=e+1$. (Thus, $F=0$.) By the induction hypothesis,
$\lar_Z$ is carried to $\lar'_Z$ by some $u_2\in U_2$. By 3.9, $\lar_W$ is carried to
$\lar'_W$ by some $u_1\in U_1$. Then $\lar$ is carried to $\lar'$ by $(u_1,u_2)\in U$.
$\sp$ {\it Case 2: $p=2$, $e$ is odd and $b_{e-2}$ is symplectic.} Take $r=e+1$. (Thus,
$F=0$.) We have $e-1\in\cl$. The sets $\cl$ attached to $\lar_Z,\lar'_Z$ are the same
as $\cl$ for $\lar,\lar'$. The quadratic forms attached to $\lar_Z$, $\lar'_Z$ and
$n\in\cl-\{e-1\}$ are the same as those attached to $\lar,\lar'$ and $n$, hence they
coincide. The quadratic forms attached to $\lar_Z,\lar'_Z$ and $n=e-1$ also coincide:
they are both $0$. Hence the Quadratic forms attached to $\lar_Z,\lar'_Z$ coincide (see
3.3(c)). The quadratic forms attached to $\lar_W,\lar'_W$ coincide: for $e-1$ they are
the same as those attached to $\lar,\lar'$ and $e-1$ and for other $n$ they are zero.
Hence the Quadratic forms attached to $\lar_W,\lar'_W$ coincide. By the induction
hypothesis, $\lar_Z$ is carried to $\lar'_Z$ by some $u_2\in U_2$. By 3.9, $\lar_W$ is
carried to $\lar'_W$ by some $u_1\in U_1$. Then $\lar$ is carried to $\lar'$ by
$(u_1,u_2)\in U$.
{\it Case 3: $p=2$, $e$ is even and $b_{e-1}$ is symplectic.} Take $r=e+1$. (Thus,
$F=0$.) The sets $\cl$ attached to $\lar_Z,\lar'_Z$ are the same as $\cl$ for
$\lar,\lar'$. The quadratic forms attached to $\lar_Z,\lar'_Z$ and $n\in\cl$ are the
same as those attached to $\lar,\lar'$ and $n$, hence they coincide. Hence the
Quadratic forms attached to $\lar_Z,\lar'_Z$ coincide. By the induction hypothesis,
$\lar_Z$ is carried to $\lar'_Z$ by some $u_2\in U_2$. By 3.9, $\lar_W$ is carried to
$\lar'_W$ by some $u_1\in U_1$. Then $\lar$ is carried to $\lar'$ by $(u_1,u_2)\in U$.
{\it Case 4: $p=2$, $e$ is even and $b_{e-1}$ is not symplectic.} Take $r=e$. The sets
$\cl$ attached to $\lar_Z,\lar'_Z$ are the same as $\cl$ for $\lar,\lar'$. The
quadratic forms attached to $\lar_Z,\lar'_Z$ and $n\in\cl$ are the same as those
attached to $\lar,\lar'$ and $n$, hence they coincide. Hence the Quadratic forms
attached to $\lar_Z,\lar'_Z$ coincide. By the induction hypothesis, $\lar_Z$ is carried
to $\lar'_Z$ by some $u_2\in U_2$. By 3.9, $\lar_W$ is carried to $\lar'_W$ by some
$u_1\in U_1$. Then $\lar$ is carried to $\lar'$ by $(u_1,u_2)\in U$.
{\it Case 5: $p=2$, $e$ is odd, $\ge3$ and $b_{e-2}$ is not symplectic.} Take $r=e$. By
3.9, $\lar_W$ is carried to $\lar'_W$ by some $u_1\in U_1$. Replacing $\lar'$ by a
translate under $(u_1,1)\in U$ we see that we may assume in addition that
$\lar_W=\lar'_W$. Let $\tW=F+NF+\do+N^{r-1}F$. Let
$W'=\{w\in W;\la w,\tW\ra=0\}=\{w\in W;\la w,\tW\ra'=0\}$. Then $W=\tW\op W'$,
orthogonal direct sum for both $\lar,\lar'$. Let $\tZ=W'\op Z$, orthogonal direct sum
for both $\lar,\lar'$. Then $V=\tW\op\tZ$, orthogonal direct sum for both $\lar,\lar'$.
Let $\lar_{\tZ},\lar'_{\tZ}$ be the restrictions of $\lar,\lar'$ to $\tZ$. Let $\tU_1$
(resp. $\tU_2$) be the analogue of $U$ for $\tW$ (resp. $\tZ$) defined in terms of $N$
and $\tW^N_*$ (resp. $\tZ^N_*$). We have naturally $\tU_1\T\tU_2\sub U$. The sets $\cl$
attached to $\lar_{\tZ}$, $\lar'_{\tZ}$ are the same as $\cl$ for $\lar,\lar'$. The
quadratic forms attached to $\lar_{\tZ},\lar'_{\tZ}$ and $n\in\cl$ are the same as
those attached to $\lar,\lar'$ and $n$, hence they coincide. Hence the Quadratic forms
attached to $\lar_{\tZ},\lar'_{\tZ}$ coincide. By the induction hypothesis,
$\lar_{\tZ}$ is carried to $\lar'_{\tZ}$ by some $\tu_2\in\tU_2$. Then $\lar$ is
carried to $\lar'$ by $(1,\tu_2)\in U$. $\sp$
This completes the proof of 3.6(c) hence also that of 3.5, 3.6, 3.7.
\subhead 3.11\endsubhead
Here is the order of the proof of the various assertions in Propositions 3.5-3.7:
3.5(a) (see 3.8); 3.7(a) (see 3.7); 3.6(a) (see 3.6); 3.7(b) (see 3.7); 3.6(b) (see
3.6); 3.5(b) (see 3.5); 3.6(c) (see 3.9, 3.10); 3.7(c) (see 3.7); 3.5(c) (see 3.5).
\subhead 3.12\endsubhead
Let $V\in\cc$ and let $\lar\in\Symp(V)$. The following result can be deduced from
\cite{\SPA, I, 2.10}.
Let $C,C_0$ be two $GL(V)$-conjugacy classes in $\Nil(V)$ such that
$C\cap\cm_{\lar}\ne\em,C_0\cap\cm_{\lar}\ne\em$ and $C$ is contained in the closure of
$C_0$ in $GL(V)$. Then $C\cap\cm_{\lar}$ is contained in te closure of
$C_0\cap\cm_{\lar}$ in $\cm_{\lar}$.
\subhead 3.13\endsubhead
Let $V\in\cc$ and let $\lar\in\Symp(V)$. Let $G=Sp(\lar)$. For any self-dual filtration
$V_*$ of $V$ and for $n\ge1$ let $E^{\lar}_{\ge n}V_*=E_{\ge n}V_*\cap\cm_{\lar}$, a
unipotent algebraic group with multiplication $T*T'=T+T'+TT'$. Let
$$\tix(V_*)=\x(V_*)\cap\cm_{\lar}=\{N\in\cm_{\lar};V^N_*=V_*\}
=\{N\in E^{\lar}_{\ge2}V_*;\bN\in\End_2^0(\gr V_*)\}$$
(see 2.9). The following three conditions are equivalent:
(i) $\tix(V_*)\ne\em$;
(ii) there exists $\nu\in\End_2^0(\gr V_*)$ which is skew-adjoint with respect to the
symplectic form on $\gr V_*$ induced by $\lar$;
(iii) $\dim\gr_nV_*=\dim\gr_{-n}V_*\ge\dim\gr_{-n-2}V_*$ for all $n\ge0$ and
$\dim\gr_{-n}V_*=\dim\gr_{-n-2}V_* \mod2$ for all $n\ge0$ even.
\nl
We have (i)$\imp$(ii) by the definition of $\tix(V_*)$; we have (ii)$\imp$(iii) by
2.3(d) and 3.1(c). Now (iii)$\imp$(ii) is easily checked. We have (ii)$\imp$(iii) by
3.5(a).
Let $\fF_{\lar}$ be the set of all self-dual filtrations $V_*$ of $V$ that satisfy
(i)-(iii). From the definitions we have a bijection
(a) $\fF_{\lar}@>\si>>D_G,V_*\m\l$
\nl
($D_G$ as in 1.1) where $\l=(G^\l_0\sps G^\l_1\sps G^\l_2\sps\do)$ is defined in terms
of $V_*$ by $G^\l_0=E_{\ge0}V_*\cap G$ and $G^\l_n=1+E_{\ge n}^{\lar}V_*$ for $n\ge1$.
The sets $\tix(V_*)$ (with $V_*\in\fF_{\lar})$ form a partition of $\cm_{\lar}$. (If
$N\in\cm_{\lar}$ we have $N\in\tix(V_*)$ where $V_*=V^N_*$.)
Let $V_*\in\fF_{\lar}$. Let $C_0$ be the unique $GL(V)$-conjugacy class in $\Nil(V)$
that contains $\x(V_*)$. We have
$E_{\ge2}^{\lar}V_*-\tix(V_*)=(E_{\ge2}V_*-\x(V_*))\cap\cm_{\lar}
=E_{\ge2}V_*\cap(\cup_CC)\cap\cm_{\lar}$
\nl
(the last equality follows from 2.9; $C$ runs over all $GL(V)$-orbits in $\Nil(V)$ such
that $C\sub\bC_0-C_0$). Using 3.12 we see that
$E_{\ge2}^{\lar}V_*-\tix(V_*)=E_{\ge2}^{\lar}V_*\cap(\cup_C(C\cap\cm_{\lar}))$
\nl
where $C$ runs over all $GL(V)$-orbits in $\Nil(V)$ such that $C\cap\cm_{\lar}\ne\em$
and $C\sub\ov{C_0\cap\cm_{\lar}}-(C_0\cap\cm_{\lar})$. We see that, if $V_*\m\l$ (as in
(a)) and $\bla$ is the $G$-orbit of $\l$ in $D_G$ then (with notation of 1.1)
$\tH^\bla$ is the union of $G$-conjugacy classes in $1+\cm_{\lar}$ contained in
$1+\bC_0$, $H^\bla$ is the union of $G$-conjugacy classes in $1+\cm_{\lar}$ contained
in $1+C_0$, $X^\l=1+\tix(V_*)=1+(E_{\ge2}^{\lar}V_*\cap C_0)$. We see that
$\fP_1-\fP_3$ hold.
\subhead 3.14\endsubhead
We preserve the setup of 3.13. Let $V_*\in\fF_{\lar}$ and let $\l\in D_G$ be the
corresponding element. Define $\lar_0\in\Symp(\gr V_*)$ as in 3.2. The map
$E_{\ge2}^{\lar}V_*@>>>\End_2^0(\gr V_*),N\m\bN$ restricts to a map
$\p:\tix(V_*)@>>>E:=\{\nu\in\End_2^0(\gr V_*);\nu\text{ skew-adjoint with respect to }
\lar_0\}$.
\nl
We show:
(a) {\it The group $E_{\ge3}^{\lar}V_*$ (see 3.13) acts freely on $\tix(V^*)$ by
$T,N\m T*N$ (see 3.13) and the orbit space of this action may be identified with $E$
via $\p$.}
\nl
We show this only at the level of sets. If $T\in E_{\ge3}^{\lar}V_*,N\in\tix(V_*)$ then
$T*N\in E_{\ge2}^{\lar}V_*$ and $T*N,N$ induce the same map in $\End_2(\gr V_*)$; hence
$T*N\in\tix(V_*)$. Thus $T,N\m T*N$ is an action of $E_{\ge3}^{\lar}V_*$ on
$\tix(V_*)$. This action is free: it is the restriction of the action of
$E_{\ge3}^{\lar}V_*$ on $E_{\ge2}^{\lar}V_*$ by left multiplication for the group
structure in 3.13. If $N,N'\in\tix(V_*)$ induce the same map in $\End_2^0(\gr V_*)$
then $N'-N\in E_{\ge3}V_*$. Set $T=(N'-N)(1+N)\i\in E_{\ge3}V_*$. Then
$(1+T)(1+N)=1+N'$ and we have automatically $T\in E^{\lar}_{\ge3}V_*$ and $T+N=N'$.
Thus the orbits of the $E_{\ge3}^{\lar}V_*$-action on $\ti\x(V^*)$ are exactly the
non-empty fibres of $\p$. It remains to show that $\p$ is surjective. This follows from
3.5(a).
Now let $N,N'\in\tix(V_*)$ be such that $\bN=\bN'=\nu\in\End_2^0(\gr V_*)$. We show:
(b) {\it there exists $g\in E_{\ge0}V_*\cap G$ such that $N'=gNg\i$.}
\nl
Assume first that $p=2$. The set $\cl\sub2\NN$ defined in 3.3 in terms of $N$ is the
same as that defined in terms of $N'$. Let $q_n:P^\nu_{-n}@>>>\kk$ be the quadratic
form defined in terms of $N$ (for $n\in\cl$) as in 3.3 and let $q'_n:P^\nu_{-n}@>>>\kk$
be the analogous quadratic form defined in terms of $N'$. From 3.3(a) we see that for
any $n\in\cl$ there exists an automorphism $h_n:P^\nu_{-n}@>>>P^\nu_{-n}$ which
preserves the symplectic form $x,y\m b_n(x,y)$ (see 3.1) and satisfies
$q'_n(x)=q_n(h_nx)$ for any $x\in P^\nu_{-n}$. There is a unique $h\in Sp(\lar_0)$ such
that $h(x)=h_n(x)$ for $x\in P^\nu_{-n},n\in\cl$, $h(x)=x$ for $x\in P^\nu_{-n}$,
$n\in\ZZ-\cl$, $h\nu=\nu h$. Let $V=\op_aV_a$ be a direct sum decomposition as in
3.2(b). Then $\End_0(V)$ is defined and we define $\tih\in\End_0(V)$ by the requirement
that for any $a$, $\tih:V_a@>>>V_a$ corresponds to $h:\gr_aV_*@>>>\gr_aV_*$ under the
obvious isomorphism $V_a@>\si>>\gr_aV_*$. Then $\tih\in E_{\ge0}V_*\cap G$ and
$\tih N\tih\i=N''$ where $N''\in E^{\lar}_{\ge2}V_*$ satisfies $\bN''=\nu$. Moreover,
the quadratic form $P^\nu_{-n}@>>>\kk$ defined as in 3.3 in terms of $N''$ (instead of
$N$) for $n\in\cl$ is $x\m h_n(x)$ that is, $q'_n$. From 3.3(c) we see that the
Quadratic form $Q_n$ defined for $n\in\cl'$ in terms of $N''$ is the same as that
defined in terms of $N'$. From 3.5(c) we see that there exists
$h'\in1+E^{\lar}_{\ge1}V_*$ such that $h'N''h'{}\i=N'$. Setting
$g=h'\tih\in E_{\ge0}V_*\cap G$ we have $gNg\i=N'$.
Next assume that $p\ne2$. From 3.5(c) we see that there exists
$g\in1+E^{\lar}_{\ge1}V_*$ such that $gNg\i=N'$. This proves (b).
We see that $\fP_6$ (hence $\fP_4$) holds.
From (a) we see that the $G^\l_0$-action on $\tix(V^*)$ (conjugation) induces an action
of $\bG^\l_0=G^\l_0/G^\l_1$ on $E$ and from (b) we see that this gives rise to a
bijection between the set of $G^\l_0$-orbits on $\tix(V^*)$ and the set of
$\bG^\l_0$-orbits on $E$. We describe this last set of orbits. We identify $\bG^\l_0$
with $\End_0(\gr V_*)\cap Sp(\lar_0)$ with the action on $E$ given by $g:\nu\m\nu'$
where $\nu'(x)=g\nu(g\i x)$ for $x\in\gr V_*$.
Let $I=\{n\in2\NN+1,\dim\gr_{-n}V_*-\dim\gr_{-n-2}V_*\in\{2,4,6,\do\}\}$. For any
subset $J\sub I$ let $E_J$ be the set of all $\nu\in E$ such that for any $n\in I$ we
have
$\{x\in\gr_{-n}V_*;\nu^{n+1}x=0,\la x,\nu^nx\ra_0\ne0\}\ne\em\lra n\in J$.
\nl
Let $\bE$ be the set of all direct sum decompositions $\gr V_*=\op_{n\ge0}W^n$ where
$W^n\in\bcc$ (see 2.1) are such that $\la W^n,W^{n'}\ra_0=0$ for $n\ne n'$ and for
$n\ge 0$, $\dim W^n_a$ is $\dim\gr_{-n}V_*-\dim\gr_{-n-2}V_*$ if $a\in[-n,n],a=n\mod2$
and is $0$ for other $a$. Define $\ph:E@>>>\bE$ by $\nu\m(W^n)$ where
$W^n=\sum_{k\ge0}\nu^kP^\nu_{-n}$. Then $\ph$ is $\bG^\l_0$-equivariant where
$\bG^\l_0$ acts on $\bE$ in an obvious way (transitively).
Let $w=(W^n)\in\bE$. Let $G^w$ be the stabilizer of $w$ in $\bG^\l_0$. Let
$E^w=\ph\i(w)$. Now $E^w$ may be identified with $\prod_{n\ge0}E^w_n$ where $E^w_n$ is
the set of all skew-adjoint elements in $\End_2^0(W^n)$ with respect to
$\lar_0|_{W^n}$. Moreover $G^w$ may be identified with $\prod_{n\ge0}G^w_n$ where
$G^w_n=\End_0(W^n)\cap Sp(\lar_0|_{W^n})$. Furthermore, we may identify
$E^w_n=E^{w1}_n\T E^{w2}_n$ where $E^{w1}_n$ consists of all sequences of isomorphisms
(c) $W^n_{-n}@>\si>>W^n_{-n+2}@>\si>>W^n_{-n+4}@>\si>>\do@>\si>>W^n_{-\d}$
\nl
($\d=0$ if $n$ is even and $\d=1$ if $n$ is odd) and $E^{w2}_n$ is the set of
non-degenerate symmetric bilinear forms $W^n_{-1}\T W^n_{-1}@>>>\kk$ (if $n$ is odd)
and is a point if $n$ is even. (This identification is obtained by attaching to
$\nu\in E^w_n$ the isomorphisms (c) induced by $\nu$ and if $n$ is odd, the bilinear
form $x,x'\m\la x,\nu x'\ra_0$ on $W^n_{-1}$.)
We claim that if $p=2$, the subsets $E_J$ are precisely the orbits of $\bG^\l_0$ on $E$
while if $p\ne2$, $E$ is a single orbit of $\bG^\l_0$. Using the transitivity of the
$\bG^\l_0$ action on $\bE$ we see that it suffices to prove: if $p=2$, the subsets
$E^w_J=E_J\cap E^w$ are precisely the $G^w$-orbits on $E$ while if $p\ne2$, $E^w$ is a
single $G^w$-orbit. If $n\n I$, $G'_n$ acts transitively on $E'_n$. If $n\in I$,
$pr_2:E^w_n@>>>E^{w2}_n$ induces a bijection between the set of $G^w_n$-orbits on
$E^w_n$ and the set of $GL(W^n_{-1})$-orbits on the set of non-degenerate symmetric
bilinear forms on $W^n_{-1}$. The last set of orbits consists of one element if $p\ne2$
and of two elements (the symplectic forms and the non-symplectic forms) if $p=2$. This
verifies our claim.
We see that the first assertion of $\fP_8$ holds.
As above, we identify $E$ with the set of triples $(w,\a,j)$ where $w\in\bE$, $\a$ is a
collection of isomorphisms as in (c) (for each $n\ge0$) and $j$ is a sequence
$(j_n)_{n\in I}$ where $j_n\in\Bil(W^n_{-1})$ is symmetric non-degenerate.
Assume that $p=2$. Let $J\sub J'\sub I$. From the previous discussion we see that the
$\bG^\l_0$-orbits on $E$ that contain $E_J$ in their closure and are contained in the
closure of $E_{J'}$ are those of the form $E_K$ where $J\sub K\sub J'$. Let
$E_{J,J'}=\cup_{K;J\sub K\sub J'}E_K$. We identify $E_J$ with the set of
$(w,\a,j)\in E$ such that $j_n$ is not symplectic for $n\in J$ and symplectic for
$n\in I-J$. We identify $E_{J,J'}$ with the set of $(w,\a,j)\in E$ such that $j_n$ is
not symplectic for $n\in J$ and symplectic for $n\in I-J'$. Let $\tE_J$ be the set of
all triples $(w,\a,\tj)$ where $w,\a$ are as above and $\tj=(\tj_n)_{n\in I}$ where for
$n\in J$, $\tj_n\in\Bil(W^n_{-1})$ is a symmetric non-symplectic non-degenerate form
and, for $n\in I-J$, $\tj_n:W^n_{-1}\T W^n_{-1}@>>>\kk$ is the square of a symplectic
non-degenerate form.
Now $E_J,E_{J,J'},\tE_J$ are naturally algebraic varieties. Define a finite bijective
morphism $\s:E_J@>>>\tE_J$ by $(w,\a,j)\m(w,\a,\tj)$ where $\tj_n=j_n$ for $n\in J$,
$\tj_n=j_n^2$ for $n\in I-J$. Define $\r:E_{J,J'}@>>>\tE_J$ by $(w,\a,j)\m(w,\a,\tj)$
where $\tj_n=j_n$ for $n\in J$ and $\tj_n(x,x')=j_n(x,x')^2+j_n(x,x)j_n(x',x')$ for
$n\in I-J$, $x,x'\in W^n_{-1}$. (To see that this is well defined, we must check that
for $n\in I-J$, the symplectic form $x,x'\m j_n(x,x')+\sqrt{j_n(x,x)j_n(x',x')}$ on
$W^n_{-1}$ is non-degenerate. Let $R$ be the radical of this symplectic form. Let
$H=\{x\in W^n_{-1};j_n(x,x)=0\}$. If $x\in R\cap H$, then $j_n(x,x')$ for all $x'$
hence $x=0$. Thus, $R\cap H=0$. Since $H$ is either $W^n_{-1}$ or a hyperplane in
$W^n_{-1}$, we see that $R\cap H$ is either $R$ or a hyperplane in $R$. It follows that
$\dim R$ is $0$ or $1$. Since $R=\dim W^n_{-1}\mod2$ we see that $\dim R$ is even.
Hence $R=0$, as required.)
Taking here $J'=I$, we see that $\fP_7$ holds. We now return to a general $J'$. We
consider the fibre $\cf$ of $\r$ at $(w,\a,\tj)\in\tE_J$. We may identify $\cf$ with
the set of all collections $(j_n)_{n\in I-J}$ where $j_n\in\Bil(W^n_{-1})$ is symmetric
non-degenerate for all $n$, $j_n$ is symplectic for $n\in I-J'$ and
$\tj_n(x,x')=j_n(x,x')^2+j_n(x,x)j_n(x',x')$ for $n\in I-J$, $x,x'\in W^n_{-1}$. Let
$\cf'$ be the set of all collections $(h_n)_{n\in I-J}$ where $h_n$ is a linear form
$W^n_{-1}@>>>\kk$, zero for $n\in I-J'$. We define a map $\cf@>>>\cf'$ by
$(j_n)_{n\in I-J}\m(h_n)_{n\in I-J}$ where $h_n(x)=\sqrt{j_n(x,x)}$ for
$x\in W^n_{-1}$. We define a map $\cf'@>>>\cf$ by $(h_n)_{n\in I-J}\m(j_n)_{n\in I-J}$
where $j_n(x,x')=\sqrt{\tj_n(x,x')}+h_n(x)h_n(x')$ for $x,x'\in W^n_{-1}$. (We show
that this is well defined. We must show that $j_n$ given by the last equality is
non-degenerate. Let $R'$ be the radical of $j_n$. Define $v\in W^n_{-1}$ by
$h_n(y)=\sqrt{\tj_n(v,y)}$ for all $y\in W^n_{-1}$. If $x\in R',y\in W^n_{-1}$, we have
$\sqrt{\tj_n(x,y)}=h_n(x)h_n(y)=h_n(x)\sqrt{\tj_n(v,y)}$ hence
$\sqrt{\tj_n(x-h_n(x)v,y)}=0$. Since $\sqrt{\tj_n}$ is non-degenerate we have
$x-h_n(x)v=0$. Hence $x=h_n(x)v=h_n(h_n(x)v)v=h_n(x)h_n(v)v$. This is $0$ since
$h_n(v)=\sqrt{\tj_n(v,v)}=0$. Thus $R'=0$.) Clearly, $\cf@>>>\cf',\cf'@>>>\cf$ are
inverse to each other. We see that $\cf$ is in natural bijection with a vector space of
dimension $\sum_{n\in J'-J}c_n$ where $c_n=\dim W^n_{-1}$. Hence if $\kk,q$ are as in
$\fP_5$, we have
$\sum_{K;J\sub K\sub J'}|E_K(\FF_q)|=|E_{J,J'}(\FF_q)|
=\prod_{n\in J'-J}q^{c_n}|E_J(\FF_q)|$.
\nl
From this we see that $|E_K(\FF_q)|=\prod_{n\in K}(q^{c_n}-1)|E_\em(\FF_q)|$ for any
$K\sub I$. Using this and $\fP_6$ we see that the second assertion of $\fP_8$ holds.
For $\kk,q$ as in $\fP_5$ we have
$|H^\bla(\FF_q)|=|X^\l(\FF_q)||G(\FF_q)/G^\l_0(\FF_q)|$,
$|X^\l(\FF_q)|=q^{\dim G^\l_3}|E(\FF_q)|$.
\nl
Hence to verify $\fP_5$ it suffices to show that $|E(\FF_q)|$ is a polynomial in $q$
with integer coefficients independent of $p$. Using the $\bG^\l_0$-equivariant
fibration $\ph:E@>>>\bE$ we see that $|E(\FF_q)|=|\bE(\FF_q)||E^w(\FF_q)|$ for any
$w\in\bE$. Since $|\bE(\FF_q)|$ is a polynomial in $q$ with integer coefficients
independent of $p$, it suffices to show that for any $w\in\bE(\FF_q)$, $|E^w(\FF_q)|$
is a polynomial in $q$ with integer coefficients independent of $p$, or that
$|E^w_n(\FF_q)|$ is a polynomial in $q$ with integer coefficients independent of $p$
for any $w\in\bE(\FF_q)$ and any $n\ge0$. Using the identification
$E^w_n=E^{w1}_n\T E^{w2}_n$ and the fact that $|E^{w1}_n(\FF_q)|$ is a polynomial in
$q$ with integer coefficients independent of $p$, we see that it suffices to show that
$|E^{w2}_n(\FF_q)|$ is a polynomial in $q$ with integer coefficients independent of
$p$. Thus it suffices to check the following statement.
{\it Let $W$ be an $\FF_q$-vector space of dimension $d$. Let $b(W)$ be the set of
non-degenerate symmetric bilinear forms $W\T W@>>>\FF_q$. Then $|b(W)|$ is a polynomial
in $q$ with integer coefficients independent of $p$.}
\nl
We argue by induction on $d$. For $d=0$ the result is obvious. Assume that $d\ge1$. We
write $|b(W)|=f(d,q)$. The set of all symmetric bilinear forms $W\T W@>>>\FF_q$ has
cardinal $q^{d(d+1)/2}$; it is a disjoint union $\sqc_Xb_X(W)$ where $X$ runs over the
linear subspaces of $W$ and $b_X(W)$ is the set of symmetric bilinear forms
$W\T W@>>>\FF_q$ with radical equal to $X$. Thus,
$q^{d(d+1)/2}=\sum_X|b_X(W)|=\sum_X|b(W/X)|=\sum_{d'\in[0,d]}g(d,d',q)f(d-d',q)$
\nl
where $g(d,d',q)=|\{X\sub W,\dim X=d'\}|$. We see that
$f(d,q)=q^{d(d+1)/2}-\sum_{d'\in[1,d]}g(d,d',q)f(d-d',q)$.
\nl
Since $g(d,d',q)$ is a polynomial in $q$ with integer coefficients independent of $p$
and the same holds for $f(d-d',q)$ with $d'\in[1,d]$ (by the induction hypothesis) it
follows that $f(d,q)$ is as required.
We see that $\fP_5$ holds.
\head 4. The group $A^1(u)$\endhead
\subhead 4.1\endsubhead
In this section we assume that $p\ge2$ and that $\fP_1$ holds. Let $u\in\cu$. According
to $\fP_1$ there is a unique $\l\in D_G$ such that $u\in X^\l$. Let
$A^1(u)=Z_{G^\l_1}(u)/Z_{G^\l_1}(u)^0$, a finite $p$-group.
The image of $A^1(u)$ in $Z_G(u)/Z_G(u)^0$ is a normal subgroup (since
$Z_G(u)=Z_{G^\l_0}(u)$, see 1.1(c), and $Z_{G^\l_1}(u)$ is normal in $Z_{G^\l_0}(u)$).
In this section we describe the finite group $A^1(u)$ in some examples assuming that
$p=2$ and $G$ is a symplectic group.
Let $n\ge 1$. Let $I=\{i\in[-n,n];i=n\mod 2\}$. Let $F\in\cc,F\ne0$. Let
$V=\op_{i\in I}F_i$ where $F_i=F$. Define $N:V@>>>V$ by $(x_i)\m(x'_i)$ where
$x'_i=x_{i-2}$ for $i\in I-\{-n\},x'_{-n}=0$. We fix $\lar_0\in\Symp(V)$ such that
$\la(x_i),(y_i)\ra_0=\sum_{i\in I}b(x_i,y_{-i})$ where $b\in\Bil(F)$ satisfies $b^*=b$,
$b$ is non-degenerate and $b\in\Symp(F)$ if $n$ is even.
Let $\lar\in\Symp(V)$ be such that $\la Nx,y\ra+\la x,Ny\ra+\la Nx,Ny\ra=0$ for
$x,y\in V$ and $\la x,y\ra=\la x,y\ra_0$ if there exists $i$ such that $x_j=0$ for
$j\ne i$ and $y_j=0$ for $j\ne-i$. We have
$\la (x_i),(y_i)\ra=\sum_{i,j\in I}b_{ij}(x_i,y_j)$ where $b_{ij}\in\Bil(F)$ are such
that
$b_{i-2,j}+b_{i,j-2}+b_{ij}=0$ if $i,j\in I-\{-n\}$,
$b_{i,-i}=b$ for all $i\in I$,
$b_{ii}\in\Symp(F)$ for all $i\in I$,
$b_{ij}^*=-b_{ji}$ for all $i,j\in I$,
\nl
(We have automatically $b_{ij}=0$ if $i+j\ge 1$.)
\nl
Let $\D'=\{T\in GL(V);TN=NT,\la x,y\ra=\la Tx,Ty\ra\qua\frl x,y\in V\}$, a subgroup of
$Sp(\lar)$; equivalently, $\D'$ is the set of linear maps $T:V@>>>V$ of the form
$T:(x_i)\m(x'_i),x'_i=\sum_{j\in I;j\le i}T_{i-j}x_j$
\nl
where $T_r\in\End(F)$ $(r\in\{0,2,4,\do,2n\})$ are such that
$$b_{ij}(x,y)=\sum_{i',j'\in I;i'\ge i,j'\ge j}b_{i'j'}(T_{i'-i}(x),T_{j'-j}y)
\tag $E_{ij}$ $$
for $i,j\in I,i+j\le0$ and $x,y\in F$. Now $(E_{ij}),(E_{i+2,j-2})$ with $i+j=2k$ are
equivalent if $(E_{ab})$ is assumed for $a+b=2k+2$ (the sum of those two equations is
just $E_{i+2,j}$). Thus the conditions that $T$ must satisfy are $E_{ii}$ and
$E_{i-2,i}$. Setting $x=y$ in these equations we obtain equations $(E_{ii}^0)$,
$(E_{i-2,i}^0)$. Note that the equation $(E_{ii}^0)$ is $0=0$ hence can be omitted; the
equation $(E_{ii})$ is a consequence of $(E_{i-2,i}^0)$ (if it is defined). Hence the
equations satisfied by the components of $T$ are as follows:
$$(E_{-2,0}^0),(E_{-2,0}),(E_{-4,-2}^0),(E_{-4,-2}),\do,(E_{-n,-n+2}^0),
(E_{-n,-n+2}),(E_{-n,-n})\tag a$$
(for $n$ even),
$$\align&(E_{-1,1}),(E_{-3,-1}^0),(E_{-3,-1}),(E_{-5,-3}^0),(E_{-5,-3}),\do,
(E_{-n,-n+2}^0),\\&(E_{-n,-n+2}),(E_{-n,-n})\tag b\endalign$$
(for $n$ odd). Assume first that $n$ is even. The solutions $T_0$ of the first equation
in (a) form an even orthogonal group, a variety with two connected components. For any
such $T_0$ the solutions $T_2$ of the second equation in (a) form an affine space of
dimension independent of $T_0$. For any $T_0,T_2$ already determined, the solutions
$T_4$ of the third equation in (a) form an affine space of dimension independent of
$T_0,T_2$. Continuing in this way we see that the solutions of the equations (a) form a
variety with two connected components. Moreover, the solutions in which $T_0$ is
specified to be $1$ form a connected variety.
Assume next that $n$ is odd and $b$ is symplectic. The solutions $T_0$ of the first
equation in (b) form a symplectic group (a connected variety). For any such $T_0$ the
solutions $T_2$ of the second equation in (b) form an affine space of dimension
independent of $T_0$. For any $T_0,T_2$ already determined, the solutions $T_4$ of the
third equation in (b) form an affine space of dimension independent of $T_0,T_2$.
Continuing in this way we see that the solutions of the equations (b) form a connected
variety. Moreover, the solutions in which $T_0$ is specified to be $1$ form a connected
variety.
One can show that, if $n$ is odd, $n\ge3$ and $b$ is not symplectic, the solutions of
the equations (b) form a variety with two connected components. Moreover, the solutions
in which $T_0$ is specified to be $1$ form a disconnected variety.
In solving the equations above we use repeatedly the statement (c) below. Let $\fQ$ be
the vector space of quadratic forms $F@>>>\kk$. Define linear maps $a_1,a_2,a_3$ as
follows:
$a_1:\End(F)@>>>\fQ(F)$ is $\t\m q,q(x)=b(x,\t(x))$;
$a_2:\{\t\in\End(F);b(\t(x),y)=b(x,\t(y)\qua\frl x,y\in F\}@>>>\Hom(F,\kk)$ is
$\t\m\th,\th(x)=\sqrt{b(x,\t(x))}$;
$a_3:\{b'\in\Bil(F);b'{}^*=b'\}@>>>\Hom(F,\kk)$ is $b'\m\th,\th(x)=\sqrt{b'(x,x)}$.
\nl
Then
(c) $a_1,a_2,a_3$ are surjective.
\nl
For $a_3$ this is clear. Consider now $a_2$. Let $\th\in\Hom(F,\kk)$. By (c) for $a_3$
we can find $b'\in\Bil(F),b'{}^*=b'$ such that $\th(x)=\sqrt{b'(x,x)}$. We can find a
unique $\t\in End(F)$ such that $b(x,\t(y))=b'(x,y)$. Then $a_2(t)=\th$. Consider now
$a_1$. Let $q\in\fQ$. Let $b^0$ be a symplectic form on $F$. We can write $b^0=d+d^*$
where $d\in\Bil(F)$. We can write $d(x,y)=b(x,\s(y))$ for some $\s\in\End(F)$. Then
$b(x,\s(y))+b(y,\s(x))=b^0(x,y)$. Apply this to the symplectic form
$b^0(x,y)=q(x+y)+q(x)+q(y)$. Then
$b(x+y,\s(x+y))+b(x,\s(x))+b(y,\s(y))=b(x,\s(y))+b(y,\s(x))=q(x+y)+q(x)+q(y)$.
\nl
Hence $b(x,\s(x))+q(x)=\th(x)^2$ for some $\th\in\Hom(F,\kk)$. By (c) for $a_2$ we can
find $\t\in\End(F)$ such that $b(x,\t(x))=\th(x)^2$. Then $b(x,\s(x))+b(x,\t(x))=q(x)$
that is $b(x,(\s+\t)(x))=q(x)$. Thus $a_1$ is surjective. This proves (c).
\subhead 4.2\endsubhead
Let $V,\lar$ be as in 3.2. Let $N\in\cm_{\lar},V_*=V_*^N$. Let $e$ be as in 2.4. We
show:
(a) {\it If $W,W'$ are $e$-special subspaces of $V$ (see 2.8) then there exists
$g\in1+E^{\lar}_{\ge1}V_*$ such that $g(W)=W'$, $gN=Ng$.}
\nl
By 2.8(b) we can find $g_1\in1+E_{\ge 1}V_*$ such that $g_1(W)=W'$, $g_1N=Ng_1$. Then
$g_1$ carries $\lar$ to a symplectic form $\lar'$ which induces the same symplectic
form as $\lar$ on $\gr V_*$ and has the same associated quadratic forms as $\lar$ (see
3.6(b)). By the proof in 3.10 (case 2 and 3) we see that there exists
$g_2\in1+E_{\ge 1}V_*$ such that $g_2(W')=W'$, $g_2N=Ng_2$ and $g_2$ carries $\lar'$ to
$\lar$. Then $g=g_2g_1$ has the required properties.
\subhead 4.3\endsubhead
Let $V,\lar,N,V_*,e$ be as in 4.2.
(a) {\it If $\la x,Nx\ra=0$ for any $x\in V_{\ge-1}$, then
$\cv:=\{g\in E^{\lar}_{\ge1}V_*;gN=Ng\}$ is connected. Hence $A^1(1+N)=\{1\}$.}
\nl
We argue by induction on $e$. Let $\cx$ be the set of all $e$-special subspaces (see
2.8) of $V$. By 2.8(b) the group $\{g\in 1+E_{\ge 1}V_*;gN=Ng\}$ acts transitively on
$\cx$. This group is connected (it may be identified as variety with the vector space
$\{\x\in E_{\ge 1}V_*;\x N=N\x\}$); hence $\cx$ is connected. By 4.2(a), $\cv$ acts
transitively on $\cx$. Since $\cx$ is connected, it suffices to show that the
stabilizer $\cv_W$ of some $W\in\cx$ in $\cv$ is connected. This stabilizer is
$\cv'\T\cv''$ where $\cv',\cv''$ are defined like $\cv$ in terms of $W,W^\pe$ instead
of $V$. By results in 4.2, $\cv'$ is connected. By the induction hypothesis applied to
$W^\pe$, $\cv''$ is connected. Hence $\cv'\T\cv''$ is connected. Hence $\cv$ is
connected.
\head 5. Study of the varieties $\cb_u$\endhead
\subhead 5.1\endsubhead
We assume that $\kk=\kk_p$.
We say that an algebraic variety $V$ over $\kk$ has the {\it purity property} if for
some/any $\FF_q$-rational structure on $V$ (where $\FF_q$ is a finite subfield of
$\kk$) with Frobenius map $F:V@>>>V$ and any $n\in\ZZ$, any complex absolute value of
any eigenvalue of $F^*:H^n_c(V,\bbq)@>>>H^n_c(V,\bbq)$ is $q^{n/2}$.
In this section we show that for certain $u\in\cu$ the varieties $\cb_u$ (see 0.1) have
the purity property. We assume that properties $\fP_1-\fP_4,\fP_6,\fP_7$ hold for $G$.
Let $\l\in D_G$. Let $\Pi^\l$ be the (finite) set of orbits for the conjugation action
of $G^\l_0$ on $\cb$. Let $\bcb=\{B\in\cb;B\sub G^\l_0\}$. For any $\co\in\Pi^\l$
define a morphism $\x^\co:\co@>>>\bcb$ by $B\m(B\cap G^\l_0)G^\l_1$. We show:
(a) {\it The fibres of $\x^\co:\co@>>>\bcb$ are exactly the orbits of $G^\l_1$ acting
on $\co$ by conjugation.}
\nl
If $B,B'\in\co$, $\x^\co(B)=\x^\co(B')$, then $B'=g\i Bg$ with $g\in G^\l_0$,
$(B'\cap G^\l_0)G^\l_1=(B\cap G^\l_0)G^\l_1=g\i(B\cap G^\l_0)G^\l_1g$. Hence
$g\in(B\cap G^\l_0)G^\l_1$. Writing $g=g'g'',g'\in B\cap G^\l_0$, $g''\in G^\l_1$, we
have $B'=g\i Bg=g''{}\i Bg''$. This proves (a).
Let $Y^\l=\{(u,B)\in X^\l\T\cb;u\in B\}$. We have a partition
$Y^\l=\cup_{\co\in\Pi^\l}Y^\l_\co$ where $Y^\l_\co=\{(u,B)\in X^\l\T\co;u\in B\}$. Let
$\co\in\Pi^\l$. We show:
(b) {\it$Y^\l_\co$ is smooth.}
\nl
Let $\tB\in\co$. Let $Y'=\{(u,g)\in X^\l\T G^\l_0;g\i ug\in\tB\cap X^\l\}$. We have a
fibration $Y'@>>>Y^\l_\co$ with smooth fibres isomorphic to $G^\l_0\cap\tB$. Hence it
suffices to show that $Y'$ is smooth. Let $Y''=(\tB\cap X^\l)\T G^\l_0$. Define
$Y'@>\si>>Y''$ by $(u,g)\m(g\i ug,g)$. It suffices to show that $Y''$ is smooth, or
that $\tB\cap X^\l$ is smooth. But $\tB\cap X^\l$ is open in $\tB\cap G^\l_2$ which is
smooth, being an algebraic group. This proves (b).
For any $\b\in\bcb$ let $\cg_\b^\co=((B\cap G^\l_2)G^\l_3)/G^\l_3$ where $B\in\co$ is
such that $\x^\co(B)=\b$. Note that $\cg_\b^\co$ is a closed connected subgroup of
$G^\l_2/G^\l_3$, independent of the choice of $B$. (To verify the last statement it
suffices, by (a), to show that for $B$ as above and $v\in G^\l_1$ we have
$(vBv\i\cap G^\l_2)G^\l_3=(B\cap G^\l_2)G^\l_3$. This follows from 1.1(b).) Now
$G^\l_0$ acts on $\bcb$ and on $G^\l_2/G^\l_3$ by conjugation. From the definitions we
see that for $g\in G^\l_0$ and $\b\in\bcb$ we have $\cg_{g\b g\i}^\co=g\cg_\b^\co g\i$.
Let $\bY^\l_\co=\{(x,\b)\in\bX^\l\T\bcb;x\in\cg_\b^\co\}$. We show:
(c) {\it$\bY^\l_\co$ is a closed smooth subvariety of $\bX^\l\T\bcb$.}
\nl
Let $\tB\in\co$. We have a fibration $X^\l\T G^\l_0@>>>\bX^\l\T\bcb$,
$(u,g)\m(\p^\l(u),\x^\co(g\tB g\i))$ with smooth fibres. It suffices to show that the
inverse image of $\bY^\l_\co$ under this fibration is a closed smooth subvariety of
$X^\l\T G^\l_0$, or that
$\{(u,g)\in X^\l\T G^\l_0;g\i ug\in X^\l\cap((\tB\cap G^\l_2)G^\l_3)\}$
\nl
is a closed smooth subvariety of $X^\l\T G^\l_0$, or that
$(X^\l\cap((\tB\cap G^\l_2)G^\l_3))\T G^\l_0$ is a closed smooth subvariety of
$X^\l\T G^\l_0$ or that $X^\l\cap((\tB\cap G^\l_2)G^\l_3)$ is a closed smooth
subvariety of $X^\l$. It is closed since $(\tB\cap G^\l_2)G^\l_3$ is closed in
$G^\l_2$. It is smooth since it is an open subset of $(\tB\cap G^\l_2)G^\l_3$ which is
smooth, being an algebraic group.
We show:
(d) {\it The morphism $a:Y^\l_\co@>>>\bY^\l_\co,(u,B)\m(\p^\l(u),\x^\co(B))$ is a
fibration with fibres isomorphic to an affine space of a fixed dimension.}
\nl
Clearly, $a$ is surjective. Let $(u,B)\in Y^\l_\co$. Let
$Z:=a\i(a(u,B))
=\{(u',B');u=u'f,B'=vBv\i\text{ for some }v\in G^\l_1,f\in G^\l_3;u'\in B'\}$.
\nl
We show only that $Z$ is isomorphic to an affine space of fixed dimension. Let
$\tZ=\{(f,v)\in G^\l_3\T G^\l_1;v\i uf\i v\in B\}$. Then $Z=\tZ/(B\cap G^\l_1)$ where
$B\cap G^\l_1$ acts freely on $\tZ$ by $b:(f,v)\m(f,vb\i)$. Since conjugation by
$G^\l_1$ acts trivially on $G^\l_2/G^\l_3$, the map $(f,v)\m(f',v),f'=u\i v\i uf\i v$
is an isomorphism
$$\align&\tZ@>>>\tZ'=\{(f',v)\in G^\l_3\T G^\l_1;uf'\in B\}=
\{(f',v)\in G^\l_3\T G^\l_1;f'\in B\}\\&=(G^\l_3\cap B)\T G^\l_1\endalign$$
(we use $u\in B$) and we have $Z=(G^\l_3\cap B)\T G^\l_1/(B\cap G^\l_1)$. Now
$G^\l_3\cap B,G^\l_1,B\cap G^\l_1$ are connected unipotent groups of dimension
independent of $B$, for $B\in\co$. (The connectedness follows from the fact that these
unipotent groups are normalized by a maximal torus of $G$ contained in $G^\l_0\cap B$.
The fact that the dimension does not depend on $B$ follows from the fact that
$G^\l_1,G^\l_3$ are normalized by $G^\l_0$.) We see that $Z$ is an affine space of
constant dimension.
We now fix $x\in\bX^\l$. Let $\Si=(\p^\l)\i(x)\sub X^\l$. Let
$\cb_\Si=\{(u,B)\in\Si\T\cb;u\in B\}$. We have $\cb_\Si=\sqc_{\co\in\Pi^\l}\co_\Si$
where $\co_\Si=\{(u,B)\in\Si\T\co;u\in B\}$. Let $\co\in\Pi^\l$. Let
$\bcb^\co_x=\{\b\in\bcb;x\in\cg^\co_\b\}$. We show:
(e) {\it$\bcb^\co_x$ is a closed subvariety of $\bcb$ and $a':\co_\Si@>>>\bcb^\co_x$,
$(u,B)\m\x^\co(B)$ is a fibration with fibres isomorphic to an affine space of a fixed
dimension.}
\nl
Let $\tB\in\co,u_0\in\Si$. We have a locally trivial fibration $G^\l_0@>>>\bcb$,
$g\m\x^\co(g\tB g\i)$. To show that $\bcb^\co_x$ is closed it suffices to show that its
inverse image under this fibration is closed in $G^\l_0$, or that
$\{g\in G^\l_0;g\i u_0g\in(\tB\cap G^\l_2)G^\l_3\}$ is closed in $G^\l_0$. This is
clear since $(\tB\cap G^\l_2)G^\l_3$ is closed in $G^\l_2$. The second assertion of (e)
follows from (d) using the cartesian diagram
$$\CD
\co_\Si@>a'>>\bcb^\co_x\\
@VVV @VVV\\
Y^\l_\co@>a>>\bY^\l_\co
\endCD$$
where the left vertical map is the obvious inclusion and the right vertical map is
$\b\m(x,\b)$.
(f) {\it If the closure of the $G^\l_0$-orbit in $G^\l_2$ of some/any $u\in\Si$ is a
subgroup $\G$ of $G^\l_2$ then $\bcb^\co_x$ is smooth.}
\nl
Let $\tB\in\co,u_0\in\Si$. As in the proof of (e) it suffices to show that
$\{g\in G^\l_0;g\i u_0g\in(\tB\cap G^\l_2)G^\l_3\}$ is smooth. This variety is a
fibration over $R=(G^\l_0-\text{conjugacy class of }u_0)\cap((\tB\cap G^\l_2)G^\l_3)$
with smooth fibres isomorphic to $Z_{G^\l_0}(u_0)$ (via $g\m g\i u_0g$). Hence it
suffices to show that $R$ is smooth. From our assumption we see that $R$ is open in
$\G\cap((\tB\cap G^\l_2)G^\l_3)$ which is smooth being an algebraic group. This proves
(f).
\mpb
Note that the hypothesis of (f) holds at least in the case where the $G^\l_0$-conjugacy
class of some/any $u\in\Si$ is open dense in $G^\l_2$. We show
(g) {\it If the hypothesis of (f) holds then $\cb_\Si$ has the purity property.}
\nl
From (e),(f) we see that $\bcb^\co_x$ is a smooth projective variety of pure dimension.
From \cite{\DE} it then follows that $\bcb^\co_x$ has the purity property. From this
and (e) we see that for $\co\in\Pi^\l$, $\co_\Si$ has the purity property. Using this
and the partition $\cb_\Si=\sqc_{\co\in\Pi^\l}\co_\Si$, we see that (g) holds.
\subhead 5.2\endsubhead
Let $\bZ(x)=\{\bg\in\bG^\l_0;\bg x=x\bg\}$. Let $\tZ(x)$ be the inverse image of
$\bZ(x)$ under $G^\l_0@>>>\bG^\l_0$. Thus we have $G^\l_1\sub\tZ(x)$ and
$\tZ(x)/G^\l_1@>\si>>\bZ(x)$. Note that the inverse image of $\bZ(x)^0$ is $\tZ(x)^0$
and we have $\tZ(x)^0/G^\l_1@>\si>>\bZ(x)^0$. Now $\tZ(x)$ acts transitively (by
conjugation) on $\Si$. (Indeed, let $u,u'\in\Si$. By $\fP_6$ we can find $g\in G^\l_0$
such that $u'=gug\i$. We have automatically $g\in\tZ(x)$.) Since $\Si$ is irreducible,
it follows that $\tZ(x)^0$ acts transitively (by conjugation) on $\Si$.
Let $u\in\Si$. Recall that $\cb_u=\{B\in\cb;u\in B\}$. Let
$Z'_G(u)=Z_G(u)\cap\tZ(x)^0$. Since $Z_G(u)\sub\tZ(x)$, see 1.1(c), we see that
$Z'_G(u)$ is a normal subgroup of $Z_G(u)$ containing $Z_G(u)^0$. Let $A'(u)$ be the
image of $Z'_G(u)$ in $A(u):=Z_G(u)/Z_G(u)^0$; this is a normal subgroup of $A(u)$. We
have $Z'_G(u)/Z_{G^\l_1}(u)@>\si>>\bZ(x)^0$. Hence
$Z'_G(u)=Z_{G^\l_1}(u)Z'_G(u)^0=Z_{G^\l_1}(u)Z_G(u)^0$. It follows that
{\it$A'(u)$ is the image of the obvious homomorphism} $A^1(u)@>>>A(u)$.
\nl
Now $Z_G(u)$ acts by conjugation on $\cb_u$; this induces an action of $A(u)$ on
$H^n(\cb_u,\bbq)$ which restricts to an $A'(u)$-action on $H^n(\cb_u,\bbq)$.
Assume that the hypothesis of 5.1(f) holds and that $A'(u)$ acts trivially
on $H^n_c(\cb_u,\bbq)$ for any $n$. We show:
(a) {\it$\cb_u$ has the purity property.}
\nl
Define $f:\cb_\Si@>>>\Si$ by $(g,B)\m g$. For any $n$, $R^nf_!(\bbq)$ is an equivariant
constructible sheaf for the transitive $\tZ(x)^0$ action on $\Si$; hence it is a local
system on $\Si$ corresponding to a representation of $A'(u)$ (the group of components
of the isotropy group of $u$ in $\tZ(x)^0$) on $H^n_c(\cb_u,\bbq)$. This representation
is trivial hence $R^nf_!(\bbq)$ is a constant local system. Since $\Si$ is an affine
space of dimension say $d$ we see that $H^a_c(\Si,R^nf_!(\bbq))$ is
$H^n_c(\cb_u,\bbq)(-d)$ if $a=2d$ and is zero if $a\ne2d$. It follows that the standard
spectral sequence
$E_2^{a,n}=H^a_c(\Si,R^nf_!(\bbq))\Rightarrow H^{a+n}_c(\cb_\Si,\bbq)$
\nl
is degenerate. Hence the purity property of $\cb_\Si$ (see 5.1(g)) implies that any
complex absolute value of any eigenvalue of the Frobenius map on
$E_2^{2d,n}=H^n_c(\cb_u,\bbq)(-d)$
\nl
is $q^{d+n/2}$. Hence any complex absolute value of any eigenvalue of the Frobenius map
on $H^n_c(\cb_u,\bbq)$ is $q^{n/2}$. This proves (a).
\subhead 5.3\endsubhead
Since the hypothesis of 5.1(f) is not satisfied in general, we seek an alternative way
to prove purity.
Let $\g$ be the $\bG^\l_0$-orbit of $x$ in $\bX^\l$. Let $\hag@>\r>>\g_1@<\s<<\g$ be as
in $\fP_7$. Let $\Xi=\r\i(\s(x))$. Let
$\bcb^\co_\Xi=\{(x',\b)\in\bY^\l_\co;x'\in\Xi\}$, a closed subvariety of $\bY^\l_\co$.
We show:
(a) {\it$\bcb^\co_\Xi$ is smooth of pure dimension.}
\nl
Let $\b_0\in\bcb$. Let $\cg_0=\cg^\co_{\b_0}$. It suffices to show that the inverse
image of $\bcb^\co_\Xi$ under the fibration $\Xi\T\bG^\l_0@>>>\Xi\T\bcb$,
$(x',\bg)\m(x',\bg\b_0\bg\i)$ (with smooth connected fibres) is smooth of pure
dimension, or that $\fS:=\{(x',\bg)\in\Xi\T\bG^\l_0;\bg\i x'\bg\in\cg_0\}$ is smooth of
pure dimension. The morphism $f:\fS@>>>\hag\cap\cg_0$, $(x',\bg)\m\bg\i x'\bg$ is
smooth with fibres of pure dimension. (We show only that for any $y\in\hag\cap\cg_0$,
the fibre $f\i(y)$ is isomorphic to $\{\bg\in\bG^\l_0;\bg x\bg\i=x\}$ which is smooth
of pure dimension. We have
$$\align&f\i(y)=\{(x',\bg)\in\Xi\T\bG^\l_0;\bg\i x'\bg=y\}\cong
\{\bg\in\bG^\l_0;\bg y\bg\i\in\Xi\}\\&=\{\bg\in\bG^\l_0;\r(\bg y\bg\i)=\s(x)\}=
\{\bg\in\bG^\l_0;\bg\s\i(\r(y))\bg\i=x\}\endalign$$
and it remains to use the transitivity of the $\bG^\l_0$-action on $\g$.) It suffices
to show that $\hag\cap\cg_0$ is empty or smooth, connected. Now $\hag$ is open in
$G^\l_2/G^\l_3$ hence $\hag\cap\cg_0$ is open in $\cg_0$ which is connected and smooth
(being an algebraic group).
We show:
(b) {\it Assume that for any $\co\in\Pi^\l$ there is a $\kk^*$-action on $\bcb^\co_\Xi$
which is a contraction to the projective subvariety $\bcb^\co_x$. Then $\cb_\Si$ has
the purity property.}
\nl
Consider an $\FF_q$-rational structure on $G$ such that $G^\l_a$ is defined over
$\FF_q$ for any $a$ and $\co,x,\Xi$ are defined over $\FF_q$. Let $\z$ be an eigenvalue
of Frobenius on $H^n(\bcb^\co_x,\bbq)$. By \cite{\DEII, 3.3.1}, any complex absolute
value of $\z$ is $\le q^{n/2}$ (since $\bcb^\co_x$ is projective). Our assumption
implies that the inclusion $\bcb^\co_x\sub\bcb^\co_\Xi$ induces for any $n$ an
isomorphism $H^n(\bcb^\co_\Xi,\bbq)@>\si>>H^n(\bcb^\co_x,\bbq)$. Hence $\z$ is also
an eigenvalue of Frobenius on $H^n(\bcb^\co_\Xi,\bbq)$. Since $\bcb^\co_\Xi$ is smooth
of pure dimension say $d$, it satisfies Poincar\'e duality; hence $q^d\z\i$ is an
eigenvalue of Frobenius on $H^{2d-n}_c(\bcb^\co_\Xi,\bbq)$. By \cite{\DEII, 3.3.1}
applied to $\bcb^\co_\Xi$, we see that any complex absolute value of $q^d\z\i$ is
$\le q^{(2d-n)/2}$ hence any complex absolute value of $\z$ is $\ge q^{n/2}$. It
follows that any complex absolute value of $\z$ is $q^{n/2}$. We see that $\bcb^\co_x$
has the purity property. (This argument is similar to one of Springer in \cite{\SP}.)
From this and 5.1(e) we see that for $\co\in\Pi^\l$, $\co_\Si$ has the purity property.
Using this and the partition $\cb_\Si=\sqc_{\co\in\Pi^\l}\co_\Si$, we see that
$\cb_\Si$ has the purity property.
\mpb
If we assume in addition that $A'(u)$ acts trivially on $H^n_c(\cb_u,\bbq)$ for any $n$
we see as in 5.2 that $\cb_u$ has the purity property.
\subhead 5.4\endsubhead
Let $V,\lar$ be as in 3.2. Assume that $p=2$ and that $G=Sp(\lar)$. Let $u\in\cu$. We
set $u=1+N,V_*=V_*^N$. Assume that
(a) $\la x,Nx\ra=0$ for any $x\in V_{\ge-1}$.
\nl
We set
$$\G=1+\{N'\in E_{\ge2}^{\lar}V_*;\la x,N'x\ra=0\qua\frl x\in V_{\ge-1}\}.$$
Now $\G$ is a subgroup of $1+E_{\ge2}^{\lar}V_*$. (Assume that $1+N',1+N''\in\tG_2$.
Let $x\in V_{\ge-1}\}$. We have $\la x,N'x\ra=0,\la x,N''x\ra=0$. We must show that
$\la x,(N'+N''+N'N'')x\ra=0$ or that $\la x,N'N''x\ra=0$. This follows from
$N'N''x\in V_{\ge3}$ and $3-1\ge1$.) Clearly, $\G$ is normal in $G^\l_0$. Since $\G$ is
a closed unipotent subgroup normalized by $G^\l_0$, it must be connected. Now
$\cj:=1+\{N'\in\tG_2;\bN'\in\End_2^0(\gr V_*)\}$
\nl
is open in $\G$ since it is the inverse image under $\G@>>>\End_2(\gr V_*),1+N'\m\bN'$
of the open subset $\End_2^0(\gr V_*)$ of $\End_2(\gr V_*)$. Also $\cj\ne\em$ since
$1+N\in\cj$. Hence $\cj$ is an open dense subset of $\G$. By results in 3.14, $\cj$ is
the $G^\l_0$-conjugacy class of $1+N$.
We see that the hypothesis of 5.1(f) holds. Using 5.2(a) we see that:
(b) {\it$\cb_u$ has the purity property for any $u\in G$ whose conjugacy class is
minimal in the unipotent piece containing it, see 1.1, and such that any Jordan block
of even size appears an even number times.}
\nl
(For such $u$, $A'(u)$ is trivial by 4.3(a).)
Alternatively, one can show that for $u$ as in (b) the method of 5.3 is applicable (the
hypothesis of 5.3(b) holds) and one obtains another proof of (b).
\widestnumber\key{DLP}
\Refs
\ref\key{\DLP}\by C.De Concini,G.Lusztig,C.Procesi\paper Homology of the zero set of a
nilpotent vector field on the flag manifold\jour J.Amer.Math.Soc.\vol1\yr1988\pages
15-34\endref
\ref\key{\DE}\by P.Deligne\paper La conjecture de Weil,I\jour Publ.Math.IHES\vol43\yr
1974\pages273-308\endref
\ref\key{\DEII}\by P.Deligne\paper La conjecture de Weil,II\jour Publ.Math.IHES\vol52
\yr1980\pages137-252\endref
\ref\key{\EN}\by H.Enomoto\paper The conjugacy classes of Chevalley groups of type
$G_2$ over finite fields of characteristic $2$ or $3$\jour J. Fac. Sci. Univ. Tokyo,I
\vol16\yr1970\pages497-512\endref
\ref\key{\KA}\by N.Kawanaka\paper Generalized Gelfand-Graev representations of
exceptional simple algebraic groups over a finite field\jour Invent.Math.\vol84\yr1986
\pages575-616\endref
\ref\key{\KO}\by B.Kostant\paper The principal three dimensional subgroup and the Betti
numbers of a complex simple Lie group\jour Amer.J.Math.\vol81\yr1959\pages973-1032
\endref
\ref\key{\LC}\by G.Lusztig\paper A class of irreducible representations of a Weyl group
\jour Proc.Kon.Nederl.Akad. (A)\vol82\yr1979\pages323-335\endref
\ref\key{\LN}\by G.Lusztig\paper Notes on unipotent classes\jour Asian J.Math.\vol1\yr
1997\pages194-207\endref
\ref\key{\LS}\by G.Lusztig and N.Spaltenstein\paper On the generalized Springer
correspondence for classical groups\inbook Algebraic groups and related topics,
Adv.Stud.Pure Math.6\publ North Holland and Kinokuniya\yr1985\pages289-316\endref
\ref\key{\MI}\by K.Mizuno\paper The conjugate classes of unipotent elements of the
Chevalley groups $E_7$ and $E_8$\jour Tokyo J.Math\vol3\yr1980\pages391-459\endref
\ref\key{\SH}\by K.Shinoda\paper The conjugacy classes of Chevalley groups of type
$F_4$ over finite fields of characteristic $2$\jour J. Fac. Sci. Univ. Tokyo,I\vol21\yr
1974\pages133-159\endref
\ref\key{\SPA}\by N.Spaltenstein\book Classes unipotentes et sous-groupes de Borel, LNM
946\publ Springer Verlag\yr1980\endref
\ref\key{\SPAII}\by N.Spaltenstein\paper On the generalized Springer correspondence for
exceptional groups\inbook Algebraic groups and related topics, Adv.Stud.Pure Math.6
\publ North Holland and Kinokuniya\yr1985\pages317-338\endref
\ref\key{\SP}\by T.A.Springer\paper A purity result for fixed point sets varieties in
flag manifolds\jour J. Fac. Sci. Univ. Tokyo,IA\vol31\yr1984\pages 271-282\endref
\ref\key{\WA}\by G.E.Wall\paper On the conjugacy classes in the unitary, symplectic and
orthogonal groups\jour J.Austral.Math.Soc.\vol3\yr1963\pages1-62\endref
\endRefs
\enddocument
|
{
"timestamp": "2005-04-03T19:28:01",
"yymm": "0503",
"arxiv_id": "math/0503739",
"language": "en",
"url": "https://arxiv.org/abs/math/0503739"
}
|
\section{Introduction}
Thompson's groups $F$, $T$ and $V$ are a remarkable family of infinite, finitely-presentable groups
studied for their own properties as well as for their connections with questions in logic,
homotopy theory, geometric group theory and the amenability of discrete groups.
Cannon, Floyd and Parry give an excellent introduction to these groups in \cite{cfp}. These three groups
can be viewed either algebraically, combinatorially, or analytically. Algebraically, each has both
finite and infinite presentations. Geometrically, an element in each group can be viewed as a {\em tree
pair diagram}; that is, as a pair of finite binary rooted trees with the same number of leaves, with a
numbering system pairing the leaves in the two trees. Analytically, an element of each group can be
viewed as a piecewise-linear self map of the unit interval:
\begin{itemize}
\item in $F$ as a piecewise linear homeomorphism,
\item in $T$ as a homeomorphism of the unit interval with the endpoints identified, and thus of $S^1$,
\item in $V$ as a right-continuous bijection which is locally orientation preserving.
\end{itemize}
Thompson's group $F$ in particular has been studied extensively. The group $F$ has a standard infinite
presentation in which every element has a unique normal form, and a standard two-generator finite
presentation. Fordham \cite{blakegd} presented a method of computing the word length of $w \in F$ with
respect to the standard finite generating set directly from a tree pair diagram representing $w$.
Regarding $F$ as a diagram group, Guba \cite{gubagrowth} also obtained an effective geometric method for
computing the word metric with respect to the standard finite generating set. Belk and Brown
\cite{belkbrown} have similar results which arise from viewing elements of $F$ as forest diagrams.
In this paper, we discuss analogues for $T$ of some properties of $F$, using all
three of the descriptions of $T$: algebriac, geometric and analytic. We begin by
desribing unique normal forms for elements which arise from their reduced tree pair descriptions.
We consider metrically how $F$ is contained as a subgroup of $T$, and show that the number of carets in a
reduced tree pair diagram representing $w \in T$ estimates the word length of $w$ with respect to a
particular generating set. Thus $F$ is quasi-isometrically embedded in $T$. Furthermore,
we show that there are families of words in $F$ which are isometrically embedded in $T$ with respect to an alternate finite generating set. The groups $T$ and $V$, unlike $F$, contain
torsion elements, and we describe how to recognize these torsion elements from their tree pair diagrams. Finally, we show that every torsion element of $T$ is conjugate to a power to a generators of $T$
and that the subgroup of rotations in $T$ is quasi-isometrically embedded.
\section{Background on Thompson's groups $F$ and $T$}
\subsection{Presentations and tree pair diagrams}
Thompson's groups $F$ and $T$ both have representations as groups of piecewise-linear homeomorphisms. The
group $F$ is the group of orientation-preserving homeomorphisms of the interval $[0,1]$, where each
homeomorphism is required to have only finitely many discontinuities of slope, called {\em breakpoints},
have slopes which are powers of two and have the coordinates of the breakpoints all lie in the set of dyadic
rationals. Similarly, the group $T$ consists of orientation-preserving homeomorphisms of the circle $S^1$
satisfying the same conditions where we represent the circle $S^1$ as the unit interval $[0,1]$ with the
two endpoints identified.
Cannon, Floyd and Parry give an excellent introduction to Thompson's groups $F$, $T$ and $V$ in
\cite{cfp}. We refer the reader to this paper for full details on results mentioned in this section.
Since more readers have some familiarity with $F$ than with $T$, we first give a very brief review of the
group $F$, and then a slightly more detailed review of $T$. Algebraically, $F$ has well known infinite
and finite presentations. With respect to the infinite presentation
$$
\langle x_i, i\geq 0\, |\,x_jx_i=x_ix_{j+1}, i<j\rangle
$$
for $F$, group elements have simple normal forms which are unique. It is easy to see that $F$ can be
generated by $x_0$ and $x_1$, which form the standard finite generating set for $F$, and yield the finite
presentation
$$
\langle x_0,x_1\,|\,[x_0x_1^{-1},x_0^{-1}x_1x_0],[x_0x_1^{-1},x_0^{-2}x_1x_0^2]\rangle.
$$
A geometric representation for an element $w$ in $F$ is a tree pair diagram, as discussed in \cite{cfp}.
A {\em tree pair diagram} is a pair of finite rooted binary trees with the same number of leaves. By
convention, the leaves of each tree are thought of as being numbered from $0$ to $n$ reading from left to
right. A node of the tree together with its two downward directed edges is called a {\em caret}. The {\em
left side} of the tree consists of the root caret, and all carets connected to the root by a path of left
edges; the {\em right side} of the tree is defined analogously. A caret is called a {\em left caret} if
its left leaf lies on the left side of the tree. A caret is called a {\em right caret} if it is not the
root caret and its right leaf lies on the right side of the tree. All other carets are called {\em
interior}. A caret is called {\em exposed} if it contains two leaves of the tree. For $w \in F$, we write
$w = (T_-,T_+)$ to express $w$ as a tree pair diagram, and refer to $T_-$ as the {\em source} tree and
$T_+$ as the {\em target} tree. These trees arise naturally from the interpretation of $F$ as a group of
homeomorphisms. Thinking of $w$ as a homeomorphism of the unit interval, the source tree represents a
subdivision of the domain into subintervals of width $1/2^n$ for varying values of $n$, and the target tree represents another such
a subdivision of the range. The homeomorphism then maps the $i^{th}$ subinterval in the domain linearly
to the $i^{th}$ subinterval in the range.
A tree pair diagram representing $w$ in $F$ is not unique. A new diagram can always be produced from a given tree pair
diagram representing $w$ simply by adding carets to the $i$th leaf of both trees. We impose a natural
reduction condition: if $w = (T_-,T_+)$ and both trees contain a caret with two exposed leaves numbered
$i$ and $i+1$, then we remove these carets, thus forming a representative for $w$ with fewer carets and
leaves.
A tree pair diagram which admits no such reductions is called a {\em reduced tree pair diagram},
and any element of $F$ is represented by a unique reduced tree pair diagram. When we write $w =
(T_-,T_+)$ below, we are assuming that the tree pair diagram is reduced unless otherwise specified.
The group $T$ also has both a finite and an infinite presentation. The infinite presentation is given by
two families of generators, $\{x_i,i\ge 0\}$, the same generators as in the infinite presentation of $F$,
a family $\{c_i,i\ge0\}$ of torsion elements, and the following relators:
\begin{enumerate}
\item $x_jx_i=x_ix_{j+1}$, if $i<j$ \item $x_kc_{n+1}=c_{n}x_{k+1}$,
if $k<n$ \item $c_nx_0=c_{n+1}^2$ \item $c_n=x_nc_{n+1}$ \item$c_n^{n+2}=1$.
\end{enumerate}
This new family of generators $c_n$ (of order $n+2$), is simple to describe. The generator $c_n$ corresponds to the
homeomorphism of the circle obtained as follows. Both domain and range can be thought of as the unit
interval with the endpoints identified. We subdivide the interval into $n+1$ subintervals by successively
halving the rightmost subinterval; or in other words inserting endpoints at $\frac{1}{2}, \frac{3}{4},
\ldots ,\frac{2^{n+1}-1}{2^{n+1}}$. Then the homeomorphism maps $[0,1/2]$ linearly to
$[\frac{2^{n}-1}{2^{n}},\frac{2^{n+1}-1}{2^{n+1}}]$, and so on around each circle. For example, the
element $c$ corresponds to the homeomorphism of $S^1$ given by
$$
c(t)=\left\{\begin{array} {ll}
\frac12t+\frac34&\text{if }0\le t<\frac12\\
2t-1&\text{if }\frac12\le t<\frac34\\
t-\frac14&\text{if }\frac34\le t\le1
\end{array}\right.
$$
Figure \ref{fig:c1c2} shows the graphs of the homeomorphisms corresponding to $c_1$ and $c_2$.
\begin{figure}[h]
\includegraphics[width=5in]{c1c2}\\
\caption{ The graphs of the homeomorphisms corresponding to the elements $c_1$ and
$c_2$.\label{fig:c1c2}}
\end{figure}
Using the first three relators, we see that only the generators $x_0$, $x_1$ and $c_1$ are required to
generate the group, since the other generators can be obtained from these three. In the following, we
will use $c$ to denote the generator $c_1$. The group $T$ is finitely presented using the following
relators,with respect to the finite generating set $\{x_0,x_1,c\}$:
\begin{enumerate}
\item $[x_0x_1^{-1},x_0^{-1}x_1x_0]=1$ \item
$[x_0x_1^{-1},x_0^{-2}x_1x_0^2]=1$ \item $x_1c_3=c_2x_2$, (that is
$x_1(x_0^{-2}cx_1^{-2})=(x_0^{-1}cx_1)(x_0^{-1}x_1x_0)$) \item $c_1x_0=c_2^2$, (that is,
$cx_0=(x_1^{-1}cx_0)^2$) \item $x_1c_2=c$, (that is, $x_1(x_0^{-1}cx_1)=c) $ \item $c^3=1$.
\end{enumerate}
As with Thompson's group $F$, we will frequently work with the more convenient infinite set of generators
when constructing normal forms for elements and performing computations in the group. We will need to
express elements with respect to a finite generating set when discussing word length.
There are two natural finite generating sets for $T$, both extending the standard
finite generating set for $F$. The first and the one that we use primarily below is
the generating set $\{x_0, x_1, c_1\}$ used in the finite presentation above. In Section \ref{isomembed} for the purposes of counting carets carefully, we also use the generating set $\{x_0, x_1, c_0\}$, which has the
advantage that the tree pair diagram for $c_0$ has only one caret, as opposed to $c_1$,
which has two carets, at the expense of slightly more complicated relators.
Just as for $F$, tree pair diagrams serve as efficient representations for elements of $T$. However,
since elements of $T$ represent homeomorphisms of the circle rather than the interval, the tree pair
diagram must also include a bijection between the leaves of the source tree and the leaves of the target
tree to fully encode the homeomorphism. Since this bijection can at most cyclically shift the leaves, it
is determined by the image of the leftmost leaf in the source tree. Since by convention this leaf in the
source tree is already thought of as leaf $0$, this information is recorded by writing a $0$ under the
image leaf in the target tree. Hence, for $w \in T$, a {\em marked tree pair diagram} representing $w$ is
a pair of finite rooted binary trees with the same number of leaves, together with a mark (the numeral 0)
on one leaf of the second tree. As usual, we write $w = (T_-,T_+)$ to express $w$ as a tree pair
diagram, and refer to $T_-$ as the {\em source} tree and $T_+$ (the one with the mark) as the {\em
target} tree. We remark that to extend this to $V$, since now the bijection of the subintervals may
permute the order in any way, the marking required on the target tree to record the bijection consists of
a number on every leaf of the target tree. Just as for $F$, there are many possible tree pair
diagrams for each element of $T$, which can be obtained by adding carets to the corresponding leaves in the source and target trees in the diagrams. However, when adding the carets, placement is guided by
the marking. The leaves of the source tree are thought of as numbered from $0$ to $n$ reading from left
to right, whereas the marking of the target tree specifies where leaf number $0$ of that tree is, and
other leaves are numbered from $1$ to $n$ reading from left to right cyclically wrapping back to the left
once you reach the rightmost leaf. With this numbering in mind, carets can be added as before to leaf $i$
of both trees. If $i \neq 0$, the mark stays where it is. Otherwise, if $i=0$, the mark on the new
target tree is placed on the left leaf of the added caret. So for $T$, we have a similar reduction
condition: if $w = (T_-,T_+)$ and both trees contain a caret with two exposed leaves numbered $i$ and
$i+1$, then we remove these carets and renumber the leaves, moving the mark if needed, thus forming a representative for $w$ with
fewer carets and leaves.
A tree pair diagram which admits no such reductions is again called a {\em reduced tree pair diagram},
and any element of $T$ is represented by a unique reduced tree pair diagram. In $T$ as well as in $F$,
when we write $w = (T_-,T_+)$ below, we are assuming that the tree pair diagram is reduced unless
otherwise specified. Checking whether or not a tree pair diagram is reduced is slightly more difficult in
$T$ than in $F$. The process of checking for possible reductions is illustrated in Figure
\ref{fig:reduction}. A marked tree pair diagram for an element of $T$ is shown on the top left of Figure
\ref{fig:reduction}. In the
top right tree pair diagram of Figure \ref{fig:reduction}, the underlying numbering of the leaves of both trees determined by the
marking is written explicitly, revealing the reducible carets. The bottom tree pair diagram shows the
resulting reduced diagram.
\begin{figure}
\includegraphics[width=5in]{reduction}\\
\caption{An example of a caret reduction in a tree pair diagram representing an element of $T$. The diagram on the top left is reducible; The two dotted carets on the top right are paired with each other, since the numbering is identitcal.
The resulting reduced diagram is shown on the bottom.\label{fig:reduction}}
\end{figure}
Note that the torsion generators $c_i$ have particularly simple tree pair diagrams. In the diagram for
$c_i$, both source and target trees consist of the root caret plus $i$ right carets. The mark $0$ is
placed on the rightmost leaf of the target tree.
\begin{figure}
\includegraphics[width=5.2in]{cs}\\
\caption{Tree pair diagrams representing the elements $c_1$, $c_2$ and $c_3$ on top, plus $c_n$ on the
bottom.\label{fig:cs}}
\end{figure}
Figure \ref{fig:cs} shows the the tree pair diagrams of the first three generators $c_1, c_2$, and $c_3$, together with a
general $c_n$. The generator $c_0$ is merely a pair of single caret trees, with the mark on the rightmost leaf of the target tree.
Whether $w \in F$ or $w \in T$, we denote the number of carets in either tree of a tree pair diagram
representing $w$ by $N(w)$. When $p$ is a word in the generators of $F$ or $T$, then $p$ represents an
element $w$ in either $F$ or $T$, and we write $N(p)$ interchangeably with $N(w)$.
\subsection{Group Multiplication in $F$ and $T$}\label{multiplication}
Group multiplication in $F$ and $T$ corresponds to composition of
homeomorphisms, which we can interpret on the level of tree pair
diagrams as well. First, we consider $u,v \in F$, where $u =
(T_-,T_+)$ and $v = (S_-,S_+)$. To compute the tree pair diagram
corresponding to the product $vu$, we create unreduced
representatives $(T'_-,T'_+)$ and $(S'_-,S'_+)$ of the two elements
in which $T'_+ = S'_-$. Then the product is represented by the
possibly unreduced tree pair diagram $(T'_-,S'_+)$. The
multiplication is written following the conventions on composition
of homeomorphisms, so the product $vu$ has as a source diagram that
of $u$, and as a target diagram that of $v$. That is, the diagram on
the left is the source of $u$ and the diagram on the right is the
target of $v$.
To multiply tree pair diagrams representing elements of $T$ we follow a similar procedure. We let $u,v
\in T$, where $u = (T_-,T_+)$ and $v = (S_-,S_+)$. To compute the tree pair diagram corresponding to the
product $vu$, we create unreduced representatives $(T'_-,T'_+)$ and $(S'_-,S'_+)$ of the two elements in
which $T'_+ = S'_-$ as trees. The product $vu$ will be represented by the pair $(T'_-,S'_+)$ of trees. To
decide which leaf in $S'_+$ to mark with the zero, we just note that it should be the leaf which is
paired with the zero leaf in $T'_-$. To identify this leaf, we find the zero leaf in $T'_+$. Since
$T'_+=S'_-$ as trees, this leaf viewed as a leaf in $S'_-$ will be labelled $m$. Then the leaf labelled
$m$ in $S'_+$ will be the new zero leaf in the tree pair diagram $(T'_-,S'_+)$ for $vu$. Alternately, we
can follow the composition in both pairs of trees to see how the leaves are paired. This newly
constructed tree pair diagram will represent $vu$ and is not necessarily reduced. For an example of this
multiplication, see Figures \ref{fig:mult1}, \ref{fig:mult2} and \ref{fig:mult3}.
\begin{figure}
\includegraphics[width=5.2in]{mult1}\\
\caption{ The tree pair diagram for sample elements $u$ and $v$ in $T$.\label{fig:mult1}}
\end{figure}
\begin{figure}
\includegraphics[width=5.2in]{mult2}\\
\caption{ Unreduced versions of $u$ and $v$ necessary for the multiplication $vu$ in $T$, with carets added to
perform the multiplication indicated with dashes. Now the target tree of $u$ has the same shape as the
source tree of $v$, allowing the composition.\label{fig:mult2}}
\end{figure}
\begin{figure}
\includegraphics[width=3in]{newmult3}\\
\caption{ The tree pair diagram representing the product $vu$
obtained from Figure \ref{fig:mult2}. The dotted carets must be
erased to find the reduced diagram.\label{fig:mult3}}
\end{figure}
\section{Words and diagrams}
\subsection{Normal forms and tree pair diagrams in $F$}
With respect to the infinite presentation for $F$ given above, every element of $F$ has a unique normal
form. Any $w$ in $F$ can be written in the form
$$w=x_{i_1}^{r_1} x_{i_2}^{r_2}\ldots x_{i_k}^{r_k}
x_{j_l}^{-s_l} \ldots x_{j_2}^{-s_2} x_{j_1}^{-s_1} $$ where $r_i, s_i >0$, $0 \leq i_1<i_2 \ldots < i_k$
and $0 \leq j_1<j_2 \ldots < j_l$. However, this expression is not unique. Uniqueness is guaranteed by
the addition of the following condition: when
both $x_i$ and $x_i^{-1}$
occur in the expression, so does at least one of $x_{i+1}$ or $x_{i+1}^{-1}$, as discussed by Brown and Geoghegan
\cite{bg:thomp}. When we refer to elements of $F$ in normal form, we mean this unique normal form.
If the normal form for $w \in F$ contains no generators with negative exponents, we refer to $w$ as a
{\em positive word} and similarly, we say a normal form represents a {\em negative word} if there are no
generators with positive exponents.
We call any word which has the form $$w=x_{i_1}^{r_1} x_{i_2}^{r_2}\ldots x_{i_k}^{r_k} x_{j_l}^{-s_l}
\ldots x_{j_2}^{-s_2} x_{j_1}^{-s_1} $$ where $r_i, s_i >0$, $0 \leq i_1<i_2 \ldots < i_k$ and $0 \leq
j_1<j_2 \ldots < j_l$, a word in {\em pq form}, where $p$ is the positive part of the normal form and $q$
the negative part. The normal form for an element of $F$ is the shortest word among all words in $pq$
form representing the given element.
To any (not necessarily reduced) tree pair diagram $(T_-,T_+)$ for an element of $F$ we may associate a
word in $pq$ form representing the element, using the {\em leaf exponents} in the target and source
trees. When the leaves of a finite rooted binary tree are numbered from left to right, beginning with
zero, the leaf exponent of leaf $k$ is the integer length of the longest path consisting only
of left edges of carets
which originates at leaf $k$ and does not reach the right side of the tree. A tree pair diagram then
gives the word
$$
x_{i_1}^{r_1}x_{i_2}^{r_2}\ldots x_{i_n}^{r_n}x_{j_m}^{-s_m}\ldots x_{j_2}^{-s_2}x_{j_1}^{-s_1}
$$
precisely when leaf $i_k$ in $T_+$ has exponent $r_k$, leaf $j_k$ in
$T_-$ has leaf exponent $s_k$, and generators which do not appear in
the word correspond to leaves with exponent zero. We think of this
word as the $pq$ factorization of the element given by the
particular tree pair diagram. We call a tree an {\em all-right tree}
if it consists of a root caret together with only right carets. Note
that if we let $R$ be the all-right tree with the same number of
carets as $T_-$ or $T_+$, then $(T_-,R)$ is a diagram for the word $q$ and
$(R,T_+)$ is a diagram for word $p$. On the other hand, any word in $pq$
form can be translated into a tree pair diagram. It can be obtained
by taking diagrams for $p$ (respectively $q$), which will have all
right source (respectively target) trees. Then, if one diagram has
fewer carets, one adds right carets to its all-right tree, and of
a corresponding path of right carets to its other tree, to make both
diagrams have exactly the same all right tree. Furthermore, under
this correspondence for $F$, reduced tree pair diagrams correspond
exactly to normal forms. Figure \ref{fig:leafexp} is an example of
this correspondence, and more details can be found in
\cite{cfp,ctcomb,blake:diss}.
\begin{figure}
\includegraphics[width=3in]{leafexp3}\\
\caption{Computing leaf exponents. The thick edges indicate edges
which contribute to non-zero leaf exponents. If a leaf labelled $i$ has
$r_i$ thick edges (a path of $r_i$ left edges
going up without reaching the right side of the tree) then the $i$-th leaf
exponent is $r_i$ and the generator appearing in the normal form is
$x_i^{r_i}$. This single tree $T$ pictured above is the target
tree of the tree pair diagram $(R,T)$, where $R$ is the all-right tree with 12 leaves,
and has leaf exponents 1,0,3,0,1,0,0,0,2,0,0, and 0 for the leaves 0-11 in order.
The tree pair diagram $(R,T)$ represents the element $x_0x_2^3x_4x_8^2$.
\label{fig:leafexp}}
\end{figure}
If an exposed caret has leaves numbered $i$ and $i+1$, then leaf $i+1$ must have leaf
exponent zero, since it is a right leaf. If both trees in a tree pair diagram have exposed carets with
leaves numbered $i$ and $i+1$, then the corresponding normal form, computed via leaf exponents, contains
the generators $x_i$ to both positive and negative powers, but no instances of the generator $x_{i+1}$.
This is precisely the situation when the normal form can be reduced by a relator of $F$. Thus the
condition that the normal form is unique is exactly the condition that the tree pair diagram is reduced.
This correspondence will be extended to elements of $T$ in the next section.
\subsection{Tree pair diagrams for elements of $T$}
We now discuss the relationship between words in $T$ and tree pair diagrams. This relationship is more
complicated in $T$ than it is in $F$.
The representation of elements of $T$ by marked tree pair diagrams suggests a way to decompose an element
of $T$ into a product of three elements: the positive and negative parts together with a torsion part in
the middle, as described in \cite{cfp}.
\begin{defn} \label{factorization}
Let the marked tree pair diagram $(T_-,T_+)$ represent $g \in T$. If~~$T_-$ and $T_+$ each have $i+1$
carets, then we let
$R$ be the all-right tree which has
$i+1$ carets. We can write $g$ as a product $pc_i^jq$, where:
\begin{enumerate}
\item $p$, a positive word in the generators of $F$, is the normal form for the element of $F$ with tree pair diagram $(R,T_+)$, ignoring the marking on $T_+$.
\item $c_i^j$ is a cyclic permutation of the leaves of $R$, with $1 \leq j \le i+2$, and
\item $q$, a negative word, is the normal form for the element of $F$ represented by $(T_-,R)$.
\end{enumerate}
Then the word $g=pc_i^jq$ is called the \emph{pcq factorization} of $g$ associated to the marked tree pair diagram $(T_-,T_+)$.
In the special case
where $g \in F \subset T$, the $pcq$ factorization will just be the usual $pq$ factorization, as we
consider the $c$ part of the word to be empty (or equivalently, we can allow the exponent $j$ in the torsion part to be zero.)
\end{defn}
Figure \ref{fig:threeparts} illustrates an example of an element of $T$ decomposed in this way.
\begin{figure}
\includegraphics[width=5.2in]{pcq2}\\
\caption{ Three tree pair diagrams representing the word $x_1 x_2 c_5^5 x_2^{-2} x_1^{-1} x_0^{-2}$
factorized as $pcq$.\label{fig:threeparts}}
\end{figure}
The following theorem follows from the existence of these decompositions, and an algebraic proof of this
result is found in \cite{cfp}.
\begin{thm}[\cite{cfp}, Theorem 5.7]\label{pcq}Any element $x\in T $ admits an expression of the form
$$
x_{i_1}^{r_1}x_{i_2}^{r_2}\ldots x_{i_n}^{r_n}\,c_i^j\,x_{j_m}^{-s_m}\ldots x_{j_2}^{-s_2}x_{j_1}^{-s_1},
$$ where $0 \leq i_1<i_2< \cdots <i_n$ and $0 \leq j_1<j_2<\cdots<j_m$ and either $1 \leq j<i+2$ or $c_i^j$ is not present.\end{thm}
We refer to any word satisfying the hypotheses of Theorem \ref{pcq} as a word in $pcq$ form for an
element of $T$ (just as words of this form with no $c_i^j$ term are called words in $pq$ form in the
group $F$). Neither proof of the existence of $pcq$ forms gives an easy explicit method for transforming
a general word in the generators $x_i^{\pm 1},c_i$ into $pcq$ form without resorting to drawing tree pair
diagrams, so we will outline an algebraic method below. We recall that the five types of relators we are using
in $T$ are:
\begin{enumerate}
\item $x_jx_i=x_ix_{j+1}$, if $i<j$ \item $x_kc_{n+1}=c_{n}x_{k+1}$,
if $k<n$ \item $c_nx_0=c_{n+1}^2$ \item $c_n=x_nc_{n+1}$ \item$c_n^{n+2}=1$
\end{enumerate}
\begin{lemma} [Pumping Lemma] The generators $x_i$ and $c_j$ of $T$
satisfy the following identities
$$
c_n^m=x_{n-m+1}c_{n+1}^m\qquad\qquad
c_n^m=c_{n+1}^{m+1}x_{m-1}^{-1}
$$
if $1\le m < n+2$.
\end{lemma}
\begin{proof} This follows immediately from the relators. For instance, for the first identity, we have that
$$
c_n^m=c_n^{m-1}c_n=c_n^{m-1}x_nc_{n+1}
$$
by an application of relator of type (4). Now, several repeated applications
of relator (2) allow the $x_n$ to switch with the $c_n^{m-1}$ to
obtain the desired result. The second identity is the first one
taking inverses, and by noticing that $c_n$ has order $n+2$, we avoid
negative exponents for the $c$.
\end{proof}
We consider a word $w \in T$ written in the generators
$\{x_i,c_j\}$, and we describe explicitly an algebraic method of rewriting
it in $pcq$ form. The idea is to first combine occurrences of multiple $c_i$ generators into a power of a single one, and then to move the $x_n$ generators to the appropriate side of it. Consider first a
subword of the original word $w$ of type
$$
c_n^m\,w(x_i)\,c_k^l,
$$
where $w(x_i)$ is a word on the generators $x_n$ only, and which may
possibly be empty. We will apply relators to reduce this subword to
a word of the form $w_1(x_i) c_j^h w_2(x_i)$, where $w_1$ has only positive powers of $x_n$ generators and $w_2$ consists of only negative ones.
By the relators of type (1), we can assume that $w$ is of the form $pq$, that
is, with all positive powers of generators on the left and in increasing order of
index, and all negative ones on the right and in decreasing order of index.
The goal is to move all the positive powers of $x_n$ generators to the left of
$c_n^m$ and all negative ones to the right of $c_k^l$. Although these moves may change the indices and powers of the $c_i$ generators, they merely change a power of a single $c_j$ generator to another power of a different single $c_k$ generator. To move all
the positive powers of generators to the left of $c_n^m$, we only need to use
relators of the type (2), assuming the index of $c$ is high enough.
If it is not, by repeated applications of the first identity of the
pumping lemma, the index can be increased arbitrarily, adding only
positive powers of generators to the {\it left\/} of $c_n^m$. When the subindex
is high enough, we can use relators of type (2) to move all positive powers of generators of
$w$ past $c_n^m$.
We note that a relator of type (3) may allow us to eliminate a occurance of $x_0$ to
the immediate right of $c_n^m$. It may be necessary
to combine the $c_i$ and $c_j$ generators obtained into a single term after this
elimination of $x_0$, as we see in an example:
$$
c_4^3x_1=c_4^2x_0c_5
$$
At this point $x_0$ cannot be moved farther, but we can use relator (3) to
obtain
$$
c_4c_5^3=x_5c_5^4
$$
with the last equality being an application of the pumping lemma to
$c_4$. We have achieved the goal of moving a positive power of a generator to the
left of $c_n^m$.
Moving the negative powers of the $x_n$ generators to the left is comparable. Using the second identity in the pumping lemma, we can
increase the index in $c_k^l$ as much as necessary to be able to
move all negative powers of the $x_n$ generators in $w$ to the right of $c_k^l$ using the relators (2) rewritten as $c_{n+1}x_{k+1}^{-1}=x_k^{-1}c_n$.
After this process, we will have a word consisting of positive powers of $x_n$
generators, two powers of $c_i$ generators, and negative powers of $x_n$ generators. We now
combine the powers of the two $c_i$ generators into a power of a single generator, by increasing
the smaller index to reach the larger. To do this, if the smaller
is on the left, we can use the first identity in the pumping lemma, and if
it is on the right, we can use the second one. This way no $x_n$ generator
will be added in between the two $c_i$ generators and after they have
the same index they can be combined into a power of a single generator. The positive powers of the $x_n$ generators now appear only to the left of the single power of the $c_i$ generator, and negative
powers of $x_n$ generators only to the right.
After repeated applications of this process to subwords of
the type $cwc$, we will have all occurrences of the $c_i$ generators
combined into a power of a single one. Our original word is now of the type
$$
w_1(x_i)\,c_n^m\,w_2(x_i),
$$
and $w_1$ and $w_2$ may again be assumed, after using relators (1), to be in $pq$ form. We only need to move the positive powers of generators in $w_2$ to the left of $c_n^m$ and the
negative powers of $x_n$ generators of $w_1$ to the right of $c_n^m$, still maintaining a power of a single $c_i$ generator in the middle. We describe above as the first step in our algorithm precisely how to do this. Furthermore, if the pumping lemma is needed to move a positive power of a generator to the left, recall that new positive positive powers of generators may appear in the word, but only to the left of the power of the $c_i$ generator.
Hence, after moving each positive power of a generator,
all positive powers of generators in the word are to the left of $c_n^m$.
We now move each negative power of a generator to the right, and notice that the only cost of this is to add more negative powers of $x_n$ generators to the right of $c_n^m$. When this is finished, the word has only positive powers of generators to the left of a power of a single $c$ and negative ones to the right.
Once the positive powers of the generators are together on the left side of the single $c$ term,
we can reorder them if necessary using relators of type (1), and similarly we can reorder the negative
part as well.
We will work an example as an illustration. Consider the word
$$
x_0^{-1}c_1x_3c_3^2x_1^{-1}
$$
The process starts by trying to move the $x_3$ to the left of
$c_1$. Since the index of $x_3$ exceeds the index of $c_1$, we cannot
apply a relator of type (2) directly. Using the pumping lemma, we write
$c_1=x_1c_2=x_1x_2c_3$. Hence our word is now the following, and
we can apply the relator of type to $c_3x_3$, obtaining:
$$
x_0^{-1}x_1x_2c_3x_3c_3^{2}x_1^{-1}=x_0^{-1}x_1x_2^2c_4c_3^2x_1^{-1}.
$$
We need to merge $c_4c_3^2$ into a single $c_i$ term. We increase the index of $c_3$ via
$c_3^2=c_4^3x_1^{-1}$ to obtain
$$
x_0^{-1}x_1x_2^2c_4c_4^3x_1^{-2}=x_0^{-1}x_1x_2^2c_4^4x_1^{-2}.
$$
The last step is to move the initial $x_0^{-1}$ to the
right side, using several relators of type (2) to obtain
$x_0^{-1}c_4^4=c_5^4x_4^{-1}$. There is no need this time to
increase the index of $c_4^4$. The
final result is
$$
x_2x_3^2c_5^4x_4^{-1}x_1^{-2}
$$
which is in $pcq$ form.
The relationship between words in $pq$ form and tree pair diagrams in $F$ is different than the
relationship between $pcq$ forms and tree pair diagrams in $T$. In $F$, every tree pair diagram has a
$pq$ factorization associated to it, and any word in $pq$ form is in fact the $pq$ factorization
associated to a (not necessarily unique) tree pair diagram. Given any word in $F$ in $pq$ form, then we can form
a tree pair diagram for this element as follows. We consider reduced tree pair diagrams for $p$ and $q$,
and construct a tree pair diagram for the product $pq$ as described in Section \ref{multiplication}. The
middle trees of the four trees involved in the product are all-right trees. The all-right trees in this
decomposition may not have the same number of carets, so in forming the diagram for $pq$ we simply
enlarge the smaller of the two of these all-right trees (as well as the other tree in that diagram).
Since only right carets are ever added during this process, all of whose leaves have leaf exponent zero,
this results in a tree pair diagram whose $pq$ factorization is precisely the word $pq$ we began with.
In $T$, the correspondence between $pcq$ factorizations and general $pcq$ words is not as
straightforward as in $F$. There is a difference between $pcq$ factorization and $pcq$ algebraic form.
Though every element has a tree pair diagram corresponding to a $pcq$ factorization associated to it, there are words in algebraic $pcq$ form which are not the $pcq$ factorizations associated to a tree pair diagram.
The difficulty arises when the tree pair diagram for $c$ does not have as many carets as those for $p$ or
$q$, as adding right carets to enlarge $c$ appropriately necessitates adding generators to the normal
forms for $p$ and $q$, so the tree pair diagram one obtains by multiplying as in $F$ will not necessarily
have the original word as its factorization. For example, the word $x_1c_1$ is in algebraic $pcq$ form,
yet it is not the $pcq$ factorization associated to any tree pair diagram. There is a different
representative for this element of $T$ which is the $pcq$ factorization associated to the reduced tree
pair diagram for this group element: $x_1c_2x_1^{-1}$. We prefer to work with words which are $pcq$
factorizations associated to tree pair diagrams, which will lead us to unique normal forms.
We can algebraically characterize the words of type $pcq$ which are $pcq$ factorizations associated to
tree pair diagrams. The important condition is that the reduced tree pair diagram for $c$ should have at
least as many carets as those for $p$ and $q$.
We say that words in $T$ with this property satisfy the {\em factorization condition}.
\begin{thm}
\label{thm:factor} For elements in $T$ which are not in $F$, the word
$$
x_{i_1}^{r_1}x_{i_2}^{r_2}\ldots x_{i_n}^{r_n}\,c_i^j\,x_{j_m}^{-s_m}\ldots x_{j_2}^{-s_2}x_{j_1}^{-s_1}
,$$ where $0 \leq i_1<i_2< \cdots < i_n$, $0\leq j_1<j_2<\cdots<j_m$, and $1 \leq j < i+2$, is the $pcq$
factorization associated to a tree pair diagram if and only if the number of carets in the reduced tree
pair diagram for $c_i^j$ is greater than or equal to the number of carets in the reduced tree pair
diagram for both of those for the words $ x_{i_1}^{r_1}x_{i_2}^{r_2}\ldots x_{i_n}^{r_n}$ or
$x_{j_m}^{-s_m}\ldots x_{j_2}^{-s_2}x_{j_1}^{-s_1} $ in $F$.
\end{thm}
\begin{proof}
Given a tree pair diagram, by construction, the $pcq$ factorization associated to it satisfies the
factorization condition. Given a word that satisfies the factorization condition, we can easily
construct the corresponding tree pair diagram as described above. The factorization condition
ensures that to perform the mulitiplication, $p \cdot c \cdot q$ as tree pair diagrams, it is only
necessary to (possibly) add carets to the tree pair diagrams for the words $p$ and $q$.
This will not alter the normal form, and thus the
diagram constructed will indeed have the original word as its $pcq$
factorization.
\end{proof}
We can compute the number of carets of a reduced tree pair diagram for a word $w \in F$ algebraically from the normal form of $w$,
as described by Burillo, Cleary and Stein in \cite{bcs}.
\begin{prop} [Proposition 2 of \cite{bcs}] \label{prop:ncarets}
Given a positive word in $w \in F$ in normal form
$$
w=x_{i_1}^{r_1}x_{i_2}^{r_2}\ldots x_{i_n}^{r_n},
$$
then the number of carets $N(w)$ in either tree of a reduced tree diagram representing $w$ is
$$
N(w)=\max\{i_k+r_k+\ldots+r_n+1\} ,\text{ for }k=1,2,\ldots,n.
$$
\end{prop}
We can always decide algebraically whether $w \in T$, written in $pcq$ form,
corresponds to a tree pair diagram. We
use Proposition \ref{prop:ncarets} to count the carets for the
positive and negative parts of the word.
The number of carets in a tree pair diagram for $c_i^j$ is
equal to $i+1$.
\section{Normal forms in $T$}
In $T$, we will declare the words in $pcq$ form which are $pcq$ factorizations associated to reduced
diagrams to be the normal forms for elements of $T$, similar to the approach used in $F$. However, it is
no longer true that these words cannot be shortened by applying a relator. As we saw with the normal
form $x_1c_2x_1^{-1}$ in $T$, a word may be the shortest word representing an element which satisfies the
factorization condition, yet there may be shorter words we can obtain by applying a relator which do not
satisfy the factorization condition.
Thus, when algebraically characterizing the normal form for elements of $T$, we restrict ourselves to
words of $pcq$ form which satisfy the factorization condition, regardless of whether or not a relator may
reduce the length of the word. We next specify algebraic conditions which characterize the $pcq$
forms that correspond to normal forms, since we have given geometric conditions in Theorem \ref{thm:factor}
\begin{thm}\label{reductions} Let $w$ be a $pcq$ factorization for an element
$g \in T$ associated to a marked tree pair diagram in which each tree has $i+1$ carets, where the $c$
part of the word is $c_i^{j}$ with $1 \leq j < i+2$. A reduction of a pair of carets from the tree pair
diagram occurs only if the word $w$ satisfies one of the following conditions:
\begin{itemize}
\item[(1)] The pair of generators $x_{k-j}$ and
$x_k^{-1}$ appear, with $j \leq k< i$, and neither of the two generators
$x_{k-j+1}$ and $x_{k+1}^{-1}$ appear. The reduction corresponds to
applying the relator
$$
x_{k-j}c_{i}^jx_k^{-1}=c_{i-1}^j
$$ after applying relators from $F$ in the $p$ and $q$ parts of the word, if necessary, to make $x_{k+j}$ and $x_k^{-1}$ adjacent to $c_i^j$.
\item[(2)] The generator $x_{i-j}$ appears, and
$x_{i-j+1}$ does not. The reduction corresponds to applying
$$
x_{i-j} c_{i}^j=c_{i-1}^j
$$ after possibly using relators from $F$ as in (1).
\item[(3)] The pair of generators $x_{k+i-j+2}$ and
$x_k^{-1}$ for $0\leq k< j-2$ appear and neither one of the generators
$x_{k+i-j+1}$ or $x_{k+1}^{-1}$ appear. The reduction corresponds to
applying $$ x_{k+i-j+2}c_{i}^jx_k^{-1}=c_{i-1}^{j-1}
$$ after possibly applying relators from $F$.
\item[(4)] The generator $x_{j-2}^{-1}$ appears, and the generator
$x_{j-1}^{-1}$ does not appear. The reduction corresponds to
$$
c_{i}^{j} x_{j-2}^{-1}=c_{i-1}^{j-1}
$$ after possibly applying relators from $F$.
\end{itemize}
\end{thm}
\begin{figure}[b]
\includegraphics[width=5.82in]{normal}\\
\caption{The four cases in Theorem \ref{reductions}, showing the two
labellings on the leaves of the trees, the cyclic labelling which
indicates the correspondence of the leaves, and the leaf exponent
labelling which indicates the corresponding generators in the normal
form. \label{fig:normal}}
\end{figure}
\begin{proof}
Let $g \in T$ be represented by a marked tree pair diagram $(T_-,T_+)$. If both trees have an exposed caret
whose leaves are identically numbered, then we call that a {\em reducible caret pair}, as it must be removed
in order to obtain the reduced tree pair diagram representing $g$. We now consider algebraic conditions
corresponding to a reducible caret in a tree pair diagram.
In the tree pair diagram $(T_-,T_+)$ for $g \in T$, there are two
ways of labelling the leaves in the target tree $T_+$. The first
labelling corresponds to the order in which the intervals in the
subdivisions determined by these trees are paired in the
homeomorphism, and is called the cyclic labelling. The cyclic
labelling gives the marked leaf in the target tree the number zero,
and the other leaves are given increasing labels from left to right
around the leaves of the tree. The second labelling ignores the
marking and puts the leaves in increasing order from left to right,
beginning with zero. The first labelling is used to determine which
leaves in $T_-$ are paired with which leaves in $T_+$, and the
second labelling is used in the computation of leaf exponents to
determine the powers of the generators that appear in the word.
Figure \ref{fig:normal} shows the labellings for
the four cases of the theorem.
Suppose that the tree pair diagram for $g \in T$ is not reduced. The four cases above correspond to the
following four possible locations of a reducible caret relative to the marked leaf in the target tree.
\begin{itemize}
\item Case (1) of the thereom corresponds to the case when the left leaf of the reducible caret is
to the left of the marked leaf in $T_+$, but the reducible caret is
not the rightmost caret in $T_-$.
\item Case (2) corresponds to the special case when the reducible caret is a right caret in $T_-$, in which case necessarily its left leaf is to the left of the marked leaf in $T_+$.
Leaf exponents from leaves of right carets will always be zero and thus right carets cannot contribute generators
to the normal form. They may still result in an exposed reducible caret, which occurs exactly in this
case, and the reduction will only affect
the $q$ part of the
normal form. \item Case (3) corresponds to the case when the left leaf of the reducible caret is either
to the right of or coincides with the marked leaf in $T_+$, but the reducible caret is not the rightmost
caret in $T_+$. \item Case (4) corresponds to the special case when the reducible caret is a right caret
in $T_+$, in which case it cannot be to the left of the marked caret in $T_+$. As in Case (2), the
exposed caret in this case is a right caret and does not contribute a generator to the normal form, but
may still be reduced. This cancellation affects only the $p$ part of the normal form.
\end{itemize}
To see that these are all the possibilities, we note that $k$, the number of the left leaf in the cyclic numbering of the reducible caret in $T_-$,
achieves all possible values in the cases above:
\begin{itemize}
\item If $0\le k<j-2$ we are in case (3).
\item If $k=j-2$ we are in case (4).
\item The case $k=j-1$ is impossible because the leaves $j-1$
and $j$ are at the two ends of the tree. With a cyclic ordering the
last and first leaves do not form a caret.
\item If $j\le k<i$ we are in case (1).
\item If $k=i$ we are in case (2).
\end{itemize}
Figure \ref{fig:normal} illustrates that these are
all the possibilities.\end{proof}
The conditions in Theorem \ref{reductions} together with the factorization condition algebraically
characterize our normal forms. The normal forms for elements in $F$ have already been characterized, so
we restrict to elements not in $F$ in our description.
\begin{thm} Any element $g\in T$ which is not an element of $F$
admits an expression of the form $pcq$ where
$$
p=x_{i_1}^{r_1}x_{i_2}^{r_2}\ldots x_{i_n}^{r_n}\qquad c=c_i^j\qquad q=x_{j_m}^{-s_m}\ldots
x_{j_2}^{-s_2}x_{j_1}^{-s_1},
$$ $0 \leq i_1<i_2< \cdots < i_n$, $0 \leq j_1<j_2<\cdots<j_m$, and $1 \leq j < i+2$.
Among all the words in this form representing an element, there is a unique one satisfying the following
conditions, which we call the normal form.
\begin{itemize}
\item The word satisfies the factorization condition, which we now state as $i+1\ge\max\{N(p),N(q)\}$.
\item The word does not admit any reductions, and thus its normal form
satisfies the following conditions:
\begin{itemize}
\item If there exists a pair of generators $x_{k-j}$ and
$x_k^{-1}$ simultaneously, for $j\le k< i$, then one of the
generators $x_{k-j+1}$ or $x_{k+1}^{-1}$ must appear as well.
\item If there is a generator $x_{i-j}$, then
$x_{i-j+1}$ must exist too.
\item If there exists a pair of
generators $x_{k+i-j+2}$ and $x_k^{-1}$ for $0\le k< j-2$, then one
of the generators $x_{k+i-j+1}$ or $x_{k+1}^{-1}$ must appear as
well.
\item If there exists a generator $x_{j-2}^{-1}$, then a generator
$x_{j-1}^{-1}$ must also appear.
\end{itemize}
\end{itemize}
\end{thm}
\begin{proof}
We claim that the conditions above precisely describe the set of unique normal forms for $T$. A $pcq$ word
satisfying the factorization condition is the $pcq$ factorization associated to a marked tree pair
diagram. However, if the $pcq$ word satisfies all four reduction conditions, we have just shown in the
previous theorem that this diagram is in fact the unique reduced diagram, and hence the word is in fact a
normal form. \end{proof}
We remark that the Pumping Lemma together with the reductions in Theorem \ref{reductions} give an
explicit way of algebraically transforming any word in the generators of $T$ into a normal form.
Given any word, we rewrite it in $pcq$ form using the process described following the Pumping Lemma. If
the resulting word does not satisfy the factorization condition, then we iterate the Pumping Lemma until
we obtain a word for which the factorization condition is satisfied. The Pumping Lemma increases the
number of carets for $c$ and the number of carets for one of the words $p$ and $q$. Once a word is
obtained which satisfies the factorization condition, there must be a corresponding tree pair diagram for
the element. Now, if the word satisfies any of the reduction conditions in Theorem \ref{reductions}, we
apply them successively using the relators described there. This method thus produces the unique normal
form.
\section{The word metric in $T$}
\subsection{Estimating the word metric}
For metric questions concerning $T$, we must consider a finite generating set instead of the one used to
obtain the normal form for elements. We now approximate the word length of an element of $T$ with
respect to the generating set $\{x_0,x_1,c_1\}$, using information contained in the normal form and the
reduced tree pair diagram. These estimates are similar to those for the estimates of word metric in $F$ with respect to the
generating set $\{x_0,x_1\}$ described \cite{burillo}, \cite{bcs}.
\begin{thm} \label{thm:D} Let $w \in T$ have normal form
$$
w=x_{i_1}^{r_1}x_{i_2}^{r_2}\ldots x_{i_n}^{r_n}\,c_i^j\,x_{j_m}^{-s_m}\ldots
x_{j_2}^{-s_2}x_{j_1}^{-s_1}.
$$
We define
$$
D(w)=\sum_{k=1}^nr_k+\sum_{l=1}^ms_l+i_n+j_m+i.
$$
Let $|w|$ denote the word metric in $T$ with respect to the generating set $\{x_0,x_1,c_1\}$. There
exists a constant $C>0$ so that for every $w \in T$,
$$
\frac{D(w)}C\le|w|\le C\,D(w)
$$
and similarly, for $N(w)$ the number of carets in the reduced tree pair diagram representing $w$,
$$
\frac{N(w)}C\le|w|\le C\,N(w).
$$
\end{thm}
\begin{proof}
These inequalities follow from the correspondence between the normal form and the tree pair diagram for
an element $w \in T$. It is clear, from Proposition \ref{prop:ncarets}, that $ N(w)\ge \sum_{k=1}^nr_k$,
$N(w)\ge \sum_{l=1}^ms_l$, $N(x)\ge i_n$, and $N(w) \ge j_m$. The inequality $ N(w)\ge i $ is clear from
the fact that $c_i$ has $i+1$ carets. These inequalities prove that
$$
D(w)\le 5\,N(w).
$$
We rewrite the generators $x_i$ and $c_j$ in terms of $x_0$, $x_1$ and $c_1$ and look at the lengths of
the resulting words to obtain the inequality
$$
|w|\le C\,D(w)
$$
for some constant $C>0$. Combining the two inequalities above, we have
$$
|w|\le C'\,N(w).
$$
To obtain lower bound on the word length, we consider the fact that the tree pair diagram for each
generator has either two or three carets. If $u$ is a word in $x_0$, $x_1$ and $c$ with length $n$, then
as these generators are multiplied together, each product may add at most $3$ carets to the tree pair
diagram. Thus the diagram for $u$ will have at most $3n$ carets. It then follows that
$$
N(w)\le 3|w|.
$$
Combining this with the above inequality, we obtain the desired bounds.
\end{proof}
We use Theorem \ref{thm:D} to show that the inclusion of $F$ in $T$ is a quasi-isometric embedding. This
means that there are constants $K>0$ and $C$ so that for any $w,z \in F$ we have
$$\frac{1}{K} d_{F}(w,z) - C \leq d_T(w,z) \leq Kd_F(w,z) + C$$
where $d_F$ and $d_T$ represent the word metric in $F$ and $T$ respectively, with regard to the
generating set $\{x_0,x_1\}$ of $F$ and $\{x_0,x_1,c_1\}$ of $T$.
When considering whether the inclusion of a finitely generated subgroup $H$ into a finitely generated
group $G$ is a quasi-isometric embedding, we can instead equivalently show that the distortion function
is bounded. The distortion function is defined by
$$h(r) = \frac{1}{r} \max \{|x|_H : x \in H \mbox{~and~} |x|_G \leq r\}.$$
Word length in $F$ is comparable to the number of carets in the reduced tree pair diagram representing
the word, by Theorem 3 of \cite{bcs} or more directly by Fordham's method \cite{blakegd}. This, combined with Theorem \ref{thm:D} easily shows that the
distortion function is bounded, and thus proves the following corollary with respect to
the generating sets $\{x_0, x_1\}$ and $\{x_0,x_1,c_1\}$ and thus all pairs of finite generating sets:
\begin{cor} \label{cor:qiemb}
The inclusion of $F$ in $T$ is a quasi-isometric embedding.
\end{cor}
\subsection{Comparing word length in $F$ and $T$\label {isomembed}}
Although Corollary \ref{cor:qiemb} shows that $F$ is quasi-isometrically embedded in $T$, in fact the
word length of many elements of $F$ does not change at all when these elements are considered as elements of
$T$, with respect to natural finite generating sets. As an example of this phenomenon, we characterize
one type of element of $F$ whose word length is unchanged when viewed as an element of $T$, using the
generating set $\{x_0,x_1\}$ for $F$ and $\{x_0,x_1,c_0\}$ for $T$. These are elements $w \in F$ for
which $N(w)$ exceeds the word length $|w|_F$. Fordham \cite{blakegd} computes $|w|_F$ by assigning an
integer weight between zero and four to each pair of carets in the tree pair diagram representing $w$. In
a given word there are at most two weights of zero. Here we investigate words in which most weights are
one. Such words, for example, are represented by tree pair diagrams with no interior carets having right
children.
\begin{thm} \label{thm:wordlength}
If $w \in F$ with $N(w) \geq |w|_F + 1$ then $|w|_T = |w|_F$, where word length if computed with respect
to the generating set $\{x_0,x_1\}$ for $F$ and $\{x_0,x_1,c_0\}$ for $T$.
\end{thm}
This theorem is proved by taking a word in the generators of $T$, and analyzing how each generator
changes the intermediate tree pair diagram as one builds up the final tree pair diagram for $w$.
Carefully controlling the process allows one to obtain an upper bound on $N(w)$ in terms of the
length of the word. If the word is actually shorter than $|w|_F$, then this bound, considered together
with the lower bound given by the hypothesis, yields a contradiction. We immediately obtain the
following corollary, since $|x_0^n|_F = |x_1^n|_F = n$, while $N(x_0^n) = n+1$ and $N(x_1^n) = n+3$.
\begin{corollary}
The elements $x_0^n$ and $x_1^n$ have word length $n$ in both $F$ and $T$ with respect to the finite
generating sets $\{x_0,x_1\}$ and $\{x_0,x_1,c_0\}$ respectively.
\end{corollary}
\section{Torsion elements}
Although the group $F$ is torsion free, both $T$ and $V$ contain torsion elements. It is easy to
construct torsion elements in $T$ or $V$ by choosing any binary tree $S$ and making any marked tree pair
diagram with $S$ as both source and target tree. If the labelling of the target tree is the same as the
labelling of the source tree, we get an unreduced representative of the identity; otherwise, we get a
non-trivial torsion element. If this is an element of $T$, the tree pair diagram has $pcq$ factorization
in which $q=p^{-1}$.
In fact, any torsion element can be represented by such a tree pair diagram, though its
reduced marked tree pair diagram may well not have the same source and target trees, corresponding to the fact
that although it has a $pcq$ word where $q=p^{-1}$, the normal form may well not have this special balanced appearance.
\begin{prop}\label{p:torsion}
If $f\in F,T$ or $V$ is a torsion element, then it can be represented by a (marked) tree pair diagram
with the same source and target trees.
\end{prop}
Before proving Proposition \ref{p:torsion}, we establish some notation which links the analytic and
algebraic representations of these groups. For $f \in F$, $T$, or $V$, if $(T_-,T_+)$ is a marked
tree pair diagram representing $f$, then it is sometimes convenient to denote the tree $T_+$ by $f(T_-)$.
The tree $T_-$ corresponds to a certain subdivision of the circle, which maps under $f$ linearly to
another subdivision of the circle. This subdivision is represented by the tree $T_+$, and the marking
describes where each subinterval of the circle is mapped. The element $f$ can be thought of as mapping
the leaves of $T_-$ to the leaves of $f(T_-)=T_+$, where the marking defines this mapping of the leaves.
If $f$ does not have a tree pair diagram in which the tree $T$ appears as the source tree, then the
symbol $f(T)$ has no meaning.
Given two rooted binary trees $T$ and $T'$, we say that $T'$ is an \emph{expansion} of $T$ if $T'$ can be
obtained from $T$ by attaching the roots of additional trees to some subset of the leaves of $T$. We
observe that if $(T, f(T))$ is a marked tree pair diagram for $f$, and $T'$ is an expansion of $T$, then
there is always a tree pair diagram $(T', f(T'))$ for $f$, and $f(T')$ is an expansion of $f(T)$. Given
two rooted binary trees $S$ and $T$, by the \emph{minimal common expansion} of $S$ and $T$ we mean the
smallest rooted binary tee which is an expansion of both $S$ and $T$. Using this language, if $(T,f(T))$
and $(S,g(S))$ are marked tree pair diagrams for $f$ and $g$ respectively, the process described in
Section 2.3 for creating a tree pair diagram for the product $gf$ could be summarized as follows. If $E$
is the minimal common expansion of $f(T)$ and $S$, then there are tree pair diagrams $(f^{-1}(E),E)$ for
$f$, $(E,g(E))$ for $g$, and $(f^{-1}(E),g(E))$ for $gf$ (with appropriate markings).
\begin{proof}
Suppose that $f$ is a torsion element. We begin by describing the construction of (marked) tree pair
diagrams $(A_n,B_n)$ for $f^n$ for every $n \geq 1$. These tree pair diagrams are constructed
inductively, viewing $f^n$ as a product $(f^{n-1})(f)$. For $n=1$, let $(A_1,B_1)$ be the reduced marked
tree pair diagram for $f$. Throughout this procedure, although markings are carefully carried through in
either $T$ or $V$, since our goal is merely to produce a tree pair diagram for $f$ with the same source
and target trees (regardless of marking), only the trees themselves are relevant for this argument. Hence
we suppress mention of any markings throughout the construction. If $k \geq 2$, suppose the marked tree
pair diagram $(A_{k-1},B_{k-1})$ for $f^{k-1}$ has been constructed. Let $E_{k-1}$ be the minimal common
expansion of the trees $A_1$ and $B_{k-1}$. Then $f^k$ has tree pair diagram
$(f^{-(k-1)}(E_{k-1}),f(E_{k-1}))$, and we let $B_k=f(E_{k-1})$ and $A_k=f^{-(k-1)}(E_{k-1})$.
By construction, $A_{k+1}$ is an expansion of $A_k$ for all $k \geq 1$. We claim also that $B_{k+1}$ is
an expansion of $B_k$ for all $k \geq 1$. For $k=1$, $E_1$ is by definition an expansion of $A_1$, which
implies that $B_2=f(E_1)$ is an expansion of $B_1=f(A_1)$. Suppose inductively that $B_k$ is an expansion
of $B_{k-1}$. Now $E_k$ is an expansion of $B_k$ and $A_1$, so $E_k$ is an expansion of $B_{k-1}$ and
$A_1$. But $E_{k-1}$ is the minimal common expansion of $B_{k-1}$ and $A_1$, so $E_k$ is an expansion of
$E_{k-1}$, which implies that $B_{k+1}=f(E_k)$ is an expansion of $B_k=f(E_{k-1})$.
Since there exists a positive integer $m$ such that $f^m$ is the identity, it follows that all tree pair
diagrams for $f^m$ must have the same source and target trees. Hence $A_m=B_m$, and then since $A_m$ is
an expansion of $A_1$, $B_m$ is an expansion of $A_1$. But since $E_{m-1}$ is the minimal common
expansion of $B_{m-1}$ and $A_1$, the fact that $B_m$ is an expansion of both $B_{m-1}$ and $A_1$ implies
that $B_m=f(E_{m-1})$ is an expansion of $E_{m-1}$. But they have the same number of carets, so in fact
$f(E_{m-1})=E_{m-1}$. In other words, the tree pair diagram $(E_{m-1}, B_n=f(E_{m-1}))$ is the desired
tree pair diagram for $f$.
\end{proof}
\begin{cor} An element of $T$ is torsion if and only if it is a conjugate of some $c_i^j$.
\end{cor}
\begin{proof} If an element is torsion, then it admits a diagram with two equal trees. The $pcq$ factorization
associated with this diagram has the form $pc_i^jp^{-1}$, where $p$ is a positive element of $F$.
\end{proof}
A particularly natural torsion subgroup is the subgroup $R$ of pure
rotations, where by a pure rotation we mean a rotation by
$d=\frac{a}{2^n}$ (where $a$ is not divisible by 2). Such pure
rotations were used in Section \ref{sec:rotationnum} to conjugate
the fixed point of a homeomorphism to 0.
This subgroup is isomorphic to the group of dyadic rational numbers modulo 1, which has a 2-adic metric
as follows: if $x=\frac{p}{2^l}$, $y= \frac{q}{2^m}$, and $z= |x-y| = \frac{r}{2^k}$, where $p, q$ and
$r$ are odd, then $d(x,y)=2^k$. With respect to this metric, the subgroup of rotations is
quasi-isometrically embedded in $T$.
\begin{prop} The subgroup $R$ of the pure rotations, with the 2-adic metric, is quasi-isometrically
embedded in $T$.
\end{prop}
\begin{proof}
We note that if $g\in T$ is the rotation by $\frac{a}{2^n}$ where $a$ is not divisible by $2$, then there
are $2^{n}-1$ carets in the reduced tree pair diagram representing $g$, so $N(g)=2^{n}-1$. Since we have
shown that the word length of $g$ in $T$ is bi-Lipschitz equivalent to $N(g)$, the proposition follows.
\end{proof}
|
{
"timestamp": "2009-09-03T16:23:44",
"yymm": "0503",
"arxiv_id": "math/0503670",
"language": "en",
"url": "https://arxiv.org/abs/math/0503670"
}
|
\section{Introduction}
Growth phenomena are ubiquitous around us. They have both very
practical applications and theoretical relevance. But they are rarely
easy to study analytically and very few rigorous or exact results
are known. In two dimensions, the description of a growing domain is
often obtained indirectly through the description of a family of
univalent holomorphic representations, leading quite generally to
equations known under the name of Loewner chains. These techniques,
based on the Riemann mapping theorem, are conceptually important but
usually far from making the problem tractable.
In the last few years, Loewner chains have been discovered which have
a large hidden symmetry -- conformal invariance -- that makes them more
amenable to an exact treatment \cite{schramm0}. These are known
under the name of Stochastic or (Schramm) Loewner evolutions
SLE.
Their mathematical elegance and simplicity is not their sole virtue.
They are also natural candidates to describe the continuum limit of an
interface in two-dimensional statistical mechanics models at
criticality. At the critical temperature and in the continuum limit,
the system is believed to be conformally invariant and physicists have
developed many powerful techniques, known under the name conformal
field theory or CFT, to deal with local questions in a conformally
invariant 2d system. However, nonlocal objects like interfaces posed
new nontrivial problems that finally SLE could attack in a systematic
way \cite{LSW,LSW:ConformalRestriction}. The connection between CFT and
SLE is now well understood
\cite{Bb:2002qn,Bb:2002tf,WernerFrie,Bb:2003kd,Bb:2003vu,Bb:2004ij}
and the interplay between the two approaches has proved fruitful.
The way SLE describes an interface deserves some comments.
As a guiding example, consider the Ising model in a simply connected
domain, say on the hexagonal lattice. Suppose that the boundary is
split in two arcs with endpoints say $a$ and $b$ and impose that on
one arc the spins are up and on the other one the spins are down. In
this situation each sample exhibits an interface. It joins the two
points where the boundary conditions change and splits the domain
in two pieces, one with all spins up on its boundary and one with all
spins down. This interface fluctuates from sample to sample. What
SLE teaches us is the following. Instead of describing the
interface between $a$ and $b$ at once, SLE views it as a curve
starting from say $a$ and growing toward $b$. And SLE describes the
distribution for the addition of an infinitesimal piece of interface
when the beginning of the interface is already known. So the
description is in terms of a growth process even if there was no
growth process to start with.
As mentioned above, the probabilistic aspects of SLE as well
as its connections with conformal field theory are now fairly well
understood. However, some fundamental questions remain, again
directly related to natural questions in the statistical mechanics
framework.
The one we shall concentrate on in this note is what happens when, due
to boundary conditions, the system contains several interfaces.
Proposals for multiple SLEs have already been made in the literature
\cite{Cardy:nSLE,D:SLEcommutation}, but our results point to a
different picture. The simplest situation is in fact when there is
only one interface but we want to deal with its two ends symmetrically
so that two growth processes will interact with each other. Remember
that standard SLE deals with the two ends of the interface
asymmetrically. This has a price : time reversed SLE is an intricate
object.
As a guiding example for more than one interface, consider again the
Ising model in a simply connected domain on the hexagonal lattice. If
one changes boundary conditions from up to down to up and so on $n=2m$
times along the boundary, each sample will exhibit $m$ interfaces,
starting and ending on the boundary at points where the boundary
conditions change, forming a so-called arch system. However, the
interfaces will fluctuate from sample to sample and so does even the
topology of the arch system. This topology, for instance, is an
observable that is trivial for a single SLE.
Our description will again be in terms of growth processes and Loewner
chains. For standard SLE, the driving parameter is a continuous
martingale and the tip of the curve separates two different states of
the system (up and down spins for Ising), leading to a well defined
boundary changing operator in statistical mechanics. The relation
between the stochastic Loewner equation and the boundary changing
operator comes via a diffusion equation that they share in common.
For multiple SLEs, we expect that for short time scales each curve
grows under the influence of an independent martingale. At its tip
stands the same boundary changing operator. But we also expect drift
terms, describing interactions between the curves.
The possibility of different arch topologies makes it even more
natural to have a description with one curve growing at each boundary
changing point so that each of them is on the same footing. So $m$
interfaces are described by $n=2m$ growth processes of ``half-interfaces''
that finally pair in a consistent way to build arches.
In statistical mechanics, each arch system has a well defined
probability to show up. The law governing this finite probability
space is again described by a Boltzmann weight which is nothing but a
partial partition function.
Our starting point is the reconsideration of the role of Boltzmann
weights and partition functions in statistical mechanics and their
simple but crucial relationship with probabilistic martingales. This
allows us to ask the question ``by what kind of stochastic
differential equations can one describe multiple SLEs ?'' by imposing
a martingale property and conformal invariance. This puts strong
constraints on the drift terms and our main result is a description
of the family of drift terms that are compatible with the basic rules
of statistical mechanics. Each drift is expressed in terms of the
partition function of the system. This partition function is given by
a sum of Boltzmann weights for configurations that satisfy certain
boundary conditions : at the starting points of the curves the
boundary conditions change. The partition function depends on the
position of these changes, so up to normalization, the partition
function is in fact a correlation function. It satisfies a number of
partial differential equations (one equation for each point) that are
related to the diffusion equations for the multi SLE process. The
solutions form a finite dimensional vector space. The positivity
constraint satisfied by physical partition functions singles out a
cone which is expected, again guided by statistical mechanics, to have
the same dimension of the underlying vector space and to be the convex
hull of a family of half
lines, so that a generic hyperplane section of the cone is a simplex.
So geometrically, the drift terms are parametrized by a cone.
Extremal drifts, i.e. drifts corresponding to extremal lines in this
cone, lead to processes for which the final pattern formed by the
growing curves is a given arch system. Drifts inside the simplex give
rise to stochastic processes where the asymptotic arch system
fluctuates from sample to sample. A crucial role to construct
martingales describing interesting events is played by the short
distance expansion in conformal field theory because this is what
tells which terms in the partition function become dominant when an
arch closes, i.e. when two driving processes of the multi SLE hit each
other.
The vector space of solution of the differential equations
for the partition functions has a famous basis indexed by Dyck
paths, which are in one to one correspondence with arch systems. But
the basis elements do not in general correspond to extremal partition
functions. We shall give a rationale for computing the matrix
elements for the change of basis and compute a number of them, but we
have no closed general formula.
We shall illustrate our proposal with concrete computations for $1$ to
$3$ interfaces with applications to percolation and the Ising model. We
shall also discuss the classical (deterministic) limit $\kappa
\rightarrow 0^+$, where only extremal drifts survive.
The notes also cover the case when a number of boundary changes are
very close to each other but the system is conditioned so that they
do not pair with each other. The details are in the main text.
Our description is rather flexible in the sense that the speed of
growth of each piece of interface can be tuned. Certain limiting cases
lead to previously known processes which are examples of
SLE$(\kappa,\underline{\rho})$ processes.
It is appropriate here to stress that many of the probabilistic
properties of the solutions of the stochastic differential equations
that we introduce are conjectural at this point. We have made some
consistency checks\footnote{For instance, Dub\'edat has derived
general ``commutation criteria'' \cite{D:SLEcommutation} for
multiple SLEs. The processes we study are a special class
satisfying commutation. This class extends vastly the special
solution found by Dub\'edat, which in our language corresponds to
self avoiding SLEs moving to infinity.} and the whole pattern is
elegant, but our confidence comes more from our familiarity with
conformal field theory and statistical mechanics.
\section{Basics of Schramm-L\"owner evolutions: Chordal SLE}
Let us briefly recall what is meant by the chordal SLE --- detailed
studies can be found in \cite{RS:BasicProperties} or \cite{W:Lectures}.
The chordal SLE process in the upper half plane $\bH$
is defined by the ordinary differential equation
\begin{equation}
\label{eq: chordal SLE}
\frac{\ud}{\ud t} g_t(z) = \frac{2}{g_t(z)-\xi_t}
\end{equation}
where the initial condition is $g_0(z) = z \in \bH$ and
$\xi_t = \sqrt{\kappa} B_t$ is a Brownian motion with variance
parameter $\kappa \geq 0$. Let $\tau_z \leq \infty$ denote the
explosion time of (\ref{eq: chordal SLE}) with initial condition
$z$ and define the hull at
time $t$ by $K_t := \overline{ \{ z \in \bH | \tau_z < t \} }$.
Then $(K_t)_{t \geq 0}$ is a family of growing hulls,
$K_s \subset K_t$ for $s<t$.
The complement $\bH \setminus K_t$ is
simply connected and $g_t$ is the unique conformal mapping
$\bH \setminus K_t \rightarrow \bH$ with
$g_t(z) = z + \order (1)$ at $z \rightarrow \infty$. One
defines the SLE trace by $\gamma_t = \lim_{\epsilon \downarrow 0}
g_t^{-1}(\xi_t + i \epsilon)$. The trace is a continuous
path in $\overline{\bH}$ and it generates the hulls in the sense
that $\bH \setminus K_t$ is the unbounded component of
$\bH \setminus \gamma_{[0,t]}$. For $\kappa \leq 4$ the trace
is a non-self-intersecting path and it doesn't hit
$\partial \bH = \bR$ for $t>0$ so $K_t = \gamma_{[0,t]}$. For
$4 < \kappa < 8$ a typical point $z \in \bH$ is swallowed, i.e.
$z \in K_t$ for large $t$ but $z \notin \gamma_{[0, \infty)}$.
In the parameter range $\kappa \geq 8$ the trace is space
filling, $\gamma_{[0, \infty)} = \overline{\bH}$. Let us point
out that no statistical mechanics models seem to correspond
to $\kappa > 8$.
In the definition of chordal SLE we took the usual parametrization of
time. From equation (\ref{eq: chordal SLE}) we see that $g_t (z) = z +
2 t z^{-1} + \Order(z^{-2})$, which means (this could be taken as a
definition) that the capacity of $K_t$ from infinity is $2 t$. Since
the capacity goes to infinity as $t \rightarrow \infty$, the hulls
$K_t$ are not contained in any bounded subset of $\overline{\bH}$.
If the parametrization of time is left arbitrary, Schramm's argument
yields :
\[ \ud g_s(z) = \frac{2\ud q_s}{g_s(z)-M_s},\]
where $M_s$ is a continuous martingale with quadratic variation
$\kappa q_s$ (an increasing function going to infinity with $s$). In
this formula, both $q_s$ and $M_s$ are random objects. The capacity of
$K_s$ is $2q_s$. But this is not really more general than eq.(\ref{eq:
chordal SLE}) which is recovered by a random time change.
\section{A proposal for multiple SLEs}
The motivations for our proposal require a good amount of background,
but the proposal and its main features themselves can be easily
stated. We gather them in this section. Some of the results are
conjectures. The rest of the paper will then be split into sections
whose purpose will be either to motivate our proposal in general, or
to prove its correctness in certain special but nontrivial cases by
explicit computations.
\subsection{The basic equations}
We propose to describe the local growth of $n$ interfaces in CFT,
labeled by an integer $i=1,\cdots,n$ and joining fixed points on the
boundary by a Loewner chain. We assume that $0 \leq \kappa <8$ in the
following. We list the set of necessary conditions and equations.
\vspace{.3cm}
\noindent \textbf{Conformal invariance}: The measure on $n$SLE is
conformally invariant. Hence it is enough to give its definition when
the domain $D$ is the upper half plane $\mathbb H$ in the
hydrodynamical normalization.
\vspace{.3cm}
\noindent \textbf{Universe}: The basic probabilistic objects
are $n$ (continuous, local) martingales $M^{(i)}_t$, $i=1,\cdots,n$
with quadratic variation $\kappa q^{(i)}_t$ absolutely continuous with
respect to $\ud t$ and vanishing cross variation, defined on an
appropriate probability space. By a time change we can and shall
assume that $\sum_i q^{(i)}_t \equiv t$.
\vspace{.2cm}
\noindent \textbf{Driving processes}: The processes $X^{(i)}_t$ are
solutions of the stochastic differential equations
\begin{equation}
\label{eq:drivepro}
\ud X^{(i)}_t=\ud M^{(i)}_t + \kappa \ud
q^{(i)}_t(\partial_{x_i}\log Z) (X^{(1)}_t,\cdots,X^{(n)}_t)+ \sum_{j
\neq i} \frac{2\ud q^{(j)}_t}{X^{(i)}_t-X^{(j)}_t}.
\end{equation}
The initial
conditions are $X_0^{(i)}=X_i$ ordered in such a way that $X_1 <
X_2< \cdots < X_n$.
\vspace{.2cm}
\noindent \textbf{Loewner chain}: The map $f_t$ uniformizing the
complement of the hulls satisfies
\begin{equation}
\label{eq:loewchain}
\ud f_t(z)=\sum_i \frac{2\ud
q^{(i)}_t}{f_t(z)-X^{(i)}_t}.
\end{equation}
The initial condition is $f_0(z)=z$. With our conventions, the total
capacity of the growing hulls at time $t$ is $2t$.
\vspace{.2cm}
\noindent \textbf{Auxiliary function}: The system depends on a function
$Z(x_1,\cdots,x_n)$ which has to fulfill the following requirements :
$i)$ $Z(x_1,\cdots,x_n)$ is defined and positive for $x_1 < x_2< \cdots
< x_n$,
$ii)$ $Z(x_1,\cdots,x_n)$ is translation invariant and homogeneous.
Its weight is $h_{n-2m}(\kappa)-nh_{1}(\kappa)$ for some
nonnegative integer $m \leq n/2$, where\footnote{A more traditional
notation for $h_{m}(\kappa)$ is $h_{1,m+1}$ in the physics literature.}
$$2\kappa h_{m}(\kappa)\equiv
m(2(m+2)-\kappa).$$
$iii)$ $Z(x_1,\cdots,x_n)$ is annihilated by the $n$ differential
operators
$${\mathcal D}_i=\frac{\kappa}{2}\partial_{x_i}^2+2\sum_{j \neq i}\left[
\frac{1}{x_j-x_i}\partial_{x_j}-\frac{h_{1}(\kappa)}{(x_j-x_i)^2}\right].$$
\vspace{.2cm}
We call this system of equations the $n$SLE system for $n$ curves
joining together the points $X_1,\cdots,X_n$ and possibly the point at
infinity. Systems for radial and dipolar versions of $n$SLE
could be defined analogously.
\subsection{Arch probabilities}
It is known from CFT that (relaxing the positivity constraint), the
solutions to $i),\;ii),\;iii)$ form a vector space of dimension
$d_{n,m}\equiv {n \choose m}-{n \choose
m-1}=(n+1-2m)\frac{n!}{m!(n-m+1)!}$.
The positive solutions form a cone and from the statistical mechanics
interpretation, we conjecture that this cone has the same dimension
and is generated by (i.e. is the convex hull of) $d_{n,m}$ half
lines (extremal lines, pure states in the sense of
statistical mechanics) so that a transverse section of the cone is a
simplex. So each solution $Z$ can be written in a unique way as a sum of
extremal states.
The numbers $d_{n,m}$ have many many combinatorial interpretations,
but the one relevant for us is the following. Draw $n+1$ points
$X_1<X_2\cdots <X_n < \infty$ ordered cyclically on the (extended)
real line bounding the upper half plane $\mathbb H$. Consider $n-m$
disjoint curves in $\mathbb H$ such that each $X_i$ is an end
point of exactly $1$ curve and $\infty$ is an end point of exactly
$n-2m$ curves. There are $d_{n,m}$ topologically inequivalent
configurations, called arch configurations when $n-2m=0$. We keep the
same name for $m\neq 0$, writing arch$_m$ configurations when
precision is needed.
Motivated by this, we claim the following :
\vspace{.3cm}
a) To each arch configuration $\alpha$ corresponds an
extremal state $Z_{\alpha}$ in the following sense : the solution of the
$n$SLE system with $Z \propto Z_{\alpha}$ can be defined up to a
(possibly infinite) time, at which the growing curves have either paired
together or joined the point at infinity and at that time the topology is
that of the arch $\alpha$ with probability one.
\vspace{.2cm}
b) One can decompose a general solution $Z$ of $i),\;ii),\;iii)$ as a sum
of $$\sum_{\alpha \,\in \, {\mathrm{arch}_m}} Z_{\alpha}.$$
\vspace{.2cm}
c) The probability that a solution of the $n$SLE system with auxiliary
function $Z$ ends in arch configuration $\alpha$ is the ratio
$$\frac{Z_{\alpha}(X_1,\cdots,X_n)}{Z(X_1,\cdots,X_n)}$$
evaluated at the initial condition $(X_1,\cdots,X_n)$.
\vspace{.3cm}
The first step toward a heuristic derivation of the above results will
be to explain how to construct martingales -- in particular
martingales associated to interfaces -- from statistical mechanics
observables in a systematic way. But we start with a few comments.
\section{First comments}
\subsection{Statistical mechanics interpretation}
To have a specific example in mind, think again of the Ising model at
the critical temperature. Let $a$ be the lattice spacing.
First, put $n=2m$ changes of boundary conditions from spins up to
spins down and so on along the boundary at points
$x_1/a,\cdots,x_n/a$. In the continuum limit when $a \rightarrow 0$
but $x_1,\cdots,x_n$ have a finite limit, the partition function
behaves like a homogeneous function $Z(x_1/a,\cdots,x_n/a)$ of weight
$0$ (when both $a$ and the $x_i$'s are rescaled) and CFT teaches us
that $Z(x_1,\cdots,x_n)$ satisfies $i),\;ii),\;iii)$ for $n=2m$. Then,
if $\alpha$ is an arch system, $Z_{\alpha}$ should be (proportional to
the continuum limit of) the partial partition function when the sum of
Boltzmann weights is performed only over the interface configurations
with topology $\alpha$.
To make generalized arch configurations, choose $n$ and $m$ with
$n\geq 2m$. Put $2n-2m$ changes of boundary conditions from spins up
to spins down and so on along the boundary, $n$ at points
$x_1/a,\cdots,x_n/a$ and $n-2m$ at $y_1/a,\cdots, y_{n-2m}/a$. Sum
only over configurations where the interfaces do not joint two
$y$-type points to each other. Take the continuum limit for the $x$'s
as before, but impose that all $y$'s go to infinity and remain at a
finite number of lattice spacings from each other. This is expected to
lead again to a partition function $Z(x_1/a,\cdots,x_n/a)$ of weight
$0$ (when both $a$ and the $x_i$'s are rescaled) and
$Z(x_1,\cdots,x_n)$ satisfies $i),\;ii),\;iii)$ for the given $n$ and
$m$. If $\alpha$ is an arch$_m$ configuration, $Z_{\alpha}$ should be
(proportional to the continuum limit of) the partial partition
function when the sum of Boltzmann weights is performed only over the
interface configurations with topology $\alpha$.
Note that the prefactor between the continuum limit finite part and
the real partition function is a power of the lattice spacing. The
power depends on $m$, so it is likely to be unphysical to use a non
homogeneous $Z$ in the $n$SLE system, mixing different values of $m$
for a fixed $n$. However, we shall later treat the example $n=2$
mixing $m=0$ and $m=1$ because it is illustrative despite the fact
that it breaks scale invariance.
\subsection{SLE as a special case of $2$SLE}
For $n=2$ the solution of $i),\;ii),\;iii)$ with $m=1$ is elementary.
Writing $x_1=a$ and $x_2=b$, one finds $Z\propto (b-a)^{(\kappa
-6)/\kappa}$. Taking the first martingale to be a Brownian and the
second one to be $0$, one retrieves the equations for SLE growing from
point $a$ to point $b$ in the hydrodynamical normalization. Let us recall
briefly why.
We start from SLE from $0$ to $\infty$. The
basic principle of conformal invariance makes the passage from this
special case to the case when SLE goes from point $a$ to point $b$ on
$\bH$ a routine task. If $u$ is any linear fractional transformation
(i.e. any conformal transformation) from $\bH$ to itself mapping $0$
to $a$ and $\infty$ to $b$, the image of the SLE trace or hull from
$0$ to $\infty$ by $u$ is by definition an SLE trace from $a$ to $b$
and this is measure preserving. The new uniformizing map is
$h_t=u\circ g_t \circ u^{-1}$ and it is readily checked that
$\frac{\ud h_t}{\ud t}$ is a rational function of $h_t$ whose precise
form can be easily computed but does not concern us.
Let us just mention that this rational function is regular everywhere
(infinity included) except for a simple pole at $h_t=u(\xi_t)$ and
has a third order zero at $h_t=u(\infty)=b$. So the map $h_t$ is
normalized in such a way that
$h_t(b+\varepsilon)=b+\varepsilon+O(\varepsilon^3)$, which is not the
hydrodynamic normalization.
But if $v_t$ is any linear fractional transformation, $v_t \circ g_t
\circ u^{-1}$ describes the same trace as $h_t=u \circ g_t \circ
u^{-1}$. As long as the trace does not separate $b$ from $\infty$,
i.e. as long as the trace has not hit the real axis in the segment
$]b,\infty[$, i.e. as long as $\infty$ is not in the hull, $v_t$ can
be adjusted in such a way that $\tilde{h}_t\equiv v_t \circ g_t \circ
u^{-1}$ is normalized hydrodynamically. Then $\ud \tilde{h}_t/\ud t$
is a function of $\tilde{h}_t$ which is regular everywhere but for a
single pole and vanishes at infinity, i.e. one can write $\ud
\tilde{h}_t/\ud t=2\mu_t/(\tilde{h}_t-\alpha_t)$. The following
computation is typical of the manipulations made with SLE (see e.g.
\cite{LSW:ConformalRestriction}). Write $(g_t \circ u^{-1})(z)=w$ an
compute from the definition
$$\frac{\ud \tilde{h}_t}{\ud t}(z)=\frac{\ud
v_t}{\ud t}(w)+ v_t^{'}(w)\frac{2}{w-\xi_t}.$$ Comparison gives
$$\frac{\ud v_t}{\ud
t}(w)=\frac{2\mu_t}{v_t(w)-\alpha_t}-\frac{2v_t^{'}(w)}{w-\xi_t}.$$
But $v_t$ is regular at $w=\xi_t$ from which one infers that
$v_t(\xi_t)=\alpha_t$ (the poles in the two terms are at the same
point) and $\mu_t=v_t^{'}(\alpha_t)^2$ (the two residues add to $0$).
Going one step further in the expansion close to $\xi_t$ yields
$\frac{\ud v_t}{\ud t}(\xi_t)= -3v_t^{''}(\xi_t).$ Ito's formula gives
$\ud \alpha_t=-3v_t^{''}(\xi_t)\ud t+v_t^{'}(\xi_t)\ud
\xi_t+\frac{\kappa}{2}v_t^{''}(\xi_t)\ud t.$ So the time change $\mu_t
\ud t =\ud s $ together with the definition $\ud \chi_s
=v_t^{'}(\xi_t)\ud \xi_t$ yields
$$\ud \alpha_t(s)=\ud
\chi_s+(\kappa-6)\frac{v_t^{''}(\xi_t)}{2v_t^{'}(\xi_t)^2}\ud s.$$
But
$v_t^{''}(w)/v_t^{'}(w)^2=2/(v_t(w)-v_t(\infty))$ because $v_t$ is a
linear fractional transformation. Finally, setting
$\tilde{h}_{t(s)}\equiv f_s$,
$\tilde{h}_{t(s)}(b)=v_{t(s)}(\infty)\equiv B_s$ and
$v_{t(s)}(\xi_{t(s)})=\alpha_{t(s)}\equiv A_s$ we can summarize
$$\frac{\ud f_s}{\ud s}=\frac{2}{f_s-A_s}\; , \quad \frac{\ud B_s}{\ud
s}=\frac{2}{B_s-A_s} \; , \quad \ud A_s= \ud \chi_s
+(\kappa-6)\frac{\ud s}{A_s-B_s},$$
where $\chi_s$ is a Brownian
motion with quadratic variation $\kappa s$, $f_0=id$, $A_0=a$,
$B_0=b$. Thus chordal SLE from $a$ to $b$ in the hydrodynamical
normalization is indeed a special case of $2$SLE.
The above equations are also a special case of SLE$(\kappa,\rho)$
$(\rho=\kappa-6)$, but it should be clear that our general
proposal goes in a different direction.
As already mentioned, the description of chordal SLE from $a$ to $b$
in the hydrodynamical normalization in fact coincides with chordal SLE
from $a$ to $b$ only up to the first time $b$ is separated from
$\infty$ by the trace. This time is infinite for $\kappa \leq 4$, but
it is finite with probability $1$ for $4 < \kappa < 8$. The most
obvious case is $\kappa=6$. The equation is nothing but the usual
chordal SLE$_6$ ending at infinity, a consequence of locality (in the
SLE sense, not in the quantum field theory sense used later). At that
time, the real chordal SLE from $a$ to $b$ swallows $\infty$, whereas
the hydrodynamically normalized version swallows $b$. The solution to
this problem is to use conformal invariance and restart the process
again in the correct domain at the time when $b$ and $\infty$ get
separated by the trace. But this is not coded in the equations.
\subsection{Making sense}
The previous example should serve as a warning. Some serious
mathematical work may have to be done even to make sense of our
conjectures, let alone prove their correctness. The problems might be
of different natures for $\kappa \leq 4$ and $4 < \kappa <8$. We
content with the following naive remarks. One of the problems is that
the arches do not have to close at the same time. It may even happen
that one of the growing curves touches the real line or another curve
in such a way that the upper half plane is split in two domains and
the one which is swallowed contains some of the growing curves.
Our putative description of $n$SLE processes can be valid in this form
only up the realization of such an event. The first thing to check
should be that the event is realized with a probability obtained by
summing $Z_{\alpha}/Z$ over all $\alpha$'s corresponding to compatible
configurations (see figure \ref{fig: compatible configurations}).
In particular, the connected
component of $\infty$ should contain at least $n_{\infty} \geq m-1$
curves for consistency, but that's not an obvious property of our
proposal.
\begin{figure}
\includegraphics[width=1.0\textwidth]{someconfigs.eps}
\caption{\emph{The probability of closing of an arch should be obtained
by summing $Z_\alpha/Z$ over all $\alpha$'s corresponding to
compatible configurations. Two compatible configurations are
portrayed in the figure.}}
\label{fig: compatible configurations}
\end{figure}
Consider the fate of the connected component of $\infty$. If
$n_1-m$ is even, conformal invariance suggests to continue the Loewner
evolution simply by suppressing the points that have been swallowed,
i.e. for the $n_1$ remaining points. If $n_1-m$ is odd, the same
should be done, but the image of the point at which one interface has
made a bridge should be included as a starting point for the
continuation of the evolution. Preferably, the function $Z$ for this new
$multi$SLE system should not be adjusted by hand to make our
conjectures correct, but should appear as a natural limit. We shall
make comments on this and give concrete illustrations later.
For the component that is
swallowed, one can use conformal invariance again to change the
normalization of the Loewner map in such a way that this component is
the one that survives and then restart a new $multi$SLE for the
appropriate number of points. This procedure may have to be iterated.
Note also that our conjectures for arch probabilities do not involve
any details on the martingales $M^{(i)}_t$. Indeed, we expect that
there is some robustness. But the precise criteria are beyond our
understanding.
\subsection{A few martingales for $n$SLEs}
Our heuristic derivation of the $n$SLE system will in particular show
that if $\tilde{Z}$ also solves $i),\;ii),\;iii)$ (even relaxing
positivity), the quotient
$$\frac{\tilde{Z}(X^{(1)}_t,\cdots,X^{(n)}_t)}
{Z(X^{(1)}_t,\cdots,X^{(n)}_t)}$$
is a local martingale. This can be proved directly using Ito's formula.
In particular,
$$\frac{Z_{\alpha}(X^{(1)}_t,\cdots,X^{(n)}_t)}
{Z(X^{(1)}_t,\cdots,X^{(n)}_t)}$$
is a local martingale bounded by
$1$, hence a martingale. On the other hand, a
standard argument shows that if $P_{\alpha}$ is the probability that
the system ends in a definite arch configuration $\alpha$ (once one
has been able to make sense of it)
$P_{\alpha}(X^{(1)}_t,\cdots,X^{(n)}_t)$ is a martingale. This is an
encouraging sign. To get a full proof, one would need to analyze the
behavior of $Z_{\alpha}(X^{(1)}_t,\cdots,X^{(n)}_t)$ when one arch
closes, or when one growing curve cuts the system in two, to get
recursively a formula that looks heuristically like
$$\frac{Z_{\alpha}(X^{(1)}_t,\cdots,X^{(n)}_t)}
{Z(X^{(1)}_t,\cdots,X^{(n)}_t)}\sim \delta_{\alpha,\alpha'}$$
if the
system forms asymptotically the arch system $\alpha'$ at large $s$.
Such a formula rests on properties of $Z_{\alpha}(x_1,\cdots,x_n)$
when some points come close together in a way reminiscent to the
formation of arch $\alpha'$ : $Z_{\alpha'}(x_1,\cdots,x_n)$ should
dominate all $Z_{\alpha}$'s, $\alpha \neq \alpha'$ in such
circumstances. In section \ref{sec:sevinterf} we shall use this to
expand explicitly the $Z_{\alpha}$'s in a basis of solutions to
$i),\;ii),\;iii)$ which is familiar from CFT, very explicitly at least
for small $n$.
\subsection{Classical limit}
Our proposal for $n$SLE has a non trivial classical limit at $\kappa
\rightarrow 0^+$. The martingales $M_t^{(i)}$ vanish in this limit,
but the $q_t^{(i)}$ remain arbitrary increasing functions. The
function $Z$ does not have a limit, but the $U_i\equiv \kappa
\partial_{x_i} \log Z$ do. They are kind of Ricatti variables for
which the equations read
$$
\frac{1}{2} \left(\partial_{x_i}U_i +
\frac{U_i^2}{\kappa}\right)+2\sum_{j \neq i}
\left(\frac{1}{x_j-x_i}\frac{U_i}{\kappa}-
\frac{6-\kappa}{2\kappa}\frac{1}{(x_j-x_i)^2}\right)=0,
$$
which have a limit when $\kappa \rightarrow 0^+$, comparable to the
classical limit of a Schroedinger equation. To summarize, the
classical limit is
$$\ud f_t(z)=\sum_i \frac{2\ud q^{(i)}_t}{f_t(z)-X^{(i)}_t}.$$
$$
\ud X^{(i)}_t=U_i(X^{(1)}_t,\cdots,X^{(n)}_t) \ud q^{(i)}_t
+\sum_{j \neq i} \frac{2\ud q^{(j)}_t}{X^{(i)}_t-X^{(j)}_t}.$$
where the auxiliary
functions $U_i(x_1,\cdots,x_n)$ are homogeneous functions of degree
$-1$ which satisfy $\partial_{x_i}U_j=\partial_{x_j}U_i$ and
$$ \frac{1}{2} U_i^2+2\sum_{j \neq i}\left(\frac{1}{x_j-x_i}U_i-
\frac{3}{(x_j-x_i)^2}\right)=0.$$
It is not too surprising that the differential equations for $Z$ have
become algebraic equations for the $U_i$'s, so that the space of
solutions which was a connected manifold for $\kappa \neq 0$
concentrates on a finite number of points in the classical limit. The
classical system, maybe with an educated guess for the $q_t^{(i)}$'s,
could be interesting for its own sake.
\subsection{Relations with other work}
Several processes involving several growing curves have appeared
in the literature.
The first proposal was made by Cardy \cite{Cardy:nSLE}. It can be formally
obtained from ours by forgetting the conditions $i),\;ii),\;iii)$ and
choosing a constant $Z$. The corresponding processes are interesting,
but the relationship with interfaces in statistical mechanics and CFT
is unclear for us.
Dub\'edat \cite{D:SLEcommutation} has derived a general criterion he calls
commutativity to constrain the class of processes that could possibly
be related to interfaces. Our proposal satisfies commutativity so they
can be viewed as a special case satisfying other relevant physical
constraints. Dub\'edat also came with a special solution of
commutativity. It corresponds to the case $m=0$ in our language. Then
the space of solutions has dimension $d_{n,0}=1$ and the corresponding
partition function is elementary:
\begin{equation}
\label{eq:dabdouwa}Z \propto \prod_{i <j} (x_j-x_i)^{2/\kappa}.
\end{equation}
\begin{figure}
\center{\includegraphics[width=0.5\textwidth]{factorizable.eps}}
\caption{\emph{The factorisable $Z$ leads to a very simple geometry.
This case has been suggested previously with a slightly different
approach.}}
\label{fig: factorizable}
\end{figure}
A single arch topology is possible, all interfaces converge to
$\infty$, see figure \ref{fig: factorizable}. Maybe this is a
good reason to call this case chordal $n$SLE.
\section{CFT background}
There was never any doubt that SLEs are related to conformal field
theories. The original approach
\cite{Bb:2002tf,Bb:2003vu,Bb:2003kd,Bb:2004ij} used the operator
formalism because if yields naturally martingale generating functions.
Here, we use the correlator approach for a change. We restrict the
presentation to a bare minimum, referring the newcomer to the many
articles, reviews and books on the subject (\cite{DMS:CFT,BPZ}). The
reader who knows too little or too much about CFT can profitably skip
this section.
Observables in CFT can be classified according to their behavior under
conformal maps. Local observables in quantum field theory are called
fields. For instance, in the Ising model, on an arbitrary (discrete)
domain, the average value of a product of spins on different (well
separated) sites can be considered. Taking the continuum limit at the
critical point, we expect that on arbitrary domains $D$ there is a
local observable, the spin. The product of two spins at nearest
neighbor points corresponds to the energy operator. In the continuum limit,
this will also lead to a local operator. In this limit, the lattice
spacing has disappeared and one can expect a definite (but
nontrivial) relationship between the energy operator and the product of two
spin fields close to each other. As on the lattice the product of two spins
at the same point is $1$, we can expect that the identity observable
also appears in such a product at short distances. Local fields come
in two types, bulk fields whose argument runs over $D$ and boundary
fields whose argument runs over $\partial D$. In this
paper, we shall not need bulk fields so we leave them aside.
The simplest conformal transformations in the upper-half plane are
real dilatations and boundary fields can be classified accordingly. It
is customary to write $\varphi_{\delta}(x)$ to indicate that in a real
dilatation by a factor $\lambda$ the field $\varphi_{\delta}(x)$ picks
a factor $\lambda^{\delta}$. By a locality argument, boundary fields
in a general domain $D$ (not invariant under dilatations) can still be
classified by the same quantum number. The number $\delta$ is called
the conformal weight of $\varphi_{\delta}$.
There are interesting
situations in which (due to degeneracies) the action of dilatations
cannot be diagonalized, leading to so called logarithmic CFT. While
this more general setting is likely to be relevant for several aspects
of SLE, we shall not need it in what follows.
Under general conformal transformations, the simplest objects in CFT
are so called primary fields. Their behavior is dictated by the
simplest generalization of what happens under dilatations. Suppose
$\varphi_{\delta_1},\cdots \varphi_{\delta_n}$ are boundary primary
fields of weights $\delta_1,\cdots,\delta_n$. If $f$ is a conformal map
from domain $D$ to a domain $D'$, CFT postulates that
$$\bra
\prod_{j=1}^n \varphi_{\delta_j}(x_j) \ket^D =
\bra
\prod_{j=1}^n \varphi_{\delta_j}(f(x_j)) \ket^{f(D)}
\prod_{j=1}^n |f'(x_j)|^{\delta_j}.
$$
Symbolically, this can be written
$f:\varphi_{\delta}(x) \rightarrow
\varphi_{\delta}(f(x))|f'(x)|^{\delta}$.
It is interesting to make a comparison of these axioms with the
previous computations relating chordal SLE from $0$ to
$\infty$ to chordal SLE from $a$ to $b$ in several
normalizations. This also involved pure kinematics.
As usual in quantum field theory, to a symmetry corresponds an
observable implementing it. In CFT, this leads to the stress tensor
$T(z)$ whose conservation equation reduces to holomorphicity. The
fact that conformal transformations are pure kinematics translates
into the fact that insertions of $T$ in known correlation functions
can be carried automatically, at least recursively. The behavior of
$T(z)$ under conformal transformations can be written as $f:T(z)
\rightarrow T(f(z))f'(z)^{2}+c/12Sf(z)$ where $Sf\equiv
(f''/f')'-1/2(f''/f')^2$ is the Schwarzian derivative and $c$ is a
conformal anomaly, a number which is the most important numerical
characteristic of a CFT. When $c=0$, $T$ is be a $(2,0)$ primary field
i.e. an holomorphic quadratic differential. When a (smooth) boundary
is present, the Schwarz reflection principle allows to extend $T$ by
holomorphicity. Holomorphicity also implies that if $O$ is any local
(bulk or boundary) observable at point $z \in D$ and $v$ is vector
field meromorphic close to $z$, the contour integral $L_vO
\equiv\oint_z dw v(w)T(w)O$ along an infinitesimal contour around $z$
oriented counterclockwise is again a local field at $z$, corresponding
to the infinitesimal variation of $O$ under the map
$f(w)=w+\varepsilon v(w)$. It is customary to write $L_n$ for
$v(w)=w^{n+1}$. It is one of the postulates of CFT that all local
fields can be obtained as descendants of primaries, i.e. by applying
this construction recursively starting from primaries. The correlation
functions of descendant fields are obtained in a routine way from
correlations of the primaries. But descendant fields do not transform
homogeneously.
When $v$ is holomorphic at $x$, $L_vO$ is a familiar object. For
instance, if $\varphi_{\delta}$ is a primary boundary field, one
checks readily that $L_n\varphi_{\delta}=0$ for $n\geq 1$,
$L_0\varphi_{\delta}=\delta \varphi_{\delta}$ and
$L_{-1}\varphi_{\delta}=\Re e \; [\partial_x\varphi_{\delta}]$. The
other descendants are in general more involved, but by definition the
stress tensor $T=L_{-2}Id$ is the simplest descendant of the identity
$Id$. It does indeed not transform homogeneously.
A primary field and its descendants form what is called a conformal
family. Not all linear combinations of primaries and descendants need
to be independent. The simplest example is the identity observable,
which is primary with weight $0$ and whose derivative along the boundary
vanishes identically\footnote{For other primary fields with the same
weight if any, this does not have to be true.}. By contour
deformation, this leads to translation invariance of correlation
functions when $D$ has translation symmetry.
The next example in order of complexity is of utmost importance for
the rest of this paper. If $(2h+1)c=2h(8h-5)$, the field
$$-2(2h+1)L_{-2}\varphi_{h}+3L_{-1}^2\varphi_{h}$$
is again a primary, i.e. it transforms homogeneously under conformal
maps. In this case, consistent CFTs can be constructed for which it
vanishes identically. This puts further constraints on correlators.
For example, when $D$ is the upper half plane, so that the Schwarz
principle extends $T$ to the full plane, the contour for $L_{-2}$ can be
deformed and shrunken at infinity. Then, for an arbitrary boundary
primary correlator one has
\begin{eqnarray}
\label{eq:sing}
\left(\frac{3}{2(2h+1)}\partial_x^2 +\sum_{\alpha=1}^{l}\left[
\frac{1}{y_{\alpha}-x}\partial_{y_{\alpha}}-
\frac{\delta_{\alpha}}{(y_{\alpha}-x)^2} \right]\right) & & \nonumber \\
& & \hspace{-3cm} \bra \varphi_{\delta}(\infty) \prod_{\alpha=1}^l
\varphi_{\delta_{\alpha}}(y_{\alpha}) \varphi_{h}(x)
\ket=0. \end{eqnarray}
It is customary to call this type of equation a null-vector equation.
Note that the primary field of weight $\delta$ sitting at $\infty$ has
led to no contribution in this differential equation. Working the
other way round, this equation valid for an arbitrary number of
boundary primary fields with arbitrary weights characterizes the field
$\varphi_{h}$ and the relation between $h$ and the central charge $c$.
The case of three points correlators is instructive. Global conformal
invariance implies that
$$\bra \varphi_{\delta}(y)\varphi_{\delta'}(y') \varphi_{h}(x) \ket
\propto |y-y'|^{h-\delta-\delta'}
|x-y|^{\delta'-h-\delta}|y'-x|^{\delta-\delta'-h}.$$
The
proportionality constant might depend on the cyclic ordering of the
three points. But if the differential equation for $\varphi_{h}$ is
used, a further constraint appears. The three point function can be
non vanishing only if
$$
3(\delta-\delta')^2-(2h+1)(\delta+\delta')= h(h-1).$$
This
computation has a dual interpretation : consider a correlation
function with any number of fields, among them a $\varphi_{\delta}(y)$
and a $\varphi_{h}(x)$. If $x$ and $y$ come very close to each other
they can be replaced by an expansion in terms of local fields. This is
called fusion. Several conformal families can appear in such an
expansion, but within a conformal family, the most singular
contribution is always from a primary. This argument applies even if
$c$ and $h$ are arbitrary. But suppose they are related as above and
the differential equation eq.(\ref{eq:sing}) is valid. This equation
is singular at $x=y$ and at leading order the dominant balance leads
to an equation where the other points are spectators. One finds that
the only conformal families that can appear are the ones whose
conformal weight $\delta'$ satisfies the fusion rule.
This is enough CFT background for the rest of this paper. We are now
in position to give the heuristic argument that leads to our main claims.
\section{Martingales from statistical mechanics}
\label{sec:StatMech}
The purpose of this section is to emphasize the intimate connection
between the basic rules of statistical mechanics and martingales. The
connection is somehow tautological, because statistical mechanics
works with partition functions, i.e. unnormalized probability
distributions, all the time. In the discrete setting, this makes
conditional expectations a totally transparent operation that one
performs without thinking and even without giving it a name. But the
following argument is, despite its simplicity and its abstract
nonsense flavor, the crucial observation that allows us to relate
interfaces in conformally invariant statistical mechanics to SLEs.
\subsection{Tautological martingales}
Consider a model of statistical mechanics with a finite but
arbitrarily large set of possible states $S$. Usually one starts with
models defined on finite grid domains so $\# S < \infty$ is natural.
To each state $s \in S$ we associate a Boltzmann
weight\footnote{Usually the Boltzmann weight is related to the energy
$H(s)$ of the state $s$ through $w(s) = \exp ( - \beta H(s))$, where
$\beta$ is the inverse temperature (a Lagrangian multiplier related
to temperature, anyway).} $w(s)$. The partition function is $Z =
\sum_{s \in S} w(s)$ so that it normalizes the Boltzmann weights to
probabilities, $\prob \{ s \} = \frac{w(s)}{Z}$. Since $S$ is finite,
we can use the power set $\sP (S) = \{ U : U \subset S \}$ as a sigma
algebra. The expected value of a random variable $\Oper : S
\rightarrow \bC$ is denoted by $\expect [\Oper] = \bra \Oper \ket =
\frac{1}{Z} \sum_{s \in S} \Oper(s) w(s)$.
Note that if $(S_\alpha)_{\alpha \in I}$ is a collection of disjoint
subsets of $S$ such that $\cup_{\alpha \in I} S_\alpha = S$, then the
collection of all unions $\sF = \{ \cup_{\alpha \in I'} S_\alpha : I'
\subset I \}$ is a sigma algebra on $S$. Conversely, since $S$ is
finite, any sigma algebra $\sF$ on $S$ is of this type.
Consider a filtration, that is an increasing family $(\sF_t)_{t \geq
0}$ of sigma algebras $\{ \emptyset , S \} \subset \sF_s \subset
\sF_t \subset \sP(S)$ for all $0 \leq s < t$. Denote the
corresponding collections of disjoint sets by
$(S^{(t)}_{\alpha})_{\alpha \in I_t}$ and define the partial partition
function $Z^{(t)}_{\alpha}\equiv \sum_{s \in S^{(t)}_\alpha} w(s)$.
The conditional expectation values
\begin{eqnarray*}
\bra \Oper \ket_t & \equiv & \expect [ \Oper | \sF_t]=
\sum_{\alpha \in I_t} \frac{\sum_{s \in S^{(t)}_\alpha}
\Oper(s) w(s)}
{\sum_{s \in S^{(t)}_\alpha} w(s)} \; \unit_{S^{(t)}_\alpha} \\
& = & \sum_{\alpha \in I_t} \Big( \frac{1}{Z^{(t)}_{\alpha}}
\sum_{s \in S^{(t)}_\alpha} \Oper(s) w(s) \Big) \;
\unit_{S^{(t)}_\alpha}
\end{eqnarray*}
are martingales by definition: for $s < t$ we have
\begin{eqnarray*}
\expect \big[ \; \expect [ \Oper | \sF_t] \; \big| \sF_s \big]
= \expect [ \Oper | \sF_s]
\end{eqnarray*}
Notice that the probability of the event $S^{(t)}_\alpha$ is
conveniently $\prob [S^{(t)}_{\alpha}] = Z^{(t)}_\alpha / Z$.
Suppose that the model is defined in a domain $D \subset \bC$ and that
there are interfaces in the model. Parametrize portions of these
interfaces touching the boundary by an arbitrary ``time'' parameter
$t$ in such a way that $n$ paths $\gamma^{(i)}_t$, $i=1,\cdots,n$
(which are pieces of the random interfaces) emerge from the boundary
at $t=0$ and are disjoint at least when $t$ is small enough, see
figure \ref{fig:statmech}. To avoid
confusion we write the time parameter $t$ now as a subscript and
continue to indicate the dependence of $s \in S$ by parenthesis, so $t
\mapsto \gamma^{(i)}_t (s)$ is a parametrization of the $i^{\textrm{th}}$ piece
of interface if the system is at state $s$. Then we can consider the
natural filtration of the interface by taking $\sF_t = \sigma (
\gamma^{(i)}_{t'} : 0 \leq t' \leq t, \; i=1,\cdots,n )$ to be the
sigma algebra generated by the random variables $\gamma^{(i)}_{t'}$ up
to time $t$.
\begin{figure}
\center{\includegraphics[width=0.6\textwidth]{statmech.eps}}
\caption{\emph{A discrete statistical mechanics model with
portions of interfaces specified.}}
\label{fig:statmech}
\end{figure}
The boundary conditions of the model are often
such that conditioning on the
$\gamma^{(i)}_{[0,t]}$, is the same as considering the model in a
smaller domain (a part of the interface removed) but with
same type of boundary conditions. Of course the position at which
the new interface should start is where the original interface
would have continued, that is the $\gamma^{(i)}_t$'s. Let $D_t$ be the
domain $D$ with the $\gamma^{(i)}_{]0,t]}$ removed.
The starting point of the next section is the input of conformal
invariance in this setup.
\subsection{Simplifying tautological martingales}
We start from the situation and notations at the end of the previous
section.
If in addition we are considering a model at
its critical point, then the continuum limit may be described
by a conformal field theory. At least for a wide class of natural
observables $\Oper$, the expectation values become CFT correlation
functions in the domain $D$ of the model
\begin{eqnarray*}
\bra \Oper \ket = \frac{\sum_{s \in S} \Oper(s) w(s)}{Z}
\longrightarrow \frac{\bra \Oper \ket^{\textrm{CFT, b.c.}}_{D}}
{\bra \unit \ket^{\textrm{CFT, b.c.}}_{D}}
\end{eqnarray*}
We need to write the correlation function of identity (proportional to
$Z$) in the denominator because the boundary conditions (b.c.) of the
model may already have led to insertions of boundary changing
operators that we have not mentioned explicitly.
The closed martingales become
\begin{eqnarray*}
\bra \Oper \ket_t = \sum_{\alpha \in I_t} \frac{1}{Z^{(t)}_\alpha}
\sum_{s \in S^{(t)}_\alpha} \Oper(s) w(s) \; \unit_{S^{(t)}_\alpha}
\longrightarrow
\frac{\bra \Oper \ket^{\textrm{CFT, b.c.}}_{D_t}}
{\bra \unit \ket^{\textrm{CFT, b.c.}}_{D_t}}
\end{eqnarray*}
where in the continuum limit $D_t$ might be $D$ with hulls (and not
only traces) removed.
For certain (but not all) observables, $\bra \Oper \ket$ is computing
a probability, which in a conformal field theory ought to be
conformally invariant. But $\bra \Oper \ket$ is written as a quotient,
and this means that the numerator and denominator should transform
homogeneously (and with the same factor) under conformal
transformations. In particular, the denominator should transform
homogeneously. This means that $\bra \unit \ket^{\textrm{CFT,
b.c.}}_{D}$ -- which depends on the position of the boundary
condition changes -- behaves like a product of boundary primary fields.
Then, by locality, for any $\Oper$, the transformation of the
numerator under conformal maps will split in two pieces: one
containing the transformations of $\Oper$ and the other one
canceling with the factor in the denominator. So we infer the
existence in the CFT of a primary boundary field, denoted by $\psi (x)$
in what follows, which implements boundary condition changes at which
interfaces anchor. Hence we may write
$$\bra \unit \ket^{\textrm{CFT, b.c.}}_{D} = \bra \psi (X^{(1)})
\cdots \psi (X^{(n)})\ket^{\textrm{CFT}}_{D}$$
and
$$\bra \Oper \ket^{\textrm{CFT, b.c.}}_{D} = \bra \Oper \psi (X^{(1)})
\cdots \psi (X^{(n)})\ket^{\textrm{CFT}}_{D}.$$
As will become clear later, there might also be one further boundary operator
anchoring several interfaces. We do not mention it explicitly here
because it will sit at a point which will not be affected
by the conformal transformations that we use.
Write the transformation of the observable $\Oper$ as $f:\Oper
\rightarrow \; ^f \!\Oper$ under a conformal map. Denote by $f_t$ a
conformal representation $f_t:D_t\rightarrow D$ and write
$f(\gamma^{(i)}_{t}) \equiv X^{(i)}_{t}$. The expression for the closed
martingale $\bra \Oper \ket_t$ can now be simplified further
\begin{equation}
\label{eq:martCFT}
\bra \Oper \ket_t \longrightarrow \frac{\bra \, ^{f_t}\Oper\, \psi
(X^{(1)}_t) \cdots \psi (X^{(1)}_t)\ket^{\textrm{CFT}}_{D}} {\bra
\psi (X^{(1)}_t) \cdots \psi (X^{(1)}_t)
\ket^{\textrm{CFT}}_{D}}.
\end{equation}
The Jacobians coming from the transformations of the boundary changing
primary field $\psi$ have canceled in the numerator and denominator.
The explicit value of the conformal weight of $\psi$ does not appear
in this formula.
Of course, we have cheated. For the actual map $f_t$ which is
singular at the $\gamma^{(i)}_{t}$'s these Jacobians are infinite. A
more proper ``derivation'' would go through a regularization but
locality should ensure that the naive formula remains valid when the
regularization is removed. Eq.(\ref{eq:martCFT}) is the starting
point of our analysis.
\section{Derivation of the proposal}
The heuristics we follow is to describe a growth process of interfaces
by a Loewner chain $f_t$ compatible with conformal invariance in that
the right hand side of eq.(\ref{eq:martCFT}) is a martingale.
\subsection{The three ingredients}
\noindent \textbf{Loewner chain}: If we use the upper half plane as a
domain, $D={\mathbb H}$, and
impose the hydrodynamic normalization, the equation for $f_t$ has to
be of the form
$$\ud f_t(z)=\sum_i \frac{2\ud q^{(i)}_t}{f_t(z)-X^{(i)}_t}$$
for some processes $X^{(i)}_t$, $i=1,\ldots ,n$. The initial condition
is $f_0(z)=z$.
\vspace{.3cm}
\noindent \textbf{Interfaces grow independently of each other on very
short time scales}: Schramm's argument deals with the case of a
single point. We expect that on very short time scales the growth
processes do not feel each other and Schramm's argument is still
valid, so that $\ud X^{(i)}_t=\ud M^{(i)}_t + F^{(i)}_t $ where the
$M^{(i)}_t$'s are $n$ (continuous,local) martingales with quadratic variation
$\kappa q^{(i)}_t$ and vanishing cross variation and $F^{(i)}_t$ is a
drift term.
\vspace{.3cm}
\noindent \textbf{The martingale property fixes the drift term}: The
drift term will be computed by imposing the martingale condition on
$\bra \Oper \ket_t$ when $\Oper$ is a product of an arbitrary number
$l$ of boundary primary fields $\Oper =\prod_{\alpha=1}^{l}
\varphi_{\delta_\alpha}(Y^{(\alpha)})$. The insertion points are away
from the boundary changing operators and $f_t$ is regular with
positive derivative there. Substitution of $^{f_t}\Oper$ in formula
(\ref{eq:martCFT}) yields
\begin{equation}
\label{eq:marto}
\bra \prod_{\alpha=1}^{l}
\varphi_{\delta_\alpha}(Y^{(\alpha)})\ket_t = \frac{\bra
\prod_{\alpha=1}^{l}
\varphi_{\delta_\alpha}(f_t(Y^{(\alpha)}))
\prod_{i=1}^{n}\psi (X^{(i)}_t) \ket^{\textrm{CFT}}_{D}}
{\bra \prod_{i=1}^{n}\psi (X^{(i)}_t) \ket^{\textrm{CFT}}_{D}}
\prod_{\alpha=1}^{l} f'_t(Y^{(\alpha)})^{\delta_\alpha}.
\end{equation}
\subsection{Computation of the Ito derivative of $\bra
\prod_{\alpha=1}^{l} \varphi_{\delta_\alpha}(Y^{(\alpha)})\ket_t$}
In formula (\ref{eq:marto}), denote respectively by $Z_t^{\varphi}$,
$Z_t$ and $J^{\varphi}_t$ the numerator, denominator and Jacobian
factor on the right hand side.
It is useful to break the computation of $\ud \bra
\prod_{\alpha=1}^{l} \varphi_{\delta_\alpha}(Y^{(\alpha)})\ket_t$ in
several steps.
\vspace{.3cm}
-- \textit{Preliminaries}. \\
Ito's formula for the $\psi$'s gives
$$\ud \psi(X^{(i)}_t)=\psi'(X^{(i)}_t)(\ud M^{(i)}_t +
F^{(i)}_t)+\frac{\kappa}{2} \psi''(X^{(i)}_t)\ud q^{(i)}_t. $$
\noindent Using
the Loewner chain for $f_t(z)$ and its derivative with respect to $z$,
one checks that
$$\ud
\left(\varphi_{\delta}(f_t(Y))f'_t(Y)^\delta\right)=f'_t(Y)^\delta
\sum_i 2\ud
q^{(i)}_t\left(\frac{\varphi'_{\delta}(f_t(Y))}{f_t(Y)-X^{(i)}_t}-
\frac{\delta\varphi_{\delta}(f_t(Y))}{(f_t(Y)-X^{(i)}_t)^2}\right).$$
\vspace{.2cm}
-- \textit{The Ito derivative of $Z_t^{\varphi}J^{\varphi}_t$}.\\
The time $t$ being given, we can simplify the notation. Set $x_i \equiv
X^{(i)}_t$ and $y_{\alpha} \equiv f_t(Y^{(\alpha)})$ and apply the
chain rule to get
\begin{eqnarray*}
\frac{\ud (Z_t^{\varphi}J^{\varphi}_t)}{J^{\varphi}_t} & = &
\left[\sum_i \left(\ud M^{(i)}_t +
F^{(i)}_t\right)\partial_{x_i}\right. \\
& & \hspace{-2cm} + \left. \sum_i \ud q^{(i)}_t\left(\frac{\kappa}{2}
\partial_{x_i}^2+2\sum_{\alpha}\left[
\frac{1}{y_{\alpha}-x_i}
\partial_{y_{\alpha}}-\frac{\delta_{\alpha}}
{(y_{\alpha}-x_i)^2}\right]\right)\right]Z_t^{\varphi}
\end{eqnarray*}
\vspace{.2cm}
-- \textit{First use of the null-vector equation : identification
of $\psi$}. \\
Let us concentrate for a moment on the familiar chordal SLE case, for
which $n=1$. The drift term $F_t^{(1)}$ is known to be zero. The
boundary conditions also change at $\infty$ (the endpoint of the
interface) and there is an operator there, that we have not written
explicitly because the notation is heavy enough. Anyway, $Z_t$ is a
two-point function with one of the fields at infinity, so it is a
constant. For chordal SLE, the drift term in the Ito derivative of the
putative martingale vanishes if and only if
$$\left(\frac{\kappa}{2}
\partial_{x}^2+2\sum_{\alpha}\left[
\frac{1}{y_\alpha-x}
\partial_{y_\alpha}-\frac{\delta_{\alpha}}
{(y_\alpha-x)^2}\right]\right)Z_t^{\varphi}=0,$$
where for simplicity we have written $x \equiv x_1$.\\
Comparison with eq.(\ref{eq:sing}) implies that $\psi$ has a vanishing
descendant at level two and has conformal weight
$h=\frac{6-\kappa}{2\kappa}\equiv h_1(\kappa)$ :
$$\psi (x)\equiv \varphi_{h_1(\kappa)}(x).$$
The central charge is
$c=\frac{(6-\kappa)(3\kappa-8)}{16\kappa}$. \\
This is of course nothing but the correlation function formalism
version of the original argument relating SLE to CFT, which was given
in the operator formalism, see \cite{Bb:2002tf}.
\vspace{.2cm}
-- \textit{Second use of the null-vector equation}.\\
Now that $\psi$ has been identified, we can return to the general
case, with an arbitrary number $n$ of growing curves. Each growing
curve has its own field $\psi$ and each field $\psi$ comes with its
differential equation, which is eq.(\ref{eq:sing}) but for $l+n-1$
spectator fields, the $l$ fields $\varphi$ and the $n-1$ other
$\psi$'s. So $Z_t^{\varphi}$ is annihilated by the $n$ differential
operators
$$
\frac{\kappa}{2}
\partial_{x_i}^2+2\sum_{\alpha}
\left[ \frac{1}{y_\alpha-x_i}
\partial_{y_\alpha}-\frac{\delta_{\alpha}}
{(y_\alpha-x_i)^2}\right] +2\sum_{j \neq i}\left[
\frac{1}{x_j-x_i}
\partial_{x_j}-\frac{h_1(\kappa)}
{(x_j-x_i)^2}\right].
$$
We can use this to get a simplified formula
$$
\ud (Z_t^{\varphi}J^{\varphi}_t)= J^{\varphi}_t \, {\mathcal
P}Z_t^{\varphi}\; , \qquad \ud Z_t={\mathcal P} Z_t$$
where ${\mathcal P}$ is the first order differential operator
$$
\sum_i \left[
\left(\ud M^{(i)}_t + F^{(i)}_t\right)\partial_{x_i}- 2\ud q^{(i)}_t
\left(\sum_{j \neq i} \left[\frac{1}{x_j-x_i}\partial_{x_j}
-\frac{h_1(\kappa)}{(x_j-x_i)^2}\right] \right)\right].
$$
The formula for $Z_t$ is just the special case $l=0$.
\vspace{.2cm}
-- \textit{Final application of Ito's formula}.\\
$$
\ud \left(\frac{Z_t^{\varphi}}{Z_t}J^{\varphi}_t
\right)=J^{\varphi}_t {\mathcal
Q}\left(\frac{Z_t^{\varphi}}{Z_t}\right)
$$
where ${\mathcal Q}$ is the first order differential operator
$$\sum_i \left[\ud M^{(i)}_t + F^{(i)}_t- \kappa \ud
q^{(i)}_t(\partial_{x_i}\log Z_t) - 2 \sum_{j \neq i} \frac{\ud
q^{(j)}_t}{x_i-x_j}\right]\partial_{x_i}$$
The martingale property is satisfied if and only if the drift terms vanish.
\subsection{Main claim}
To summarize, we have shown that the system
$$\ud f_t(z)=\sum_i \frac{2\ud q^{(i)}_t}{f_t(z)-X^{(i)}_t} \quad ,
\quad \ud X^{(i)}_t=\ud M^{(i)}_t + F^{(i)}_t$$
admits conditioned
correlation functions from CFT as martingales if and only if
$$F^{(i)}_t=\kappa \ud
q^{(i)}_t(\partial_{x_i}\log Z_t) + 2 \sum_{j \neq i} \frac{\ud
q^{(j)}_t}{x_i-x_j}.$$
where $Z_t$ is a partition function.
It is under this condition that it describes the growth of $n$
interfaces in a way compatible with statistical mechanics and
conformal field theory.
\vspace{.3cm}
In fact, we have used a special family of correlators. But the same
argument applies to all operators (hence the ``if'' part). Of special
interest in the sequel will be the case when $\Oper$ is a topological
observable, for instance taking value $1$ if the interface forms a
given arch system and $0$ otherwise. No Jacobian is involved for such
observables and the numerator looks again like a partition function.
\subsection{The moduli space}
From the definition of $Z_t$ as a correlation of primary fields with
null descendants at level $2$, it is clear that properties
$i),\;ii),\;iii)$ are satisfied, except maybe for the quantization of
the possible scaling dimensions of $Z_t$, to which we turn now.
This is standard material from CFT and we include it here for
completeness.
The correlator $\bra \varphi_{h_{\infty}} (\infty)\psi(x_1)
\cdots \psi(x_n)\ket$ on the real line satisfies $n$
differential equations. We shall recall why the space of simultaneous
solutions which have global conformal invariance has dimension
$${n \choose m}-{n \choose m-1}=(n+1-2m)\frac{n!}{m!(n-m+1)!}$$
if $h_{\infty}= h_{n-2m}(\kappa)$ for some nonnegative integer
$m \leq n/2$ and has dimension $0$ otherwise. This will end
the derivation of our proposal and match the counting of arches.
At the end of the background on conformal invariance, we mentioned
fusion rules: when $\varphi_{h_{1}(\kappa)}(x)$ and a
$\varphi_{h_{j}(\kappa)}(y)$ are brought close together, they can be
expanded in a basis of local operators that can be grouped in
conformal families. We also recalled why the weight $h'$ of the
primaries in each conformal family had to satisfy
$3(h_{j}(\kappa)-h')^2-(2h_{1}(\kappa)+1)(h_{j}(\kappa)+h')=
h_{1}(\kappa)(h_{1}(\kappa)-1),$ so that only two conformal families
can appear in a fusion with $\varphi_{h_{1}(\kappa)}$. The two
conformal weights are easily found to
be $h'=h_{j\pm 1}(\kappa)$. Furthermore, $h_0(\kappa)=0$ and one can
show that the corresponding field has to be the boundary identity
operator. By global conformal invariance, the only local operator with
a non vanishing one point correlator is the identity and boundary two
point functions vanish unless the two local fields have the same
conformal weight. This takes care of the counting and selection rules
for the $n=0,1$ cases.
One proceeds by recursion. The points are ordered $x_1< x_2 \cdots <
x_n$. If $n\geq 2$ then move $x_2$ close the $x_1$ (for instance by a
global conformal transformation) and fuse to get an expansion for
local fields at $x_1$ say. Only the conformal families of
$\varphi_{h_{1\pm 1}(\kappa)}$ appear. If $n=2$ this fixes the weight
of the field at $\infty$. If $n \geq 3$, iterate. This leads
immediately to the selection rules mentioned above : the field at
infinity has to be a $\varphi_{h_{n-2m}(\kappa)}$. The dimension is
nothing but the number of path of $n$ steps $\pm 1$ from $0$ to $n-2m$
on the nonnegative integers, a standard combinatorial problem whose
answer is ${n \choose m}-{n \choose m-1}$. The efficient way to do the
counting is by the reflection principle. The possible outcomes of each
fusion can be encoded in a so-called Bratelli diagram:
\begin{equation}
\label{eq: fusion diagram}
\left. \begin{array}{ccccccccccc}
& & & & & & & & & & \cdots\\
& & & & & & & & & \nearrow & \\
& & & & & & & & h_{4}(\kappa) & & \\
& & & & & & & \nearrow & & \searrow & \\
& & & & & & h_{3}(\kappa) & & & & \cdots \\
& & & & & \nearrow & & \searrow & & \nearrow & \\
& & & & h_{2}(\kappa) & & & & h_{2}(\kappa) & & \\
& & & \nearrow & & \searrow & & \nearrow & & \searrow & \\
& & h_{1}(\kappa) & & & & h_{1}(\kappa) & & & & \cdots \\
& \nearrow & & \searrow & & \nearrow & & \searrow & & \nearrow & \\
h_{0}(\kappa) & & & & h_{0}(\kappa) & & & & h_{0}(\kappa) & & \\
& & \quad & & \quad & & \quad & & \quad & & \\
& & \textrm{$1$SLE}& & \textrm{$2$SLE} & & \textrm{$3$SLE} & & \textrm{$4$SLE} & & \cdots \\
\end{array} \right.
\end{equation}
This is totally parallel to the discussion of composition of $n$ spins
$1/2$ for the representation theory of the Lie algebra of rotations.
The multiplicity is exactly one when $m=0$ which corresponds to the
partition function (\ref{eq:dabdouwa}) and to the insertion of the
operator $\varphi_{h_{n}(\kappa)}$ at infinity, toward which the $n$
interfaces run.
What is not proved here is
that the different paths lead to a basis of solutions of the $n$
partial differential equations, but it is true. Each path corresponds
to a succession of choices of a single conformal family, one at each
fusion step. Let us mention in advance that multi SLE processes, i.e.
the consideration of multiple interfaces, will lead to the definition
of another basis with a topological interpretation.
\section{Multiple SLEs describing several interfaces}
\label{sec:sevinterf}
\subsection{Double SLEs}
\label{sec: double SLE}
The case of double SLEs is instructive and simple to analyze.
Although double SLEs is sometimes interesting for its own sake,
the purpose of this section is to give easy examples to guide
the study of the general case.
\subsubsection{2SLEs and Bessel processes}
To specify the process we have to specify the partition function $Z$.
There are only two possible choices corresponding to two different
type of boundary conditions, or alternatively to two different fields
inserted at infinity:
\begin{eqnarray*}
\bra h_\infty|\psi(X_1)\psi(X_2) | 0 \ket
& = & \const \times (X_1 - X_2)^{\Delta}
\end{eqnarray*}
where the exponent is $\Delta = h_\infty - 2 h_{1}(\kappa)$ and the
constant will be fixed to $1$ from now on. According to CFT fusion
rules, $h_\infty$ can only be either $h_{2}(\kappa) =
\frac{8-\kappa}{\kappa}$ or $h_{0}(\kappa)=0$. The exponent becomes
$\Delta = 2/\kappa$ or $\Delta = \frac{\kappa-6}{\kappa}$
respectively, so that we have two basic choices for $Z$:
$$
Z_0\equiv (X_1-X_2)^{(\kappa-6)/\kappa}\quad {\rm or}\quad
Z_2 \equiv (X_1-X_2)^{2/\kappa}
$$
As we shall see, choosing $Z_0$ selects configurations with no
curve ending at infinity -- so that we are actually describing
standard chordal SLE joining to the two initial positions of $X_1$ and
$X_2$ -- while choosing $Z_2$ selects configurations with two curves
emerging from the initial positions of $X_1$ and $X_2$ and ending both
at infinity.
Up to normalizing the quadratic variation by $dq^{(i)}_t= a_i dt$ so
that the martingales $M^{(i)}$ are simply
$dM^{(i)}_t=\sqrt{\kappa a_i}dB_t^{(i)}$
with $dB_t^{(i)}$ two independent normalized Brownian motions,
our double SLE equations become~:
\begin{eqnarray*}
df_t(z) & = & \frac{2 a_1\;\ud t}{f_t(z) - X^{(1)}_t}
+ \frac{2 a_2\;\ud t}{f_t(z) - X^{(2)}_t} \\
d X^{(1)}_t & = & \sqrt{a_1 \kappa} \; d B^{(1)}_t +
\frac{2 a_2 + \kappa \Delta a_1}{X^{(1)}_t - X^{(2)}_t} \; \ud t \\
d X^{(2)}_t & = & \sqrt{a_2 \kappa} \; d B^{(2)}_t +
\frac{2 a_1 + \kappa \Delta a_2}{X^{(2)}_t - X^{(1)}_t} \; \ud t
\end{eqnarray*}
It describes two curves emerging from points $X_1= X^{(1)}_0$ and
$X_2=X^{(2)}_0$ at speeds parametrized by $a_1$ and $a_2$.
Up to an irrelevant translation, the process is actually driven by the
difference $Y_t=X^{(1)}_t-X^{(2)}_t$. Up to a time change, $ds =
\kappa (a_1 + a_2)dt$, this is a Bessel process,
\begin{eqnarray*}
dY_s = \ud \tilde{B}_s +
\frac{\Delta + 2/\kappa}{Y_s} \; \ud s,
\end{eqnarray*}
of effective dimension $d_{\rm eff}=1 + 2\Delta + 4/\kappa$. For
$h_\infty = h_{2}(\kappa)$ (i.e. $\Delta=2/\kappa$) the dimension is
$d_{\rm eff}=1+ 8/\kappa$ and for $h_\infty = 0$
(i.e. $\Delta=(\kappa-6)/\kappa$) it is $d_{\rm eff}=3 - 8/\kappa$.
In the physically interesting parameter range $\kappa < 8$,
the former is $> 2$ and the latter is $< 2$.
Recall now that a Bessel process is recurrent (not recurrent) if its
effective dimension is less (greater) than $2$.
Thus, the driving processes $X^{(i)}_t$ hit each other almost
surely in the case $h_{\infty} = 0$ and they don't hit (a.s.)
in the case $h_\infty =h_{2}(\kappa)$. Since the hitting of driving
processes means hitting of the SLE traces, this teaches us that case
$h_\infty=0$ describes a single curve joining $X_1$ and $X_2$ while case
$h_\infty=h_{2}(\kappa)$ describes two curves converging toward infinity.
Notice that previous results are independent of $a_1$ and $a_2$,
provided their sum does not vanishes.
We also observe that setting $a_1 = 1$ and $a_2 = 0$ (or vice versa)
one recovers an SLE$(\kappa; \kappa \Delta)$. Recall that if
$h_\infty = 0$ then $\rho = \kappa \Delta = \kappa - 6$ corresponds
to an ordinary chordal SLE from $X_1$ to $X_2$. Our
double SLEs with $h_\infty = 0$ corresponds to one chordal SLE seen
from both ends and the fact that the tips of the traces hit is
natural. The other case, $h_\infty = h_{2}(\kappa)$ corresponds to
$\rho = \kappa \Delta = 2$ and since the driving processes can not
hit, the process can be defined for all $t \geq 0$. Assuming
that $\int_0^\infty ( a_1 + a_2) \ud t = \infty$, the
capacity of the hulls grow indefinitely and (at least one of) the
SLE traces go to infinity.
The two possible geometries are illustrated in figure \ref{fig: 2SLEs}.
\begin{figure}
\includegraphics[width=1.0\textwidth]{doubleSLEs.eps}
\caption{\emph{The two geometries for 2SLE: on the left is the case
$h_\infty = 0$ and on the right $h_\infty = h_2 (\kappa)$.}}
\label{fig: 2SLEs}
\end{figure}
\subsubsection{A mixed case for 2SLE}
\label{sec: mixed 2SLE}
Because of its simplicity, we use double SLE as a testing ground for
mixed correlation functions. So we consider the sum
$$ Z = \lambda Z_0 + \mu Z_2$$
with both $\lambda$ and $\mu$ positive. As already mentioned, the
interpretation of $Z$ as the continuum limit of partition functions of
lattice models is unclear since $Z_0$ and $Z_2$ do not scale the same way.
We nevertheless study it to illustrate ways of
computing (arch or geometry) probabilities.
As one may expect, we no longer have an almost sure global geometry
but rather nontrivial probabilities for the two geometries: either no
curve at infinity or two curves converging there.
Let $\tau = \inf \{ t \geq 0 : X_t^{(1)} = X_t^{(2)} \}$ be the
stopping time which indicates the hitting of the driving processes --
and thus of the two curves. We can define the driving processes as
solutions of the 2SLE system on the (random) time interval $t \in [0,
\tau)$. At the stopping time we define $f_\tau(z) = \lim_{s \uparrow
\tau} f_s(z)$ for such $z \in \bH$ that the limit exists and stays
in the half plane $\bH$. The hull $K_\tau$ is defined as the set where
the limit doesn't exist or hits $\bdry \bH$.
The question of geometry is answered by the knowledge of whether
the two traces hit, that is whether $\tau < \infty$ or not.
Thus we again consider the difference
$Y_t = X^{(1)}_t - X^{(2)}_t$, whose Ito derivative is now~:
\begin{eqnarray*}
\ud Y_t & = & \sqrt{\kappa} \ud \tilde B_t + \frac{2}{Y_t}(a_1+a_2)\ud t
+ \frac{(\kappa - 6) \lambda Y_t^{\frac{\kappa-6}{\kappa}} +
2 \mu Y_t^{\frac{2}{\kappa}}}{Y_t(\lambda Y_t^{\frac{\kappa-6}{\kappa}} +
\mu Y_t^{\frac{2}{\kappa}} )} \; (a_1+a_2)\ud t
\end{eqnarray*}
with $\tilde B_t=\sqrt{a_1}B^{(1)}_t-\sqrt{a_2}B^{(2)}_t$ is a
Brownian motion, so that after a time change, $\ud s= (a_1+a_2)\ud t$,
the result doesn't depend on $a_1$ or $a_2$. The last drift term
comes from the derivative of $\log Z$.
One might for example try to find the distribution of $\tau$ by its
Laplace transform $\expect_{Y_0 = y} [ e^{-\beta \tau} ] =
f_\beta(y)$. By Markov property,
\begin{eqnarray*}
\expect [ e^{-\beta \tau} | \sF_t] = e^{-\beta t} f_\beta (Y_t)
\end{eqnarray*}
is a closed martingale on $t \in [0, \tau)$ so requiring its Ito drift
to vanish leads to the differential equation
\begin{eqnarray*}
\Big( -\frac{\beta}{a_1 + a_2} + \big( \frac{2}{y} + \frac{(\kappa-6)
\lambda + 2 \mu y^{(8-\kappa)/\kappa}}{\lambda
+ \mu y^{(8-\kappa)/\kappa}} \big) \partial_y
+ \frac{\kappa}{2} \partial_y^2 \Big) f_\beta (y) = 0
\end{eqnarray*}
The result depends only on $\beta / (a_1 + a_2)$. We conclude that
the distribution of $(a_1 + a_2) \tau$,
the capacity of the final hull $K_\tau$,
is independent of the speeds of
growth $a_1$ and $a_2$. Also the result depends on $\lambda$ and
$\mu$ only through $\mu / \lambda$.
In particular we want to take $\beta \downarrow 0$ to compute
the probability that the traces hit. Constant functions solve the
differential equation but another linearly independent solution
has the correct boundary values $f_0 (0)=1$ and $f_0 (\infty)=0$,
namely
\begin{eqnarray*}
\prob_{Y_0 = y} [\tau < \infty] = \lim_{\beta \downarrow 0}
\expect_{Y_0 = y} [e^{-\beta \tau}]
= \frac{\lambda}{\lambda+{\mu} y^{(8-\kappa)/\kappa}}
\end{eqnarray*}
As expected on general ground, this is the fraction of the two
partition functions $\lambda Z_0$ and $Z = \lambda Z_0 + \mu Z_2$.
\subsection{Triple and/or quadruple SLEs}
We will give a few more of examples of multiple SLEs.
Certain triple and quadruple SLEs are the scaling limits of
interfaces in percolation and Ising model with rather natural
boundary conditions. These models will be considered in
section \ref{sec: Ising}. Here we study triple and quadruple
SLEs for their own sake. We restrict ourselves to $\kappa<8$.
\subsubsection{3SLE (pure) configurations}
Partition functions with $n=3$ have only two possible scaling
behaviors depending whether the weight $h_\infty$ of the field at
infinity equals either to $h_{3}(\kappa)=\frac{3(10-\kappa)}{2\kappa}$
or to $h_{1}(\kappa)=\frac{6-\kappa}{2\kappa}$. This follows from CFT
fusion rules. For reasons already explained we shall not mixed them.
The case $h_\infty=h_{3}(\kappa)$ is the simplest.
There is only one possible partition function with this scaling,
namely
$$ [(X_2-X_1)(X_3-X_1)(X_3-X_2)]^{2/\kappa}$$
It is expected to correspond to configurations with three curves starting
at initial positions $X_1,\ X_2$ and $X_3$ and converging toward infinity.
The case $h_\infty=h_{1}(\kappa)$ is more interesting since the space of
such partition functions is of dimension two and coincides with the
space of conformal block with 4 insertions of boundary operators
$\psi$, with one localized at $X_4=\infty$:
$$
\langle \psi(X_4)\psi(X_3)\psi(X_2)\psi(X_1)\rangle
$$
We assume the points to be ordered $X_1<X_2<X_3<X_4$. The
associated process should describe a family of two curves joining any
pair of adjacent points without crossing. There are thus two possible
topologically distinct geometries: either the curves join the pairs
$[X_1X_2]$ and $[X_3X_4]$ or they join $[X_4X_1]$ and $[X_2X_3]$,
see figure \ref{fig: 3SLEs}. As expected, the number of topologically distinct
configuration equals that of conformal blocks, namely two. Notice
that the last process is the same as a 4SLE but with the speed $a_4$
vanishing, see figure \ref{fig: 4SLEs}.
\begin{figure}
\includegraphics[width=1.0\textwidth]{tripleSLE.eps}
\caption{\emph{For $h_\infty=h_1(\kappa)$ the curves of 3SLE join either
$[X_1 X_2]$ and $[X_3 X_4]$ (on the left) or $[X_4 X_1]$ and
$[X_2 X_3]$ (on the right).}}
\label{fig: 3SLEs}
\end{figure}
\begin{figure}[b]
\includegraphics[width=1.0\textwidth]{zigzig.eps}
\caption{\emph{Arch configurations for four SLE processes in
an arbitrary domain.}}
\label{fig: 4SLEs}
\end{figure}
By conformal invariance we may normalize the points so that $X_1=0$,
$X_2=x$, $X_3=1$ and $X_4=\infty$ with $0<x<1$. We have two distinct
topological configurations and we thus have to identify the two
corresponding pure partition functions. This is will be done by
specifying the way the partition functions behave when points are
fused together. By construction these partition functions may be
written as correlation functions
$$Z(x)=\langle\psi(\infty)\psi(1)\psi(x)\psi(0)\rangle$$
so that their behavior when points are fused are governed by CFT
fusion rules. As a consequence, $Z(x)$ behave either as
$x^{\frac{\kappa-6}{\kappa}}$ or as $x^{\frac{2}{\kappa}}$ as $x\to 0$.
We select the pure partition functions $Z_I$ and $Z_{II}$ by demanding that:
\begin{eqnarray}
Z_I(x) &=& x^{\frac{\kappa-6}{\kappa}}\times [1+\cdots],\quad
~~~~~~~~~~~~ {\rm as}\ x\to0
\label{bdrypure}\\
&=& (1-x)^{\frac{2}{\kappa}}\times [{\rm const.}+\cdots],\quad
~ {\rm as}\ x\to 1 \nonumber
\end{eqnarray}
and $Z_{II}(x)=Z_I(1-x)$ so that
\begin{eqnarray*}
Z_{II}(x) &=& x^{\frac{2}{\kappa}}\times [{\rm const.}+\cdots],\quad
~~~~~~~~~~ {\rm as}\ x\to 0 \\
&=& (1-x)^{\frac{\kappa-6}{\kappa}}\times [1+\cdots],\quad
~~~~~~ {\rm as}\ x\to 1
\end{eqnarray*}
$Z_I$ will turn out to be the pure partition function for configurations in
which the curves join the pairs $[0x]$ and $[1\infty]$ while $Z_{II}$
will turn out to correspond to the configurations $[x1]$ and $[\infty 0]$.
The rationale behind these conditions consists in imposing that the
pure partition function possesses the leading singularity, with
exponent $(6-\kappa)/\kappa$, when $x$ is approaching the point allowed
by the configuration but has subleading singularity, with exponent
$2/\kappa$, when $x$ is approaching the point forbidden by the
configuration.
This set of conditions uniquely determines the functions $Z_I$ and
$Z_{II}$. These follows from CFT rules but may also be checked by
explicitly solving the differential equation that these functions
satisfy. Writing $Z(x)=x^{2/\kappa}(1-x)^{2/\kappa}\; G(x)$ yields,
$$ \kappa^2 x(1-x) G''(x) + 8\kappa (1-2x) G'(x) -
4(12-\kappa) G(x)=0 $$
so that $G(x)$ is an hypergeometric function and
$$
Z_{II}(x) = {\rm const.} x^{2/\kappa}(1-x)^{2/\kappa}\;
F(\frac{4}{\kappa}, \frac{12-\kappa}{\kappa};\frac{8}{\kappa}|x)
$$
with the constant chosen to normalize $Z_I$ as above.
Using this explicit formula one may verify that $Z_I(x)$ is
effectively a positive number for any $x\in[0;1]$ so it has all
expected properties to be a pure partition function.
For $\kappa=4$, $Z_I(x)=\sqrt{(1-x)/x}$
and for $\kappa=2$, $Z_I(x)=(1-x^2)/x^2$.
\subsubsection{Arch probabilities}
Let us now compute the probabilities for having one of the two
topologically distinct configurations: either $(I)$ with curves
joining either $[0x]$ and $[1\infty]$ or $(II)$ with curves joining
$[x1]$ and $[\infty0]$ as we just discussed. We shall proceed blindly,
but the reader should beware that there are subtleties involved. What
is computed is the probability for certain $X^{(i)}_t$'s to hit each
other. What happens at the level of hulls and how the process should
be properly continued is not investigated, but is expected to yield
the announced probability for arch configuration.
We consider a generic partition function $Z$ which is the sum of the
pure partition functions $Z_I$ and $Z_{II}$:
$$
Z(x)= p_I Z_I(x) + p_{II}Z_{II}(x)
$$
with $p_I$ and $p_{II}$ positive. To specify the 3SLE (or 4SLE)
process we need the partition function $Z(X_1,X_2,X_3,X_4)$ which is
recover from $Z(x)$ by conformal transformation~:
$$
Z(X_1,X_2,X_3,X_4)= [(X_4-X_2)(X_3-X_1)]^\frac{\kappa-6}{\kappa}\;Z(X)
$$
with $X$ the harmonic ratio of the four points $X_1$, $X_2$, $X_3$ and $X_4$~:
$$X=\Big(\frac{ X_1-X_2 }{ X_1-X_3 }\Big)\Big(
\frac{ X_4-X_3 }{ X_4-X_2 }\Big).$$
Let $M_I(x)$ and $M_{II}(x)=1-M_I(x)$ be defined by
$$
M_I(x) \equiv p_I Z_I(x)/Z(x)\quad,\quad
M_{II}(x)\equiv p_{II} Z_{II}(x) / Z(x)
$$
By construction the processes $t\to M_I(X_t)$ and $t\to M_{II}(X_t)$,
with $X_t$ the harmonic ratio of the four moving points, are local
martingales. Since both $Z_I$ and $Z_{II}$ are positive, $M_I$ are
$M_{II}$ are bounded local martingales and thus are martingales.
Let $\tau$ be the stopping time given by the first instant at which a
pair of points $X_t^{(i)}$ coincide. Then, in configuration $(I)$ we
have $\lim_{t\nearrow \tau} X_t=0$ while $\lim_{t\nearrow \tau} X_t
=1$ in configuration $(II)$. Since, for $\kappa<8$, $M_I(x)$ is such
that $\lim_{x\to 0} M_I(x)= 1$ but $\lim_{x\to 1}M_I(x)=0$, we obtain
that $M_I$ evaluated at the stopping time $\tau$ is the characteristic
function for events with the topological configuration $(I)$, ie:
\begin{eqnarray*}
\lim_{t\nearrow \tau} M_I(X_t) &=& {\bf 1}_{{\rm config.} (I)} \\
\lim_{t\nearrow \tau} M_{II}(X_t) &=& {\bf 1}_{{\rm config.} (II)}
\end{eqnarray*}
Since $M_I$ and $M_{II}$ are martingales, we get the probability of
occurrence of configurations of topological type $(I)$:
\begin{eqnarray}
\prob [{\rm config.} (I)]=M_I(X_{t=0})= \frac{p_I
Z_I(x)}{p_IZ_I(x)+p_{II}Z_{II}(x)}
\label{crossprob}
\end{eqnarray}
and similarly for the probabilities of having configuration $(II)$.
As expected they are ratios of partition functions.
\subsection{Applications to percolation and Ising model}
\label{sec: Ising}
We are now ready to give an application of triple (or
quadruple) SLE to percolation and Ising model.
Exploration processes in critical percolation are
described by SLEs with $\kappa=6$, as proved in
\cite{SS: percolation}.
Interfaces of spin clusters in critical Ising model is believed to
correspond to $\kappa = {3}$ while interfaces of Fortuin-Kasteleyn
clusters -- which occur in a high temperature expansion of the Ising
partition function -- are expected to correspond to the dual value
$\kappa=16/3$.
What we have in mind are these statistical models, defined on the
upper half plane, with boundary condition changing operators at the four
points $0,\ x,\ 1$ and $\infty$. They change the boundary condition
from open to closed (or vice versa) in percolation $(\kappa=6)$ and
from plus to minus (or vice versa) for Ising model $(\kappa=3)$.
To apply previous results on 4SLE processes to these situations, we
have to specify the partition functions $Z(x)$, or equivalently, we
have to specify the value of $p_I$ and $p_{II}$. This is done by
noticing that these models are left-right symmetric so that for $x=1/2$
there is equal probability to find configuration $(I)$ or $(II)$.
Since $Z_I(1/2)=Z_{II}(1/2)$, we have $p_I=p_{II}=1$, so that the
total partition function is $Z(x)=Z_I(x)+Z_{II}(x)$ and the
probability of occurrence of configuration $(I)$ for any $0<x<1$ is
now:
$$
\prob [{\rm config.} (I)]= \frac{
Z_I(x)}{Z_I(x)+Z_{II}(x)}\quad,\quad Z_{II}(x)=Z_I(1-x)
$$
\vspace{.2cm}
-- Percolation corresponds to $\kappa=6$. The boundary changing operator
$\psi$ has dimension $0$. The pure partition function $Z_I$ has a
simple integral representation:
$$
Z_I(x)_{\rm perco}= \frac{\Gamma(2/3)}{\Gamma(1/3)^2}\ \int_x^1 \ud
s\ s^{-2/3}(1-s)^{-2/3}.
$$
By construction $Z_{II}(x)=Z_I(1-x)$
also possesses a simple integral representation but, most importantly,
it is such that the total partition function is constant,
$Z(x)=Z_I(x)+Z_{II}(x)=1$, as expected for percolation. As a
consequence we find:
$$
\prob [{\rm config.} (I)]_{\rm perco}=
\frac{\Gamma(2/3)}{\Gamma(1/3)^2}\ \int_x^1 \ud s\ s^{-2/3}(1-s)^{-2/3}
$$
This is nothing but Cardy percolation crossing formula.
\vspace{.2cm}
-- Ising spin clusters correspond to $\kappa=3$. The boundary changing
operator $\psi$ has dimension $1/2$ and may thus be identified with a
fermion on the boundary. However the pure partition functions do not
correspond to the free fermion conformal block. By solving the
differential equation with the appropriate boundary
condition we get:
$$
Z_I(x)_{\rm spin\ Ising}= \mathrm{const}
\frac{1-x+x^2}{x(1-x)}\int_x^1\ud y\frac{(y(1-y))^{2/3}}{(1-y+y^2)^2}
$$
The total partition function $Z_I(x)+Z_I(1-x)$ is proportional to
$\frac{1-x+x^2}{x(1-x)}$, which is the free fermion result.
Hence, the Ising configuration probabilities are~:
$$
\prob [{\rm config.} (I)]_{\rm spin\ Ising}= \int_x^1\ud
y\frac{(y(1-y))^{2/3}}{(1-y+y^2)^2}\; \Big/ \int_0^1\ud
y\frac{(y(1-y))^{2/3}}{(1-y+y^2)^2}
$$
This is nothing but a new -- and previously unknown -- Ising crossing
formula.
\vspace{.2cm}
-- FK Ising clusters correspond to $\kappa=16/3$. The operator $\psi$ has
then dimension $1/16$. The pure partition function are given by:
$$
Z_I(x)_{\rm FK\ Ising} = \frac{(1-x)^{3/8}}{x^{1/8}(1+\sqrt{x})^{1/2}}
$$
and the crossing probabilities by:
\begin{eqnarray*}
\prob [{\rm config.} (I)]_{\rm FK\ Ising} =
\frac{\sqrt{(1-x) + (1-x)^{3/2}}}{\sqrt{x + x^{3/2}}
+ \sqrt{(1-x) + (1-x)^{3/2}}}
\end{eqnarray*}
The other critical random cluster (or Potts) models with $0 \leq Q
\leq 4$ have $Q = 4 \cos^2 \big( \frac{4 \pi}{\kappa} \big)$,
$4 \leq \kappa \leq 8$ and it is straightforward to obtain
explicit crossing formulas involving only hypergeometric functions.
\subsection{nSLEs and beyond}
We now comment on how to compute multiple arch probabilities for
general nSLEs. This section only aims at giving some hints on how to
generalize previous computations. So it shall be sketchy. It is clear
that the key point is to identify the pure partition functions -- once
this is done the rest is routine. As exemplified above by
eq.(\ref{bdrypure}) this is linked to CFT fusions. The rules there were
that, for a given arch system, fusing two points linked by an arch
produces the dominant singularity which means that the two boundary
operators are fused on the identity operator, whereas fusing two
points not linked by an arch produces the subleading singularity which
means the fusion of the two boundary fields on the identity should
vanish. In general there could be a whole hierarchy of arches with
arches in the interior of others, i.e. with a family of self-surrounding
arches, the next encircling the previous.
So we are lead to propose the following rules.\\
For a given arch configuration:\\
--- The most interior pair of adjacent pair of points, say
$X_i,X_{i+1}$ in a family of self-surrounding arches fused into the
identity operator, so that the pure partition function evaluated at
$X_i\simeq X_{i+1}$ should be proportional to
$(X_{i+1}-X_i)^\frac{\kappa-6}{\kappa}$ times the pure partition
function associated to the arch system with the interior arch
$[X_iX_{i+1}]$ removed. Symbolically~:
$$
Z_{\rm pure}(\cdots, X_i\simeq X_{i+1},\cdots)\simeq {\rm const.}\
(X_{i+1}-X_i)^\frac{\kappa-6}{\kappa}\times
Z_{{\rm pure}\setminus[X_iX_{i+1}]}(\cdots,\cdots)
$$
for $X_i$ and $X_{i+1}$ linked by an arch.\\
\noindent
--- The fusion on the identity of any pair of adjacent points
not linked by an arch should vanish, so that the fusion of this
pair of points produces the subleading singularity. Symbolically~:
$$
Z_{\rm pure}(\cdots, X_i\simeq X_{i+1},\cdots)\simeq {\rm const.}\
(X_{i+1}-X_i)^\frac{2}{\kappa} +\cdots
$$
for $X_i$ and $X_{i+1}$ for not linked by an arch.
We do not have a complete proof that these rules fully determine the
pure partition functions but we checked it on a few cases, see figure
\ref{fig:rules}.
\begin{figure}
\includegraphics[width=1.0\textwidth]{quadrupleSLE.eps}
\caption{\emph{Illustration of the fusion rules corresponding to
arch configurations.}}
\label{fig:rules}
\end{figure}
Here are a few samples. We shall give the relation between the pure
partition and the CFT conformal blocks indexed by the corresponding
Bratelli diagram. For $n=4$, we may have the following arch systems
$[X_1X_2][X_3X_4]$ or $[X_1[X_2X_3]X_4]$. (A given geometrical
configuration may correspond to different arch systems depending at
which location we open the closed boundary. But they are all
equivalent to these two up to an order preserving relabeling of the
points. For instance $[X_4X_1][X_2X_3]$ is equivalent to
$[X_1[X_2X_3]X_4]$.) Applying the previous rules we get:
\begin{eqnarray*}
Z_{[X_1[X_2X_3]X_4]} &=&
\langle {}_{[h_0{}]}\psi(X_1)
{}_{[h_1{}]}\psi(X_2)
{}_{[h_2{}]}\psi(X_3)
{}_{[h_1{}]}\psi(X_4)
{}_{[h_0{}]}\rangle
\\
Z_{[X_1X_2][X_3X_4]}&=&
\langle {}_{[h_0{}]}\psi(X_1)
{}_{[h_1{}]}\psi(X_2)
{}_{[h_0{}]}\psi(X_3)
{}_{[h_1{}]}\psi(X_4)
{}_{[h_0{}]}\rangle \\
& + & \omega \;
\langle {}_{[h_0{}]}\psi(X_1)
{}_{[h_1{}]}\psi(X_2)
{}_{[h_2{}]}\psi(X_3)
{}_{[h_1{}]}\psi(X_4)
{}_{[h_0{}]}\rangle,
\end{eqnarray*}
where the indices $h_{m}$, $m=0,1,\cdots$ refer to the corresponding
points in the Bratelli diagram, i.e. to the weights $h_{m}(\kappa)$ of
the intermediate Virasoro modules. The coefficient $\omega$ is fully
determined, in terms of CFT fusion coefficients, by demanding that the
fusion of $X_2$ and $X_3$ on the identity vanishes.
One may go on and solve for the pure partition functions in few
other cases. A particularly simple example with $n=6$ is given by~:
\begin{eqnarray*}
Z_{[X_1[X_2[X_3X_4]X_5]X_6]} & = & \\
& & \hspace{-5cm}
\langle {}_{[h_0{}]}\psi(X_1)
{}_{[h_1{}]}\psi(X_2)
{}_{[h_2{}]}\psi(X_3)
{}_{[h_3{}]}\psi(X_4)
{}_{[h_2{}]}\psi(X_5)
{}_{[h_1{}]}\psi(X_6)
{}_{[h_0{}]}\rangle
\end{eqnarray*}
As can be seen on these examples, there is no simple relation between
arch systems and Bratelli diagrams and the change of basis for one
to the other is quite involved. The only simple rule we find is that
the pure partition function for a unique family of self-surrounding
arches is a pure conformal block corresponding to a unique Bratelli
diagram.
|
{
"timestamp": "2005-03-21T17:02:45",
"yymm": "0503",
"arxiv_id": "math-ph/0503024",
"language": "en",
"url": "https://arxiv.org/abs/math-ph/0503024"
}
|
\section{Introduction}
In a series of papers, Mweene has developed a generalized description of
angular momentum which contains the standard results in a certain limit. He
has thereby obtained new generalized expressions for the operators and
eigenvectors for spin 1/2[1-4], spin 1[5], spin 3/2[6], spin 2[7] and spin
5/2[8]. Applying this approach to angular momentum addition, he has shown
how the standard results for various states that arise from combining two
values of angular momentum come about from a consideration of probability
amplitudes for measurements on these systems. This had led to generalized
results for angular momentum addition which also reduce to the standard
results in an appropriate limit[9-10].
In a further development of the approach, Mweene has shown that the usual
spherical harmonics are just special forms of more generalized quantities
and he has obtained the generalized spherical harmonics for the case $l=1$%
[11]. In this paper, we give the generalized spherical harmonics for $l=2$.
This paper is organized as follows. This Introduction is followed in Section
2 by a brief review of the theory underlying the work. Section 3 contains
the derivation of the generalized spherical harmonics and their probability
amplitudes. Section 4 is a discussion of some of the properties of these
quantities. The Discussion and Conclusion in Section 5 closes the paper.
\section{Theoretical Background}
This work is inspired by the interpretation of quantum mechanics due to
Land\'e[12-15]. According to Land\'e, if a quantum system possesses three
sets of observables $A$, $B$ and $C$ with respective eigenvalue spectra $A_1$%
, $A_2,$...,$A_N$, $B_1$, $B_2$, ...,$B_N$ and $C_1$, $C_2$, ....,$C_N$ -
where $N$ is the multiplicity of each spectrum, which is necessarily the
same for each observable[13] - then three sets of probability amplitudes can
be defined for the system. One set, which is denoted by $\psi (A_i,C_j),$
relates to measurement of the observable $C$ if the system is in a state
corresponding to eigenvalues of the observable $A$: thus $\left| \psi
(A_i;C_j)\right| ^2$ gives the probability for obtaining the value $C_j$ if
the initial state corresponds to the eigenvalue $A_i$ of the observable $A$.
The second set $\chi (A_i;B_j)$ relates to measurement of $B$ when the
system is in a state corresponding to the eigenvalue $A$. Finally, the set $%
\phi (B_i;C_j)$ describes measurements of $C$ when the initial state belongs
to the observable $B$. Since the three sets of probability amplitudes belong
to one system, they are interdependent, and the law of interdependence
is[12,13]
\begin{equation}
\psi (A_i;C_j)=\sum_j\chi (A_i;B_j)\phi (B_j;C_n) \label{on1}
\end{equation}
Another aspect of Land\'e's interpretation of quantum mechanics is that
every eigenfunction or wave function of a quantum system is first and
foremost a probability amplitude and that every such probability amplitude
connects two well-defined states - one corresponding to the state in which
the system is before a measurement, and the other to the state that comes
about as a result of the measurement[12,13]. It is therefore always possible
to identify an initial and a final state for any wave function or
eigenfunction.
Mweene has argued that for an eigenfunction resulting from solution of a
differential eigenvalue equation, the initial state corresponds to the
eigenvalue while the final state corresponds to the eigenvalue defined by
the continuous variable in terms of which the differential operator is
defined[16]. For the Schr\"odinger equation, the eigenfunctions $\psi
_{E_i}(x)$ should really be written as $\psi (E_i;x)$ to emphasize that the
initial state in the probability amplitude corresponds to the eigenvalue $%
E_i $ while the final state corresponds to the eigenvalue $x$. Another
example comes from the solution of Legendre's equation. The spherical
harmonics $Y_{lm}(\theta ,\varphi )$ should really be written as $%
Y(l,m;\theta ,\varphi )$ to emphasize that in this case the initial state is
defined by the eigenvalues $m\hbar $ and $l(l+1)\hbar ^2$ while the final
state corresponds to the angular position $(\theta ,\varphi ).$ Since $%
m\hbar $ is an angular momentum projection, it must be defined with respect
to some axis. Owing to the absence of another set of angles in the
expressions of the spherical harmonics which could define this direction, it
must be the $z$ axis[11]. But since an axis of quantization can be chosen
arbitrarily, it is possible to define spherical harmonics with respect to
any other direction as the axis of initial quantization. The functions
resulting from this are the generalized spherical harmonics and have already
been worked out for the case $l=1$. In this work, we obtain them for $l=2$.
\section{Generalized Spherical Harmonics}
\subsection{Probability Amplitudes}
The generalized spherical harmonics connect states of angular momentum
projection in the arbitrary direction $\widehat{\mathbf{a}}$ defined by the
polar angles $(\theta ^{\prime },\varphi ^{\prime })$ to states of the
angular position $(\theta ,\varphi ).$ We denote them by $Y(l,m^{(\widehat{%
\mathbf{a}})};\theta ,\varphi )$ . To derive them, we use the probability
addition law Eq. (\ref{on1}). We start off by writing
\begin{equation}
Y(l,m^{(\widehat{\mathbf{a}})};\theta ,\varphi )=\sum_j\chi (l,m^{(\widehat{%
\mathbf{a}})};B_j)\phi (B_j;\theta ,\varphi ) \label{th34}
\end{equation}
If we choose the observable $B$ carefully, we should find that both the
probability amplitudes $\chi (l,m^{(\widehat{\mathbf{a}})};B_j)$ and $\phi
(B_j;\theta ,\varphi )$ are known. If $B$ is chosen to be the spin
projection with respect to the $z$ direction, it is found that
\begin{equation}
\phi (B_j;\theta ,\varphi )=Y(l,m^{(\widehat{\mathbf{k}})};\theta ,\varphi )
\label{th34a}
\end{equation}
are the standard spherical harmonics, while $\chi (l,m_i^{(\widehat{\mathbf{a%
}})};l,m_f^{(\widehat{\mathbf{k}})})$ are just spin probability amplitudes
connecting states such that the initial one corresponds to the spin
projection being $m_i\hbar $ in the direction $\widehat{\mathbf{a}}$ while
the final state corresponds to the spin projection being $m_f\hbar $ along
the $z $ axis. These have already been worked out[7] and are as given below.
If the initial spin projection in the direction $\widehat{\mathbf{a}}$ is $%
2\hbar $, these probability amplitudes are
\begin{equation}
\chi (2,2^{(\widehat{\mathbf{a}})};2,2^{(\widehat{\mathbf{k}})})=\cos ^4%
\frac{\theta ^{\prime }}2e^{-2i\varphi ^{\prime }} \label{fi53}
\end{equation}
\begin{equation}
\chi (2,2^{(\widehat{\mathbf{a}})};2,1^{(\widehat{\mathbf{k}})})=2\sin \frac{%
\theta ^{\prime }}2\cos ^3\frac{\theta ^{\prime }}2e^{-i\varphi ^{\prime }}
\label{fi54}
\end{equation}
\begin{equation}
\chi (2,2^{(\widehat{\mathbf{a}})};2,0^{(\widehat{\mathbf{k}})})=\sqrt{6}%
\sin ^2\frac{\theta ^{\prime }}2\cos ^2\frac{\theta ^{\prime }}2
\label{fi55}
\end{equation}
\begin{equation}
\chi (2,2^{(\widehat{\mathbf{a}})};2,(-1)^{(\widehat{\mathbf{k}})})=2\sin ^3%
\frac{\theta ^{\prime }}2\cos \frac{\theta ^{\prime }}2e^{i\varphi ^{\prime
}} \label{fi56}
\end{equation}
and
\begin{equation}
\chi (2,2^{(\widehat{\mathbf{a}})};2,(-2)^{(\widehat{\mathbf{k}})})=\sin ^4%
\frac{\theta ^{\prime }}2e^{2i\varphi ^{\prime }} \label{fi57}
\end{equation}
If the initial spin projection in the direction $\widehat{\mathbf{a}}$ is $%
\hbar ,$ the probability amplitudes are
\begin{equation}
\chi (2,1^{(\widehat{\mathbf{a}})};2,2^{(\widehat{\mathbf{k}})})=2\sin \frac{%
\theta ^{\prime }}2\cos ^3\frac{\theta ^{\prime }}2e^{-2i\varphi ^{\prime }}
\label{fi58}
\end{equation}
\begin{equation}
\chi (2,1^{(\widehat{\mathbf{a}})};2,1^{(\widehat{\mathbf{k}})})=-(3\sin ^2%
\frac{\theta ^{\prime }}2-\cos ^2\frac{\theta ^{\prime }}2)\cos ^2\frac{%
\theta ^{\prime }}2e^{-i\varphi ^{\prime }} \label{fi59}
\end{equation}
\begin{equation}
\chi (2,1^{(\widehat{\mathbf{a}})};2,0^{(\widehat{\mathbf{k}})})=-\sqrt{6}%
\cos \frac{\theta ^{\prime }}2\sin \frac{\theta ^{\prime }}2\cos \theta
^{\prime } \label{si60}
\end{equation}
\begin{equation}
\chi (2,1^{(\widehat{\mathbf{a}})};2,(-1)^{(\widehat{\mathbf{k}})})=(3\cos ^2%
\frac{\theta ^{\prime }}2-\sin ^2\frac{\theta ^{\prime }}2)\sin ^2\frac{%
\theta ^{\prime }}2e^{i\varphi ^{\prime }} \label{si61}
\end{equation}
and
\begin{equation}
\chi (2,1^{(\widehat{\mathbf{a}})};2,(-2)^{(\widehat{\mathbf{k}})})=-2\sin ^3%
\frac{\theta ^{\prime }}2\cos \frac{\theta ^{\prime }}2e^{2i\varphi ^{\prime
}} \label{si62}
\end{equation}
If the initial spin projection in the direction $\widehat{\mathbf{a}}$ is $%
0, $the probability amplitudes are
\begin{equation}
\chi (2,0^{(\widehat{\mathbf{a}})};2,2^{(\widehat{\mathbf{k}})})=\sqrt{6}%
\sin ^2\frac{\theta ^{\prime }}2\cos ^2\frac{\theta ^{\prime }}%
2e^{-2i\varphi ^{\prime }} \label{si63}
\end{equation}
\begin{equation}
\chi (2,0^{(\widehat{\mathbf{a}})};2,1^{(\widehat{\mathbf{k}})})=-\sqrt{6}%
\sin \frac{\theta ^{\prime }}2\cos \frac{\theta ^{\prime }}2\cos \theta
^{\prime }e^{-i\varphi ^{\prime }} \label{si64}
\end{equation}
\begin{equation}
\chi (2,0^{(\widehat{\mathbf{a}})};2,0^{(\widehat{\mathbf{k}})})=\frac
12(2\cos ^2\theta ^{\prime }-\sin ^2\theta ^{\prime }) \label{si65}
\end{equation}
\begin{equation}
\chi (2,0^{(\widehat{\mathbf{a}})};2,(-1)^{(\widehat{\mathbf{k}})})=\sqrt{6}%
\sin \frac{\theta ^{\prime }}2\cos \frac{\theta ^{\prime }}2\cos \theta
^{\prime }e^{i\varphi ^{\prime }} \label{si66}
\end{equation}
and
\begin{equation}
\chi (2,0^{(\widehat{\mathbf{a}})};2,(-2)^{(\widehat{\mathbf{k}})})=\sqrt{6}%
\sin ^2\frac{\theta ^{\prime }}2\cos ^2\frac{\theta ^{\prime }}2e^{2i\varphi
^{\prime }} \label{si67}
\end{equation}
If the initial spin projection in the direction $\widehat{\mathbf{a}}$ is $%
-\hbar ,$ the probability amplitudes are
\begin{equation}
\chi (2,(-1)^{(\widehat{\mathbf{a}})};2,2^{(\widehat{\mathbf{k}})})=2\cos
\frac{\theta ^{\prime }}2\sin ^3\frac{\theta ^{\prime }}2e^{-2i\varphi
^{\prime }} \label{si68}
\end{equation}
\begin{equation}
\chi (2,(-1)^{(\widehat{\mathbf{a}})};2,1^{(\widehat{\mathbf{k}})})=-(3\cos
^2\frac{\theta ^{\prime }}2-\sin ^2\frac{\theta ^{\prime }}2)\sin ^2\frac{%
\theta ^{\prime }}2e^{-i\varphi ^{\prime }} \label{si69}
\end{equation}
\begin{equation}
\chi (2,(-1)^{(\widehat{\mathbf{a}})};2,0^{(\widehat{\mathbf{k}})})=\sqrt{6}%
\sin \frac{\theta ^{\prime }}2\cos \frac{\theta ^{\prime }}2\cos \theta
^{\prime } \label{se70}
\end{equation}
\begin{equation}
\chi (2,(-1)^{(\widehat{\mathbf{a}})};2,(-1)^{(\widehat{\mathbf{k}}%
)})=(3\sin ^2\frac{\theta ^{\prime }}2-\cos ^2\frac{\theta ^{\prime }}2)\cos
^2\frac{\theta ^{\prime }}2e^{i\varphi ^{\prime }} \label{se71}
\end{equation}
and
\begin{equation}
\chi (2,(-1)^{(\widehat{\mathbf{a}})};2,(-2)^{(\widehat{\mathbf{k}}%
)})=-2\sin \frac{\theta ^{\prime }}2\cos ^3\frac{\theta ^{\prime }}%
2e^{2i\varphi ^{\prime }} \label{se72}
\end{equation}
Finally, if the initial spin projection in the direction $\widehat{\mathbf{a}%
}$ is $-2\hbar ,$ the probability amplitudes are
\begin{equation}
\chi (2,(-2)^{(\widehat{\mathbf{a}})};2,2^{(\widehat{\mathbf{k}})})=\sin ^4%
\frac{\theta ^{\prime }}2e^{-2i\varphi ^{\prime }} \label{se73}
\end{equation}
\begin{equation}
\chi (2,(-2)^{(\widehat{\mathbf{a}})};2,1^{(\widehat{\mathbf{k}})})=-2\cos
\frac{\theta ^{\prime }}2\sin ^3\frac{\theta ^{\prime }}2e^{-i\varphi
^{\prime }} \label{se74}
\end{equation}
\begin{equation}
\chi (2,(-2)^{(\widehat{\mathbf{a}})};2,0^{(\widehat{\mathbf{k}})})=\sqrt{6}%
\sin ^2\frac{\theta ^{\prime }}2\cos ^2\frac{\theta ^{\prime }}2
\label{se75}
\end{equation}
\begin{equation}
\chi (2,(-2)^{(\widehat{\mathbf{a}})};2,(-1)^{(\widehat{\mathbf{k}}%
)})=-2\sin \frac{\theta ^{\prime }}2\cos ^3\frac{\theta ^{\prime }}%
2e^{i\varphi ^{\prime }} \label{se76}
\end{equation}
and
\begin{equation}
\chi (2,(-2)^{(\widehat{\mathbf{a}})};2,(-2)^{(\widehat{\mathbf{k}})})=\cos
^4\frac{\theta ^{\prime }}2e^{2i\varphi ^{\prime }} \label{se77}
\end{equation}
The ordinary spherical harmonics $Y_{2m}(\theta ,\varphi )=Y(2,m^{(\widehat{%
\mathbf{k}})};\theta ,\varphi )$ for $l=2$ are
\begin{equation}
Y(2,2^{(\widehat{\mathbf{k}})};\theta ,\varphi )=\sqrt{\frac{15}{32\pi }}%
\sin ^2\theta e^{2i\varphi } \label{se78}
\end{equation}
\begin{equation}
Y(2,1^{(\widehat{\mathbf{k}})};\theta ,\varphi )=-\sqrt{\frac{15}{8\pi }}%
\sin \theta \cos \theta e^{i\varphi } \label{se79}
\end{equation}
\begin{equation}
Y(2,0^{(\widehat{\mathbf{k}})};\theta ,\varphi )=\sqrt{\frac 5{16\pi }}%
(3\cos ^2\theta -1) \label{ei80}
\end{equation}
\begin{equation}
Y(2,(-1)^{(\widehat{\mathbf{k}})};\theta ,\varphi )=\sqrt{\frac{15}{8\pi }}%
\sin \theta \cos \theta e^{-i\varphi } \label{ei81}
\end{equation}
\begin{equation}
Y(2,(-2)^{(\widehat{\mathbf{k}})};\theta ,\varphi )=\sqrt{\frac{15}{32\pi }}%
\sin ^2\theta e^{-2i\varphi } \label{ei82}
\end{equation}
Using Eq. (\ref{th34}), the generalized spherical harmonics for $l=2$ are
found to be
\begin{eqnarray}
Y(2,2^{(\mathbf{\hat a})};\theta ,\varphi ) &=&\sqrt{\frac{15}{32\pi }}%
\{\sin ^2\theta (\cos ^4\frac{\theta ^{\prime }}2e^{2i(\varphi -\varphi
^{\prime })}+\sin ^4\frac{\theta ^{\prime }}2e^{-2i(\varphi -\varphi
^{\prime })}) \nonumber \label{eq52} \\
&&+\sin 2\theta \sin \theta ^{\prime }(-\cos ^2\frac{\theta ^{\prime }}%
2e^{i(\varphi -\varphi ^{\prime })}+\sin ^2\frac{\theta ^{\prime }}%
2e^{-i(\varphi -\varphi ^{\prime })}) \nonumber \\
&&+\frac 12\sin ^2\theta ^{\prime }(3\cos ^2\theta -1)\} \label{ei83}
\end{eqnarray}
\begin{eqnarray}
Y(2,1^{(\mathbf{\hat a})};\theta ,\varphi ) &=&\sqrt{\frac{15}{32\pi }}%
\{\sin \theta ^{\prime }\sin ^2\theta (\cos ^2\frac{\theta ^{\prime }}%
2e^{2i(\varphi -\varphi ^{\prime })}-\sin ^2\frac{\theta ^{\prime }}%
2e^{-2i(\varphi -\varphi ^{\prime })}) \nonumber \\
&&\ -\sin 2\theta [(3\sin ^2\frac{\theta ^{\prime }}2-\cos ^2\frac{\theta
^{\prime }}2)\cos ^2\frac{\theta ^{\prime }}2e^{i(\varphi -\varphi ^{\prime
})} \nonumber \\
&&\ +(3\cos ^2\frac{\theta ^{\prime }}2-\sin ^2\frac{\theta ^{\prime }}%
2)\sin ^2\frac{\theta ^{\prime }}2e^{-i(\varphi -\varphi ^{\prime })}]
\nonumber \\
&&\ -\frac 12\sin 2\theta ^{\prime }(3\cos ^2\theta -1)\} \label{ei84}
\end{eqnarray}
\begin{eqnarray}
Y(2,0^{(\mathbf{\hat a})};\theta ,\varphi ) &=&\sqrt{\frac{45}{256\pi }}%
\{\sin ^2\theta ^{\prime }\sin ^2\theta (e^{2i(\varphi -\varphi ^{\prime
})}+e^{-2i(\varphi -\varphi ^{\prime })}) \nonumber \label{eq54} \\
&&+\sin 2\theta \sin 2\theta ^{\prime }(e^{i(\varphi -\varphi ^{\prime
})}+e^{-i(\varphi -\varphi ^{\prime })}) \nonumber \\
&&+\frac 23(3\cos ^2\theta -1)(2\cos ^2\theta ^{\prime }-\sin ^2\theta
^{\prime })\} \label{ei85}
\end{eqnarray}
\begin{eqnarray}
Y(2,(-1)^{(\mathbf{\hat a})};\theta ,\varphi ) &=&\sqrt{\frac{15}{32\pi }}%
\{\sin \theta ^{\prime }\sin ^2\theta (\sin ^2\frac{\theta ^{\prime }}%
2e^{2i(\varphi -\varphi ^{\prime })}-\cos ^2\frac{\theta ^{\prime }}%
2e^{-2i(\varphi -\varphi ^{\prime })}) \nonumber \label{eq55} \\
&&\ +\sin 2\theta [(3\cos ^2\frac{\theta ^{\prime }}2-\sin ^2\frac{\theta
^{\prime }}2)\sin ^2\frac{\theta ^{\prime }}2e^{i(\varphi -\varphi ^{\prime
})} \nonumber \\
&&\ +(3\sin ^2\frac{\theta ^{\prime }}2-\cos ^2\frac{\theta ^{\prime }}%
2)\cos ^2\frac{\theta ^{\prime }}2e^{-i(\varphi -\varphi ^{\prime })}]
\nonumber \\
&&\ +\frac 12\sin 2\theta ^{\prime }(3\cos ^2\theta -1)\} \label{ei86}
\end{eqnarray}
\begin{eqnarray}
Y(2,(-2)^{(\mathbf{\hat a})};\theta ,\varphi ) &=&\sqrt{\frac{15}{32\pi }}%
\{\sin ^2\theta (\sin ^4\frac{\theta ^{\prime }}2e^{2i(\varphi -\varphi
^{\prime })}+\cos ^4\frac{\theta ^{\prime }}2e^{-2i(\varphi -\varphi
^{\prime })}) \nonumber \label{eq56} \\
&&\ +\sin 2\theta \sin \theta ^{\prime }(\sin ^2\frac{\theta ^{\prime }}%
2e^{i(\varphi -\varphi ^{\prime })}-\cos ^2\frac{\theta ^{\prime }}%
2e^{-i(\varphi -\varphi ^{\prime })}) \nonumber \\
&&\ +\frac 12\sin ^2\theta ^{\prime }(3\cos ^2\theta -1)\} \label{ei87}
\end{eqnarray}
\subsection{Probability Amplitudes for the $x^{\prime }$ Direction}
The results we have presented refer to the direction $\widehat{\mathbf{a}}$
as the direction of initial quantization. We may think of the vector $%
\widehat{\mathbf{a}}$ as defining a new $z$ axis, which we denote by $%
z^{\prime }$, since in the limit $\theta ^{\prime }=\varphi ^{\prime }=0,$
the results corresponding to it reduce to those for the $z$ axis. This $%
z^{\prime }$ axis corresponds to a new coordinate system in which the unit
vector in the $x^{\prime }$ direction is $\widehat{\mathbf{u}}$ and that in
the $y^{\prime }$ direction is $\widehat{\mathbf{v}}$[11]. From the results
for the $\widehat{\mathbf{a}}$ or $z^{\prime }$ axis, we can obtain the
probability amplitudes and probabilities densities for the $x^{\prime }$
direction by applying the transformation $\theta ^{\prime }\rightarrow
\theta ^{\prime }-\pi /2$ to them[3,5]. We are justified in associating the
results so obtained with the $x^{\prime }$ axis since in the limit $\theta
^{\prime }=\varphi ^{\prime }=0, $ they reduce to those for the $x$
direction. When we make these argument changes, $\widehat{\mathbf{a}}$
becomes $\widehat{\mathbf{u}}$. Applying this prescription to the
generalized spherical harmonics, we obtain the results
\begin{eqnarray}
Y(2,2^{(\widehat{\mathbf{u}})};\theta ,\varphi ) &=&\sqrt{\frac{15}{128\pi }}%
\{\frac 12\sin ^2\theta [(1+\sin \theta ^{\prime })^2e^{2i(\varphi -\varphi
^{\prime })} \nonumber \\
&&+(1-\sin \theta ^{\prime })^2e^{-2i(\varphi -\varphi ^{\prime })}]
\nonumber \\
&&\ +\sin 2\theta \cos \theta ^{\prime }[(1+\sin \theta ^{\prime
})e^{i(\varphi -\varphi ^{\prime })} \nonumber \\
&&-(1-\sin \theta ^{\prime })e^{-i(\varphi -\varphi ^{\prime })}]+\cos
^2\theta ^{\prime }(3\cos ^2\theta -1)\} \label{ni99a}
\end{eqnarray}
\begin{eqnarray}
Y(2,1^{(\widehat{\mathbf{u}})};\theta ,\varphi ) &=&\sqrt{\frac{15}{128\pi }}%
\{-\sin ^2\theta \cos \theta ^{\prime }[(1+\sin \theta ^{\prime
})e^{2i(\varphi -\varphi ^{\prime })} \nonumber \\
&&-(1-\sin \theta ^{\prime })e^{-2i(\varphi -\varphi ^{\prime })}] \nonumber
\\
&&\ -\sin 2\theta [(1-2\sin \theta ^{\prime })(1+\sin \theta ^{\prime
})e^{i(\varphi -\varphi ^{\prime })} \nonumber \\
&&+(1+2\sin \theta ^{\prime })(1-\sin \theta ^{\prime })e^{-i(\varphi
-\varphi ^{\prime })}] \nonumber \\
&&\ +\sin 2\theta ^{\prime }(3\cos ^2\theta -1)\} \label{ni99b}
\end{eqnarray}
\begin{eqnarray}
Y(2,0^{(\widehat{\mathbf{u}})};\theta ,\varphi ) &=&\sqrt{\frac{15}{128\pi }}%
\{\cos ^2\theta ^{\prime }\sin ^2\theta (e^{2i(\varphi -\varphi ^{\prime
})}+e^{-2i(\varphi -\varphi ^{\prime })}) \nonumber \\
&&\ -\sin 2\theta \sin 2\theta ^{\prime }[e^{i(\varphi -\varphi ^{\prime
})}+e^{-i(\varphi -\varphi ^{\prime })}] \nonumber \\
&&\ +\frac 23(2\sin ^2\theta ^{\prime }-\cos ^2\theta ^{\prime })(3\cos
^2\theta -1)\} \label{ni99c}
\end{eqnarray}
\begin{eqnarray}
Y(2,(-1)^{(\widehat{\mathbf{u}})};\theta ,\varphi ) &=&\sqrt{\frac{15}{%
128\pi }}\{-\sin ^2\theta \cos \theta ^{\prime }[(1-\sin \theta ^{\prime
})e^{2i(\varphi -\varphi ^{\prime })} \nonumber \\
&&-(1+\sin \theta ^{\prime })e^{-2i(\varphi -\varphi ^{\prime })}] \nonumber
\\
&&\ +\sin 2\theta [(1+2\sin \theta ^{\prime })(1-\sin \theta ^{\prime
})e^{i(\varphi -\varphi ^{\prime })} \nonumber \\
&&+(1-2\sin \theta ^{\prime })(1+\sin \theta ^{\prime })e^{-i(\varphi
-\varphi ^{\prime })}] \nonumber \\
&&\ -\sin 2\theta ^{\prime }(3\cos ^2\theta -1)\} \label{ni99d}
\end{eqnarray}
\begin{eqnarray}
Y(2,(-2)^{(\widehat{\mathbf{u}})};\theta ,\varphi ) &=&\sqrt{\frac{15}{%
128\pi }}\{\frac 12\sin ^2\theta [(1-\sin \theta ^{\prime })^2e^{2i(\varphi
-\varphi ^{\prime })} \nonumber \\
&&+(1+\sin \theta ^{\prime })^2e^{-2i(\varphi -\varphi ^{\prime })}
\nonumber \\
&&\ -\sin 2\theta \cos \theta ^{\prime }[(1-\sin \theta ^{\prime
})e^{i(\varphi -\varphi ^{\prime })} \nonumber \\
&&-(1+\sin \theta ^{\prime })e^{-i(\varphi -\varphi ^{\prime })}]+\cos
^2\theta ^{\prime }(3\cos ^2\theta -1)\} \label{ni99e}
\end{eqnarray}
\subsection{Probability Amplitudes for the $y^{\prime }$ Direction}
The prescription for obtaining the probability amplitudes and probability
densities corresponding to $y^{\prime }$ is to set $\theta ^{\prime }=\pi /2$%
, $\varphi ^{\prime }\rightarrow \varphi ^{\prime }-\pi /2$ in the
expressions corresponding to the $z^{\prime }$ direction. As well as
transforming the unit vector $\widehat{\mathbf{a}}$ to the unit vector $%
\widehat{\mathbf{v}}$, this yields the probability amplitudes:
\begin{eqnarray}
Y(2,2^{(\widehat{\mathbf{v}})};\theta ,\varphi ) &=&-\sqrt{\frac{15}{128\pi }%
}\{\frac 12\sin ^2\theta (e^{2i(\varphi -\varphi ^{\prime })}+e^{-2i(\varphi
-\varphi ^{\prime })}) \nonumber \\
&&+i\sin 2\theta [e^{i(\varphi -\varphi ^{\prime })}+e^{-i(\varphi -\varphi
^{\prime })}]-(3\cos ^2\theta -1)\} \label{ni99k}
\end{eqnarray}
\begin{eqnarray}
Y(2,1^{(\widehat{\mathbf{v}})};\theta ,\varphi ) &=&\sqrt{\frac{15}{128\pi }}%
\{\sin ^2\theta (-e^{2i(\varphi -\varphi ^{\prime })}+e^{-2i(\varphi
-\varphi ^{\prime })}) \nonumber \\
&&\ +i\sin 2\theta [e^{i(\varphi -\varphi ^{\prime })}-e^{-i(\varphi
-\varphi ^{\prime })}]\} \label{ni99l}
\end{eqnarray}
\begin{eqnarray}
Y(2,0^{(\widehat{\mathbf{v}})};\theta ,\varphi ) &=&-\sqrt{\frac{45}{256\pi }%
}\{\sin ^2\theta (e^{2i(\varphi -\varphi ^{\prime })}+e^{-2i(\varphi
-\varphi ^{\prime })}) \nonumber \\
&&\ \ +\frac 23(3\cos ^2\theta -1)\} \label{ni99m}
\end{eqnarray}
\begin{eqnarray}
Y(2,(-1)^{(\widehat{\mathbf{v}})};\theta ,\varphi ) &=&\sqrt{\frac{15}{%
128\pi }}\{\sin ^2\theta (-e^{2i(\varphi -\varphi ^{\prime
})}+e^{-2i(\varphi -\varphi ^{\prime })}) \nonumber \\
&&\ \ +i\sin 2\theta [e^{i(\varphi -\varphi ^{\prime })}-e^{-i(\varphi
-\varphi ^{\prime })}]\} \label{ni99n}
\end{eqnarray}
\begin{eqnarray}
Y(2,(-2)^{(\widehat{\mathbf{v}})};\theta ,\varphi ) &=&\sqrt{\frac{15}{%
128\pi }}\{-\frac 12\sin ^2\theta (e^{2i(\varphi -\varphi ^{\prime
})}+e^{-2i(\varphi -\varphi ^{\prime })}) \nonumber \\
&&\ +i\sin 2\theta [e^{i(\varphi -\varphi ^{\prime })}+e^{-i(\varphi
-\varphi ^{\prime })}]+3\cos ^2\theta -1\} \nonumber \\
&& \label{ni99o}
\end{eqnarray}
We emphasize that the unit vectors $\widehat{\mathbf{u}}$, $\widehat{\mathbf{%
v}}$ and $\widehat{\mathbf{a}}$ define a system of mutually orthogonal
coordinate axes.
\section{ General Properties of the Generalized Spherical harmonics}
The generalized quantities presented here reduce to the standard quantities
in the limit $\theta ^{\prime }=\varphi ^{\prime }=0$, which corresponds to
the arbitrary vector $\widehat{\mathbf{a}}$ pointing in the direction of the
$z$ axis. Thus, in this limit, we get
\begin{equation}
Y(2,m^{(\mathbf{\hat a})};\theta ,\varphi )\rightarrow Y_{2m}(\theta
,\varphi ) \label{ni99u}
\end{equation}
A property of special interest with regard to the ordinary spherical
harmonics is their behaviour under the parity operation $\mathbf{r}%
\rightarrow -\mathbf{r,}$ a reflection in the origin. Under this operation,
the spherical polar coordinates $(r,\theta ,\varphi )$ transform thus: $%
r\rightarrow r,$ $\theta \rightarrow \pi -\theta ,,\;\phi \rightarrow \phi
+\pi $ . Thus if $\rho $ is the parity operator defined by
\begin{equation}
\rho \Psi (\mathbf{r})=\Psi (-\mathbf{r}). \label{hu100}
\end{equation}
then
\begin{equation}
\rho Y(l,m^{(\widehat{\mathbf{k}})};\theta ,\varphi )=Y(l,m^{(\widehat{%
\mathbf{k}})};\pi -\theta ,\varphi +\pi ) \label{hu101}
\end{equation}
As is well-known however,
\begin{equation}
Y(l,m^{(\widehat{\mathbf{k}})};\pi -\theta ,\varphi +\pi )=(-1)^lY(l,m^{(%
\widehat{\mathbf{k}})};\theta ,\varphi ) \label{hu102}
\end{equation}
so that $Y(l,m^{(\widehat{\mathbf{k}})};\theta ,\varphi )$ has even parity
if $l $ is even and odd parity if $l$ is odd.
The generalized spherical harmonics have the form
\begin{equation}
Y(l,m^{(\widehat{\mathbf{a}})};\theta ,\varphi )=\sum_jc_jY(l,m_j^{(\widehat{%
\mathbf{k}})};\theta ,\varphi ) \label{hu103}
\end{equation}
where $c_j=\chi (l,m_i^{(\widehat{\mathbf{a}})};l,m_f^{(\widehat{\mathbf{k}}%
)})$ is a constant with respect to the angles $(\theta ,\varphi ).$ Hence,
\begin{equation}
\rho Y(l,m^{(\widehat{\mathbf{a}})};\theta ,\varphi )=\sum_jc_j\rho
Y(l,m_j^{(\widehat{\mathbf{k}})};\theta ,\varphi )=(-1)^lY(l,m^{(\widehat{%
\mathbf{a}})};\theta ,\varphi ) \label{hu104}
\end{equation}
Thus, the generalized spherical harmonics have the same parity as the
corresponding standard spherical harmonics.
The generalized spherical harmonics for value of l can be shown to be
orthonormal:
\begin{equation}
\iint Y^{*}(l,m^{\prime (\widehat{\mathbf{a}})};\theta ,\varphi )Y(l,m^{(%
\widehat{\mathbf{a}})};\theta ,\varphi )d\Omega =\delta _{m^{\prime }m}
\label{hu105}
\end{equation}
Thus, since
\begin{equation}
Y(l,m^{(\widehat{\mathbf{a}})};\theta ,\varphi )=\sum_j\chi (l,m;l,m_j^{(%
\widehat{\mathbf{k}})})Y(m_j^{(\widehat{\mathbf{k}})};\theta ,\varphi )
\label{hu106}
\end{equation}
and
\begin{equation}
Y^{*}(l,m^{\prime (\widehat{\mathbf{a}})};\theta ,\varphi )=\sum_{j^{\prime
}}\chi ^{*}(l,m^{\prime (\widehat{\mathbf{a}})};l,m_{j^{\prime }}^{(\widehat{%
\mathbf{k}})})Y^{*}(m_{j^{\prime }}^{(\widehat{\mathbf{k}})};\theta ,\varphi
) \label{hu107}
\end{equation}
the overlap integral is
\begin{eqnarray}
I &=&\iint \sum_{j^{\prime }}\chi ^{*}(l,m^{\prime (\widehat{\mathbf{a}}%
)};l,m_{j^{\prime }}^{(\widehat{\mathbf{k}})})Y^{*}(l,m_{j^{\prime }}^{(%
\widehat{\mathbf{k}})};\theta ,\varphi ) \nonumber \\
&&\times \sum_j\chi (l,m^{(\widehat{\mathbf{a}})};l,m_j^{(\widehat{\mathbf{k}%
})})Y(l,m_j^{(\widehat{\mathbf{k}})};\theta ,\varphi )d\Omega \\
&=&\sum_{j^{\prime }}\sum_j\chi ^{*}(l,m^{\prime (\widehat{\mathbf{a}}%
)};l,m_{j^{\prime }}^{(\widehat{\mathbf{k}})})\chi (l,m^{(\widehat{\mathbf{a}%
})};l,m_j^{(\widehat{\mathbf{k}})}) \nonumber \\
&&\times \iint Y^{*}(l,m_{j^{\prime }}^{(\widehat{\mathbf{k}})};\theta
,\varphi )Y(l,m_j^{(\widehat{\mathbf{k}})};\theta ,\varphi )d\Omega \\
&=&\sum_{j^{\prime }}\sum_j\chi ^{*}(l,m^{\prime (\widehat{\mathbf{a}}%
)};l,m_{j^{\prime }}^{(\widehat{\mathbf{k}})})\chi (l,m^{(\widehat{\mathbf{a}%
})};l,m_j^{(\widehat{\mathbf{k}})})\delta _{m_jm_{j^{\prime }}} \nonumber \\
&=&\sum_j\chi ^{*}(l,m^{\prime (\widehat{\mathbf{a}})};l,m_j^{(\widehat{%
\mathbf{k}})})\chi (l,m;l,m_j^{(\widehat{\mathbf{k}})}) \nonumber \\
&=&\delta _{m^{\prime }m} \label{hu108}
\end{eqnarray}
In the proof, we have used the result
\begin{equation}
\sum_j\chi ^{*}(l,m^{\prime (\widehat{\mathbf{a}})};l,m_j^{(\widehat{\mathbf{%
k}})})\chi (l,m;l,m_j^{(\widehat{\mathbf{k}})})=\delta _{m^{\prime }m}
\label{hu109}
\end{equation}
which is just the orthonormality relation for the spin probability
amplitudes.
\section{Discussion and Conclusion}
This work has extended the derivation of the new generalized spherical
harmonics to the case $l=2$. The expressions for the functions have been
derived, as well as the corresponding probability densities for the $%
z^{\prime }$ direction. By means of simple transformations the corresponding
expressions for the $x^{\prime }$ and $y^{\prime }$ directions have been
obtained.
Now, for the case $l=1$, it has been shown that the generalized spherical
harmonics satisfy the eigenvalue equation[11]
\begin{equation}
L_{(\widehat{\mathbf{a}})}Y(1,m^{(\widehat{\mathbf{a}})};\theta ,\varphi
)=m\hbar Y(1,m^{(\widehat{\mathbf{a}})};\theta ,\varphi ) \label{hu110}
\end{equation}
where
\begin{equation}
L_{(\widehat{\mathbf{a}})}=i\hbar \{\sin \theta ^{\prime }\sin (\varphi
-\varphi ^{\prime })\frac \partial {\partial \theta }+[\sin \theta ^{\prime
}\cot \theta \cos (\varphi -\varphi ^{\prime })-\cos \theta ^{\prime }]\frac
\partial {\partial \varphi }\} \label{hu111}
\end{equation}
We note that $L_{(\widehat{\mathbf{a}})}$ can also be written as $%
L_{z^{\prime }}$ since as argued in the section on probability amplitudes
and probability densities, it is convenient to think of the vector $\widehat{%
\mathbf{a}}$ as defining a new $z$ direction, denoted by $z^{\prime }$.
It is expected that all generalized spherical harmonics satisfy the
eigenvalue equation, Eq. (\ref{hu110}). This is tedious to prove in
practice, and has not been done for the present case $l=2$. This will be
tackled in the near future, since it is an important part of the proof of
the correctness of the philosophy underlying this work.
\section{References}
1. Mweene H. V., ''Derivation of Spin Vectors and Operators From First
Principles'', quant-ph/9905012
2. Mweene H. V., ''Generalized Spin-1/2 Operators and Their Eigenvectors'',
quant-ph/9906002
3. Mweene H. V., ''Alternative Forms of Generalized Vectors and Operators
for Spin 1/2'', quant-ph/9907031
4. Mweene H. V., ''Spin Description and Calculations in the Land\'e
Interpretation of Quantum Mechanics'', quant-ph/9907033
5. Mweene H. V., ''Vectors and Operators for Spin 1 Derived From First
Principles'', quant-ph/9906043
6. Mweene H. V., Unposted results on spin 3/2 systems.
7. Mweene H. V., ''Generalized Probability Amplitudes for Spin Projection
Measurements on Spin 2 Systems'', quant-ph/0502005
8. Mweene H. V., Unposted results on spin 5/2 systems.
9. Mweene H. V., ''New Treatment of Systems of Compounded Angular
Momentum'', quant-ph/9907082.
10. Mweene H. V., ''Derivation of Standard Treatment of Spin Addition From
Probability Amplitudes'', quant-ph/0003056
11. Mweene H. V., ''Generalized Spherical Harmonics'', quant-ph/0211135
12. Land\'e A., ''From Dualism To Unity in Quantum Physics'', Cambridge
University Press, 1960.
13. Land\'e A., ''New Foundations of Quantum Mechanics'', Cambridge
University Press, 1965.
14. Land\'e A., ''Foundations of Quantum Theory,'' Yale University Press,
1955.
15. Land\'e A., ''Quantum Mechanics in a New Key,'' Exposition Press, 1973.
16. Mweene H. V., ''Proposed Differential Equation for Spin 1/2'', \textit{%
Proc. Third Int. Workshop on Contemporary Problems in Mathematical Physics,
Cotonou 2003}, ed. J. Govaerts, M. N. Hounkounnou and A. Z. Msezane (World
Scientific, 2004), quant-ph/0411060.
\end{document}
|
{
"timestamp": "2005-03-05T07:21:54",
"yymm": "0503",
"arxiv_id": "quant-ph/0503059",
"language": "en",
"url": "https://arxiv.org/abs/quant-ph/0503059"
}
|
\section{1. Introduction}
Quantum algorithms provide elegant opportunities to harness available quantum resources and perform certain computational
tasks more efficiently than classical devices. The idea that a quantum computer
could simulate the physical behavior of a quantum system as well as perform computation, attracted immediate attention \cite{preskill,ss}.
The theory of such quantum computers is now well understood and several quantum algorithms like Deutsch-Jozsa (DJ) algorithm \cite{deu},
Grover's search algorithm \cite{grover}, Shor's prime factorization algorithm \cite{shor}, Hogg's algorithm \cite{hogg},Bernstein-Vazirani
problem \cite{vazi} and quantum counting \cite{count1} have been developed . All these algorithms start from a well-defined initial state
and perform computation by a sequence of reversible logic gates. After computation, the final state of the system gives the output.
Various methods are being examined for building a quantum information processing (QIP) device which is coherent and unitary \cite{bou}.
Nuclear Magnetic Resonance has emerged as a leading candidate
for implementation of various quantum computational problems on a small number of qubits
\cite{cory97,chuang97,cory98,djchu,djjo,grochu,grojo,ka1,ka,jcp,nat,ranapra2,ijqi,ranapra1,ranabirtomo}.
Quantum adiabatic algorithms provide an alternative method for computing \cite{ad1,ad2}. In this method the
computation is done by evolving the system under a Hamiltonian for a given amount of time.
Such algorithms start from a suitable input ground state and by evolution under a slowly time-varying Hamiltonian, reach the
desired output state. Quantum adiabatic algorithms have been efficiently applied to solve various optimization problems
\cite{ad3,ad4,ad5,ad6}. Chuang {\it et al.} have demonstrated the implementation of a quantum adiabatic algorithm by solving the MAX-CUT \cite{garey}
problem on a three qubit system by NMR \cite{chu} . In these algorithms, the condition for adiabaticity is fulfilled globally by using only
the minimum energy gap between the ground state and the first excited state for calculating the time of evolution. This method of evolution
is not efficient in some cases such as adiabatic Grover's search algorithm and adiabatic \dj algorithm as they result in a complexity
O(N) (N is the size of the data set), which is as good as their classical algorithms. However, these algorithms can be
improved by application of local adiabatic evolution, where the adiabatic condition is fulfilled at each instant of time. This technique
has been adopted theoretically by Roland and Cerf \cite{cerf} for the adiabatic Grover's search algorithm and by S. Das
{\it et al.} for adiabatic \dj algorithm \cite{das} yielding a complexity O($\sqrt{N}$). Experimental implementation of
adiabatic Grover's search algorithm based on the proposal of Roland and Cerf and adiabatic \dj algorithm of S. Das {\it et al.}, is
reported here. Section 2 contains an introduction to adiabatic algorithms. Section 3 discusses the adiabatic version of the Grover's search
algorithm proposed by Roland and Cerf and its NMR implementation. Section 4 discusses the adiabatic \dj algorithm and its NMR
implementation. Section 5 contains the experimental results, on a 2-qubit system, for both these algorithms. To the best of our knowledge
this is the first experimental implementation of adiabatic Grover's search and adiabatic \dj algorithms.
\section{2. Adiabatic Algorithm}
The adiabatic theorem of quantum mechanics states that when a system is evolved under a slowly time varying Hamiltonian, it stays
in its instantaneous ground state \cite{me}. This fact is used in solving certain
computational problems \cite{ad3,ad4,ad5,ad6}. The problem to be solved is encoded in a final Hamiltonian ($H_F$),
whose ground state is not easy to find. Adiabatic algorithms start with the ground state of a beginning Hamiltonian
($H_B$) which is easy to construct and whose ground state is also easy to prepare. The ground state of $H_B$, which is a superposition
of all the eigenstates of H$_F$, is evolved under a time varying Hamiltonian $H(s)$. $H(s)$ is a linear interpolation of the beginning
Hamiltonian $H_B$ and the final Hamiltonian $H_F$ such that
\begin{eqnarray}
H(s)= (1-s)H_B + s H_F, \hspace{3cm}\mbox{where}\;\;\; 0\leq s \leq 1. \label{hs}
\end{eqnarray}
The parameter $s=t/T_{total}$, where $T_{total}$ is the total time of evolution and $t$ varies from 0 to $T_{total}$. After evolution under
the Hamiltonian $H(s)$ for a time $T_{total}$, the system is in the ground state of $H_F$ with a probability $(1-\varepsilon^2)^2$,
provided the evolution rate satisfies,
\begin{eqnarray}
\frac{\underset{0\leq s \leq 1}{max}\left|\left<1;s\left|\frac{dH(s)}{dt}\right| 0;s\right>\right|}{g_{min}^{2}} \leq \varepsilon, \label{r1} \label{epsilon}
\end{eqnarray}
and the parameters of the algorithm are chosen to make $\varepsilon \ll$1 \cite{ad1}.
The numerator in Eq. 2 is the transition amplitude between the ground state and the first excited state of {\it H(s)}, and the denominator
is the square of the smallest energy gap $(g_{min})$ between them. Ideally the time of evolution ($T_{total}$) must be infinite. However as
long as the gap is finite, for any finite and positive $\varepsilon$, the time of evolution can be finite. The time of evolution
of the algorithm is determined by the minimum energy gap between the ground state and the first excited state.
In the adiabatic case the time of evolution determines the complexity of the algorithms (that is how long
it takes for the task to be completed), which can then be compared to the complexity of the discrete algorithms in classical and
quantum paradigms. The time of evolution is measured in units of natural time scale associated with the system, $\bar T$ which is
O($\hbar /{\bar E}$) where $\bar E$ is the fundamental energy scale associated with the physical system used to construct the states.
\cite{das}.
\indent In the actual implementation, the Hamiltonian $H(s)$ is discretized into $M+1$ steps as
$H(\frac{m}{M})$ where m goes from $0\rightarrow M$ \cite{chu,wvd}. Thus the time varying
Hamiltonian $H(s)$ goes from beginning Hamiltonian to final Hamiltonian in M+1 steps. As the total number of steps increase, the evolution
becomes more and more adiabatic \cite{chu}. The evolution operator for the m$^{\rm th}$ step is given by \cite{chu}
\begin{eqnarray}
U_m=e^{-i[(1-\frac{m}{M})H_B + \frac{m}{M}H_F]\Delta t},
\end{eqnarray}
where $ \Delta t=T/(M+1)$. The total evolution is given by,
\begin{eqnarray}
U=\prod_{m=0}^M U_m.
\end{eqnarray}
Since, $H_B$ and $H_F$ do not commute in general, the evolution operator of Eq. 3 is approximated to first order in $\Delta$t, by the use of
the Trotter's formula \cite{chu} as
\begin{eqnarray}
U_m \approx e^{-iH_{B}(1-\frac{m}{M})\frac{\Delta t}{2}}\cdot e^{-iH_{F}\frac{m}{M}\Delta t}\cdot e^{-iH_{B}(1-\frac{m}{M})\frac{\Delta t}{2}}. \label{eq:trot}
\end{eqnarray}
Thus in each step only a small evolution of the system from ground state of ${\rm H_B}$ towards the ground state of ${\rm H_F}$ takes place.
\section{3. Grover's search algorithm}
Suppose we are given an unsorted database of N items and one of those items is marked. To search for the marked item classically, it would
require on an average N/2 queries. However using quantum resources, the algorithm prescribed by Grover \cite{grover} performs the same
search with O($\sqrt{N}$) queries. The algorithm starts with an equal superposition of states, representing the items,
repeatedly flips the amplitude of the marked state (done by the oracle) followed by the flip of the
amplitudes of all the states about the mean. The number of times this process is repeated determines the complexity of the algorithm and
this scales with the size of the database as O($\sqrt{N}$). \\
\indent In the adiabatic version, the system is evolved under a time dependent Hamiltonian which is a linear interpolation of H$_B$ and
H$_F$. As n qubits are used to label a database of size N (=2$^n$), the resulting Hilbert space is of dimension N. The basis
states in this space are $\vert i\rangle$ where i=0,$\cdots$,N. H$_B$ is chosen such that the ground state is a linear
superposition of all the basis states. Therefore for a 2-qubit case,
\begin{eqnarray}
\vert\psi_B\rangle &=& \frac{1}{2}\left(\vert 00\rangle + \vert 01\rangle + \vert 10\rangle + \vert 11\rangle\right). \label{groinistate}\\
H_B &=& I - \vert\psi_B\rangle\langle\psi_B\vert, \cr
&=& I - \frac{1}{4}\begin{pmatrix}1&1&1&1 \cr 1&1&1&1 \cr 1&1&1&1 \cr 1&1&1&1\end{pmatrix}.\label{grohb}
\end{eqnarray}
The Final Hamiltonian has the marked state $\vert\psi_F \rangle$ as the ground state.
\begin{eqnarray}
H_F &=& I - \vert\psi_F \rangle\langle\psi_F \vert. \label{grohf}
\end{eqnarray}
The rate at which the interpolating Hamiltonian H(s) (given by Eq. \ref{hs}) changes from $H_B$ to $H_F$ depends on the condition,
\begin{eqnarray}
\left|\frac{ds}{dt}\right| \leq \varepsilon \frac{g^{2}(s)}{\left|\langle\frac{dH}{ds}\rangle\right|}. \label{adcon}
\end{eqnarray}
Following Roland and Cerf \cite{cerf}, t is obtained as a function of s as,
\begin{eqnarray}
t=\frac{1}{2\varepsilon}\frac{N}{\sqrt{N-1}}\left[arctan\{\sqrt{N-1}\left(2s-1\right)\}+arctan\sqrt{N-1}\right].
\end{eqnarray}
Taking t$'= \varepsilon t$ and on inverting the above function, s(t$'$) is obtained as
\begin{eqnarray}
s(t') = \frac{1}{2}\left[\{ \frac{1}{\sqrt{N-1}}tan\left(\frac{2\sqrt{N-1}t'}{N} - arctan\sqrt{N-1} \right)\}+1\right]. \label{stprime}
\end{eqnarray}
The plot of this function for N=4 (for a 2 qubit case) is given in Fig. 1. In the experiment the time of
evolution is varied according to Eq. \ref{stprime}. It has been shown by Roland and Cerf \cite{cerf} that with this
adiabatic evolution, the complexity of the algorithm is O($\sqrt{N}$).
\section{3.1. Experimental Implementation}
The NMR Hamiltonian for a weakly coupled two-spin system is :
\begin{eqnarray}
{\mathcal H}= -\omega_1 I_{z1} - \omega_2 I_{z2} +2\pi J_{12}I_{z1}I_{z2}. \label{nmrham}
\end{eqnarray}
where $\omega_1$ and $\omega_2$ are Larmour frequencies and $J_{12}$ the indirect spin-spin coupling. The beginning Hamiltonian for a
2-qubit Grover's algorithm as stated in Eq. \ref{grohb}, written in terms of spin-half operators, is
\begin{eqnarray}
{\mathcal H}_B = \frac{3}{4} I - \frac{1}{2} \{I_{x1} + I_{x2} + 2 I_{x1}I_{x2}\}.
\end{eqnarray}
The identity term does not cause any evolution of the state and so it can be omitted, yielding the beginning Hamiltonian without
the negative sign and the factor half as:
\begin{eqnarray}
\tilde{\mathcal H}_B = I_{x1} + I_{x1} + 2I_{x1}I_{x2} \label{nmrhb}
\end{eqnarray}
The evolution under $\tilde{\mathcal H}_B$ can be simulated by a free evolution under the Hamiltonian ${\mathcal H}$ of Eq. \ref{nmrham} between two
$\pi$/2 pulses with appropriate phases.
\begin{eqnarray}
e^{i\frac{\pi}{2}\left(I_{y1} + I_{y2}\right)}\cdot e^{i{\mathcal H}T}\cdot e^{-i\frac{\pi}{2}\left(I_{y1} + I_{y2}\right)}
&=&e^{i\left(\omega_1 I_{x1} + \omega_2 I_{x2} + 2JI_{x1}I_{x2}\right)T}\cr &=& e^{i{\mathcal H}'T} \label{hbevol}
\end{eqnarray}
Let the state $\vert 00\rangle$ be the marked state. The final Hamiltonian is,
\begin{eqnarray}
H_{F}^{\vert 00\rangle} = I - \begin{pmatrix}1&0&0&0\cr 0&0&0&0\cr 0&0&0&0\cr 0&0&0&0\cr\end{pmatrix}
\end{eqnarray}
In terms of spin operators the final Hamiltonian is,
\begin{eqnarray}
{\mathcal H}_{F}^{\vert 00\rangle} = \frac{3}{4} I - \frac{1}{2}\left[I_{z1} + I_{z2} + 2I_{z1}I_{z2}\right]
\end{eqnarray}
The final Hamiltonian keeping the spin operator terms only and without the negative sign and the factor half is
\begin{eqnarray}
\tilde{\mathcal H}_{F}^{\vert 00\rangle} = I_{z1} + I_{z2} + 2I_{z1}I_{z2} \label{grohf00}
\end{eqnarray}
Similarly the final Hamiltonian for other states being marked, in terms of the spin-half operators, is
\begin{eqnarray}
\tilde{\mathcal H}_{F}^{\vert 01\rangle}&=&I_{z1} - I_{z2} - 2I_{z1}I_{z2} \label{grohf01} \\
\tilde{\mathcal H}_{F}^{\vert 10\rangle}&=&-I_{z1} + I_{z2} - 2I_{z1}I_{z2} \label{grohf10} \\
\tilde{\mathcal H}_{F}^{\vert 11\rangle}&=&-I_{z1} - I_{z2} + 2I_{z1}I_{z2} \label{grohf11}
\end{eqnarray}
\indent The schematic representation of the experiment for the adiabatic Grover's algorithm in a two qubit system [consisting of a $^1$H
spin and a $^{13}$C spin] is shown in Fig. 2a. The experiment is divided into three parts. The
first part (preparation part) consists of preparation of pseudo-pure state (PPS) followed by equal superposition. The second part is the
adiabatic evolution, and the third part is the tomography of the resultant state. The pulse programme for the
preparation of PPS and equal superposition is shown in Fig. 2b. The PPS is prepared by the method of spatial averaging \cite{du}. After
preparing PPS, equal superposition of states is obtained by application of the Hadamard gate on both the qubits. The Hadamard gate is
implemented by $(\pi/2)_y$ -pulses, followed by $\pi_x$ -pulses on both proton and carbon spins (Fig. 2b) \cite{grochu}.
The next stage consists of adiabatic evolution which has been carried out in the present work in 60 steps. Each step of the
adiabatic evolution (Figs. 2c, 2d, 2e and 2f) consists of evolution under the final Hamiltonian for a time $\tau$ sandwiched between two
evolutions under the beginning Hamiltonian for a time (T-$\tau$)/2. T is the total evolution time for one step and
is equal to 1/$\pi$J. The value of $\tau (= s\times \frac{1}{\pi J})$ varies from 0 to T takes place as `s' increases from
0 to 1 according to Eq. \ref{stprime}, in 60 steps. The pulse sequence for the beginning Hamiltonian is a free evolution of the system
juxtaposed between two $\pi$/2 pulses with appropriate phases on each of the spins (the part marked as H$_B$ in Figs. 2c-2f).
The pulse sequence for the final Hamiltonian depends on the marked state as stated in Eqs. \ref{grohf01}-\ref{grohf11}.
If the state $\vert 00\rangle$ is the marked state, then the pulse sequence for the implementation of the final Hamiltonian is a free
evolution of the system under the NMR Hamiltonian juxtaposed between two $\pi$ pulses on each of the spins (Fig. 2c). Similarly, if the
state $\vert 01\rangle$ is marked the pulse sequence for the final Hamiltonian is a free evolution of the system between two $\pi$ pulses
on the spin 1 (Fig. 2d), if the state $\vert 10\rangle$ is marked then the pulse sequence is a free evolution between two $\pi$ pulses on
the spin 2 (Fig. 2e) and if the state $\vert 11\rangle$ is marked, then the pulse sequence simulating the final Hamiltonian is just a free
evolution of the system under the NMR Hamiltonian (Fig. 2f).
\indent The third stage of the experiment is the tomography of the final density matrix after the adiabatic evolution. The density matrix
of a 2-spin system is a 4$\times$4 matrix consisting of 6 independent off-diagonal complex elements (the remaining 6 are their complex
conjugates), and the four diagonal elements which are the populations of the various levels. The diagonal elements are measured by
90$^o$ pulses on each qubit preceded by a gradient pulse.
The six off-diagonal elements consist of four single quantum (SQ), one double quantum (DQ) and one zero quantum (ZQ) coherences. The real
and the imaginary SQ, DQ and ZQ coherences in terms of the spin operators are;
\begin{eqnarray}
{\rm SQ}^{real}_{i} &=& I_{ix}\pm 2(I_{ix}I_{jz}), \cr
{\rm SQ}^{imag}_{i} &=& I_{iy}\pm 2(I_{iy}I_{jz}), \cr
{\rm DQ}^{real} &=& 2(I_{ix}I_{jx} - I_{iy}I_{jy}), \cr
{\rm DQ}^{imag} &=& 2(I_{iy}I_{jx} + I_{ix}I_{jy}), \cr
{\rm ZQ}^{real} &=& 2(I_{ix}I_{jx} + I_{iy}I_{jy}), \cr
{\rm ZQ}^{imag} &=& 2(I_{iy}I_{jx} - I_{ix}I_{jy}),
\end{eqnarray}
where i$\neq$ j = 1,2 represents the qubits. Although the single quantum terms are directly observable, for proper scaling,
all the off-diagonal elements are observed by a common protocol of two experiments;
\begin{eqnarray}
{\rm A:}\hspace{3cm}\left(\frac{\pi}{2}\right)^{i}_{\phi_1}\left(\theta\right)^{j}_{\phi_2} \longrightarrow &G_{z}& \longrightarrow\left(\frac{\pi}{2}\right)^{i}_{y}, \\
{\rm B:}\hspace{3cm}\left(\frac{\pi}{2}\right)^{i}_{\phi_1}\left(\theta\right)^{j}_{\phi_2} \longrightarrow &G_{z}& \longrightarrow\left(\pi
\right)^{j}\left(\frac{\pi}{2}\right)^{i}_{y}.
\end{eqnarray}
where $\theta$ denotes the pulse angle, $\phi_1$, $\phi_2$ the pulse phases and $G_z$ a gradient pulse. The first two pulses of the
experiment A (depending on the pulse angle $\theta$ and the pulse phases $\phi_1$ and $\phi_2$) convert terms like
$I_{i\alpha}+2I_{i\alpha}I_{j\beta}$ into diagonal terms given by $I_{iz}+2I_{iz}I_{jz}$, where $\alpha$ and
$\beta$ denote the x, y, or z component of the spin operators of the first and the second qubit respectively. The gradient destroys all the
transverse magnetization retaining only the longitudinal terms. The last pulse converts the retained longitudinal magnetization
$I_{iz}+2I_{iz}I_{jz}$ into observable terms $I_{ix}+2I_{ix}I_{jz}$. Thus the magnitude of $I_{i\alpha}+2I_{i\alpha}I_{j\beta}$ is mapped on
to $I_{ix}+2I_{ix}I_{jz}$ which is then observed. In experiment B, a $\pi$-pulse is applied on the spin `j' just before the $\pi$/2 pulse on
the spin `i'. This creates the observable term $I_{ix}-2I_{ix}I_{jz}$. The sum and difference of the two experiments yields 2$I_{i\alpha}$
and 2$I_{i\alpha}I_{j\beta}$ respectively. Six different experiments are needed to be performed to map the whole density matrix (real and
imaginary). The various pulse angles and phases required during the experiment, and the resultant terms that are observed due to them are
given in Table I. Experiments I and II yield the SQ, and experiments III-VI yield the ZQ and DQ coherences.
\section{4. Deutsch-Jozsa Algorithm}
The \dj Algorithm determines whether a binary function $f(x)$,
\begin{eqnarray*}
f(x\vert x\in\{ 0,1\}^n) \rightarrow \{0,1\},
\end{eqnarray*}
is Constant or Balanced \cite{dja}.A constant function implies that the function has the same value 0 or 1 for all $x$. A balanced
function implies that the function {\it`f'} is 0 for half the values of $x$ and 1 for the other half . For a two qubit case the constant and
the balanced functions are given in Table II.
In the adiabatic version of the \dj algorithm, the beginning Hamiltonian and its ground state, for a two qubit system, is given by
Eq. \ref{grohb} and Eq. \ref{groinistate} respectively. The final Hamiltonian is given by Eq. \ref{grohf} and the
ground state of the final Hamiltonian for two qubits is of the form \cite{das};
\begin{eqnarray}
\vert\psi_F\rangle &=& \alpha\vert00\rangle + \frac{\beta}{\sqrt 3}\left(\vert01\rangle + \vert10\rangle + \vert11\rangle \right), \label{djfinstate}
\end{eqnarray}
where
\begin{eqnarray}
\alpha &=& \frac{1}{4}\left|(-1)^{f(00)}+(-1)^{f(01)}+(-1)^{f(10)}+(-1)^{f(11)}\right|, \cr \beta^2 &=& 1-\alpha^2. \label{ab}
\end{eqnarray}
From Eq. \ref{ab} it is seen that when $\alpha=1$ the function $f$ is constant, and when $\alpha=0$ then it
is balanced. Thus $\alpha$ is chosen depending on whether the function to be encoded in the final Hamiltonian is constant or balanced.
\indent Using Eqs. \ref{hs},\ref{grohb},\ref{grohf},\ref{djfinstate} and \ref{ab} the matrix for the interpolating Hamiltonian($H(s)$) can be written as \cite{das};
\begin{eqnarray}
H(s)=I-\frac{1-s}{4}
\begin{pmatrix}
1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1
\end{pmatrix} -\frac{s}{3}
\begin{pmatrix}
3\alpha & 0 & 0 & 0 \\
0 & \beta & \beta & \beta \\
0 & \beta & \beta & \beta \\
0 & \beta & \beta & \beta
\end{pmatrix}. \label{hsmatrix}
\end{eqnarray}
S. Das {\it et al.\;} have shown that on evolution under the Hamiltonian $H(s)$ takes the initial state $\vert\psi_B\rangle$ to the
solution state $\vert\psi_{F}\rangle$ \cite{das}. In the next section we describe an NMR implementation of the above algorithm.
\section{4.1. NMR Implementation}
The adiabatic \dj algorithm also, is implemented on the 2-qubit system. The beginning Hamiltonian in terms of the spin-half
operators is the same as given in Eq. \ref{nmrhb}, and its implementation has been discussed in section 3.1. \\
\indent The final Hamiltonian, obtained from Eqs. \ref{grohb}, \ref{grohf}, \ref{djfinstate} and \ref{ab}, for constant case ($\alpha$=1) yields,
\begin{align}
H_{F}^{c} = I - \begin{pmatrix}1&0&0&0 \cr 0&0&0&0 \cr 0&0&0&0 \cr 0&0&0&0\end{pmatrix},\\
\intertext{and for balanced case ($\alpha$=0) yields,}
H_{F}^{b} = I - \frac{1}{3}\begin{pmatrix}0&0&0&0 \cr 0&1&1&1 \cr 0&1&1&1 \cr 0&1&1&1\end{pmatrix}.
\end{align}
The above final Hamiltonians in terms of spin-half operators can be written respectively as,
\begin{align}
{\mathcal H}_F^{c} = \frac{3}{4}I - \frac{1}{2}&(I_{z1} + I_{z2} + 2I_{z1}I_{z2}), \intertext{and,}
{\mathcal H}_F^{b} = \frac{3}{4}I -\frac{1}{3}&\biggl[-\frac{1}{2}(I_{z1} + I_{z2} + 2I_{z1}I_{z2}) + 2(I_{x1}I_{x2} + I_{y1}I_{y2}) \cr & + I_{x1} + I_{x2} - 2(I_{x1}I_{z2} + I_{z1}I_{x2})\biggr].
\end{align}
As the identity does not cause any evolution of the state we consider only the spin operator terms. Thus the final Hamiltonian keeping only
the spin operators (dropping the minus sign), for the constant case, can be written as
\begin{eqnarray}
\tilde{{\mathcal H}}^{c}_F &=& \frac{1}{2}\{I_{z1} + I_{z2} + 2I_{z1}I_{z2}\}, \label{nmrhcf}
\end{eqnarray}
and for the balanced case as
\begin{eqnarray}
\tilde{{\mathcal H}}^{b}_F = - \frac{1}{6}(I_{z1} &+ I_{z2} + 2I_{z1}I_{z2})+\frac{2}{3}(I_{x1}I_{x2}+I_{y1}I_{y2}) \cr
&+\frac{1}{3}I_{x1}+\frac{1}{3}I_{x2}-\frac{2}{3}(I_{x1}I_{z2}+I_{z1}I_{x2}). \label{nmrhbf}
\end{eqnarray}
The signs of Eqs. \ref{nmrhb}, \ref{nmrhcf} and \ref{nmrhbf} are changed for consistency.
Since the various terms in Eq. \ref{nmrhbf} do not commute, the evolution under this Hamiltonian would require a complex pulse sequence in
NMR. However, we have found that by keeping only the diagonal terms in the Eq. \ref{nmrhbf}, the pulse sequence simplifies considerably with
the information regarding the balanced nature of the problem still encoded in it. This truncated final Hamiltonian for the balanced case
is given by;
\begin{eqnarray}
(\tilde{\mathcal H}^{b}_{F})^{trunc} = -\frac{1}{6}(I_{z1} + I_{z2} + 2I_{z1}I_{z2}) \label{nmrhbftrunc}
\end{eqnarray}
The opposite signs of Eq. \ref{nmrhcf} and Eq. \ref{nmrhbftrunc} distinguish the constant and the balanced case.
\indent In the following we show that the balanced nature of the \dj problem is still encoded in $(\tilde{\mathcal H}_{F}^{b})^{trunc}$.
Substituting $\alpha=0$ and $\beta=1$ and dropping the off-diagonal terms from the last part of Eq. \ref{hsmatrix} , we obtain
\begin{eqnarray}
\tilde{H}^{b}(s) =I-\frac{1-s}{4} \begin{pmatrix}
1&1&1&1\\1&1&1&1\\1&1&1&1\\1&1&1&1
\end{pmatrix}
-\frac{s}{3}\begin{pmatrix}
0&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1
\end{pmatrix}.
\end{eqnarray}
The eigenvalues of this Hamiltonian are:
\begin{eqnarray}
\lambda_0 &=&\frac{1}{6}\left[3+2s-\sqrt{9 + s(7s-15)}\right], \\
\lambda_1 &=&\frac{1}{6}\left[3+2s+\sqrt{9 + s(7s-15)}\right], \\
\lambda_2 &=&\lambda_3 = 1-\frac{s}{3}.
\end{eqnarray}
The values of $\lambda_0$, $\lambda_1$, $\lambda_2$ and $\lambda_3$ as a function of `s' are plotted in Fig. 3.
$\lambda_0$ is the ground state. As `s' increases from 0, $\lambda_0$ continues to be the ground state and becomes the ground state
of the final Hamiltonian in the limit $s\rightarrow 1$. The eigenvectors corresponding to $\lambda_0$,
$\lambda_1$, $\lambda_2$ and $\lambda_3$ are respectively obtained as;
\begin{eqnarray}
v_0 \!=\! \begin{pmatrix}
\frac{3-s-2\sqrt{9-15s+7s^2}}{3(s-1)} \cr 1\cr 1\cr 1
\end{pmatrix},\;
v_1 \!=\!
\begin{pmatrix}
\frac{3-s+2\sqrt{9-15s+7s^2}}{3(s-1)} \cr 1\cr 1\cr 1
\end{pmatrix},\;
v_2 \!=\!
\begin{pmatrix}
0 \cr -1 \cr 0 \cr 1
\end{pmatrix},\;
v_3 \!=\!
\begin{pmatrix}
0 \cr -1 \cr 1 \cr 0
\end{pmatrix},
\end{eqnarray}
The final state to which the system converges after the evolution is
\begin{eqnarray}
\underset{s\rightarrow 1}{lim}\;v_0 = \begin{pmatrix}0\cr 1\cr 1\cr 1 \end{pmatrix},
\end{eqnarray}
which is the desired output state.
The energy gap between the ground state and the states corresponding to
$\lambda_2$ and $ \lambda_3$ goes to zero as $s\rightarrow 1$ as shown in Fig. 3. However, there is no transition from
$\lambda_0$ to $\lambda_2$, $\lambda_3$ as the transition amplitude given by the numerator in
Eq. \ref{epsilon} is zero in these cases. Therefore the transition amplitude from the ground state
$\lambda_0$ to the next excited state $\lambda_1$ is relevant for calculation of $s(t)$. The minimum energy gap between $\lambda_0$ and
$\lambda_1$, needed in Eq. \ref{epsilon}, is obtained for $s \simeq 1$ as seen in Fig. 3. Since the algorithm
is implemented using local adiabatic evolutions we need to change $s(t)$ such that the adiabatic condition \cite{cerf}
\begin{eqnarray}
\frac{ds}{dt} \leq \varepsilon \frac{\left|g(s)\right|^2}{\left|\left<\frac{dH}{ds}\right>\right|},
\end{eqnarray}
is met at each time interval. Here g(s) is the energy gap between the ground state and the first excited state, given by
$\frac{1}{3}\sqrt{9 -15s + 7s^2}$ and $\left|\left< dH/ds\right>\right| = H_F - H_B$. The Hamiltonian is evolved at a rate that is a
solution of
\begin{eqnarray}
\frac{ds}{dt} = \varepsilon \frac{\left|g(s)\right|^2}{\left|H_F - H_B\right|} \label{dsdt}
\end{eqnarray}
On integrating Eq. \ref{dsdt}, we obtain {\it t} as a function of {\it s}.
\begin{eqnarray}
t=\frac{1}{\varepsilon}\frac{14s-15}{2\sqrt{3}\sqrt{7s^2 -15s + 9}} + k,\label{eq:t}
\end{eqnarray}
where the constant of integration $k=\frac{5}{\varepsilon 2\sqrt 3}$ to obey $s=0$ at $t=0$.
Inverting this function we obtain $s(t')$ as
\begin{eqnarray}
s(t')=\frac{3}{14}\left[ 5 - \frac{\sqrt{225 + 24t\left( 55\sqrt{3} - 183t +60\sqrt{3}t^2 - 18t^3\right)}}{3 + 20\sqrt{3}t - 12t^2}\right]
\end{eqnarray}
where $t'$ is $\varepsilon t$.
The plot of $s$ as a function of $t'$ is shown in Fig. 4. From Figs. 3 and 4 it is seen that the rate of change of $s$ (and hence of
the Hamiltonian) is fast when the energy gap between $\lambda_0$ and $\lambda_1$ is large, and slow when the gap is small.
In practice the time of evolution for $H_B$ and $H_F$ is given by $(1-s)\times T$ and $s\times T$ respectively,
where $T$ is 1/$\pi$J and $s$ is varied from 0 to 1 according to Eq. \ref{eq:t}. In our implementation, the $t^{'}$ interval for which
$s$ varies from 0 to 1 is divided in 80 equal steps, and the corresponding values of s for each step (calculated from Eq. 48) are
substituted in the evolution time of $H_B$ and $H_F$. \\
\indent On integrating Eq. 46 from s=0 to s=1, we get the total time of evolution
\begin{eqnarray}
T_{total}=\frac{1}{\varepsilon}\frac{2}{\sqrt{3}}{\bar T}. \label{ttotal}
\end{eqnarray}
T$_{total}$ is given in the units of ${\bar T}$ which is the time scale associated with the physical system used \cite{das}.
The time scale associated with evolution under the NMR Hamiltonian is $\sim 10^{-3} s$. The total time of evolution of the experiment
($T_{total}$) is given by 80$\times$T, where T is the time for one step (see Fig. 5b).
For the choice $\varepsilon\sim 10^{-2}$, T $\sim 60\times 10^{-3}$s in our case.
\section{4.2. Experimental Implementation}
The experimental implementation of adiabatic \dj algorithm on a 2-qubit system [consisting of a $^1$H spin and a $^{13}$C spin]
also consists of three parts namely preparation, adiabatic evolution and tomography of the final density matrix. The preparation of the
pseudo pure state (PPS) and making of equal superposing of states as well as the tomography of the final states has already been discussed
in section 3. So we only describe the method of implementation of the final Hamiltonian for the \dj algorithm.\\
\indent The pulse sequence for the implementation of the constant case final
Hamiltonian ($\tilde{\mathcal H}_{F}^{c}$) is given in Fig. 5b. The beginning Hamiltonian is implemented by a free evolution juxtaposed
between $\pi$/2 pulses with required phases (Fig. 5b). The implementation of the final Hamiltonian for the constant case is a free evolution
under the NMR Hamiltonian of Eq. 14, juxtaposed between two $\pi$-pulses as shown in Fig. 5b. In the balanced case the implementation of the
beginning Hamiltonian is same as in Fig. 5b. However, the implementation of the final Hamiltonian $(\tilde{\mathcal H}_{F}^{b})^{trunc}$ is
done in two parts [Fig. 5c]. The first part is a free evolution under the Hamiltonian given in
Eq. 14 [${\rm T_f}$ period in Fig. 5c]. The operator corresponding to such an evolution for time $\tau$ will be of the form;
\begin{eqnarray}
e^{i\pi J(-I_{z1}-I_{z2}+2I_{z1}I_{z2})\tau}. \label{tf}
\end{eqnarray}
In the second evolution of $2\tau$, the chemical shifts are refocused so that the system evolves only under its scalar coupling Hamiltonian
$2\pi JI_{z1}I_{z2}$.
Just before and after the evolution $\pi$-pulses with appropriate phases are put on each of the spins to flip the sign of the corresponding
spin operator [${\rm T_j}$ period in Fig. 5c]. The operator for the sequence of two pulses with an intermediate evolution for $2\tau$
is of the form
\begin{eqnarray}
e^{-i(I_{x1})\pi}\cdot e^{i\pi J(2I_{z1}I_{z2})2\tau}\cdot e^{i(I_{x1})\pi}= e^{-i\pi J(2I_{z1}I_{z2})2\tau}. \label{tj}
\end{eqnarray}
As these two evolutions given in Eq. \ref{tf} and Eq. \ref{tj} commute, the effective evolution for the 3$\tau$ period is:
\begin{eqnarray}
e^{i\pi J(-I_{z1}-I_{z2}+2I_{z1}I_{z2})\tau} \cdot e^{-i\pi J(2I_{z1}I_{z2})2\tau} =e^{i\pi J(-I_{z1}-I_{z2}-2I_{z1}I_{z2})\tau}.\label{tftj}
\end{eqnarray}
Thus the evolution during ${\rm T_j}$ cancels the J-evolution during ${\rm T_f}$ and adds a minus sign to it, yielding the effective
Hamiltonian of Eq. \ref{tftj} and an effective evolution time of $\tau$. An evolution time of $\tau$=1/$\pi$J implements the full
Hamiltonian of Eq. 39 as required for adiabatic evolution. Overall the cycle time for each step for the balanced
case is increased to T$+2\tau$.\\
\section{5. Experimental Results}
The experiments have been carried out using carbon-13 labeled chloroform ($\rm ^{13}CHCl_3$) where the two spins $^1$H and $^{13}$C form
the two qubit system. The proton spin represents the first qubit and carbon-13 the second. The sample of $\rm ^{13}CHCl_3$ was dissolved in
the solvent CDCl$_3$ and the experiments were performed at room temperature in a magnetic field of 11.2 Tesla. At this field the $^{1}\rm H$
resonance frequency is 500.13 MHz and the $^{13}\rm C$ resonance frequency is 125.76 MHz. During the
entire experiment, the transmitter frequencies of $^{1}\rm H$ and $^{13}\rm C$ are set at a value $J/2$ away from resonance to achieve the
condition $\omega_1=\omega_2=\pi$J. The equilibrium spectra of the two qubits are shown in Fig. 6a, and the spectrum corresponding to
$\vert 00\rangle$ PPS is shown in Fig. 6b.
To quantify the experimental result we calculate the {\it average absolute deviation} \cite{nures} of each element of the experimentally
obtained density matrix from each element of the theoretically predicted density matrix given by,
\begin{equation}
\Delta x=\frac{1}{N^2}\sum^N_{i,j=1}\vert x_{i,j}^{T} - x_{i,j}^{E} \vert \label{error}
\end{equation}
where N=$2^n$ (n being the number of qubits), $x_{i,j}^T$ is $(i,j)^{th}$ element of the theoretically predicted density matrix and
$x_{i,j}^E$ is $(i,j)^{th}$ element of the experimentally obtained density matrix.
\section{5.1. Grover's Search Algorithm}
The experimental spectra corresponding to the implementation of Grover's search algorithm on the above two qubit system are given
in Fig. 7. the spectra given in Figs. 7a(i-iv) contain the reading of populations after respectively searching states $\vert 00\rangle$,
$\vert 01\rangle$,$\vert 10\rangle$ and $\vert 11\rangle$. The population spectra are obtained by
application of a gradient followed by a $\pi$/2 pulse. Depending on the final state, the population spectra consist of one single spectral
line for each spin. These correspond to, $\vert 00\rangle$ $\rightarrow$ $\vert 01\rangle$ and $\vert 00\rangle$ $\rightarrow$
$\vert 10\rangle$transition when the searched state is $\vert 00\rangle$ (Fig. 7a-i); $\vert 01\rangle$ $\rightarrow$ $\vert 00\rangle$ and
$\vert 01\rangle$ $\rightarrow$ $\vert 11\rangle$ when the search state $\vert 01 \rangle$ (Fig. 7a-ii); $\vert 10\rangle$ $\rightarrow$
$\vert 00\rangle$ and $\vert 10\rangle$ $\rightarrow$ $\vert 11\rangle$ when the search state $\vert 10 \rangle$ (Fig. 7a-iii);
$\vert 11\rangle$ $\rightarrow$ $\vert 01\rangle$ and $\vert 11\rangle$ $\rightarrow$ $\vert 10\rangle$ when the search state
$\vert 11 \rangle$ (7a-iv). The coherence spectra in Fig. 7b have been obtained by observing the searched state without application of
any r.f. pulses. The absence of any signal in the spectra confirms that there is no single quantum coherences after the search.
To check for the absence of zero quantum and double quantum coherences as well, the entire density matrix has been tomographed.
Fig 8a shows the theoretical and the experimental density matrices after the adiabatic evolution, when state $\vert 00\rangle$ has been
searched. The mean deviation of the experimentally obtained density matrix from the theoretically predicted one (calculated using
Eq. \ref{error}) is 2.49$\%$. Similarly Figs. 8b, 8c and 8d contain the
theoretically predicted and experimentally obtained density matrices when the states $\vert 01\rangle$, $\vert 10\rangle$ and
$\vert 11\rangle$ have been searched. The mean deviation of the experimental density matrices from their theoretically predicted
counterparts are 1.92$\%$, 1.89$\%$ and 1.97$\%$ respectively.
\section{5.2. Deutsch-Jozsa Algorithm}
{\bf{5.2.1 \em Constant case}}\\
\noindent For the constant case (Eq. \ref{ab}), the state expected after the evolution (using the pulse sequence
given in Fig. 5b) is $\vert00\rangle$. The density matrix consists of population in $\vert 00\rangle$ state and no coherences.
The spectrum corresponding to the population for such a state, obtained by application of a gradient followed by $\pi$/2 pulses on each
of the spins, consists of one single quantum coherence in each spin (`Population spectrum' in Fig. 9a). The spectrum for coherence,
observed without application of any pulses on any of the spins, has a near absence of any signal (`Coherence spectrum' in Fig. 9a).
Further confirmation of the final state is done by the tomography of the complete density matrix. The
Fig. 10 shows the tomography of the experimental and theoretically predicted density matrices of the final state for the constant case.
The mean deviation of the experimental density matrix from the theoretical one is 5.28$\%$ \\
\indent{\bf{5.2.2 \em Balanced case}}\\
\indent
For the balanced case [Eq. 31, $\alpha$=0 and $\beta$=1], the state expected after the evolution (using the pulse sequence of Fig. 5c)
is $\frac{1}{\sqrt{3}}\left(\vert01\rangle +\vert10\rangle +\vert11\rangle\right)$. The theoretical density matrix of the final state
is given in Fig 11(a). This state theoretically has three diagonal elements, one SQ coherence of each qubit and a ZQ coherence between
the two qubits, all of equal intensity. This state is confirmed by the spectra shown in Fig. 9b and the density matrix in Fig. 11(b).
The mean deviation of the experimentally obtained density matrix from the theoretically predicted one is 17.2$\%$.
It is seen that in the density matrix obtained from experiment, the SQ coherence of $^{13}$C (second
qubit) and the ZQ coherence between $^{13}$C and $^1$H have significantly reduced intensity, compared to the theoretically expected
values.
There are three sources of error in adiabatic algorithms. $\varepsilon$ gives a measure of the first source of error. Theoretically
the total time of evolution in adiabatic algorithms should be infinite. However, in practice the evolution is terminated once the
state is supposed to have been reached with sufficiently high probability given by $(1-\varepsilon^2)^2$ which in our case (for
$\varepsilon =10^{-2}$)is obtained to be 99.98$\%$. The second source of error is due to neglect of O($\Delta t^3$) terms in the
Trotter's Formula (Eq. \ref{eq:trot}). The maximum error introduced due to this is $\approx$ 0.92 $\%$ which can be safely neglected.
The third source of error is due to decoherence effects arising from the interaction of the spins with their
surroundings. To study decoherence, the relaxation times T$_1$ and T$_2$ of $^{1}$H and $^{13}$C were measured. The T$_{2}$ for SQ
coherences were measured by CPMG sequence. For the measurement of ZQ and DQ coherence decay rate, the term I$_{1x}$I$_{2x}$ was created
and its relaxation rate was measured by CPMG sequence. The T$_{2}$ of SQ coherence of $^{1}$H was found to be 3.4 s and for $^{13}$C it
was found to be 0.29 s. The
decay rate of I$_{1x}$I$_{2x}$ term was found to be 0.19 s. The T$_{1}$ for $^{1}$H and $^{13}$C measured from the initial part of the
inversion recovery experiment was found to be 21 s for $^1$H and 16s for $^{13}$C. Using these measured values of T$_{1}$ and T$_{2}$ the
simulation for the balanced case was repeated including relaxation using Bloch's equations \cite{ernst}. Significant decay of the carbon
coherences was observed. The mean deviation of the of the experimental density matrix form the theoretical density matrix including
relaxation is found to be 8.0$\%$. \\
\indent The observed mean deviation between the theoretically expected and the experimentally obtained density matrices for the Grover's
search and the constant case of the \dj are small ($<$ 2$\%$ and $<$ 6$\%$ respectively) while that for the balanced case of the \dj is
large ($\sim$ 17$\%$). In the first two cases, the results are encoded in the diagonal elements of the density matrix, which are
attenuated by the spin lattice relaxation, the times for which are large ($>$ 16 sec). On the other hand, in the balanced \dj case,
there are off-diagonal elements as well which are attenuated by spin-spin relaxation, the times for which are small ($<$ 4 sec for $^1$H
and $<$ 0.3 sec for $^{13}$C). The decoherence times thus have a large effect in this case. A correction for the decoherence has improved
the mean deviation considerably (reduced to $\sim$ 8$\%$), confirming the succesful implementation of these algorithms.
\section{6. Conclusion}
In this paper we have demonstrated the experimental implementation of Grover's search and \dj algorithms by using local adiabatic
evolution in a two-qubit quantum computer by nuclear magnetic resonance technique. We have suggested a different Hamiltonian for the
adiabatic \dj algorithm which is diagonal in the computational basis and hence easier to implement by NMR. To the best of our knowledge
this is the first experimental implementation of these two algorithms by adiabatic evolution.We believe that this work will provide
impetus to solving other problems by adiabatic evolution.\\
\underline{Acknowledgment}:
The authors thank K.V. Ramanathan for useful discussions. The use of DRX-500 NMR spectrometer funded by the Department of
Science and Technology (DST), New Delhi, at the NMR Research Centre (formerly Sophisticated Instruments Facility),
Indian Institute of Science, Bangalore, is gratefully acknowledged.
AK acknowledges ``DAE-BRNS" for senior scientist support and DST for a research grant for
``Quantum Computing by NMR".
|
{
"timestamp": "2005-06-30T18:51:21",
"yymm": "0503",
"arxiv_id": "quant-ph/0503060",
"language": "en",
"url": "https://arxiv.org/abs/quant-ph/0503060"
}
|
\section{Introduction}
Melting of the DNA duplex is the process by which two DNA strands unbind
upon heating. The nature of this transition has been studied for
decades \cite{Wartell85a,Hwa97a,Breslauer99a}. For short DNA with fewer
than 12--14 base pairs, melting and hybridization can be described by
a two-state model as an equilibrium between single- and
double-stranded DNA \cite{Crothers00a,CantorII}. For long and
heterogeneous DNA, the melting curve exhibits a multi-step behavior
consisting of plateaus with different sizes separated by sharp jumps.
Although much of the thermodynamic properties of the melting of free
DNA are known, DNA melting in a constrained space, such
as on surfaces, is still poorly understood \cite {Magnasco02a}. DNA
molecules functionalized with gold nanoparticles provide a model
system for such study.
The sequence-specific hybridization properties of DNA
have been used for self-assembly of nanostructures
and for highly sensitive DNA detection \cite{Mirkin96a,Kiang03a}.
Previous work relies on
a linker DNA \cite{Mirkin96a,Kiang03a,Kiang05a,Kiang05b}, and it
has been suggested that entropic cooperativity plays an important
role in the sharp phase transition of such DNA-linked nanoparticle
assembly systems. On the other hand, most simulations do not
explicitly incorporate linker DNA \cite{Stroud03a}, and the results
cannot be directly compared to experimental data. Here we
synthesized a system that eliminated the usage of a linker DNA
and found that the melting transitions of these direct-linked
gold particles exhibit distinct behavior from those connected
via a linker DNA.
\section{Experimental Procedures}
Sample was prepared according to the procedures described
in \cite{Kiang03a}. Briefly,
DNA-capped gold nanoparticles were prepared by conjugating gold
colloidal nanoparticles with thiol-modified DNA.
The configuration of the DNA used in different experiments is
illustrated in Fig.~\ref{fig:fig1}.
We prepared four sets of samples with different DNA lengths and sequences.
In sample I, the gold particles are connected through a 24-base DNA linker;
in sample II, the gold particles are directly connected via 12-base
complementary DNA on gold particles; in sample III, the gold
particles are directly connected via 12- and 18-base DNA; in sample IV,
the gold particles are directly connected via 18-base DNA.
\begin{figure}[h]
\begin{center}
\epsfig{file=fig1.eps,height=3.0in,clip=}
\end{center}
\caption{DNA sequences used to form DNA-linked gold nanoparticles.
Sample I is connected through a linker DNA. The line between bases
A and G in the probe DNA sequences indicates that there is no
chemical bond between these two bases. Sample II-IV are directly
connected through surface-attached DNA with spacings of 12, 18, and
24 DNA bases between particles.}
\label{fig:fig1}
\end{figure}
The aggregates of DNA-linked gold colloids were allowed to stand at 4
$^\circ$C for several days for aggregation. Optical spectroscopy was
used to study the phase transition of the DNA-linked gold colloids,
since DNA bases have strong absorption in the UV region
\cite{Crothers00a,CantorII}. We monitor the thermal melting by
measuring the extinction at 260 nm while slowly heating the solution
containing aggregates. The solution was heated
from 25 to 75 $^\circ$C at a rate of 0.5 $^\circ$C/min.
All spectra were taken with a PerkinElmer Lambda 45 spectrophotometer
equipped with a peltier temperature controller, magnetic stirrer, and
a temperature probe. The recorded temperature of the sample was
measured by a temperature probe.
\section{Results and Discussion}
Fig.~\ref{fig:fig2} shows the the melting curves of sample I (a)
and sample II (b). The melting curves of corresponding DNA in solution
are also shown. The melting temperature of DNA duplex attached
to gold particle surfaces is lower than that of free DNA, and the melting
transition is much sharper. However, direct comparison of melting temperatures
between these two systems is difficult, since the DNA compositions
are different for these two systems. The system without linker appears to have
a lower melting temperature, perhaps due to the short distance between
gold particles (12 base versus 24 base).
\begin{figure}[tb]
\begin{center}
\epsfig{file=fig2.eps,width=\columnwidth,clip=}
\end{center}
\caption{Melting curves of gold nanoparticles connected (a) with a DNA
linker (sample I), and (b) via direct hybridization of complementary
surface-attached DNA (sample II). The corresponding free DNA melting
curves are also shown.}
\label{fig:fig2}
\end{figure}
To study how the melting depends on the spacing between gold particles,
we prepared gold particles capped with 18 base DNA (Sample III),
which is composed of identical sequence to the 12 base DNA in
Sample II plus a 6 base DNA spacer (see Fig.~\ref{fig:fig1}).
The difference between sample II and III is the spacer DNA length,
which alter the spacing between gold particles.
The melting temperature is 62 $^\circ$C for sample IV
versus 32 $^\circ$C for sample II (see Fig~\ref{fig:fig3}),
which suggests that increased particle spacing leads to higher
melting temperature of the assemblies.
\begin{figure}[tb]
\begin{center}
\epsfig{file=fig3.eps,height=2.0in,clip=}
\end{center}
\caption{Melting curves of 12/12 (sample~II), 12/18 (sample~III),
18/18 (sample~IV), and a combination of 12/12, 12/18, and
18/18 (Sample II~+~III~+~IV). The mixed system shows multi-step melting
at temperatures corresponding to the $T_m$'s, within the experimental uncertainty,
for Sample~II and Sample~III.}
\label{fig:fig3}
\end{figure}
To introduce disorder, we mixed the 12 and 18 base DNA capped gold
particles in one solution. Thus, the 12 base DNA are allowed to
hybridize with either 12 or 18 base DNA, and the 18 base to 18 or 12
base DNA. This combination allows three possible duplex formations:
12/12 (sample I), 12/18 (sample II), and 18/18 (sample III)
hybridization in one solution and possibly in one aggregate. Note
that in all three base pairing only 12 bases are complementary, and
the only variable is the non-pairing DNA spacer length, which controls
the inter-particle distance. Since the duplexes with higher melting
temperatures are more stable, we expect to see more of those duplexes
forming. Indeed, Fig.~\ref{fig:fig3} shows that the most abundant
duplex is the 18/18 combination, followed by 12/18, with
almost no 12/12 duplex formed.
The multi-step melting is an unusual phenomenon in DNA-capped gold particle
assembly. For the system connected by either 24 or 30 base DNA
linker, where the 30 base linker differs from the 24 base by an extra 6
base spacer in the middle of the linker,
heating the assembly results in a single melting temperature $T_m$.
The $T_m$ of the system with spacer is higher (37~$^\circ$C) than that
without spacer (33~$^\circ$C). When equal amounts of linkers with
and without spacer are present in the solution, the system has a $T_m$
in between the high $T_m$ (37~$^\circ$C) and low $T_m$ (33~$^\circ$C) systems.
The $T_m$ of the mixed system (36.5~$^\circ$C) is much closer to the
more stable system (37~$^\circ$C), as illustrated
in Fig.~\ref{fig:fig4}a. However, free DNA
with the same sequences exhibits different trend in $T_m$
(see Fig.~\ref{fig:fig4}b), where a linker with spacer results in
lower melting temperature than that without spacer.
The finding suggests that the
inter-particle distance plays an important role in determining the $T_m
$ in the nanoparticle system.
On the other hand, the multi-step melting in
the system without linker DNA suggests most clusters are composed of either
12/18 connections or 18/18 connectionis and few with both connected in the
same cluster, unlike the systems with linker DNA.
The abundance of clusters of a given connection is
related to its stability. We speculate that once the cluster nucleate
with a certain type of connection (defined here by the DNA length, hence the
interparticle spacing) only the same type of
connection is allowed to grow. Further studies are needed to determine
whether this phenomenon is kinetics or thermodynamics driven.
\begin{figure}[tb]
\begin{center}
\epsfig{file=fig4.eps,width=\columnwidth,clip=}
\end{center}
\caption{Melting curves of DNA duplex containing spacer for (a) nanoparticle
assembly, and (b) free DNA.}
\label{fig:fig4}
\end{figure}
\section{Summary}
In summary, we have studied the thermal denaturation
of DNA strands attached to gold nanoparticle surfaces. In the DNA-capped
gold nanoparticle systems, the interactions are complex, involving
DNA-DNA interactions and particle-particle interactions. The DNA are
constrained to a gold particle surface and often exhibit interesting
behaviors not seen by DNA in free solution. The multi-step melting
of systems with different spacers is unique to the systems directly
linked with DNA that are attached to gold nanoparticles.
$^*$To whom correspondence should be addressed, email: chkiang@rice.edu.
|
{
"timestamp": "2005-03-10T23:19:10",
"yymm": "0503",
"arxiv_id": "physics/0503090",
"language": "en",
"url": "https://arxiv.org/abs/physics/0503090"
}
|
\section{Introduction}
A well known theorem of Szemer\'edi \cite{SZ} states that every
dense subset of integers contains long arithmetic progressions. A
different, but somehow related result of Freiman \cite{FR} says
that if the sumset of a finite set of numbers $A$ is small, i.e.
$|A+A|\leq C|A|,$ then $A$ is the subset of a (not very large)
generalized arithmetic progression. Balog and Szemer\'edi proved
in \cite{BSZ} that a similar structural statement holds under
weaker assumptions. (For correct statements and details, see
\cite{NA}). As a corollary of their result, Freiman's
theorem, and Szemer\'edi's theorem about $k$-term arithmetic
progressions, Balog and Szemer\'edi proved Theorem 1 below.
The goal of this paper is to present a simple,
purely combinatorial proof of this assertion.
Let $A$ be a set of numbers and $G$ be a graph such that the
vertex set of $G$ is $A.$ The {\em sumset of $A$ along $G$} is
\[
A+_GA = \{a+b: a,b \in A \text{ and } (a,b) \in E(G)\}.
\]
\begin{theorem} For every $c,K,k >0$ there is a threshold
$n_0=n_0(c,K,k)$ such that if $|A|=n\geq n_0$, $|A+_GA|\leq K|A|$,
and $|E(G)|\geq cn^2$, then $A$ contains a $k$-term arithmetic
progression. \end{theorem}
\section{Lines and hyperplanes}
There are arrangements of $n$ lines on the Euclidean plane such that
the maximum number of points incident with at least three lines is
${n^2\over 6}.$ Not much is known about the
structure of arrangements where the number of such points is close
to the maximum, say $cn^2$, where $c$ is a positive constant.
Nevertheless, the following is true.
\begin{lemma} \label{lemma:line} For every $c>0$ there is a threshold
$n_0=n_0(c)$ and a positive $\delta =\delta (c)$ such that, for any
set of $n\geq n_0$ lines $L$ and any set of $m\geq cn^2$ points
$P$, if every point is incident to three lines, then there are at
least $\delta n^3$ triangles in the arrangement. (A triangle is a
set of three distinct points from $P$ such that any two are
incident to a line from $L.$) \end{lemma}
{\bf Proof}.\ This lemma follows from the following theorem of Ruzsa and
Szemer\'edi \cite{RSZ}.
\begin{theorem} \cite{RSZ} Let $G$ be a graph on $n$ vertices. If $G$ is the
union of $cn^2$ edge-disjoint triangles, then $G$ contains at
least $\delta n^3$ triangles, where $\delta$ depends on $c$ only.
\end{theorem}
To prove Lemma 1, let us construct a graph where $L$ is the vertex
set, and two vertices are adjacent if and only if the
corresponding lines cross at a point of $P$. This graph is the
union of $cn^2$ disjoint triangles, every point of $P$ defines a
unique triangle, so we can apply Theorem 2.$\square$\vspace{.8cm}
The result above suffices to prove Theorem 1 for 3-term
arithmetic progressions. But for larger values of $k$, we need a
generalization of Lemma 1.
\begin{lemma} \label{lemma:plane} For every $c>0$ and $d\geq 2$, there
is a threshold $n_0=n_0(c,d)$ and a positive $\delta =\delta
(c,d)$ such that, for any set of $n\geq n_0$ hyperplanes $L$ and
any set of $m\geq cn^d$ points $P$, if every point is incident to
$d+1$ hyperplanes, then there are at least $\delta n^{d+1}$
simplices in the arrangement. (A simplex is a set of $d+1$
distinct points from $P$ such that any $d$ are incident to a
hyperplane from $L.$) \end{lemma}
Lemma 2 follows from the Frankl-R\"odl conjecture \cite{FRR}, the
generalization of Theorem 2. The
$d=3$ case was proved in \cite{FRR} and the conjecture has been
proved recently by Gowers \cite{GO} and independently by Nagle,
R\"odl, Schacht, and Skokan \cite{NRS},\cite{RS}. For details, how
Lemma 2 follows from the Frankl-R\"odl conjecture, see \cite{SO}.
\section{The $k=3$ case}
Let $A$ be a set of numbers and $G$ be a graph such that the vertex
set of $G$ is $A.$ We define the {\em difference-set of $A$ along
$G$} as
\[
A-_GA = \{a-b: a,b \in A \text{ and } (a,b) \in E(G)\}.
\]
\begin{lemma} For every $\epsilon ,c,K >0$ there is a number $D=D(\epsilon
,c,K)$ such that if $|A+_GA|\leq K|A|$ and $|E(G)|\geq c|A|^2$,
then there is a graph $G'\subset G$ such that $|E(G')|\geq
(1-\epsilon)|E(G)|$ and $|A-_{G'}A|\leq D|A|$. \end{lemma}
{\bf Proof}.\ Let us consider the arrangement of points given by a subset of
the Cartesian product $A\times A$ and the lines $y=a$, $x=a$ for
every $a\in A$, and $x+y=t$ for every $t\in A+_GA.$ The pointset
$P$ is defined by $(a,b)\in P$ iff $(a,b)\in E(G).$ By Lemma 1,
the number of triangles in this arrangement is $\delta n^3.$ The
triangles here are right isosceles triangles. We say that a
point in $P$ is {\em popular} if the point is the right-angle
vertex of at least $\alpha n$ triangles. Selecting
$\alpha={{\delta (\epsilon c)}\over {\epsilon c}}$, where $\delta
(\cdot)$ is the function from Lemma 1, all but at most $\epsilon
cn^2$ points of $P$ are popular.
A $t\in A-A$ is {\em popular} if $|\{(a,b):a-b=t; a,b\in A\}|\geq
\alpha n.$ The number of popular $t$s is at most $Dn$, where $D$
depends on $\alpha$ only. $A\times A$ is a Cartesian product,
therefore every triangle can be extended to a square adding one
extra point from $A\times A$. Every popular point $p$ is the
right-angle vertex of at least $\alpha n$ triangles. Therefore $p$
is incident to a line $x-y=t$, where $t$ is popular, because this
line contains at least $\alpha n$ ``fourth" vertices of squares
with $p$. $\square$\vspace{.8cm}
{\bf Proof of Theorem 1, case $k=3.$} Let us apply Lemma 1 to the
pointset $P'$ defined by $(a,b)\in P'$ iff $(a,b)\in E(G')$ and
the lines are $y=a$ for every $a\in A$, $x-y=t$ for every $t\in
A-_{G'}A$, and $x+y=s$ for every $s\in A+_GA.$ By Theorem 2, if
$|A|$ is large enough, then there are triangles in the
arrangement. The vertices of such triangles are vertices from
$P'\subset A\times A.$ The vertical lines through the vertices
form a 3-term arithmetic progression and therefore $A$ contains
$\delta n^2$ 3-term arithmetic progressions, where $\delta > 0$
depends on $c$ only. $\square$\vspace{.8cm}
\section{The general, $k>3$, case}
Following the steps of the proof for $k=3$, we prove the general
case by induction on $k.$ We prove the following theorem, which
was conjectured by Erd\H os and proved by Balog and Szemer\'edi in
\cite{BSZ}. Theorem 3, together with the $k=3$ case, gives a proof
of Theorem 1.
\begin{theorem} For every $c>0$ and $k>3$ there is an $n_0$ such that, if $A$
contains at least $c|A|^2$ 3-term arithmetic progressions and
$|A|\geq n_0$, then $A$ contains a $k$-term arithmetic progression.
\end{theorem}
Instead of triangles, we must consider simplices. Set $k=d$.
In the $d$-dimensional space we show that $A\times \cdots \times A$,
the $d$-fold Cartesian product of $A$, contains a simplex in which the
vertices' first coordinates form a $(d+1)$-term arithmetic
progression.
The simplices we are looking for are homothetic\footnote{Here we
say that two simplices are homothetic if the corresponding facets
are parallel.} images of the simplex $S_d$ whose vertices are listed
below:
\begin{displaymath}
\begin{array}{c}
(0, 0, 0,0, \ldots ,0,0)\\
(1, 1,0,0, \ldots ,0,0)\\
(2, 0, 1,0, \ldots ,0,0)\\
(3, 0,0,1, \ldots ,0,0)\\
\vdots\\
(d-1, 0, \ldots ,1,0)\\
(d, 0,0,0, \ldots ,0,0).
\end{array}
\end{displaymath}
\indent An important property of $S_d$ is that its facets can be pushed into
a ``shorter" grid. The facets of $S_d$ are parallel to hyperplanes,
defined by the origin $(0,0,0,0,\ldots ,0,0)$, and some
$(d-1)$-tuples of the grid
$$\{0,1,2,\ldots ,d-1\}\times \{-1,0,1\}\times\{0,1\}^{d-2}.$$
For example, if $d=3$, then the facets are
\begin{displaymath}
\begin{array}{c}
\{(0,0,0),(1,1,0),(2,0,1)\}\\
\{(0,0,0),(1,1,0),(3,0,0)\}\\
\{(0,0,0),(2,0,1),(3,0,0)\}\\
\{(1,1,0),(2,0,1),(3,0,0)\},
\end{array}
\end{displaymath}
and the corresponding parallel planes in
$$\{0,1,2\}\times
\{-1,0,1\}\times\{0,1\}$$ are the planes incident to the triples
\begin{displaymath}
\begin{array}{c}
\{(0,0,0),(1,1,0),(2,0,1)\}\\
\{(0,0,0),(1,1,0),(2,0,0)\}\\
\{(0,0,0),(2,0,1),(2,0,0)\}\\
\{(0,0,0),(1,-1,1),(2,-1,0)\}.
\end{array}
\end{displaymath}
\indent In general, if a facet of $S_d$ contains the origin and the ``last
point" $(d, 0,0,0, \ldots ,0,0),$ then if we replace the later one by
$(d-1, 0,0,0, \ldots ,0,0)$, the new $d$-tuples define the same hyperplane.
The remaining facet $f$, given by
\begin{displaymath}
\begin{array}{c}
(1, 1,0,0, \ldots ,0,0)\\
(2, 0, 1,0, \ldots ,0,0)\\
(3, 0,0,1, \ldots ,0,0)\\
\vdots\\
(d-1, 0, \ldots ,1,0)\\
(d, 0,0,0, \ldots ,0,0),
\end{array}
\end{displaymath}
is parallel to the hyperplane through the vertices of $f-(1,1,0,0,
\ldots ,0,0),$
\begin{displaymath}
\begin{array}{c}
(0, 0,0,0, \ldots ,0,0)\\
(1, -1, 1,0, \ldots ,0,0)\\
(2, -1,0,1, \ldots ,0,0)\\
\vdots\\
(d-2, -1, \ldots ,1,0)\\
(d-1,-1,0,0, \ldots ,0,0).
\end{array}
\end{displaymath}
\indent In a homothetic copy of the grid $$T_d=\{0,1,2,\ldots ,d-1\}\times
\{-1,0,1\}\times\{0,1\}^{d-2},$$ the image of the origin is called
the {\em holder} of the grid.
As the induction hypothesis, let us suppose that Theorem 3 is true
for a $k\geq 3$ in a stronger form, providing that the number of
$k$-term arithmetic progressions in $A$ is at least $c|A|^2.$
Then the number of distinct homothetic copies of $T_d$ in
$\mathbb{A}_d=\underbrace{A\times \ldots \times A}_d$ is at least
$c'|A|^{d+1}$ ($c'$ depends on $c$ only). Let us say that a point
$p\in \mathbb{A}_d$ is \emph{popular} if $p$ is the holder of at least
$\alpha |A|$ grids. If $p$ is popular, then for any facet of
$S_d$, $f$, the point $p$ is the element of at least $\alpha |A|$
$d$-tuples, similar and parallel to $f.$ If $\alpha$ is small
enough, then at least $\gamma |A|^d$ points of $\mathbb{A}_d$ are
popular, where $\gamma$ depends on $c$ and $\alpha$ only.
A hyperplane $H$ is \emph{$\beta$-rich} if it is incident to many points,
$|H\cap \mathbb{A}_d|\geq \beta |A|^{d-1}.$ For every facet of
$S_d$, $f$, let us denote the set of $\beta$-rich hyperplanes
which are parallel to $f$ by $\mathcal{H}_f.$
\begin{lemma} For some choice of $\beta$, at least half of the
popular points are incident to $d+1$ $\beta$-rich hyperplanes,
parallel to the facets of $S_d.$ \end{lemma}
Suppose to the contrary that for a facet $f$, more than
${\gamma\over 2d} |A|^d$ popular points are not incident to
hyperplanes of $\mathcal{H}_f.$ Then more than
\begin{equation}
\alpha |A|{\gamma\over 2d} |A|^d={\gamma\alpha\over 2d} |A|^{d+1}
\end{equation}
\noindent $d$-tuples, similar and parallel to $f$, are not covered by
$\mathcal{H}_f.$ Let us denote the hyperplanes incident to the
``uncovered" $d$-tuples by $L_1,L_2,\ldots ,L_m$, and the number of
points on the hyperplanes by $\mathcal{L}_1,\mathcal{L}_2,\ldots
,\mathcal{L}_m.$ A simple result of Elekes and Erd\H os
\cite{EE},\cite{EL} implies that hyperplanes with few points cannot
cover many $d$-tuples.
\begin{theorem} \cite{EE} The number of homothetic copies of $f$ in $L_i$ is
at most $c_d\mathcal{L}_i^{1+1/(d-1)}$, where $c_d$ depends on $d$
only. \end{theorem}
The inequalities
$$\sum_{i=1}^m\mathcal{L}_i\leq |A|^d, \text{ and } \mathcal{L}_i\leq \beta |A|^{d-1}.$$
lead us to the proof of Lemma 4.
The number of $d$-tuples covered by $L_i$s is at most
$$c_d\sum_{i=1}^m\mathcal{L}_i^{1+1/(d-1)}\leq c_d{{|A|^d}\over
{\beta |A|^{d-1}}}(\beta |A|^{d-1})^{1+1/(d-1)}=c_d\beta^{1/(d-1)}|A|^{d+1}.$$
If we compare this bound to (1), and choose $\beta$
such that ${\gamma\alpha\over 2d}=c_d\beta^{1/(d-1)}$,
then at least half of the popular points are covered by $d+1$
$\beta$-rich hyperplanes parallel to the facets of $S_d.$
$\square$\vspace{.8cm}
Finally we can apply Lemma 2 with the pointset $P$ of ``well-covered"
popular points of $\mathbb{A}_d$ and with the sets of hyperplanes
$L=\bigcup_{f\subset S_d}\mathcal{H}_f.$ The number of points is
at least ${\gamma\alpha\over 2} |A|^{d}$. For a given $f,$
$|\mathcal{H}_f|\leq {{|A|^d}\over{\beta |A|^{d-1}}}=|A|/\beta.$
The number of hyperplanes in $L$ is at most $(d+1)|A|/\beta.$ By
Lemma 2, we have at least $\delta '|A|^{d+1}$ homothetic copies of
$S_d$ in $\mathbb{A}_d.$ Let us project them onto $x_1$, the first
coordinate axis. Every image is a $(k+1)$-term arithmetic
progression, and the multiplicity of one image is at most
$|A|^{d-1}.$ Therefore there are at least $\delta '|A|^2$
$(k+1)$-term arithmetic progressions in $A.$
$\square$\vspace{.8cm}
\section{$G_n=K_n$}
When the full sumset $A+A$ is small then it is easier to prove
that $A$ contains long arithmetic progressions. We can use the
following Pl\"unecke type inequality \cite{PL,RU,NA}.
\begin{theorem}
Let $A$ and $B$ be finite subsets of an abelian group such that
$|A|=n$ and $|A+B|\leq \delta n$. Let $k\geq 1$ and $l\geq 1.$
Then
$$|kB-lB|\leq \delta^{k+l}n.$$
\end{theorem}
It follows from the inequality, that for any dimension $d$ and
$d$-dimensional integer vector $\vec{v}=(x_1,\ldots ,x_d), x_i\in
\mathbb{Z}$, there is a $c>0$ depending on $d,\delta$ and $\vec{v}$
such that the following holds: \emph{If $|A+A|\leq \delta |A|$, then
$\mathbb{A}_d$ can be covered by $c|A|$ hyperplanes with the same
normal vector $\vec{v}$}. Using this, we can define our
hyperplane-point arrangement, with the hyperplanes parallel to the
facets of $S_d$ containing at least one point of $\mathbb{A}_d$,
and the pointset of the arrangement is $\mathbb{A}_d.$ Then we
do not have to deal with rich planes and popular points, and we can
apply Lemma 2 directly.
|
{
"timestamp": "2005-03-28T23:54:29",
"yymm": "0503",
"arxiv_id": "math/0503649",
"language": "en",
"url": "https://arxiv.org/abs/math/0503649"
}
|
\section*{Introduction}
Predicting the three-dimensional structure of a protein from its
amino acid sequence is an essential step toward the
thorough bottom-up understanding of complex biological phenomena.
Recently, much progress has been made in developing
so-called \emph{ab initio} or \emph{de novo} structure prediction
methods\cite{BonneauANDBaker2001}.
In the standard approach to such \emph{de novo} structure predictions,
a protein is represented as a physical object in three-dimensional (3D) space,
and the global minimum of free energy surface is sought with a given
force-field or a set of scoring functions. In the minimization process,
structural features predicted from the amino acid sequence
may be used as restraints to limit the conformational space to be sampled.
Such structural features include so-called one-dimensional (1D) structures of
proteins.
Protein 1D structures are 3D structural features
projected onto strings of residue-wise structural assignments
along the amino acid sequence\cite{Rost2003}.
For example, a string of secondary structures is a 1D structure.
Other 1D structures include (solvent)
accessibilities\cite{LeeANDRichards1971}, contact
numbers\cite{KinjoETAL2005} and recently introduced residue-wise contact
orders\cite{KinjoANDNishikawa2005}.
The contact number, also referred to as coordination number or Ooi
number\cite{NishikawaANDOoi1980},
of a residue is the number of contacts that the residue makes with
other residues in the native 3D structure, while the residue-wise
contact order of a
residue is the sum of sequence separations between that residue and
contacting residues.
We have recently shown that it is possible to reconstruct the native
3D structure of a protein from a set of three types of native 1D structures,
namely secondary structures (SS), contact numbers (CN), and residue-wise
contact orders (RWCO)\cite{KinjoANDNishikawa2005}.
Therefore, these 1D structures contain rich information regarding the
corresponding 3D structure, and their accurate prediction may be very helpful
for 3D structure prediction.
In our previous study\cite{KinjoETAL2005}, we have developed a simple linear
method to predict contact numbers from amino acid sequence. In that method,
the use of multiple sequence alignment
was shown to improve the prediction accuracy, achieving an average
correlation coefficient of 0.63 between predicted and observed contact
numbers per protein. There, we used amino acid frequency table obtained from
the HSSP\cite{HSSP} multiple sequence alignment.
In this paper, we extend the previous method by introducing a new framework
called critical random networks (CRNs), and apply it to the prediction of
secondary structure and residue-wise contact order in addition to contact
number prediction. In this framework, a state vector of a large dimension
is associated with each site of a target sequence.
The state vectors are connected via random nearest-neighbor interactions.
The value of the state vectors are determined by solving an equation of
state. Then a 1D quantity of each site is predicted as a linear function
of the state vector of the site as well as the corresponding local PSSM segment.
This approach was inspired by the method of echo state networks
(ESNs) which has been recently developed and successfully applied
to complex time series analysis\cite{Jaeger2001,JaegerANDHaas2004}.
Unlike ESNs which treat infinite series of input signals in one direction
(from the past to the future), CRNs treat finite systems incorporating
both up- and downstream information at the same time. Also,
the so-called echo state property is not imposed to a network,
but the system is instead set at a critical point of the network.
As the input to CRNs-based prediction, we employ position-specific
scoring matrices (PSSMs) generated by PSI-BLAST\cite{AltschulETAL1997}.
By the combination of PSSMs and CRNs, accurate prediction of
SS, CN and RWCO have been achieved.
Currently, almost all the accurate methods for one-dimensional structure
predictions combine some kind of sophisticated machine-learning approaches
such as neural networks and support vector machines with PSSMs. The method
presented here is no exception.
This trend raises a question as to what extent the machine-learning
approaches are effective. In this study, we address this question by comparing
the CRNs-based method with a purely linear method based on PSSMs. Although
not so good as the CRNs-based method, the linear predictions are of
surprisingly high quality. This result suggests that, although not
insignificant, the effect of the machine-learning approaches is relatively
of minor importance while the use of PSSMs is the most significant ingredient
in one-dimensional structure prediction. The problem of how to effectively
extract meaningful information from the amino acid sequence beyond that
provided by PSSMs requires yet further studies.
\section*{Materials and Methods}
\subsection*{Definition of 1D structures}
\paragraph{Secondary structures (SS)}
Secondary structures were defined by the DSSP program\cite{DSSP}.
For three-state SS prediction, the simple encoding scheme was employed.
That is, $\alpha$ helices ($H$), $\beta$ strands ($E$), and other structures
(``coils'') defined by DSSP were encoded as $H$, $E$, and $C$, respectively.
For SS prediction, we introduce feature variables $(y_i^H, y_i^E, y_i^C)$
to represent each type of secondary structures at the $i$-th residue position,
so that $H$ is represented as $(1,-1,-1)$, $E$ as $(-1,1,-1)$, and $C$ as
$(-1,-1,1)$.
\paragraph{Contact numbers (CN)}
Let $C_{i,j}$ represent the contact map of a protein. Usually, the contact
map is defined so that $C_{i,j} = 1$ if the $i$-th and $j$-th residues are in
contact by some definition, or $C_{i,j} = 0$, otherwise. As in our
previous study, we slightly modify the definition using a sigmoid function.
That is,
\begin{equation}
C_{i,j} = 1/\{1+\exp[w(r_{i,j} - d)]\}
\end{equation}
where $r_{i,j}$ is the distance between $C_{\beta}$ ($C_{\alpha}$
for glycines) atoms of the $i$-th and $j$-th residues, $d = 12$\AA{} is a
cutoff distance, and $w$ is a sharpness parameter of the sigmoid function
which is set to 3\cite{KinjoETAL2005,KinjoANDNishikawa2005}. The rather
generous cutoff length of 12\AA{} was shown to optimize the prediction
accuracy\cite{KinjoETAL2005}. The use of the sigmoid function enables us to
use the contact numbers in molecular dynamics
simulations\cite{KinjoANDNishikawa2005}.
Using the above definition of the contact map, the contact number of the
$i$-th residue of a protein is defined as
\begin{equation}
n_i = \sum_{j:|i-j|>2}C_{i,j}. \label{eq:defcn}
\end{equation}
The feature variable $y_i$ for CN is defined as $y_i = n_i / \log L$ where
$L$ is the sequence length of a target protein. The normalization
factor $\log L$ is introduced because we have observed that the contact
number averaged over a protein chain is roughly proportional to $\log L$,
and thus division by this value removes the size-dependence of predicted
contact numbers.
\paragraph{Residue-wise contact orders (RWCO)}
RWCOs were first introduced in Kinjo and Nishikawa\cite{KinjoANDNishikawa2005}.
Using the same notation as contact numbers (see above),
the RWCO of the $i$-th residue in a protein structure is defined by
\begin{equation}
o_i = \sum_{j:|i-j|>2}|i-j|C_{i,j}. \label{eq:defrwco}
\end{equation}
The feature variable $y_i$ for RWCO is defined as $y_i = o_i / L$ where
$L$ is the sequence length. Due to the similar reason as CN, the normalization
factor $L$ was introduced to remove the size-dependence of the predicted
RWCOs (the RWCO averaged over a protein chain is roughly proportional to the
chain length).
\subsection*{Linear regression scheme}
The input to the prediction scheme we develop in this paper is a
position-specific scoring matrix (PSSM) of the amino acid sequence of
a target protein.
Let us denote the PSSM by $U = (\mathbf{u}_1, \cdots , \mathbf{u}_{L})$
where $L$ is the sequence length of the target protein and
$\mathbf{u}_i$ is a 20-vector containing the scores of 20 types of
amino acid residues at the $i$-th position:
$\mathbf{u}_i = (u_{1,i}, \cdots , u_{20,i})^{t}$.
When predicting a type of 1D structures, we first predict the feature
variable(s) for that type of 1D structures [i.e., $y_i = y_i^H$, etc. for SS,
$n_i/\log L$ for CN, and $o_i/L$ for RWCO], and then
the feature variable is converted to the target 1D structure.
Prediction of the feature variable $y_i$ can be considered as a mapping
from a given PSSM $U$ to $y_i$. More formally, we are going to
establish the functional form of the mapping $F$ in $\hat{y}_{i} = F(U,i)$
where $\hat{y}_{i}$ is the predicted value of the feature variable $y_i$.
In our previous paper, we showed that CN can be predicted to a moderate
accuracy by a simple linear regression scheme with a
local sequence window\cite{KinjoETAL2005}.
Accordingly, we assume that the function $F$ can be decomposed into
linear ($F_l$) and nonlinear ($F_n$) parts: $F = F_{l} + F_{n}$.
The linear part is expressed as
\begin{equation}
F_l(U,i) = \sum_{m=-M}^{M}\sum_{a=1}^{21}D_{m,a}u_{a,i+m}
\label{eq:lin}
\end{equation}
where $M$ is the half window size of the local PSSM segment around
the $i$-th residue, and $\{D_{m,a}\}$ are the weights to be trained.
To treat N- and C-termini separately, we introduced
the ``terminal residue'' as the 21st kind of amino acid residue.
The value of $u_{21,i+m}$ is set to unity if $i+m<0$ or $i+m>L$, or to zero
otherwise. The ``terminal residue'' for the central residue ($m=0$) serves
as a bias term and is always set to unity.
To establish the nonlinear part, we first introduce an $N$-dimensional
``state vector'' $\mathbf{x}_i = (x_{1,i}, \cdots , x_{N,i})^{t}$
for the $i$-th sequence position where the dimension $N$ is a free parameter.
The value of $\mathbf{x}_i$ is determined by solving the equation of state
which is described in the next subsection. For the moment, let us assume
that the equation of state has been solved, and denote
the solution by $\mathbf{x}_{i}^{*}$. The state
vector can be considered as a function of the whole PSSM $U$
(i.e., $\mathbf{x}_{i}^{*} = \mathbf{x}_{i}^{*}(U)$), and
implicitly incorporates nonlinear and long-range effects. Now, the nonlinear
part $F_n$ is expressed as a linear projection of the state vector:
\begin{equation}
F_n(U,i) = \sum_{k=1}^{N}E_{k}x_{k,i}^{*}(U)
\label{eq:nonlin}
\end{equation}
where $\{E_{k}\}$ are the weights to be trained.
In summary, the prediction scheme is expressed as
\begin{equation}
\hat{y}_{i} = \sum_{m=-M}^{M}\sum_{a=1}^{21}D_{m,a}u_{a,i+m}
+ \sum_{k=1}^{N}E_{k}x_{k,i}^{*}(U) \label{eq:pred0}
\end{equation}
Regarding $\mathbf{u}_{i-M}, \cdots, \mathbf{u}_{i+M}$ and
$\mathbf{x}_{i}^{*}$ as independent variables, Eq. \ref{eq:pred0} reduces to
a simple linear regression problem for which the optimal weights $\{D_{m,a}\}$
and $\{E_k\}$ are readily determined by using a least squares method.
For CN or RWCO predictions, the predicted feature variable can be easily
converted to the corresponding 1D quantities by multiplying by
$\log L$ or $L$, respectively.
For SS prediction, the secondary structure $\hat{s}_i$ of the $i$-th residue
is given by $\hat{s}_i = \mathrm{arg}\max_{s\in \{H, E, C\}}y_i^s$.
\subsection*{Critical random networks and the equation of state}
We now describe the equation of state for the system of state vectors.
We denote $L$ state vectors along the amino acid sequence by
$\mathbf{X} =
(\mathbf{x}_{1}, \cdots , \mathbf{x}_{L}) \in \mathbf{R}^{N\times L}$,
and define a nonlinear mapping
$g_i : \mathbf{R}^{N\times L} \to \mathbf{R}^{N}$ for $i = 1, \cdots , L$ by
\begin{equation}
g_i(\mathbf{X}) = \tanh \left[\beta W(\mathbf{x}_{i-1}+\mathbf{x}_{i+1})+\alpha V \mathbf{u}_{i}\right]
\end{equation}
where $\beta$ and $\alpha$ are positive constants, $W$ is an $N\times N$
block-diagonal orthogonal random matrix, and $V$ is an $N\times 21$ random
matrix (a unit bias term is assumed in $\mathbf{u}_i$).
The hyperbolic tangent function ($\tanh$) is applied element-wise.
We impose the boundary conditions as
$\mathbf{x}_0 = \mathbf{x}_{L+1} = \mathbf{0}$.
In this equation, the term containing $W$ represents nearest-neighbor
interactions along the sequence. The amino acid sequence information is
taken into account as an external field in the form of
$\alpha{}V\mathbf{u}_{i}$. Next we define a mapping
$G : \mathbf{R}^{N\times L} \to \mathbf{R}^{N\times L}$ by
\begin{equation}
G(\mathbf{X}) = (g_{1}(\mathbf{X}), \cdots , g_{L}(\mathbf{X})).
\end{equation}
Using this mapping $G$, the equation of state is defined as
\begin{equation}
\mathbf{X} = G(\mathbf{X}). \label{eq:fixpoint}
\end{equation}
That is, the state vectors are determined as a fixed point of the mapping
$G$. More explicitly, Eq. \ref{eq:fixpoint} can be expressed as
\begin{equation}
\mathbf{x}_{i} = \tanh \left[\beta W(\mathbf{x}_{i-1}+\mathbf{x}_{i+1})+\alpha V \mathbf{u}_{i}\right], \label{eq:eos}
\end{equation}
for $i = 1, \cdots, L$. That is, the state vector $\mathbf{x}_i$ of the site
$i$ is determined by the interaction with the state vectors of the neighboring
sites $i-1$ and $i+1$ as well as with the `external field' $\mathbf{u}_i$ of
the site. The information of the external field at each site is propagated
throughout the whole amino acid sequence via the nearest-neighbor interactions.
Therefore, solving Eq. (\ref{eq:eos}) means finding the state vectors that
are consistent with the external field as well as the nearest-neighbor
interactions, and each state vector in the obtained solution
$\{\mathbf{x}_i\}$ self-consistently embodies the information of the whole
amino acid sequence in a mean-field sense.
For $\beta < 0.5$, it can be shown
that $G$ is a contraction mapping in $\mathbf{R}^{N\times L}$
(with an appropriate norm defined therein).
And hence, by the contraction mapping principle\cite{TakahashiNLFA}, the
mapping $G$ has a unique fixed point independently of the strength $\alpha$
of the external field.
When $\beta$ is sufficiently smaller than 0.5,
the correlation between two state vectors, say $\mathbf{x}_{i}$
and $\mathbf{x}_{j}$, is expected to decay exponentially as a function of
the sequential separation $|i-j|$.
On the other hand, for $\beta > 0.5$, the number of the fixed points
varies depending on the strength of the external field $\alpha$.
In this regime, we cannot reliably solve the equation of
state (Eq.\ref{eq:fixpoint}).
In this sense, $\beta = 0.5$ can be considered as a critical
point of the system $\mathbf{X}$.
From an analogy with critical phenomena of physical
systems\cite{Goldenfeld1992} (note the formal similarity of Eq. \ref{eq:eos}
with the mean field equation of the Ising model), the correlation length
between state vectors is expected to diverge, or become long when the
external field is finite but small.
We call the system defined by Eq. \ref{eq:eos}
with $\beta = 0.5$ a critical random network (CRN).
The equation of state (Eq. \ref{eq:eos}) is parameterized by two random
matrices $W$ and $V$, and consequently, so is the predicted feature variables
$\hat{y}_{i}$. Following a standard technique of statistical
learning such as neural networks\cite{Haykin}, we may improve the prediction
accuracy by averaging $\hat{y}_{i}$ obtained by multiple CRNs with
different pairs of $W$ and $V$. This averaging operation reduces the prediction
errors due to the random fluctuations in the estimated parameters.
We employ such an ensemble prediction with 10 sets of random matrices $W$ and
$V$ in the following. The use of a larger number of random matrices for
ensemble predictions improved the prediction accuracies slightly, but the
difference was insignificant.
\subsection*{Numerics}
Here we describe the value of the free parameters used, and a numerical
procedure to solve the equation of state.
The half window size $M$ in the linear part of Eq. \ref{eq:pred0} is
set to 9 for SS and CN predictions, and to 26 for RWCO prediction.
These values are found to be optimal in preliminary studies\cite{KinjoETAL2005,
KinjoANDNishikawa2005b}.
Regarding the dimension $N$ of the state vector, we have found that $N=2000$
gives the best result after some experimentation, and this value is
used throughout. Using the state vector of a large dimension as 2000, it is
expected that various properties of amino acid sequences can be extracted and
memorized. If the dimension is too large, overfitting may occur, but we did
not find such a case up to $N=2000$. Therefore, in principle, the state vector
dimension could be even larger (but the computational cost becomes a problem).
Each element in the $N\times 21$ random matrix $V$ in Eq. \ref{eq:eos} is
obtained from a uniform distribution in the range [-1, 1] and the strength
parameter $\alpha$ is set to 0.01.
Here and in the following, all random numbers were generated by the Mersenne
twister algorithm\cite{MersenneTwister}.
The $N\times N$ random matrix $W$ is obtained in the following manner.
First we generate a random block diagonal matrix $A$ whose block sizes
are drawn from a uniform distribution of integers 2 to 20 (both inclusive),
and the values of the block elements are drawn
from the standard Gaussian distribution (zero mean and unit variance).
By applying singular value decomposition, we have $A = U\Sigma V^{t}$
where $U$ and $V$ are orthogonal matrices and $\Sigma$ is a diagonal
matrix of singular values.
We set $W = UV^{t}$ which is orthogonal as well as block diagonal.
To solve the equation of state (Eq. \ref{eq:eos}), we use a simple functional
iteration with a Gauss-Seidel-like updating scheme.
Let $\nu$ denote the stage of iteration.
We set the initial value of the state vectors (with $\nu = 0$) as
\begin{equation}
\mathbf{x}_{i}^{(0)} = \tanh \left[\alpha V \mathbf{u}_{i}\right].\label{eq:init_eos}
\end{equation}
Then, for $i = 1, \cdots , L$ (in increasing order of $i$), we update
the state vectors by
\begin{equation}
\mathbf{x}_{i}^{(2\nu+1)} \gets \tanh \left[W(\mathbf{x}_{i-1}^{(2\nu+1)}+\mathbf{x}_{i+1}^{(2\nu)})+\alpha V \mathbf{u}_{i}\right].
\label{eq:feos}
\end{equation}
Next, we update them in the reverse order. That is, for $i = L, \cdots , 1$
(in decreasing order of $i$),
\begin{equation}
\mathbf{x}_{i}^{(2\nu+2)} \gets \tanh \left[W(\mathbf{x}_{i-1}^{(2\nu+1)}+\mathbf{x}_{i+1}^{(2\nu+2)})+\alpha V \mathbf{u}_{i}\right].
\label{eq:beos}
\end{equation}
We then set $\nu \gets \nu + 1$, and iterate Eqs. (\ref{eq:feos}) and (\ref{eq:beos}) until $\{\mathbf{x}_{i}\}$ converges. The convergence criterion is
\begin{equation}
\sqrt{\sum_{i=1}^{L}\left\|\mathbf{x}_{i}^{(2\nu+2)}-\mathbf{x}_{i}^{(2\nu+1)}\right\|_{\mathbf{R}^{N}}^{2}/{NL}}<10^{-7}
\end{equation}
where $\left\|\cdot\right\|_{\mathbf{R}^{N}}$ denotes the Euclidean norm.
Convergence is typically achieved within 100 to 200 iterations for one protein.
\subsection*{Preparation of training and test sets}
We use the same set of proteins as used in our preliminary
study\cite{KinjoANDNishikawa2005b}. In this set, there are 680 protein domains
selected from the ASTRAL database\cite{ASTRAL},
each of which represents a superfamily from one of all-$\alpha$, all-$\beta$,
$\alpha/\beta$, $\alpha+\beta$ or ``multi-domain'' classes of the SCOP database
(release 1.65, December 2003)\cite{SCOP}. Conversely, each SCOP superfamily
is represented by only one of the protein domains in the data set.
Thus, no pair of protein domains in the data set are expected to
be homologous to each other.
For training the parameters and testing the prediction accuracy, 15-fold
cross-validation is employed. The set of 680 proteins is randomly
divided into two groups: one consisting of 630 proteins (training set),
and the other consisting of 50 proteins (test set). For each training set,
the regression parameters $\{D_{m,a}\}$ and $\{E_{i}\}$ are determined, and
using these parameters, the prediction accuracy is evaluated for the
corresponding test set.
This procedure was repeated for 15 times with different random divisions,
leading to 15 pairs of training and test sets. In this way, there is some redundancy in the training and test sets although each pair of these sets share no
proteins in common. But this raises no problem since our objective is to
estimate the average accuracy of the predictions. A similar validation procedure was also employed by Petersen et al.\cite{PetersenETAL2000}
In total, 750 ($= 15\times 50$) proteins were tested over which
the averages of the measures of accuracy (see below) were calculated.
\subsection*{Preparation of position-specific scoring matrix}
To obtain the position-specific scoring matrix (PSSM) of a protein,
we conducted ten iterations of PSI-BLAST\cite{AltschulETAL1997} search
against a customized sequence database with the E-value cutoff of
0.0005\cite{TomiiANDAkiyama2004}. The sequence database was compiled from the
DAD database provided by DNA Data Bank of Japan\cite{DDBJ2005}, from which
redundancy was removed by the program CD-HIT\cite{CD-HIT} with 95\% identity
cutoff. This database was subsequently filtered by the program
PFILT used in the PSIPRED program\cite{Jones1999}.
We use the position-specific scoring matrices (PSSM) rather than the frequency
tables for the prediction.
\subsection*{Measures of accuracy}
For assessing the quality of SS predictions, we mainly use $Q_3$ and
$SOV$ (the 1999 revision)\cite{SOV99}. The $Q_3$ measure quantifies the percentage of correctly predicted residues, while the $SOV$ measure evaluates the
segment overlaps of secondary structural elements of predicted and native
structures. Optionally, we use $Q_s$ and $Q_s^{pre}$
(with $s$ being $H$, $E$, or $C$) and Matthews' correlation coefficient $MC$.
The $Q_s$ is defined by the percentage of correctly predicted SS type $s$
out of the native SS type $s$, and $Q_s^{pre}$ is defined by the percentage
of correctly predicted SS type $s$ out of the predicted SS type $s$.
For CN and RWCO predictions, we use two measures for evaluating the prediction
accuracy.
The first one is the correlation coefficient ($Cor$) between the observed
($n_{i}$) and predicted ($\hat{n}_{i}$) CN or RWCO\cite{KinjoETAL2005}.
The second is the RMS error normalized by the standard deviation of the
native CN or RWCO ($DevA$)\cite{KinjoETAL2005}.
While $Cor$ measures the quality of relative values, $DevA$ measures that of
absolute values of the predicted CN or RWCO.
Note that the measures $Q_3$, $SOV$, $Cor$ and $DevA$ are defined for a
single protein chain. In practice, we average these quantities over the
proteins in the test sets to estimate the average accuracy of prediction.
On the other hand, per-residue measures, $Q_s$, $Q_s^{pre}$ and $MC$,
were calculated using all the residues in the test data sets, rather than
on a per-protein basis.
\section*{Results}
We examine the prediction accuracies for SS, CN, and RWCO in turn.
The main results are summarized in Table \ref{tab:summ} and Figure \ref{fig:histo}. Finally, in order to examine the effect of nonlinear terms, we verify
the prediction results obtained using only linear terms (Eq. \ref{eq:lin}).
\begin{table}
\caption{\label{tab:summ}Summary of average prediction accuracies.}
\begin{center}
\begin{tabular}[h]{ll}\hline
Struct. & Accuracy \\\hline
SS & $Q_3$ = 77.8; $SOV$ = 77.3\\
CN & $Cor$ = 0.726; $DevA$ = 0.707\\
RWCO& $Cor$ = 0.601; $DevA$ = 0.881\\\hline
\end{tabular}
\end{center}
\end{table}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=7cm]{./histos.eps}
\end{center}
\caption{\label{fig:histo}Histograms of accuracy measure obtained by
ensemble predictions using 10 critical random networks. (a) $Q_3$ for
secondary structure prediction; (b) $Cor$ for contact number prediction;
(c) $Cor$ for residue-wise contact order prediction.}
\end{figure}
\subsection*{Secondary structure prediction}
The average accuracy of secondary structure prediction achieved by
the ensemble CRNs-based approach is $Q_3=77.8$\% and $SOV=77.3$
(Table \ref{tab:summ}). This is comparable to the current state-of-the-art
predictors such as PSIPRED\cite{Jones1999}. The results in
terms of per-residue accuracies ($Q_s$ and $Q_s^{pre}$) are listed in
Table \ref{tab:ss}.
The values of $Q_s$ suggest that the present method
underestimates $\alpha$ helices ($H$) and, especially, $\beta$ strands ($E$)
compared to coils $C$.
However, when a residue is predicted as being $H$ or $E$, the probability
of the correct prediction is rather high, especially for $E$
($Q_E^{pre} =$ 79.9\%).
The histogram of $Q_3$ (Figure \ref{fig:histo}a) shows that the peak
of the histogram resides well beyond $Q_3$ = 80\%, and that
only 20\% of the predictions exhibit $Q_3$ of less than 70\%. These
observations demonstrate the capability of the CRNs-based prediction schemes.
\begin{table}
\caption{\label{tab:ss}Summary of per-residue accuracies for SS predictions.}
\begin{center}
\begin{tabular}[h]{lrrr}\hline
measure & $H$ & $E$ & $C$ \\\hline
$Q_s$ & 78.4 & 61.9 & 84.6 \\
$Q_s^{pre}$ & 81.9 & 79.9 & 74.3\\
$MC$ & 0.704 & 0.636 & 0.602 \\\hline
\end{tabular}
\end{center}
\end{table}
\subsection*{Contact number prediction}
Using an ensemble of CRNs, a correlation coefficient ($Cor$) of 0.726 and
normalized RMS error ($DevA$) of 0.707 was achieved for CN predictions on
average (Table \ref{tab:summ}). This result is a significant improvement over
the previous method\cite{KinjoETAL2005} which yielded
$Cor=0.627$ and $DevA = 0.941$. The median of the distribution of $Cor$
(Figure \ref{fig:histo}b) is 0.744, indicating that the majority of
the predictions are of very high accuracy.
We have also examined the dependence of prediction accuracy on the structural
class of target proteins (Table \ref{tab:cnhisto}).
Among all the structural classes, $\alpha/\beta$ proteins are predicted most
accurately with $Cor=$ 0.757 and $DevA =$ 0.668. The accuracy for other
classes do not differ qualitatively although all-$\beta$ proteins are predicted
slightly less accurately.
\begin{table}
\caption{\label{tab:cnhisto}Summary of CN predictions for each SCOP class$^a$.}
\begin{center}
\begin{tabular}{lrrrrr}\hline
range$^b$ &\multicolumn{5}{c}{SCOP class$^c$}\\
($Cor$) & a & b & c & d & e\\\hline
(-1,0.5] & 8 & 6 & 3 & 14 & 1 \\
(0.5,0.6] & 19 & 25 & 8 & 19 & 1 \\
(0.6,0.7] & 29 & 29 & 22 & 54 & 3 \\
(0.7,0.8] & 62 & 66 & 76 & 85 & 10 \\
(0.8,0.9] & 43 & 38 & 57 & 67 & 3 \\
(0.9,1.0] & 1 & 0 & 0 & 1 & 0 \\
total & 162 & 164 & 166 & 240 & 18 \\\hline
average $Cor$ & 0.721 & 0.712 & 0.757 & 0.728 & 0.722\\
average $DevA$ & 0.715 & 0.726 & 0.668 & 0.717 & 0.705\\
\hline
\end{tabular}
\end{center}
$^a$ The number of occurrences of $Cor$ for the proteins in the test sets,
classified according to the SCOP database; average values of $Cor$ and $DevA$
are also listed for each class.\\
$^b$ The range ``$(x,y]$'' denotes $x < Cor \leq y$.\\
$^c$ a: all-$\alpha$; b: all-$\beta$; c: $\alpha / \beta$; d: $\alpha + \beta$;
e: multi-domain.
\end{table}
\subsection*{Residue-wise contact order prediction}
For RWCO prediction, the average accuracy was such that $Cor$ = 0.601 and
$DevA$ = 0.881. Although these figures appear to be poor compared to those
of the CN prediction described above, they are yet statistically significant.
The distribution of $Cor$ appears to be rather dispersed
(Figure \ref{fig:histo}c), indicating that the prediction accuracy
strongly depends on the characteristics of each target protein.
In a similar manner as for CN, we also examined the dependence of prediction
accuracy on the structural class of target
proteins (Table \ref{tab:rwcohisto}).
In this case, we have found a notable dependence of prediction accuracy on
structural classes. The best accuracy is obtained for $\alpha+\beta$
proteins with $Cor = $ 0.629 and $DevA = $ 0.832. For these proteins,
the distribution of $Cor$ also shows good tendency in that the fraction of
poor predictions is relatively small (e.g., 14\% for $Cor <$ 0.5).
Interestingly, all-$\beta$ proteins also show good accuracies but
all-$\alpha$ proteins are particularly poorly predicted. These observations
suggest that the correlation between amino acid sequence and RWCO is
strongly dependent on the structural class of the target protein.
However, the rather dispersed distribution of $Cor$ for each class
(Table \ref{tab:rwcohisto}) also suggests that there are more detailed
effects of the global context on the accuracy of RWCO prediction.
\begin{table}
\caption{\label{tab:rwcohisto}Summary of RWCO predictions for each SCOP class$^a$}
\begin{center}
\begin{tabular}{lrrrrr}\hline
range &\multicolumn{5}{c}{SCOP class}\\
($Cor$) & a & b & c & d & e\\\hline
(-1,0.5] & 58 & 31 & 46 & 34 & 6 \\
(0.5,0.6] & 29 & 37 & 31 & 56 & 4 \\
(0.6,0.7] & 41 & 27 & 33 & 65 & 5 \\
(0.7,0.8] & 24 & 47 & 40 & 72 & 3 \\
(0.8,0.9] & 10 & 22 & 16 & 13 & 0 \\
total & 162 & 164 & 166 & 240 & 18 \\\hline
average $Cor$ & 0.549 & 0.620 & 0.595 & 0.629 & 0.564\\
average $DevA$ & 0.981 & 0.869 & 0.857 & 0.832 & 0.957\\
\hline
\end{tabular}
\end{center}
$^a$See Table \ref{tab:cnhisto} for notations.
\end{table}
\subsection*{Purely linear predictions with PSSMs}
Almost all the modern methods for 1D structure prediction make use of PSSMs
in combination with some kind of machine-learning techniques such as
feed-forward or recurrent neural networks or support vector machines.
The present study is no exception. Curiously, machine-learning approaches
have become so widespread that no attempt appears to have been made to test
simplest linear predictors based on PSSMs.
In this subsection, we present results of 1D predictions using only the linear
terms (Eq. \ref{eq:lin}) but without CRNs. In this prediction scheme,
input is a local segment of a PSSM generated by PSI-BLAST, and a feature
variable is predicted by a straight forward linear regression.
As can be clearly seen in Table \ref{tab:lin}, the results of the linear
predictions are surprisingly good although not as good as with CRNs.
For example, in SS prediction, the purely linear scheme achieved
$Q_3$ = 75.2\% which is lower than that of the CRNs-based scheme by only
3.6\%. Although this is of course a large difference in a statistical sense,
there may not be a discernible difference when individual predictions are
concerned. (However, the improvement in the $SOV$ measure by using CRNs is
quite large, indicating that the nonlinear terms in CRNs are indeed able to
extract cooperative features.) It is widely accepted that the upper limit of
accuracy ($Q_3$) of SS prediction based on a local window of a single
sequence is less than 70\%\cite{CrooksANDBrenner2004}.
Therefore, more than 5\% of the increase in $Q_3$ is brought simply
by the use of PSSMs.
Similar observations also hold for CN and RWCO predictions
(Table \ref{tab:lin}). In case of CN prediction, we have previously
obtained $Cor$ = 0.555 by a simple linear method
with single sequences\cite{KinjoETAL2005}. Therefore, the effect of
PSSMs is even more dramatic than SS prediction. This may be due to the fact
that the most conspicuous feature of amino acid sequences conserved among
distant homologs (as detected by PSI-BLAST)
is the hydrophobicity of amino acid residues\cite{KinjoANDNishikawa2004},
which is closely related to contact numbers.
Of course, the improvement by the use of PSSMs is largely made possible by
the recent increase of amino acid sequence
databases\cite{PrzybylskiANDRost2002}.
\begin{table}
\caption{\label{tab:lin}Summary of prediction accuracies using only linear terms.}
\begin{center}
\begin{tabular}[h]{ll}\hline
Struct. & Accuracy \\\hline
SS & $Q_3$ = 75.2; $SOV$ = 72.7\\
CN & $Cor$ = 0.701; $DevA$ = 0.735\\
RWCO& $Cor$ = 0.584; $DevA$ = 0.902\\\hline
\end{tabular}
\end{center}
\end{table}
\subsection*{The significance of criticality}
The condition of criticality ($\beta = 0.5$ in Eq. \ref{eq:eos}) is expected
to enhance the extraction of the long-range correlations of an amino acid
sequence, thus improving the prediction accuracy. To confirm this
point, we tested the method by setting $\beta = 0.1$ so that the
network of state vectors is not at the critical point any more (otherwise
the prediction and validation schemes were the same as above).
The prediction accuracies obtained by these non-critical random networks
were $Q_3 = 76.7$\% and $SOV = 76.6$ for SS,
$Cor = 0.716$ and $DevA = 0.719$ for CN,
and $Cor = 0.589$ and $DevA = 0.897$ for RWCO.
These values are inferior to those obtained by the critical random networks
(Table \ref{tab:summ}), although slightly better than the purely linear
predictions (Table \ref{tab:lin}).
Therefore, compared to the non-critical random networks, the critical random
networks can indeed extract more information from amino acid sequence and
improve the prediction accuracies.
\section*{Discussion}
\subsection*{Comparison with other methods}
Regarding the framework of 1D structure prediction, the critical random
networks are most closely related to bidirectional recurrent neural
networks (BRNNs)\cite{BaldiETAL1999}, in that both can treat a whole amino
acid sequence rather than only a local window segment. The main differences
are the following. First, network weights between input and hidden
layers as well as those between hidden units are trained in BRNNs,
whereas the corresponding weights in CRNs (random matrices $V$ and $W$,
respectively, in Eq. \ref{eq:eos}) are fixed. Second, the output layer is
nonlinear in BRNNs but linear in CRNs. Third, the network components that
propagate sequence information from N-terminus to C-terminus are decoupled
from those in the opposite direction in BRNNs, but they are coupled in CRNs.
Regarding the accuracy of SS prediction, BRNNs\cite{PollastriETAL2002b} and
CRNs exhibit comparable results of $Q_3 \approx$ 78\%.
However, a standard local window-based approach using feed-forward neural
networks can also achieve this level of accuracy\cite{Jones1999}.
Thus, the CRNs-based method is not a single best predictor, but may serve as
an addition to consensus predictions.
Although BRNNs have been also applied to CN prediction\cite{PollastriETAL2002},
contact numbers are predicted as 2-state categorical data (buried or exposed)
so that the results cannot be directly compared. Nevertheless, we can convert
CRNs-based real-value predictions into 2-state predictions. By using the same
thresholds for
the 2-state discretization as Pollastri et al.\cite{PollastriETAL2002}
(i.e., the average CN for each residue type),
we obtained $Q_2 =$ 75.6\% per chain (75.1\% per residue),
and Matthews' correlation coefficient $MC =$ 0.503
whereas those obtained by BRNNs are $Q_2 =$ 73.9\% (per residue)
and $MC =$ 0.478.
Therefore, for 2-state CN prediction, the present method yields more
accurate results.
Since the present study is the very first attempt to predict RWCOs, there are
no alternative methods to compare with. However, the comparison of
CRNs-based methods for SS and CN predictions with other methods suggests that
the accuracy of the RWCO prediction presented here may be the best
possible result using any of the statistical learning methods currently
available for 1D structure predictions.
\subsection*{Possibilities for further improvements}
In the present study, we employed the simplest possible architecture for CRNs
in which different sites are connected via nearest-neighbor interactions.
A number of possibilities exist for the elaboration of the architecture.
For example, we may introduce short-cuts between distant sites to treat
non-local interactions more directly. Since the prediction accuracies
depend on the structural context of target proteins (Tables \ref{tab:cnhisto}
and \ref{tab:rwcohisto}), it may be also useful to include more global
features of amino acid sequences such as the bias of amino acid composition
or the average of PSSM components. These possibilities are to be pursued in
future studies.
\section*{Conclusion}
We have developed a novel method, CRNs-based regression, for predicting
1D protein structures from amino acid sequence. When combined
with position-specific scoring matrices produced by PSI-BLAST, this method
yields SS predictions as accurate as the best current predictors, CN
predictions far better than previous methods, and RWCO predictions
significantly correlated with observed values. We also examined
the effect of PSSMs on prediction accuracy, and showed that most
improvement is brought by the use of PSSMs although the further improvement
due to the CRNs-based method is also significant.
In order to achieve a qualitatively yet better predictions, however, it seems
necessary to take into account other, more global, information than is
provided by PSSMs.
\section*{Acknowledgments}
The authors thank Motonori Ota for critical comments on an early version of
the manuscript, and Kentaro Tomii for the advice on the use of PSI-BLAST.
Most of the computations were carried out at the supercomputing facility of
National Institute of Genetics, Japan. This work was supported in part by a
grant-in-aid from the MEXT, Japan.
The source code of the programs for the CRNs-based prediction
as well as the lists of protein domains used in this study are available at
\verb|http://maccl01.genes.nig.ac.jp/~akinjo/crnpred_suppl/|.
|
{
"timestamp": "2005-10-20T10:02:43",
"yymm": "0503",
"arxiv_id": "q-bio/0503032",
"language": "en",
"url": "https://arxiv.org/abs/q-bio/0503032"
}
|
\section{Introduction}
Inevitably, the entanglement of a multi-partite quantum
state becomes degraded with time due to experimental and
environmental noise. The influence of noise on bipartite entanglement
is a problem in the theory of open systems \cite{Yu-Eberly06B}, as
well as of practical importance in any application using quantum
features of information \cite{CM}.
The topic of evolution of quantum coherence in the presence of noise
sits between two well-investigated problems. One of these is the
relaxation toward steady-state of one-body coherence of a simple
quantum system (spin, atom, exciton, quantum dot, etc.) in contact
with a much larger reservoir \cite{Slichter78}. The other is the
newer two-body problem where the evolution of the disentanglement of
the system from its environment is of interest. It is generally
understood that the latter decoherence occurs much more rapidly than
the former.
Recently, a practical problem that includes parts of both has drawn
attention -- the survival of the joint entanglement of two small
systems with each other while each is exposed to a local noisy
environment. Their rapid disentanglement from their environments is
supposed not to be observed, but their disentanglement from each
other is considered interesting and potentially important.
We have shown in a specific instance of such bipartite
disentanglement of qubits \cite{Yu-Eberly02,Yu-Eberly03} that
entanglement is lost in a very different way compared to the usual
one-body decoherence measured by the decay of off-diagonal elements
of the density matrix of either qubit system separately. More
surprisingly, we have shown \cite{Yu-Eberly04,Yu-Eberly05} that the
presence of either pure vacuum noise or even classical noise can
cause entanglement to decay to zero in a finite time, an effect
that is labelled ``entanglement sudden death'' (ESD). In the last few years the
issue of such entanglement decoherence has been discussed in a number
of distinct contexts such as qubit pairs \cite{Yu-Eberly02,
Yu-Eberly03, Yu-Eberly04, Lucamarini-etal04, Jakobczyk-Jamroz04,
Tolkunov-etal05, Ban06, Ban-Shibata06,Malinovsky06, Glendinning-etal,
An-etal06, Liang06, Roszak-Machnikowski06, Jamroz06, Ficek-Tanas06,
Solenov05}, finite spin chains \cite{Pratt-Eberly01, Kamta-Starace02,
WangJ-etal05, Khveshchenko03, Grigorenko-Khveshchenko05, Pratt04a},
multipartite systems \cite{Carvalho-etal04, Carvalho-etal05},
decoherence dynamics in adiabatic entanglement \cite{Sun01},
entanglement transfer \cite{Lamata-etal06}, and open quantum systems
\cite{Diosi03, Dodd-Halliwell04, Dodd04,Zyczkowski02}, to name a few.
In addition, a proposal for the direct measurement of finite-time
disentanglement in a cavity QED context has been made recently
\cite{Santos-etal06}.
In this paper we report several steps that we expect will assist
further understanding of this complex and fundamental topic. We focus
on the smallest and simplest non-trivial situations, in order to help
expose consequences that are dynamically fundamental, as opposed to
ones originating simply in one or another kind of complexity. For
greatest utility, this more or less mandates that results should be
analytic rather than numeric. We will treat two qubits prepared in a
mixed state as an information time-evolution question
in the presence of noises. For this purpose, solutions of the
appropriate Born-Markov-Lindblad master equations can be obtained
\cite{BMLexamples} and we will use a Kraus operator approach
throughout the paper \cite{Kraus}.
The focus will be maintained strictly on the way information itself
evolves by considering the entanglement of two quantum systems $A$
and $B$ exposed to local noises but completely isolated from
interacting with each other. We will examine evolution toward
complete disentanglement in a class of commonly occurring bipartite
density matrices (which we call ``X'' states) and establish: (a) that
X-state character is robust, i.e., an X state remains an X state in its
evolution under the most common noise influences, (b) as Werner
states are a subclass of X states, we will show that there exisits a
critical Werner fidelity below which termination of
entanglement must occur in a finite time, and (c) that there are
purely local operations that can sometimes be used to alter the survival
dynamics of bipartite entanglement. We show that ESD will customarily
occur, but that in some cases it can be avoided by applying
appropriate local operations initially. We will illustrate all of
these in the following sections.
The paper is organized as follows: In Sec. \ref{modelsection}, we
present two-qubit models where the bipartite system is coupled to
external sources of phase-damping and amplitude-damping noises.
Explicit time-dependent solutions in terms of Kraus operators are
given. Sec. \ref{concurrence/standard} deals with concurrence, the
chosen measure of entanglement, and the defining character of an
X state. In Sec. \ref{decoherence}, the evolution of a Werner state
toward decoherence is discussed, as an important example of X-state
behavior under the influence of noise. We find a new fidelity
boundary below which entanglement sudden death (ESD) must occur for
all Werner states. In the following Sec. \ref{depolar} we derive the
ESD that is encountered with depolarizing noise for the X states. In
Sec. \ref{fragile}, we show that in some cases it is feasible to
transform a short-lived state into a long-lived state by applying
specified local operations initially, and we conclude in Sec.
\ref{conc}.
\section{Models}
\label{modelsection}
\noindent
The non-interacting quantum systems $A$ and $B$ and their separate
reservoirs labeled $a$ and $b$ are assumed to follow the same evolution route
separately. We use the familiar Hamiltonian (for qubit $A$ say):
\begin{equation}} \newcommand{\eeq}{\end{equation}
\label{model}
H^A_{\rm tot}= H^A_{\rm at} + H^a_{\rm res} + H^{\rm Aa}_{\rm int},
\eeq
where:
\begin{equation}} \newcommand{\eeq}{\end{equation}
H^A_{\rm at} =
\frac{1}{2}\om_A \si^A_z \quad {\rm and} \quad
H^a_{\rm res} = \sum_{{k}}\om_{k}a_{ k}^\dag a_{ k}
\label{eq1}
\eeq
and for exposure to phase and amplitude noises the interaction Hamiltonians
$H_{\rm int}$ are given by
\begin{equation}} \newcommand{\eeq}{\end{equation}
H^{Aa}_{\rm ph-int} = \sum_{{k}} \si^A_z (f_{ k}a^\dag_{k}
+ f_{ k}^*a_{ k}),
\label{eq3}
\eeq
and
\begin{equation}} \newcommand{\eeq}{\end{equation}
\label{intam}
H^{Aa}_{\rm am-int} = \sum_{{ k}} ( g_{ k} \si^A_- a^\dag_{k}
+ g_{k}^* \sigma^A_+a_{ k}).
\eeq
Here the $a_{ k}$ are bosonic reservoir coordinates satisfying $[a_{
k}, a_{ k'}]=\delta_{ k, k'},$ the $g_{ k}$ are broadband coupling
constants, and the $\si^A$s denote the usual Pauli matrices for qubit
$A$. The same forms hold for $B$, with a set of bosonic reservoir
coordinates $b_k$. $\{A,a\}$ and $\{B, b\}$ are
completely independent, and no
decoherence-free joint subspaces are available. As remarked, these
total Hamiltonians provide well-known solvable qubit-reservoir
interactions, but we are interested in the evolution of joint
information as a consequence of the completely separate interactions.
We consider qubits $A$ and $B$ prepared in a mixed state. For this purpose, solutions of the
appropriate Born-Markov-Lindblad equations can be obtained via
several routes, and we find the Kraus operator form \cite{Kraus}
convenient for our purpose. Given an initial state $\rho$ (pure or
mixed) for two qubits $A$ and $B$, its evolution can be written
compactly as
\begin{equation}} \newcommand{\eeq}{\end{equation} \label{Kraus}
\rho(t) = \sum_\mu K_\mu(t)\rho(0) K_\mu^\dag(t),
\eeq
where the so-called Kraus operators $K_\mu$ satisfy $\sum_\mu
K_\mu^\dag K_\mu = 1$ for all $t$. Obviously, the Kraus operators
contain the complete information about the system's dynamics.
In the case of dephasing noise one has following compact Kraus operators:
\begin{eqnarray}
\label{k1}E_1
&=&\left(\begin{array}{clcr}
\gamma_A && 0\\
0 && 1\\
\end{array}
\right)\otimes \left(
\begin{array}{clcr}
\gamma} \newcommand{\ka}{\kappa_B & 0\\
0 & 1\\
\end{array}
\right),\\
E_2&=&\left(\begin{array}{clcr}
\gamma_A && 0\\
0 && 1\\
\end{array}
\right)\otimes\left(
\begin{array}{clcr}
0 & 0 \\
0 & \om_B\\
\end{array}
\right),\\
E_3&=& \left(
\begin{array}{clcr}
0 & 0\\
0 & \om_A \\
\end{array}
\right) \otimes \left(
\begin{array}{clcr}
\gamma} \newcommand{\ka}{\kappa_B & 0\\
0 & 1\\
\end{array}
\right),\\
E_4 &=& \left(
\begin{array}{clcr}
0 & 0\\
0 & \om_A\\
\end{array}
\right)\otimes \left(
\begin{array}{clcr}
0 & 0 \\
0 & \om_B\\
\end{array}
\right),\label{k5}
\end{eqnarray}
where the time-dependent Kraus matrix elements are $$\gamma_A(t) =
\exp{(-\Gamma^A_{\rm ph} t/2)} \quad {\rm and} \quad \om_A(t) =
\sqrt{1-\gamma^2_A(t)},$$ where $\Gamma^A_{\rm ph}$ is the phase
damping rate of qubit A. We use the similar expressions
$\gamma_B(t)$ and $\om_B(t)$ for qubit B, and will take
$\Gamma^A_{\rm ph} = \Gamma^B_{\rm ph}=\Gamma_{\rm ph}$ for greatest
simplicity.
Similarly, the Kraus operators for zero-temperature amplitude noise
are given by
\begin{eqnarray}
\label{k10}
F_1&=&\left(\begin{array}{clcr}
\gamma_A & 0\\
0 & 1\\
\end{array}
\right)\otimes\left(
\begin{array}{clcr}
\gamma_B & 0 \\
0 & 1\\
\end{array}
\right),\label{e1}\\
F_2&=&\left(
\begin{array}{clcr}
\gamma_A & 0 \\
0 & 1\\
\end{array}
\right)\otimes\left(
\begin{array}{clcr}
0 & 0 \\
\omega_B & 0\\
\end{array}
\right),\label{e2}\\
F_3&=& \left(\begin{array}{clcr}
0 & 0 \\
\om_A & 0\\
\end{array}
\right)\otimes\left(
\begin{array}{clcr}
\gamma_B & 0 \\
0 & 1\\
\end{array}
\right),\label{e3}\\
F_4 &=& \left(\begin{array}{clcr}
0 & 0) \\
\omega_A & 0 \\
\end{array}
\right)\otimes\left(
\begin{array}{clcr}
0 & 0 \\
\omega_B & 0 \\
\end{array}
\right),\label{e4}
\end{eqnarray}
and the time-dependent Kraus matrix elements are defined similarly as in
the dephasing model, e.g., $\gamma_A(t)=\exp \left(-\Gamma^A_{\rm am
} t/2\right)$, etc. With the above explicit solutions of the models,
we are able to compute the degree of entanglement of the two qubits
in temporal evolution.
\section{Decoherence Measure and X States}
\label{concurrence/standard}
In order to describe the dynamic evolution of quantum entanglement we
use Wootters' concurrence \cite{Wootters}. Any entropy-based measure
of entanglement will yield the same conclusion about bipartite
separability. Concurrence varies from $C=0$ for a separable state to
$C=1$ for a maximally entangled state. For any two qubits, the
concurrence may be calculated explicitly from their density matrix
$\rho$ for qubits $A$ and $B$:
\begin{equation}} \newcommand{\eeq}{\end{equation}
\label{definationc}
C(\rh)=\max\{0,\sqrt{\lam_1}-\sqrt{\lam_2}-\sqrt{\lam_3}-\sqrt{\lam_4}\,\,\},
\eeq
where the quantities $\lam_i$ are the eigenvalues in decreasing order
of the matrix $\zeta$:
\begin{equation}} \newcommand{\eeq}{\end{equation} \zeta=\rho(\sigma_y\otimes
\sigma_y)\rho^*(\sigma_y\otimes \sigma_y),
\label{concurrence}
\eeq
where $\rh^*$ denotes the complex
conjugation of $\rh$ in the standard basis $|+ +\rangle, |+ -\rangle,
|- +\rangle, |- -\rangle$ and
$\si_y$ is the Pauli matrix expressed in the same basis as:
\begin{equation}
\si_y= \left(\begin{array}{clcr}
0 & -i\\
i & 0 \\
\end{array}
\right).
\end{equation}
In the following we will examine the evolution of entanglement under
noise-induced relaxation of a class of important bipartite density
matrices which are defined below. Since a density matrix in this
class only contains non-zero elements in an ``X" formation, along
the main diagonal and anti-diagonal, we call them ``X states":
\begin{equation}
\label{e.oldrho}
\rho^{AB} =\left(
\begin{array}{clcr}
a & 0 & 0 & w\\
0 & b & z & 0 \\
0 & z^* & c & 0\\
w^* & 0 & 0 & d
\end{array} \right).
\end{equation}
where $a+b+c+d = 1$.
Such a simple matrix is actually not unusual. Experience shows that
this X mixed state arises naturally in a wide variety of
physical situations (see \cite{WangJ-etal05,Pratt04a,standard2}). We
particularly note that it includes pure Bell states as well as the
well-known Werner mixed state \cite{Werner} as special cases. Unitary
transforms of it extend its domain even more widely, as we will
explain below.
The mixed states defined here not only are rather common but also
have the property that they often retain the X form under noise
evolution. This may be expected for phase noise, which can only give
time dependence to the off-diagonal matrix elements. The interaction
Hamiltonian and Kraus operators for amplitude (e.g.,
quantum vacuum) noise evolution are different, and evolution under
amplitude noise is more elaborate, affecting all six non-zero
elements (see \cite{Yu-Eberly04}), but robust form-invariance during
evolution is easy to check. This very simple finding applies to a
wide array of realistic noise sources.
For the X state defined in (\ref{e.oldrho}), concurrence
\cite{Wootters} can be easily computed as
$$C(\rho^{AB})=2\max\{0, |z|-\sqrt{ad}, |w|-\sqrt{bc}\}.$$
\section{Evolution to Decoherence of the Werner State}
\label{decoherence}
\noindent
Now we examine decoherence evolution under first phase damping
and then amplitude damping. Within the set of X matrices, let us
focus now on a Werner state \cite{Werner,curious}:
\begin{equation}} \newcommand{\eeq}{\end{equation} \label{werner}
\rho_W=\frac{1-F}{3}I_4 +
\frac{4F-1}{3}|\Psi^-\rangle\langle \Psi^-|,
\eeq
whose matrix elements can be matched to those of the X state
$\rho^{AB}$ easily. $F$ is termed fidelity, and $1 \ge F \ge
\frac{1}{4}$. We will begin by obtaining the time-dependence of
entanglement for the Werner state. Under phase noise the only time
dependence is in $z$:
\begin{equation}} \newcommand{\eeq}{\end{equation} \label{zphase}
z(t) = \frac{1-4F}{6} \gamma^2(t), \quad {\rm with} \quad \gamma(t)
\equiv e^{-\Gamma_{\rm ph}t/2}.
\eeq
\begin{figure} [htbp]
\vspace*{13pt}
\centerline{\epsfig{file=FigW-phs2.eps, width=8.2cm}}
\vspace*{13pt}
\fcaption{\label{fig2} Phase noise causes $\rho_W$ to
disentangle completely in finite time for all Werner states except in
the limiting case of a pure Bell state. The graph shows $C(t)$ vs. $F$
and $\Gamma_{\rm ph} t$.}
\end{figure}
The results are shown in Fig. \ref{fig2}. In particular we note the
occurrence of ESD, in which concurrence non-smoothly goes to zero at
a finite time (and remains zero). This is apparent for all initial
$F<1$. It has been noted already for quantum vacuum noise qubit
decoherence \cite{Yu-Eberly04} and for disentanglement of continuous
joint states \cite{Diosi03, Dodd-Halliwell04, Dodd04}. The analytic
expressions above make it clear why this is so. Since the matrix
elements $a$ and $d$ are fixed, as $z$ decays it must become less
than $\sqrt{ad}$ at a specific time $\tau_{\rm ph}$, which can be
easily determined to be given by
\begin{equation}} \newcommand{\eeq}{\end{equation} \label{phaseratio}
\frac{\tau^{\rm ph}}{\tau_0} = {\ln}
\left[\frac{4F-1}{2-2F}\right],\,\,\,\, 1>F>\frac{1}{2}
\eeq
where $\tau_0 = 1/\Gamma_{\rm ph}$ marks the $1/e$ point in the purely
exponential decay of the underlying individual qubits.
Next we consider Werner state evolution under amplitude noise, and we
find from the appropriate Kraus operators given in
(\ref{k10}-\ref{e4}) \cite{Yu-Eberly04}
that the following time dependences specify $\rho_W(t)$ at any time:
\begin{equation}} \newcommand{\eeq}{\end{equation}
z(t)=\frac{1-4F}{6} \gamma} \newcommand{\ka}{\kappa^2,
\eeq
\begin{equation}} \newcommand{\eeq}{\end{equation}
a(t)=\frac{1-F}{3} \gamma} \newcommand{\ka}{\kappa^4,
\eeq
\begin{equation}} \newcommand{\eeq}{\end{equation}
b(t)=\frac{2F+1}{6} \gamma} \newcommand{\ka}{\kappa^2 +\frac{1-F}{3}\gamma} \newcommand{\ka}{\kappa^2\om^2,
\eeq
\begin{equation}} \newcommand{\eeq}{\end{equation}
c(t)=\frac{2F+1}{6} \gamma} \newcommand{\ka}{\kappa^2 +\frac{1-F}{3}\gamma} \newcommand{\ka}{\kappa^2\om^2, \quad {\rm and}
\eeq
\begin{equation}} \newcommand{\eeq}{\end{equation}
d(t)=\frac{1-F}{3} +\frac{2F+1}{3}\om^2 + \frac{1-F}{3}\om^4.
\eeq
In principle the time-dependent $\gamma$ and $\omega$ parameters
could be different for qubits $A$ and $B$, but we again take them
identical and write $\gamma} \newcommand{\ka}{\kappa = \exp[-\Gamma_{\rm am} t/2]$ and $\om^2 = 1
- \gamma} \newcommand{\ka}{\kappa^2$, where we use $\Gamma_{\rm am}$ to denote the upper level
decay rate of the qubits.
\begin{figure} [htbp]
\vspace*{13pt}
\centerline{\epsfig{file=FigW-ampl2.eps, width=8.2cm}}
\vspace*{13pt}
\fcaption{\label{fig1} In the presence of amplitude
noise there is long-lived concurrence of Werner states only for
sufficiently high fidelity, $F > F_{\rm c} \simeq 0.714$. The graph
shows the critical fidelity boundary in plotting $C(t)$ vs.
$\Gamma_{\rm am} t$.}
\end{figure}
Sudden death of Werner state entanglement appears here also, but with the
important added element that sudden death from amplitude noise occurs
only for a low range of fidelity values. Our result shows that for
all initial $F$ above the critical fidelity $F_{\rm c} \simeq 0.714$
entanglement remains finite for all time, and has an infinitely long
smooth decay, faster than but similar to, the decay of single-qubit
coherence. This is shown in the plot in Fig.~\ref{fig1}.
\section{Bistability Decoherence}
\label{depolar}
\noindent
In this section, we discuss the entanglement decoherence of a
representative X matrix under bistability noise, by which we mean
noise that induces incoherent random transfer back and forth between
the two qubit states when they are energetically degenerate. Examples
occur in bistable systems of all kinds, for example in semiconductor
junctions or double-well electron potentials in photonic crystals. A
physically different example is polarization of photons in optical
fiber with indeterminately random local birefringence. In all of
these cases we can speak of the effect as arising from exposure to an
infinite-temperature reservoir. In that case the
population-equalizing up-transfer and down-transfer rates will be
denoted $\Gamma_{\rm eq}$ and taken the same for the two qubits.
Given these remarks, our basic model with Hamiltonians (\ref{model})
and (\ref{intam}) still allows a useful Kraus representation and the
Kraus matrices are given by:
\begin{eqnarray}
G_1&=& \frac{1}{\sqrt 2}\left(
\begin{array}{cc}
\gamma(t) & 0\\
0 & 1\\
\end{array}
\right), \label{G1}\\
G_2 &=& \frac{1}{\sqrt 2} \left(
\begin{array}{cc}
0 & 0 \\
\om(t) & 0\\
\end{array}
\right), \label{G2}\\
G_3 &=& \frac{1}{\sqrt 2} \left(
\begin{array}{cc}
1 & 0 \\
0 & \gamma(t)\\
\end{array}
\right), \label{G3}\\
G_4 &=& \frac{1}{\sqrt 2} \left(
\begin{array}{cc}
0 & \omega(t) \\
0 & 0 \\
\end{array}
\right), \label{G4}
\end{eqnarray}
where now
$$\gamma(t) = \exp{(-\Gamma_{\rm eq} t/2)} \quad {\rm and} \quad
\om(t) = \sqrt{1-\gamma^2(t)}.$$
State equalization presents the extreme opposite case from vacuum
noise, in the sense that strong equalization treats both qubit states
equally incoherently, whereas vacuum noise induces incoherent decay
into just the energetically lower of the two states.
One finds that the X form of the density matrix (\ref{e.oldrho}) is
still preserved under state equalizing noise and so at time $t$ it
retains the X form (we set $w=0$ for simplicity):
\begin{equation}
\label{sol} \rho(t) = \left(
\begin{array}{clcr}
a(t) & 0 & 0 & 0 \\
0 & b(t) & z(t) & 0 \\
0 & z(t) & c(t) & 0\\
0 & 0 & 0 & d(t)
\end{array} \right).
\end{equation}
We assume that the two qubits are affected by two identical local
depolarization noises, and in this case the time-dependent matrix
elements are given by the following:
\begin{eqnarray}
4a(t) &=& \gamma} \newcommand{\ka}{\kappa^4a + a+\om^2 (b + c) + \om^4d\nonumber\\
&&+ 2\gamma} \newcommand{\ka}{\kappa^2a + \gamma} \newcommand{\ka}{\kappa^2\om^2( b+c),\\
4b(t) &=& 2\gamma} \newcommand{\ka}{\kappa^2b +\gamma} \newcommand{\ka}{\kappa^2\om^2 (a+d) \nonumber\\
&&+ b+\gamma} \newcommand{\ka}{\kappa^4 b+\om^2(a+d) + \om^4c,\\
4c(t) &=& 2\gamma} \newcommand{\ka}{\kappa^2c +\gamma} \newcommand{\ka}{\kappa^2\om^2 (a +d) \nonumber\\
&&+ c+\om^2(d + a) + \om^4 b+\gamma} \newcommand{\ka}{\kappa^4 c,\\
4d(t) &=& d+\om^2(b + c) + \om^4 a + \gamma} \newcommand{\ka}{\kappa^4 d\nonumber\\
&&+ 2\gamma} \newcommand{\ka}{\kappa^2 d + \gamma} \newcommand{\ka}{\kappa^2\om^2(b+c),\\
z(t) &=& \gamma} \newcommand{\ka}{\kappa^2 z.
\end{eqnarray}} \newcommand{\lam}{\lambda
Some algebraic examination shows that this result also leads to ESD,
i.e., bistable equalization leads all entangled X states
(\ref{e.oldrho}) to become separable states in a finite time.
It is easy to check that when $t \to \infty$,
\begin{equation}} \newcommand{\eeq}{\end{equation}
z \to z(\infty) = 0,
\eeq
\begin{equation}} \newcommand{\eeq}{\end{equation}
\{a, b, c, d\} \to \{\frac{1}{4}, \frac{1}{4}, \frac{1}{4},\frac{1}{4}\},
\eeq
and the asymptotic separability arising from the equivalence of all
diagonal elements is just a special case of the general theorem by
Zyczkowski, et al.\cite{Lewenstein98}.
\section{Fragile and Robust Initial Entangled States}
\label{fragile}
\noindent
It is known that entangled states evolve differently under different
environmental noise influences if special symmetries exist. For
example, in the case of collective dephasing noise (see, e.g.,
\cite{Yu-Eberly02}), there may exist decoherence-free subspaces in
which the entangled states are well protected against interaction
with the noise. For the models presented here the noises influence
each qubit independently, so there are no decoherence-free subspaces
and there is no such protection from ESD available. However, we now
show that it is still possible to avoid sudden death by using
appropriate local initial preparations. To illustrate this, we
consider another mixed state within the category of the X matrix
defined in (\ref{e.oldrho}):
\begin{equation}} \newcommand{\eeq}{\end{equation} \label{tildewerner}
\tilde\rho_W=\frac{1-F}{3}I_4 +
\frac{4F-1}{3}|\Phi^-\rangle\langle \Phi^-|,
\eeq
where $|\Phi^-\rangle = (|++\rangle - |--\rangle)/\sqrt{2},$
is a Bell state. In matrix form at any $t>0$ we can write
\begin{equation} \label{tilderho}
\tilde\rho_W(t)=
\left( \begin{array}{clcr}
a(t) & 0 & 0 & w(t) \\
0 & b(t) & 0 & 0 \\
0 & 0 & c(t) & 0\\
w^*(t) & 0 & 0 & d(t)
\end{array} \right).
\end{equation}
It is easy to compute the concurrence of this mixed state: $C = 2
\max\{0, |w|-\sqrt{bc}\}$.
Consider the time dependences for $\tilde\rho_W$ obtained from the
amplitude-noise Kraus operators as before. It is easy to check that the
sudden death condition for $\tilde\rho_W$'s concurrence is now:
\begin{equation}} \newcommand{\eeq}{\end{equation} \label{ConcurtildeW}
\frac{4F-1}{6}\gamma^2 \ =\ \frac{1-F}{3}\gamma^2 +
\frac{2F+1}{6}\gamma^2 \omega^2 ,
\eeq
which is satisfied at a finite $t$ for any value of fidelity $F$.
That is, here there is no range of ``protected" fidelity values under
amplitude noise, as was the case for the other form of X state
in Fig. ~\ref{fig1}. Indefinite survival is impossible in this case,
similar to what the plot in Fig.~\ref{fig2} shows for phase noise.
However, this result has important implications related to survival.
One easily shows that $\tilde\rho_W$ is closely related to the
earlier Werner state $\rho_W$, which does have a range of protected
fidelity values. In fact $\rho_W$ and $\tilde\rho_W$ are unitary
transforms of each other under a {\em purely local} transformation
operator: $U = i\sigma_x^A \otimes I_B$. This shows that survival
against noise of initial mixed state entanglement can in a wide range
of situations be dramatically improved by a simple local unitary
operation (here changing $\tilde\rho_W$ into $\rho_W$), even while
the degree of entanglement is not changed. Intuitively, it is easy
to see that the noise influence represented by the Kraus operators
varies for different matrix elements of a bipartite density matrix.
Although local operations cannot change the degree of entanglememt,
it is possible that local operations can rearrange the matrix
elements of the X states such that the resulting density matrix is
more robust (or fragile) than the original one. Therefore, ESD may
be manipulated by preparatory transformation that is purely local.
\section{Concluding Remarks}
\label{conc}
\noindent
In summary, we have examined quantitatively via fully analytic
expressions the non-local decoherence properties of a wide range of
mixed states. We have used relatively simple Kraus operators to do
this. We have also shown that three well understood physical noise
types (phase noise, amplitude noise, and state-equalizing noise) do
not alter the form of the X mixed state during evolution, although
entanglement survival may be long or short. In particular, we have
established that Werner states are subject to the sudden death
effect, and have specified a new critical fidelity boundary below
which sudden death must occur. Surprisingly, we have found that
Werner states are more robust under pure amplitude noise (e.g.,
spontaneous emission) than under pure phase noise even though
amplitude noise is in a sense more disruptive than dephasing, as the
former causes diagonal and off-diagonal relaxation and the latter
off-diagonal relaxation alone. Moreover, we have shown that
bistability decoherence can cause all the X states to disentangle
completely in finite time. In addition, we have shown that in some
cases an initial mixed state's entanglement can be preserved under
subsequent noisy evolution, i.e., sudden death avoided, by an initial
local unitary operation. Such local operations may offer a useful
tool in entanglement control when the duration of entangled states is
crucial in the processes of quantum state storage and preparation.
Finally, we note that while we have found interesting and unexpected
features of Werner and other X mixed states in the presence of common
noise sources, our Kraus operators treated all of them as white
(Markovian) noises. It will be an important theoretical challenge to extend these
results to the case of non-Markovian environmental influences
\cite{Glendinning-etal,Yonac-etal06,Hu-etal00-05}.
\nonumsection{Acknowledgements}
\noindent
We acknowledge financial support from NSF Grant
PHY-0456952 and ARO Grant W911NF-05-1-0543.
We also thank C. Broadbent for assistance with Figs. 1 and 2.
\nonumsection{References}
\noindent
|
{
"timestamp": "2007-01-02T14:50:15",
"yymm": "0503",
"arxiv_id": "quant-ph/0503089",
"language": "en",
"url": "https://arxiv.org/abs/quant-ph/0503089"
}
|
\section{Introduction}
There has been an accrued interest on quantum plasmas, motivated by applications in ultra small electronic devices \cite{Markowich}, dense astrophysical plasmas \cite{Chabrier}-\cite{Jung} and laser plasmas \cite{Kremp}. Recent developments involves quantum corrections to Bernstein-Greene-Kruskal equilibria \cite{Luque}, quantum beam instabilities \cite{Anderson}-\cite{Haas3}, quantum ion-acoustic waves \cite{Haas4}, quantum corrections to the Zakharov equations \cite{Garcia, Haas5}, modifications on Debye screening for quantum plasmas with magnetic fields \cite{Shokri1}, quantum drift waves \cite{Shokri2}, quantum surface waves \cite{Shokri3}, quantum plasma echoes \cite{Manfredi1}, the expansion of a quantum electron gas into vacuum \cite{Mola} and the quantum Landau damping \cite{Suh}. In addition, quantum methods have been used for the treatment of classical plasma problems \cite{Fedele, Bertrand}.
One possible approach to charged particle systems where quantum effects are relevant is furnished by quantum hydrodynamics models. In fact, hydrodynamic formulations have appeared in the early days of quantum mechanics \cite{Madelung}. More recently, the quantum hydrodynamics model for semiconductors has been introduced to handle questions like negative differential resistance as well as resonant tunneling phenomena in micro-electronic devices \cite{Gardner1}-\cite{Gardner3}. The derivation and application of the quantum hydrodynamics model for charged particle systems is the subject of a series of recent works \cite{Manfredi2}-\cite{Degond}. In classical plasmas physics, fluid models are ubiquitous, with their applications ranging from astrophysics to controlled nuclear fusion \cite{Nicholson, Bittencourt}. In particular, magnetohydrodynamics provides one of the most useful fluid models, focusing on the global properties of the plasma. The purpose of this work is to obtain a quantum counterpart of magnetohydrodynamics, starting from the quantum hydrodynamics model for charged particle systems. This provides another place to study the way quantum physics can modify classical plasma physics. However, it should be noted that the quantum hydrodynamic model for charged particle systems was build for non magnetized systems only. To obtain a quantum modified magnetohydrodynamics, this work also offer the appropriated extension of the quantum hydrodynamics model to the cases of non zero magnetic field.
The paper is organized as follows. In Section 2, the equations of quantum hydrodynamics are obtained, now allowing for the presence of magnetic fields. The approach for this is based on a Wigner equation with non zero vector potentials. Defining macroscopic quantities like charge density and current through moments of the Wigner function, we arrive at the desired quantum fluid model. In Section 3, we repeat the well known steps for the derivation of magnetohydrodynamics, now including the quantum corrections present in the quantum hydrodynamic model. This produces a quantum magnetohydrodynamics set of equations. In Section 4, a simplified set of quantum magnetohydrodynamics is derived, yielding a quantum version of the generalized Ohm's law. In addition, the infinity conductivity case is shown to imply an ideal quantum magnetohydrodynamic model. In this ideal case, there is the presence of quantum corrections modifying the transport of momentum and the equation for the electric field. Section 5 studies the influence of the quantum terms on the equilibrium solutions. Exact solutions are found for translational invariance. Section 6 is devoted to the conclusions.
\section{Quantum Hydrodynamics in the Presence of Magnetic Fields}
For completeness, we begin with the derivation of the Wigner-Maxwell system providing a kinetic description for quantum plasmas in the presence of electromagnetic fields.
For notational simplicity, we first consider a quantum hydrodynamics model for non zero magnetic fields in the case of a single species plasma. Extension to multi-species plasmas is then straightforward. Our starting point is a statistical mixture with $N$ states described by the wave functions $\psi_\alpha = \psi_{\alpha}({\bf r},t)$, each with probability $p_\alpha$, with $\alpha = 1 \dots N$. Of course, $p_\alpha \geq 0$ and $\sum_{\alpha=1}^{N}p_\alpha = 1$. The wave functions obey the Schr\"odinger equation,
\begin{equation}
\label{e1}
\frac{1}{2m}(-i\hbar\nabla - q{\bf A})^{2}\,\psi_\alpha + q\phi\,\psi_\alpha = i\hbar\frac{\partial\psi_\alpha}{\partial t} \,.
\end{equation}
Here we consider charge carriers of mass $m$ and charge $q$, subjected to possibly self-consistent scalar and vector potentials $\phi = \phi({\bf r},t)$ and ${\bf A} = {\bf A}({\bf r},t)$ respectively. For convenience in some calculations, we assume the Coulomb gauge, $\nabla\cdot{\bf A} = 0$.
From the statistical mixture, we construct the Wigner function $f = f({\bf r},{\bf p},t)$ defined as usual from
\begin{equation}
\label{e2}
f({\bf r},{\bf p},t) = \frac{1}{(2\pi\hbar)^3}\sum_{\alpha=1}^{N}p_{\alpha}\int\,d{\bf s}\,\psi_{\alpha}^{*}({\bf r}+\frac{\bf s}{2})\,e^{\frac{i{\bf p}\cdot{\bf s}}{\hbar}}\,\psi_{\alpha}({\bf r}-\frac{\bf s}{2}) \,.
\end{equation}
After some long but simple calculations involving the Schr\"odinger equations for each $\psi_\alpha$ and the choice of the Coulomb gauge, we arrive at the following integro-differential equation for the Wigner function,
\begin{eqnarray}
\label{e3}
&\strut& \frac{\partial f}{\partial t} + \frac{\bf p}{m}\cdot\nabla\,f = \\ &\strut& \frac{iq}{\hbar(2\pi\hbar)^3}\int\int d{\bf s}\,d{\bf p}'\,e^{\frac{i({\bf p}-{\bf p}')\cdot{\bf s}}{\hbar}}\,[\phi({\bf r}+\frac{\bf s}{2})-\phi({\bf r}-\frac{\bf s}{2})]\,f({\bf r},{\bf p}',t) + \nonumber
\\
&\strut& \frac{iq^2}{2\hbar m(2\pi\hbar)^3}\int\int d{\bf s}\,d{\bf p}'\,e^{\frac{i({\bf p}-{\bf p}')\cdot{\bf s}}{\hbar}}\,[A^{2}({\bf r}+\frac{\bf s}{2})-A^{2}({\bf r}-\frac{\bf s}{2})]\,f({\bf r},{\bf p}',t) + \nonumber \\
&\strut& \frac{q}{2m(2\pi\hbar)^3}\,\,\nabla\cdot\int\int d{\bf s}\,d{\bf p}'\,e^{\frac{i({\bf p}-{\bf p}')\cdot{\bf s}}{\hbar}}\,[{\bf A}({\bf r}+\frac{\bf s}{2})-{\bf A}
({\bf r}-\frac{\bf s}{2})]\,f({\bf r},{\bf p}',t) \nonumber \\
&-& \frac{iq}{\hbar m(2\pi\hbar)^3}\,\,{\bf p}\cdot\int\int d{\bf s}\,d{\bf p}'\,e^{\frac{i({\bf p}-{\bf p}')\cdot{\bf s}}{\hbar}}\,[{\bf A}({\bf r}+\frac{\bf s}{2})-{\bf A}({\bf r}-\frac{\bf s}{2})]\,f({\bf r},{\bf p}',t) \,. \nonumber
\end{eqnarray}
All macroscopic quantities like charge and current densities can be found taking appropriated moments of the Wigner function. This is analogous to classical kinetic theory, where charge and current densities are obtained from moments of the one-particle distribution function. Alternatively, we could have started from the complete many body wave function, defined a many body Wigner function and then obtained a quantum Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy. With some closure hypothesis, in this way we arrive at a integro-differential equation for the one-particle Wigner function, which has to be supplemented by Maxwell equations. This is the Wigner-Maxwell system, which plays, in quantum physics, the same role the Vlasov-Maxwell system plays in classical physics. When the vector potential is zero, it reproduces the well known Wigner-Poisson system \cite{Drummond, Klimontovich}. In addition, in the formal classical limit when $\hbar \rightarrow 0$, the Wigner equation (\ref{e3}) goes to the Vlasov equation,
\begin{equation}
\label{e4}
\frac{\partial f}{\partial t} + {\bf v}\cdot\nabla f + \frac{q}{m}({\bf E} + {\bf v}\times{\bf B})\cdot\frac{\partial f}{\partial{\bf v}} = 0 \,,
\end{equation}
where ${\bf v} = ({\bf p}-q{\bf A})/m$, ${\bf E} = - \nabla\phi - \partial{\bf A}/\partial t$ and ${\bf B} = \nabla\times{\bf A}$. However, notice that a initially positive definite Wigner function can evolve in such a way it becomes negative in some regions of phase space. Hence, it can not be considered as a true probability function. Nevertheless, all macroscopic quantities like charge, mass and current densities can be obtained from the Wigner function through appropriated moments.
Equation (\ref{e3}) coupled to Maxwell equations provides a self-consistent kinetic description for a quantum plasma. As long as we know, it has been first obtained, with a different notation, in the work \cite{Arnold}. It has been rediscovered in \cite{Materdey}, in the case of homogeneous magnetic fields. Wigner functions appropriated to non zero magnetic fields have also been discussed, for instance, in \cite{Carruthers}-\cite{Bialynicki}, without the derivation of an evolution equation for the Wigner function alone. More recently, a different transport equation for Wigner functions appropriated to non zeros magnetic field and spin has been obtained in \cite{Saikin}. The starting point of this latter development, however, is the Pauli and not the Schr\"odinger equation as here and \cite{Arnold}. Finally, relativistic models for self-consistent charged particle systems with spin can be found in \cite{Masmoudi}.
Most of the works dealing with quantum charged particle systems prefer to work with the wave functions and not directly with the Wigner function, as in \cite{Kumar}. The impressive form of (\ref{e3}) seems to support this approach. Indeed, probably (\ref{e3}) can be directly useful only in the linear or homogeneous magnetic field cases. This justifies the introduction of alternative descriptions. At the coast of the loss of some information about kinetic phenemena like Landau damping, we can simplify our model adopting a formal hydrodynamic formulation. Define the fluid density
\begin{equation}
\label{e5}
n = \int\,d{\bf p}\,f \,,
\end{equation}
the fluid velocity
\begin{equation}
\label{e6}
{\bf u} = \frac{1}{mn}\int\,d{\bf p}\,({\bf p}-q{\bf A})\,f
\end{equation}
and the pressure dyad
\begin{equation}
\label{e7}
{\bf P} = \frac{1}{m^2}\int\,d{\bf p}\,({\bf p} - q{\bf A})\otimes({\bf p} - q{\bf A})\,f - n{\bf u}\otimes{\bf u} \,.
\end{equation}
We could proceed to higher order moments of the Wigner function, but (\ref{e5})-(\ref{e7}) are sufficient if we do not want to offer a detailed description of energy transport.
Taking the appropriated moments of the Wigner equation (\ref{e3}) and using the definitions (\ref{e5})-(\ref{e7}), we arrive at the following quantum hydrodynamic model,
\begin{eqnarray}
\label{e8}
\frac{\partial n}{\partial t} &+& \nabla\cdot(n{\bf u}) = 0 \,, \\
\label{e9}
\frac{\partial{\bf u}}{\partial t} &+& {\bf u}\cdot\nabla{\bf u} = - \frac{1}{n}\nabla\cdot{\bf P} + \frac{q}{m}({\bf E} + {\bf u}\times{\bf B}) \,.
\end{eqnarray}
Equations (\ref{e8})-(\ref{e9}) does not show in an obvious way any quantum effects, since $\hbar$ is not explicitly present there. To found the hidden quantum effects, we follow mainly the style of references \cite{Manfredi2}, \cite{Gasser1} and \cite{Lopez}, but now allowing for magnetic fields. In the definition (\ref{e2}) of the Wigner function, consider the decomposition
\begin{equation}
\label{e10}
\psi_\alpha = \sqrt{n_\alpha}\,\,\,e^{iS_{\alpha}/\hbar} \,,
\end{equation}
for real $n_\alpha = n_{\alpha}({\bf r},t)$ and $S_\alpha = S_{\alpha}({\bf r},t)$. Evaluating, the integral for the pressure dyad, we get a decomposition in terms of ``classical" ${\bf P}^C$ and ``quantum" ${\bf P}^Q$ contributions,
\begin{equation}
\label{e11}
{\bf P} = {\bf P}^C + {\bf P}^Q \,,
\end{equation}
where
\begin{eqnarray}
\label{a1}
{\bf P}^C &=& m\sum_{\alpha=1}^{N}{p_{\alpha}n_\alpha}({\bf u}_{\alpha} - {\bf u})\otimes({\bf u}_{\alpha} - {\bf u}) + \\ &+&
m\sum_{\alpha=1}^{N}{p_{\alpha}n_\alpha}({\bf u}^{o}_{\alpha} - {\bf u}^{o})\otimes({\bf u}^{o}_{\alpha} - {\bf u}^{o}) \,, \nonumber \\
\label{a2}
{\bf P}^Q &=& - \frac{\hbar^{2}n}{4m}\nabla\otimes\nabla\,\ln\,n \,.
\end{eqnarray}
In the definitions of classical pressure dyad ${\bf P}^C$, we considered the kinetic fluid velocity associated to the wave function $\psi_\alpha$,
\begin{equation}
\label{e15}
{\bf u}_{\alpha} = \frac{\nabla S_\alpha}{m} \,,
\end{equation}
and the kinetic fluid velocity associated to the statistical mixture,
\begin{equation}
\label{e16}
{\bf u} = \sum_{\alpha=1}^{N}\,\frac{p_{\alpha}n_\alpha}{n}{\bf u}_{\alpha} \,.
\end{equation}
In a similar way, the second term at the right hand side of equation (\ref{a1}) is constructed in terms of ${\bf u}_{\alpha}^o$, the osmotic fluid velocity associated to the wave function $\psi_\alpha$,
\begin{equation}
\label{e17}
{\bf u}_{\alpha}^o = \frac{\hbar}{2m}\frac{\nabla n_\alpha}{n_\alpha} \,,
\end{equation}
and ${\bf u}^o$, the osmotic fluid velocity associated to the statistical mixture,
\begin{equation}
\label{e18}
{\bf u}^o = \sum_{\alpha=1}^{N}\,\frac{p_{\alpha}n_\alpha}{n}{\bf u}_{\alpha}^o \,.
\end{equation}
We also observe that in terms of the fluid density $n_\alpha$ of the state $\alpha$ the density $n$ of the statistical mixture is given by
\begin{equation}
\label{e19}
n = \sum_{\alpha=1}^{N}\,p_{\alpha}n_\alpha \,.
\end{equation}
Notice that ${\bf P}^C$ a faithful classical pressure dyad, since it comes from dispersion of the velocities, vanishing for a pure state. Indeed, the classical pressure dyad is the sum of a kinetic part, arising from the dispersion of the kinetic velocities, and a osmotic part, arising from the dispersion of the osmotic velocities. However, ${\bf P}^C$ is not strictly classical, since it contains $\hbar$ through the osmotic velocities. In a sense, however, it is ``classical", since it comes from statistical dispersion of the velocities.
In most cases, it suffices to take some equation of state for ${\bf P}^C$. For simplicity, from now on we assume a diagonal, isotropic form $P_{ij} = \delta_{ij}P$, where $P = P(n)$ is a suitable equation of state. Certainly, strong magnetic fields have to be treated more carefully, since they are associated to anisotropic pressure dyads. However, since we are mainly interested on the role of the quantum effects, we disregard such possibility here.
Now inserting the preceding results for the pressure dyad into the momentum transport equation (\ref{e9}), we obtain the suggestive equation
\begin{equation}
\label{e20}
\frac{\partial{\bf u}}{\partial t} + {\bf u}\cdot\nabla{\bf u} = - \frac{1}{mn}\nabla P + \frac{q}{m}({\bf E} + {\bf u}\times{\bf B}) + \frac{\hbar^2}{2m^2}\nabla\left(\frac{\nabla^{2}\sqrt{n}}{\sqrt{n}}\right) \,.
\end{equation}
The equation of continuity (\ref{e8}) and the force equation (\ref{e20}) constitute our quantum hydrodynamic model for magnetized systems. All the quantum effects are contained in the last term of the equation (\ref{e20}), the so called Bohm potential. In comparison with standard fluid models for charged particle systems, the Bohm potential is the only quantum contribution, and the rest of the paper is devoted to study its consequences for magnetohydrodynamics.
\section{Quantum Magnetohydrodynamics Model}
The equations from the last Section were written for a single species charged particle system. Now we generalize to a two species system. Consider electrons with fluid density $n_e$, fluid velocity ${\bf u}_e$, charge $-e$, mass $m_e$ and pressure $P_e$. In an analogous fashion, consider ions with fluid density $n_i$, fluid velocity ${\bf u}_i$, charge $e$, mass $m_i$ and pressure $P_i$. Proceeding as before, now starting from the Wigner equations for electrons and ions, we get the following bipolar quantum fluid model,
\begin{eqnarray}
\label{e21}
\frac{\partial n_e}{\partial t} + \nabla\cdot(n_{e}{\bf u}_e) &=& 0 \,, \\
\label{e22}
\frac{\partial n_i}{\partial t} + \nabla\cdot(n_{i}{\bf u}_i) &=& 0 \,, \\
\label{e23}
\frac{\partial{\bf u}_e}{\partial t} + {\bf u}_{e}\cdot\nabla{\bf u}_e &=& - \frac{\nabla P_{e}}{m_{e}n_{e}} - \frac{e}{m_e}({\bf E} + {\bf u}_{e}\times{\bf B}) + \nonumber \\ &+& \frac{\hbar^2}{2m^{2}_{e}}\nabla\left(\frac{\nabla^{2}\sqrt{n_e}}{\sqrt{n_e}}\right) - \nu_{ei}({\bf u}_e - {\bf u}_{i}) \,,\\
\label{e24}
\frac{\partial{\bf u}_i}{\partial t} + {\bf u}_{i}\cdot\nabla{\bf u}_i &=& - \frac{\nabla P_{i}}{m_{i}n_{i}} + \frac{e}{m_i}({\bf E} + {\bf u}_{i}\times{\bf B}) + \nonumber \\ &+& \frac{\hbar^2}{2m^{2}_{i}}\nabla\left(\frac{\nabla^{2}\sqrt{n_i}}{\sqrt{n_i}}\right) - \nu_{ie}({\bf u}_i - {\bf u}_{e})\,.
\end{eqnarray}
In the equations (\ref{e23}-\ref{e24}), we have added some often used phenomenological terms to take into account for the momentum transport by collisions. The coefficients $\nu_{ei}$ and $\nu_{ie}$ are called collision frequencies for momentum transfer between electrons and ions \cite{Nicholson, Bittencourt}. For quasineutral plasmas, global momentum conservation in collisions imply $m_{e}\nu_{ei} = m_{i}\nu_{ie}$, so that $\nu_{ie} \ll \nu_{ei}$ when the ions are much more massive than electrons \cite{Nicholson, Bittencourt}.
Equations (\ref{e21})-(\ref{e24}) have to be supplemented by Maxwell equations,
\begin{eqnarray}
\label{e25}
\nabla\cdot{\bf E} &=& \frac{\rho}{\varepsilon_0} \,,\\
\label{e26}
\nabla\cdot{\bf B} &=& 0 \,,\\
\label{e27}
\nabla\times{\bf E} &=& - \frac{\partial\bf B}{\partial t} \,,\\
\label{e28}
\nabla\times{\bf B} &=& \mu_{0}{\bf J} + \mu_{0}\varepsilon_{0}\frac{\partial\bf E}{\partial t} \,,
\end{eqnarray}
where the charge and current densities are given respectively by
\begin{equation}
\label{e29}
\rho = e\,(n_i - n_{e}) \,, \quad {\bf J} = e\,(n_{i}{\bf u}_i - n_{e}{\bf u}_{e}) \,.
\end{equation}
Equations (\ref{e21}-\ref{e29}) constitute our complete quantum hydrodynamic model, allowing for magnetic fields. When ${\bf B} \equiv 0$, it goes to the well known quantum hydrodynamic model for bipolar charged particle systems.
Several possibilities of study are open starting from (\ref{e21}-\ref{e29}). Here we are interested in obtaining equations analogous to the classical magnetohydrodynamic equations. In some places, for the sake of clarity and to point exactly for the new contributions of quantum nature, we repeat some well known steps in the derivation of classical magnetohydrodynamics. To proceed in this direction, define the global mass density
\begin{equation}
\label{e30}
\rho_m = m_{e}n_e + m_{i}n_i
\end{equation}
and the global fluid velocity
\begin{equation}
\label{e31}
{\bf U} = \frac{m_{e}n_{e}{\bf u}_{e} + m_{i}n_{i}{\bf u}_{i}}{m_{e}n_{e} + m_{i}n_{i}}\,.
\end{equation}
With these definitions and proceeding like in any plasma physics book \cite{Nicholson, Bittencourt}, we obtain the following equations for $\rho_m$ and ${\bf U}$,
\begin{eqnarray}
\label{e32}
\frac{\partial\rho_m}{\partial t} + \nabla\cdot(\rho_{m}{\bf U}) &=& 0 \,,\\
\rho_{m}(\frac{\partial{\bf U}}{\partial t} + {\bf U}\cdot\nabla{\bf U}) &=& - \nabla\cdot{\bf\Pi} +
\rho{\bf E} + {\bf J}\times{\bf B} + \nonumber \\
\label{e33}
&+& \frac{\hbar^{2}n_{e}}{2m_{e}}\nabla\left(\frac{\nabla^{2}\sqrt{n_{e}}}{\sqrt{n_{e}}}\right) + \frac{\hbar^{2}n_{i}}{2m_{i}}\nabla\left(\frac{\nabla^{2}\sqrt{n_{i}}}{\sqrt{n_{i}}}\right) \,,
\end{eqnarray}
for
\begin{equation}
\label{e34}
{\bf\Pi} = P\,{\bf I} + \frac{m_{e}m_{i}n_{e}n_{i}}{\rho_m}({\bf u}_e - {\bf u}_{i})\otimes({\bf u}_e - {\bf u}_{i}) \,,
\end{equation}
where $P = P_{e} + P_{i}$ and where ${\bf I}$ is the identity matrix. In equations (\ref{e33}-\ref{e34}), the electronic and ionic densities are defined in terms of the mass and charge densities according to
\begin{equation}
\label{e35}
n_e = \frac{1}{m_i + m_{e}}\,\,(\rho_m - \frac{m_{i}}{e}\rho) \,, \quad n_i = \frac{1}{m_i + m_{e}}\,\,(\rho_m + \frac{m_{e}}{e}\rho) \,.
\end{equation}
We can simplify (\ref{e33}) considerably assuming, as usual, quasi-neutrality ($\rho = 0$ so that $n_e = n_i$), $P_e = P_i = P/2$ and neglecting $m_e$ in comparison to $m_i$ whenever possible. In addition, disregarding the last term at the right hand side of (\ref{e34}), we obtain
\begin{equation}
\label{e36}
\frac{\partial{\bf U}}{\partial t} + {\bf U}\cdot\nabla{\bf U} = -\frac{1}{\rho_m}\nabla P + \frac{1}{\rho_m}{\bf J}\times{\bf B} + \frac{\hbar^{2}}{2m_{e}m_{i}}\nabla\left(\frac{\nabla^{2}\sqrt{\rho_m}}{\sqrt{\rho_m}}\right) \,.
\end{equation}
Under the same assumptions and following the standard derivation of magnetohydrodynamics \cite{Nicholson, Bittencourt}, we obtain the following equation for the current ${\bf J}$,
\begin{equation}
\label{e37}
\frac{m_{e}m_{i}}{\rho_{m}e^2}\frac{\partial{\bf J}}{\partial t} - \frac{m_{i}\nabla P}{\rho_{m}e} = {\bf E} + {\bf U}\times{\bf B} - \frac{m_{i}}{\rho_{m}e}\,{\bf J}\times{\bf B} - \frac{\hbar^{2}}{2e m_{e}}\nabla\left(\frac{\nabla^{2}\sqrt{\rho_m}}{\sqrt{\rho_m}}\right) - \frac{1}{\sigma}\,{\bf J} \,,
\end{equation}
where $\sigma = \rho_{m}e^{2}/(m_{e}m_{i}\nu_{ei})$ is the longitudinal electrical conductivity. Equation (\ref{e37}) is the quantum version of the generalized Ohm's law \cite{Nicholson, Bittencourt}. The continuity equation (\ref{e32}), the force equation (\ref{e36}), the quantum version of the generalized Ohm's law (\ref{e37}), an equation of state for $P$, plus Maxwell equations, provides a full system of quantum magnetohydrodynamic equations. However, it is probably still complicated and in the next section we propose some approximations in the same spirit of those of classical magnetohydrodynamics.
\section{Simplified and Ideal Quantum Magnetohydrodynamic Equations}
Usually \cite{Nicholson, Bittencourt}, the left-hand side of the equation (\ref{e37}) is neglected in the cases of slowly varying processes and small pressures. Also, for slowly varying and high conductivity problems , the displacement current can be neglected in Amp\`ere's law. Finally, we assume an equation of state appropriated for adiabatic processes. This provides a complete system of simplified quantum magnetohydrodynamic equations, which we collect here for convenience,
\begin{eqnarray}
\label{e38}
\frac{\partial\rho_m}{\partial t} &+& \nabla\cdot(\rho_{m}{\bf U}) = 0 \,,\\
\label{e39}
\frac{\partial{\bf U}}{\partial t} &+& {\bf U}\cdot\nabla{\bf U} = - \frac{1}{\rho_m}\nabla P + \frac{1}{\rho_m}{\bf J}\times{\bf B} + \frac{\hbar^{2}}{2m_{e}m_{i}}\nabla(\frac{\nabla^{2}\sqrt{\rho_m}}{\sqrt{\rho_m}}) \,,\\
\label{e40}
\nabla P &=& V_{s}^{2}\nabla\rho_m \,,\\
\label{e41}
\nabla&\times&{\bf E} = - \frac{\partial{\bf B}}{\partial t} \,,\\
\label{e42}
\nabla&\times&{\bf B} = \mu_{0}{\bf J} \,,\\
\label{e43}
{\bf J} &=& \sigma[{\bf E} + {\bf U}\times{\bf B} - \frac{m_{i}}{\rho_{m}e}\,{\bf J}\times{\bf B} - \frac{\hbar^{2}}{2e m_{e}}\nabla(\frac{\nabla^{2}\sqrt{\rho_m}}{\sqrt{\rho_m}})] \,.
\end{eqnarray}
In equation (\ref{e40}), $V_s$ is the adiabatic speed of sound of the fluid. Gauss law can be regarded as the initial condition for Faraday's law. Also notice that the Hall term ${\bf J}\times{\bf B}$ at (\ref{e43}) is often neglected in magnetohydrodynamics.
Inserting (\ref{e40}) into (\ref{e39}), we are left with a system of 13 equations for 13 unknowns, namely, $\rho_m$ and the components of ${\bf U}, {\bf J}, {\bf B}$ and ${\bf E}$.
This is our quantum magnetohydrodynamics model. In comparison to classical magnetohydrodynamics, the difference of the present model rests on the presence of two quantum corrections, the last terms at equations (\ref{e39}) and (\ref{e43}).
In the ideal magnetohydrodynamics approximation, we assume an infinite conductivity and neglect the Hall force at (\ref{e43}). This provides the following ideal quantum magnetohydrodynamics model,
\begin{eqnarray}
\label{e44}
{\bf E} = - {\bf U}\times{\bf B} &+& \frac{\hbar^{2}}{2e m_{e}}\nabla(\frac{\nabla^{2}\sqrt{\rho_m}}{\sqrt{\rho_m}}) \,, \\
\rho_{m}(\frac{\partial{\bf U}}{\partial t} + {\bf U}\cdot\nabla{\bf U}) &=& - \nabla P + \frac{1}{\mu_0}(\nabla\times{\bf B})\times{\bf B} + \nonumber \\ \label{e45} &+& \frac{\hbar^{2}\rho_m}{2m_{e}m_{i}}\nabla(\frac{\nabla^{2}\sqrt{\rho_m}}{\sqrt{\rho_m}}) \,,\\
\label{e46}
\frac{\partial{\bf B}}{\partial t} &=& \nabla\times({\bf U}\times{\bf B}) \,,
\end{eqnarray}
supplemented by the continuity equation (\ref{e38}) and the equation of state (\ref{e40}).
Taking into account (\ref{e40}), equations (\ref{e45}-\ref{e46}) plus the continuity equation provides a system of 7 equations for 7 unknowns, namely, $\rho_m$ and the components of ${\bf U}$ and ${\bf B}$. This is our ideal quantum magnetohydrodynamics model. In comparison to classical ideal magnetohydrodynamics, the difference of the present model rests on the presence of a quantum correction, the last term at equation (\ref{e45}). Interestingly, taking the curl of (\ref{e44}) makes disappear one of the quantum correction terms present in the non ideal quantum magnetohydrodynamics. This leads to a dynamo equation (\ref{e46}) identical to that of classical magnetohydrodynamics. Consequently, for infinite conductivity the magnetic field lines are still frozen to the fluid, even allowing for the quantum corrections proposed here. In fact, even for finite conductivity, the diffusion of magnetic field lines is described by the same diffusion equation as that of classical magnetohydrodynamics. This comes from the fact that the quantum correction disappear after taking the curl of both sides of (\ref{e43}), neglecting the Hall term and assuming a constant $\sigma$ as usual. However, a further quantum correction on the electric field still survives through (\ref{e44}).
In order to obtain a deeper understanding of the importance of quantum effects, we propose the following rescaling for our ideal quantum magnetohydrodynamic equations,
\begin{eqnarray}
\bar{\rho}_m &=& \rho_{m}/\rho_0 \,, \quad \bar{\bf U} = {\bf U}/V_A \,, \quad \bar{\bf B} = {\bf B}/B_0 \,, \nonumber \\
\label{e48}
\bar{\bf r} &=& \Omega_{i}{\bf r}/V_A \,, \quad \bar{t} = \Omega_{i}t \,,
\end{eqnarray}
where $\rho_0$ and $B_0$ are the equilibrium mass density and magnetic field. In addition, $V_A = (B_{0}^{2}/(\mu_{0}\rho_{0}))^{1/2}$ is the Alfv\'en velocity and $\Omega_i = eB_{0}/m_i$ is the ion cyclotron velocity. We justify the chosen rescaling in the following way. In magnetohydrodynamics, the Alf\'en velocity provides a natural velocity scale. Also, since we deal with low frequency problems, $\Omega_{i}^{-1}$ is a reasonable candidate for a natural time scale. These velocity and time scales induces the length scale $V_{A}/\Omega_{i}$, as shown in (\ref{e48}).
Applying the rescaling (\ref{e48}) to the ideal quantum magnetohydrodynamic equations, we obtain the following non dimensional model,
\begin{eqnarray}
\label{e49}
\frac{\partial\bar{\rho}_m}{\partial t} &+& \nabla\cdot(\bar{\rho}_{m}\bar{\bf U}) = 0 \,,\\
\bar{\rho}_{m}(\frac{\partial\bar{\bf U}}{\partial t} + \bar{\bf U}\cdot\nabla\bar{\bf U}) &=& - \frac{V_{s}^2}{V_{A}^2}\nabla\bar{\rho}_m + (\nabla\times\bar{\bf B})\times\bar{\bf B} + \nonumber \\ \label{e50} &+& \frac{H^{2}\bar{\rho}_m}{2}\nabla(\frac{\nabla^{2}\sqrt{\bar{\rho}_m}}{\sqrt{\bar{\rho}_m}}) \,,\\
\label{e51}
\frac{\partial\bar{\bf B}}{\partial t} &=& \nabla\times(\bar{\bf U}\times\bar{\bf B}) \,,
\end{eqnarray}
where
\begin{equation}
\label{e52}
H = \frac{\hbar\Omega_i}{\sqrt{m_{e}m_{i}}\,\,V_{A}^{2}}
\end{equation}
is a non dimensional parameter measuring the relevance of quantum effects. Numerically, using M.K.S. units, we have $H = 3.42 \times 10^{-30} \,\,n_{0}/B_{0}$, where $n_0$ is the ambient particle density. While for ordinary plasmas $H$ is negligible, for dense astrophysical plasmas \cite{Chabrier}-\cite{Jung}, with $n_0$ about $10^{29} - 10^{34}\,\, m^{-3}$, $H$ can be of order unity or more. Hence, in dense astrophysical plasmas like the atmosphere of neutron stars or the interior of massive white dwarfs, quantum corrections to magnetohydrodynamics can be of experimental importance. Similar comments apply to our non ideal quantum magnetohydrodynamics model. However, even for moderate $H$ quantum effects can be negligible if the density is slowly varying in comparison with some typical length scale, due to the presence of a third order derivative at the Bohm potential. This is in the same spirit of the Thomas-Fermi approximation.
\section{Quan\-tum Ideal Magnetostatic E\-qui\-li\-brium}
There is a myriad of developments based on classical magnetohydrodynamics (linear and nonlinear waves, dynamo theory and so on) and we shall not attempt to reproduce all the quantum counterparts of these subjects in the framework of our model. We will be restricted to just one subject, namely the construction of exact equilibria for ideal quantum magnetohydrodynamics, with no attempt to study the important question of the stability of the equilibria.
Assuming that ${\bf U} = 0$ and that all quantities are time-independent, the ideal quantum magnetohydrodynamic equations (\ref{e44}-\ref{e46}) becomes
\begin{eqnarray}
\label{e53}
{\bf E} &=& \frac{\hbar^{2}}{2e m_{e}}\nabla(\frac{\nabla^{2}\sqrt{\rho_m}}{\sqrt{\rho_m}}) \,, \\
\label{e54}
\nabla P &=& \frac{1}{\mu_0}(\nabla\times{\bf B})\times{\bf B} +
\frac{\hbar^{2}\rho_m}{2m_{e}m_{i}}\nabla(\frac{\nabla^{2}\sqrt{\rho_m}}{\sqrt{\rho_m}}) \,.
\end{eqnarray}
According to (\ref{e53}), the equilibrium solutions of ideal quantum magnetohydrodynamics are not electric field free any longer. In addition, equation (\ref{e54}) has an quantum correction that invalidate the classical magnetic surface equation for ${\bf B}\cdot\nabla{\bf B} = 0$, namely $P + B^{2}/(2\mu_{0}) =$ cte.
Equation (\ref{e54}) together with an equation of state is the key for the search of equilibrium solutions. We will try to follow, as long as possible, the strategy of reference \cite{Hamabata} for classical magnetostatic equilibria. Inspired by well known classical solutions \cite{Hamabata}, assume a translationally invariant solution of the form
\begin{eqnarray}
\label{e55}
P &=& P(r,\varphi) \,, \quad \rho_m = \rho_{m}(r,\varphi) \,, \\
\label{e56}
{\bf B} &=& \nabla A(r,\varphi)\times\hat{z} + B_{z}(r,\varphi)\hat{z} \,,
\end{eqnarray}
using cylindrical coordinates and where $A = A(r,\varphi)$ and $B_{z} = B_{z}(r,\varphi)$ as well as the pressure and the mass density are functions of $(r,\varphi)$ only.
Substituting the proposal (\ref{e55}-\ref{e56}) into (\ref{e54}), we get, for the radial and azimuthal components of this equation,
\begin{equation}
\label{e57}
\nabla(P + \frac{B_{z}^2}{2\mu_0}) = - \frac{1}{\mu_0}\,\nabla A\,\,\nabla^{2}A + \frac{\hbar^{2}\rho_{m}}{2m_{e}m_{i}}\,\,\nabla(\frac{\nabla^{2}\sqrt{\rho_m}}{\sqrt{\rho_m}}) \,,
\end{equation}
while, for the $z$ component, the result is
\begin{equation}
\label{e58}
\frac{\partial(B_{z},A)}{\partial(r,\varphi)} = 0 \,.
\end{equation}
In (\ref{e58}) and in what follows, we used the definition of Jacobian,
\begin{equation}
\label{e59}
\frac{\partial(B_{z},A)}{\partial(r,\varphi)} =
\left(\matrix{\frac{\partial\,B_z}{\partial r} & \frac{\partial\,B_z}{\partial \varphi}\cr
\frac{\partial\,A}{\partial r} & \frac{\partial\,A}{\partial \varphi}\cr}\right) \,.
\end{equation}
From (\ref{e58}), we obtain
\begin{equation}
\label{e60}
B_z = B_{z}(A) \,.
\end{equation}
Taking into account (\ref{e57}) and the fact that $B_z$ is a function of $A$, it follows that
\begin{equation}
\label{e61}
\frac{\partial(P,A)}{\partial(r,\varphi)} = \frac{\hbar^{2}\rho_{m}}{2m_{e}m_{i}}\frac{\partial(\nabla^{2}\sqrt{\rho_{m}}\,\,/\sqrt{\rho_m}\,\,
,A)}{\partial(r,\varphi)} \,.
\end{equation}
In the classical limit $\hbar \rightarrow 0$, the right hand of (\ref{e61}) vanishes, implying just the functional relationship $P = P(A)$. In the present work, we still postulate
\begin{equation}
\label{e62}
P = P(A) \,,
\end{equation}
so that, from (\ref{e61}), we have
\begin{equation}
\label{e63}
\frac{\nabla^{2}\sqrt{\rho_{m}}}{\sqrt{\rho_{m}}} = F(A) \,,
\end{equation}
where $F = F(A)$ is an arbitrary function.
The last equation is a distinctive feature of ideal quantum magnetohydrodynamic equilibrium. Indeed, (\ref{e63}) would not be necessary if $\hbar = 0$ in (\ref{e61}). Hence, even if $\hbar$ is not present in (\ref{e63}), this equation has a quantum nature, with important implications in what follows. The reason why $\hbar$ does not appear in (\ref{e63}) is that it factor at the right hand side of (\ref{e61}).
From (\ref{e62}) and some subjacent equation of state, $P = P(\rho_{m})$, we deduce
\begin{equation}
\label{e64}
\sqrt{\rho_{m}} = G(A) \,,
\end{equation}
for some function $G = G(A)$. Plugging this into (\ref{e63}), the result is
\begin{equation}
\label{e65}
\frac{G'}{G}\,\nabla^{2}A + \frac{G''}{G}\,(\nabla\,A)^2 = F(A) \,,
\end{equation}
where the prime denotes derivation with respect to $A$.
Coming back to (\ref{e57}), we obtain
\begin{equation}
\label{e66}
\nabla^{2}A = \mu_{0}[- K'(A) + \frac{\hbar^{2}}{2m_{e}m_{i}}\,\,G^{2}F'(A)] \,,
\end{equation}
where we have defined
\begin{equation}
\label{e67}
K = K(A) = P(A) + \frac{B_{z}^{2}(A)}{2\mu_0} \,.
\end{equation}
Recapitulating, we have three four functions of $A$ to be stipulated, namely $F$, $G$, $K$ and $P$. However, $A$ satisfy two different equations, (\ref{e65}) and (\ref{e66}). Once $A$ is found, all other quantities (pressure, mass density, electromagnetic field) comes as consequences.
A reasonable choice is to take $G$ as a linear function of $A$, since then (\ref{e65}) becomes linear in the derivatives. Hence, let
\begin{equation}
\label{e68}
G = k_{1}A + k_2 \,, \quad k_1 \neq 0 \,,
\end{equation}
for numerical constants $k_1$ and $k_2$. We take $k_1 \neq 0$ since $k_1 = 0$ would imply $F = 0$, making disappear the quantum correction at (\ref{e66}). With the choice (\ref{e68}), the couple (\ref{e65}-\ref{e66}) becomes
\begin{eqnarray}
\label{e69}
\nabla^{2} A &=& \frac{1}{k_1}\,(k_{1}A + k_{2})\,F(A) \,, \\
\label{e70}
\nabla^{2} A &=& \mu_{0}\,[-K'(A) + \frac{\hbar^{2}}{2m_{e}m_{i}}\,(k_{1}A+k_{2})^{2}\,F'(A)] \,.
\end{eqnarray}
The right hand sides of (\ref{e69}) and (\ref{e70}) should coincide, implying
\begin{equation}
\label{e71}
K'(A) = \frac{\hbar^{2}}{2m_{e}m_{i}}\,(k_{1}A + k_{2})^{2}\,F'(A) - \frac{1}{\mu_{0}k_{1}}\,(k_{1}A+k_{2})\,F(A) \,.
\end{equation}
The last equation define $K$ up to an unimportant numerical constant.
Equation (\ref{e69}) is the key equation for our translationally invariant magnetostatic equilibria. For a given $F(A)$ and solving (\ref{e69}) for $A$, all other quantities follows for a known equation of state. Indeed, knowing $A$ we can construct the radial and azimuthal components of the magnetic field through (\ref{e56}) and the mass density from (\ref{e64}). From the mass density and the equation of state, we obtain the pressure $P$. Proceeding, equation (\ref{e71}) yields $K(A)$ and then the $z$ component of the magnetic field through (\ref{e67}). Finally, the electric field follows from (\ref{e53}) and the current density from the curl of the magnetic field. The free ingredients to be chosen to construct explicitly the exact solution are the function $F(A)$ and the equation of state, and the numerical constants $k_1$ and $k_2$. Other possibilities can be explored if we do not restrict to linear $G(A)$ functions as in (\ref{e68}), but then $A$ will not satisfy an linear in the derivatives equation.
\subsection{An Explicit Exact Solution}
An interesting case of explicit solution is provided by the choice
\begin{equation}
\label{e72}
F(A) = \frac{k_{1}\,B_{0}\,(1-\varepsilon^{2}k)}{k_{1}A + k_{2}}\,\,e^{-2kA/B_{0}} \,,
\end{equation}
where $B_0$ is an arbitrary constant magnetic field, $k$ is an arbitrary constant with dimensions of an inverse length and $0 \leq \varepsilon < 1$. With the choice (\ref{e72}), the equation (\ref{e69}) traduces into the Liouville equation,
\begin{equation}
\label{e73}
\nabla^{2}A = (1-\varepsilon^{2})\,B_{0}\,k\,e^{-2kA/B_{0}} \,,
\end{equation}
which admits the exact cat eye solution
\begin{equation}
\label{e74}
A = \frac{B_{0}}{k}\,\,\ln[\cosh(kr\,cos\varphi) + \varepsilon\,\cos(kr\sin\varphi)] \,.
\end{equation}
All other relevant quantities follows from this exact solution following the recipe just stated. The mass density, from (\ref{e64}), is
\begin{equation}
\label{e75}
\rho_m = [\frac{k_{1}B_{0}}{k}\,\,\ln(\cosh(kr\,\cos\varphi) + \varepsilon\,\cos(kr\sin\varphi)) + k_{2}]^2 \,,
\end{equation}
while the radial and azimuthal components of the magnetic field follows from (\ref{e56}),
\begin{eqnarray}
\label{e76}
B_r &=& - \frac{B_{0}\,[\sin\varphi\,\sinh(kr\,\cos\varphi) + \varepsilon\cos\varphi\sin(kr\,\sin\varphi)]}{[\cosh(kr\,\cos\varphi) + \varepsilon\cos(kr\,\sin\varphi)]} \,, \\
\label{e77}
B_\varphi &=&
- \frac{B_{0}\,[\cos\varphi\,\sinh(kr\,\cos\varphi) - \varepsilon\sin\varphi\sin(kr\,\sin\varphi)]}{[\cosh(kr\,\cos\varphi) + \varepsilon\cos(kr\,\sin\varphi)]} \,.
\end{eqnarray}
Assuming an adiabatic equation of state, $P = V_{s}\rho_m$, we get, from (\ref{e67}),
\begin{eqnarray}
\label{e78}
B_{z}^2 &=& B_{0}^2 - 2\mu_{0}\,V_{s}^{2}\,(k_{1}A+k_{2})^{2} + \\ &+&(1 - \varepsilon^{2})\,k^{2}\,e^{-2A}\,\,[1 + \mu_{0}k_{1}\,(k_{1}+k_{2})\hbar^{2}/m + \mu_{0}\,\hbar^{2}\,k_{1}^{2}\,A/m] \,,
\end{eqnarray}
with $A$ given by the cat eye solution (\ref{e74}). If desired, the electric field and the current density can then be calculated via (\ref{e53}) and Amp\`ere's law respectively.
In figure 1, we show the contour plot of the function $A$ given by (\ref{e74}), while in figure 2 we show the corresponding mass density. The parameters chosen were $B_0 = 1$, $k = 1$, $\varepsilon = 0.9$, $k_1 = 1$ and $k_2 = 0$. These graphics shows coherent, periodic patterns resembling quantum periodic solutions arising in other quantum plasma systems \cite{Manfredi2}. Similar graphics can be easily obtained for the electromagnetic field and other macroscopic quantities derivable from the cat eye solution (\ref{e74}).
\section{Conclusion}
In this work, we have obtained a quantum version of magnetohydrodynamics starting from a quantum hydrodynamics model with nonzero magnetic fields. In view of its simplicity, this magnetic quantum hydrodynamics model seems to be an attractive alternative to the Wigner magnetic equation of Section 2. The infinite conductivity approximation leads to an ideal quantum magnetohydrodynamics. For very dense plasmas and not to strong magnetic fields, the quantum corrections to magnetohydrodynamics can be relevant, as apparent from the parameter $H$ derived in Section 4. Under a number of suitable assumptions, we have derived some exact translationally invariant quantum ideal magnetostatic solutions. More general quantum ideal magnetostatic equilibria can be conjectured, in particular for axially symmetric situations. In addition, we have left a full investigation of linear waves to future works.
\vskip 1cm
\noindent{\bf Acknowledgments}\\
We thanks the Brazilian agency Conselho Nacional de
Desenvolvimento Cien\-t\'{\i}\-fi\-co e Tecn\'ologico (CNPq) for
financial support.
|
{
"timestamp": "2005-03-02T11:18:42",
"yymm": "0503",
"arxiv_id": "physics/0503021",
"language": "en",
"url": "https://arxiv.org/abs/physics/0503021"
}
|
\section{Introduction}
Unitary quantum mechanics (that is, quantum mechanics without collapse
of the wave function) has local interactions: the quantum state of a
system (e.g.\,a qubit, or a spacetime region in quantum field theory) is
affected only by influences which propagate via the quantum states of
its immediate past light cone.\footnote{In QFT, this is a consequence of
the requirement that spacelike separated observables must commute.}
As conventionally presented, though, QM does not have local
\emph{states}: if $S_1$ and $S_2$ are systems with quantum states $\ensuremath{\rho}_1$ and
$\ensuremath{\rho}_2$, then because of entanglement the state of the composite
system $S_1 \times S_2$ is not necessarily $\ensuremath{\rho}_1 \otimes \ensuremath{\rho}_2$.
Deutsch and Hayden\cite{deutschhayden} argue that this `state nonlocality'
is an artifact of the normal way in which we represent quantum states,
and that it disappears in an alternative formalism which they propose.
Their formalism is derived from the Heisenberg picture of quantum
mechanics, in which the unitary time evolution is applied to the
observables rather than to the state vector. In the normal understanding of
that formalism, though, the state vector is still taken to express the
physical state of the system (via its role in calculating expectation values) and the
algebra of observable quantities is regarded as mathematical `superstructure',
used to help us to calculate those observables.
Deutsch and Hayden reverse this `normal understanding'. They regard the
state vector \ket{0} as fixed, once and for all and independent of the physical
state of the system, and they regard the state of a quantum system as
literally given by the associated observables (so that
the state of a qubit, for instance, is given by the triple of
Heisenberg picture operators $S_x, S_y, S_z$ pertaining to the spin
observables of that qubit). The dynamics of this theory are given by
\begin{equation} \label{truedyn}\ensuremath{\frac{\dr{}}{\dr{t}}}\op{X}_i= \frac{-i}{\hbar}\comm{\op{H}(\op{X}_1, \ldots
\op{X}_n)}{\op{X}_i}\end{equation}
(where $\op{X}_1, \ldots \op{X}_n$ are the observables of the theory).
It is easy to see that the theory is local in both the interaction and the
state senses, apparently vindicating Deutsch and Hayden's claims.
\section{Quantum gauge transformations}
Suppose $\op{V}(t)$ is a function from times to unitary operators, and
suppose that for each $t$, $\op{V}(t)\ket{0}=\exp(-i \theta) \ket{0}$
(for arbitrary phase factor $\theta$). Then if the state is represented,
according to Deutsch and Hayden, by observables $\op{X}_1, \ldots
\op{X}_n$, suppose that we make the transformation
\begin{equation} \op{X}_i(t) \longrightarrow
\op{X}_i'(t)=\opad{U}(t)\op{X}_i(t)\op{U}(t).\end{equation}
If $\op{V}(t)$ is not a constant then this changes the dynamics to
\begin{equation} \label{gaugedyn}\ensuremath{\frac{\dr{}}{\dr{t}}}\op{X}'_i = \frac{-i}{\hbar}
\comm{\op{H}(\op{X}'_1, \ldots\op{X}'_n)}{\op{X}'_i} + \frac{-i}{\hbar}
\comm{\opad{V}(t)\ensuremath{\frac{\dr{}}{\dr{t}}}\op{V}(t)}{\op{X}'_i}.\end{equation}
It does not, however, change anything observable, since everything
observable is given by the expectation values of observables with
respect to \ket{0}, and clearly
\begin{equation} \matel{0}{\op{X}'_i}{0}=\matel{0}{\op{X}_i}{0}.\end{equation}
To understand the significance of these `quantum gauge transformations',
it is useful to consider an analogous example: electromagnetism in the
context of the Aharonov-Bohm effect \cite{aharonovbohm}. Recall: the
electromagnetic potential \vctr{A} couples to electron wavefunctions
via the rule
\begin{equation} \op{P}\longrightarrow \op{P}+e\vctr{A}.\end{equation}
If an electron beam is split, passed on either side of a solenoid, and
recombined, there will be interference between the beams, and as the field in the
solenoid is varied the interference fringes will shift by an amount
proportional to the line integral of \vctr{A} around the electron's
path. This occurs despite the fact that the magnetic field outside the
solenoid is zero or nearly so.
The A-B effect
makes clear that the electromagnetic potential \vctr{A}, and not just
the fields \vctr{E} and \vctr{B}, must be regarded as physically
significant; however, all observable quantities (including the A-B
effect itself) are invariant under gauge transformations
\begin{equation} \vctr{A}\longrightarrow \vctr{A}'=\vctr{A}+ \nabla f\end{equation}
for arbitrary smooth functions $f$ (along with an associated
transformation of the wavefunction).
It is generally accepted that the correct response to this observation
is to regard gauge-equivalent \vctr{A}s as describing the same physical
situation, so as not to burden our theory with massive indeterminism
(caused by the possibility of arbitrary \emph{time-dependent}
gauge transformations) and with an excess of unobservable properties
(caused by the fact that the observable data \emph{right now} only fixes
the state up to a gauge transformation).
However, this does come with a price: if we identify gauge-equivalent
vector potentials then our theory has non-local states in the sense
described above. For while the Aharonov-Bohm vector potential cannot be
gauge-transformed to zero everywhere, it can be in any region which does not
completely enclose the solenoid. Since a region which \emph{does}
enclose the solenoid can be decomposed into regions which do not, it
follows that whether the solenoid-enclosing region induces an A-B effect
is not determined by the properties of its parts.
The loop representation of \vctr{A} makes this state
non-locality manifest. We replace \vctr{A} with the \emph{loop phases}
\begin{equation} C_\gamma = \int_\gamma \vctr{A} \cdot \dr{x}\end{equation}
where $\gamma$ is any closed loop. \vctr{A} is fixed up to gauge
transformations by the $C_\gamma$, and $\vctr{B}_i$ is given at a point \vctr{x} by
the loop phase for an infinitesimal loop in a plane perpendicular to
$\vctr{e}_i$. A loop which encloses the solenoid cannot be expressed as
the sum of loops which do not enclose the solenoid, so the loop
representation has nonlocal states.
\section{Lessons for quantum mechanics}
The same arguments which lead us to identify gauge-equivalent vector
potentials should lead us to identify gauge-equivalent quantum states.
Specifically:
\begin{enumerate}
\item The possibility of time-dependent quantum gauge transformations
makes it undetermined which dynamical equations give the true dynamics
for the quantum state: is it (\ref{truedyn}) or some (\ref{gaugedyn})?
(\ref{truedyn}) is somewhat simpler, but it is unclear whether this is
sufficient: after all, in electromagnetism
\begin{equation} \Box A_\mu =0\end{equation}
is a somewhat simpler choice of dynamics than those given by many other
gauges, but this does not lead us to regard it as the `true' dynamics.
\item Even time-independent gauge transformations make the state grossly
underdetermined by observable data. Provided that $\op{V}\ket{0}=\exp(-i
\theta) \ket{0}$, nothing whatever --- no observable data, no
theoretical considerations --- can tell us that the physical state is
given by $\op{X}_1, \ldots \op{X}_n$ rather than
$\opad{V}\op{X}_1\op{V}, \ldots \opad{V}\op{X}_n\op{V}.$
\end{enumerate}
(There is also a more `philosophical' concern: in a physical theory we
would normally prefer that what is `observable' (\mbox{i.\,e.\,}, the expectation
values derived from \ket{0}) would emerge from a physical analysis of measurement,
rather than by \emph{fiat}.)
This suggests that we should identify Deutsch-Hayden states which differ
only by a gauge transformation. But if we do so, we return to the usual
representation of quantum states! For two Deutsch-Hayden states are
gauge-equivalent if and only if they have the same expectation values
--- and of course the expectation values of all possible measurements on
a given quantum system are encoded in that system's density operator. So
if we do identify gauge-equivalent states, we are again left with a
theory whose states are non-local.
\section{Conclusion}
Deutsch and Hayden's proposal secures locality of states only at the
cost of a gauge freedom closely analogous to the gauge freedom of
electromagnetism. However, in quantum mechanics as in electromagnetism, to avoid
problems of indeterminism and state underdetermination it is necessary
to identify gauge-equivalent states. In quantum mechanics as in
electromagnetism, if we do make this identification then it leads to
nonlocality of states.
Deutsch and Hayden argue \cite[p.\,1772]{deutschhayden} that if a theory is
local according to any formulation, then it is local period. But their
version of quantum mechanics is only a new formulation if we do indeed
identify gauge-equivalent states. If not, it is not a `new formulation': it is a new
\emph{theory} --- with novel properties such as associating many distinct states
to the same in-principle-observable data ---
albeit one which has the same observational consequences as the old
theory. (Deutsch has himself insisted on this distinction in his more
foundational work, for instance in discussing the de Broglie-Bohm
interpretation \cite{deutschlockwood}). It is a new theory which is
genuinely local, but which pays an unacceptably high price for that
locality.
We conclude that Deutsch and Hayden's proposal is best understood as a
gauge theory whose gauge-independent physical properties are given by
the normal quantum formalism. As such, although it may well give
important insights into quantum-information issues such as information
flow(for a detailed analysis of this point see \cite{timpson}), it does not achieve the goal of showing that quantum mechanics is
completely local. Rather, quantum mechanics has only local interactions, but has
nonlocal states.
|
{
"timestamp": "2005-03-16T13:25:40",
"yymm": "0503",
"arxiv_id": "quant-ph/0503149",
"language": "en",
"url": "https://arxiv.org/abs/quant-ph/0503149"
}
|
\section{Preliminaries}\label{secprelim}
In this article,
we are working in the setting
of infinite-dimensional differential
calculus known as Keller's $C^\infty_c$-theory,
based on smooth maps in the sense
of Michal-Bastiani
(see \cite{BED},
\cite{Ham}, \cite{Mic}, \cite{Mil}
for further information).
\begin{defn}
Let $E$, $F$ be locally convex spaces and $f\!:U\to F$
be a mapping, defined on an open subset~$U$ of~$E$.
We say that $f$ is {\em of class~$C^0$\/}
if~$f$ is continuous.
If~$f$ is a continuous map such that
the two-sided directional derivatives
\[
df(x,v)=\lim_{t\to 0} {\textstyle
\frac{1}{t}\left( f(x+tv)-f(x)\right)}
\]
exist for all $(x,v)\in U\times E$,
and the map $df\!: U\times E\to F$
so defined is continuous,
then $f$ is said to be
{\em of class~$C^1$\/}. Recursively, given $k\in {\mathbb N}$
we call~$f$ a mapping of class~$C^{k+1}$
if it is of class~$C^1$ and $df$ is of class~$C^k$
on the open subset $U\times E$
of $E\times E$.
We set $d^{k+1}f:=d(d^kf)=d^k(df)\!:
U\times E^{2^{k+1}-1}\to F$
in this case. The function~$f$ is called
{\em smooth\/} (or of class $C^\infty$)
if it is of class~$C^k$ for each $k\in {\mathbb N}_0$.
\end{defn}
\begin{defn}
Let $M$ be a finite-dimensional, $\sigma$-compact
smooth manifold and~$E$ be a locally convex topological vector space.
We equip the vector space $C^\infty(M,E)$ of
$E$-valued smooth mappings~$\gamma$ on~$M$
with the topology of uniform convergence of
$\partial^\alpha(\gamma\circ \kappa^{-1})$
on compact subsets of~$V$,
for each chart $\kappa\!:M\supseteq U\to V\subseteq {\mathbb R}^d$
of~$M$ and multi-index $\alpha\in {\mathbb N}_0^d$
(where $d:=\dim(M)$).
Given a compact subset $K\subseteq M$, we equip the
vector subspace
$C^{\,\infty}_K(M,E):=\{\gamma\in C^\infty(M,E)\!:
\gamma|_{M\backslash K}=0\}$ of $C^\infty(M,E)$ with the induced
topology. We give
$C^\infty_c(M,E):=\bigcup_K C^{\, \infty}_K(M,E)={\displaystyle \lim_{\longrightarrow}}\,
C^{\,\infty}_K(M,E)$\vspace{-.8 mm}
the locally convex direct limit topology.
We abbreviate
$C^\infty_c(M):=C^\infty_c(M,{\mathbb R})$,
$C^\infty(M):=C^\infty(M,{\mathbb R})$,
and $C^{\, \infty}_K(M):= C^{\,\infty}_K(M,{\mathbb R})$.
Further details can be found,
e.g., in~\cite{GCX}.
\end{defn}
\section{Example of a discontinuous mapping on {\boldmath
$C^\infty_c({\mathbb R})$}}\label{secline}
We show that the map
$f\!: C^\infty_c({\mathbb R})\to C^\infty_c({\mathbb R})$,
$\gamma\mapsto \gamma\circ \gamma-\gamma(0)$
is discontinuous,
although its restriction to
$C^\infty_{[-n,n]}({\mathbb R})$ is smooth, for each $n\in {\mathbb N}$.\\[2mm]
The following
fact is essential
for our constructions.
It follows from \cite[Cor.\,3.13]{KaM}
and is also a special case
of~\cite[Prop.\,11.3]{ZOO}.
For the convenience of the reader,
we offer a direct, elementary proof
as an appendix.
\begin{la}\label{La1}
The composition map
\[
\Gamma\!: C^\infty({\mathbb R}^n,{\mathbb R}^m)\times
C^\infty(M,{\mathbb R}^n)\to C^\infty(M,{\mathbb R}^m)\,,\qquad
\Gamma(\gamma,\eta)\, :=\, \gamma\circ \eta
\]
is smooth,
for each finite-dimensional, $\sigma$-compact
smooth manifold~$M$ and $m,n\in {\mathbb N}_0$.\nopagebreak\hspace*{\fill}$\Box$
\end{la}
For the following proof,
recall that
the sets
\[
{\textstyle {\cal V}(k,e):=\left\{
\gamma\in C^\infty_c({\mathbb R})\!:\;
(\forall n\in {\mathbb Z})\,
(\forall j\in \{0,\ldots, k_n\})\,
(\forall x\in [n-\frac{1}{2},n+\frac{1}{2}])\;
|\gamma^{(j)}(x)|<\varepsilon_n\right\}}
\]
form a basis of open
zero-neighbourhoods
for the topology on $C^\infty_c({\mathbb R})$,
where $k=(k_n)\in ({\mathbb N}_0)^{\mathbb Z}$ and $e=(\varepsilon_n)\in ({\mathbb R}^+)^{\mathbb Z}$
(cf.\ \cite[\S\,II.1]{Sch};
see \cite[Prop.\,4.8]{GCX}).
\begin{prop}\label{prototype}
$f\!: C^\infty_c({\mathbb R})\to C^\infty_c({\mathbb R})$,
$\gamma\mapsto \gamma\circ \gamma-\gamma(0)$
has the following properties:
\begin{itemize}
\item[\rm (a)]
The restriction of~$f$ to a map
$C^\infty_{[-n,n]}({\mathbb R})\to C^\infty_c({\mathbb R})$
is smooth $($and hence continuous$)$, for each $n\in {\mathbb N}$.
\item[\rm (b)]
$f$ is discontinuous at $\gamma=0$.
\end{itemize}
\end{prop}
\begin{proof}
(a) Fix $n\in {\mathbb N}$;
we have to show
that $f|_{C^\infty_{[-n,n]}({\mathbb R})}
\!: C^\infty_{[-n,n]}({\mathbb R})\to C^\infty_c({\mathbb R})$
is smooth.
The image of this map being contained
in the closed vector subspace
$C^\infty_{[-n,n]}({\mathbb R})$
of $C^\infty_c({\mathbb R})$,
which also
is a closed vector subspace
of $C^\infty({\mathbb R})$ (with the same induced topology),
it suffices to show
that the map $C^\infty_{[-n,n]}({\mathbb R})\to
C^\infty({\mathbb R})$,
$\gamma\mapsto \gamma\circ \gamma-\gamma(0)$
is smooth
(see \cite[Prop.\,1.9]{SEC} or \cite[La.\,10.1]{BGN}).
Now $\gamma\mapsto \gamma(0)$
being a continuous linear (and thus smooth)
map, it suffices to show that
$C^\infty_{[-n,n]}({\mathbb R})\to C^\infty({\mathbb R})$,
$\gamma\mapsto \gamma\circ \gamma$ is smooth.
This readily follows from Lemma~\ref{La1}.
(b) Consider the zero-neighbourhood
$V:={\cal V}((|n|)_{n\in {\mathbb Z}}, (1)_{n\in {\mathbb Z}})$
in $C^\infty_c({\mathbb R})$.
Let $k=(k_n)\in ({\mathbb N}_0)^{\mathbb Z}$
and $e=(\varepsilon_n)\in ({\mathbb R}^+)^{\mathbb Z}$ be arbitrary.
We show that $f({\cal V}(k,e))\not\subseteq V$.
Since $f(0)=0$,
this entails
that~$f$ is discontinuous
at $\gamma=0$.
It is easy
to construct a function
$h\in C^\infty_c({\mathbb R})$
such that $\mbox{\n supp}(h)\subseteq \;
]{- \frac{1}{2}},\frac{1}{2}[$
and $h(x)=x^{k_0+1}$
for all $x\in [-\frac{1}{4},\frac{1}{4}]$.
Then $rh\in {\cal V}(k,e)$ for some
$r>0$. For $m\in {\mathbb N}$,
we define
$h_m\in C^\infty_c({\mathbb R})$
via
\[
h_m(x):= \frac{r}{m^{k_0}}h(mx).\]
Then $\mbox{\n supp}(h_m)\subseteq \; ]{-\frac{1}{2m}},\frac{1}{2m}[$
and thus $h_m\in {\cal V}(k,e)$
since, for all $j=0,\ldots, k_0$
and $x\in [-\frac{1}{2},\frac{1}{2}]$,
we have $|h_m^{(j)}(x)|=\frac{rm^j}{m^{k_0}}|h^{(j)}(mx)|<\varepsilon_0$.
We now choose $n\in {\mathbb N}$ such that
$n\geq k_0+2$.
It is easy to construct a
function $\psi\in C^\infty_c({\mathbb R})$
such that $\psi(x)=x-n$
for $x$ in some neighbourhood of~$n$ in~${\mathbb R}$,
and $\mbox{\n supp}(\psi)\subseteq \;]n-\frac{1}{2}, n+\frac{1}{2}[$.
Then $\phi:=s \cdot \psi\in {\cal V}(k,e)$
for suitable $s>0$.
Choosing $s$ small enough,
we may assume that
$\mbox{\n im}(\phi)\subseteq
[-1,1]$.
The supports of $\phi$ and $h_m$
being disjoint,
we easily deduce from $\phi,h_m\in {\cal V}(k,e)$
that also $\gamma_m:=\phi+ h_m\in {\cal V}(k,e)$.
Then $\gamma_m(0)=0$,
and since $\mbox{\n im}(\phi)\subseteq [-1,1]$,
we have
$f(\gamma_m)(x)
= (h_m\circ \phi)(x)$
for all $x\in W:=\;]n-\frac{1}{2}, n+\frac{1}{2}[$.
For $x\in W$ sufficiently close to~$n$,
we have $\phi(x)=s\cdot (x-n)\in [-\frac{1}{4m},\frac{1}{4m}]$
and thus
$f(\gamma_m)(x)= r\cdot m\cdot s^{k_0+1}\cdot (x-n)^{k_0+1}$,
whence $f(\gamma_m)^{(k_0+1)}(n)=r\cdot m\cdot s^{k_0+1}
\cdot(k_0+1)!\,$.
Thus $f(\gamma_m)\not\in V$ for all
$m\in {\mathbb N}$
such that
$r\cdot m\cdot s^{k_0+1}\cdot (k_0+1)!\geq 1$,
and so $f({\cal V}(k,e))\!\not\subseteq \!V$.
As $k$ and $e$ were arbitrary,
(b)\,follows.
\end{proof}
Note that $\mbox{\n supp}(f(\gamma))\subseteq \mbox{\n supp}(\gamma)$
here, for all $\gamma\in C^\infty_c({\mathbb R})$.
\begin{rem}
Although the
map~$f$ from Proposition~\ref{prototype}
is discontinuous and thus not smooth in the
Michal-Bastiani sense,
it is easily seen to be smooth
in the sense of convenient
differential calculus (as any map~$f$
on a ``regular'' countable strict direct
limit $E={\displaystyle \lim_{\longrightarrow}}\, E_n$\vspace{-1.3 mm} of complete
locally convex spaces,
all of whose restrictions $f|_{E_n}$
are smooth).\footnote{Regularity means that every
bounded subset of~$E$ is contained and bounded
in some~$E_n$.}
\end{rem}
\section{Discontinuous mappings on
{\boldmath $C^\infty_c(M,E)$}}\label{gencase}
In this section,
we generalize our discussion of $C^\infty_c({\mathbb R})$
from Section~\ref{secline}
to the spaces $C^\infty_c(M,E)={\displaystyle \lim_{\longrightarrow}}\, C^\infty_K(M,E)$\vspace{-.8 mm}
of compactly supported
smooth mappings
on a $\sigma$-compact
finite-dimensional
smooth manifold~$M$
with values in
a locally convex space~$E$.
We show:
\begin{prop}\label{propzero}
If $E\not=\{0\}$,
the manifold $M$ is non-compact,
and $\dim(M)>0$,
then there exists a mapping
$f\!: C^\infty_c(M,E)\to C^\infty_c(M,{\mathbb R})$
such that
\begin{itemize}
\item[\rm (a)]
The restriction of $f$ to $C^\infty_K(M,E)$
is smooth, for each compact subset
$K$ of~$M$.
\item[\rm (b)]
$f$ is discontinuous at~$0$.
\end{itemize}
In particular,
the locally convex direct limit topology on
$C^\infty_c(M,E)={\displaystyle \lim_{\longrightarrow}}\, C^\infty_K(M,E)\vspace{-.8 mm}$ is properly
coarser than the topology making
$C^\infty_c(M,E)$ the direct limit of the spaces $C^\infty_K(M,E)$
in the category of topological spaces.
\end{prop}
Instead of proving this
proposition directly, we establish an analogous result
for spaces of sections in bundles of locally
convex spaces, which is no harder to prove.
Noting that the function space $C^\infty_c(M,E)$
is topologically isomorphic to
the space $C^\infty_c(M,M\times E)$
of compactly supported smooth sections
in the trivial bundle $\mbox{\rm pr}_M\!:M\times E\to M$,
clearly Proposition~\ref{propzero}
is covered by the ensuing discussions for vector
bundles. For background material
concerning bundles of locally convex spaces
and the associated spaces of sections,
the reader is referred to~\cite{SEC}
(or also \cite[Appendix~F]{ZOO}).
For the present purposes, we recall:
if $\pi\!: E\to M$ is a smooth bundle of locally convex
spaces over the finite-dimensional, $\sigma$-compact smooth manifold~$M$,
with typical fibre
the locally convex space~$F$,
then one considers on the space $C^\infty(M,E)$ of all smooth
sections the initial topology with respect
to the family of mappings
\[
\theta_\psi\!: C^\infty(M,E)\to C^\infty(U,F),\;\;\;
\theta_\psi(\sigma):=\sigma_\psi:=\mbox{\rm pr}_F\circ \psi\circ \sigma|_U^{\pi^{-1}(U)}
\, ,
\]
which take a smooth section~$\sigma$ to its
local representation $\sigma_\psi\!: U\to F$
with respect to the local trivialization
$\psi\!: \pi^{-1}(U)\to U\times F$ of~$E$.
Given a compact subset~$K\subseteq M$,
the subspace $C^{\,\infty}_K(M,E)\subseteq C^\infty(M,E)$
of sections vanishing off~$K$ is equipped
with the induced topology,
and $C^\infty_c(M,E):=\bigcup_K C^{\, \infty}_K(M,E)={\displaystyle \lim_{\longrightarrow}}
\, C^{\,\infty}_K(M,E)$\vspace{-1.3 mm}
is given the locally convex direct limit topology.
\begin{thm}\label{fingeruebung}
Let $M$ be a $\sigma$-compact,
non-compact, finite-dimensional smooth
manifold of dimension $\dim(M)>0$,
and $\pi\!: E\to M$ be a smooth bundle
of locally convex spaces over~$M$,
whose typical fibre is a locally convex
topological vector space~$F\not=\{0\}$.
Then there exists a discontinuous
mapping
$f\!: C^\infty_c(M,E)\to C^\infty_c(M,{\mathbb R})$
whose restriction to
$C^{\,\infty}_K(M,E)$ is
smooth, for each compact
subset~$K$ of~$M$.
\end{thm}
\begin{proof}
Let $d:=\dim(M)$.
Since~$M$ is non-compact,
there exists a sequence $(U_n)_{n\in {\mathbb N}_0}$
of mutually disjoint
coordinate neighbourhoods $U_n\subseteq M$
diffeomorphic to ${\mathbb R}^d$
such that local trivializations
$\psi_n\!:\pi^{-1}(U_n)\to U_n\times F$
of~$E$ exist, and such that
every compact subset of~$M$
meets only finitely many of the sets~$U_n$.
We define
\[
\theta_{\psi_n}\!:
C^\infty_c(M,E)\to C^\infty(U_n,F),\;\;\;\;
\theta_{\psi_n}(\sigma):=\sigma_{\psi_n}:=
\mbox{\rm pr}_F\circ \psi_n\circ \sigma|_{U_n}^{\pi^{-1}(U_n)}\,.
\]
By definition of the topology on $C^\infty_c(M,E)$,
the linear maps $\theta_{\psi_n}$
are continuous.
For each $n\in{\mathbb N}_0$, let
$\kappa_n\!: U_n\to {\mathbb R}^d$ be a $C^\infty$-diffeomorphism;
define $x_n:=\kappa_n^{-1}(0)$.
We choose a function
$h\in C^\infty_c({\mathbb R}^d,{\mathbb R})$
such that $h|_{[-1,1]^d}=1$;
we define $h_n\in C^\infty_c(M,{\mathbb R})$ via
$h_n(x):=h(\kappa_n(x))$ if $x\in U_n$,
$h_n(x):=0$ if $x\in M\,\backslash\, U_n$.
Let $K_n:=\mbox{\n supp}(h_n)\subseteq U_n$.
We choose a continuous linear functional
$0\not= \lambda\in F'$,
and pick $v\in F$ such that $\lambda(v)=1$.
Note that $A:=\bigcup_{n\in {\mathbb N}} K_n$
is closed in~$M$, the sequence $(K_n)_{n\in{\mathbb N}}$
of compact sets being locally finite.
Let $\mu\!:{\mathbb R}\times F\to F$ be the
scalar multiplication.
The eventual definition of the mapping~$f$
we are looking for will involve
the map $\Phi\!: E\to M\times {\mathbb R}$, defined via
\begin{equation}\label{dfn1}
\Phi|_{\pi^{-1}(U_n)}:= (\pi|_{\pi^{-1}(U_n)},
\lambda\circ \mu\circ ((h_n\circ \pi)|_{\pi^{-1}(U_n)},
\mbox{\rm pr}_F\circ \psi_n))
\end{equation}
for $n\in {\mathbb N}$,
and
$\Phi|_{E\backslash \pi^{-1}(A)}:=(\pi|_{E\backslash \pi^{-1}(A)}, 0)$.
Note that $\Phi$ is well-defined
as the function in Equation\,(\ref{dfn1})
coincides with $(\pi,0)$
on the set $\bigcup_{n\in {\mathbb N}} \pi^{-1}(U_n \,\backslash\, A)$.
Also note that $\Phi$ is a fibre-preserving mapping
from~$E$ into the trivial bundle $M\times {\mathbb R}$.
Furthermore,
it is readily verified that~$\Phi$
is a smooth.
By \cite[Thm.\,5.9]{SEC}
(or \cite[Rem.\,F.25\,(a)]{ZOO}),
the pushforward
\[
C^\infty_c(M,\Phi)\!: C^\infty_c(M,E)\to C^\infty_c(M,M\times {\mathbb R}),\;\;\;\;
\sigma\mapsto \Phi\circ \sigma\]
is smooth.
For later use,
we introduce the continuous linear map
\[
\Lambda\; :=\; \theta_{\text{id}_{M\times {\mathbb R}}}\!:
C^\infty_c(M,M\times {\mathbb R})\to C^\infty(M,{\mathbb R})\, .
\]
Let $\iota\!:{\mathbb R}\to {\mathbb R}^d$
denote the embedding $t\mapsto (t,0,\ldots, 0)$.
The mapping~$f$ to be constructed will also involve
the map
$\Psi\!: C^\infty_c(M,E)\to C^\infty({\mathbb R},{\mathbb R})$ defined via
\[
\Psi:=
C^\infty({\mathbb R},\lambda)
\circ C^\infty(\kappa_0^{-1}\circ \iota ,F)\circ \theta_{\psi_0},
\]
where the pullback
$C^\infty(\kappa_0^{-1}\circ \iota, F)\!:
C^\infty(U_n,F)\to
C^\infty({\mathbb R},F)$,
$\gamma\mapsto \gamma\circ \kappa_0^{-1}\circ \iota$
and the pushforward $C^\infty({\mathbb R},\lambda)\!: C^\infty({\mathbb R},F)\to C^\infty({\mathbb R},
{\mathbb R})$, $\gamma\mapsto \lambda\circ \gamma$
are
continuous linear mappings and thus smooth,
by \cite[La.\,3.3, La.\,3.7]{GCX}.
Being a composition of smooth maps,
$\Psi$
is smooth.
We now define the desired map $f\!:C^\infty_c(M,E)\to
C^\infty_c(M,{\mathbb R})$ via
\[
f\, :=\, \Gamma\circ (\Psi,\Lambda\circ C^\infty_c(M,\Phi)) \;
- \;\lambda\circ \mbox{\n ev}_{x_0}\circ \theta_{\psi_0}
\]
(co-restricted from $C^\infty(M,{\mathbb R})$ to $C^\infty_c(M,{\mathbb R})$),
where
\[
\Gamma\!: C^\infty({\mathbb R},{\mathbb R})\times C^\infty(M,{\mathbb R})\to C^\infty(M,{\mathbb R}),\;\;\;\;
\Gamma(\gamma,\eta):=\gamma\circ \eta\]
denotes composition,
and $\mbox{\n ev}_{x_0}\!: C^\infty(U_0,F)\to F$
the evaluation map
$\gamma\mapsto \gamma(x_0)$. Here $\lambda\circ \mbox{\n ev}_{x_0}\circ
\theta_{\psi_0}$ is a continuous
linear map and thus smooth.
Explicitly, for $\sigma\in C^\infty_c(M,E)$
\begin{eqnarray*}
f(\sigma)(x) &=&
\Big(\lambda\circ\sigma_{\psi_0}\circ\kappa_0^{-1}\circ\iota\Big)
\bigl( \lambda\bigl( h_n(x)\, \sigma_{\psi_n}(x)\bigr)\bigr)\\
&=&
\lambda\Big( \sigma_{\psi_0} \bigl(\kappa_0^{-1}(
h_n(x)\cdot
\lambda(\sigma_{\psi_n}(x)),\; 0)\bigr)\Big)
\;-\;\lambda(\sigma_{\psi_0}(x_0))
\end{eqnarray*}
if $x\in U_n$ ($n\in{\mathbb N}$),
whereas $f(\sigma)(x)=0$ if $x\in M\,\backslash\, A$.\vspace{1.3mm}
{\em Claim\/}: {\em
The restriction of~$f$ to $C^\infty_K(M,E)$ is smooth,
for each compact subset~$K$ of~$M$.}\\
To see this, note that
$f(C^{\,\infty}_K(M,E)\subseteq C^{\,\infty}_K(M,{\mathbb R})$,
where $C^{\, \infty}_K(M,{\mathbb R})$ is a closed vector
subspace of $C^\infty(M,{\mathbb R})$ and
$C^\infty_c(M,{\mathbb R})$.
Thus, it suffices to show
that $f|_{C^\infty_K(M,E)}$ is smooth
as a map into $C^\infty(M,{\mathbb R})$
(\cite[Prop.\,1.9]{SEC}, or \cite[La.\,10.1]{BGN}).
But this follows from the Chain Rule,
as $\Gamma$ is smooth by Lemma~\ref{La1}
and also the other constituents of~$f$ are smooth.\vspace{1.3mm}
{\em Claim\/}: {\em $f$ is discontinuous at the zero-section
$\sigma=0$.}
To see this, consider the
set~$V$ of all $\gamma\in C^\infty_c(M,{\mathbb R})$
such that, for all $n\in {\mathbb N}$
and multi-indices $\alpha\in {\mathbb N}_0^d$
of order $|\alpha|\leq n$,
we have $|\partial^\alpha(\gamma\circ \kappa_n^{-1})(0)|<1$.
It is easily verified that $V$ is a symmetric,
convex zero-neighbourhood
in $C^\infty_c(M,{\mathbb R})$.
Let $U$ be any convex zero-neighbourhood
in $C^\infty_c(M,E)$;
we claim that $f(U)\not\subseteq V$.
To see this, set $L_n:=
\kappa_n^{-1}([-1,1]^d)$
for $n\in {\mathbb N}_0$.
Then
\[
\rho_n\!:
C^\infty_{L_n}(M,E)\to C^\infty_{[-1,1]^d}({\mathbb R}^d,F),\;\;\;\;
\sigma\mapsto \sigma_{\psi_n}\circ \kappa_n^{-1}
\]
is a topological isomorphism
(cf.\ \cite[La.\,3.9, La.\,3.10]{SEC}
or \cite[La.\,F.9, La.\,F.15]{ZOO})
whose inverse gives rise
to a topological embedding $j_n\!: C^\infty_{[-1,1]^d}({\mathbb R}^d,F)
\to C^\infty_c(M,E)$.
The linear mapping $\phi\!:{\mathbb R}\to F$, $t\mapsto tv$
gives rise to a continuous linear map
$C^\infty_{[-1,1]^d}({\mathbb R}^d,\phi)\!:C^\infty_{[-1,1]^d}({\mathbb R}^d,{\mathbb R})
\to C^\infty_{[-1,1]^d}({\mathbb R}^d,F)$,
$\gamma\mapsto \phi\circ \gamma$.
Then $W_n:=(j_n\circ C^\infty_{[-1,1]^d}({\mathbb R}^d,\phi))^{-1}(\frac{1}{2}U)$
is a convex zero-neighbourhood
in $C^\infty_{[-1,1]^d}({\mathbb R}^d,{\mathbb R})$.
Thus, there exists $k_n\in {\mathbb N}_0$
and $\varepsilon_n>0$ such that $W_{k_n,\varepsilon_n}\subseteq W_n$,
where $W_{k_n,\varepsilon_n}$
is the set of
all $\gamma\in C^\infty_{[-1,1]^d}({\mathbb R}^d,{\mathbb R})$
such that $\sup\{|\partial^\alpha \gamma(x)|\!:
x\in [-1,1]^d\}<\varepsilon_n$
for all $\alpha\in {\mathbb N}_0^d$
such that $|\alpha|\leq k_n$.
We let $g\in C^\infty_{[-1,1]^d}({\mathbb R}^d,{\mathbb R})$
be a function such that
$g(y_1,\ldots,y_d)=y_1^{{k_0}+1}$
for all $y=(y_1,\ldots, y_d)\in [-\frac{1}{2},\frac{1}{2}]^d$.
Then $rg\in W_{k_0,\varepsilon_0}$ for some $r>0$.
It is clear from the definition of $W_{k_0,\varepsilon_0}$
that then also $\gamma_m\in W_{k_0,\varepsilon_0}$
for all $m\in {\mathbb N}$,
where
\[
\gamma_m\!:{\mathbb R}^d\to{\mathbb R}\, ,\quad
\gamma_m(y_1,\ldots,y_d):= \frac{r}{m^{k_0}}\,g(my_1,y_2,\ldots, y_d)\, .
\]
Thus $\tau_m:= j_0(\phi\circ \gamma_m)\in \frac{1}{2}U$.
Let $\ell:=k_0+1$;
we easily find $\eta\in W_{k_\ell,\varepsilon_\ell}$
such that, for suitable $s>0$, we have
$\eta(y)=s\cdot y_1$ for $y=(y_1,\ldots, y_d)$
in some zero-neighbourhood in~${\mathbb R}^d$. We
define $\tau:=j_\ell(\phi\circ \eta)\in \frac{1}{2}U$.
Then $\sigma_m:=\tau_m+\tau\in U$
by convexity of~$U$.
Consider $g_m:=f(\sigma_m)\circ \kappa_\ell^{-1}\!:
{\mathbb R}^d\to {\mathbb R}$.
For $y\in [-1,1]^d$
sufficiently close to~$0$, we have $\eta(y)=sy_1$
and $m|\eta(y)|\leq\frac{1}{2}$. Thus
\[
g_m(y)=\gamma_m(\eta(y),0,\ldots,0)
=r\cdot m\cdot s^{k_0+1}\cdot y_1^{k_0+1},
\]
entailing
that
$\frac{\partial^{k_0+1} g_m}{\partial y_1^{k_0+1}}(0)
=r\cdot m\cdot s^{k_0+1}\cdot (k_0+1)!\,$.
Hence $f(\sigma_m)\not\in V$
for each $m\in {\mathbb N}$
such that
$r\cdot m\cdot s^{k_0+1}\cdot (k_0+1)!\geq 1$.
We have shown that $f(U)\not\subseteq V$
for any $0$-neighbourhood~$U$
in $C^\infty_c(M,E)$, although
$f(0)=0$. Thus $f$ is discontinuous
at $\sigma=0$.
\end{proof}
\section{Further examples}\label{secmisc}
We describe various pathological bilinear mappings.
\begin{prop}
Let ${\mathbb K}\in\{{\mathbb R},{\mathbb C}\}$. The pointwise multiplication map
\[
\mu\!:C^\infty({\mathbb R},{\mathbb K})\times C^\infty_c({\mathbb R},{\mathbb K})\to C^\infty_c({\mathbb R},{\mathbb K}),\;\;\;\;
\mu(\gamma,\eta):=\gamma\cdot\eta
\]
is a hypocontinuous bilinear $($and thus sequentially
continuous$)$ mapping on the locally convex direct limit
\[
C^\infty({\mathbb R},{\mathbb K})\times C^\infty_c({\mathbb R},{\mathbb K})=
{\displaystyle \lim_{\longrightarrow}}\, (C^\infty({\mathbb R},{\mathbb K})\times C^\infty_{[-n,n]}({\mathbb R},{\mathbb K}))\, ,\]
whose restriction to $C^\infty({\mathbb R},{\mathbb K})\times C^\infty_{[-n,n]}({\mathbb R},{\mathbb K})$
is continuous bilinear and thus ${\mathbb K}$-analytic,
for each $n\in {\mathbb N}$.
However, $\mu$ is discontinuous.
\end{prop}
\begin{proof}
Using the Leibniz Rule for the differentiation
of products of functions,
it is easily verified that~$\mu$ is
separately continuous.\footnote{Alternatively,
we can obtain the assertion as a special
case of \cite[Cor.\,2.7]{SEC}
or \cite[La.\,4.5\,(a) and Prop.\,4.19\,(d)]{ZOO},
combined with the locally convex direct limit property.}
The spaces $C^\infty({\mathbb R},{\mathbb K})$ and $C^\infty_c({\mathbb R},{\mathbb K})$
being barrelled,
this entails that~$\mu$ is hypocontinuous
and thus sequentially continuous
\cite[Thm.\,41.2]{Tre}.
The restriction of~$\mu$ to
$C^\infty({\mathbb R},{\mathbb K})\times C^\infty_{[-n,n]}({\mathbb R},{\mathbb K})$
is a sequentially continuous
bilinear mapping on a product of metrizable
spaces and therefore continuous.
To see that $\mu$ is discontinuous,
consider the zero-neighbourhood
\[
W:=\{\gamma\in C^\infty_c({\mathbb R},{\mathbb K})\!: \; (\forall x\in {\mathbb R})\;\,|\gamma(x)|<1\}
\]
in $C^\infty_c({\mathbb R},{\mathbb K})$.
If $U$ is any zero-neighbourhood
in $C^\infty({\mathbb R},{\mathbb K})$
and $V$ any zero-neighbourhood in $C^\infty_c({\mathbb R},{\mathbb K})$,
then there exists a compact subset
$K$ of~${\mathbb R}$ such that
\[
(
\forall \gamma\in C^\infty({\mathbb R},{\mathbb K}))\;\;\;
\gamma|_K=0\; \Rightarrow\; \gamma\in U.
\]
Pick any $x_0\in {\mathbb R}\, \backslash\, K$.
There is a function $\phi\in C^\infty_c({\mathbb R},{\mathbb K})$
such that $\phi(x_0)\not=0$ and $\mbox{\n supp}(\phi)\subseteq
{\mathbb R}\,\backslash\, K$.
Then $r\phi\in V$ for some $r>0$,
and $t\phi\in U$ for all $t\in {\mathbb R}$.
Choosing $t\geq \frac{1}{r\cdot|\phi(x_0)|^2}$,
we have $(t\phi,r\phi)\in U\times V$ but
$|\mu(r\phi,t\phi)(x_0)|=rt|\phi(x_0)|^2\geq 1$,
entailing that $\mu(U\times V)\not\subseteq W$.
Thus $\mu$ is discontinuous at~$(0,0)$.
\end{proof}
Another instructive example is the following
(compare also the examples in \cite{DaW}):
\begin{example}
Let $E_1\subset E_2\subset \cdots$ be a strictly ascending
sequence of Banach spaces, such that $E_{n+1}$
induces the given topology on~$E_n$.
Set $E:={\displaystyle \lim_{\longrightarrow}}\, E_n$\vspace{-.8mm}
and $F:=E'_b$. For example,
we can take $E_n:=L^2[-n,n]$,
in which case $E=L^2_{\mbox{\n \footnotesize comp}}({\mathbb R})$ and
$F=L^2_{\mbox{\n \footnotesize loc}}({\mathbb R})={\displaystyle \lim_{\longleftarrow}}\, L^2[-n,n]$.
Then
$A_n:=F\times E_n\times {\mathbb K}\times {\mathbb K}$
is a Fr\'{e}chet space (and reflexive
in the example $E_n=L^2[-n,n]$).
The evaluation map $E_n'\times E_n\to {\mathbb R}$ being continuous
as~$E_n$ is a Banach space,
it is easy to see that~$A_n$
becomes a unital associative
topological algebra via
\begin{equation}\label{formull}
(\lambda_1,x_1,z_1,c_1)\cdot(\lambda_2,x_2,z_2,c_2)
\,:=\,
\bigl(
c_1\lambda_2+c_2\lambda_1,\, c_1x_2+c_2x_1,\,c_1z_2+ \lambda_1(x_2)+z_1c_2,
\, c_1c_2 \bigr)\, .
\end{equation}
The
multiplication can be visualized by considering
$(\lambda,x,z,c)\!\in \!A_n$ as the 3-by-3 matrix
{\scriptsize
\[
\left(
\begin{array}{ccc}
c & \lambda & z\\
0 & c & x \\
0 & 0 & c
\end{array}
\right).
\]
}The topological algebras~$A_n$
are very well-behaved:
they have open groups of units,
and inversion is a ${\mathbb K}$-analytic map.
We can also use Formula\,(\ref{formull})
to define a multiplication map
$\mu\!: A\times A\to A$
turning the direct limit locally convex space
$A:=F\times E\times {\mathbb K}\times {\mathbb K}={\displaystyle \lim_{\longrightarrow}}\, A_n$\vspace{-.8 mm}
into a unital, associative algebra.
However, although
the restriction of~$\mu$
to $A_n\times A_n$ is a continuous bilinear map
for each~$n\in{\mathbb N}$,
$\mu\!: A\times A={\displaystyle \lim_{\longrightarrow}}\, (A_n\times A_n)\to A$\vspace{-1.3 mm}
is discontinuous (since the evaluation map
$E'_b\times E\to{\mathbb R}$ is discontinuous,
the space~$E$ not being normable).
We refer to \cite[Section~10]{Glo}
for more details.
\end{example}
|
{
"timestamp": "2005-03-18T20:04:28",
"yymm": "0503",
"arxiv_id": "math/0503387",
"language": "en",
"url": "https://arxiv.org/abs/math/0503387"
}
|
\section{\bf Introduction}
Invariant structures on homogeneous manifolds are traditionally
one of the most important objects in differential geometry,
specifically, in Hermitian geometry. Some remarkable classes of
almost Hermitian structures such as K\"ahler, nearly K\"ahler,
Hermitian structures etc. are well known and intensively used in
geometry and a number of applications. In particular, a special
role is played by a significant class of invariant nearly K\"ahler
structures based on the canonical almost complex structure on
homogeneous $3$-symmetric spaces (see \cite{S2}, \cite{WG},
\cite{G2}, \cite{Ki1}). It should be mentioned that the canonical
almost complex structure on such spaces became an effective tool
and a remarkable example in some deep constructions of
differential geometry and global analysis such as homogeneous
structures (\cite{TV}, \cite{Sat}, \cite{Ki4}, \cite{GV},
\cite{LV}, \cite{AG} etc.), Einstein metrics (\cite{SW},
\cite{SY}), holomorphic and minimal submanifolds (\cite{Sal1},
\cite{Sal2}), real Killing spinors (\cite{Gru}, \cite{BFGK},
\cite{Ka}).
The concept of generalized Hermitian geometry created in the 1980s
(see, for example, \cite{Ki2}, \cite{Ki7}) is a natural
consequence of the development of Hermitian geometry and the
theory of almost contact structures with many applications. One of
its central objects is the metric $f$-structures of the classical
type $(f^3+f=0)$, which include the class of almost Hermitian
structures. Many important classes of metric $f$-structures such
as K\"ahler, Killing, nearly K\"ahler, Hermitian $f$-structures
and some others were introduced and intensively investigated in
various aspects (see \cite{Ki2}, \cite{Ki5}, \cite{Ki7}, \cite{KL}
etc.). Specifically, Killing and nearly K\"ahler $f$-structures
became natural generalizations of classical nearly K\"ahler
structures in Hermitian geometry. However, this theory had not
provided new invariant examples of its own up to the recent
period, and so the lack of these examples was becoming all the
more noticeable.
There has recently been a qualitative change in the situation,
related to the complete solution of the problem of describing
canonical structures of classical type on regular $\Phi$-spaces
\cite{BS2}. A rich collection of canonical $f$-structures has been
discovered (including almost complex structures) leading to the
presentation of wide classes of invariant examples in generalized
Hermitian geometry (see \cite{B4}-\cite{B7}, \cite{C3} and
others). In particular, nearly K\"ahler $f$-structures were
provided with a remarkable class of their own invariant examples
(see \cite{B6}, \cite{B7}). This has ensured a continuation of the
classical results of J.A.Wolf, A.Gray, V.F.Kirichenko and others.
As to Killing $f$-structures, it is really an essential problem to
find proper non-trivial invariant examples of these structures.
Moreover, the possibilities for constructing such examples are
fairly limited (see \cite{B4}).
The main goals of this paper are
(i) to give a brief survey on invariant structures in generalized
Hermitian geometry and
(ii) to characterize all invariant $f$-structures on the flag
manifold $SU(3)/T_{max}$ in the sense of generalized Hermitian
geometry, in particular, to present first invariant examples of
Killing $f$-structures.
Sections 2-4 are mostly of survey character. In Section 2, we
collect some basic notions and results on homogeneous regular
$\Phi$-spaces and canonical affinor structures. In particular, a
precise description of all canonical structures of classical types
on homogeneous $k$-symmetric spaces is included. Besides, the
exact formulae for these structures and the relationship between
them on 4- and 5-symmetric spaces are presented.
In Section 3, we recall the main classes of almost Hermitian
structures following the Gray-Hervella division of almost
Hermitian manifolds into sixteen classes (see \cite{GH}). Besides,
we select particular results related to invariant almost Hermitian
structures.
Further, in Section 4, we describe main classes of metric
$f$-structures in generalized Hermitian geometry. Here we also
formulate the recent results on invariant nearly K\"ahler,
$G_1f$-, Hermitian, and Killing $f$-structures. In this
consideration, the canonical $f$-structures on homogeneous 4- and
5-symmetric spaces are especially important.
Finally, in Section 5, we examine in detail all invariant
$f$-structures on the complex flag manifold $SU(3)/T_{max}$ with
respect to all invariant Riemannian metrics. We discuss belonging
these structures to the main classes of metric $f$-structures
above mentioned. In particular, invariant non-trivial Killing
$f$-structures together with the corresponding Riemannian metrics
are first presented.
\section{\bf Homogeneous regular $\Phi$-spaces and canonical affinor structures}
Here we briefly formulate some basic definitions and results
related to regular $\Phi$-spaces and canonical affinor structures
on them. More detailed information can be found in \cite{BS2},
\cite{B10}, \cite{WG}, \cite{Ko}, \cite{F}, \cite{S1}, \cite{S2}.
Let $G$ be a connected Lie group, $\Phi$ its (analytic)
automorphism. Denote by $G^{\Phi}$ the subgroup of all fixed
points of $\Phi$ and $G_o^{\Phi}$ the identity component of
$G^{\Phi}$. Suppose a closed subgroup $H$ of $G$ satisfies the
condition $$G_o^{\Phi}\subset{H}\subset{G^{\Phi}}.$$ Then $G/H$ is
called a {\it homogeneous $\Phi$-space}.
Homogeneous $\Phi$-spaces include homogeneous symmetric spaces
$(\Phi^2=id)$ and, more general, {\it homogeneous $\Phi$-spaces of
order $k$} $(\Phi^k=id)$ or, in the other terminology, {\it
homogeneous $k$-symmetric spaces} (see \cite{Ko}).
For any homogeneous $\Phi$-space $G/H$ one can define the mapping
$$
S_o = D\colon\ G/H \to G/H,\ xH\to \Phi (x) H.
$$
It is known \cite{S1} that $S_o$ is an analytic diffeomorphism of
$G/H$. $S_o$ is usually called a "symmetry" of $G/H$ at the point
$o=H$. It is evident that in view of homogeneity the "symmetry"
$S_p$ can be defined at any point $p\in G/H$. More exactly, for
any $p=\tau(x)o=xH,\ q=\tau(y)o=yH$ we put
$$
S_p=\tau(x)\circ S_o \circ \tau(x^{-1}).
$$
It is easy to show that
$$
S_p(yH)=x\Phi(x^{-1})\Phi(y)H.
$$
Thus any homogeneous $\Phi$-space is equipped with the set of
symmetries $\{S_p\mid p\in G/H\}$. Moreover, each $S_p$ is an
analytic diffeomorphism of the manifold $G/H$ (see \cite{S1}).
Note that there exist homogeneous $\Phi$-spaces that are not
reductive. That is why so-called regular $\Phi$-spaces first
introduced by N.A.Stepanov \cite{S1} are of fundamental
importance.
Let $G/H$ be a homogeneous $\Phi$-space, $\mathfrak{g}$ and
$\mathfrak{h}$ the corresponding Lie algebras for $G$ and $H$,
$\varphi=d{\Phi}_e$ the automorphism of $\mathfrak{g}$. Consider
the linear operator $A=\varphi-id$ and the Fitting decomposition
$\mathfrak{g}=\mathfrak{g}_0\oplus\mathfrak{g}_1$ with respect to
$A$, where $\mathfrak{g}_0$ and $\mathfrak{g}_1$ denote $0$-
and $1$-component of the decomposition respectively. Further, let
$\varphi=\varphi_{s}\:\varphi_{u}$ be the Jordan decomposition,
where $\varphi_{s}$ and $\varphi_{u}$ is a semisimple and
unipotent component of $\varphi$ respectively,
$\varphi_{s}\:\varphi_{u}=\varphi_{u}\:\varphi_{s}$. Denote by
$\mathfrak{g}^{\gamma}$ a subspace of all fixed points for a
linear endomorphism $\gamma$ in $\mathfrak{g}$. It is clear that
$\mathfrak{h}=\mathfrak{g}^{\varphi}=Ker\,A$,
$\mathfrak{h}\subset\mathfrak{g}_0$,
$\mathfrak{h}\subset\mathfrak{g}^{\varphi_s}$.
{\bf\noindent Definition 1} {\rm (\cite{S1}, \cite{BS2},
\cite{B10}, \cite{F}). A homogeneous $\Phi$-space $G/H$ is called
a {\it regular $\Phi$-space} if one of the following equivalent
conditions is satisfied:
\begin{enumerate}
\item $\mathfrak{h}=\mathfrak{g}_0$; \item
$\mathfrak{g}=\mathfrak{h}\oplus{A}\mathfrak{g}$; \item The
restriction of the operator $A$ to ${A}\mathfrak{g}$ is
non-singular; \item $A^2X=0\Longrightarrow{A}X=0$ for all
$X\in\mathfrak{g}$. \item The matrix of the automorphism $\varphi$
can be represented in the form \\ $\left(\begin{array}{cc}
E & 0 \\
0 & B
\end{array}\right),$ where the matrix $B$ does not admit the eigenvalue $1$.
\item $\mathfrak{h}=\mathfrak{g}^{\varphi_s}$.
\end{enumerate}}
\noindent We recall two basic facts:
\newpage
\begin{theorem}{\rm(\cite{S1})}
\begin{itemize}
\item Any homogeneous $\Phi$-space of order $k$ $(\Phi^k\ = \
id)$ is a regular $\Phi$-space. \item Any regular $\Phi$-space is
reductive. More exactly, the Fitting decomposition
\begin{equation}\label{f1}
\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{m}, \
\mathfrak{m}=A\mathfrak{g}
\end{equation}
is a reductive one.
\end{itemize}
\end{theorem}
Decomposition (\ref{f1}) is the {\it canonical reductive
decomposition} corresponding to a regular $\Phi$-space $G/H$, and
$\frak{m}$ is the {\it canonical reductive complement}.
We also note that for any regular $\Phi$-space $G/H$ each point
$p=xH\in G/H$ is an isolated fixed point of the symmetry $S_p$
(see \cite{S1}).
Decomposition (\ref{f1}) is obviously $\varphi$-invariant. Denote
by $\theta$ the restriction of $\varphi$ to $\mathfrak{m}$. As
usual, we identify $\mathfrak{m}$ with the tangent space
$T_o(G/H)$ at the point $o=H$. It is important to note that the
operator $\theta$ commutes with any element of the linear isotropy
group $Ad(H)$ (see \cite{S1}). It also should be noted (see
\cite{S1}) that
$$
(dS_o)_o=\theta.
$$
An {\it affinor structure} on a manifold is known to be a tensor
field of type $(1,1)$ or, equivalently, a field of endomorphisms
acting on its tangent bundle. Suppose $F$ is an invariant affinor
structure on a homogeneous manifold $G/H$. Then $F$ is completely
determined by its value $F_o$ at the point $o$, where $F_o$ is
invariant with respect to $Ad(H)$. For simplicity, we will denote
by the same manner both any invariant structure on $G/H$ and its
value at $o$ throughout the rest of the paper.
{\bf\noindent Definition 2} {\rm (\cite{BS1},\cite{BS2}). An
invariant affinor structure $F$ on a regular $\Phi$-space $G/H$ is
called {\em canonical} if its value at the point $o=H$ is a
polynomial in $\theta$.
Denote by $\mathcal A(\theta)$ the set of all canonical affinor
structures on a regular $\Phi$-space $G/H$. It is easy to see that
$\mathcal A(\theta)$ is a commutative subalgebra of the algebra
$\mathcal A$ of all invariant affinor structures on $G/H$.
Moreover, $$dim\ \mathcal A(\theta)=deg\ \nu\ \le\ dim\ G/H,$$
where $\nu$ is a minimal polynomial of the operator $\theta$. It
is evident that the algebra $\mathcal A(\theta)$ for any
symmetric $\Phi$-space $(\Phi^2=id)$ consists of scalar
structures only, i.e. it is isomorphic to $\mathbb{R}$. As to
arbitrary regular $\Phi$-space $(G/H,\Phi)$, the algebraic
structure of its commutative algebra $\mathcal A(\theta)$ has been
recently completely described (see \cite{B9}).
It should be mentioned that all canonical structures are, in
addition, invariant with respect to the "symmetries" $\{S_p\}$ of
$G/H$(see \cite{S1}). Moreover, from $(dS_o)_o=\theta$ it follows
that the invariant affinor structure $F_p=(dS_p)_p, p\in G/H$
generated by the symmetries $\{S_p\}$ belongs to the algebra
$\mathcal A(\theta)$.
The most remarkable example of canonical structures is the
canonical almost complex structure
$J=\frac{1}{\sqrt{3}}(\theta-\theta^2)$ on a homogeneous
$3$-symmetric space (see \cite{S2}, \cite{WG}, \cite{G2}). It
turns out that it is not an exception. In other words, the algebra
$\mathcal A(\theta)$ contains many affinor structures of classical
types.
In the sequel we will concentrate on the following affinor
structures of classical types:
{\it almost complex structures} $J$ $(J^2=-1)$;
{\it almost product structures} $P$ $(P^2=1)$;
{\it $f$-structures} $(f^3+f=0)$ \cite{Y};
$f$-structures of hyperbolic type or, briefly, {\it
$h$-structures} $(h^3-h=0)$ \cite{Ki2}. \\ Clearly, $f$-structures
and $h$-structures are generalizations of structures $J$ and $P$
respectively.
All the canonical structures of classical type on regular
$\Phi$-spaces were completely described
\cite{BS1},\cite{BS2},\cite{B8}. In particular, for homogeneous
$k$-symmetric spaces, precise computational formulae were
indicated. For future reference we select here some results.
Denote by $\tilde s$ (respectively, $s$) the number of all
irreducible factors (respectively, all irreducible quadratic
factors) over $\mathbb{R}$ of a minimal polynomial $\nu$.
\begin{theorem}(\cite{BS1},\cite{BS2},\cite{B8})
Let $G/H$ be a regular $\Phi$-space.
\begin{enumerate}
\item The algebra $\mathcal{A}(\theta )$ contains precisely $2^{\tilde s}$
structures $P$.
\item $G/H$ admits a canonical structure $J$ if and only if
$s=\tilde{ s}$. In this case $\mathcal{ A}(\theta )$ contains
$2^s$ different structures $J$.
\item $G/H$ admits a canonical $f$-structure if and only if
$s\ne 0$. In this case ${\mathcal A} (\theta )$ contains $3^s-1$
different $f$-structures. Suppose $s={\tilde s}$. Then $2^s$
$f$-structures are almost complex and the remaining $3^s-2^s-1$
have non-trivial kernels.
\item The algebra $\mathcal{A}(\theta)$ contains $3^{\tilde{s}}$
different $h$-structures. All these structures form a
(commutative) semigroup in $\mathcal{A}(\theta)$ and include a
subgroup of order $2^{\tilde{s}}$ of canonical structures $P$.
\end{enumerate}
\end{theorem}
Further, let $G/H$ be a homogeneous $k$-symmetric space. Then
$\tilde{s}=s+1$ if $-1\in \, spec\,\theta$, and $\tilde{s}=s$ in
the opposite case. We indicate explicit formulae enabling us to
compute all canonical $f$-structures and $h$-structures. We shall
also use the notation $$u= \left\{\begin{array}{ccc}
n & {\text if} & k=2n+1 \\
n-1 & {\text if} & k=2n
\end{array}\right..$$
\begin{theorem}(\cite{BS1},\cite{BS2},\cite{B8})
Let $G/H$ be a homogeneous $\Phi$-space of order $k$.
\begin{enumerate}
\item All non-trivial canonical $f$-structures on $G/H$ can be
given by the operators
$$f=\frac{2}{k}\sum_{m=1}^u\left(\sum_{j=1}^u\zeta_j\sin{
\frac{2\pi m j}{k}}\right)\left(\theta^m-\theta^{k-m}\right),$$
where $\zeta_j\in\{-1;0 ;1\},\;j=1,2,\ldots,u$, and not all
coefficients $\zeta_j$ are zero. In particular, suppose that
$-1\notin \, spec\,\theta$. Then the polynomials $f$ define
canonical almost complex structures $J$ iff all
$\zeta_j\in\{-1;1\}$. \item All canonical $h$-structures on $G/H$
can be given by the polynomials
$h=\sum\limits_{m=0}^{k-1}a_m\theta^m$, where:
\begin{enumerate}
\item if $k=2n+1$, then $$a_m=a_{k-m}=\frac{2}{k}\sum_{j=1}^u\xi_j
\cos{\frac{2\pi m j}{k}};$$ \item if $k=2n$, then $$a_m=a_{k-m}=
\frac{1}{k}\left(2\sum_{j=1}^u\xi_j \cos{\frac{2\pi m j}{k}} +
(-1)^m\xi_n \right)$$
\end{enumerate}
Here the numbers $\xi_j$ take their values from the set $\{-1;0
;1\}$ and the polynomials $h$ define canonical structures $P$ iff
all $\xi_j\in\{-1;1\}$.
\end{enumerate}
\end{theorem}
We now particularize the results above mentioned for homogeneous
$\Phi$-spaces of orders $3$, $4$, and $5$ only. Note that there
are no fundamental obstructions to considering of higher orders
$k$.
\begin{corol}\label{c1}(\cite{BS2},\cite{B8})
Let $G/H$ be a homogeneous $\Phi$-space of order $3$. There are
(up to sign) only the following canonical structures of classical
type on $G/H$: $$J=\frac{1}{\sqrt{3}}(\theta-\theta^2),\;P=1.$$
\end{corol}
We note that the existence of the structure $J$ and its properties
are well known (see \cite{S2},\cite{WG},\cite{G2},\cite{Ki1}).
\begin{corol}\label{c2}(\cite{BS2},\cite{B8})
On a homogeneous $\Phi$-space of order $4$ there are (up to sign)
the following canonical classical structures:
$$P=\theta^2,\;f=\frac12(\theta-
\theta^3),\;h_1=\frac12(1-\theta^2),\;h_2=\frac12(1+\theta^2).$$
The operators $h_1$ and $h_2$ form a pair of complementary
projectors: $h_1+h_2=1$, $h_1^2=h_1$, $h_2^2=h_2$. Moreover, the
following conditions are equivalent:
\begin{enumerate}
\item $-1\notin spec\,\theta$; \item the structure $P$ is trivial
$P=-1$; \item the $f$-structure is an almost complex structure;
\item the structure $h_1$ is trivial $(h_1=1)$; \item the
structure $h_2$ is null.
\end{enumerate}
\end{corol}
General properties of the canonical structures $P$ and $f$ on
homogeneous $4$-symmetric spaces were investigated in \cite{BD}.
\begin{corol}\label{c3}(\cite{BS1},\cite{BS2},\cite{B8})
There exist (up to sign) only the following canonical structures
of classical type on any homogeneous $\Phi$-space of order $5$:
\begin{gather*}
P=\frac{1}{\sqrt{5}}(\theta-\theta^2-\theta^3+\theta^4);\\
J_1=\alpha(\theta-\theta^4)-\beta(\theta^2-\theta^3);\quad
J_2=\beta(\theta-\theta^4)+\alpha(\theta^2-\theta^3);\\
f_1=\gamma(\theta-\theta^4)+\delta(\theta^2-\theta^3);\quad
f_2=\delta(\theta-\theta^4)-\gamma(\theta^2-\theta^3);\\
h_1=\frac12(1+P);\quad h_2=\frac12(1-P);
\end{gather*}
where $\alpha=\frac{\sqrt{5+2\sqrt5}}{5}$;\
$\beta=\frac{\sqrt{5-2\sqrt5}}{5}$;\
$\gamma=\frac{\sqrt{10+2\sqrt5}}{10}$;\
$\delta=\frac{\sqrt{10-2\sqrt5}}{10}$.
Besides, the following relations are satisfied:
\begin{gather*}
J_1\,P=J_2;\quad f_1\,P=J_1\,h_1=J_2\,h_1=f_1;\quad h_1\,P=h_1;\quad h_2\,P=-h_2;\\
f_2\,P=J_2\,h_2=-J_1\,h_2=-f_2;\quad f_1\,f_2=h_1\,h_2=0;\quad h_1+h_2=P.
\end{gather*}
In addition, the following conditions are equivalent:
\begin{enumerate}
\item $spec\ \theta$ consists of two elements; \item the structure
$P$ is trivial; \item the structures $J_1$ and $J_2$ coincide (up
to sign); \item one of the structures $f_1$ and $f_2$ is null,
while the other is an almost complex structure coinciding with one
of the structures $J_1$ and $J_2$; \item one of the structures
$h_1$ and $h_2$ is trivial, while the other is null.
\end{enumerate}
\end{corol}
We note that for the first time the canonical structure $P$ on
homogeneous $5$-symmetric spaces was introduced and studied in
\cite{BC}. Other canonical structures on these spaces were later
studied in \cite{C1}-\cite{C3}.
It should be also mentioned that in the particular case of
homogeneous $\Phi$-spaces of any odd order $k=2n+1$ the method of
constructing invariant almost complex structures was described in
\cite{Ko}. It can be easily seen that all these structures are
canonical in the above sense.
\section{\bf Almost Hermitian structures}
We briefly recall some notions of Hermitian geometry including the
main classes of almost Hermitian structures.
Let $M$ be a smooth manifold, $\frak{X} (M)$ the Lie algebra of
all smooth vector fields on $M$, $d$ the exterior differentiation
operator. An {\it almost Hermitian structure} on $M$ (briefly,
{\it $AH$-structure}) is a pair $(g,J)$, where
$g=\langle\cdot,\cdot\rangle$ is a (pseudo)Riemannian metric on
$M$, $J$ an almost complex structure such that $\langle JX, JY
\rangle = \langle X, Y \rangle$ for any $X,Y\in\frak{X}(M)$. It
follows immediately that the tensor field $\Omega (X,Y)=\langle X,
JY \rangle$ is skew-symmetric, i.e. $(M,\Omega)$ is an almost
symplectic manifold. $\Omega$ is usually called a {\it fundamental
form} (the {\it K\"{a}hler form}) of an $AH$-structure $(g,J)$.
Further, denote by $\nabla$ the Levi-Civita connection of the
metric $g$ on $M$. We recall below some main classes of
$AH$-structures together with their defining properties (see, for
example, \cite {GH}):
\begin{tabbing}
Kill f123 \= $f$-structure of class $G_1$, or -structure \= $T(X,X)=0,$ i.e.
$\frak{X}(M)$ is an anticommutative $Q$-algebra \kill
{\bf K} \> {\it K\"{a}hler structure}: \> $\nabla J=0$; \\
{\bf H} \> {\it Hermitian structure}: \> $\nabla_X(J)Y-\nabla_{JX}(J)JY=0$; \\
{\bf G$_1$} \> {$AH$-structure of class $G_1$,} or \>
$\nabla_X(J)X-\nabla_{JX}(J)JX=0$; \\
\> $G_1$-{\it structure}: \> \\
{\bf QK} \> {\it quasi-K\"{a}hler structure}: \> $\nabla_X(J)Y+\nabla_{JX}(J)JY=0$;\\
{\bf AK} \> {\it almost K\"{a}hler structure}: \> $d\,\Omega=0$;\\
{\bf NK} \> {\it nearly K\"{a}hler structure,}
\> $\nabla_{X}(J)X=0$.\\
\> or $NK$-{\it structure}: \>
\end{tabbing}
It is well known (see, for example, \cite {GH}) that
$$
{\bf K}\subset{\bf H}\subset{\bf G_1};\;\;{\bf K}\subset{\bf
NK}\subset{\bf G_1};\;\;{\bf NK}={\bf G_1}\cap{\bf QK};\;\;{\bf
K}={\bf H}\cap{\bf QK}.
$$
As usual, we will denote by $N$ the Nijenhuis tensor of an almost
complex structure $J$, that is,
$$
N(X,Y)=\frac14([JX,JY]-J[JX,Y]-J[X,JY]-[X,Y])
$$
for any $X,Y\in\frak{X}(M)$. Then the condition $N=0$ is
equivalent to the integrability of $J$ (see, for example, \cite
{KN}). Moreover, an almost Hermitian structure $(g,J)$ belongs to
the class {\bf H} if and only if $N=0$ (see, for example, \cite
{GH}).
As was already mentioned, the role of homogeneous almost Hermitian
manifolds is particularly important "because they are the model
spaces to which all other almost Hermitian manifolds can be
compared" (see \cite {G3}). A wealth of examples for the most
classes above noted, both of general and specific character, can
be found in \cite {WG}, \cite {G2}, \cite {G3}, \cite {Ki1} and
others. In particular, after the detailed investigation of the
6-dimensional homogeneous nearly K\"{a}hler manifolds
V.F.Kirichenko proved \cite {Ki1} that naturally reductive
strictly nearly K\"{a}hler manifolds $SO(5)/U(2)$ and
$SU(3)/T_{max}$ are not isometric even locally to the
6-dimensional sphere $S^6$. These examples gave a negative answer
to the conjecture of S.Sawaki and Y.Yamanoue (see \cite {SaYa})
claimed that any $6$-dimensional strictly $NK$-manifold was a
space of constant curvature. It should be noted that the canonical
almost complex structure $J=\frac1{\sqrt{3}}(\theta-\theta^2)$ on
homogeneous $3$-symmetric spaces plays a key role in these and
other examples of homogeneous $AH$-manifolds.
Let $G$ be a connected Lie group, $H$ its closed subgroup, $g$ an
invariant (pseudo-)Riemannian metric on the homogeneous space
$G/H$. Denote by $\mathfrak{g}$ and $\mathfrak{h}$ the Lie
algebras corresponding to $G$ and $H$ respectively. Suppose that
$G/H$ is a reductive homogeneous space,
$\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{m}$ the reductive
decomposition of the Lie algebra $\mathfrak{g}$. As usual, we
identify $\mathfrak{m}$ with the tangent space $T_o(G/H)$ at the
point $o=H$. Then the invariant metric $g$ is completely defined
by its value at the point $o$. For convenience we denote by the
same manner both any invariant metric $g$ on $G/H$ and its value
at $o$.
Recall that $(G/H,g)$ is {\it naturally reductive} with respect to
a reductive decomposition
$\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{m}$ \cite{KN} if
$$g([X,Y]_{\mathfrak{m}},Z)=g(X,[Y,Z]_{\mathfrak{m}})$$ for all
$X,Y,Z\in\mathfrak{m}$. Here the subscript $\mathfrak{m}$ denotes
the projection of $\mathfrak{g}$ onto $\mathfrak{m}$ with respect
to the reductive decomposition.
We select here some known results closely related to the main
subject of our future consideration.
\begin{theorem} \label{t1}(\cite{AG})
Any invariant almost Hermitian structure on a naturally reductive
space $(G/H,g)$ belongs to the class {\bf G$_1$}.
\end{theorem}
\begin{theorem} \label{t2}(\cite{WG}, \cite{G2})
A homogeneous $3$-symmetric space $G/H$ with the canonical almost
complex structure $J$ and an invariant compatible metric $g$ is a
quasi-K\"{a}hler manifold. Moreover, $(G/H,J,g)$ belongs to the
class {\bf NK} if and only if $g$ is naturally reductive.
\end{theorem}
\begin{theorem} \label{t3}(\cite{Ma}, \cite{G4}, \cite{Ki6})
A $6$-dimensional strictly nearly K\"{a}hler manifold is Einstein.
\end{theorem}
Note that the latter result was obtained in \cite{Ki6} as a
particular case of the general approach based on investigating
nearly K\"{a}hler manifolds of constant type.
\section{\bf Metric $f$-structures and homogeneous manifolds}
The concept of the {\it generalized Hermitian geometry} created in
the 1980s (see, for example, \cite{Ki2}, \cite{Ki7}, \cite{Ki8})
was a natural consequence of the development of Hermitian geometry
and the theory of almost contact metric structures with many
applications. One of its central objects is the metric
$f$-structures of classical type, which include the class of
almost Hermitian structures. We recall here some basic notions.
An {\it $f$-structure} on a manifold $M$ is known to be a field of
endomorphisms $f$ acting on its tangent bundle and satisfying the
condition $f^3+f=0$ (see \cite{Y}). The number $r=dim\;Im\:f$ is
constant at any point of $M$ (see \cite{St}) and called a {\it
rank} of the $f$-structure. Besides, the number
$dim\;Ker\:f=dim\:M-r$ is usually said to be a {\it deficiency} of
the $f$-structure and denoted by $def\:f$.
Recall that an $f$-structure on a (pseudo)Riemannian manifold
$(M,g=\langle\cdot,\cdot\rangle)$ is called a {\it metric
$f$-structure}, if $\langle {fX}, Y \rangle +\langle X, fY \rangle
=0$, \;$X,Y\in\frak{X}(M)$ (see \cite{Ki2}). In the case the
triple $(M,g,f)$ is called a {\it metric $f$-manifold}. It is
clear that the tensor field $\Omega(X,Y)=\langle X, fY \rangle$ is
skew-symmetric, i.e. $\Omega$ is a $2$-form on $M$. $\Omega$ is
called a {\it fundamental form} of a metric $f$-structure
\cite{Ki7}, \cite{Ki2}. It is easy to see that the particular
cases $def\;f=0$ and $def\; f=1$ of metric $f$-structures lead to
almost Hermitian structures and almost contact metric structures
respectively.
Let $M$ be a metric $f$-manifold. Then
$\frak{X}(M)=\mathcal{L}\oplus\mathcal{M}$, where
$\mathcal{L}=Im\;f$ and $\mathcal{M}=Ker\;f$ are mutually
orthogonal distributions, which are usually called the {\it first}
and the {\it second fundamental distributions} of the
$f$-structure respectively. Obviously, the endomorphisms $l=-f^2$
and $m=id+f^2$ are mutually complementary projections on the
distributions $\mathcal{L}$ and $\mathcal{M}$ respectively. We
note that in the case when the restriction of $g$ to $\mathcal{L}$
is non-degenerate the restriction $(F,g)$ of a metric
$f$-structure to $\mathcal{L}$ is an almost Hermitian structure,
i.e. $F^2=-id,\;\langle FX, FY \rangle=\langle X, Y
\rangle,\;X,Y\in\mathcal{L}$.
A fundamental role in the geometry of metric $f$-manifolds is
played by the {\it composition tensor} $T$, which was explicitly
evaluated in \cite{Ki7}:
\begin{equation}\label{eqT}
T(X,Y)=\frac14{f}(\nabla_{fX}(f){fY}-\nabla_{f^2X}(f){f^2Y}),
\end{equation}
where $\nabla$ is the Levi-Civita connection of a
(pseudo)Riemannian manifold $(M,g)$, \; $X,Y\in\frak{X}(M)$.
Using this tensor $T$, the algebraic structure of a so-called {\it
adjoint $Q$-algebra} in $\frak{X}(M)$ can be defined by the
formula:
$$
X\ast{Y}=T(X,Y).
$$
It gives the opportunity to introduce some classes of metric
$f$-structures in terms of natural properties of the adjoint
$Q$-algebra (see \cite{Ki2}, \cite{Ki7}).
We enumerate below the main classes of metric $f$-structures
together with their defining properties:
\begin{tabbing}
Kill f123 \= $f$-structure of class $G_1$, or -structure \= $T(X,X)=0,$ i.e.
$\frak{X}(M)$ is an anticommutative $Q$-algebra \kill
{\bf Kf} \> {\it K\"{a}hler $f$--structure}: \> $\nabla f=0$; \\
{\bf Hf} \> {\it Hermitian $f$--structure}: \> $T(X,Y)=0,$ i.e.
$\frak{X}(M)$ is\\
\> \> an abelian $Q$-algebra;\\
{\bf G$_1$f} \> {$f$-structure of class $G_1$,} or \> $T(X,X)=0,$
i.e.
$\frak{X}(M)$ is \\
\> $G_1f$-{\it structure}: \> an anticommutative $Q$-algebra;\\
{\bf QKf} \> {\it quasi-K\"{a}hler $f$--structure}: \> $\nabla_X f +T_X
f=0$;\\
{\bf Kill f} \> {\it Killing $f$-structure}: \> $\nabla_X (f) X=0$;\\
{\bf NKf} \> {\it nearly K\"{a}hler $f$-structure,}
\> $\nabla_{fX}(f)fX=0$.\\
\> or $NKf$-{\it structure}: \>
\end{tabbing}
The classes {\bf Kf}, {\bf Hf}, {\bf G$_1$f}, {\bf QKf} (in more
general situation) were introduced in \cite{Ki2} (see also
\cite{Si}). Killing $f$-manifolds {\bf Kill f} were defined and
studied in \cite{Gr1}, \cite{Gr2}. The class {\bf NKf} was first
determined in \cite{B1} (see also \cite{B6}, \cite{B7}).
The following relationships between the classes mentioned are
evident:
$$
{\bf Kf}={\bf Hf}\cap{\bf QKf};\;\;\; {\bf Kf}\subset{\bf
Hf}\subset{\bf G_1f};\;\;\; {\bf Kf}\subset{\bf
Kill\;f}\subset{\bf NKf}\subset{\bf G_1f.}
$$
It is important to note that in the special case $f=J$ we obtain
the corresponding classes of almost Hermitian structures (see
\cite{GH}). In particular, for $f=J$ the classes {\bf Kill f} and
{\bf NKf} coincide with the well-known class {\bf NK} of {\it
nearly K\"ahler structures}.
{\bf Remark 1.} Killing $f$-manifolds are often defined by
requiring the fundamental form $\Omega$ to be a Killing form, i.e.
$d\Omega=\nabla\Omega$ (see \cite{Gr1}, \cite{KL}). It is not hard
to prove that the definition is equivalent to the above condition
$\nabla_{X}(f)X=0$.
Indeed, in accordance with \cite{YB}, $\Omega$ is a Killing
$2$-form if and only if $\nabla\Omega$ is a $3$-form. Further,
using \cite{KN}, we have
\begin{equation}\label{eq1}
(\nabla\Omega)(X,Y;Z)=Z\:\Omega(X,Y)-\Omega(\nabla_{Z}X,Y)-\Omega(X,\nabla_{Z}Y).
\end{equation}
It follows that $\nabla\Omega$ is always skew-symmetric in
arguments $X$ and $Y$. Using the fact, it is easy to see that
$\nabla\Omega$ is a $3$-form if and only if $\nabla\Omega$ is
skew-symmetric in $Y$ and $Z$:
\begin{equation}\label{eq2}
(\nabla\Omega)(X,Y;Z)=-(\nabla\Omega)(X,Z;Y).
\end{equation}
Taking into account formula (\ref{eq1}) and the definition of
$\Omega$, condition (\ref{eq2}) can be written in the form:
$$
Z\langle X, fY \rangle-\langle \nabla_{Z}X, fY \rangle-\langle X,
f\nabla_{Z}Y \rangle+Y\langle X, fZ \rangle-\langle \nabla_{Y}X,
fYZ \rangle-\langle X, f\nabla_{Y}Z \rangle=0.
$$
Since $\nabla$ is the Levi-Civita connection, in particular, we
have:
$$
Z\langle X, Y \rangle=\langle \nabla_{Z}X, Y \rangle+\langle X,
\nabla_{Z}Y \rangle.
$$
It follows that the previous formula can be written in the form:
$$
\langle X, \nabla_{Z}fY -f\nabla_{Z}Y \rangle+\langle X,
\nabla_{Y}fZ -f\nabla_{Y}Z \rangle=0.
$$
Using the formula $\nabla_{X}(f)Y=\nabla_{X}fY-f\nabla_{X}Y$, we
get:
$$
\langle X, \nabla_{Z}(f)Y + \nabla_{Y}(f)Z \rangle=0.
$$
It implies that $\nabla_{Z}(f)Y + \nabla_{Y}(f)Z=0$ for any
$Y,Z\in\frak{X}(M)$. This is obviously equivalent to the condition
$\nabla_{X}(f)X=0,\;X\in\frak{X}(M)$. {\hfill $\square$}
Now we dwell on invariant metric $f$-structures on homogeneous
spaces.
Any invariant metric $f$-structure on a reductive homogeneous
space $G/H$ determines the orthogonal decomposition
$\mathfrak{m}=\mathfrak{m}_1\oplus\mathfrak{m}_2$ such that
$\mathfrak{m}_1=Im\:f$, $\mathfrak{m}_2=Ker\:f$.
As it was already noted (see Section 3), the main classes of
almost Hermitian structures are provided with the remarkable set
of invariant examples. It turns out that there is also a wealth of
invariant examples for the basic classes of metric $f$-structures.
These invariant metric $f$-structures can be realized on
homogeneous $k$-symmetric spaces with canonical $f$-structures. We
select here some results in this direction. More detailed
information can be found in \cite{B1}-\cite{B7}, \cite{C3},
\cite{Li}.
\begin{theorem} \label{t4}(\cite{B5})
Any invariant metric $f$-structure on a naturally reductive space
$(G/H,g)$ is a $G_{1}f$-structure.
\end{theorem}
As a special case $(Ker\:f=0)$, it follows Theorem \ref{t1}.
\begin{theorem} \label{t5}(\cite{B5})
Let $(G/H,g,f)$ be a naturally reductive space with an invariant
metric $f$-structure that satisfies the condition
$[\frak{m}_1,\frak{m}_1]\subset\frak{m}_2\oplus\frak{h}$. Then
$G/H$ is a Hermitian $f$-manifold.
\end{theorem}
We note that Theorem \ref{t5} is also valid for arbitrary
invariant (pseudo)Riemanni\-an metric $g$ compatible with an
invariant $f$-structure on a reductive homogeneous space $G/H$
(see \cite{BV}).
Theorems \ref{t4} and \ref{t5} can be effectively provided with a
large class of examples. In particular, for a semi-simple group
$G$, the invariant (pseudo)Riemannian metric $g$ generated by the
Killing form on any regular $\Phi$-space $G/H$ is naturally
reductive with respect to the canonical reductive decomposition
$\frak{g}=\frak{h}\oplus\frak{m}$ (see \cite{S1}). Moreover, all
canonical structures $f$ and $J$ on homogeneous naturally
reductive $k$-symmetric spaces are compatible with such a metric,
i.e. $f$ is a metric $f$-structure, $J$ is an almost Hermitian
structure (see \cite{B1}, \cite{B10}). In what follows, we mean by
a naturally reductive decomposition the canonical reductive
decomposition for a regular $\Phi$-space $G/H$. To sum up, we have
\begin{theorem} \label{t6}(\cite{B5})
Let $(G/H,g)$ be a naturally reductive $k$-symmetric space. Any
canonical metric $f$-structure on $G/H$ is a $G_{1}f$-structure,
and any canonical almost Hermitian structure $J$ is a
$G_{1}$-structure.
\end{theorem}
\begin{theorem} \label{t7}(\cite{B6},\cite{B7})
Let $G/H$ be a regular $\Phi$-space, $g$ a naturally reductive
metric on $G/H$ with respect to the canonical reductive
decomposition $\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{m}$, $f$ a
metric canonical $f$-structure on $G/H$. Suppose the $f$-structure
satisfies the condition $f^2=\pm\theta\:f$. Then $(G/H,g,f)$ is a
nearly K\"ahler $f$-manifold.
\end{theorem}
\begin{corol} (\cite{B6},\cite{B7})
Let $(G/H,g)$ be a naturally reductive homogeneous $\Phi$-space
of order $k=4n,\:n\ge{1}$. If $\{i,-i\}\ \subset\ spec\ \theta$,
then there exists a non-trivial canonical $NKf$-structure on
$G/H$.
\end{corol}
We stress the particular role of homogeneous $4$- and
$5$-symmetric spaces.
\begin{theorem} \label{t8}(\cite{B3}-\cite{B7})
The canonical $f$-structure $f=\frac12(\theta-\theta^3)$ on any
naturally reductive $4$-symmetric space $(G/H,g)$ is both a
Hermitian $f$-structure and a nearly K\"ahler $f$-structure.
Moreover, the following conditions are equivalent:
1) $f$ is a K\"ahler $f$-structure; 2) $f$ is a Killing
$f$-structure; 3) $f$ is a quasi-K\"ahler $f$-structure; 4) $f$ is
an integrable $f$-structure; 5)
$[\frak{m}_1,\frak{m}_1]\subset\frak{h}$; 6)
$[\frak{m}_1,\frak{m}_2]=0$; 7) $G/H$ is a locally symmetric
space: $[\frak{m},\frak{m}]\subset\frak{h}$.
\end{theorem}
\begin{theorem} \label{t9}(\cite{B2}-\cite{B5}, \cite{B7}, \cite{C3})
Let $(G/H,g)$ be a naturally reductive $5$-symmetric space, $f_1$
and $f_2$, $J_1$ and $J_2$ the canonical structures on this space.
Then $f_1$ and $f_2$ belong to both classes {\bf Hf} and {\bf
NKf}. Moreover, the following conditions are equivalent:
1) $f_1$ is a K\"ahler $f$-structure; 2) $f_2$ is a K\"ahler
$f$-structure; 3) $f_1$ is a Killing $f$-structure; 4) $f_2$ is a
Killing $f$-structure; 5) $f_1$ is a quasi-K\"ahler $f$-structure;
6) $f_2$ is a quasi-K\"ahler $f$-structure; 7) $f_1$ is an
integrable $f$-structure; 8) $f_2$ is an integrable $f$-structure;
9) $J_1$ and $J_2$ are $NK$-structures; 10)
$[\frak{m}_1,\frak{m}_2]=0$ (here $\frak{m}_1=Im\:f_1=Ker\:f_2,
\frak{m}_2=Im\:f_2=Ker\:f_1$); 11) $G/H$ is a locally symmetric
space: $[\frak{m},\frak{m}]\subset\frak{h}$.
\end{theorem}
It should be mentioned that Riemannian homogeneous $4$-symmetric
spaces of classical compact Lie groups were classified and
geometrically described in \cite{J}. The similar problem for
homogeneous $5$-symmetric spaces was considered in \cite{TX}. By
Theorem \ref{t8} and Theorem \ref{t9}, it presents a collection of
homogeneous $f$-manifolds in the classes {\bf NKf} and {\bf Hf}.
Note that the canonical $f$-structures under consideration are
generally non-integrable.
Besides, there are invariant $NKf$-structures and $Hf$-structures
on homogeneous spaces $(G/H,g)$, where the metric $g$ is not
naturally reductive. The example of such a kind can be realized on
the $6$-dimensional Heisenberg group $(N,g)$. As to details
related to this group, we refer to \cite{Ka1}, \cite{Ka2},
\cite{TV}.
\begin{theorem} \label{t10}(\cite{B5}-\cite{B7})
The 6-dimensional generalized Heisenberg group $(N,g)$ with
respect to the canonical $f$-structure
$f=\frac12(\theta-\theta^3)$ of a homogeneous $\Phi$-space of
order $4$ is both $Hf$- and $NKf$-manifold. This $f$-structure is
neither Killing nor integrable on $(N,g)$.
\end{theorem}
{\bf Remark 2.} Theorems \ref{t8} and \ref{t10}, in particular,
illustrate simultaneously the analogy and the difference between
the canonical almost complex structure $J$ on homogeneous
$3$-symmetric spaces $(G/H,g,J)$ and the canonical $f$-structure
on homogeneous $4$-symmetric spaces $(G/H,g,f)$ (see Theorem
\ref{t2}).
Let us also remark that the 6-dimensional generalized Heisenberg
group $(N,g)$ is an example of solvable type. In Section 5, we
present $NKf$-structures with non-naturally reductive metrics in
the case of semi-simple type.
Finally, we briefly discuss the existence problem for invariant
Killing $f$-structu\-res. By Theorems \ref{t8} and \ref{t9}, the
canonical $f$-structures on naturally reductive $4$- and
$5$-symmetric spaces are never strictly (i.e. non-K\"ahler)
Killing $f$-structures. Moreover, we recall the following general
result:
\begin{theorem} \label{t11}(\cite{B4})
Let $(G/H,g,f)$ be a naturally reductive Killing $f$-manifold.
Then the following relations hold:
$$
[\mathfrak{m}_1,\mathfrak{m}_1]\subset\mathfrak{m}_1\oplus\mathfrak{h},
\quad
[\mathfrak{m}_2,\mathfrak{m}_2]\subset\mathfrak{m}_2\oplus\mathfrak{h},
\quad [\mathfrak{m}_1,\mathfrak{m}_2]\subset\mathfrak{h}.
$$
In particular, both the fundamental distributions of the Killing
$f$-structure generate invariant totally geodesic foliations on
$G/H$.
\end{theorem}
By the results in \cite{Gr1} and Theorem \ref{t11}, it follows
\begin{corol}(\cite{B4})
There are no non-trivial (i.e. $def\:f>0$) invariant Killing
$f$-structures of the so-called fundamental type (see \cite{Gr1})
on naturally reductive homogeneous spaces $(G/H,g)$.
\end{corol}
This fact is a wide generalization of the similar result of
A.Gritsans obtained for Riemannian globally symmetric spaces.
Besides, it shows a substantial difference between invariant
Killing $f$-structures and invariant $NK$-structures.
In Section 5, we will indicate, in particular, first examples of
invariant Killing $f$-structures.
\section{\bf Invariant $f$-structures on the complex \\flag manifold $\bf M=SU(3)/T_{max}$}
In this Section, we will consider all invariant $f$-structures on
the flag manifold $M=SU(3)/T_{max}$. Note that invariant almost
complex structures (i.e. $f$-structures of maximal rank $6$) on
this space were investigated in \cite{G3}, \cite{AGI1},
\cite{AGI2} and many other papers.
The homogeneous manifold $SU(3)/T_{max}$ is known to be an
important example in many branches of differential geometry and
beyond. In particular, $M=SU(3)/T_{max}$ is a Riemannian
homogeneous $3$-symmetric space not homeomorphic with the
underlying manifold $M$ of any Riemannian symmetric space (see
\cite{LP1}). Further, $M$ is a homogeneous $k$-symmetric space for
any $k\ge3$. Moreover, $M$ is a naturally reductive Riemannian
homogeneous space that is {\it non-commutative} (see \cite{J2}).
It means that the algebra of invariant differential operators
$\mathcal{D}(SU(3)/T_{max})$ is not commutative (see \cite{H1}).
It follows that $M=SU(3)/T_{max}$ is not even a {\it weakly
symmetric space} (see, for example, \cite{Vi}).
Besides, $M$ is the twistor space for the projective space
$\mathbb{C}P^2$ (see, for example, \cite{Be}, Chapter 13). It was
a key point for constructing the first examples of $6$-dimensional
Riemannian manifolds admitting a real Killing spinor (see
\cite{BFGK}). More exactly, the flag manifold $M=SU(3)/T_{max}$
with the nearly K\"ahler structure $(g,J)$ just possesses a real
Killing spinor (see \cite{BFGK}, \cite{Gru}). Moreover, using the
duality procedure for this space $SU(3)/T_{max}$, one can
effectively construct pseudo-Riemannian homogeneous manifolds with
the real Killing spinors (see \cite{Ka}).
Let $\Phi=I(s)$ be an inner automorphism of the Lie group $SU(3)$
defined by the element $s=diag\:(\varepsilon,\overline
\varepsilon,1)$, where $\varepsilon$ is a primitive third root of
unity. Then the subgroup $H=G^{\Phi}$ of all fixed points of
$\Phi$ is of the form:
$$
G^{\Phi}=\{ diag(e^{i\beta_1}, e^{i\beta_2},e^{i\beta_3})|
\beta_1+\beta_2+\beta_3=0,\;\beta_j\in\mathbb{R}\}.
$$
Obviously, $G^{\Phi}$ is isomorphic to $T^2=T_{max}$ diagonally
imbedded into $SU(3)$. It means that the flag manifold
$M=SU(3)/T_{max}$ is a homogeneous $3$-symmetric space defined by
the automorphism $\Phi$.
Consider the canonical reductive decomposition
$\frak{g}=\frak{h}\oplus\frak{m}$ of the Lie algebra
$\frak{g}=\frak{s}\frak{u}(3)$ for the homogeneous $\Phi$-space
$M$. Using the notations in \cite{R1}, we obtain:
$$\frak{g}=\frak{su}(3)=\left\{\left(\begin{array}{ccc}
\alpha_1 & a & \overline{c} \\
-\overline{a} & \alpha_2 & b \\
-c & -\overline{b} & \alpha_3
\end{array}\right) \left| \begin{array}{l}
\alpha_1,\alpha_2, \alpha_3\in Im\,\mathbb{C}, \\
a,b,c\in\mathbb{C}, \\
\alpha_1+\alpha_2+\alpha_3=0
\end{array}\right.\right\}=$$
$$=E(\alpha_1,\alpha_2, \alpha_3)\oplus
D(a,b,c)=\frak{h}\oplus\frak{m}.$$
If we put $X=D(a,b,c), Y=D(a_1,b_1,c_1),
Z=E(\alpha_1,\alpha_2,\alpha_3)$, then the Lie brackets can be
briefly indicated (see \cite{R2}):
$$ [X,Y]=D(\overline{bc_1-b_1c}, \overline{ca_1-c_1a},
\overline{ab_1-a_1b}) -$$ $$
2E(Im(a\overline{a_1}+\overline{c}c_1),Im(\overline{a}a_1+b\overline{b_1}),
Im(c\overline{c_1}+\overline{b}b_1) ) ,$$
$$[Z,X]=D(\alpha_1 a-a\alpha_2, \alpha_2 b-b\alpha_3,
\alpha_3c-c\alpha_1).$$
Further, we put
$\frak{m}=\frak{m}_1\oplus\frak{m}_2\oplus\frak{m}_3,$ where
$$\frak{m}_1 = \{X\in\frak{su}(3)| X=D(a,0,0), a\in\mathbb{C}\},$$
$$\frak{m}_2 =\{X\in\frak{su}(3)| X=D(0,b,0), b\in\mathbb{C}\},$$
$$\frak{m}_3 = \{X\in\frak{su}(3)| X=D(0,0,c), c\in\mathbb{C}\}.$$
Using the Killing form of the Lie algebra $\frak{su}(3)$, we
define an invariant inner product on $\frak{m}$:
$$
g_o(X,Y)=\langle X, Y\rangle_o=-\frac12Re\:tr\:XY.
$$
Then (see \cite{R1})
$\frak{g}=\frak{h}\oplus\frak{m}_1\oplus\frak{m}_2\oplus\frak{m}_3$
is $\langle\cdot,\cdot\rangle_o$-orthogonal decomposition
satisfying the following relations:
$$
[\frak{h},\frak{m}_j]\subset\frak{m}_j,\;[\frak{m}_j,\frak{m}_j]\subset\frak{h},\;
[\frak{m}_j,\frak{m}_{j+1}]\subset\frak{m}_{j+2},
$$
where $j=1,2,3$ and the index $j$ should be reduced by modulo $3$.
Besides, the $H$-modules $\frak{m}_j$ are pairwise non-isomorphic.
Now we turn to invariant Riemannian metrics on $M$. Taking into
account the well-known one-to-one correspondence between
$G$-invariant Riemannian metrics on $G/H$ and $Ad(H)$-invariant
inner products on $\frak{m}$ (see \cite{KN}), we will make use of
the following fact:
\begin{lemma}(\cite{R1})
Any $SU(3)$-invariant Riemannian metric
$g=\langle\cdot,\cdot\rangle$ on the flag manifold
$M=SU(3)/T_{max}$ can be written in the form
$$
g=\langle\cdot,\cdot\rangle=
\lambda_1\langle\cdot,\cdot\rangle_{o{|\frak{m}_1\times\frak{m}_1}}
+
\lambda_2\langle\cdot,\cdot\rangle_{o{|\frak{m}_2\times\frak{m}_2}}
+
\lambda_3\langle\cdot,\cdot\rangle_{o{|\frak{m}_3\times\frak{m}_3}},
$$
where $\lambda_j>0,\;j=1,2,3.$
\end{lemma}
A triple $(\lambda_1,\lambda_2,\lambda_3)$ is called \cite{R1} a
{\it characteristic collection} of a Riemannian metric $g$ above
mentioned . Considering Riemannian metrics up to homothety, one
can assume that
$(\lambda_1,\lambda_2,\lambda_3)=(1,t,s),\;t>0,s>0.$ For
convenience we will denote this correspondence in the following
way: $g=(\lambda_1,\lambda_2,\lambda_3)$ or $g=(1,t,s).$
We also recall the following result:
\begin{theorem} \label{t12}(\cite{ZW},\cite{AN},\cite{R1})
There are exactly (up to homothety) the following invariant
Einstein metrics on the flag manifold $SU(3)/T_{max}$ $:$
$$(1,1,1), (1,2,1), (1,1,2), (2,1,1).$$
\end{theorem}
Let $\alpha$ be the Nomizu function (see \cite{N}) of the
Levi-Civita connection $\nabla$ for an invariant Riemannian metric
$g=\langle\cdot,\cdot\rangle$ on a reductive homogeneous space
$G/H$. Then
\begin{equation}\label{eq4}
\alpha(X,Y)=\frac12[X,Y]_{\frak{m}}+U(X,Y),\;\;\;X,Y\in\frak{m},
\end{equation}
where $U:\frak{m}\times\frak{m}\to\frak{m}$ is a symmetric
bilinear mapping determined by the formula (see\cite{KN}):
$$
2\langle U(X,Y),Z\rangle=\langle X,[Z,Y]_{\frak{m}}\rangle+
\langle [Z,X]_{\frak{m}},Y\rangle.
$$
For our case in these notations we have
\begin{lemma}(\cite{W},\cite{R2})
For the Levi-Civita connection of a Riemannian metric
$g=(\lambda_1,\lambda_2,\lambda_3)$ on the flag manifold
$SU(3)/T_{max}$ the following conditions are satisfied$:$
$$
U(X,Y)=0, \;\;if\;\; X,Y\in\frak{m}_j,\,j\in\{1,2,3\};
$$
$$ U(X,Y)= -
(2\lambda_j)^{-1}(\lambda_{j+1}-\lambda_{j+2})[X,Y], \;\;if\;\;
X\in\frak{m}_{j+1},Y\in\frak{m}_{j+2},
$$
where $j=1,2,3$ and the numbers $j$ are reduced by modulo $3$.
\end{lemma}
Let us now turn to invariant $f$-structures on $M=SU(3)/T_{max}$.
Keeping the above notations, any invariant $f$-structure on $M$
can be expressed by the mapping
\begin{equation}\label{eqf}
f:D(a,b,c)\rightarrow D(\zeta_1ia,\zeta_2ib,\zeta_3ic),
\end{equation}
where $\zeta_{j}\in\{1,0,-1\},\;j=1,2,3$, $i$ is the imaginary
unit. We will call the collection $(\zeta_1,\zeta_2,\zeta_3)$ a
{\it characteristic collection} of the invariant $f$-structure and
for convenience denote $f=(\zeta_1,\zeta_2,\zeta_3).$ Obviously,
all invariant $f$-structures on $M$ pairwise commute.
If we agree to consider $f$-structures up to sign, then there are
the following invariant $f$-structures on $M=SU(3)/T_{max}$:
1) {\it invariant $f$-structures of rank $6$ (invariant almost
complex structures)}:
$$
J_1=(1,1,1),\;\;J_2=(1,-1,1),\;\;J_3=(1,1,-1),\;\;J_4=(1,-1,-1).
$$
2) {\it invariant $f$-structures of rank $4$}:
$$
\begin{array}{ccc}
f_1=(1,1,0),\;\;\;\;f_2=(1,0,1),\;\;\;\;f_3=(0,1,1),\\
f_4=(1,-1,0),\;\;\;\;f_5=(1,0,-1),\;\;\;\;f_6=(0,1,-1).
\end{array}
$$
3) {\it invariant $f$-structures of rank $2$}:
$$
f_7=(1,0,0),\;\;\;\;f_8=(0,1,0),\;\;\;\;f_9=(0,0,1).
$$
Our description of all invariant $f$-structures and all invariant
Riemannian metrics evidently implies that any invariant
$f$-structure $f=(\zeta_1,\zeta_2,\zeta_3)$ is a metric
$f$-structure with respect to any invariant Riemannian metric
$g=(\lambda_1,\lambda_2,\lambda_3)$. In particular,
$J_j,\;j=1,2,3,4$ are invariant almost Hermitian structures with
respect to all invariant Riemannian metrics
$g=(\lambda_1,\lambda_2,\lambda_3)$.
Now we are able to investigate all invariant $f$-structures in the
sense of generalized Hermitian geometry, i.e. the special classes
{\bf Kf}, {\bf NKf}, {\bf Kill f}, {\bf Hf}, {\bf G$_1$f}.
A key point of our consideration belongs to the expression
$\nabla_X(f)Y$. Using formula (\ref{eq4}), we get:
$$
\begin{array}{cc}
\nabla_X(f)Y=\nabla_X fY-f\nabla_X Y=\alpha(X,fY)-f
\alpha(X,Y)\\
=\frac12 ([X,fY]_{\frak{m}}-f[X,Y]_{\frak{m}})+U(X,fY)-fU(X,Y).
\end{array}
$$
As a result, we can obtain:
\begin{multline}\label{123}
\nabla_X(f)Y=\frac12 D(A,B,C), \text{where}\\
A=\overline{i((\zeta_1+\zeta_3)(1+s-t)bc_1+(\zeta_1+\zeta_2)(s-t-1)b_1c)},\\
B=\overline{i((\zeta_2+\zeta_1)(1+\frac{1-s}{t})ca_1+(\zeta_2+\zeta_3)(\frac{1-s}{t}-1)c_1a)},\\
C=\overline{i((\zeta_3+\zeta_2)(\frac{t-1}{s}+1)ab_1+(\zeta_3+\zeta_1)(\frac{t-1}{s}-1)a_1b)}.
\end{multline}
\subsection{\bf K\"ahler $f$-structures}
K\"ahler $f$-structures are defined by the condition
$\nabla_X(f)Y=0$ (see Section 4). Using formula (\ref{123}), this
condition is equivalent to the following system of equations:
\begin{equation}\label{syst1}
\left\{\begin{array}{c}
(\zeta_1+\zeta_3)(s-t+1)=0 \\
(\zeta_1+\zeta_2)(s-t-1)=0 \\
(\zeta_2+\zeta_3)(s+t-1)=0 \
\end{array}\right.
\end{equation}
Solving system (\ref{syst1}) for all invariant $f$-structures, we
obtain the following result:
\begin{propos}
The flag manifold $M=SU(3)/T_{max}$ admits the following invariant
K\"ahler $f$-structures with respect to the corresponding
invariant Riemannian metrics only:
$$
\begin{array}{lll}
J_2=(1,-1,1),&g_t=(1,t,t-1),&t>1;\\
J_3=(1,1,-1),&g_t=(1,t,t+1),&t>0;\\
J_4=(1,-1,-1),&g_t=(1,t,1-t),&0<t<1.
\end{array}
$$
In particular, there are no invariant K\"ahler $f$-structures of
rank $2$ and $4$ on $M$.
\end{propos}
We note that the result is known for invariant almost complex
structures (see \cite{G3},\cite{AGI2}). We can also observe that
for each of K\"ahler $f$-structures $J_2,J_3,J_4$ the
corresponding $1$-parameter set $g_t$ of invariant Riemannian
metrics contains exactly one Einstein metric excluding the
naturally reductive metric $g=(1,1,1)$ (see Theorem \ref{t12}).
Taking into account Theorem \ref{t2}, the latter fact implies that
the structures $J_2,J_3,J_4$ cannot be realized as the canonical
almost complex structures $J=\frac1{\sqrt3}(\theta-\theta^2)$ for
some homogeneous $\Phi$-spaces of order $3$.
In addition, Lie brackets relations for the subspaces $\frak{m}_j,
\;j=1,2,3$ imply that all invariant $f$-structures of rank $2$ and
$4$ are non-integrable. It immediately follows that these
$f$-structures cannot be K\"ahler $f$-structures.
\subsection{\bf Killing $f$-structures}
The defining condition for Killing $f$-structures can be written
in the form $\nabla_X(f)X=0$ (see Section 4). From (\ref{123}), it
follows
\begin{multline*}
\nabla_X(f)X=\frac12 D(A_0,B_0,C_0), \text{where}\\
A_0=\overline{ibc((\zeta_1+\zeta_3)(1+s-t)+(\zeta_1+\zeta_2)(s-t-1))},\\
B_0=\overline{ica((\zeta_2+\zeta_1)(1+\frac{1-s}{t})+(\zeta_2+\zeta_3)(\frac{1-s}{t}-1))},\\
C_0=\overline{iab((\zeta_3+\zeta_2)(\frac{t-1}{s}+1)+(\zeta_3+\zeta_1)(\frac{t-1}{s}-1))}.
\end{multline*}
It easy to show that the condition $\nabla_X(f)X=0$ is equivalent
to the following system of equations:
$$
\left\{\begin{array}{c}
(\zeta_1+\zeta_3)(s-t+1)+(\zeta_1+\zeta_2)(s-t-1)=0 \\
(\zeta_1+\zeta_2)(s-t-1)+(\zeta_2+\zeta_3)(s+t-1)=0 \
\end{array}\right.
$$
Analyzing this system for all invariant $f$-structures, we obtain
the following result:
\begin{propos}
All invariant strictly Killing (i.e. non-K\"ahler) $f$-structures
on the flag manifold $M=SU(3)/T_{max}$ and the corresponding
invariant Riemannian metrics (up to homothety) are indicated
below:
$$
\begin{array}{ll}
J_1=(1,1,1),&g=(1,1,1);\\
f_1=(1,1,0),&g=(3,3,4);\\
f_2=(1,0,1),&g=(3,4,3);\\
f_3=(0,1,1),&g=(4,3,3).\
\end{array}
$$
In particular, there are no invariant Killing $f$-structures of
rank $2$ on $M$.
\end{propos}
Note the structure $J_1$ is a well-known non-integrable nearly
K\"ahler structure on a naturally reductive space $M$ (see
\cite{G2}, \cite{G3}, \cite{Ki1}, \cite{AGI2} and others). The
structures $f_1,f_2,f_3$ present first invariant non-trivial
Killing $f$-structures \cite{B11}. The important feature of these
structures is that the corresponding invariant Riemannian metrics
are not Einstein (see Theorem \ref{t12}). It illustrates a
substantial difference between non-trivial strictly Killing
$f$-structures and strictly $NK$-structures at least in the
$6$-dimensional case (see Theorem \ref{t3}).
{\bf Remark 3.} It is interesting to note that all strictly
Killing $f$-structures above indicated are canonical
$f$-structures for suitable homogeneous $\Phi$-spaces of the Lie
group $SU(3)$. We already mentioned that $M=SU(3)/T_{max}$ is a
homogeneous $k$-symmetric space for any $k\ge 3$. It means $M$ as
an underlying manifold could be generated by various automorphisms
$\Phi$ of the Lie group $SU(3)$. In particular, $J_1$ is the
canonical almost complex structure
$J=\frac1{\sqrt{3}}(\theta-\theta^2)$ for the homogeneous
$\Phi$-space of order $3$, where $\Phi=I(s),\;
s=diag\:(\varepsilon,\overline
\varepsilon,1),\;\varepsilon=\sqrt[3]{1}$ (see the beginning of
this Section). Further, if we consider the automorphism
$\Phi_1=I(s_1),\; s_1=diag\:(i,-i,1)$, where $i=\sqrt[4]{1}$ is
the imaginary unit, then $M$ is a homogeneous $\Phi_1$-space of
order $4$. The corresponding canonical $f$-structure
$f=\frac12(\theta_1-\theta_{1}^3)$ for this $\Phi_1$-space just
coincides (up to sign) with the $f$-structure $f_3=(0,1,1)$. The
structures $f_1$ and $f_2$ can be obtained in the similar way.
Moreover, all the structures $f_1,f_2,f_3$ and $f_7,f_8,f_9$ can
be realized as canonical $f$-structures for suitable homogeneous
$\Phi$-spaces of order $5$.
We also note that all $f$-structures $f_1,f_2,f_3$ are just the
restrictions of the structure $J_1$ onto the corresponding
distributions $\frak{m}_p\oplus\frak{m}_q,\;p,q\in\{1,2,3\}.$
\subsection{\bf Nearly K\"ahler $f$-structures}
Using (\ref{123}), we can easily obtain:
\begin{multline*}
\nabla_{fX}(f)fX=\frac12 D(\hat A,\hat B,\hat C), \text{where}\\
\hat A=\overline{-i\zeta_2\zeta_3bc((\zeta_1+\zeta_3)(1+s-t)+(\zeta_1+\zeta_2)(s-t-1))},\\
\hat B=\overline{-i\zeta_1\zeta_3ca((\zeta_2+\zeta_1)(1+\frac{1-s}{t})+
(\zeta_2+\zeta_3)(\frac{1-s}{t}-1))},\\
\hat C=\overline{-i\zeta_1\zeta_2ab((\zeta_3+\zeta_2)(\frac{t-1}{s}+1)+
(\zeta_3+\zeta_1)(\frac{t-1}{s}-1))}.
\end{multline*}
It follows that the condition $\nabla_{fX}(f)fX=0$ is reduced to
the following system of equations:
$$
\left\{\begin{array}{c}
\zeta_2\zeta_3((\zeta_1+\zeta_3)(s-t+1)+(\zeta_1+\zeta_2)(s-t-1))=0 \\
\zeta_1\zeta_3((\zeta_2+\zeta_1)(1+t-s)+(\zeta_2+\zeta_3)(1-s-t))=0 \\
\zeta_1\zeta_2((\zeta_3+\zeta_2)(t+s-1)+(\zeta_3+\zeta_1)(t-s-1))=0 \
\end{array}\right.
$$
Consideration of this system implies
\begin{propos}\label{p3}
The only invariant strictly nearly K\"ahler (i.e. non-K\"ahler)
$f$-structure of rank $6$ on the flag manifold $M=SU(3)/T_{max}$
is the nearly K\"ahler structure $J_1=(1,1,1)$ with respect to the
naturally reductive metric $g=(1,1,1)$.
Invariant strictly nearly K\"ahler $f$-structures of rank $4$ and
the corresponding invariant Riemannian metrics (up to homothety)
on $M$ are:
$$
\begin{array}{lll}
f_1=(1,1,0),&g_s=(1,1,s),&s>0;\\
f_2=(1,0,1),&g_t=(1,t,1),&t>0;\\
f_3=(0,1,1),&g_t=(1,t,t),&t>0.\
\end{array}
$$
The invariant $f$-structures $f_7,f_8,f_9$ of rank $2$ on $M$ are
strictly $NKf$-structures with respect to all invariant Riemannian
metrics $g=(1,t,s),\;t,s>0$.
\end{propos}
First we notice that the structures $f_1,f_2,f_3$ and
$f_7,f_8,f_9$ provide invariant examples of $NKf$-structures with
non-naturally reductive metrics on the homogeneous space
$M=SU(3)/T_{max}$, which belongs to a semi-simple type.
We can also observe that for any invariant strictly
$NKf$-structure on $M$ there exists at least one (up to homothety)
corresponding Einstein metric. More exactly, for these
$NKf$-structures of rank $6$, $4$, and $2$ there are (up to
homothety) $1$, $2$, and $4$ Einstein metrics respectively (see
Theorem \ref{t12}). In a certain degree, it is a particular
analogy with the result of Theorem \ref{t3}. This particular fact
and some related general results lead to the following conjecture,
which seems to be plausible:
{\bf Conjecture}. For any strictly nearly K\"ahler $f$-structure
on a $6$-dimensional manifold there exists at least one
corresponding Einstein metric.
{\bf Remark 4.} The invariant $f$-structures $f_4,f_5,f_6$ on the
flag manifold $M=SU(3)/T_{max}$ cannot be canonical $f$-structures
for all homogeneous $\Phi$-spaces of orders $4$ and $5$ of the Lie
group $SU(3)$. It evidently follows by comparing the results in
Theorem \ref{t8}, Theorem \ref{t9}, and Proposition \ref{p3}.
\subsection{\bf Hermitian $f$-structures}
First let us calculate the composition tensor $T$ (see formula
(\ref{eqT})) for arbitrary invariant $f$-structure on
$(M=SU(3)/T_{max},g=(1,t,s))$. Combining (\ref{123}) and
(\ref{eqf}), we can obtain:
\begin{multline}\label{T}
T(X,Y)=\frac18 D(\check A,\check B,\check C), \text{where}\\
\check A=-\zeta_1\zeta_2\zeta_3(1+\zeta_2\zeta_3)((\zeta_1+\zeta_3)(1+s-t)\overline{bc_1}+
(\zeta_1+\zeta_2)(s-t-1)\overline{b_1c}),\\
\check B=-\zeta_1\zeta_2\zeta_3(1+\zeta_1\zeta_3)((\zeta_2+\zeta_1)(1+\frac{1-s}{t})\overline{ca_1}+
(\zeta_2+\zeta_3)(\frac{1-s}{t}-1)\overline{c_1a}),\\
\check C=-\zeta_1\zeta_2\zeta_3(1+\zeta_1\zeta_2)((\zeta_3+\zeta_2)(\frac{t-1}{s}+1)\overline{ab_1}+
(\zeta_3+\zeta_1)(\frac{t-1}{s}-1)\overline{a_1b}).
\end{multline}
We recall that the defining property for a Hermitian $f$-structure
is the condition $T(X,Y)=0$. Now from (\ref{T}), we get the
following result:
\begin{propos}\label{p4}
The invariant $f$-structures $J_2,J_3,J_4$ and $f_1,\dots,f_9$ are
Hermitian $f$-structures with respect to all invariant Riemannian
metrics $g=(1,t,s)$,\newline$t,s>0$ on the flag manifold
$M=SU(3)/T_{max}$.
\end{propos}
Notice that the almost complex structure $J_1=(1,1,1)$ is
non-integrable. It agrees with the fact that $J_1$ is not a
Hermitian $f$-structure for each Riemannian metric. While we
stress that all $f$-structures $f_1,\dots,f_9$ of rank $4$ and $2$
are non-integrable, but they are Hermitian $f$-structures.
\subsection{\bf G$_1$f-structures}
Finally, we consider the condition $T(X,X)=0$, which is the
defining property for $G_1f$-structures. Using (\ref{T}) and
taking into account Propositions \ref{p3} and \ref{p4}, we get
\begin{propos}\label{p5}
The flag manifold $M=SU(3)/T_{max}$ does not admit invariant
strictly $G_1f$-structures (i.e. neither $NKf$-structures nor
$Hf$-structures). In particular, there are no invariant strictly
$G_1$-structures $J$ (i.e. neither nearly K\"ahler nor Hermitian)
on $M$.
\end{propos}
|
{
"timestamp": "2005-03-24T12:52:20",
"yymm": "0503",
"arxiv_id": "math/0503533",
"language": "en",
"url": "https://arxiv.org/abs/math/0503533"
}
|
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